# Sign changes of a product of Dirichlet characters and Fourier coefficients of Hecke eigenforms

## TL;DR

This paper proves that for almost all primes not dividing the level, certain sequences involving Fourier coefficients of Hecke eigenforms and Dirichlet characters exhibit infinitely many sign changes, highlighting oscillatory behavior.

## Contribution

It establishes new results on sign changes of Fourier coefficients modulated by Dirichlet characters for Hecke eigenforms, extending understanding of their oscillatory properties.

## Key findings

- Sequences have infinitely many sign changes for almost all primes p not dividing N.
- Sign change results hold for sequences involving Fourier coefficients and Dirichlet characters.
- Similar sign change results are obtained for specific subsequences when j is odd.

## Abstract

Let $f\in S_k(\Gamma_{0}(N))$ be a normalized Hecke eigenform of even integral weight $k$ and level $N$. Let $j\ge1$ be a positive integer. We prove that for almost all primes $p$, $p\nmid N$, and for all characters $\chi_{0}=\pm 1\pmod N$, the sequence $\left(\chi_{0}(p^{nj})a(p^{nj})\right)_{n\in\N}$ has infinitely many sign changes. We also obtain a similar result for the sequence $\left(a(p^{j(1+2n)})\right)_{n\in\N}$ when $j$ is odd.