Sums of Fourier coefficients involving theta series and Dirichlet characters

Abstract

Let $f$ be a holomorphic or Maass cusp forms for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ with normalized Fourier coefficients $\hat{I}{»}_f(n)$ and \bna r_{\ell}(n)=\#\left\{(n_1,\cdots,n_{\ell})\in \mathbb{Z}^2:n_1^2+\cdots+n_{\ell}^2=n\right\}. \ena Let $\ddot{I}‡$ be a primitive Dirichlet character of modulus $p$, a prime. In this paper, we are concerned with obtaining nontrivial estimates for the sum \bna \sum_{n\geq1}λ_f(n)r_{\ell}(n)χ(n)w\left(\frac{n}{X}\right) \ena for any $\ell \ge 3$, where $w(x)$ be a smooth function compactly supported in $[1/2,1]$.