Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

TL;DR

This paper investigates the simultaneous sign changes and distribution properties of Fourier coefficients of pairs of cusp forms, including both integral and half-integral weights, providing new insights into their behavior at prime power indices.

Contribution

It introduces new results on simultaneous sign changes and equidistribution for Fourier coefficients of pairs of cusp forms, extending to half-integral weights.

Findings

Proves simultaneous sign change results for Fourier coefficients of cusp forms.

Establishes equidistribution results for these coefficients.

Analyzes analogous behavior for half-integral weight forms.

Abstract

We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.