TITLE: Lower bound of Hecke eigenvalues of Maass form
QUESTION [6 upvotes]: If $f$ is a Maass form and $p$-Hecke eigenvalue (i.e. Hecke eigenvalue of usual Hecke operator $T_p$) of $f$ is $\lambda_f(p)$, do we know anything about lower bound of the sum$$S(x) = \sum_{x\le p\le 2x}|\lambda_f(p)|^2?$$
To avoid Confusion $$(T_pf)(z)=\frac{1}{\sqrt{p}}\left[\sum_{b=0}^{p-1}f\left(\frac{z+b}{p}\right)+f(pz)\right]$$
Any referecne would be highly helpful.

REPLY [3 votes]: Following up on @Idoneal's comment, one can prove something a bit more precise.  In particular, $S(x)\asymp\frac{x}{\log x}$, provided that $\log x\gg\log(\lambda N)$, where $f$  has level $N$ and Laplace eigenvalue $\lambda$. I'm assuming Matt Young's normalization and that the central character is unitary; also, I'm writing $f\asymp g$ to mean that both $f=O(g)$ and $g=O(f)$.  This result follows from work of Motohashi https://arxiv.org/abs/1209.4140.  A little extra effort is needed to get the claimed range, but all the necessary tools are in this paper.