My title requests something more general than I actually require right now, so I would settle for an answer to something more specific (details below) but I would like to understand the more general concept as well.

When we mean a subgroup of $PSL_2(\mathbb{Z})$, a concise way of describing a congruence subgroup of level $N$ would be: a subgroup containing the kernel of the natural projection map $PSL_2(\mathbb{Z})\rightarrow PSL_2(\mathbb{Z}/N\mathbb{Z})$, where $N$ is minimal.

I am looking to understand how this concept generalizes if instead of subgroups of $PSL_2(\mathbb{Z})$, we are discussing subgroups of $PSL_2(\mathcal{O})$ where $\mathcal{O}$ is the ring of integers of a number field. But for now, I'm interested in the case where $\mathcal{O}$ is the ring of integers of $\mathbb{Q}(\sqrt{-d})$ for $d\in\mathbb{N}$ squarefree, and where the level is $N=2$. Is there a concise definition in this case, analogous to the one given above for the modular case?

It seems that the first thing to do here is use an integral basis to deal with the fact that the entries are no longer rational integers. How exactly escapes me, and then I wonder (departing from the $N=2$ case for a moment) would this then introduce the possibility of reducing modulo multiples of these basis elements? For instance is there such thing as a level $\frac{1+i\sqrt{3}}{2}$ congruence subgroup of $PSL_2(\mathcal{O}_3)$? (If that question is too ignorant, feel free to ignore it and address the one in the last paragraph).

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