bounds for Fourier coefficients of non-holomorphic automorphic forms of weight 2

Question

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 $a_{k^2+k+1}$.