Non-vanishing of automorphic $L$-functions of prime power level (joint papers with O.G. Balkanova)

Iwaniec and Sarnak showed that at the minimum 25% of $L$-values associated to holomorphic newforms of fixed even integral weight and level $N \rightarrow \infty$ do not vanish at the critical point when $N$ is square-free and $\phi(N)\sim N$. We extend the given result to the case of prime power level $N=p^{\nu}$, $\nu\geqslant 2$. The proof is based on asymptotic evaluation of twisted moments
$$ M_1(l,u,v)=\sum_{f \in H_{2k}^{*}(N)}^{h}\lambda_f(l)L_{f}\bigl(\tfrac{1}{2}+u+v\bigr), $$

$$ M_2(l,u,v)=\sum_{f \in H_{2k}^{*}(N)}^{h}\lambda_f(l)L_{f}\bigl(\tfrac{1}{2}+u+v\bigr),L_{f}\bigl(\tfrac{1}{2}+u-v\bigr), $$
and the technique of mollification.