Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms

For any large prime $q$, $x\le 1$ and any real $k\ge 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{χ\in X_q^}} \Big| \sum_{n\leq x} χ(n)λ(n)\Big|^{2k}, \end{equation} where $X_q^{\ast }$ denotes the set of primitive Dirichlet characters modulo $q$ and $\lambda (n)$ the Fourier coefficients of a fixed modular form. The bound we obtain is sharp up to a constant factor under the generalized Riemann Hypothesis.