Polynomial interpolation of modular forms for Hecke groups

TL;DR

This paper explores polynomial interpolation of modular forms for Hecke groups, expressing these polynomials via Fourier expansions and divisor sums, and investigates their roots in relation to Lehmer's conjecture through numerical experiments.

Contribution

It extends previous work by explicitly computing polynomials associated with modular forms for Hecke groups and analyzing their roots in connection with Lehmer's question.

Findings

Polynomials are expressed in terms of Hauptmoduln and divisor sums.

Roots of the polynomials relate to Lehmer's conjecture on Ramanujan's tau function.

Numerical experiments support theoretical insights into root distributions.

Abstract

Extending work of J. Raleigh, we compute polynomials $P_{n,F}(x)$ associated to certain families $F = \{f_m\}_{m = 3, 4, ...}$ of modular forms for Hecke groups $G(\lambda_m)$ with the property that $P_{n,F}(m)$ is the $n^{th}$ coefficient in the Fourier expansion of $f_m$. We express the $P_{n,F}$ in terms of the Fourier expansions of well-known Hauptmoduln, or in terms of certain divisor-sums. By studying the complex roots of the $P_n$, we relate them to Lehmer's question about Ramanujan's tau function. We review the theory of triangle functions and Hecke's theory of modular forms in order to establish a basis for our code, some of which originates in the dissertation of J. Leo. The article is an account of numerical experiments; the only theorems in it belong to work by others that we review as described above.