Second moment of $\text{GL(3)}\times \text{GL(2)}$ $L$--functions

For $M_1$ and $M_2$ two distinct primes, let $H_k^{\star }(M_1{M}_2,\psi )$ denote the set of primitive newforms of level $M_1{M}_2$, weight $k\ge 3$ and Nebentypus $\psi$ of conductor $M_1$. Let $\pi$ be a fixed $SL(3,\mathbb{Z})$ Hecke cusp form. We prove a Lindelöf–consistent upper bound for the second moment $$\text{sideset}{}^h\sum _{f\in H_k^{\star }(M_1{M}_2,\psi )}\mathrm{\mid }L(1\mathrm{/}2,\pi \times f){\mathrm{\mid }}^2{\ll }_{\pi ,\epsilon }{M}_1^{1+\epsilon }$$ in the range $M_2\le M_1^{1+\epsilon }$.