Magma V2.7 includes a preview of a major new initiative in modular forms and modular curves. The basis for the constructions is a finite-rank Hecke module equipped with the action of Hecke operators. The first examples of Hecke modules come from modular symbols. Other constructions include divisor groups of supersingular points on modular curves. A model for the supersingular points is provided either by supersingular elliptic curves following Mestre and Oesterlé or by quaternion ideal theory, the latter having deep connections to the theory of Shimura curves, generalizing modular curves.