On the computation of modular forms on noncongruence subgroups

TL;DR

This paper introduces two improved computational methods for modular forms on noncongruence subgroups, enabling faster high-precision calculations and algebraic identification of Fourier coefficients, especially for genus zero cases.

Contribution

It presents enhanced algorithms for computing modular forms on noncongruence subgroups, including a faster version of Hejhal's method and a genus zero Belyi map approach.

Findings

Hejhal's method becomes up to 100 times faster with improved precision techniques.

High-precision Fourier coefficients can be effectively used to identify algebraic forms.

Efficient computation of Belyi maps facilitates modular form construction for genus zero subgroups.

Abstract

We present two approaches that can be used to compute modular forms on noncongruence subgroups. The first approach uses Hejhal's method for which we improve the arbitrary precision solving techniques so that the algorithm becomes about up to two orders of magnitude faster in practical computations. This allows us to obtain high precision numerical estimates of the Fourier coefficients from which the algebraic expressions can be identified using the LLL algorithm. The second approach is restricted to genus zero subgroups and uses efficient methods to compute the Belyi map from which the modular forms can be constructed.