Modular Symbols for ${\mathrm{\Gamma }}_0(N)$

Introduction

This repository contains supplemental code to my Part III essay on modular symbols.

main.py contains some classes and functions involved in computing with modular symbols. In particular:

• m = ModularSymbols(k, N) constructs the space ${\mathbb{M}}_k({\mathrm{\Gamma }}_0(N))$
• s = m.cuspidal_subspace() computes the subspace of cuspidal modular symbols ${\mathbb{S}}_k({\mathrm{\Gamma }}_0(N))$
• s.T_matrix(n) computes the matrix of the action of $T_n$ on ${\mathbb{S}}_k({\mathrm{\Gamma }}_0(N))$

The above functions are used in the function cusp_forms to compute a basis for $S_k({\mathrm{\Gamma }}_0(N))$.

See demo.ipynb for example usage of cusp_forms.

Requirements

The sympy package is required to run the code.

References

Stein, W. (2007). Modular forms, a computational approach (Graduate studies in mathematics; v. 79). Providence, R.I.: American Mathematical Society.

Merel, L. (1994). Universal Fourier expansions of modular forms. In: Frey, G. (eds) On Artin's Conjecture for Odd 2-dimensional Representations. Lecture Notes in Mathematics, vol 1585. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074110