# How to Write a Modular Form In Terms of a Set of Generators?

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In this post, I will explain an algorithmic way to write any modular form $f \in \mathcal{M}_*(\Gamma)$ as a polynomial in the generators of the graded ring $\mathcal{M}_*(\Gamma)$ ($\Gamma = \Gamma_0(N), \Gamma_1(N)$ or $\mathrm{SL}_2(\mathbb{Z})$).

First it is important to mention that for any congruence subgroup $\Gamma \leq \mathrm{SL}_2(\mathbb{Z})$, the graded ring of modular forms $\mathcal{M}_*(\Gamma)$ is finitely generated (but not necessarily freely). This is not trivial and it was proven in the paper Les schémas de modules de courbes elliptiques [DR73, Thm 3.4]. In fact, they prove that it is finitely generated as a $\mathbb{Z}$-algebra.

Let’s start with an (easy) example. Consider the weight 12 normalized Eisenstein serie for $\mathrm{SL}_2(\mathbb{Z})$ denoted $E_{12}$ (by “normalized” I mean that the first Fourier coefficient is $1$). This modular form lives in the graded ring of modular forms for the full modular group:

$E_{12} \in \mathcal{M}_*(\mathrm{SL}_2(\mathbb{Z})) \cong \mathbb{C}[E_4, E_6].$

Thus, there exist a unique polynomial $P(X, Y) \in \mathbb{C}[X, Y]$ such that

$\label{poly} E_{12} = P(E_4, E_6)$

Next, we claim that $P$ must be homogeneous of weight $12$ in $E_4$ and $E_6$. In other words, if $X = x^4$ and $Y = y^6$ then $P(x^4, y^6)$ must be of degree $12$ and respect the following relation: $P((\lambda x)^4, (\lambda y)^6) = \lambda^{12}P(x^4, y^6)$ for every $\lambda\in \mathbb{C}$. This is simply because $E_{12}$ is of weight $12$ and the product of two modular forms of weight $k_1$ and $k_2$ respectively is a modular form of weight $k_1 + k_2$. Since the only homogeneous monomials of weight $12$ are $E_4^3$ and $E_6^2$ the problem reduces to finding two complex numbers $a$ and $b$ such that

$\label{E12_pol} E_{12} = a E_4^3 + b E_6^2.$

To determine $a$ and $b$, we will need a tool called the Sturm bound of a modular forms space. This bound tell us that a modular form is determined only by a finite number of coefficients in its $q$-expansion. Here’s the general statement:

## Theorem.

Let $f\in \mathcal{M}_k(\mathrm{SL}_2(\mathbb{Z}))$ with $q$-expansion $\sum_{n\geq 0} a_n q^n$. Suppose that $a_i = 0$ for $0 \leq i \leq \lfloor k/12 \rfloor$. Then $f = 0$. The number $\mathrm{SB}_k := \lfloor k/12 \rfloor$ is called the Sturm bound of $\mathcal{M}_k(\mathrm{SL}_2(\mathbb{Z}))$

This theorem is a corollary of the valence formula. Moreover, a more general version for congruences subgroups also exist (see for example this page)

Using the Sturm bound, we can now solve our initial problem by following these steps:

1. Compute the sturm bound of $\mathcal{M}_{12}(\mathrm{SL}_2(\mathbb{Z}))$: $~~ \mathrm{SB}_{12} = \lfloor 12/12 \rfloor = 1$
2. Compute the $q$-expansion of $E_{12}$, $E_4^3$ and $E_6^2$ up to precision $q^{\mathrm{SB}_{12}+1}$:

$E_{12} = 1 + \frac{65520}{691}q + O(q^2);$ $E_4^3 = 1 + 720q + O(q^2);$ $E_6^2 = 1 - 1008q + O(q^2).$
3. Solve the resulting linear system given by $C_{E_{12}} = a C_{E_{4}^3} + b C_{E_6^2}$ where $C_{E_k^n}$ is the vector formed by the coefficients of $E_k^n$ up to precision $q^{\mathrm{SB}_{12}+1}$:

$\begin{pmatrix} 1 \\ \frac{65520}{691} \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 720 & -1008 \end{pmatrix} \begin{pmatrix} a \\ b \end{pmatrix}$

The unique solution to this linear system is given by $a = 441/691$ and $b = 250/691$. Therefore, the solution to our initial problem is:

$E_{12} = \frac{441}{691}E_{4}^3 + \frac{250}{691} E_{6}^{2}$

The same approach can be used for any given modular forms $f\in \mathcal{M}_k$. An advantage of this method is that it is computational in the sense that it can be implemented into a computer program.

## Sagemath Implementation

The algorithm showcased in the example above was implemented into SageMath. The idea was to create two new methods: to_polynomial and from_polynomial. The first method takes a GradedModularFormElement and returns its polynomial form:

The second method takes a polynomial and returns the associated GradedModularFormElement:

The implementation also works for congruence subgroups as SageMath knows how to compute the generators for these rings.

It is important to note here that, in the case where the group is not $\mathrm{SL}_2(\mathbb{Z})$, then there might be some relations between the generators. So the methods to_polynomial and from_polynomial are not necessarily the inverses of each other. This is illustrated by this example:

## Source Code

Here is the SageMath implemenation of the method to_polynomial:

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