  Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for a
square-free integer $N>6$. For any Eisenstein maximal ideal $\mathfrak{m}$ of
the Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}]\neq 0$.
To prove this, we calculate the index of an Eisenstein ideal $\mathcal{I}$
contained in $\mathfrak{m}$ by computing the order of a cuspidal divisor
annihilated by $\mathcal{I}$.
