Sign-change of the Fourier coefficients of a Hauptmodul for $\Gamma_{0}(2)$

TL;DR

This paper proves that the Fourier coefficients of a specific modular function related to $ ext{SL}_2( ext{Z})$ exhibit a sign-change property, contributing to the understanding of their oscillatory behavior.

Contribution

It establishes the sign-change property of Fourier coefficients for a Hauptmodul associated with $ ext{Gamma}_0(2)$, a new result in modular form theory.

Findings

Fourier coefficients change sign infinitely often

Provides proof for sign oscillation in specific modular functions

Enhances understanding of Fourier coefficient behavior in modular forms

Abstract

In this short note, we aim to prove that the Fourier coefficients of the modular function $ \frac{\eta^{24}(\tau)}{\eta^{24}(2\tau)} $ possess a sign-change property.