Twisted Hecke L-values and period polynomials

Abstract

Let f 1 , . . . , f d be an orthogonal basis for the space of cusp forms of even weight 2k on Γ 0 (N). Let L( f i , s) and L( f i , χ , s) denote the L-function of f i and its twist by a Dirichlet character χ , respectively. In this note, we obtain a "trace formula" for the values L( f i , χ , m)L( f i , n) at integers m and n with 0 < m, n < 2k and proper parity. In the case N = 1 or N = 2, the formula gives us a convenient way to evaluate precisely the value of the ratio L( f , χ , m)/L( f , n) for a Hecke eigenform f .