Bounds for short character sums for $GL(2) \times GL(3)$ twists

Abstract

Let $\pi$ be a $SL(3,\mathbb{Z})$ Hecke Maass-cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass-cusp form with normalized Fourier coefficients $\lambda_{\pi}(r,n) \text{ and }\lambda_{f}(n)$ respectively and $\chi$ be any non-trivial character mod $p$ where $p$ is a prime. Then we have $$S_{\pi ,f,\chi }(N)\ll_{\pi , f, \epsilon} N^{3/4}p^{11/16 +{\eta /4}}(Np)^\epsilon .$$