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 Gamma(0)(N). Let (f(i), s) and L(f(i), chi, s) denote the L-function of f(i) and its twist by a Dirichlet character chi, respectively. In this note, we obtain a "trace formula" for the values L(f(i), chi, m)<(L(f(i), n))over bar> 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, chi, m)/L(f, n) for a Hecke eigenform f. (C) 2009 Elsevier Inc. All rights reserved.