What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$? I have been trying to learn about congruence groups.  Here is an example:
\begin{eqnarray*} \Gamma\big(1+2i\big) &=& \text{SL}_2\big(\mathbb{Z}[i]\big)(1+2i)  \\ \\
&=&  \left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) 
: ad-bc = 1 \text{ and }
\left( \begin{array}{cc} a & b \\ c & d \end{array} \right)  \equiv 
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)  \pmod{1+2i} 
  \right\} \\ \\
&\subseteq & \text{SL}_2\big(\mathbb{Z}[i]\big) \end{eqnarray*}
While the proof is uses the theory of algebraic groups, we can prove that is finitely generated.  This particular case is look elementary, these are $2 \times 2$ invertible matrices, $\mathbb{Z}[i]$ is a Eucliean domain, and we can solve $ad-bc = 1$ by finding two primes (e.g. $6+i $ or  $ 2+3i$) and looking for their greatest common divisor. 
Since this group is finitely generated, how can I find a generating set?  What are the generators? Even computer code would be helpful.

In order to specify what I am looking for here is the result for $\text{SL}_2(\mathbb{Z})$: 
$$ \text{SL}_2(\mathbb{Z}) \simeq \big\{  S,T : S^2 = 1,\; (ST)^3 = 1 \big\}  \quad\text{with}\quad 
S = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array} \right)
\text{ and } 
T = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right)$$
At least over $\mathbb{Z}$ there an exact sequence relating the congruence groups to the special linear group over finite fields:
$$ 1 \to \Gamma(N) \to \Gamma \to \text{PSL}_2\big(\mathbb{Z}/N\mathbb{Z}\big) \to 1 $$
And so it's likely that congruence groups of $\text{SL}_2(\mathbb{Z})$ should have finite presentations.  Even if I do something slightly more inefficient and use the exact sequence.

Also 


*

*Are congruence subgroups of the modular group finitely presented?
 A: $PSL_2(\mathbb{O}_d)$ acts on the upper half-space model of hyperbolic 3-space in a nice way, namely the quotient can be viewed as a finite volume 3-orbifold. Since all principal congruence subgroups are finite index in $PSL_2(\mathbb{O}_d)$, all principal congruence subgroups are finitely generated and correspond to some (finite-sided/geometrically finite) finite volume orbifold cover of $H^3/PSL_2(\mathbb{O}_d)$. (There are probably more direct ways of showing this, but might help frame this discussion.) 
As the OP notes in the question, after specifying a $d$, one can draw a direct analogy between  group, and $PSL_2(\mathbb{Z})$ acting on $H^2$. There is natural question, "Which principal congruence subgroups of $PSL_2(\mathbb{Z})$ are genus zero (aka fill to a sphere)?" One 3d analog of this question is "Which principal congruence subgroups of $PSL_2(\mathbb{O}_d)$ are homeomorphic to $S^3\setminus L$ for some link $L$ in $S^3$?" 
Happily, over a series of papers the subsets of collection of Baker, Goerner and Reid provide a complete solution to this question. More relevantly, $PSL_2(\mathbb{Z}[i])(2+i)$ is one of the links they study.
Proposition 3.1 and Lemma 3.2 of
Baker, Mark D.; Reid, Alan W., Principal congruence link complements, Ann. Fac. Sci. Toulouse, Math. (6) 23, No. 5, 1063-1092 (2014). ZBL1322.57015.http://www.numdam.org/item/AFST_2014_6_23_5_1063_0
discusses this question directly. It gives a six generator presentation of  $PSL_2(\mathbb{Z}[i])(2+i)$. The index of this group is confirmed using a magma computation. Also the generating set is minimal as the abelianization of this group has rank 6. Selecting the correct coset representative from each of these elements in $PSL_2(\mathbb{Z}[i])$ should give you group you are looking for. 
Some of the other papers in there line of attack are here if you are interested:
Baker, Mark D., Link complements and the Bianchi modular groups, Trans. Am. Math. Soc. 353, No. 8, 3229-3246 (2001). ZBL0986.20049.
Baker, Mark D., Link complements and integer rings of class number greater than one, Topology ’90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 55-59 (1992). ZBL0768.57005.
Görner, Matthias, Regular tessellation link complements, Exp. Math. 24, No. 2, 225-246 (2015). ZBL1319.57002.
 Baker, Mark D.,  Goerner, Matthias, and  Reid, Alan W., All principal congruence link groups arXiv preprint arXiv:1802.01275 (2018).
Note that as the title suggests, the last paper gives a complete classification of principal congruence link complements. Of course, the methods of these papers which make a good guess for what the generators of the principal congruence subgroups are can be adjusted, especially if the pair $(d,I)$ is not on their lists. In this case, one would definitely need more than just a trace 2 generator corresponding to each cusp. Nevertheless, the magma code can be adapted to more general searches for generating sets of principal congruence groups of $PSL_2(\mathbb{O}_d)$. 
