  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 S_{w+2}(\Gamma_0(N)) by means of Bernoulli
polynomials. When N=2, from these explicit formulas we obtain new bases for
S_{w+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
S_{w+2}(\Gamma_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
S_{w+2}(\Gamma_0(2)).
