Rankin--Selberg coefficients in arithmetic progressions modulo prime powers

Let $\epsilon >0$ be given. For prime power moduli $q=p^k$ with $k\ge 2$ and $p\mathrm{\ne }3$, and assuming the Ramanujan–Petersson conjecture for ${\text{GL}}_2$ Maass forms, we prove that the Rankin–Selberg coefficients $\{{\lambda }_f(n{)}^2{\}}_{n\ge 1}$ have a level of distribution $\theta =2\mathrm{/}5+3\mathrm{/}305-\epsilon$ in arithmetic progressions $n\equiv aq$.