Rankin-Selberg coefficients in large arithmetic progressions

Let $(\lambda_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta>0$, under the Ramanujan-Petersson conjecture for $\rm GL_2$ Maass forms, the Rankin-Selberg coefficients $(\lambda_f(n)^2)_{n\geq 1}$ admit a level of distribution $\theta=2/5+1/260-\eta$ in arithmetic progressions.