Upper Bounds for low moments of twisted Fourier coefficients of modular forms

Abstract

For any large prime $q$, $1\le x\le q$ and any real $0\le k\le 1$, we prove an upper bound for the following $2k$-th moment $$\sum _{\begin{array}{c}\chi \mathrm{mod}q\end{array}}|\sum _{n\le x}\chi (n)\lambda (n){|}^{2k},$$ where $\lambda (n)$ denotes the Fourier coefficients of a fixed modular form. In particular, our result implies that $$\frac{1}{q-1}\sum _{\begin{array}{c}\chi \mathrm{mod}q\end{array}}|\sum _{n\le x}\chi (n)\lambda (n)|=o(\sqrt{x}),$$ when both $x$ and $q/x$ tend to infinity with $q$.

21 pages