We give a general conjecture concerning the existence of Eisenstein congruences between weight \(k\ge 3\) newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character \(\chi \) of conductor N. The special cases \(M=1\) and \(M=p\) prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.
