A generalization of Cohen- Eisenstein series and Shimura liftings and some congruences between Cusp forms and Eisenstein series

In this paper we introduce some modular forms of half-integral weight on congruence group Гo(4N) withN an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we can prove that the restriction of Shimura lifting on Eisenstein spaceEk+1/2+(4N,χl) gives an isomorphism fromEk+1/2+(4N,χl) toE2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.