The **Fricke involution** $W_N$ is the involution of modular curve $X_{0}(N)$ given by $z\mapsto \frac{-1}{Nz}$.

The Fricke involution is also an operation on the group $\Gamma_0(N)$ given by conjugation by $\begin{pmatrix}0&-1\\N&0\end{pmatrix}$. That action extends to an an action on the modular forms with trivial character on $\Gamma_0(N)$.

The Fricke involution is the product of all the Atkin-Lehner involutions $W_Q$ for $Q \Vert N$. As a consequence, forms which are eigenforms for the Hecke operators are also eigenforms for $W_N$. Note: the Fricke involution also acts on some spaces with non-trivial character but in those cases it does not commute with all Hecke operators.

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2019-05-01 10:34:30

**Referred to by:**

- cmf.minus_space
- cmf.plus_space
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (line 90)
- lmfdb/classical_modular_forms/templates/cmf_newform_list.html (line 26)
- lmfdb/classical_modular_forms/templates/cmf_space.html (line 136)
- lmfdb/classical_modular_forms/web_space.py (line 89)
- lmfdb/modular_forms/maass_forms/maass_waveforms/views/mwf_main.py (line 297)
- 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 43)

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

- 2019-05-01 10:34:30 by David Farmer (Reviewed)
- 2019-04-29 10:08:05 by Andrew Sutherland (Reviewed)
- 2019-04-28 22:13:51 by David Farmer (Reviewed)
- 2019-04-28 22:11:39 by David Farmer (Reviewed)
- 2019-04-28 22:04:05 by David Farmer
- 2019-04-28 22:03:44 by David Farmer
- 2019-02-01 19:52:28 by John Voight (Reviewed)

**Differences**(show/hide)