A lower-bound for the square-mean of Fourier coefficients of cusp forms at primes argument There is a basis question which puzzles me for a while. The question is the following:
Let $X\ge 2,$ and $\lambda(n)$ be the $n$-th Fourier coefficient of a $GL(2)$ newform of prime level $N>1$, with $N\ll X$ and the trivial nebentypus.
If one could show
$$\sum_{p\le X} \lambda^2(p)\gg X^{1-\varepsilon},  \quad \tag{1}$$
where the implied constant does not depend on the level $N$?
My understanding is that, note that $\lambda^2(p)=1+\lambda(p^2)$ if $p\nmid N$, so that the sum in (1) becomes $$\pi(X)+\frac{1}{N}-1+\sum_{p\le X} \lambda(p^2),$$ while this guy, $\lambda(p^2)$, may be viewed as the Fourier coefficient of the the symmetric-square
lift of $f$, i.e., $\text{sym}^2f$, which is however a $GL(3)$ Maass form by Gelbart and Jacquet's theory. One may thus show, by appealing to Theorem 5.13 of I-K's book, the following
$$\sum_{p\le X}\lambda(p^2) =\pi(X)+O\left\{X\exp\left(- \frac{c\log X}{\sqrt{\log X}+\log N}\right) \right \} $$
for some computable constant $c>0$, where the implied $O$-constant is absolute. Thus one may deduce that the sum in (1) equals
$$\pi(X)+\left\{X\exp\left(- \frac{c\log X}{\sqrt{\log X}+\log N}\right) \right \}\quad \tag{2}.$$ But it seems one fails to show that the error-term in (2) is $\gg X^{1-\varepsilon}$; for example, if one takes $N=X^{\delta}$ for some $\delta<1$, we find the error-term is $\gg X$. It seems one cannot achieve a power-saving.
If any expert leans something on this topic, please show a guide. Thanks in advance! And thanks for your time.
 A: Assuming that the spectral parameter is absolutely bounded (which you seem to implicitly state), the best that one can achieve with existing tools is the following:  There exist absolute and effectively computable constants $c_1,c_2,c_3>0$ such that if $x>N^{c_1}$, then
$\displaystyle c_2 \frac{x}{\log x} \leq \sum_{p\leq x}|\lambda(p)|^2\leq c_3 \frac{x}{\log x}$.
In some regards, this is a direct generalization of Linnik's bound on the least prime in an arithmetic progression.  As such, the only way I'd know to prove this would involve using a log-free zero density estimate (as Linnik did).  This result is proved in a significantly broader context here.
ADDED:  In view of GH from MO's comment, I should specify that I am assuming that the cusp form under consideration is in fact a newform.  If you want to assume that $x$ is at least a polynomial in $N$ (and not an even larger function of $N$), then it is unclear (at least to me) what one can say otherwise.
A: Old results of Iwaniec-Kohnen-Sengupta give non-vanishing of $\lambda_f(p)$ for $p < N^{1/2 - \delta}$ and some small $\delta > 0$ so this is the best range that you could hope for give the current technology.
