Mixed moment of $GL(2)$ and $GL(3)$ $L$-functions in the level aspect

In this paper, we establish an asymptotic formula of mixed moments of $L(1/2,f)$ and $L(1/2,\mathrm{sym}^2 f)$ for cusp forms $f$ of weight $k$, prime level $q$ and real primitive nebentypus $χ$ modulo $q$ in the level aspect with a power-saving error term. As an application, we deduce the simultaneous non-vanishing of $L(1/2, f)$ and $L(1/2,\mathrm{sym}^2 f)$ for sufficiently large prime levels.