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

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