$K$ is a real quadratic number field with ring of integers $\mathcal O_K$.
In Zagiers Modular Forms Associated to Real Quadratic Fields from 1975 at page 6 he introduces a surjective map $$m : S\to T$$ where $T$ is given by $$T=\{ (\mu,\nu)\in (\mathfrak a \times \mathfrak a \setminus \{(0,0)\})/\mathcal{O}_K^\times : \delta(\mu,\nu)=\mathfrak a \}$$ where $\mathfrak a$ is a fraktional ideal representing its ideal class. For $\mathfrak a=\mathcal O_K$ it is possible to regard $T$ as the quotient $$\Gamma_\infty \backslash \Gamma_K$$ $\Gamma_K$ being $\operatorname{SL}_2(\mathcal O_K)$ and $\Gamma_\infty$ the subgroup stabilizing the cusp $\infty$. The identification maps an element of $\Gamma_\infty \backslash \Gamma_K$ to its lower row.
I was wondering if there is such an identification for other fraktional ideals as well. Is there some subgroup $\Gamma \subset \operatorname{SL}_2(K)$ such that you can identify $T$ with $\Gamma_\infty \backslash \Gamma$?