The **Fricke involution** is the Atkin-Lehner involution $w_N$ on the space $S_k(\Gamma_0(N))$ (induced by the corresponding involution on the modular curve $X_0(N)$).

For a newform $f \in S_k^{\textup{new}}(\Gamma_0(N))$, the sign of the functional equation satisfied by the L-function attached to $f$ is $i^{-k}$ times the eigenvalue of $W_N$ on $f$. So, for example when $k=2$, the signs swap, and the analytic rank of $f$ is even when $w_N f = -f$ and odd when $w_N f = +f$.

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2021-08-04 12:28:20

**Referred to by:**

- cmf.atkin-lehner
- cmf.minus_space
- cmf.plus_space
- ec.q.101.a1.top
- ec.q.131.a1.top
- ec.q.37.a1.top
- ec.q.43.a1.top
- ec.q.53.a1.top
- ec.q.61.a1.top
- ec.q.65.a1.bottom
- ec.q.65.a1.top
- ec.q.79.a1.top
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- g2c.10609.a.10609.1.top
- g2c.11449.a.11449.1.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/main.py (line 859)
- lmfdb/classical_modular_forms/templates/cmf_newform_common.html (line 105)
- lmfdb/classical_modular_forms/templates/cmf_space.html (line 136)
- lmfdb/classical_modular_forms/web_space.py (line 93)
- lmfdb/maass_forms/main.py (line 225)
- lmfdb/maass_forms/templates/maass_form.html (line 26)

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

- 2021-08-04 12:28:20 by John Voight (Reviewed)
- 2021-08-04 12:13:50 by John Voight
- 2021-08-03 14:37:49 by John Voight
- 2021-08-03 11:09:26 by John Voight (Reviewed)
- 2021-08-03 10:46:51 by Andrew Sutherland
- 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)

**Differences**(show/hide)