  Let $p\geq 5$ be prime. For elliptic modular forms of weight 2 and level
$\Gamma_0(N)$ where $N>6$ is squarefree, we bound the depth of Eisenstein
congruences modulo $p$ (from below) by a generalized Bernoulli number with
correction factors and show how this depth detects the local non-principality
of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to
give an infinite class of examples where the Eisenstein ideal is not locally
principal. Lastly, we illustrate these results with explicit computations and
give an interesting commutative algebra application related to Hilbert--Samuel
multiplicities.
