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).



Definition of level N congruence subgroup of an arithmetic group, useful for computations
Thank you to @Qiaochu Yuan and @Alex B. for the kick in the right direction. Here is the answer to this question for the case introduced in the third paragraph. Hopefully it benefits someone other than me.


Let $K$ be a number field, let $\mathcal{O}$ be its ring of integers, and let $\mathfrak{I}\vartriangleleft\mathcal{O}$. The **principle congruence subgroup of level $\mathfrak{I}$** in $PSL\_2(\mathcal{O})$ is the kernel of the natural projection map $PSL\_2(\mathcal{O})\rightarrow PSL\_2(\mathcal{O}/\mathfrak{I})$. A **congruence subgroup of level $\mathfrak{I}$** in $PSL\_2(\mathcal{O})$ is a subgroup containing the principle congruence subgroup of level $\mathfrak{I}$, where $\mathfrak{I}$ is maximal with respect to inclusion.


In the case where $K=\mathbb{Q}(\sqrt{-d})$ for $d\in\mathbb{N}$ squarefree, $PSL\_2(\mathcal{O})$ is called a **Bianchi group** and denoted $\Gamma\_d$. Also the principle congruence subgroup of level $\mathfrak{I}$ in $\Gamma\_d$ is denoted $\Gamma\_d(\mathfrak{I})$, and can be written explicitly as
$P\Big\{
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix}\in P^{-1}\Gamma\_d\;\Big|\;a\equiv\_{\mathfrak{I}} d\equiv\_{\mathfrak{I}} 1,\; b\equiv\_{\mathfrak{I}} c\equiv\_{\mathfrak{I}} 0 \Big\}$, where $P$ is just the natural projection map from $SL\_2$ to $PSL\_2$.


In the case of general arithmetic groups, we could apply similar definitions once we fix a matrix representation for the group. To define these objects without the matrix representation can of course be done, but in light of the question's focus on performing explicit computations, those definitions are most likely less useful.


