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

Let f   be a normalized holomorphic Hecke newform of weight k≤Kk≤K and level q≤Qq≤Q with trivial nebentypus. We give the approximate formulas for the first moments of L(1/2,f⊗g)L(1/2,f⊗g) and L′(1/2,f⊗g)L′(1/2,f⊗g), where g   runs over Hl(N,χN)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)L(s,f⊗g) and L′(s,f⊗g)L′(s,f⊗g), where g   runs over Hl(N,χN)Hl(N,χN).