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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 our main theorems, we will also give an affirmative answer to a speculation of Imamoḡlu and Kohnen on a basis of S w+2 (Γ 0 (2)).2000 Mathematics Subject Classification. Primary 11F25; Secondary 11F11, 11F67.

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