Sums of Fourier coefficients involving theta series and Dirichlet characters

Let $f$ be a holomorphic or Maass cusp forms for $\mathrm{S}{\mathrm{L}}_2(\mathbb{Z})$ with normalized Fourier coefficients ${\lambda }_f(n)$ 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 $\chi$ 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 $\mathrm{\ell }\ge 3$, where $w(x)$ be a smooth function compactly supported in $[1\mathrm{/}2,1]$.