Mean values of second moments of $L$-series of modular forms averaged over all twists by characters mod $q$

This is joint work with Min Lee and Nikos Diamantis. Let $f,g$ be modular forms of even weight $k$ for $SL(2,Z)$ with Fourier coefficients $a(n), b(n)$.  We find a meromorphic  continuation and two different spectral expansions for the double shifted multiple Dirrichlet series

$\sum_{n, h\geq 1} \frac{a(n+h) b(n) \sigma_{1-2v}(h)}{n^{s+k-1} h^{u}}$.

We then relate this to a smoothed average over $q$ close to $Q$ of

$$\sum_{\substack{\chi\bmod{q}, \\ \text{ primitive}}} L(1/2, f\times \overline{\chi}) L(1/2, g\times \chi).$$

and obtain a main term with a very sharp error term.   Such a formula has been found for the special case of $q$ prime, in 2018, by Blomer, Fouvry, Kowalski, Michel, Milićević, and Sawin.   The presence of small prime divisors of $q$ make such a result very difficult for specific composite $q$, making an average over $q$ an attractive alternative.