On quadratic character twists of Hecke \(L\)-functions attached to cusp forms of varying weights at the central point (Q2717766)

Let \(F_{2k}\) denote the set of normalized elliptic Hecke cusp forms of even weight \(2k\). Let \(L(f,D,s)\) be the twist of the Hecke \(L\)-series of \(f\in F_{2k}\) with the quadratic character \((\frac D*)\), where \(D\) is a fundamental discriminant. Then the authors show that NEWLINE\[NEWLINE\sum_{f\in F_{2k}}L(f,D,k)\ll_{\varepsilon,D}k^{1+\varepsilon}\qquad(k\to\infty,(-1)^kD>0).NEWLINE\]NEWLINE Under the stronger assumption \(L(f,D,k)\ll_{\varepsilon,D}k^\varepsilon\) for some \(0 <\varepsilon <1\) they moreover derive NEWLINE\[NEWLINE\#\{f\in F_{2k}\mid L(f, D, k)\neq 0\}\gg_{\varepsilon,D}k^{1-\varepsilon}/\log k.NEWLINE\]