The sporadic group J2, Hauptmodul and Belyi map

TL;DR

This paper introduces a new algorithm for approximating modular forms and applies it to compute the hauptmodul and Belyi map associated with a genus zero subgroup linked to the second Janko group, revealing their explicit forms.

Contribution

The paper presents a novel algorithm for calculating Fourier coefficients of modular forms and explicitly determines the Belyi map and hauptmodul for a specific subgroup related to J2.

Findings

Explicit Belyi map and hauptmodul computed

Field of definition of the Belyi map identified

Fourier coefficients of the hauptmodul obtained

Abstract

Determining Fourier coefficients of modular forms of a finite index noncongruence subgroups of the modular group is still a non-trivial task. In this brief note we describe a new algorithm to reliably calculate an approximation for a modular form of a given weight. As an example we calculate the hauptmodul and Belyi map of a genus zero subgroup of the modular group defined via a canonical homomorphism by the second Janko group. Our main result is the field of definition of its Belyi map, the explicit Belyi map and the Fourier coefficients of its hauptmodul.