The kernel of a rational Eisenstein prime at non-squarefree level

Let $\ell \geq 5$ be a prime and let $N$ be a non-squarefree integer not divisible by $\ell$. For a rational Eisenstein prime $\mathfrak{m}$ of the Hecke ring $\mathbb{T}(N)$ of level $N$ acting on $J_0(N)$, we precisely compute the dimension of the kernel $J_0(N)[\mathfrak{m}]$ under a mild assumption. In the case of level $qr^2$ which violates our mild assumption, we propose a conjecture based on Sage computations. Assuming this conjecture, we complete our computation in all the remaining cases.