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