The Hecke Operators on Sk(Γ1(N))

In this paper we describe the space of the cusp forms with the weight k for the congruence subgroup Γ1(N), Sk (Γ1(N)), using Eichler-Shimura isomorphism, Shapiro lemma and the theory of group cohomology. An algorithm computing an integral basis of Sk (Γ1(N)) is developed. Applying the Hecke operators on the basis we can determine newforms in Sk (Γ1(N)).