Lower Bounds for Higher Moments of the Twisted L-functions

Let f be a cuspidal holomorphic or Maass Hecke eigenform for the modular group SL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{SL}_{2}(\mathbb{Z})$$\end{document}. Let L(s,f⊗χ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L(s,f\otimes\chi)$$\end{document} denote the twisted L-function associated to f twisted by a primitive Dirichlet character χ modulo q. In this paper, we obtain a lower bound for the 2kth moment of central values of the twisted L-functions.