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 $χ$ modulo $C$, we prove that the $χ$-part of the group $J_0(DC)_\text{tor}$ of torsion points of $J_0(DC)$ coincides with the $χ$-part of its cuspidal subgroup, away from those primes of bad reduction or where possible congruences between oldforms and newforms occur.

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