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 represented by the matrix $W_N=\begin{pmatrix}0&-1\\N&0\end{pmatrix}$; this matrix normalizes the group $\Gamma_0(N)$ and therefore induces an (involutive) operator on the space of cusp forms $S_k(\Gamma_0(N))$ of weight $k$ with trivial character.

The Fricke involution is the product of all the Atkin-Lehner involutions $W_Q$ for $Q \parallel N$. 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 John Voight on 2020-10-29 11:17:58

**Referred to by:**

- cmf.minus_space
- cmf.plus_space
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- ec.q.131.a1.top
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- ec.q.43.a1.top
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- ec.q.65.a1.bottom
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- g2c.10609.a.10609.1.top
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- g2c.27889.a.27889.1.top
- g2c.36481.a.36481.1.top
- g2c.4489.a.4489.1.top
- g2c.5329.b.5329.1.top
- g2c.8281.a.8281.1.top
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (line 96)
- 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 90)
- lmfdb/maass_forms/templates/maass_form.html (line 26)
- lmfdb/maass_forms/templates/maass_search_results.html (line 17)

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

- 2020-10-29 11:17:58 by John Voight (Reviewed)
- 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)

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