Twisted periods of modular forms

Let $S_k$ denote the space of cusp forms of weight $k$ and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $\chi$ mod $D$, we introduce twisted periods $r_{t,\chi}$ on $S_k$. 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.