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

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-25 11:26:46

**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/bianchi_modular_form.py (line 666)
- lmfdb/bianchi_modular_forms/bianchi_modular_form.py (line 729)
- lmfdb/bianchi_modular_forms/bianchi_modular_form.py (line 737)
- lmfdb/bianchi_modular_forms/bianchi_modular_form.py (line 749)
- lmfdb/bianchi_modular_forms/bianchi_modular_form.py (line 767)
- lmfdb/bianchi_modular_forms/templates/bmf-browse.html (lines 21-26)
- lmfdb/bianchi_modular_forms/templates/bmf-field_dim_table.html (lines 13-18)
- lmfdb/bianchi_modular_forms/templates/bmf-field_dim_table.html (line 99)
- lmfdb/bianchi_modular_forms/templates/bmf-newform.html (lines 24-25)
- lmfdb/bianchi_modular_forms/templates/bmf-search_results.html (line 14)
- lmfdb/bianchi_modular_forms/templates/bmf-space.html (line 22)

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

- 2020-10-25 11:26:46 by Andrew Sutherland (Reviewed)
- 2020-10-23 17:44:02 by John Voight (Reviewed)
- 2020-10-23 17:43:39 by John Voight
- 2020-10-23 17:43:00 by John Voight
- 2020-10-23 17:42:32 by John Voight
- 2020-10-23 17:37:02 by John Voight
- 2020-10-18 17:19:24 by Andrew Sutherland
- 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)