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

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 $${\displaystyle \sum _{\begin{array}{c}\chi q\end{array}}\mid \sum _{n\le x}\chi (n)\lambda (n){\mid }^{2k},}$$ where $\lambda (n)$ denotes the Fourier coefficients of a fixed modular form. In particular, our result implies that $${\displaystyle \frac{1}{q-1}\sum _{\begin{array}{c}\chi q\end{array}}\mid \sum _{n\le x}\chi (n)\lambda (n)\mid =o(\sqrt{x}),}$$ when both $x$ and $q\mathrm{/}x$ tend to infinity with $q$.