Modular symbols, Eisenstein series, and congruences

Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k >= 2 and of the same level N, both eigenfunctions of the Hecke operators, and both normalized so that a(1) (f) = a(1)(E) = 1. The main result we prove is that when E and f are congruent mod a prime p (which we take in this paper to be a prime of (Q) over tilde lying over a rational prime p > 2), the algebraic parts of the special values L(E, x, j) and L(f, x, j) satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions,