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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 $\eta(\tau)=q^{1/24}\prod\limits_{n=0}^{\infty}(1-q^n)$ is the Dedekind eta function with $q=e^{2\pi i\tau}$ and $Im \tau>0$. I want to know the bound for $a_{k^2+k+1}$.

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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

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