  Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be
an even integer. We use deformation theory of pseudo-representations to study
the completion of the Hecke algebra acting on the space of cuspidal modular
forms of weight $k$ and level $\Gamma_0(\ell)$ at the maximal Eisenstein ideal
containing $p$. We give a necessary and sufficient condition for the
$\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of
vanishing of the cup products of certain global Galois cohomology classes. We
also recover some of the results proven by Wake and Wang-Erickson for $k=2$
using our methods. In addition, we prove some $R=\mathbb{T}$ theorems under
certain hypothesis.
