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 $\Gamma_0(\mathcal{N})$.

**Knowl status:**

- Review status: reviewed
- Last edited by Holly Swisher on 2019-04-30 17:20:43

**Referred to by:**

- dq.mf.bianchi.extent
- mf.bianchi.cm
- mf.bianchi.hecke_algebra
- mf.bianchi.newform
- mf.bianchi.sign
- mf.bianchi.spaces
- lmfdb/bianchi_modular_forms/templates/bmf-browse.html (lines 23-28)
- lmfdb/bianchi_modular_forms/templates/bmf-browse.html (line 69)
- lmfdb/bianchi_modular_forms/templates/bmf-field_dim_table.html (lines 13-18)
- lmfdb/bianchi_modular_forms/templates/bmf-field_dim_table.html (line 90)
- lmfdb/bianchi_modular_forms/templates/bmf-newform.html (line 24)
- lmfdb/bianchi_modular_forms/templates/bmf-search_results.html (line 11)
- lmfdb/bianchi_modular_forms/templates/bmf-search_results.html (line 76)
- lmfdb/bianchi_modular_forms/templates/bmf-space.html (line 22)

**History:**(expand/hide all)

- 2019-08-31 21:22:19 by Andrew Sutherland
- 2019-04-30 17:20:43 by Holly Swisher (Reviewed)
- 2019-04-30 17:01:02 by Holly Swisher
- 2018-12-13 12:57:10 by Alex J. Best

**Differences**(show/hide)