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.



Lower bound of Hecke eigenvalues of Maass form
If $x$ is large enough, then Rankin-Selberg theory will show that $S(x) \gg x^{1-\varepsilon}$. However, if $x$ is not large enough, then it is unknown how to obtain a lower bound for $S(x)$. In particular, it is unknown how to show that $S(x) \neq 0$. A good starting point for this is Chapter 13 of Iwaniec's book, Topics in Classical Automorphic Forms.


