Twisted periods of modular forms

Let Sₖ denote the space of cusp forms of weight k and level one. For 0≤ t≤ k-2 and primitive Dirichlet character χ mod D , we introduce twisted periods rt,χ on Sₖ . We show that for a fixed natural number n , if k is sufficiently large relative to n and D , then any n periods with the same twist but different indices are linearly independent. We also prove that if k is sufficiently large relative to $D,$ then any n periods with the same index but different twists mod D are linearly independent. These results are achieved by studying the trace of the products and Rankin–Cohen brackets of Eisenstein series of level D with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda’s conjecture implies a non-vanishing result on twisted central L -values of normalized Hecke eigenforms.