The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point

Abstract: We prove an asymptotic formula with a power-saving error term for the second moment $$ \sum_{q \in \mathcal{Q}} \; \; \sideset{}{^{\flat}\sum_{χ\bmod} q} \left| L\big( \tfrac{1}{2} , χ\big) L\big(\tfrac{1}{2},f\otimes χ)\right|² $$ of the twisted ${\rm GL}(2)$ {$L$}-function and the Dirichlet {$L$}-function at the central point under the assumption of Selberg's eigenvalue conjecture. Here $f$ is a fixed Hecke holomorphic cusp form for $\mathrm{SL}(2,\mathbb{Z})$ and the sum over $χ$ runs over all primitive even Dirichlet characters modulo $q$, and $$\mathcal{Q}=\left{q=q_1 q_2\asymp Q: q_1 \leq Q_1, q_2 \leq Q_2,\ Q_2\asymp Q^{{δ_1},(q,6)=1,(q_1,q_2)=1} \right}$$ with $0<δ_1<0.0004$.