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 $j\ge1$ 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.