On effective determination of cusp forms by L-values, level aspect

Abstract

Let f be a normalized holomorphic Hecke newform of weight k≤K and level q≤Q with trivial nebentypus. We give the approximate formulas for the first moments of L(1/2,f⊗g) and L′(1/2,f⊗g), where g runs over Hl(N,χN), the normalized Hecke eigen-basis of holomorphic cusp forms of weight l and level N with nebentypus χN=(N⋅). As an application, we obtain some quantitative results that f is uniquely determined by the central values of L(s,f⊗g) and L′(s,f⊗g), where g runs over Hl(N,χN).