Modular forms in Pari/GP

Abstract

We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of noneigenforms, and compute general Petersson scalar products.

1 Introduction

Three packages exist which allow computations on classical modular forms: Sage, Magma, and Pari/GP, the latter being available since the spring of 2018. The first two packages are based on modular symbols, while the third is based on trace formulas. This difference is not so important (although the efficiency of certain computations can vary widely from one package to another), but at present the Pari/GP package is the only one which is routinely able to perform a number of computations on modular forms such as expansions at cusps, evaluation near the real axis, evaluation of L-functions of noneigenforms, computation of general Petersson scalar products.

The method used for these more advanced commands is based on the one hand on a theorem of Borisov–Gunnells [1, 2] stating that with known exceptions (which can easily be circumvented) spaces of modular forms are generated by products of two Eisenstein series, and on the other hand on tedious computations on the expansions of these Eisenstein series. None of this is completely original, but it took us several months before obtaining a satisfactory implementation. In addition, note that we do not need the Borisov–Gunnells theorem (in fact in the beginning we were not even aware of their work) since we can always check whether products of two Eisenstein series generate the desired spaces (which they sometimes do not in weight 2, but this can be circumvented).

This paper is divided into three parts. In the first part (Sects. 2, 3, 4, 5, 6), we describe the theoretical tools used in the construction of modular form spaces. In the short second part (Sect. 7), we give some implementation details. In the third somewhat lengthy part (Sects. 8, 9), we give some sample commands and results obtained using the package, with emphasis on the advanced commands not available elsewhere.

2 Construction of spaces of integral weight $k\ge 2$

2.1 Introduction

We decided from the start to restrict to spaces of classical modular forms, and more precisely to the usual spaces $M_k({\varGamma }_0(N),\chi )$ where $\chi$ is a Dirichlet character modulo N of suitable parity, including $k=1$ and k half-integral. In addition to this full modular form space, we also want to construct the space of cusp forms $S_k({\varGamma }_0(N),\chi )$, the space of Eisenstein series ${{\mathscr {E}}}_k({\varGamma }_0(N),\chi )$, (so that $M_k({\varGamma }_0(N),\chi )={{\mathscr {E}}}_k({\varGamma }_0(N),\chi )\oplus S_k({\varGamma }_0(N),\chi )$), the space of newforms $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$, and the space of oldforms $S_k^{\mathrm {old}}({\varGamma }_0(N),\chi )$ (so that $S_k({\varGamma }_0(N),\chi )=S_k^{\mathrm {old}}({\varGamma }_0(N),\chi )\oplus S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$).

Other finite index subgroups of ${{\mathrm{SL}}}_2(\mathbb {Z})$ could be considered, as well as other subspaces of $M_k({\varGamma }_0(N),\chi )$, such as Skoruppa–Zagier’s certain space [7], but to limit the amount of work we have restricted ourselves to the above. In this section, we assume that $k \ge 2$ is an integer and defer half-integral weights and weight 1 to later sections.

2.2 Construction of ${{\mathscr {E}}}_k({\varGamma }_0(N),\chi )$

The construction of the space of Eisenstein series is easy and based on a theorem apparently first published by Weisinger [9]. Recall that if $\chi _1$ and $\chi _2$ are two primitive characters modulo $N_1$ and $N_2$, respectively, we define for $k>2$ the Eisenstein series

and if $k=1$ or $k=2$, one defines $G_k$ by analytic continuation to $s=0$ of the corresponding series where $(c\tau +d)^k$ is multiplied by $|c\tau +d|^{2s}$ (“Hecke’s trick”). Then, $G_k(\chi _1,\chi _2)$ belongs to $M_k({\varGamma }_0(N_1N_2),\chi _1\chi _2)$, except when $k=2$ and $\chi _1$ and $\chi _2$ are trivial characters, in which case there is a nonanalytic term in $1/\mathfrak {I}(\tau )$.

We introduce the following useful notation: if $\chi$ is any Dirichlet character, we denote by $\chi _f$ the primitive character equivalent to $\chi$, and we denote by 1 the trivial character modulo 1. Weisinger’s theorem (slightly corrected for $k=2$) is as follows:

Theorem 1

• 1. For $k\ge 3$ or for $k=2$ and $\chi$ a nontrivial character, a basis of the space ${{\mathscr {E}}}_k({\varGamma }_0(N),\chi )$ of Eisenstein series is given by the $G_k(\chi _1,\chi _2;m\tau )$, where $(\chi _1,\chi _2)$ ranges over pairs of primitive characters as above such that $(\chi _1\chi _2)_f=\chi _f$ and $N_1N_2\mid N$, and m ranges over all divisors of $N/(N_1N_2)$.

• 2. For $k=2$ and $\chi$ a trivial character, a basis of ${{\mathscr {E}}}_2({\varGamma }_0(N))$ is given by the same functions as in (1) except that if $(\chi _1,\chi _2)=(1,1)$ we replace $G_k(\chi _1,\chi _2;m\tau )$ by $G_2(\chi _1,\chi _2;m\tau )-G_2(\chi _1,\chi _2;\tau )/m$ and exclude $m=1$.

• 3. For $k=1$, a basis of ${{\mathscr {E}}}_1({\varGamma }_0(N),\chi )$ is given by the same functions as in (1), except that we restrict to $\chi _1$ being an even character.

[Note that since in weight 1 (and only in weight 1) the characters $\chi _1$ and $\chi _2$ play a symmetrical role, we could instead restrict to $\chi _2$ being an even character.]

Thanks to this theorem, it is immediate to construct a basis of ${{\mathscr {E}}}_k({\varGamma }_0(N),\chi )$. However, this is not the whole story. Indeed, one can easily compute the Fourier expansion at infinity of $G_k(\chi _1,\chi _2;\tau )$ and (after suitable normalization) the coefficients belong to the large cyclotomic field $\mathbb {Q}(\zeta _{o_1},\zeta _{o_2})$, where $o_i$ denotes the order of the character $\chi _i$ and $\zeta _n$ denotes a primitive nth root of unity. It is in fact possible to obtain a basis whose Fourier coefficients are in the smaller cyclotomic field $\mathbb {Q}(\zeta _o)$, where o is the order of $\chi$. For this, we introduce the following notation:

Definition 2

• 1. ${{\mathrm{Tr}}}_{1,2}$ will denote the trace map from $\mathbb {Q}(\zeta _{o_1},\zeta _{o_2})$ to $\mathbb {Q}(\zeta _o)$.

• 2. We will say that two primitive characters $\chi$ and $\chi '$ modulo N are equivalent and write $\chi \sim \chi '$ if there exists j coprime to the order of $\chi$ such that $\chi '=\chi ^j$.

Theorem 3

Let $d_{1,2}=[\mathbb {Q}(\zeta _{o_1},\zeta _{o_2}):\mathbb {Q}(\zeta _o)]$ be the degree of the field extension and let $\alpha _{1,2}$ be such that $\mathbb {Q}(\zeta _{o_1},\zeta _{o_2})=\mathbb {Q}(\zeta _o)(\alpha _{1,2})$. A basis for the space ${{\mathscr {E}}}_k({\varGamma }_0(N),\chi )$ is given by the ${{\mathrm{Tr}}}_{1,2}(\alpha _{1,2}^jG_k(\chi _1,\chi _2;m\tau ))$ where $0\le j<d_{1,2}$ and $(\chi _1,\chi _2,m)$ are as in the previous theorem (with the suitable modification when $k=2$) except that $\chi _1$ is only chosen up to equivalence.

Thus, we indeed obtain a basis of the Eisenstein space whose Fourier expansions have coefficients in the smaller field $\mathbb {Q}(\zeta _o)$, for instance in $\mathbb {Q}$ if $\chi$ is trivial or a quadratic character.

2.3 Construction of $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$ and of $S_k({\varGamma }_0(N),\chi )$ when $k\ge 2$

Here, the Pari/GP package differs from the others. (Note that we do not claim that this is a better choice). First recall the Eichler–Selberg trace formula on ${\varGamma }_0(N)$. For every n including those not coprime to N, one defines a Hecke operator T(n) on $M_k({\varGamma }_0(N),\chi )$ by the formula

$$\begin{aligned} T(n)(f)(\tau )=\dfrac{1}{n}\sum _{\begin{array}{c} ad=n\\ \gcd (a,N)=1 \end{array}}\chi (a)a^k\sum _{b\bmod d}f\left( \dfrac{a\tau +b}{d}\right) . \end{aligned}$$

Because of the condition $\gcd (a,N)=1$ (which is irrelevant if $\gcd (n,N)=1$), it is important to note that when $\gcd (n,N)>1$, the operator T(n) depends on the level N of the underlying space, so should more properly be denoted $T_N(n)$. Equivalently, if we always consider $\chi$ as a Dirichlet character modulo N, so such that $\chi (a)=0$ when $\gcd (a,N)>1$, we can omit the condition $\gcd (a,N)=1$.

An important formula, due to Selberg and Eichler, gives the trace of T(n) on $S_k({\varGamma }_0(N),\chi )$:

Theorem 4

Let $\chi$ be a Dirichlet character modulo N and let $k\ge 2$ be an integer such that $\chi (-1)=(-1)^k$. For all $n\ge 1$, including those not coprime to N, we have

$$\begin{aligned} {{\mathrm{Tr}}}_{S_k({\varGamma }_0(N),\chi )}(T(n))=A_1-A_2-A_3+A_4, \end{aligned}$$

where the different contributions $A_i$ are as follows:

$$\begin{aligned} A_1=n^{k/2-1}\chi (\sqrt{n})\dfrac{k-1}{12}N\prod _{p\mid N}\left( 1+\dfrac{1}{p}\right) , \end{aligned}$$

where it is understood that $\chi (\sqrt{n})=0$ if n is not a square (including when $\chi$ is a trivial character).

$$\begin{aligned} A_2=\sum _{\begin{array}{c} t\in \mathbb {Z}\\ t^2-4n<0 \end{array}}\dfrac{\rho ^{k-1}-\overline{\rho }^{k-1}}{\rho -\overline{\rho }}\sum _{f^2\mid (t^2-4n)}\dfrac{h((t^2-4n)/f^2)}{w((t^2-4n)/f^2)}\mu (t,\gcd (N,f),n), \end{aligned}$$

with

$$\begin{aligned} \mu (t,g,n)=g\prod _{\begin{array}{c} p\mid N\\ p\not \mid N/g \end{array}}\left( 1+\dfrac{1}{p}\right) \sum _{\begin{array}{c} x\bmod N\\ x^2-tx+n\equiv 0\pmod {Ng} \end{array}}\chi (x), \end{aligned}$$

where $\rho$ and $\overline{\rho }$ are the roots of the polynomial $X^2-tX+n$, in other words, $\rho +\overline{\rho }=t$ and $\rho \overline{\rho }=n$, and for $d<0$, h(d) and w(d) are the class number and number of roots of unity of the quadratic order of discriminant d.

where:

• means that the term $d=n^{1/2}$, if present, must be counted with coefficient 1 / 2,

• ${{\mathfrak {f}}}(\chi )$ is the conductor of $\chi$,

• $x_1$ is defined modulo ${{\mathrm{lcm}}}(c,N/c)=N/\gcd (c,N/c)$ by the Chinese remainder congruences $x_1\equiv d\pmod {c}$ and $x_1\equiv n/d\pmod {N/c}$,

• $\phi$ is Euler’s totient function.

$A_4=0$ if either $k>2$ or if $k=2$ and $\chi$ is not the trivial character, and otherwise, if $k=2$ and $\chi$ is trivial then

$$\begin{aligned} A_4=\sum _{\begin{array}{c} t\mid n\\ \gcd (n/t,N)=1 \end{array}}t. \end{aligned}$$

We emphasize that in all the above formulas $\chi (x)=0$ if $\gcd (x,N)>1$, i.e., $\chi$ is always considered as a character modulo N.

From this, a nontrivial application of the Möbius inversion formula (explained to us by J. Bober, A. Booker, and M. Lee) allows us to compute the trace on the new space $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$. To simplify notation, denote by ${{\mathrm{Tr}}}(N,n)$ (resp., ${{\mathrm{Tr}}}^{\mathrm {new}}(N,n)$) the trace of $T_N(n)$ on $S_k({\varGamma }_0(N),\chi )$ (resp., $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$). We introduce the following definitions:

Definition 5

• 1. We define the multiplicative arithmetic function $\beta (n)$ on prime powers by $\beta (p)=-2$, $\beta (p^2)=1$, and $\beta (p^a)=0$ for $a\ge 3$.

• 2. For $m\ge 1$, we define the multiplicative arithmetic functions $\beta _m(n)$ on prime powers by $\beta _m(p^a)=\beta (p^a)$ if $p\not \mid m$ and $\beta _m(p^a)=\mu (p^a)$ if $p\mid m$, where $\mu$ is the usual Möbius function.

• 3. An integer N is said to be squarefull if for all primes $p\mid N$ we have $p^2\mid N$.

Theorem 6

Let $\chi _N$ be a Dirichlet character modulo N of conductor ${{\mathfrak {f}}}\mid N$, and $k\ge 2$ be an integer such that $\chi _N(-1)=(-1)^k$. Denote as above by $\chi _{{{\mathfrak {f}}}}$ the primitive character modulo ${{\mathfrak {f}}}$ equivalent to $\chi _N$. Finally, write $N=N_1N_2$ with $\gcd (N_1,N_2)=1$, $N_1$ squarefree and $N_2$ squarefull. We have

$$\begin{aligned} {{\mathrm{Tr}}}^{\mathrm {new}}(N,n)=\sum _{{{\mathfrak {f}}}\mid M\mid N}\sum _{\begin{array}{c} d\mid \gcd (M/{{\mathfrak {f}}},N_1)\\ d^2\mid n \end{array}}\chi _{{{\mathfrak {f}}}}(d)d^{k-1}\beta _{n/d^2}(N/M){{\mathrm{Tr}}}(M/d,n/d^2). \end{aligned}$$

The point of this theorem is the following: set

$$\begin{aligned} {{\mathscr {T}}}^{\mathrm {new}}(N)=\sum _{n\ge 1}{{\mathrm{Tr}}}^{\mathrm {new}}(N,n)q^n. \end{aligned}$$

Then, ${{\mathscr {T}}}^{\mathrm {new}}(N)$ is equal to the sum of the normalized eigenforms in $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$, hence a simple argument shows that the $T(n){{\mathscr {T}}}^{\mathrm {new}}(N)$ generate $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$, so we simply construct these forms until the dimension of the space they generate is equal to the dimension of the full new space (equal to ${{\mathrm{Tr}}}^{\mathrm {new}}(N,1)$).

Once we have obtained a basis for the space $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$, it is immediate to obtain a basis of $S_k({\varGamma }_0(N),\chi )$ thanks to the relation

$$\begin{aligned} S_k({\varGamma }_0(N),\chi )=\bigoplus _{{{\mathfrak {f}}}\mid M\mid N}\bigoplus _{d\mid N/M}B(d)S_k^{\mathrm {new}}({\varGamma }_0(M),\chi _{{{\mathfrak {f}}}}), \end{aligned}$$

where B(d) is the usual expanding operator $\tau \mapsto \mathrm{d}\tau$.

The old space is given by the same formula but restricting to $M<N$:

$$\begin{aligned} S_k^{\mathrm {old}}({\varGamma }_0(N),\chi )=\bigoplus _{\begin{array}{c} {{\mathfrak {f}}}\mid M\mid N \\ M<N \end{array}}\bigoplus _{d\mid N/M}B(d)S_k^{\mathrm {new}}({\varGamma }_0(M),\chi _{{{\mathfrak {f}}}}), \end{aligned}$$

Note that one could think of using directly the trace formula on the full cuspidal space (Theorem 4), but experiment and complexity analysis both show that, in addition to being much less canonical, it would also be less efficient. (Since on the one hand the $A_i$ to be computed are eventually the same, and on the other hand linear algebra’s cost is superlinear in the dimension, it is more costly to work in a direct sum than in each subspace independently.)

3 Construction of modular forms of half-integral weight

Recall that modular form spaces of half-integral weight $M_k({\varGamma }_0(N),\chi )$ with $k\in 1/2+\mathbb {Z}$ are defined only when $4\mid N$ and $\chi$ is an even character. In weight 1 / 2, a beautiful theorem of Serre–Stark asserts that the space $M_{1/2}({\varGamma }_0(N),\chi )$ is spanned by unary theta series, and the theorem also specifies the cuspidal subspace $S_{1/2}({\varGamma }_0(N),\chi )$. Thus, we consider the construction of the spaces $S_k({\varGamma }_0(N),\chi )$ and $M_k({\varGamma }_0(N),\chi )$ when $4\mid N$, $k\ge 3/2$ is a half-integer, and $\chi$ is an even character.

Recall the standard theta series

$$\begin{aligned} \theta (\tau )=\sum _{n\in \mathbb {Z}}q^{n^2}=1+2\sum _{n\ge 1}q^{n^2}\in M_{1/2}({\varGamma }_0(4)). \end{aligned}$$

Because of the well-known product expansion

$$\begin{aligned} \theta (\tau )=\prod _{n\ge 1}(1-q^{2n})(1+q^{2n-1})^2, \end{aligned}$$

it is clear that $\theta$ does not vanish on the upper half plane ${\mathfrak {H}}$, and it does not vanish at the cusps $i\infty$ and 0 of ${\varGamma }_0(4)$. On the other hand, since the cusp 1 / 2 is irregular, $\theta$ necessarily vanishes at the cusp 1 / 2. Applying ${\varGamma }_0(4)$, we see that $\theta (\tau )=0$ if and only if $\tau$ is a cusp of the form a / b with $\gcd (a,b)=1$ and $b\equiv 2\pmod 4$.

It follows that $\theta (2\tau )=0$ if and only if $\tau$ is a cusp of the form a / b with $b\equiv 4\pmod 8$. In particular, we see the essential fact that $\theta (\tau )$ and $\theta _2(\tau )=\theta (2\tau )$ have no common zeros in the completed upper half plane (we say that they are coprime forms).

This allows us to construct the desired modular form spaces of half-integral weight as follows. Let $k\in \mathbb {Z}+1/2$ and say we want to construct $M_k({\varGamma }_0(N),\chi )$. If $f\in M_k({\varGamma }_0(N),\chi )$, then $f\theta \in M_{k+1/2}({\varGamma }_0(N),\chi ')$, where $\chi '=\chi$ if $k+1/2\equiv 0\pmod 2$ otherwise $\chi '=\chi \chi _{-4}$, where , and similarly

$$\begin{aligned} f\theta _2\in M_{k+1/2}({\varGamma }_0(N'),\chi ')\supset M_{k+1/2}({\varGamma }_0(N),\chi '), \end{aligned}$$

where $N'=N$ if $8\mid N$, and $N'=2N$ otherwise. By the preceding section, we know how to construct a basis B of $M_{k+1/2}({\varGamma }_0(N'),\chi ')$.

Now the forms $g_1=f\theta$ and $g_2=f\theta _2$ which are both in that space satisfy $g_1\theta _2=g_2\theta$. This equality can be solved by simple linear algebra on the basis B, and once a basis of $(g_1,g_2)$ is found, one recovers f as the quotient $g_1/\theta$ (or $g_2/\theta _2$). Since the level $N'$ is at most twice the initial level N, this gives an efficient method for computing $M_k({\varGamma }_0(N),\chi )$. To compute the cuspidal space $S_k({\varGamma }_0(N),\chi )$, simply replace all the $M_{k+1/2}$ by $S_{k+1/2}$.

Note that we have a check on the correctness of the result by using a theorem of Oesterlé and the second author [3] which gives the dimensions of $M_k({\varGamma }_0(N),\chi )$ and $S_k({\varGamma }_0(N),\chi )$ when $k\in 1/2+\mathbb {Z}$.

The reader has certainly noticed that we do not speak of the old/new space, nor of the Eisenstein space. The construction of the old/new space is better performed in the so-called Kohnen $+$-space and is implemented in the package, but we do not explain the details here.

It should be possible to construct the Eisenstein space explicitly in a manner analogous to Weisinger’s theorem, but as far as the authors are aware this has been done only when N / 4 is squarefree.

4 Construction of modular forms of weight 1

Although in principle it is algorithmically just as simple to construct modular forms of weight 1 as modular forms of half-integral weight, it is more difficult to do it efficiently.

A first method which comes to mind is again to use two coprime forms. In fact, we can again use $\theta$ and $\theta _2$ as in the previous section, but to stay in the realm of integral weight forms it is preferable to use the weight 1 coprime forms $\theta ^2$ and $\theta _2^2$. This works exactly in the same way that we used for half-integral weight, but the main efficiency loss is due to the level: since $\theta _2^2\in M_1({\varGamma }_0(8),\chi _{-4})$, the level of $f\theta _2^2$ will be $N' = {{\mathrm{lcm}}}(8,N)$. In the half-integral case, we always had $4\mid N$, so $N'$ was at most 2N, but here if for instance N is odd, $N'=8N$ so we are required to work in a space of weight 2 forms and level 8 times larger, which is prohibitive since the complexity is at least proportional to the cost of linear algebra in dimension $N'$.

We can search for other coprime forms. For instance, J. Bober (personal communication) suggests to use two specific Eisenstein series of weight 1 and levels 3 and 4, respectively. We would then need to work in level ${{\mathrm{lcm}}}(12,N)$, which can be lower than ${{\mathrm{lcm}}}(8,N)$ when $3\mid N$ for instance.

However, to our knowledge the most efficient method to construct spaces of modular forms of weight 1, and the one which is implemented in the Pari/GP package, is the use of Schaeffer’s Hecke stability theorem. This theorem essentially states the following: if V is a finite-dimensional vector space of meromorphic modular functions over ${\varGamma }_0(N)$ with character $\chi$, and if V is stable by any single Hecke operator T(n) with n coprime to N, then V is in fact a space of holomorphic modular forms.

Since Eisenstein series of weight 1 are just as explicit as in higher weight, it is sufficient to construct the cuspidal space $S_1({\varGamma }_0(N),\chi )$, and to do so we proceed as follows. Let ${{\mathscr {E}}}_1({\varGamma }_0(N),\overline{\chi })$ be the space of Eisenstein series of weight 1 and conjugate character. Note that if $f\in S_1({\varGamma }_0(N),\chi )$ and $E\in {{\mathscr {E}}}_1({\varGamma }_0(N),\overline{\chi })$, then $fE\in S_2({\varGamma }_0(N))$. Consider the (finite dimensional) space

$$\begin{aligned} W=\bigcap _{E\in B_1({\varGamma }_0(N),\overline{\chi })}\dfrac{S_2({\varGamma }_0(N))}{E}, \end{aligned}$$

where $B_1({\varGamma }_0(N),\overline{\chi })$ is a basis of ${{\mathscr {E}}}_1({\varGamma }_0(N),\overline{\chi })$. It is clear that $S_1({\varGamma }_0(N),\chi )\subset W$, and sometimes W is 0 (for instance if $N\le 22$), so we are done. In general, this is not the case, so we apply Schaeffer’s theorem. It is easy to show that the maximal stable subspace of W under the action of T(n) (for some fixed n coprime to N) is exactly equal to the desired space $S_1({\varGamma }_0(N),\chi )$.

All the above operations (intersections of spaces and finding maximal stable subspaces) are elementary linear algebra, but can be extremely expensive in particular when the values of the character $\chi$ lie in a large cyclotomic field. Even when $\chi$ has small order, the computations suffer from intermediate coefficient explosion, whereas we expect the final result to have tiny dimension. We thus use modular algorithms and perform the computations in various finite fields before lifting the final result.

Note that in the actual implementation we first look for dihedral forms, i.e., forms coming from Hecke Grössencharacters of quadratic fields. Once these forms computed, the orders of the possible characters for the so-called exotic forms are much more limited.

5 Elementary computations on modular forms

Note that since the construction of our modular form spaces ultimately boils down to the computation of the trace forms ${{\mathscr {T}}}^{\mathrm {new}}$, modular forms are always implicitly given by their Fourier expansion at infinity, which can be unfortunate for some applications.

Nonetheless, a large number of standard operations can be done on modular forms represented in this way: first, elementary arithmetic operations such as products, quotients, linear combinations, derivatives, and second, specifically modular operations such as the action of Hecke operators, of the expanding operator B(d), Rankin–Cohen brackets, twisting, etc.

Several limitations immediately come to mind: the action of the Atkin–Lehner operators can be described explicitly only when the level is squarefree. One can evaluate numerically a modular form by summing its q-expansion, but only if |q| is not too close to 1, i.e., if $\mathfrak {I}(\tau )$ is not too small. (In small levels, one can use the action of ${\varGamma }_0(N)$ to increase $\mathfrak {I}(\tau )$.) One can compute the Fourier expansion at other cusps than infinity, but only if there exists an Atkin–Lehner involution sending infinity to that cusp, which will not always be the case in nonsquarefree level. We will see below how these limitations can be lifted by the use of the Borisov–Gunnells theorem.

An important operation is splitting: once the new space $S_k^{\mathrm {new}}({\varGamma }_0(N),\chi )$ has been constructed, we want to compute the basis of normalized Hecke eigenforms. This is done by simple linear algebra after factoring the characteristic polynomials of a sufficient number of elements of the Hecke algebra. Note that it is in general sufficient to use the T(p) themselves, but it may happen that one needs more complicated elements. For instance, to split the space $S_2^{\mathrm {new}}({\varGamma }_0(512))$, no amount of T(n) will be sufficient (the characteristic polynomials will always have square factors), but one needs to use in addition for instance the operator $T(3)+T(5)$.

Among the other elementary computations, note that modular forms can arise naturally from several different sources: forms associated with elliptic curves defined over $\mathbb {Q}$, forms coming from theta functions of lattices (possibly with a spherical polynomial), eta quotients, or forms coming from natural L-functions whose gamma factor is ${\varGamma }_{\mathbb {C}}(s)=2(2\pi )^{-s}{\varGamma }(s)$.

6 Advanced computations on modular forms

We now come to the more advanced functions of the package, which are up to now not available elsewhere. In view of the limitations above, the basic stumbling block is the computation of the Fourier expansions of a modular forms at cusps other than $i\infty$.

For $f\in M_k({\varGamma }_0(N),\chi )$, we want more generally to be able to compute the Fourier expansion of $f|_k\gamma$ for any $\gamma \in {\varGamma }$. (In fact, it is trivial to generalize the construction to any $\gamma \in M_2^+(\mathbb {Q})$, and this is done in the package but will not be explained here.) We can assume that k is an integer: indeed the expansion of $\theta |_k\gamma$ is known, and we can compute the expansion of $f\theta$ when f has half-integral weight. We recall that if , then

$$\begin{aligned} f|_k\gamma (\tau )=(c\tau +d)^{-k}f\left( \dfrac{a\tau +b}{c\tau +d}\right) . \end{aligned}$$

First, what precisely do we mean by “Fourier expansion” ? It is easy to show that there exists an integer $w\le N$ and a rational number $\alpha \in [0,1[\cap \mathbb {Q}$ such that¹

$$\begin{aligned} f|_k\gamma (\tau )=q^{\alpha }\sum _{n\ge 0}a_{\gamma }(n)q^{n/w}, \end{aligned}$$

where we always use the convention that $q^x=e^{2\pi ix\tau }$ for $x\in \mathbb {Q}$. More precisely, one can always choose $w=N/\gcd (N,c^2)$, the width of the cusp $a/c=\gamma (i\infty )$, and $\alpha$ is the unique number in [0, 1[ such that $\chi (1+acw)=e^{2\pi i\alpha }$. Note in passing that the denominator of $\alpha$ divides $\gcd (N,c^2)/\gcd (N,c)$, and by definition that $\alpha$ is nonzero if and only if the cusp a / c is so-called irregular for the space $M_k({\varGamma }_0(N),\chi )$.

The basic idea is as follows. Consider the space generated by products of two Eisenstein series $E_1$ and $E_2$, chosen of course such that $E_1E_2\in M_k({\varGamma }_0(N),\chi )$, including the trivial Eisenstein series 1 of weight 0. Experiments show that usually this space is the whole of $M_k({\varGamma }_0(N),\chi )$, and in fact the only exceptions seem to be in weight $k=2$. In fact, it is a theorem of Borisov–Gunnells [1, 2] that this observation is indeed true and that the exceptions occur only in weight 2 when there exists an eigenform f such that $L(f,1)=0$, for instance modular forms attached to elliptic curves of positive rank. (So $N=37$ is the smallest level for which there is an exception.)

Assume for the moment that we are in a space generated by products of two Eisenstein series. Since these are so completely explicit, it is possible by a tedious computation to obtain the expansions of $E|\gamma$, hence of all our forms. In the special case (occurring only in weight 2) where the space is not generated by products of two Eisenstein series, we simply multiply by some known Eisenstein series E so as to be in larger weight, do the computation there, and finally divide by the expansion of $E|\gamma$. This is what we do in any case in weight 1 and in half-integral weight.

This may sound straightforward, but as mentioned at the beginning, it required several months of work first to obtain the correct formulas and second to write a reasonably efficient implementation. In fact, we had to make a choice. With our current choice of Eisenstein series (which may change in the future), the coefficients of the Eisenstein series $E|\gamma$ lie in a very large cyclotomic field, at worst $\mathbb {Q}(\zeta _{{{\mathrm{lcm}}}(N,\phi (N))})$. It is possible that we can reduce this considerably, but for now we do not know how to do this at least in a systematic way. Handling such large algebraic objects is extremely costly, so we chose instead to work with approximate complex numbers with sufficient accuracy, and if desired to recognize the algebraic coefficients at the end using the LLL algorithm. This is reflected in the command that we will explain below to compute these expansions.

Note, however, the following theorem, communicated to us by F. Brunault and M. Neururer whom we heartily thank:

Theorem 7

Let $f\in M_k({\varGamma }_0(N),\chi )$, denote by $M\mid N$ the conductor of $\chi$, and assume that the coefficients of the Fourier expansion of f at infinity all lie in a number field K. Then if , the Fourier coefficients $a_{\gamma }(n)$ of $f|_k\gamma$ lie in the field $K(\zeta _u)$, where $u={{\mathrm{lcm}}}(N/\gcd (N,CD),M/\gcd (M,BC))$.

Note that the Fourier coefficients of $f|_k\gamma$ can live in a smaller number field or even in a smaller cyclotomic extension than that predicted by the theorem.

Once solved the problem of computing expansions of $f|_k\gamma$, essentially all of the limitations mentioned in Sect. 5 disappear, we can evaluate a modular form even very near the real axis (or at cusps), we can compute the action of the Atkin–Lehner operators in nonsquarefree level, we can compute general period integrals involving modular forms and in particular modular symbols such as

$$\begin{aligned} \int _a^b(X-\tau )^{k-2}f(\tau )\,\mathrm{d}\tau \end{aligned}$$

when $k \ge 2$ is integral, we can numerically evaluate L-functions of modular forms which are not necessarily eigenforms at an arbitrary $s\in \mathbb {C}$, and we can compute general Petersson scalar products thanks to the following theorem similar to Haberland’s:

Theorem 8

Let $k\ge 2$ be an integer, let f and g in $S_k({\varGamma }_0(N),\chi )$ be two cusp forms, let

$$\begin{aligned} {\varGamma }=\bigsqcup _{j=1}^r{\varGamma }_0(N)\gamma _j \end{aligned}$$

be a right coset decomposition, and define $f_j=f|_k\gamma _j$ and $g_j=g|_k\gamma _j$. Finally, for any function h and a and b in the completed upper-half-plane set

$$\begin{aligned} I_n(a,b,h)=\int _a^b \tau ^nh(\tau )\,\mathrm{d}\tau , \end{aligned}$$

the integral being taken along a geodesic arc from a to b. Then, we have

$$\begin{aligned} 6r(2i)^{k-1}\langle {f,g}\rangle _{{\varGamma }_0(N)}=\sum _{j=1}^r\sum _{n=0}^{k-2} (-1)^n\left( {\begin{array}{c}k-2\\ n\end{array}}\right) I_{k-2-n}(0,i\infty ,f_j)\overline{I_n(-1,1,g_j)}. \end{aligned}$$

More generally, when at each cusp at least one of the two forms vanishes, there exists a similar formula which we do not give here.

On the other hand, this theorem cannot be applied in weight $k=1$ or when $k\in 1/2+\mathbb {Z}$. However, recent work by Collins [5] using a formula of Nelson [6] allows to compute Petersson products in these cases, although less efficiently in general, and this is also implemented in the package.

7 Implementation issues

Modular form spaces can be represented by a basis, in echelon form or not, together with suitable linear algebra precomputations allowing fast recognition of elements of the space (recall the notion of Sturm bound, which tells us that if the Fourier coefficients at infinity of two modular forms belonging to the same space are equal up to some effective bound, the forms are identical).

The main problem is the representation of modular forms themselves. Because of our choice of using trace formulas, we must in some way represent the forms by their Fourier expansion at infinity. Since we do not want to specify in advance the number of coefficients that we want, a first approach would be to say that a modular form is a program which, given some L, outputs the Fourier coefficients up to $q^L$. This would be quite inefficient because of the action of Hecke operators: for instance, let p be a prime not dividing the level N, and assume that $f(\tau )=\sum _{n\ge 0}a(n)q^n$. Then, $(T(p)f)(\tau )=\sum _{m\ge 0}b(m)q^m$ with $b(m)=a(pn)+p^{k-1}\chi (p)a(n/p)$, where the last term occurs only if $p\mid n$. Thus, if we want L Fourier coefficients of T(p)f, we need on the one hand the coefficients a(m) for $m\le L/p$, but also the coefficients a(m) for $p\mid m$ and $m\le pL$. All the other Fourier coefficients a(m) for $m\le pL$ with $p\not \mid m$ are not needed, so it would be a waste to compute them if it can be avoided.

Thus, we modify our first approach, and we say that a modular form is a program which, given some L and step size d, outputs the Fourier coefficients of $q^m$ for $m\le dL$ and $d\mid m$ (hence all the coefficients if $d=1$). Such a program will define a modular form in our package.

Note that such representation may look like magic: for instance, in a GP session, type D=mfDelta() which creates the Ramanujan ${\varDelta }$ function. The output is only one and a half line long and contains mostly trivial information. Nonetheless, this information is sufficient to compute the Fourier expansion to millions of terms if desired, using the fundamental function mfcoefs(D,n) since internally the small information calls a much more sophisticated program which computes the expansion. (Note that in this specific case it is faster to compute the expansion directly using the product formula for the delta function than to use the modular form package).

Additional implementation comments: since the trace formula involves computing the class numbers h(D) for $D<0$, we use a cache method for those: we first precompute a reasonable number of such class numbers, then if it is not sufficient, we precompute again a larger number, and so on. We do similar caching for the factorizations and divisors of integers.

We also need to represent Dirichlet characters $\chi$. Since the most frequent are (trivial or) quadratic characters modulo D, such a character will simply be represented by the number D (so $D=1$ or omitted completely when $\chi$ is the trivial character). More general characters can be represented in several Pari/GP compatible ways, but the preferred way is to use the Conrey numbering which we do not explain here, so that a general Dirichlet character modulo N, primitive or not, is represented by Mod(a,N), where $\gcd (a,N)=1$.

8 Pari/GP commands

8.1 Commands involving only modular forms

As already mentioned above, the first basic command is mfcoefs(f,n) which gives the vector of Fourier coefficients $[a(0),a(1),\cdots ,a(n)]$. We have chosen to give a vector and not a series first because it is more compact, and second because the series variable (in principle q) could conflict with other user variables. Of course nothing prevents the user from defining his own function

if the variable q is ok.² For simplicity, we will use this user-defined function in the examples.

There are a number of predefined modular forms: in addition to the Ramanujan ${\varDelta }$ function, we have mfEk(k), the normalized Eisenstein series for the full modular group $E_k$, more generally mfeisenstein(k,chi1,chi2) for general Eisenstein series, mfEH(k) for Eisenstein series over ${\varGamma }_0(4)$ in half-integral weight, mfTheta(chi), the unary theta function associated with the Dirichlet character chi (if chi is omitted, the standard theta series), as well as modular forms coming from preexisting mathematical objects such as mffrometaquo, eta quotients, mffromell, modular cusp form of weight 2 associated with an elliptic curve over $\mathbb {Q}$, mffromqf, modular form associated with a quadratic form, with an optional spherical polynomial, and mffromlfun, modular form associated with an L-function having factor at infinity equal to ${\varGamma }_{\mathbb {C}}(s)=2\cdot (2\pi )^{-s}{\varGamma }(s)$.

All the standard arithmetic operations are implemented. For instance, the modular form $F=E_4({\varDelta }^2+E_{24})$ is obtained by the following commands:

Note that there is no mfadd, mfsub, or mfscalmul functions since they can be emulated by mflinear, which creates arbitrary linear combinations of forms:

Note also that the internal representation is an expression tree in direct Polish notation. In fact, since after a number of operations you may have forgotten what your modular form is, there is a function mfdescribe which essentially outputs this representation. For instance, applying to our above example:

The mfdescribe command is not to be confused with the mfparams command which gives a short description of modularity and arithmetic properties of the form. The above command shows that $F\in M_{28}({\varGamma }_0(1),1)=M_{28}({\varGamma })$ and that the number field generated by the Fourier coefficients of F is $\mathbb {Q}[y]/(y) = \mathbb {Q}$.

Other modular operations are available which work directly on forms, such as mfbd (expansion $\tau \mapsto \mathrm{d}\tau$), mfderivE2 (Serre derivative), mfbracket (Rankin–Cohen bracket). But most operations need an underlying modular form space, even the Hecke operators, as we have seen above.

8.2 Commands on modular form spaces

The basic command which creates a basis of a modular form space is mfinit, analogous to the other Pari/GP commands such as nfinit, bnfinit, ellinit. This command takes two parameters. The first is a vector [N,k,CHI] (or simply [N,k] if CHI is trivial), where N is the level, k the weight, and CHI the character in the format briefly described above. The second parameter is a code specifying which space we want (0 for the new space, 1 for the cuspidal space, or omitted for the full space for instance). A simpler command with exactly the same parameters is mfdim which gives the dimension.

The first command creates the space $S_2^{\mathrm {new}}({\varGamma }_0(26))$ (no character), and the second commands gives the q-expansions of the basis elements:

These are of course not the eigenforms. To obtain the latter:

Note that eigenforms can be defined over a relative extension of $\mathbb {Q}(\chi )$:

There are two ways to better see them:

The first command gives the number fields over which the eigenforms are defined. Here, there is only one eigenform and only one field $\mathbb {Q}[y]/(y^2+y-1)$. The second command “lifts” the coefficients to $\mathbb {Q}[y]$, so the result is much more legible. The third command “embeds” the eigenform in all possible ways in $\mathbb {C}$: indeed, even though (in the present example) there is only one formal eigenform, the space is of dimension two, so there are two eigenforms, given numerically as the last result.

Unavoidably, when nonquadratic characters occur, we can obtain even more complicated output. We have chosen to represent formal values of a character (which are in a cyclotomic field) with the variable letter “t,” but it must be understood that contrary to eigenforms this corresponds to a single canonical embedding: for instance

means in fact $e^{2\pi i/5}$, not some other fifth root of unity.

The first command shows that the eigenforms will have coefficients in a quadratic extension of a quadratic extension, hence in the quartic field $\mathbb {Q}[y,t]/(t^2+1, y^2-3t)$. The second command lifts the q-expansion to $\mathbb {Q}[y,t]$. To obtain the expansion over the quartic field, which is isomorphic to $\mathbb {Q}[y]/(y^4+9)$, we write

Note that the variable y actually stands for the same algebraic number in this block and the previous one, but rnfequation does not guarantee this in general.

8.3 Miscellaneous commands

(all outputs edited for clarity). These examples are self-explanatory. Note that the basis we compute for our modular form spaces is essentially random and does not guarantee that the matrices attached to Hecke or Atkin-Lehner operators have integral coefficients. They will in general have coefficients in $\mathbb {Q}(\chi )$ (up to normalizing Gauss sums in the case of $W_Q$); of course, their characteristic polynomials have coefficients in $\mathbb {Z}[\chi ]$.

Since in both cases the basis of mf can be given explicitly, this gives explicit formulas for the number of representations of an integer as a sum of r squares for $r=4$ and $r=10$, respectively (this can be done for $1\le r\le 8$ and $r=10$).

Modular forms can be evaluated numerically (even when the imaginary part is very small, see below), as well as their L-functions:

This equality is a consequence of the theory of complex multiplication, and in particular of the Lerch, Chowla–Selberg formula.

The command lfunmf creates the L-function attached to ${\varDelta }$, and lfunmfspec gives the corresponding special values in the interval [1, 11], which are rational numbers times two periods $\omega ^+$ (for the odd integers) and $\omega ^-$ (for the even integers).

L is now an L-function in the sense of the L-function package of Pari/GP and can be handled as such. For instance:

outputs in 50 ms the plot of the Hardy function associated with ${\varDelta }$ on the critical line $\mathfrak {R}(s)=6$ from height 0 to 50:

Similarly, we can compute zeros:

Note the nontrivial fact that lfunmf is completely general and computes the L-function attached to any modular form, eigenform or not (although the computation is indeed more efficient when the function detects an eigenform): this makes use of the “advanced” features that we will explain below.

A very useful command is mfeigensearch, which searches for rational eigenforms (hence with trivial or quadratic character) in a given range.

The first command asks for all rational eigenforms of levels between 1 and 60, of weight 2 such that $a(2)=-1$ and $a(3)=-3$. The mfparams command shows that there are two such forms, one in level 53 the other in level 58. The last command gives the beginning of their Fourier expansions, which of course agree up to the coefficient of $q^3$.

Note that the functions are returned as a black box allowing to compute an arbitrary number of Fourier coefficients, which need not be specified in advance. We could just as well have written mfser(x,1000) if we wanted 1000 coefficients.

There exists also the more straightforward mfsearch command which simply searches for a rational modular form with given initial coefficients:

This tells us that there exist exactly two forms of weight 3 and level $N\le 30$ in the cuspidal space (code 1) whose Fourier expansion begins by $q+2q^2+\cdots +8q^8+O(q^9)$; the last command shows their Fourier expansion up to $q^{10}$.

8.4 Weight 1 examples

Almost all of the commands given up to now (with the exception of lfunmf for noneigenforms and mfatkininit for nonsquarefree levels) are direct (although sometimes complicated) applications of the trace formula and linear algebra over cyclotomic fields. We now come to more advanced aspects of the package.

As already mentioned, constructing modular forms of weight 1 is more difficult than in higher weight, but they are fully implemented in the package.

This command uses the joker character 0 (which is available for all weights but especially useful in weight 1): it asks to output information about $S_1({\varGamma }_0(148,\chi ))$ for all Galois equivalence classes of characters, but only for nonzero spaces. Here, it gives us the Conrey labels of three characters modulo 148, such as Mod(105, 148), of respective orders 4, 6 and 18. The other two integers are of course also important and give the dimension of the space and the dimension of the subspace generated by the dihedral forms. Let us look at the first space, which contains an exotic (nondihedral) form:

This tells that the projective image of the Galois representation associated to the (unique) eigenform in mf is isomorphic to $S_4$, so is “exotic”. This is the lowest possible level for which it occurs (the smallest exotic $A_4$ is in level 124, already found long ago by J. Tate, and the smallest exotic $A_5$ is in level 633 found only a few years ago by K. Buzzard and A. Lauder:

(The first eigenform is of dihedral type $D_5$, and the second has exotic type $A_5$.) Note that this computation only requires 5 seconds.

This answers an old question of Serre who conjectured the existence of exotic $A_5$ forms of prime level $p\equiv 3\pmod 4$ with quadratic character : $p=2083$ is the smallest such prime.

Typical nonexotic examples:

Three eigenforms with projective image isomorphic to $D_3$, $D_5$, and $D_{15}$.

8.5 Half-integral weight examples

This returns the Shimura lift of f of weight $2k-1 = 4$ in . The Kohnen $+$-space as well as the known bijections between it and spaces of integral weight are implemented, as well as the new space and the eigenforms in the Kohnen space. We refer the reader to the manual for details.

9 Advanced examples

9.1 Expansions of $F|_k\gamma$ and applications

As mentioned at the beginning, we now give a number of examples which we believe are not possible (at least in general) with other packages. The basic function which allows all the remaining advanced examples to work is the computation of the Fourier expansion of $F|_k\gamma$ for any $\gamma \in {\varGamma }$. We begin with a simple example:

This requires a few explanations: the mfslashexpansion command asks for 6 terms of the Fourier expansion of $F|_k\gamma$ with , and the flag 1 which follows asks to give the result in algebraic form if possible (set the flag to 0 for complex floating point approximations). Thus, as for values of characters, the variable “t” which is printed is canonically $e^{2\pi i/16}$, root of ${\varPhi }_{16}(t)=t^8+1$. The params components are $\alpha$ and w, and mean that the “q” in the expansion should be understood as $e^{2\pi i\tau /w}$ (so here $q=e^{2\pi i\tau /8}$), and the expansion should be multiplied by $q^{\alpha }$ (here by 1).

Here is a more complicated example which illustrates this:

Thus, $w=9$ so $q=e^{2\pi i\tau /9}$, and $\alpha =1/18$ so the expansion must be multiplied by $e^{2\pi i\tau /18}$. Note that the constant coefficient $-t^4/54 - t/54 \in \mathbb {Q}(t)/(t^6 + t^3 + 1)$ in the expansion is nonzero, so necessarily $\alpha >0$ otherwise F would not be a cusp form.

Of course, we can have “raw” expansions with approximate complex coefficients:

(Here, we did not ask for params since we already know it from the previous computations.) Recall that all this is possible thanks to the expression of F as a linear combination of products of two Eisenstein series. More generally, for $\gamma$ in $M_2^+(\mathbb {Q})$, the result would be expressed as $u^{k/2}f(\tau + v)$ and $f(q) = q^\alpha \sum _{m\ge 0} a_m q^{m/w}$ as above for some rational numbers u and v chosen so as to minimize the field of definition of the $a_m$ (see the manual).

Atkin–Lehner operators are an important special case:

(here $C=8^{-1/2}$). This is a difficult case for the Atkin–Lehner operators since first the level is not squarefree, and second the character, here 8 which represents , is not defined modulo $N/Q=32/32$.

The result involves a normalizing constant C given above, essentially a Gauss sum, such that the expansion of $C\cdot F|_k W_Q$ has the same field of coefficients as F. A similar question for the more general $F|_k\gamma$ is answered by Theorem 7, but the given value may not be optimal. In general, $C\cdot F|_k W_Q$ belongs to a different space than F (the Nebentypus becomes $\overline{\chi _Q} \chi _{N/Q}$ in integral weight, and a similar formula holds in nonintegral weight). The matrix M expresses $C \cdot F_i |_k W_Q$ when $(F_i)$ is a basis of mf in terms of a basis of that other space mfB. The operator can also be applied to an individual form:

This returns $G = C\cdot F|_k W_{32}$.

9.2 Numerical applications

It is now easy to compute period polynomials

$$\begin{aligned} P(F,X)=\int _0^{i\infty }(X-\tau )^{k-2}F(\tau )\,\mathrm{d}\tau \end{aligned}$$

for modular forms of integral weight $k\ge 2$. Indeed, in addition to the Fourier expansion at infinity, this only requires the expansion of $F|_kW_N$ with . In particular, we can compute in complete generality special values and periods.

More generally, we can compute numerically general period polynomials and modular symbols

$$\begin{aligned} \int _{s_1}^{s_2}(X-\tau )^{k-2}(F|_k\gamma )(\tau )\,\mathrm{d}\tau \end{aligned}$$

for any cusps $s_1$ and $s_2$ and any $\gamma \in M_2^+(\mathbb {Q})$, or even for any $s_1$ and $s_2$ in the completed upper half plane ${\mathfrak {H}}$.

Thanks to these symbols, we can also compute general Petersson products of modular forms of integral weight $k\ge 2$ by using Haberland-type formulas such as Theorem 8:

We can also evaluate a form near the real axis: for forms over the full modular group ${\varGamma }$ or small index, we can use modular transformations in the group to significantly increase imaginary parts. In general, this is not possible, but now it is easy since we can always use the whole of ${\varGamma }$: in fact, we can always reduce to $\mathfrak {I}(\tau )\ge 1/(2N)$, which is almost optimal:

Note that |ev(m)| seems to tend to infinity with m, but with a pronounced “dip” around $m=7$ and $m=8$. This is not specific to the modular form F but probably to the diophantine approximation properties of $1/\pi$, and in particular to its very close convergent 113 / 355.