Newform Eisenstein Congruences of Local Origin

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.