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

Abstract

Let $f\in S_k({\Gamma }_0(N))$ be a normalized Hecke eigenform of even integral weight $k$ and level $N$. Let $j\ge 1$ be a positive integer. We prove that for almost all primes $p$, $p\nmid N$, and for all characters ${\chi }_0=\pm 1(\mathrm{mod}N)$, the sequence ${\left({\chi }_0(p^{nj})a(p^{nj})\right)}_{n\in \mathbb{N}}$ has infinitely many sign changes. We also obtain a similar result for the sequence ${\left(a(p^{j(1+2n)})\right)}_{n\in \mathbb{N}}$ when $j$ is odd.