  Let $D$ be an odd square-free positive integer and $C$ a divisor of $D$. For
any quadratic character $\chi$ modulo $C$, we prove that the $\chi$-part of the
group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the
$\chi$-part of its cuspidal subgroup, away from those primes of bad reduction
or where possible congruences between oldforms and newforms occur.
