Experimental finding of modular forms for noncongruence subgroups

TL;DR

This paper employs experimental and computational techniques to discover modular forms for non-congruence subgroups and explores their connection to congruence subgroup forms through the Atkin--Swinnerton-Dyer correspondence, also generalizing a key criterion.

Contribution

It introduces a computational approach to identify modular forms for non-congruence subgroups and generalizes Ligozat's eta-quotient criterion for modular functions.

Findings

Identified new modular forms for specific non-congruence subgroups

Established links between non-congruence and congruence subgroup forms via Atkin--Swinnerton-Dyer correspondence

Generalized Ligozat's eta-quotient criterion

Abstract

In this paper we will use experimental and computational methods to find modular forms for non-congruence subgroups, and the modular forms for congruence subgroups that they are associated with via the Atkin--Swinnerton-Dyer correspondence. We also prove a generalization of a criterion due to Ligozat for an eta-quotient to be a modular function.