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

TL;DR

This paper asymptotically evaluates a smoothed sum involving quadratic twists of Fourier coefficients of modular forms, providing insights into their mean square behavior over square-free integers.

Contribution

It introduces a new asymptotic evaluation of quadratic twists of Fourier coefficients, extending understanding of their mean square distribution.

Findings

Asymptotic formula for the sum over square-free integers

Insights into the distribution of Fourier coefficients under quadratic twists

Extension of previous results on modular form coefficients

Abstract

In this paper, we evaluate asymptotically a smoothed version of the sum \[ \displaystyle \sideset{}{^*}\sum_{d \leq X} \left( \sum_{n \leq Y} \lambda_f(n)\Big(\frac{8d}{n}\Big)\right)^2, \] where $\Big ( \frac{8d}{\cdot}\Big )$ 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 $f$.