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, let $K$ be an imaginary quadratic field; we suppose that $K$ has class number $1$ for simplicity. For a Bianchi modular form over $K$, 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 says that $F$ has "level $\mathcal{N}$" as an abbreviation for "level $\Gamma_0(\mathcal{N})$". The level label is the ideal label of $\mathcal N$ and the level norm is its norm (a positive integer).

Note these levels are $\mathbf{GL}_2$-levels. It is also possible to consider $\mathbf{SL}_2$-levels, which are subgroups of $\SL_2(\mathcal{O}_K)$: for example, $\Gamma_0(\mathcal{N})\cap\SL_2(\mathcal{O}_K)$. Since such groups are smaller, the corresponding spaces of Bianchi forms are larger, containing as a subspace the forms on $\Gamma_0(\mathcal{N})$.