Moments of quadratic twists of modular \(L\)-functions (Q6579144)

Let \(f\) be a modular form of weight \(\kappa\equiv 0\: \text{mod}\: 4\) for the full modular group and suppose that \(f\) is a Hecke eigenform. For \(d\) a fundamental discriminant, let \(\chi_d (\cdot) = (\frac{d}{.})\) denote the primitive quadratic character with conductor \(|d|\). Then, \(f \otimes \chi_d \) is a primitive Hecke eigenform of level \(|d|^2\), with \(L\)-function denoted by \(L(s, f\otimes \chi_d)\). The purpose of the paper under review is to study \N\[\NM(k)=\mathop{{\sum}^*}_{\substack{0<8d<X\\ (d,2)=1}} L\left(\frac{1}{2}, f\otimes \chi_{8d}\right)^k,\N\]\Nwhere \(d\) is odd and the sum \(\sum^*\) is taken over squarefree integers. It is proven unconditionally that \N\[\NM(2)\sim C_f X\log X, \N\]\Nwhere \(C_f\) is as an explicit constant. This result was previously known conditionally on GRH by the work of \textit{K. Soundararajan} and \textit{M. Young} [J. Eur. Math. Soc. 12, No. 5, 1097--1116 (2010; Zbl 1213.11165)].