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