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\begin{document}
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\title{Tutorial for Modular Forms in Pari/GP}
\author{Henri Cohen}
\maketitle
\smallskip
\section{Introduction}
Three packages are available to work with modular forms and related functions
in \kbd{Pari/GP}. The first one is the $L$-function package, which has been
available since 2.9.0 (2015), and computes with general motivic
$L$-functions, and in particular with $L$-functions attached to Dirichlet
characters, Hecke characters, Artin representations, and modular forms. The
name of most functions in this package begins with \kbd{lfun}, such as
\kbd{lfuninit}.
The second is the modular symbol package, whose primary aim
is not so much to compute modular form spaces and modular forms, but to
compute $p$-adic $L$-functions attached to modular forms. The name of most
functions in this package begins with \kbd{ms}, such as \kbd{msinit}.
The third package is the modular forms package, whose aim is to compute in
the standard spaces $M_k(\G_0(N),\chi)$ with $k$ integral or half-integral,
both with modular form \emph{spaces} and individual modular \emph{forms}. The
name of most functions in this package begins with \kbd{mf}, such as
\kbd{mfinit}. The goal of the present manual is to describe this package in
view of a guide for a new user, so will essentially be a tutorial,
although we include a reference guide at the end.
\medskip
We can work on five subspaces of $M_k(\G_0(N),\chi)$, through a
corresponding \emph{space flag} in the commands: the cuspidal \emph{new
space} $S_k^{\new}(\G_0(N),\chi)$ (flag = 0), the full cuspidal space
$S_k(\G_0(N),\chi)$ (flag = 1), the old space $S_k^{\text{old}}(\G_0(N),\chi)$
(flag = 2, probably of little use), the space generated by all Eisenstein
series ${\cal E}_k(\G_0(N),\chi)$ (flag = 3), and finally the full space
including the Eisenstein part $M_k(\G_0(N),\chi)$ (flag = 4, which can be
omitted since it is the default). Note that although it can be defined, we have
not included the space $M_k^{\new}$, nor the ``certain space'' of
Zagier--Skoruppa. In the half-integral weight case, we have included
only the full cuspidal space and the full space (flags $1$ and $4$), as
well as the Kohnen's $+$-space and the corresponding newspace and eigenforms
when $N$ is squarefree.
Note in particular that the package includes the computation of modular
forms of weight $k=1$ and of half-integral weight.
\medskip
The modular forms themselves are represented in a special internal format
which the user need not worry about and which basically is a recipe to
compute successive Fourier coefficients at infinity: if $F$ is a GP modular
form, \kbd{mfcoefs}$(F, 10)$ will give you the Fourier coefficients at infinity
from $a(0)$ to $a(10)$ of the modular form corresponding to $F$ as a row
\emph{vector} (if you want a power series expansion, use the GP function
\kbd{Ser}, see below). Many operations are available on such objects, but the
most important thing the user needs to know is that the number of Fourier
coefficients need not be specified in advance: the command
\kbd{mfcoefs}$(F,n)$ is valid for any integer $n$. We will of course explain
the details of this below.
\medskip
Finally, note that we may roughly divide the complexity of available functions
into three levels:
\begin{enumerate}
\item The first level includes all the basic modular form and modular spaces
creation and operations. Many functions are very fast, but some are quite
time-consuming for very different reasons; first those dealing with forms and
spaces involving Dirichlet characters of large order; second finding eigenforms
when the splitting using the Hecke algebra is difficult; and third modular
spaces of weight $1$ when there exist ``exotic'' forms. Reasonable levels
(for low weight) can go up to a few thousands. The critical parameter is
actually the dimension of the underlying modular form space where linear
algebra needs to be performed, so the time complexity is at least
proportional to $(N\times k)^3$, and in fact more than that due to
coefficient explosion in the base field $\Q(\chi)$.
\item The second level needs technical information about spaces generated by
products of two Eisenstein series, and is quite expensive. But it allows to
perform computations which would be almost impossible otherwise, such as
Fourier expansions of $f|_k\gamma$ (hence at any cusp), numerical
evaluation of modular forms at any point in the upper half-plane (even close
to the real axis) or $L$-functions attached to an arbitrary form. We have
included a caching method, so that once a single such computation is
performed in a given space, the needed technical data is stored and
no longer needs to be computed so that all subsequent calls are much faster.
Reasonable levels (for low weight) can go up to one thousand, say.
Again, the critical parameter is the space dimension but this time the
linear algebra is performed in large degree cyclotomic fields, even when the
Nebentypus is trivial. Compared to the first level, we lose at least
a factor $\Lambda(N)$ (the exponent of the multiplicative group $(\Z/N\Z)^*$)
in the time complexity, which gets as large as $N-1$ if $N$ is prime.
\item The third level, which uses the second level functions, allows more
numerical computations such as period polynomials, modular symbols,
Petersson products, etc\dots The time complexity does not increase much
since it is dominated by the second level.
\end{enumerate}
\section{Creation of Modular Forms}
In \kbd{Pari/GP} modular forms can be created in three different ways:
\begin{itemize}\item As \emph{basic modular forms}, i.e., forms attached (or
not) to different mathematical objects, and which are of so frequent use
that we have implemented them so that the user has them at his disposal.
Examples: \kbd{mfDelta} (Ramanujan's delta), \kbd{mfEk}
(Eisenstein series of weight $k$ on the full modular group; of course
we also have more general Eisenstein series), \kbd{mffrometaquo}
(eta quotients), \kbd{mffromell} (modular form attached to an elliptic
curve over $\Q$), \kbd{mffromqf} (modular form attached
to a lattice).
\item From existing forms by applying \emph{operations}.
Examples: multiplication/division, linear combination,
derivation and integration, Serre derivative, RC-brackets, Hecke and
Atkin--Lehner operations, expansion and diamond operators, etc\dots
\item Through the creation of the modular form \emph{spaces}:
typically, if only \kbd{mf=mfinit} is applied, then a
basis of forms is obtained by the command \kbd{mfbasis(mf)}.
The command \kbd{mfeigenbasis(mf)} produces the canonical basis of
eigenforms.\end{itemize}
\section{A First Session: working with Leaves}
This is now a tutorial session. We will see sample commands as we go along.
\begin{verbatim}
? D = mfDelta(); V = mfcoefs(D, 8)
% = [0, 1, -24, 252, -1472, 4830, -6048, -16744, 84480]
\end{verbatim}
The command \kbd{mfcoefs(D,n)} gives the vector of Fourier coefficients (at
infinity) $[a(0),a(1),\dots,a(n)]$ (note that there are $n+1$ coefficients).
This is a compact representation, but if you prefer power series you
can use \kbd{Ser(V,q)} (convert a vector into a power series).
\begin{verbatim}
% = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6\
- 16744*q^7 + 84480*q^8 + O(q^9)
\end{verbatim}
(This simple-minded recipe only works when the form has rational
coefficients. Make sure to use \kbd{q = varhigher("q")} first if the form has
non-rational algebraic coefficients to avoid problems with variable priorities.)
Similarly
\begin{verbatim}
? E4 = mfEk(4); E6 = mfEk(6); apply(f->mfcoefs(f,3),[E4,E6])
% = [[1, 240, 2160, 6720], [1, -504, -16632, -122976]]
? E43 = mfpow(E4, 3); E62 = mfpow(E6, 2);
? DP = mflinear([E43, E62], [1, -1]/1728);
? mfcoefs(DP, 6)
% = [0, 1, -24, 252, -1472, 4830, -6048]
? mfisequal(D, DP)
% = 1
\end{verbatim}
Self-explanatory. Note that there is a command \kbd{mfcoef(F, n)} (without
the final ``s'') which simply outputs the coefficient $a(n)$.
A final example of the same type:
\begin{verbatim}
? F = mffrometaquo([1,2; 11,2]); mfcoefs(F,10)
% = [0, 1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
? G = mffromell(ellinit("11a1"))[2];
? mfisequal(F, G)
% = 1
\end{verbatim}
Here, \kbd{mffrometaquo} takes as argument a matrix representing an
\emph{eta quotient}, here $\eta(1\times\tau)^2\eta(11\times\tau)^2$.
The second component of the \kbd{mffromell} output is the modular form
associated to the elliptic curve by modularity.
\section{A Second Session: Modular Form Spaces}
In the first session, we have seen a few preinstalled modular forms (that
we can call \emph{leaves}), and a number of operations on them. All reasonable
operations have been implemented (if some are missing, please tell us). We are
now going to work with \emph{spaces} of modular forms.
\begin{verbatim}
? mf = mfinit([1,12]); L = mfbasis(mf); #L
% = 2
? mfdim(mf)
% = 2
\end{verbatim}
This creates the full space of modular forms of level $1$ and weight $12$.
This space is created thanks to an almost random basis that one can obtain
using \kbd{mfbasis}, and we see either by asking for the number of elements
of \kbd{L} or by using the command \kbd{mfdim}, that it has dimension $2$,
not surprising. We can see it better by writing:
\begin{verbatim}
? mfcoefs(L[1],6)
% = [691/65520, 1, 2049, 177148, 4196353, 48828126, 362976252]
? mfcoefs(L[2],6)
% = [0, 1, -24, 252, -1472, 4830, -6048]
\end{verbatim}
or simply
\begin{verbatim}
? mfcoefs(mf,6) \\ apply mfcoefs to mfbasis elements
% =
[691/65520 0]
[ 1 1]
[ 2049 -24]
[ 177148 252]
[ 4196353 -1472]
[ 48828126 4830]
[362976252 -6048]
\end{verbatim}
Note two things: first, the Eisenstein series are given before the cusp forms
(this may change, but for now this is the case), and second, the Eisenstein
series is normalized so that it is the coefficient $a(1)$ which is equal to
$1$, and not $a(0)$. In particular, here at least, it is a normalized
Hecke eigenform.
If we want to work only in the cuspidal space $S_{12}(\G)$, we simply use
the flag $1$, such as:
\begin{verbatim}
? mf = mfinit([1,12], 1); L = mfbasis(mf); #L
% = 1
? mfcoefs(L[1],6)
% = [0, 1, -24, 252, -1472, 4830, -6048]
\end{verbatim}
Let us now look at higher dimensional cases. In the following example, we
consider the \emph{new space} (flag = $0$), although in the present case
this is the same as the cuspidal space:
\begin{verbatim}
? mf = mfinit([35,2], 0); L = mfbasis(mf); #L
% = 3
? for (i = 1, 3, print(mfcoefs(L[i], 10)))
[0, 3, -1, 0, 3, 1, -8, -1, -9, 1, -1]
[0, -1, 9, -8, -11, -1, 4, 1, 13, 7, 9]
[0, 0, -8, 10, 4, -2, 4, 2, -4, -12, -8]
\end{verbatim}
These are essentially random cusp forms. Usually, you want the eigenforms:
this is obtained by the function \kbd{mfeigenbasis} (note in passing that
\kbd{B=mfeigenbasis(mf)} adds components to \kbd{mf}, so that the next
call is instantaneous). You can ask for the defining number fields with the
command \kbd{mffields}. Note that these commands act only on the new space,
but the package also accepts the spaces that contain it (such as the cuspidal
space or the full space, but not the old space), although the result is only
about the new space.
\begin{verbatim}
? mffields(mf)
% = [y, y^2 - y - 4]
? L = mfeigenbasis(mf); #L
% = 2
? mfcoefs(L[1],10)
% = [0, 1, 0, 1, -2, -1, 0, 1, 0, -2, 0]
? mfcoefs(L[2],4)
% = [Mod(0, y^2 - y - 4), Mod(1, y^2 - y - 4),\
Mod(-y, y^2 - y - 4),Mod(y - 1, y^2 - y - 4),\
Mod(y + 2, y^2 - y - 4)]
? lift(mfcoefs(L[2],10))
% = [0, 1, -y, y - 1, y + 2, 1, -4, -1, -y - 4, -y + 2, -y]
\end{verbatim}
The command \kbd{mffields} gives the polynomials in the variable $y$ defining
the number field extensions on which the eigenforms are defined. Here, one of
the fields is $\Q$, the other is $\Q(\sqrt{17})$. To obtain the eigenforms,
we use \kbd{mfeigenbasis}, and there are only two and not three, since the
one defined on $\Q(\sqrt{17})$ goes together with its conjugate. Asking
directly \kbd{mfcoefs(L[2],4)} gives the coefficients as \kbd{polmods}, not
easy to read, so it is usually preferable to \emph{lift} them, giving the
last command, where in the output we must of course remember that $y$ stands
for \emph{one} of the two roots of $y^2-y-4=0$, i.e., $(1\pm\sqrt{17})/2$.
In fact, for some numerical computations, we really need the
coefficients of the eigenform embedded in $\C$, and not just as abstract
algebraic numbers (in our case of trivial character, they will be in $\R$).
This is why a few functions (most notably \kbd{mfeval} and \kbd{lfunmf})
will return a \emph{vector} of results and not a scalar when called on such
a form.
For instance, here is a little GP script which computes the numerical
expansion of a modular form instead of the expansion in \kbd{polmods}:
\begin{verbatim}
mfcoefsembed(F,n) = mfembed(F, mfcoefs(F,n));
\end{verbatim}
Note that this produces a vector of expansions when the eigenforms
are defined over an extension, i.e. $[\Q(F):\Q(\chi)] > 1$, one per conjugate
form.
\begin{verbatim}
? mfcoefsembed(L[2],5) \\ two conjugate forms
%4 = [[0, 1, 1.5615..., -2.5615..., 0.43844..., 1],
[0, 1, -2.561...,, 1.5615...,, 4.5615..., 1]]
\end{verbatim}
The first eigenform found above is \emph{rational}, hence by the modularity
theorem there exists up to isogeny a unique elliptic curve to which it
corresponds. We check this by writing
\begin{verbatim}
? [mf,F] = mffromell(ellinit("35a1")); mfcoefs(F, 10)
% = [0, 1, 0, 1, -2, -1, 0, 1, 0, -2, 0]
? mfisequal(F, L[1])
% = 1
\end{verbatim}
For a more typical example (still with no character):
\begin{verbatim}
? [ mfdim([96,2], flag) | flag <- [0..4] ]
% = [2, 9, 7, 15, 24]
\end{verbatim}
This gives us the dimensions of the new space, the cuspidal space,
the old space, the space of Eisenstein series, and the whole space of
modular forms.
Just for fun, we write (recall that the default is the full space):
\begin{verbatim}
? mf = mfinit([96,2]); L = mfbasis(mf);
? for (i = 12, 15, print(mfcoefs(L[i], 15)))
[23/24, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24]
[31/24, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24]
[47/24, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24]
[95/24, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24]
\end{verbatim}
Apparently, these four Eisenstein series differ only by their constant
term, which is of course not possible. Indeed:
\begin{verbatim}
? F = mflinear([L[14],L[12]],[1,-1]); mfcoefs(F, 50)
% = [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\
0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 0]
? G = mfhecke(mf, F, 24); mfcoefs(G, 12)
% = [1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96]
? mftobasis(mf, G)
% = [0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\
0, 0, 0, 0, 0, 0]~
? 24*mfcoefs(L[5], 12)
% = [1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96]
\end{verbatim}
The first command shows that the Eisenstein series differ on their $n$-th
Fourier coefficient for $n=0$, $24$, and $48$, and the second command applies
the Hecke operator $T_{24}$ (sometimes denoted $U_{24}$) to the difference,
whose effect is to replace $a(n)$ by $a(24n)$, giving the much more
compact output of $G$. The last commands show that $G$ is equal to
$24$ times the fifth Eisenstein series \kbd{L[5]}.
\begin{verbatim}
? mf=mfinit([96,2],0); mffields(mf)
% = [y, y]
? L = mfeigenbasis(mf); for(i=1, 2, print(mfcoefs(L[i], 16)))
[0, 1, 0, 1, 0, 2, 0, -4, 0, 1, 0, 4, 0, -2, 0, 2, 0]
[0, 1, 0, -1, 0, 2, 0, 4, 0, 1, 0, -4, 0, -2, 0, -2, 0]
? Fa = mffromell(ellinit("96a1"))[2]; mfcoefs(Fa, 16)
% = [0, 1, 0, 1, 0, 2, 0, -4, 0, 1, 0, 4, 0, -2, 0, 2, 0]
? Fb = mffromell(ellinit("96b1"))[2]; mfcoefs(Fb, 16)
% = [0, 1, 0, -1, 0, 2, 0, 4, 0, 1, 0, -4, 0, -2, 0, -2, 0]
\end{verbatim}
The \kbd{mffromell} function returns a triple \kbd{[mf,F,C]},
where \kbd{mf} is the modular form cuspidal space to which \kbd{F} belongs,
\kbd{F} is the rational eigenform corresponding to the elliptic curve by
modularity, and \kbd{C} is the vector of coefficients of \kbd{F} on the
basis in \kbd{mf}, which we recall is usually not a basis of eigenforms
(otherwise \kbd{F} would belong to this basis).
Note also that \kbd{Fa} and \kbd{Fb} are twists of one another:
\begin{verbatim}
? mfisequal(mftwist(Fa, -4), Fb)
% = 1
\end{verbatim}
\section{Interlude: Dirichlet characters}
There are many ways to represent multiplicative characters on $(\Z/N\Z)^*$ in
\kbd{Pari/Gp}, we will list them by increasing order of sophistication,
restricting to characters with complex values:
\begin{itemize}
\item A quadratic character $(D/.)$ (Kronecker symbol) is described by
the integer $D$. For instance $1$ is the trivial character.
\item There is a (non-canonical but fixed) bijection between $(\Z/N\Z)^\times$
and its character group, via \emph{Conrey labels}. So \kbd{Mod}$(a,N)$
represents a character whenever $a$ is coprime to $N$. This makes it easy
to loop on all characters without worrying too much about which is which.
In this labeling, \kbd{Mod(1,N)} is the trivial character, and characters
are multiplied/divided by performing the corresponding operation on their
Conrey labels.
\item The finite abelian group $G = (\Z/N\Z)^*$ is written
$$G = \bigoplus_{i\leq n}\; (\Z/d_i\Z) \cdot g_i,$$
with $d_n \mid \dots \mid d_2 \mid d_1$ (SNF condition), all $d_i > 0$, and
$\prod_i d_i = \phi(N)$. The SNF condition makes the $d_i$ unique, but the
generators $g_i$, of respective order $d_i$, are definitely not unique. The
$\oplus$ notation means that all elements of $G$ can be written uniquely as
$\prod_i g_i^{n_i}$ where $n_i \in \Z/d_i\Z$. The $g_i$ are the so-called
\emph{SNF generators} of $G$. The command \kbd{znstar}$(N)$ outputs the SNF
structure (group order, $d_i$ and $g_i$), but $G = \kbd{znstar}(N, 1)$ is
needed to initialize a group we can work with: most importantly we can now
solve discrete logarithm problems and decompose elements on the $g_i$.
A character on the abelian group $\oplus (\Z/d_j\Z) g_j$ is given by a row
vector $\chi = [a_1,\ldots,a_n]$ of integers $0\leq a_i < d_i$ such that
$\chi(g_j) = e(a_j / d_j)$ for all $j$, with the standard notation $e(x) :=
\exp(2i\pi x)$. In other words, $\chi(\prod g_j^{n_j}) = e(\sum a_j n_j /
d_j)$. In this encoding $[0,\dots,0]$ is the trivial character. Of course
a character $\chi$ must always be given as a \emph{pair} $[G,\chi]$,
since $\chi$ is meaningless without knowledge of the $(g_i)$ or the $(d_i)$.
\end{itemize}
The command \kbd{znchar}$(S)$ converts a datum describing a character to the
third form $[G,\chi]$. The command \kbd{znchartokronecker} converts a
character of order $\leq 2$ to the first form $(D/.)$, and functions such
as \kbd{zncharconductor}, \kbd{znchartoprimitive}, and \kbd{zncharinduce}
allow to restrict or extend characters between different $(\Z/M\Z)^*$.
Note the important fact that it is necessary to give the two arguments $G$ and
$\chi$ separately to these functions, for instance \kbd{zncharconductor(G,chi)}
(and not \kbd{zncharconductor([G,chi])}).
Functions such as \kbd{charmul}, \kbd{chardiv}, \kbd{charpow},
\kbd{charorder} or \kbd{chareval} apply to more general abelian characters
than characters on $(\Z/N\Z)^\times$, whence the prefix \kbd{char} instead of
\kbd{znchar}.
\section{A Third Session: Nontrivial Characters}
Recall that a nontrivial character can be represented either by a discriminant
$D$ (not necessarily fundamental), the character being the Legendre--Kronecker
symbol $(D/n)$, or by its \emph{Conrey label} in $(\Z/N\Z)^\times$, for
instance \kbd{Mod(161,633)} (which has order 42, as \kbd{znorder} tells us).
Defining modular form spaces with character is as simple as without:
we replace the parameters $[N,k]$ by $[N,k,\chi]$.
Instead of \kbd{mf=mfinit([35,2])}, one can write
\kbd{mf=mfinit([35,2,5], 0)}, where 5 is the quadratic character $(5/.)$. Thus:
\begin{verbatim}
? mf = mfinit([35,2,5],0); mffields(mf)
% = [y^2 + 1]
? F = mfeigenbasis(mf)[1]; lift(mfcoefs(F, 10))
% = [0, 1, 2*y, -y, -2, -y - 2, 2, -y, 0, 2, -4*y + 2]
\end{verbatim}
where in the last output $y$ is equal to one of the two roots of $y^2+1=0$,
i.e., $\pm i$.
Working with nontrivial characters allows us in particular to work with odd
weights, and in particular in weight $1$:
\begin{verbatim}
? mf = mfinit([23,1,-23], 0); mfdim(mf)
% = 1
? F = mfbasis(mf)[1]; mfcoefs(F, 16)
% = [0, 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1]
? mfgaloistype(mf,F)
% = 6
\end{verbatim}
The last output means that the image in $\PSL_2(\C)$ of the projective
representation associated to $F$ is of type $D_3$. Note that an ''exotic''
representation is given by a negative number, opposite of the cardinality
of the projective image.
Since this form is of dihedral type, it can be obtained via theta functions.
Indeed:
\begin{verbatim}
? F1 = mffromqf([2,1; 1,12])[2]; V1 = mfcoefs(F1, 16)
% = [1, 2, 0, 0, 2, 0, 4, 0, 4, 2, 0, 0, 4, 0, 0, 0, 2]
? F2 = mffromqf([4,1; 1,6])[2]; V2 = mfcoefs(F2, 16)
% = [1, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 0, 4, 2, 0, 0, 4]
? (V1 - V2)/2
% = [0, 1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1]
? mfisequal(F, mflinear([F1, F2], [1, -1]/2))
% = 1
\end{verbatim}
Here we were lucky in that we ``knew'' that the correct character was
$(-23/n)$. But what if we did not know this ? The first observation is
that modular form spaces corresponding to Galois conjugate characters
are isomorphic ($\chi$ is Galois conjugate to $\chi'$ if $\chi'=\chi^m$
for some $m$ coprime to the order of $\chi$). Thus, it is sufficient
to find a representative of each equivalence class, and this is given by
the \kbd{GP} commands \kbd{G=znstar(N,1); chargalois(G)}, where $N$ is the
level of the desired character (note that $N$ will not necessarily be
the conductor of the characters). This exactly outputs a list of representative
of each equivalence class (do not for now try to understand the details of
this command, nor the fact that \kbd{chargalois} and \kbd{znstar} have
optional parameters). However, this is not quite yet what we want.
Although only for efficiency, we want characters with the same parity
as the weight, otherwise the corresponding modular form spaces will be $0$.
This is achieved by the \kbd{GP} command \kbd{zncharisodd(G,chi)} which
does what you think it does. Let us first do this for $N=23$: we write
\begin{verbatim}
? G = znstar(23, 1);
? L = [chi | chi<-chargalois(G), zncharisodd(G,chi)]; #L
% = 2
? [mfdim([23,1,[G,chi]], 0) | chi <- L ]
% = [0, 1]
? [charorder(G,chi) | chi <- L]
% = [22, 2]
\end{verbatim}
This tells us that (up to Galois conjugation) there are two possible odd
characters, one, of order $22$, giving a $0$-dimensional space, the other
being the quadratic character given above. Note that \kbd{chargalois}
returns (orbits of) characters attached to an arbitrary abelian finite group
$G$ while \kbd{mfinit} expects a \emph{pair} \kbd{[G,chi]} for some
\kbd{znstar} $G$, as written above.
When doing long explorations with all characters of a certain level, it
is preferable to use \emph{wildcards}. For instance, instead of the above
one can write:
\begin{verbatim}
? mfall = mfinit([23,1,0], 0); #mfall
% = 1
? mf = mfall[1]; mfdim(mf)
% = 1
? mfparams(mf)
% = [23, 1, -23, 0]
\end{verbatim}
This does not exactly give us the same information: the third parameter $0$
in the first command asks for \emph{all} nonempty spaces of level $23$ and
weight $1$, and the program tells us that there is only one, of dimension $1$.
The last command \kbd{mfparams} outputs \kbd{[N,k,CHI,space]}, so here tells
us that the corresponding character is the Kronecker--Legendre symbol $(-23/n)$.
Using wildcards, let us explore levels in certain ranges: we write
\begin{verbatim}
wt1exp(lim1,lim2)=
{ my(mfall,mf,chi,v);
for (N = lim1, lim2,
mfall = mfinit([N,1,0], 0); /* use wildcard */
for (i=1, #mfall,
mf = mfall[i];
chi = mfparams(mf)[3]; /* nice format: D or Mod(a,N) */
[ print([N,chi,-t]) | t<-mfgaloistype(mf), t < 0 ]
)
);
}
\end{verbatim}
For instance, \kbd{wt1exp(1,230)} outputs in 4 seconds
\begin{verbatim}
[124, Mod(87, 124), 12]
[133, Mod(83, 133), 12]
[148, Mod(105, 148), 24]
[171, Mod(94, 171), 12]
[201, Mod(104, 201), 12]
[209, Mod(197, 209), 12]
[219, Mod(8, 219), 12]
[224, Mod(95, 224), 12]
[229, Mod(122, 229), 24]
[229, Mod(122, 229), 24]
\end{verbatim}
Thus, the smallest exotic $A_4$ form is in level $124$ and the smallest $S_4$
form is in level $148$. Note that in level $229$, we have two (non Galois
conjugate) eigenforms of type $S_4$.
If we type \kbd{wt1exp(633,633)}, in 6 seconds we obtain \kbd{[633, Mod(107,
633), 60]}, and this level is indeed the lowest level for which there exists
a type $A_5$ form. The character orders are obtained either as
\kbd{znorder(chi)} (since all the \kbd{chi} are \kbd{intmods}), or
using the general construction
\begin{verbatim}
[G,v] = znstar(chi);
ord = charorder(G,v)
\end{verbatim}
where we first convert \kbd{chi} to a general abelian character in
$[G,\chi]$ format.
\section{Leaf Functions}
Although we have already seen most of these functions in the first session,
we repeat some of examples here.
\subsection{Functions Created from Scratch}
We now start a slightly more systematic exploration of the available functions.
We begin by \emph{leaf functions}, i.e., functions created from scratch or
from a given mathematical object.
\begin{verbatim}
? D = mfDelta(); mfcoefs(D, 5)
% = [0, 1, -24, 252, -1472, 4830]
? E4 = mfEk(4); mfcoefs(E4, 5)
% = [1, 240, 2160, 6720, 17520, 30240]
? E6 = mfEk(6);
? D2 = mflinear([mfpow(E4, 3), mfpow(E6, 2)], [1, -1]/1728);
? mfisequal(D, D2)
% = 1
\end{verbatim}
Self-explanatory. More complicated Eisenstein series:
\begin{verbatim}
? E3 = mfeisenstein(1, 1, -3); mfcoefs(E3, 10)
% = [1/6, 1, 0, 1, 1, 0, 0, 2, 0, 1, 0]
? E4 = mfeisenstein(5, -4, 1); mfcoefs(E4, 10)
% = [5/4, 1, 1, -80, 1, 626, -80, -2400, 1, 6481, 626]
? H2 = mfEH(5/2); mfcoefs(H2,10)
% = [1/120, -1/12, 0, 0, -7/12, -2/5, 0, 0, -1, -25/12, 0]
\end{verbatim}
The \kbd{mfeisenstein(k,c1,c2)} command generates the Eisenstein series of weight
$k$ and characters \kbd{c1} and \kbd{c2}. The \kbd{mfEH(k)} command is specific
to half-integral weight $k$ and generates the Cohen--Eisenstein series of
weight $k$.
\begin{verbatim}
? T = mfTheta(); mfcoefs(T,16)
% = [1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2]
? mf = mfinit([4, 5, -4]); mftobasis(mf, mfpow(T, 10))
% = [64/5, 4/5, 32/5]
? B = mfbasis(mf); apply(mfdescribe, B)
% = ["F_5(1, -4)", "F_5(-4, 1)", "TR^new([4, 5, -4, y])"]
? mfisCM(B[3])
% = -4
\end{verbatim}
Here, we compute the coefficients of $\th^{10}$ on the basis of \kbd{mf}
(we know of course the level, weight, and character). We then apply
the \kbd{mfdescribe} function, which tells us that the first two forms in
the basis are Eisenstein series, and the third one is some trace form
on the cuspidal new space. However, the last command says that this third
basis element is a \emph{CM form}, so that its coefficients can be computed
just as fast as those of Eisenstein series, so that there does exist
an explicit formula for the number of representations as a sum of ten
squares.
Keeping the above sessions, we can also write:
\begin{verbatim}
? mftobasis(mf, mfpow(H2, 2))
% = [1/18000, 1/18000, -3/2000]~
\end{verbatim}
\subsection{Functions Created from Mathematical Objects}
\begin{verbatim}
? [mf,F,co] = mffromell(ellinit("26b1")); co
% = [1/2, 1/2]~
? mfcoefs(F,10)
% = [0, 1, 1, -3, 1, -1, -3, 1, 1, 6, -1]
\end{verbatim}
This creates the modular form attached by modularity to the second isogeny
class of elliptic curves over $\Q$ for conductor $26$. The result is a
3-component vector: \kbd{mf} is the modular form space, $F$ the modular form,
and \kbd{co} are the coefficients of $F$ on the basis of \kbd{mf}.
Similarly, there are functions \kbd{mffromlfun} (from $L$-functions attached
to eigenforms), \kbd{mffromqf} (from quadratic forms) and \kbd{mffrometaquo}:
\begin{verbatim}
? F = mffrometaquo([1, 2; 11, 2]); mfcoefs(F, 10)
% = [0, 1, -2, -1, 2, 1, 2, -2, 0, -2, -2]
? F = mffrometaquo([1, 2; 2, -1]); mfparams(F)
% = [16, 1/2, 1, y]
? mfcoefs(F, 10)
% = [1, -2, 0, 0, 2, 0, 0, 0, 0, -2, 0]
\end{verbatim}
The \kbd{mfparams} command tells us that $F\in M_{1/2}(\G_0(16))$.
\section{Atkin, Hecke and Expanding Operators}
\begin{verbatim}
? mf = mfinit([96,4], 0); mfdim(mf)
% = 6
? M = mfheckemat(mf, 7)
% =
[0 0 0 372 696 0]
[0 0 36 0 0 -96]
[0 27/5 0 -276/5 -276/5 0]
[1 0 -12 0 0 62]
[0 0 1 0 0 -16]
[0 -3/5 0 14/5 -16/5 0]
? P = charpoly(M)
% = x^6 - 1456*x^4 + 209664*x^2 - 2985984
? factor(P)
% =
[x - 36 1]
[x - 12 1]
[ x - 4 1]
[ x + 4 1]
[x + 12 1]
[x + 36 1]
\end{verbatim}
Note a few things: first, the matrix of the Hecke operator $T(7)$ does not
have integral coefficients. Indeed, recall that the basis of modular forms
in \kbd{mf} is mostly random, so there is no reason for the matrix to be
integral. On the other hand, since the eigenvalues of Hecke operators are
algebraic integers, the characteristic polynomial of $T(7)$ must be monic
with integer coefficients. As it happens, it factors completely into
linear factors to the power $1$, so all the eigenvalues of $T(7)$ are in
fact in $\Z$: this immediately shows that the splitting will be entirely
rational and the eigenforms with integer coefficients. Let's check:
\begin{verbatim}
? mffields(mf)
% = [y, y, y, y, y, y]
? L = mfeigenbasis(mf); for(i=1,6,print(mfcoefs(L[i],16)))
[0, 1, 0, 3, 0, 10, 0, 4, 0, 9, 0, -20, 0, 70, 0, 30, 0]
[0, 1, 0, 3, 0, 2, 0, 12, 0, 9, 0, 60, 0, -42, 0, 6, 0]
[0, 1, 0, 3, 0, -14, 0, -36, 0, 9, 0, -36, 0, 54, 0, -42, 0]
[0, 1, 0, -3, 0, 10, 0, -4, 0, 9, 0, 20, 0, 70, 0, -30, 0]
[0, 1, 0, -3, 0, 2, 0, -12, 0, 9, 0, -60, 0, -42, 0, -6, 0]
[0, 1, 0, -3, 0, -14, 0, 36, 0, 9, 0, 36, 0, 54, 0, 42, 0]
\end{verbatim}
We see that of the six eigenforms, the last three are twists of the first
three.
There also exists the command \kbd{G=mfhecke(mf,F,n)}, which given a modular
form $F$ in \kbd{mf}, outputs the modular form $T(n)F$.
\begin{verbatim}
? mf=mfinit([96,6],0); mffields(mf)
% = [y, y, y, y, y, y, y^2 - 31, y^2 - 31]
? mfatk = mfatkininit(mf,3);
% factor(charpoly(mfatk[2]/mfatk[3]))
% =
[x - 1 5]
[x + 1 5]
\end{verbatim}
This requires a little explanation: the command \kbd{mfatkininit(mf,3)}
computes a number of quantities necessary to work with the Atkin--Lehner
operator $W_3$ in the space \kbd{mf}. The main part of the result is
the second component, which is essentially the matrix of $W_3$ on the
basis of \kbd{mf}, and which is guaranteed to have exact coefficients
(here rational). However in the general case, the matrix of $W_3$
is equal to \kbd{mfatk[2]/mfatk[3]}, where \kbd{mfatk[3]} may be an
inexact complex number. For now you need not worry about the first component.
Thus, the eigenvalues (or possibly the pseudo-eigenvalues) must be of modulus
$1$, and in the case of a quadratic character defined modulo $N/Q$, they
are equal to $\pm1$ in even weight, to $\pm i$ in odd weight. Here,
$1$ and $-1$ both occur $5$ times. However, this does not tell us which
eigenvalues correspond to each eigenspace. For this, we do the following:
\begin{verbatim}
? mfatkineigenvalues(mf,3)
% = [[-1], [-1], [-1], [1], [1], [1], [-1, -1], [1, 1]]
? mf=minit([96,3,-3],0); mffields(mf)
% = [y^4 + 8*y^2 + 9, y^4 + 4*y^2 + 1]
? mfatkineigenvalues(mf,32)
% = [[I, -I, -I, I], [-I, I, I, -I]]
? mfatkineigenvalues(mf,3)
% = [[a, -conj(a), -a, conj(a)], [b, -conj(b), conj(b), -b]]
\end{verbatim}
The first command tells us that in the six rational eigenspaces, the first
three have eigenvalue $-1$, the other three $+1$, and in the eigenspaces
of dimension $2$, the first eigenspace has both eigenvalues $-1$, the
second both $+1$. As is seen from the next lines, it is of course not
necessary for the eigenvalues of $W_Q$ in the same eigenspace to be equal.
In the next two commands, we are now in a case where the character is
non trivial and the weight odd. The eigenvalues are now $\pm i$, and not
equal in the same eigenspace.
Finally, the last command is a case where the character is not defined modulo
$N/Q=96/3=32$, so we only have pseudoeigenvalues, which are simply of
absolute value $1$ by Atkin--Lehner theory. Here, $a$ and $b$ are
complicated complex numbers and \kbd{conj} denotes the complex conjugate
(using the \kbd{algdep} command, one can check that $a$ is a root of
$9x^4+10x^2+9=0$ and $b$ is a root of $3x^4-2x^2+3=0$.
Note that when the character is (trivial or) quadratic and defined modulo
$N/Q$ the output is always rounded, but otherwise, the eigenvalues are given
as approximate complex numbers.
As for the Hecke operators, there exists an \kbd{mfatkin} command, whose
syntax is \kbd{mfatkin(mfatk, F)}, where \kbd{mfatk} is the output of
an \kbd{mfatkininit} command and $F$ is in the space \kbd{mfatk}, and which
outputs the modular form $F|_kW_Q$, where $Q$ is implicit in \kbd{mfatk}.
Finally note the \kbd{mfbd} expanding command which computes $B(d)F$:
\begin{verbatim}
? E4 = mfEk(4); mfcoefs(E4,6)
% = [1, 240, 2160, 6720, 17520, 30240, 60480]
? F = mfbd(E4,2); mfcoefs(F,6)
% = [1, 0, 240, 0, 2160, 0, 6720]
\end{verbatim}
\section{Algebraic Functions on Modular Forms}
Here we give examples of functions on modular forms which do not involve
any approximate numerical computation. We have already mentioned the most
important ones: \kbd{mfhecke}, \kbd{mfatkin}, and \kbd{mfbd}.
\begin{verbatim}
? E4 = mfEk(4); F = mfderivE2(E4); mfcoefs(F,5)
% = [-1/3, 168, 5544, 40992, 177576, 525168]
? E6 = mfEk(6); mfisequal(F, mflinear([E6], [-1/3]))
% = 1
? G = mfbracket(E4, E6, 1); mfcoefs(G,5)
% = [0, -3456, 82944, -870912, 5087232, -16692480]
? mfisequal(G, mflinear([mfDelta()], [-3456]))
% = 1
\end{verbatim}
\medskip
In the first commands, we compute the Serre derivative of $E_4$, and
check that it is equal to $-E_6/3$. The name \kbd{mfderivE2} of course
comes from the fact that the Serre derivative involves the quasi-modular
Eisenstein series $E_2$. Note that there exists the function \kbd{mfderiv}
(including to negative order, corresponding to integration), which is
provided for the user's convenience for certain computations, but whose
output is outside the range of modular forms.
The second computation checks that the first Rankin--Cohen bracket of
$E_4$ and $E_6$ is a multiple of $\Delta$.
You may complain that it is heavy to write an \kbd{mflinear} command as above
simply to compute a scalar multiple of a form. But nothing prevents you from
defining in a script that you read at the beginning of your session:
\begin{verbatim}
mfscalmul(F,s)=mflinear([F],[s]);
mfadd(F,G)=mflinear([F,G],[1,1]);
mfsub(F,G)=mflinear([F,G],[1,-1]);
\end{verbatim}
There also exist the natural operations on modular forms \kbd{mfmul},
\kbd{mfdiv} (which may result in modular functions, i.e., with poles),
and \kbd{mfpow}. There is also a function \kbd{mfshift} (multiply or divide
by a power of $q$), but which again takes us outside the range of modular
forms.
\begin{verbatim}
? E4 = mfEk(4); F = mftwist(E4, -3); mfcoefs(F, 7)
% = [0, 240, -2160, 0, 17520, -30240, 0, 82560]
? mfparams(F)
% = [9, 4, 1, y]
? mf = mfinit([4,5,-4], 1); F = mfbasis(mf)[1]; mfcoefs(F, 10)
% = [0, 1, -4, 0, 16, -14, 0, 0, -64, 81, 56]
? mfisCM(F)
% = -4
? G = mftwist(F, -4); mfcoefs(G, 10)
% = [0, 1, 0, 0, 0, -14, 0, 0, 0, 81, 0]
? mfparams(G)
% = [16, 5, -4, y]
? mfconductor(mfinit(G, 1), G)
% = 8
\end{verbatim}
This session illustrates a number of important issues concerning
\emph{twisting}. In the first commands, we twist $E_4$ by the quadratic
character $-3$ (in the present implementation, only twisting by quadratic
characters is allowed), and we see that the resulting form has level
$9=(-3)^2$. Fine. In the next command, we compute the unique form
in $S_4(\G_0(5),\chi_{-4})$, and see that it has CM by $\Q(\sqrt{-4})$.
However, note that the form is not equal to the form twisted by the
character $\chi_{-4}$ (only the coefficients of $q^n$ with $n$ prime to $4$
are equal, the others vanish). The \kbd{mfparams} command tells us that
the twisted form has level $16=(-4)^2$. However, the final command tells
us that in fact it has level $8$: \kbd{mfconductor} gives the smallest
level on which the form is defined.
\begin{verbatim}
? mf = mfinit([96,2], 1); L = mfbasis(mf);
? apply(x->mfconductor(mf,x), L)
% = [24, 48, 96, 32, 96, 48, 96, 96, 96]
? apply(x->mftonew(mf,x)[1][1..2], L)
% = [[24, 1], [24, 2], [24, 4], [32, 1], [32, 3],\
[48, 1], [48, 2], [96, 1], [96, 1]]
\end{verbatim}
Here we compute the full cuspidal space $S_2(\G_0(96))$, of dimension $9$,
and we ask which is the lowest level on which each form in the basis
is defined. This list shows that there is one form $F_1$ in level $24$
which, by applying $B(d)$ with $d=2$ and $d=4$ gives a form of level $48$
and one of level $96$. Then a form $F_2$ in level $32$, by applying $B(3)$
gives a form of level $96$, a form $F_3$ in level $48$, by applying $B(2)$
gives a form of level $96$, and finally two genuine forms of level $96$
(so that the dimension of the newspace is equal to $2$, which we can check
by typing \kbd{mfdim([96,2],0)}).
The last command \kbd{mftonew} checks all this; look at the precise description
of the command.
\section{Cusps and Cosets}
Recall that in the present version of the package, the only congruence
subgroup that is considered is $\G_0(N)$, so when we consider cusps in
the geometrical sense, they are cusps of $\G_0(N)$, and cosets are
right cosets of $\G_0(N)$ in $\G$, so that $\G=\bigsqcup_j\G_0(N)\ga_j$.
The function \kbd{mfcusps(N)} gives the list of all (equivalence classes of)
cusps of $\G_0(N)$, \kbd{mfcuspwidth(N,cusp)} gives the width of the cusp;
these are linked to the \emph{geometry}. On the other hand, the notion
of \emph{regularity} of a cusp is linked to the specific modular form space,
and the function \kbd{mfcuspisregular([N,k,CHI],cusp)} determines if the cusp
is regular or not:
\begin{verbatim}
? C = mfcusps(108)
% = [0, 1/2, 1/3, 2/3, 1/4, 1/6, 5/6, 1/9, 2/9, 1/12,\
5/12, 1/18, 5/18, 1/27, 1/36, 5/36, 1/54, 1/108]
? [mfcuspwidth(108,c) | c<-C]
% = [108, 27, 12, 12, 27, 3, 3, 4, 4, 3, 3, 1, 1, 4,\
1, 1, 1, 1]
? NK = [108,3,-4];
? [mfcuspisregular(NK,c) | c<-C]
% = [1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1]
? [c | c<-C, !mfcuspisregular(NK,c)]
% = [1/2, 1/6, 5/6, 1/18, 5/18, 1/54]
\end{verbatim}
The first command list the $18$ cusps of $\G_0(108)$ (\kbd{mfnumcusps(108)}
gives this directly, useful if there are thousands of cusps and you do not
want them explicitly), the second command prints their widths, and the last
commands show that the cusps $1/2$, $1/6$, $5/6$, $1/18$, $5/18$, and $1/54$
are irregular in the space $M_3(\G_0(108),\chi_{-4})$, and the others are
regular.
There is another command \kbd{mfcuspval} having to do with cusps, but this
will be mentioned later.
\medskip
\begin{verbatim}
? C = mfcosets(4)
% = [[0, -1; 1, 0], [1, 0; 1, 1], [0, -1; 1, 2],\
[0, -1; 1, 3], [1, 0; 2, 1], [1, 0; 4, 1]]
? mftocoset(4, [1, 1; 2, 3], C)
% = [[-1, 1; -4, 3], 5]
\end{verbatim}
The \kbd{mfcosets(N)} command lists all right cosets of $\G_0(N)$ in $\G$.
Note that in the present implementation the trivial coset is always the
last one, and is represented by the matrix $[1,0;N,1]$, but since this
may change one must be careful.
The \kbd{mftocoset(N,M,C)} command gives a two-component
vector $[\ga,i]$, where $\ga\in\G_0(N)$ is such that $M=\ga\cdot C[i]$.
\section{The mfslashexpansion command}
We now give examples of the use of advanced features of the package, which
use inexact complex arithmetic. However in many cases the results are known
algebraic numbers and, if asked to do so, the function gives them exactly.
This command returns the Fourier expansion at infinity of $f |_k \gamma$,
for $\gamma\in \GL_2(\Q)^+$. It returns a vector $v$ of coefficients, which
can only be interpreted together with three extra parameters
$\al\in \Q_{\geq 0}$, $w \in \Z_{\geq 1}$ and a
$2\times 2$ upper triangular matrix $A=[a,b;0,d]$
(equal to the identity if $\gamma\in\PSL_2(\Z)$). We have
$f |_k \gamma = F |_k A $, with
$$F(\tau) = q^\al \sum_{n\geq 0} v[n] q^{n/w}$$
and $q = e(\tau)$. Of course, $F |_k A = (a/d)^{k/2} F(\tau + b/d)$ so the
exact expansion is easily inferred from the returned one, whereas the chosen
encoding allows to compute the coefficients $v[n]$ in a smaller number field
than if we had included all the constants into $v$. It is important to note
that the three parameters $\al, w, A$ only depend on the modular form
space and $\gamma$, but not on the form $f$.
\begin{verbatim}
? mf = mfinit([4,6]); B = mfbasis(mf);
? for (i=1, #B, \
print( mfslashexpansion(mf,B[i],[1,0;2,1],5,1,&P) ))
\\ we don't print P which is [0, 1, [1,0;0,1]] in all cases
[-1/504, 1, 33, 244, 1057, 3126]
[-1/504, 0, 1, 0, 33, 0]
[-1/32256, -1/64, 33/64, -61/16, 1057/64, -1563/32]
[0, -1, 0, 12, 0, -54]
? R = mfslashexpansion(mf,B[1],[0,-1;4,0],5,1,&P); [P,R]
% = [[0, 1, [1,0;0,1]], [-8/63, 0, 0, 0, 64, 0]]
? R = mfslashexpansion(mf,B[1],[0,-1;1,0],5,1,&P); [P,R]
% = [[0, 4, [1,0;0,1]], [-1/504, 0, 0, 0, 1, 0]]
? mf=mfinit([4,7,-4]); B=mfbasis(mf);
? for (i=1, #B, \
print( mfslashexpansion(mf,B[i],[1,0;2,1],5,1,&P) ))
\\ we don't print P which is [1/2, 1, [1,0;0,1]] in all cases
[1/64, 91/8, 7813/32, 7353/4, 530713/64, 221445/8]
[1, -728, 15626, -117648, 530713, -1771560]
[1/16, -15/2, 5/8, 75, -231/16, -465/2]
[2, 0, 20, 0, -462, 0]
? mfslashexpansion(mf,B[1],[0,-1;4,0],5,1,&P)
% = [Mod(61/256*t, t^2 + 1), Mod(-1/64*t, t^2 + 1),
Mod(-1/64*t, t^2 + 1), Mod(91/8*t, t^2 + 1),
Mod(-1/64*t, t^2 + 1), Mod(-7813/32*t, t^2 + 1)]
? P
% = [0, 1, [1, 0; 0, 1]]
? R=mfslashexpansion(mf,B[1],[0,-1;4,0],5,0)
% = [0.23828125000000000000000000000000000000*I,\
-0.015625000000000000000000000000000000000*I,\
-0.015625000000000000000000000000000000000*I,\
11.375000000000000000000000000000000000*I,\
-0.015625000000000000000000000000000000000*I,\
-244.15625000000000000000000000000000000*I]
? bestappr(R)
% = [61/256*I, -1/64*I, -1/64*I, 91/8*I, -1/64*I, -7813/32*I]
\end{verbatim}
Here are some detailed explanations. The first space is $M_6(\G_0(4))$,
of dimension $4$. We ask for $1+5$ terms of the Fourier expansion of
$F|_6\ga$ for all $F$ in the given basis, and $\ga=[1,0;2,1]$, which is one
of the possible Fourier expansions at the cusp $1/2$. The last parameter
$P$ contains $[\al,w,A]$ after the call and the $1$ means we want an exact
algebraic expression (a $0$ as used in the last example means we expect
floating point complex numbers).
We obtain the $4$ desired expansions. In the next commands, we do the same
for the first basis element and $\ga=[0,-1;4,0]$, which is the
\emph{Fricke involution}, and corresponds to the cusp $0$. The next
command, which does essentially the same computation, uses $\ga=[0,-1;1,0]$,
and now $P=[0,4, [1,0;0,1]]$ which tells us that the expansion is in powers of
$q^{1/4}$.
The next example is the space $M_7(\G_0(4),\chi_{-4})$, also of
dimension $4$, and we ask the same thing. Now $P$ tells us that $\al = 1/2$
and $w = 1$ so we now have for instance
$$B[1]|_7\ga = (1/64)q^{1/2} + (91/8)q^{3/2}+\cdots$$
In the next command, we expand $B[1]|_7\ga$ with $\ga$ equal to the Fricke
involution; and finally we check numerically by setting the one-to-last
parameter to $0$ and obtain the expansion as raw complex numbers. We
recognize them immediately using the \kbd{bestappr} command.
Note that in the special case (like here) where $\ga$ is a Fricke (or more
generally an Atkin--Lehner) involution, we can proceed otherwise to obtain
the expansion:
\begin{verbatim}
? mfatk = mfatkininit(mf,4); C = mfatk[3]
% = -1.000000000000000000000000000*I
? F = mfatkin(mfatk, B[1]); mfcoefs(F, 6)
% = [61/256, -1/64, -1/64, 91/8, -1/64, -7813/32, 91/8]
\end{verbatim}
This tells us that the true expansion of $F|_7W_4$ is the expansion that
is output divided by the constant $C$, so we recover the previous
expansion.
\smallskip
It is important to see what affects the timing and correctness of the
\kbd{mfslashexpansion} command. The following session gives typical examples:
\begin{verbatim}
? mf = mfinit([496,4],0); F = mfbasis(mf)[1]; mfdim(mf)
time = 329 ms.
% = 45
? mfslashexpansion(mf,F,[1,0;3,1],5,0,&P);
time = 1,316 ms.
? mfslashexpansion(mf,F,[1,0;3,1],5,1,&P);
time = 51,136 ms.
? mf = mfinit([503,4],0); F = mfbasis(mf)[1]; mfdim(mf)
time = 1,505 ms.
% = 125
? sizebyte(mf)
% = 5123352
? mfslashexpansion(mf,F,[1,0;3,1],5,0,&P);
time = 32,640 ms.
? sizebyte(mf)
% = 18216400
? mfslashexpansion(mf,F,[1,0;3,1],5,0,&P);
time = 6,504 ms.
\end{verbatim}
We omit the numerical outputs since they have no significance for the present
discussion. We notice several things:
\begin{itemize}
\item First, the time for rationalization (flag $1$) in the first example
is extremely large: $51$ seconds instead of $1.3$. The reason for this is
that the width of the corresponding cusp (here $1/3$) is equal to $P[2]=496$,
and the program must recognize algebraic numbers in the large cyclotomic
field $\Q(\zeta_{496})$ which takes a huge amount of time. In fact, at the
default accuracy of $38D$, the result is certainly wrong.
\item Second, the time depends enormously on the dimension: the expansion
for dimension $125$ is $25$ times slower than for dimension $45$, of course
not surprising, but it must be taken into account.
\item Third, and most importantly, the last command shows the cache effect:
exactly the same instruction now requires only $6.5$ seconds instead of
$32.6$. This is because, behind the scenes, the first \kbd{mfslashexpansion}
precomputed a number of quantities which it stored in your variable \kbd{mf}:
in fact, the \kbd{sizebyte} commands show that, after the first expansion,
the size of \kbd{mf} has more than tripled.
\end{itemize}
\section{Analytic Commands}
The existence of the \kbd{mfslashexpansion} command allows us to do many
useful things. In fact, already the \kbd{mfatkininit} and \kbd{mfatkin}
commands would not be possible without it. Immediate applications are the
\kbd{mfcuspval} command which computes the valuation at cusps, and the
\kbd{mfeval} command, which in addition to computing values in the upper-half
plane (see below), also computes values at the cusps:
\begin{verbatim}
? T = mfTheta(); mf=mfinit(T);C=mfcusps(mf)
% = [0, 1/2, 1/4]
? [ mfcuspval(mf,T,c) | c<-C ]
% = [0, 1/4, 0]
? mfeval(mf, T, C) \\ or [mfeval(mf,T,c) | c<-C]
% = [1/2 - 1/2*I, 0, 1]
\end{verbatim}
More sophisticated is the computation of numerical periods, and more
generally of \emph{symbols}
$$\int_{s_1}^{s_2}(X-\tau)^{k-2}F|_k\ga(\tau)\,d\tau\;,$$
where $s_1$ and $s_2$ are two cusps (e.g., $s_1=0$, $s_2=\infty$):
\begin{verbatim}
? mf = mfinit([96,4],0); [F1] = mfbasis(mf);
? FS1 = mfsymbol(mf,F1);
time = 2,272 ms
? mfsymboleval(FS1,[0,oo])
% = 2.0968669678226579060336519703627002478*I*x^2\
+ 0.36368580656317635568444277442842940073*x\
- 0.049315736834713109138297211986510643780*I
? mfsymboleval(FS1,[1,5/2])
% = 4.1937339356453158120673039407254004956*I*x^2\
+ (0.72737161312635271136888554885685880147\
- 14.678068774758605342235563792538901735*I)*x\
+ (-1.2729003229711172448955497104995029026\
+ 15.103654043044843600467382361156555509*I)
? mfsymboleval(FS1,[1,2],[0,-1;1,0])
% = (0.54552870984476453352666416164264410111\
+ 2.5224522361088961642654705389803540222*I)*x^2\
+ (-0.72737161312635271136888554885685880148\
- 6.2906009034679737181009559110881007434*I)*x\
+ 4.1937339356453158120673039407254004956*I
\end{verbatim}
The general strategy for computing these quantities is first to do a
precomputation which only involves \kbd{mf} and the form $F$ using
\kbd{mfsymbol}, which can take a few seconds, but afterwards all the
computations are instantaneous.
Note that if you only want the period polynomial from $0$ to $\infty$ use
\kbd{mfperiodpol(mf,F1)} which gives the same answer as before but in only
$20$ ms.
You may also use \kbd{mfsymboleval} in two other ways, but note that in
this case the precomputation is not used so the computation may be slow:
\begin{verbatim}
? mf=mfinit([96,6],0);F=mfbasis(mf)[1];
? FS=mfsymbol(mf,F);
time = 9,761 ms.
? mfsymboleval(FS,[I,oo])
% = 0.0029721...*I*x^4 + 0.0137806...*x^3 + ... + 0.0061009...
? mfsymboleval(FS,[I,2*I])
% = 0.0029665...*I*x^4 + 0.0137326...*x^3 + ... + 0.0059760...
? mfsymboleval(FS,[I/10000,I])
% = 46.363730...*I*x^4 + 3.8815894...*x^3 + ... + 0.0183869...
? -x^4*subst(mfsymboleval(FS,[I,10000*I],[0,-1;1,0]),x,-1/x)
% = 46.363730...*I*x^4 + 3.8815894...*x^3 + ... + 0.0183869...
? mfsymboleval([mf,F],[I,oo])
% = 0.0029721...*I*x^4 + 0.0137806...*x^3 + ... + 0.0061009...
? mfsymboleval([mf,F],[I,2*I])
% = 0.0029665...*I*x^4 + 0.0137326...*x^3 + ... + 0.0059760...
\end{verbatim}
These examples illustrate four points:
\begin{enumerate}\item Computing an \kbd{mfsymbol}
may be rather long ($9.8$ seconds in this example), although as already
mentioned, subsequent computations of symbols \emph{between cusps} will then
be instantaneous.
\item As the next three commands show, \kbd{mfsymboleval}
also accepts paths with endpoints in the upper half-plane. Although we
have tried to optimize the computation, in certain cases (but not in
the above example) when one of the endpoints is close to the real line
the computation may be slow.
\item The next command shows the use of the extra parameter $\ga$ which
asks to integrate $F|_k\ga$ instead of $F$, here with \kbd{ga=[0,-1;1,0]}.
This allows to perform the same computation with endpoints which are away
from the real line. This is essentially what is done \emph{automatically} by
\kbd{mfsymboleval}.
\item The last two commands show a special format which avoids doing the
longish \kbd{mfsymbol} computation: the results are obtained almost
instantaneously \emph{without} using symbols. The price to pay in using this
``cheaper'' format is that the endpoints of the path cannot be cusps other
than \kbd{oo}.
\end{enumerate}
\begin{verbatim}
? mf = mfinit([96,4],0); [F1,F2] = mfbasis(mf);
? FS1 = mfsymbol(mf,F1); FS2 = mfsymbol(mf,F2);
? mfpetersson(FS1)
% = 0.00061471684149817788924091516302517391826
? mfpetersson(FS2)
% = 0.0055324515734836010031682364672265652647
? mfpetersson(FS1, FS2)
% = 1.5879887877319313665 E-40 + 7.652958013165934297 E-42*I
\end{verbatim}
Same remark: once the \kbd{mfsymbols} \kbd{FS1} and \kbd{FS2} initialized,
all the Petersson product computations (as well as others) are essentially
immediate. Note that since neither \kbd{F1} nor \kbd{F2} are eigenforms,
there is no reason for their Petersson product to vanish. To prove it does
we do as follows:
\begin{verbatim}
? BE = mfeigenbasis(mf);
? M = Mat([mftobasis(mf,f) | f<-BE]); M^(-1)
% =
[1 3 10 4 -20 70]
[1 3 2 12 60 -42]
[1 3 -14 -36 -36 54]
[1 -3 10 -4 20 70]
[1 -3 2 -12 -60 -42]
[1 -3 -14 36 36 54]
\end{verbatim}
On the other hand, it is immediate to see that \kbd{BE[i+3]} is a twist of
\kbd{BE[i]} and that as a consequence their Petersson square are equal. It
follows from the shape of the above matrix that the Petersson scalar product
of $B[i]$ with $B[j]$ will vanish when the corresponding scalar product of
the corresponding columns vanish, hence for $(i,j)=(1,2)$, $(1,4)$, $(1,5)$,
$(2,3)$, $(2,6)$, $(3,4)$, $(3,5)$, $(4,6)$, and $(5,6)$.
\smallskip
Note that \kbd{mfpetersson} can also be used for two noncuspidal forms, as
long as the Petersson product converges. Consider the following example:
\begin{verbatim}
? mf = mfinit([12,5,-3]); cusps = mfcusps(mf);
? E1 = mfeisenstein(5,1,-3); [mfcuspval(mf,E1,c) | c<-cusps]
% = [0, 0, 1, 0, 1, 1]
? E2 = mfeisenstein(5,-3,1); [mfcuspval(mf,E2,c) | c<-cusps]
% = [1/3, 1/3, 0, 1/3, 0, 0]
? P(mf) =
{ my(E1S = mfsymbol(mf,E1));
my(E2S = mfsymbol(mf,E2));
mfpetersson(E1S,E2S); }
? P(mf)
% = -1.8848216716468969562647734582232071466 E-5\
- 1.9057659114817512165 E-43*I
? mf3 = mfinit([3,5,-3]); P(mf3)
time = 16 ms.
? mf96 = mfinit([96,5,-3]); P(mf96)
time = 3,521 ms.
\end{verbatim}
The first commands create two Eisenstein series of weight $5$,
$E_5(1,\chi_{-3})$ and $E_5(\chi_{-3},1)$, which belong to
$M_5(\G_0(3),\chi_{-3})$. In the next commands, we look at the larger space of
level $12$ and compute the valuations of $E_1$ and $E_2$ at the six cusps of
$\G_0(12)$. We see that at these six cusps one of the two Eisenstein series
vanishes, so the Petersson product will converge, and is computed in the
next command. In the last commands we compute the same product but in level
$3$ and level $96$; because of the normalization, we obtain essentially the
same result (not given), but of course the times are very different: $0.016$
seconds in level $3$ and $3.5$ seconds in level $96$.
\medskip
There are two more important numerical functions: evaluating a modular form
at a point in the upper half plane, and evaluating the corresponding
$L$-function. We begin by a trivial example:
\begin{verbatim}
? E4 = mfEk(4); mf = mfinit(E4); mfeval(mf,E4,I)
% = 1.4557628922687093224624220035988692874
? 3*gamma(1/4)^8/(2*Pi)^6
% = 1.4557628922687093224624220035988692874
\end{verbatim}
This is of course a trivial computation, simply sum the $q$-expansion. The
fact that the value of a modular form with rational coefficients such as
$E_4$ at a \emph{CM point} such as $i$ has an explicit expression is a
consequence of complex multiplication.
\begin{verbatim}
? mf = mfinit([12,4],1); F = mfbasis(mf)[1];
? mfeval(mf, F, 1/Pi + 10^(-6)*I)
% = -89811.049350396250531782882568405506024\
- 58409.940965200894541585402642924371696*I
? mfeval(mf, F, 1/Pi + 10^(-7)*I)
% = 4.8212468504661113183253396691813292261 E-52\
+ 6.7885262281520647908871247541561415340 E-52*I
\end{verbatim}
Several remarks are in order.
\begin{enumerate}
\item We are evaluating a modular form very near the
real axis. If the form was in level $1$ such as $E_4$ above, we could use a
modular transformation to reduce to the evaluation in the fundamental domain
of $\G$, which would be very fast. Here we do something similar but
more sophisticated.
\item Contrary to most examples, the result at height $10^{-7}$ is not a
numerical approximation of $0$, the exact value is indeed as printed to
the given accuracy.
\item It is amusing to see the large oscillations of the value: at height
$10^{-6}$ the value is still in the $10^5$ range, and at $10^{-7}$ it is
in the $10^{-52}$ range. Of course it must eventually tend to $0$ since
$F$ is a cusp form (for $E_4$ it would tend to infinity).
\item When applying \kbd{mfeval} at a \emph{cusp} (as above for
\kbd{mfTheta()}), the result is
the value at the cusp, but is in general \emph{not} equal
to the limit of the value of the modular form when the argument tends to
the cusp, since this limit is often infinite for a non-cusp form.
\end{enumerate}
Note that when dealing with \emph{eigenforms}, which may have several
embeddings into $\C$, the result will have several components, one for each
embedding:
\begin{verbatim}
? mf = mfinit([23,2],0); F=mfeigenbasis(mf)[1];
? mfeval(mf,F,I)
% = [0.0018695834459685012330841605500720163964,\
0.0018618146628840767703527958851699552194]
\end{verbatim}
More generally, this embedding problem affects all numerical functions.
Continuing the above example:
\begin{verbatim}
? mfparams(F)
% = [23, 2, 1, y^2 - y - 1]
? mfslashexpansion(mf,F,[0,-1;1,0],5,1)
% = [0, -1/23, 1/23*y, -2/23*y + 1/23, -1/23*y + 1/23, 2/23*y]
? FS = mfsymbol(mf,F); mfpetersson(FS,FS)
% =
[0.00394889657400250316885... -1.0827196147167250830 E-40]
[-1.2120247024777595243 E-40 0.00564425429876478351015...]
\end{verbatim}
The $y$ in the second result is thus understood to be \emph{one} of
the roots of the polynomial $y^2-y-1$, and the result of \kbd{mfpetersson}
is a $2\times 2$ \emph{diagonal} matrix because of the two embeddings of $F$.
\smallskip
The other important evaluation function is that of the $L$-function
attached to a modular form. In fact, the modular form package only
creates (in a clever way) the $L$-function, all the rest of the work
is done by the $L$-function package. Note the important fact that
the modular form need not be an eigenform or even stable under the
Fricke involution.
\begin{verbatim}
? E4 = mfEk(4); mf=mfinit(E4); LE = lfunmf(mf,E4);
? lfun(LE, 2) / Pi^2
% = -3.3333333333333333333333333333333333333
? lfun(LE, 0)
% -1
? D = mfDelta(); mf=mfinit(D); LD = lfunmf(mf,D);
? lfunlambda(LD, 3)/lfunlambda(LD, 5)
% = 1.5555555555555555555555555555555555556
? lfunlambda(LD, 1)/lfunlambda(LD, 3)
% = 2.3444283646888567293777134587554269175
? bestappr(%)
% = 1620/691
? mf = mfinit([23,2],0); F = mfbasis(mf)[1]; L = lfunmf(mf,F);
? lfun(L, 2)
% = 1.5959983753450272580976413437480171832
? G = mfeigenbasis(mf)[1]; M = lfunmf(mf,G);
? apply(x->lfun(x,I),M)
% = [-0.15856033373254740657327844579672155664\
+ 0.79671369922504818377602680344686311969*I,\
-0.10230278816509023908993775663030712037\
+ 0.65954223983092583287784522268295299513*I]
\end{verbatim}
Note that the constant term $a(0)$ is ignored by the $L$-function, but
can be recovered thanks to the formula $a(0)=-L(F,0)$.
The last commands illustrate first the fact that the $L$-functions can be
computed for non-eigenforms ($F$ is not an eigenform), and second that
if there are several embeddings, the \kbd{lfunmf} function returns a
vector of \kbd{lfunmf}, one for each embedding.
Another illustration of the $L$-function package:
\begin{verbatim}
? LIN = lfuninit(LD, [6, 6, 50]);
? ploth(t = 0, 50, lfunhardy(LIN, t))
\end{verbatim}
%\includegraphics[width=\textwidth]{pari3.pdf}
\medskip
\section{The mfeigensearch and mfsearch commands}
The last commands that we want to illustrate are \emph{searching} commands,
The idea is simple: you believe that you have a modular form, but you do not
know its level, weight, character, or field of definition of its
coefficients, but only a number of its Fourier coefficients, perhaps only
modulo $p$, and you would like to find forms which ``match'' your given form.
In this degree of generality, the search space is too wide. We have therefore
decided to reduce the generality, so as to make the search more reasonable.
Note that this will probably vary with the different versions of the program,
so what is described here may be more restrictive than future versions. In
the present implementation, we assume that the form we are looking for has
rational coefficients, so that its character is (trivial or) quadratic.
The \kbd{mfsearch} command does this naively but is likely to be more
efficient than taylor-made scripts:
\begin{verbatim}
? V = mfsearch([60,2],[0,1,2,3,4,5,6], 1); #V
time = 5 ms.
% = 3
? V = mfsearch([[1..60],2],[0,1,2,3,4,5,6], 1); #V
time = 40 ms.
% = 5
? [ mfparams(f) | f<-V ]
% = [[56, 2, 8, y], [58, 2, 1, y],
[60, 2, 1, y], [60, 2, 12, y], [60, 2, 60, y]]
? [ print(mfcoefs(f,10)) | f<-V ]
[0, 1, 2, 3, 4, 5, 6, -6, -4, -7, -20]
[0, 1, 2, 3, 4, 5, 6, -34, 37, 22, 7]
[0, 1, 2, 3, 4, 5, 6, 20, 0, -27, -6]
[0, 1, 2, 3, 4, 5, 6, -170/9, -272/9, -11/3, -134/9]
[0, 1, 2, 3, 4, 5, 6, 200/13, -304/13, -435/13, -278/13]
\end{verbatim}
This command looks for all forms, first in level $60$ then in level $1$ to $60$
and weight $2$ whose first coefficients are $[0,1,2,3,4,5,6]$, the final $1$
is optional and specifies the \emph{space} (in \kbd{mfinit} sense) where the
search is performed, here the cuspidal space (by default the full space). It
returns a list of $3$ forms in level $60$ and $5$ in total.
The \kbd{mfeigensearch} command is more interesting. We look for is a
cuspidal \emph{eigenform} whose field of definition is $\Q$, so that its
Fourier coefficients are integers, and its character is (trivial or)
quadratic. An example is as follows:
\begin{verbatim}
? AP = [[2,2], [3,-1]] \\ a(2) = 2 and a(3) = -1
? L = mfeigensearch([[1..120],4], AP); #L
% = 2
? [f,g] = L; [mfparams(f), mfparams(g)]
% = [[26, 4, 1, y], [118, 4, 1, y]]
? mfcoefs(f, 10)
% = [0, 1, 2, -1, 4, 17, -2, -35, 8, -26, 34]
? mfcoefs(g, 10)
% = [0, 1, 2, -1, 4, -13, -2, -27, 8, -26, -26]
\end{verbatim}
The first command asks for all forms as above in weight $4$ and level from
$1$ up to $120$, such that $a(2)=2$ and $a(3)=-1$. The answer is that there
are two forms, which we call $f$ and $g$. We compute their levels ($26$ and
$118$ respectively), notice they have trivial character, and list their
Fourier coefficients up to $10$ and we see that indeed $a(2)=2$ and $a(3)=-1$
in both cases.
To specify the coefficients that we want there are a number of ways. The
simplest, as above, is to give the list of pairs of integers $[p,a(p)]$.
For instance:
\begin{verbatim}
? L = mfeigensearch([[1..80],2], [[2,2], [7,-3]]); #L
% = 1
? F = L[1]; mfparams(F)
% = [75, 2, 1, y]
? mfcoefs(F, 12)
% = [0, 1, 2, -1, 2, 0, -2, -3, 0, 1, 0, 2, -2]
\end{verbatim}
The coefficient $a(p)$ may also be given as an \kbd{intmod} \kbd{Mod}$(a,m)$
then one looks for a match for $a(p)$ modulo $m$. For instance, we come
back to our first example:
\begin{verbatim}
? AP5 = [[2,Mod(2,5)], [3,Mod(-1,5)]]; \\ now modulo 5
? L=mfeigensearch([[1..120], 4], AP); #L
% = 3
? [ mfparams(f)[1] | f <- L ]
% = [26, 26, 118]
? [F1,F2] = L; \\ let's consider the first two
? mfcoefs(F1, 10)
% = [0, 1, 2, -1, 4, 17, -2, -35, 8, -26, 34]
? mfcoefs(F2, 10)
% = [0, 1, 2, 4, 4, -18, 8, 20, 8, -11, -36]
? F = mflinear([F1, F2], [-1, 1]);
? content(mfcoefs(F, mfsturm([26,4])+1))
% = 5
\end{verbatim}
Working modulo $5$, we now find that there is an extra eigenform satisfying
our criteria, and perhaps surprisingly, again in level $26$. The first,
\kbd{F1}, is the one found above, with $a(2)=2$ and $a(3)=-1$. The second,
\kbd{F2}, has $a(2)=2$ but $a(3)=4\equiv-1\pmod5$.
But we can go further and see that this is not a simple coincidence:
the next command shows that both eigenforms seem to be congruent modulo $5$,
at least up to $a(10)$. In fact they are indeed congruent modulo $5$:
to prove this, we use the fact that the basic Sturm bound (the one obtained
using \kbd{mfsturm([N,k])}, not \kbd{mfsturm(mf)}) is also valid modulo $p$.
Since all coefficients are congruent up to the Sturm bound, they are
congruent for all $n$.
\section{Half-Integral Weight Functions}
\subsection{General Functions}
Many of the commands that we have seen, and most importantly the
\kbd{mfinit} and \kbd{mfdim} command, can be used verbatim in the case
of modular forms of half-integral weight, sometimes with small differences.
\begin{itemize}
\item Two functions created from mathematical objects can give forms
of half-integral weight, \kbd{mfetaquo} and \kbd{mffromqf}.
\item Leaf functions created from scratch are \kbd{mfTheta}, which gives the
standard Jacobi theta function of weight $1/2$, and \kbd{mfEH}, which gives
the Cohen--Hurwitz Eisenstein series of half-integral weight.
\end{itemize}
\begin{verbatim}
? F = mffrometaquo([2,5;1,-2;4,-2]); Ser(mfcoefs(F,10),q)
% = 1 + 2*q + 2*q^4 + 2*q^9 + O(q^11)
? T = mfTheta(); mfisequal(F,T)
% = 1
? F = mffromqf(2*matid(3))[2]; Ser(mfcoefs(F,5),q)
% = 1 + 6*q + 12*q^2 + 8*q^3 + 6*q^4 + 24*q^5 + O(q^6)
? mfisequal(F, mfpow(T,3))
% = 1
\end{verbatim}
\begin{itemize}\item The only spaces which are \emph{directly} available
by \kbd{mfinit} and \kbd{mfdim} are the full cuspidal space and the full
modular form space. The new space can be defined in some cases but indirectly,
using Kohnen's theory, see below.
\item The only Hecke operators $T(n)$ which are nonzero are those where
$n$ is a square (we have not programmed the $T(p)$ with $p$ dividing the
level).
\end{itemize}
\subsection{Specific Functions}
The most important specific function in half-integral weight is
\kbd{mfshimura}, which computes the Shimura lift corresponding to a
discriminant $D$ (1 by default) and also returns an \kbd{mf} space
containing the lift:
\begin{verbatim}
? mf=mfinit([60,5/2],1); F=mfbasis(mf)[1];
? D = [1,5,8,12,13,17,21];
? for (i=1, #D, \
[mf2,G] = mfshimura(mf,F,D[i]); print(mfcoefs(G,10)))
[0, 1, 2, 0, 2, -1, -2, 6, 6, -3, -10]
[0, 0, 0, -1, 0, 0, 20, 0, 0, -2, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 24, 0, 0, 0, 0, -48]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 1, 0, 3, 52, -5, -120, 14, -156, 3, 0]
? mfdescribe(mf)
% = "S_5/2(G_0(60, 1))"
? mfdescribe(mf2)
% = "S_4(G_0(30, 1))"
\end{verbatim}
Two things to notice: first the image can be identically $0$.
Second, the program takes some time (20 seconds for the above), because
computing a Shimura image takes time proportional to $D^4$.
\smallskip
The other specific functions are related to the Kohnen $+$-space. Continuing
the above example:
\begin{verbatim}
? K=mfkohnenbasis(mf); matsize(K)
% = [14, 4]
? K[,1]
% = [-1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]~
? F=mflinear(mf,K[,1]);
? Ser(mfcoefs(F,35),q)
% = -q + 2*q^4 - 4*q^16 + 3*q^21 - 6*q^24 + 5*q^25 + O(q^36)
\end{verbatim}
The first command shows that although the dimension of the cuspidal space
is $14$, that of the Kohnen $+$-space is $4$; if desired, the corresponding
modular forms can be obtained by \kbd{mflinear(mf,K[,j])} for each $j$ as
done in the next command. The Fourier expansion of $F$ given
on the last line shows that the only nonzero coefficients of $q^n$ occur
when $n\equiv0,1\pmod4$. Continuing:
\begin{verbatim}
? [mf2,FS]=mfshimura(mf,F); mfparams(FS)
% = [15, 4, 1, y]
? [mf2,FS]=mfshimura(mf,mfbasis(mf)[1]); mfparams(FS)
% = [30, 4, 1, y]
\end{verbatim}
These commmands show that the image of an element of the Kohnen $+$-space has
level $15=60/4$, while the second shows that the image of a random form has
level $30=60/2$.
\smallskip
A final command related to the Kohnen $+$-space is \kbd{mfkohnenbijection}.
This allows, in half-integral weight, to compute the new space, its
splitting, and the eigenforms:
\begin{verbatim}
? [mf3,M,K,shi] = mfkohnenbijection(mf);
? M * mfheckemat(mf3,11) * M^(-1)
% =
[ 48 24 24 24]
[ 0 32 0 -20]
[-48 -72 -40 -72]
[ 0 0 0 52]
? mf30 = mfinit(mf3,0); B0 = mfbasis(mf30); #B0
% = 2
? BNEW = [mflinear(mf, K * M * mftobasis(mf3,f)) | f<-B0];
? BE = mfeigenbasis(mf30);
? BEIGEN = [mflinear(mf, K * M * mftobasis(mf3,f)) | f<-BE ];
? Ser(mfcoefs(BEIGEN[1],24),q)
% = q + q^4 - 3*q^9 - 5*q^16 + 6*q^21 + 3*q^24 + O(q^25)
? Ser(mfcoefs(BEIGEN[2],24),q)
% = q^5 + q^8 - 3*q^12 - 4*q^17 + 3*q^20 + O(q^25)
? mfcoefs(BEIGEN[1],10^4);
time = 7,532 ms.
\end{verbatim}
The \kbd{mfkohnenbijection} command computes a square matrix $M$ giving
a Hecke-module isomorphism from the space $S_{2k-1}(\G_0(N),\chi^2)$ to the
Kohnen $+$-space $S_k^+(\G_0(4N,\chi))$. Note that this
makes sense only when $N$ is squarefree.
Thus, $M$ allows to transport all problems from the ``difficult'' space
$S=S_k^+(\G_0(4N),\chi)$ to the ``easy'' space $S_{2k-1}(\G_0(N),\chi^2)$.
For instance, the next command (essentially instantaneous) gives the matrix of
the Hecke operator $T(121)$ on $S$; a direct implementation using the action
of $T(121)$ would take $10.6$ seconds.
The vector \kbd{BNEW} computed afterwards gives a basis of the Kohnen new
space $S_k^{+,\text{new}}(\G_0(4N),\chi)$, here of dimension $2$.
The vector \kbd{BEIGEN} computed in a similar way contains the eigenfunctions
of this new space. The example of \kbd{BEIGEN[2]} shows that, contrary to
the integral weight case, these eigenfunctions can have vanishing coefficient
of $q^1$. Note that we \emph{know} by construction that the image of
\kbd{BEIGEN[j]} by any Shimura lift is a multiple of \kbd{BE[j]} (with the
same index $j$).
\smallskip
The above construction of the new space and the eigenforms being so useful,
a specific function exists for this purpose: instead of all the above,
simply write \kbd{[mf30,BNEW,BEIGEN]=mfkohneneigenbasis(mf,bij)}. Here
\kbd{BNEW} and \kbd{BEIGEN} will be matrices whose columns are the
coefficients of a basis of the Kohnen new space and of the eigenforms
respectively, and \kbd{mf30} the corresponding new space of integral weight.
\section{Reference Manual for the Package}
We give a brief description in alphabetical order of all the functions
specific to the package. To use the package, it is sometimes necessary to
use functions on characters or functions of the \kbd{lfun} package, but
those will not be described here.
Note that when a modular form $F$ can be embedded in $\C$ in several ways
(typically for eigenforms), some functions give a vector (or even a matrix
for bilinear operations) of results, one for each embedding: this occurs
specifically for \kbd{lfunmf}, \kbd{mfeval}, \kbd{mfmanin}, \kbd{mfpetersson},
\kbd{mfsymboleval}. This will not always be specified.
\def\f{\medskip\noindent}
\noindent\kbd{getcache()}: returns technical information about auto-growing
caches.
\f\kbd{lfunmf(mf,$\{F\}$)}: creates the $L$-function associated to $F$, for use
in the \kbd{lfun} package, where $F$ need not be an eigenform. If $F$ is
omitted, output all $L$-functions associated to the eigenforms. If $F$ (or
the eigenforms) have several embeddings in $\C$, output the vector of the
corresponding \kbd{lfunmf}.
\f\kbd{mfatkin(mfatk, F)}: computes $F|_k W_Q$, where $Q\Vert N$, where
\kbd{mfatk} must have been initialized by \kbd{mfatk=mfatkininit(mf,Q)}.
\f\kbd{mfatkineigenvalues(mf, Q)}: \kbd{mf} being a cuspidal or new space and
$Q$ a primitive divisor of $N$, output the vector of Atkin--Lehner
eigenvalues or pseudo-eigenvalues for each Galois eigenspace.
\f\kbd{mfatkininit(mf,Q)}: initialization function for the \kbd{mfatkin}
function. The output is \kbd{[mfb,M,C,mf]}, where $C$ is a complex constant,
$M/C$ is the matrix of the Atkin--Lehner operator $W_Q$ from the space
\kbd{mf} to the space \kbd{mfb} (set equal to $0$ if equal to \kbd{mf}). The
matrix $M$ is guaranteed to be with exact coefficients (rational or
\kbd{polmods}).
\f\kbd{mfbasis(mf,$\{\text{space}=4\}$)}: gives a basis of the space of
modular forms \kbd{mf}, either output by an \kbd{mfinit} command, in which
case \kbd{space} is ignored, or \kbd{mf=[N,k,CHI]} (use \kbd{mfeigenbasis}
for the eigenforms).
\f\kbd{mfbd(F,d)}: gives $B(d)(F)$, $B(d)$ expanding operator.
\f\kbd{mfbracket(F,G,$\{m=0\}$)}: $m$th Rankin--Cohen bracket of $F$ and $G$.
\f\kbd{mfcoef(F,n)}: $n$th Fourier coefficient $a(n)$ of $F$.
\f\kbd{mfcoefs(F,n)}: vector $[a(0),a(1),...,a(n)]$ of the Fourier coefficients
of $F$ up to $n$. If $F$ is a modular form \emph{space}, give the matrix
whose columns are the vectors of the Fourier coefficients of the basis.
\f\kbd{mfconductor(mf,F)}: smallest $M$ such that $F$ belongs to
$M_k(\G_0(M),\chi)$.
\f\kbd{mfcosets(N)}: list of right cosets of $\G$ modulo $\G_0(N)$. In the
present implementation, the trivial coset is the last and represented by
the matrix $[1,0;N,1]$. $N$ can also be an \kbd{mf}.
\f\kbd{mfcuspisregular(NK,cusp)}: \kbd{NK} being $[N,k,\chi]$ or an \kbd{mf},
determine if the cusp is regular or not.
\f\kbd{mfcusps(N)}: list of cusps of $\G_0(N)$. $N$ can also be an \kbd{mf}.
\f\kbd{mfcuspval(mf,F,cusp)}: valuation of modular form $F$ at \kbd{cusp}, which
can be a rational number or \kbd{oo}.
\f\kbd{mfcuspwidth(N,cusp)}: width of \kbd{cusp} in $\G_0(N)$. $N$ can also be
an \kbd{mf}.
\f\kbd{mfDelta()}: Ramanujan's Delta function of weight $12$.
\f\kbd{mfderiv(F,$\{m=1\}$)}: $m$th derivative $q.d/dq$ of $F$, where
$m$ can be negative, corresponding to integration (the constant term
is then set to $0$ by convention). The result is only quasi-modular.
\f\kbd{mfderivE2(F,$\{m=1\}$)}: $m$th Serre derivative $q.d/dq F-kE_2F/12$.
\f\kbd{mfdescribe(F,$\{\&G\}$)}: $F$ being a modular form or a modular form
space, gives a human-readable description of $F$. If the address of $G$ is
given, put in it the vector of parameters of the outmost operator defining
$F$ (empty vector if $F$ is a leaf or a modular form space).
\f\kbd{mfdim(mf,$\{\text{space}=4\}$)}: dimension of the space \kbd{mf},
where \kbd{mf} can also be of the form $[N,k,\chi]$ in which case \kbd{space}
is taken into account. \kbd{mf} can also be of the form $[N,k,0]$, where $0$
is a wildcard, in which case it gives detailed information for each character
$\chi$ for which the corresponding space of level $N$, weight $k$ and given
character is nonzero: each result is of the form
\kbd{[order,Conrey,dim,dimdih]}, where \kbd{Conrey} is the Conrey label for
the character, \kbd{order} is its order, \kbd{dim} is the dimension of the
corresponding space, and \kbd{dimdih}, which is computed only in weight $1$,
is the dimension of the subspace of dihedral forms.
\f\kbd{mfdiv(F,G)}: division of \kbd{F} by \kbd{G}.
\f\kbd{mfEH(k)}: $k$ being half-integral, gives the Cohen--Eisenstein series
of weight $k$ on $\G_0(4)$.
\f\kbd{mfeigenbasis(mf)}: \kbd{mf} containing the new space, gives
(in some order) the basis of normalized eigenforms.
\f\kbd{mfeigensearch(NK,AP)}: search for normalized eigenforms with
integer coefficients in spaces specified by \kbd{NK}, satisfying conditions
satisfied by \kbd{AP}. \kbd{NK} is a pair $[N,k]$, the search being in
level $N$ and weight $k$ with trivial or quadratic character; the parameter
$N$ may be replaced by a vector of allowed levels. \kbd{AP} is a list of pairs
$[[p_1,a(p_1)],...,[p_n,a(p_n)]]$, where $a(p)$ is either an integer or an
\kbd{intmod} (match modulo $a(p)\kbd{.mod}$).
\f\kbd{mfeisenstein(k,$\{\chi_1\}$,$\{\chi_2\}$)}: Eisenstein series
$E_k(\chi_1)$ or
$E_k(\chi_1,\chi_2)$, normalized so that $a(1)=1$ (so \kbd{mfeisenstein(k)}
without any character argument is equal to \kbd{mfEk(k)} multiplied by
$-B_k/(2k)$).
\f\kbd{mfEk(k)}: Eisenstein series $E_k$ for the full modular group normalized
so that $a(0)=1$, including for $k=2$.
\f\kbd{mfeval(mf,F,vtau)}: evaluation of $F$ at the point \kbd{vtau} (or a
vector of points) in the completed upper half-plane. If $F$ is an eigenform
with several embeddings in $\C$, evaluate at each embedding.
\f\kbd{mffields(mf)}: \kbd{mf} containing the new space, gives the list of
relative polynomials defining the number field extensions for all the Galois
orbits of the eigenforms. \kbd{mf} can also be a modular form, in which case
the result is the number field extension of $\Q(\chi)$ in which the Fourier
coefficients of \kbd{mf} lie.
\f\kbd{mffromell(e)}: \kbd{e} being an elliptic curve defined over $\Q$ in
\kbd{ellinit} format, gives \kbd{[mf,F,coe]}, where \kbd{F} is the eigenform
corresponding to \kbd{e} by modularity, \kbd{mf} the corresponding new space,
and \kbd{coe} the coefficients of \kbd{F} on the basis of \kbd{mf}.
\f\kbd{mffrometaquo(eta,$\{\text{flag}=0\}$)}: \kbd{eta} being a matrix
representing an eta quotient, gives the corresponding modular form or
function. If the result is not a modular form or function, return an error if
\kbd{flag=0}, or $0$ otherwise. If the result has negative valuation,
normalize to valuation $0$.
\f\kbd{mffromlfun(L)}: \kbd{L} being the $L$-function of a self-dual modular
form with rational coefficients, for instance a rational eigenform, retun
\kbd{[NK,space,v]}, where \kbd{mf = mfinit(NK,space)} is a modular form space
containing the form and \kbd{mftobasis(mf,v)} yields the coefficients of
\kbd{F} on the basis of \kbd{mf}.
\f\kbd{mffromqf(Q,$\{P\}$)}: \kbd{Q} being an even integral quadratic form of
even dimension and \kbd{P} an optional homogeneous spherical polynomial with
respect to \kbd{Q}, gives \kbd{[mf,F,coe]}, where \kbd{F} is the theta
function associated to \kbd{Q} and \kbd{P}, \kbd{mf} the corresponding space,
and \kbd{coe} the coefficients of \kbd{F} on the basis of \kbd{mf}.
\f\kbd{mfgaloistype(mf,$\{F\}$)}: \kbd{mf} being either $[N,1,\chi]$ or
a new or cuspidal space of weight $1$ forms, outputs the type of the projective
representations attached to all the eigenforms in \kbd{mf}, or only that of
\kbd{F} if it is given. The output is $2n$ for $D_n$, or $-12$, $-24$, $-60$
for $A_4$, $S_4$, $A_5$.
\f\kbd{mfhecke(mf,F,n)}: Computes $T(n)(f)$, where $T(n)$ is the $n$th Hecke
operator. Note that the level which is used is that of the modular form space
\kbd{mf}, not that of $F$ if it is different.
\f\kbd{mfheckemat(mf,n)}: matrix of $T(n)$ on the space \kbd{mf}.
\f\kbd{mfinit(NK,$\{\text{space}=4\}$)}: create the space of modular forms
associated to $NK=[N,k,\chi]$ or $NK=[N,k]$. Codes for \kbd{space} is $0$,
new space, $1$ cuspidal space, $2$ old space, $3$ space of Eisenstein series,
$4$ full space $M_k$ (default). $NK$ can also be of the form $NK=[N,k,0]$,
where $0$ is a wildcard, in which case it gives the vector of all nonzero
\kbd{mfinit} for each Galois orbit of characters $\chi$.
\f\kbd{mfisCM(F)}: returns $0$ if $F$ does not have complex multiplication,
and the CM discriminant(s) if it does. Note that in weight $1$ $F$ may have
two CM discriminants, which occurs iff its galoistype is $D_2$.
\f\kbd{mfisequal(F,G,$\{\text{lim}=0\}$)}: Are $F$ and $G$ equal, or at least
are their first \kbd{lim+1} Fourier coefficients equal ?
\f\kbd{mfkohnenbasis}(mf): \kbd{mf} being a cuspidal space of half-integral
weight and level $4N$ with $N$ squarefree, computes a basis $B$ of the Kohnen
$+$-space as a matrix whose columns are the coefficients of $B$ on the basis
of \kbd{mf}.
\f\kbd{mfkohnenbijection}(mf): \kbd{mf} being a cuspidal space of
half-integral weight, computes \kbd{[mf2,M,K,shi]}, where \kbd{M} is a matrix
giving a Hecke-module isomorphism from the cuspidal space \kbd{mf2} of weight
$2k-1$ and level $N$ to the Kohnen $+$-space of weight $k$ and level $4N$,
the columns of the matrix \kbd{K} are the coefficients of the Kohnen $+$-space
on the basis of \kbd{mf}, and \kbd{shi} gives technical information about
which linear combination of Shimura lifts has been chosen.
\f\kbd{mfkohneneigenbasis}(mf,bij): \kbd{mf} being a cuspidal space of
half-integral weight and \kbd{bij} the output of \kbd{mfkohnenbijection(mf)},
computes a triple \kbd{[mf0,Bnew,Beigen]}, where \kbd{Bnew} and \kbd{Beigen}
are matrices whose columns are the coefficients of a basis of the Kohnen
new space and of the eigenforms on the basis of \kbd{mf} respectively, and
\kbd{mf0} is the corresponding new space of integral weight $2k-1$.
\f\kbd{mflinear(vecF,vecL)}: linear combination of the forms in \kbd{vecF}
with coefficients in \kbd{vecL}. Forms must have the same weight and
character, but not necessarily the same level. This function is used
for simpler operations such as
\begin{verbatim}
mflinear([F],[s]) \\ scalar multiplication
mflinear([F,G],[1,1]) \\ addition
mflinear([F,G],[1,-1]) \\ subtraction
\end{verbatim}
If \kbd{vecF=mfbasis(mf)}, it is better to write \kbd{mflinear(mf,vecL)}
instead, since coefficient computations will be faster.
\f\kbd{mfmanin(FS)}: $FS$ being a modular symbol associated to an eigenform,
returns $[[P^+,P^-],[\omega^+,\omega^-,r]]$ where the $P^{\pm}$ are the
even/odd polynomials of special values, the $\omega^{\pm}$ the
corresponding periods, and $r=\Im(\omega^+\overline{\omega^-})/<F,F>$.
\f\kbd{mfmul(F,G)}: product of the modular forms $F$ and $G$.
\f\kbd{mfnumcusps(N)}: number of cusps of $\G_0(N)$.
\f\kbd{mfparams(F)}: returns parameters $[N,k,\chi,P]$ of the modular form $F$,
where $K$ is the polynomial defining the number field containing the
coefficients of $F$ (e.g., $y$ if $F$ is rational), or $[-1,-1,-1,0]$ if
it is not defined. If $F$ is a modular form space, returns $[N,k,\chi,space]$.
\f\kbd{mfperiodpol(mf,F,$\{\text{parity}=0\}$)}: period polynomial of the
form $F$; if the \kbd{parity} argument is $1$ or $-1$, return the even/odd
period polynomial.
\f\kbd{mfperiodpolbasis(k,$\{\text{parity}=0\}$)}: basis of period polynomials of weight
$k$ for the full modular group, even/odd ones if \kbd{parity} is $1$ or $-1$.
\f\kbd{mfpetersson(FS,$\{\text{GS}=\text{FS}\}$)}: $FS$ and $GS$ being the modular symbols associated
to $F$ and $G$ with \kbd{mfsymbol}, computes the Petersson product of $F$ and
$G$ with the usual normalization $1/[\G:\G_0(N)]$. (Petersson square if $GS$
is omitted.)
\f\kbd{mfpow(F, n)}: Modular form $F$ to the power $n$.
\f\kbd{mfsearch([N,k], V, $\{\text{space}=4\}$)}: search for \emph{rational}
modular forms of weight~$k$ and level $N$ in the specified modular
form spaces whose Fourier expansion up to the length of $V$ exactly matches
$V$. The output is a list of forms. The parameter $N$ may be replaced
by a list of allowed levels, e.g. \kbd{[$N_1$..$N_2$]} for all levels
between $N_1$ and $N_2$.
\f\kbd{mfshift(F,m)}: \kbd{F} divided by $q^m$, omitting the remainder if there
is one, where $m$ can be positive or negative. The result is usually not
a modular form.
\f\kbd{mfshimura(mf,F,$\{D = 1\}$)}: $F$ being a modular form of
half-integral weight $k\ge3/2$ and $D$ a discriminant, return
\kbd{[mf2,FS,v]}, where \kbd{FS} is the corresponding Shimura lift of integral
weight $2k-1$, \kbd{mf2} the corresponding modular form space and \kbd{v} the
coefficients of \kbd{FS} on the basis of \kbd{mf2}. By extension, $D$ can also
be a positive squarefree integer.
\f\kbd{mfslashexpansion(mf,f,g,n,flrat,$\{\&P\}$)}: compute the Fourier
expansion of $f|_k g$ to order $n$, where $f$ is a form in \kbd{mf} and
$g\in M_2^+(\Q)$. If \kbd{flrat} is set, try to ``rationalize''
(error if unsuccessful). If the output is $[a(0),...,a(n)]$ and the
optional $P$ contains parameters $[\al,w,A]$, then $f|_k g = F|_k A$
where
$F(\tau) = q^{\al}\sum_{0\le j\le n}a(j)q^{j/w}$, with $q=\exp(2\pi i\tau)$.
$A$ is always upper triangular and usually the identity, so that $F|_k A$
is immediate to compute.
\f\kbd{mfspace(mf,$\{F\}$)}: type of modular space \kbd{mf} if $F$ is omitted,
or of a modular form $F$ in \kbd{mf}: result is $0$ for new, $1$ for cuspidal,
$2$ for old, $3$ for full, $4$ for Eisenstein, and $-1$ if form is not in
the space.
\f\kbd{mfsplit(mf,$\{\text{dimlim}=0\}$,$\{\text{flag}=0\}$)}: compute the
eigenforms in \kbd{mf}, and limit the dimension of each Galois orbit to
\kbd{dimlim} if set. \kbd{flag} is used to avoid some long computations (see
doc). The space \kbd{mf} \emph{must} contain the new space. Note that the
result is only a two-component vector \kbd{vF,vK}, where \kbd{vF} is a vector
of eigenforms and \kbd{vK} the corresponding number fields, but is \emph{not}
similar to the output of an \kbd{mfinit} command.
\f\kbd{mfsturm(mf)}: If \kbd{mf} is a space, true Sturm bound of \kbd{mf}, i.e.,
largest valuation at infinity of a nonzero form. If \kbd{mf} is $[N,k,\chi]$,
only an upper bound.
\f\kbd{mfsymbol(mf,F)}: initialize data for working with integrals related
to $F$ such as \kbd{mfsymboleval}, \kbd{mfpetersson}, and \kbd{mfmanin}.
\f\kbd{mfsymboleval(FS,path,$\{\ga\}$)}: $FS$ being the modular symbol
assocated to some form $F$ and \kbd{path} being $[s_1,s_2]$ where $s_1$ and
$s_2$ are cusps or points in the upper half-plane, evaluate the symbol on the
path, i.e., compute the polynomial
$\int_{s_1}^{s_2}(X-\tau)^{k-2}F(\tau)\,d\tau$. If $\ga\in GL_2^+(\Q)$ is
given, replace $F$ by $F|_k\ga$. If the integral diverges, the result will be
either a rational function or a polynomial of degree $d>k-2$.
\f\kbd{mftaylor(F,n,$\{\text{fl}=0\}$)}: for now, only for $F\in M_k(\G)$ and
at the point $i$. Compute the first $n$ Taylor coefficients of $F$ around
$i$; if \kbd{fl} is set compute in fact $p_n$ such that
$$f(\tau)=(2i/(\tau+i))^k\sum_{n\ge0}p_n((\tau-i)/(\tau+i))^n\;.$$
\f\kbd{mfTheta($\{\chi\}$)}: unary theta series corresponding to the primitive
Dirichlet character $\chi$, thus in weight $1/2$ (resp., $3/2$)
if $\chi$ is even (resp., odd).
\f\kbd{mftobasis(mf,F,$\{\text{flag}=0\}$)}: coefficients of form $F$ on the
basis in \kbd{mf}. If \kbd{flag} is set, do not return an error if $F$ does
not belong to \kbd{mf} or not enough coefficients.
\f\kbd{mftocoset(N,M,L)}: $L$ being the list of cosets output by
\kbd{L=mfcosets(N)} and $M$ being in $\SL_2(\Z)$, output a pair
$[\ga,i]$ such that $M=\ga L[i]$, where $\ga\in \G_0(N)$.
\f\kbd{mftonew(mf,F)}: Decompose $F$ is the cuspidal space \kbd{mf} as
a sum of $B(d)G_M$ where $G_M\in S_k^{\new}(\G_0(M),\chi)$ and $dM\mid N$,
return the vector of $[M,d,G]$.
\f\kbd{mftraceform(NK,$\{\text{space}=0\}$)}: gives the trace form
corresponding to $NK=[N,k,\chi]$ and \kbd{space} (only the new space and the
cuspidal space).
\f\kbd{mftwist(F,D)}: twist of the form $F$ by the quadratic character
$(D/n)$.
\end{document}