Rankin--Selberg coefficients in arithmetic progressions modulo prime powers

Abstract

Let $\epsilon >0$ be given. For prime power moduli $q=p^k$ with $k\ge 2$ and $p\ne 3$, and assuming the Ramanujan--Petersson conjecture for $\GL_2$ Maass forms, we prove that the Rankin--Selberg coefficients $\{{\lambda }_f(n{)}^2{\}}_{n\ge 1}$ have a level of distribution $\theta =2/5+3/305-\epsilon$ in arithmetic progressions $n\equiv a\mathrm{mod}q$.