A Note On Signs Of Fourier Coefficients Of Two Cusp Forms

TL;DR

This paper investigates the sign patterns of Fourier coefficients of two cusp forms, extending previous results to specific congruence subgroups and sparse sequences of coefficients, revealing new sign behavior insights.

Contribution

It compares the signs of Fourier coefficients of two cusp forms for $\Gamma_0(N)$ and sparse sequences, expanding understanding of sign changes in modular forms.

Findings

Sign patterns of Fourier coefficients are analyzed for $\Gamma_0(N)$.

Sign change behavior is studied for sparse sequences of coefficients.

Results extend previous work on simultaneous sign changes.

Abstract

Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $\Gamma_0(\mathit{N})$ where the coefficients lie in an arithmetic progression. Next we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group.