Quadratic torsion subgroups of modular Jacobian varieties

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.