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 Г_o(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 spaceE _k+1/2 ⁺ (4N,χl) gives an isomorphism fromE _k+1/2 ⁺ (4N,χl) toE _2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.