In his landmark paper “Modular forms and the Eisenstein ideal,” Mazur studied congruences modulo a prime p between the Hecke eigenvalues of an Eisenstein series and the Hecke eigenvalues of cusp forms, assuming these modular forms have weight 2 and prime level N.

He asked about generalizations to squarefree levels N. I will present some work on such generalizations, which is joint with Preston Wake and Catherine Hsu.