Computing points on bielliptic modular curves over fixed quadratic fields

We present a Mordell-Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb{Q}(\sqrt{d}))$ for $N\in \{53,61,65,79,83,89,101,131\}$ and $\mid d\mid <100$.