Asymptotic Formulas and Sign-changes of Coefficients of Hauptmoduls on $Γ_0(p)$

Liuquan Wang

Published 2021-08-09Version 1

We study the coefficients of the hauptmodul $(\eta(\tau)/\eta(p\tau))^{24/(p-1)}$ on $\Gamma_0(p)$ with $p\in \{2,3,5,7,13\}$, where $\eta(\tau)$ is the Dedekind eta function. We provide asymptotic formulas for its coefficients and give explicit characterizations of its signs. For the cases $p=2$ or $3$, we also give alternative proof for its sign-change properties without using the asymptotic formulas.