On the higher mean over arithmetic progressions of Fourier coefficients of cusp forms

Let $\lambda_f(n)$ be the $n$th normalized Fourier coefficient of a holomorphic or Maass cusp form $f$ for $\mathrm{SL(2,\mathbb{Z})}$. We establish the asymptotic formula for the summatory function $$ \sum_{\substack{n\leq x \\ n\equiv l \,({\rm mod}\, q)}}|\lambda_f(n)|^{2j} $$ as $x\rightarrow \infty,$ where $q$ grows with $x$ in a definite way and $j=2,3,4$.