bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2 Are there any results about the bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2? More precisely, let $k$ be a positive integer and $m=4/k$. Write
\begin{equation*}
\sum\limits_{n=1}^{\infty}a_nq^n=\eta(k\tau)^2\eta(2k\tau)^{1+m}\eta(4k\tau)^{3-3m}\eta(8k\tau)^{2m-2},
\end{equation*}
where $\eta(\tau)=q^{1/24}\prod\limits_{n=1}^{\infty}(1-q^n)$ is the Dedekind eta function with $q=e^{2\pi i\tau}$ and $Im \tau>0$. I want to know the bound for $a_{k^2+k+1}$.

Non-holomorphic automorphic forms are called Maass forms. There are nontrivial bounds avalaible and an analogue of the Ramanujan conjecture for modular forms is expected.
This goes under the name Ramanujan-Petersson conjecture.
The state of art is conserved in the introduction here
Blomer, Brumley - The role of the Ramanujan conjecture in analytic number theory, Bulletin AMS 50 (2013), 267-320
