The modular group and its congruence subgroups has an outer automorphism of order two, which is given by the reflection in the imaginary axis: \( z=x+iy \mapsto -\bar{z}=-x+iy \). This map belongs to $\mathrm{PGL}(2,\mathbb{Z})$ and can be represented by the matrix $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$.

A Maass form $f$ is said to be **even** if $f(-\bar{z})=f(z)$ and **odd** if $f(-\bar{z})=-f(z)$.

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2019-05-01 11:24:23

**Referred to by:**

- mf.maass.mwf.fourierexpansion
- lmfdb/modular_forms/maass_forms/maass_waveforms/views/mwf_main.py (line 295)
- lmfdb/modular_forms/maass_forms/maass_waveforms/views/mwf_main.py (line 463)
- lmfdb/modular_forms/maass_forms/maass_waveforms/views/templates/mwf_browse_all_eigenvalues.html (line 35)
- lmfdb/modular_forms/maass_forms/maass_waveforms/views/templates/mwf_browse_graph.html (line 9)

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

- 2019-05-01 11:24:23 by David Farmer (Reviewed)
- 2019-05-01 11:23:40 by David Farmer
- 2018-12-19 06:35:05 by Alex J. Best

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