A formula for Fourier coefficients of certain eta-quotients and their expansions as Eisenstein series

TL;DR

This paper derives a uniform formula for Fourier coefficients of specific eta-quotients and expresses them as Eisenstein series, advancing understanding of modular forms with particular multiplier systems.

Contribution

It provides a comprehensive list of 113 eta-quotients, a closed-form formula for their Fourier coefficients, and their expansions as Eisenstein series, using advanced modular form techniques.

Findings

Explicit Fourier coefficient formula for eta-quotients

List of 113 holomorphic eta-quotients of integral weight

Expansion of eta-quotients as linear combinations of Eisenstein series

Abstract

We give a list of $113$ holomorphic eta-quotients of integral weight ($66$ of which are primitive) and provide a uniform closed formula for their Fourier coefficients $c(l)$ where $l\equiv1\bmod{m}$ with some fixed $m\mid24$. The proof involves Wohlfahrt's extension of Hecke operators and a dimension formula for spaces of modular forms of general multiplier system. We further provide the expansions of these eta-quotients as linear combinations of standard Eisenstein series.