Moments of L-Functions Attached to the Twist ofModular Form by Dirichlet Characters

Let $f(z)$ be a holomorphic cusp form of weight $kappa$ with respect to the fullmodular group $SL_{2}(mathbb{Z})$. Let $L(s,f)$ be the automorphic$L$-function associated with $f(z)$ and $chi$ be a Dirichletcharacter modulo $q$. In this paper, the authors prove thatunconditionally for $k=frac{1}{n}$ with $nin mathbb{N}$,M_{k}(q,f)=sumlimits_{{chi({rm mod},q)}atop {chineqchi_0}}Big|LBig(frac{1}{2},fotimeschiBig)Big|^{2k}ll_kphi(q)(logq)^{k^2}, and the result also holds for any real number $00$ and any large prime $q$.