On the Computation of General Vector-valued Modular Forms

TL;DR

This paper introduces a new algorithm for computing Fourier expansions of vector-valued modular forms of weight at least 2, broadening computational capabilities beyond existing methods and enabling applications like Moonshine and Jacobi forms.

Contribution

The paper presents a novel algorithm that directly computes Fourier expansions of general vector-valued modular forms, extending beyond prior algorithms limited to specific cases.

Findings

Algorithm successfully computes Fourier expansions for a wide class of vector-valued modular forms.

Implementation improves runtime performance through permutation group techniques.

The approach generalizes classical cusp expansion methods for modular forms.

Abstract

We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least $2$ associated with representations whose kernel is a congruence subgroup. It complements two available algorithms that are limited to inductions of Dirichlet characters and to Weil representations, thus covering further applications like Moonshine or Jacobi forms for congruence subgroups. We examine the calculation of invariants in specific representations via techniques from permutation groups, which greatly aids runtime performance. We explain how a generalization of cusp expansions of classical modular forms enters our implementation. After a heuristic consideration of time complexity, we relate the formulation of our algorithm to the two available ones, to highlight the compromises between level of generality and performance that each them makes.