Level of a Bianchi modular form (reviewed)

The level of a Bianchi modular form $F$ is the discrete subgroup $\Gamma$ of $\PSL(2,\C)$ such that $F|_k\gamma = F$ for every $\gamma \in \Gamma$, where $k$ is the weight of $F$.

More precisely, for a Bianchi modular form over the imaginary quadratic field $K$, the level $\Gamma$ is a congruence subgroup of the Bianchi group $\GL(2,\mathcal{O}_K)$. The most common levels are those of the form \[ \Gamma_0(\mathcal{N}) = \left\{\begin{pmatrix} a &b \\ c&d \end{pmatrix}\in\GL(2,\mathcal{O}_K)\mid c\in\mathcal{N}\right\} \] where $\mathcal{N}$ is an integral ideal of $\mathcal{O}_K$. In this case one often says that $F$ has "level $\mathcal{N}$" as an abbreviation for "level $\Gamma_0(\mathcal{N})$".

Note these levels are $\GL_2$-levels. It is also possible to consider $\SL_2$-levels, which are subgroups of $\SL(2,\mathcal{O}_K)$. The most common such levels are those of the form $\Gamma_0(\mathcal{N}\cap\SL(2,\mathcal{O}_K))$. Since this is a smaller group, the space of Bianchi forms at such a level is larger, and contains the forms on $\Gamma_0(\mathcal{N})$.