Period polynomials and explicit formulas for Hecke operators on Γ₀(2)

Abstract

Abstract Let S w +2 (Γ 0 ( N )) be the vector space of cusp forms of weight w + 2 on the congruence subgroup Γ 0 ( N ). We first determine explicit formulas for period polynomials of elements in S w +2 (Γ 0 ( N )) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for S w +2 (Γ 0 (2)), and extend the Eichler–Shimura–Manin isomorphism theorem to Γ 0 (2). This implies that there are natural correspondences between the spaces of cusp forms on Γ 0 (2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on S w +2 (Γ 0 (2)). As an application of main theorems, we will also give an affirmative answer to a speculation of Imamo g lu and Kohnen on a basis of S w +2 (Γ 0 (2)).