On Mazur's Eisenstein ideal

In his landmark 1976 paper "Modular curves and the Eisenstein ideal," Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed some questions: how many cusp forms of a given level are congruent to the Eisenstein series? Is a similar theory available for squarefree level N? In joint work with Preston Wake, we give answers to these questions.