Twisted Moments of Rankin-Selberg $L$-functions in the Prime-Power Level Aspect

We compute the twisted first and second moments of the shifted central values of the Rankin-Selberg $L$-functions given by $L(\frac{1}{2}+\omega ,f\otimes g)$ as $f$ varies over primitive forms of prime power level $p^{\nu }$ with $\nu \ge 3$. Here $\omega$ is a bounded shift and $g$ is a fixed primitive form of level relatively prime to $p$.