Mean squares of quadratic twists of the Fourier coefficients of modular forms

Abstract

In this paper, we evaluate asymptotically a smoothed version of the sum \[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} λ_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \] where $\left(\frac{8d}{\cdot }\right)$ is the Kronecker symbol, $\sideset{}{^*}\sum$ denotes a sum over positive odd square-free integers and ${\lambda }_f(n)$ are Fourier coefficients of a given modular form $f$.

8 pages