Let $N$ be a positive integer, and let $Q$ be a positive divisor of $N$ satisfying $\gcd(Q,N/Q)=1$. Then there exist $x,y,z,t \in \Z$ for which the matrix \[ W_Q=\left( \begin{matrix} Qx & y \\ Nz & Qt\end{matrix} \right) \] has determinant $Q$. The matrix $W_Q$ normalizes the group $\Gamma_0(N)$, and for any weight $k$ it induces a linear operator $w_Q$ on the space of cusp forms $S_k(\Gamma_0(N))$ that commutes with the Hecke operators $T_p$ for all $p \nmid Q$ and acts as its own inverse.

The linear operator $w_Q$ does not depend on the choice of $x,y,z,t$ and is called the **Atkin-Lehner involution** of $S_k(\Gamma_0(N))$. Any cusp form $f$ in $S_k(\Gamma_0(N))$ which is an eigenform for all $T_p$ with $p \nmid N$ is also an eigenform for $w_Q$, with eigenvalue $\pm 1$.

The matrix $W_Q$ induces an automorphism of the modular curve $X_0(N)$ that is also denoted $w_Q$.

In the case $Q=N$, the Atkin-Lehner involution $w_N$ is also called the Fricke involution.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Voight on 2021-08-04 12:13:07

**Referred to by:**

- cmf.atkin_lehner_dims
- cmf.fricke
- ec.local_root_number
- ec.q.118.a1.top
- ec.q.123.b1.top
- ec.q.141.d1.top
- ec.q.142.a1.top
- ec.q.143.a1.top
- ec.q.155.c1.top
- ec.q.57.a1.top
- ec.q.58.a1.top
- ec.q.65.a1.bottom
- ec.q.65.a2.top
- ec.q.77.a1.top
- ec.q.91.a1.top
- g2c.1225.a.6125.1.top
- g2c.363.a.11979.1.top
- g2c.450.a.2700.1.top
- g2c.450.a.36450.1.top
- g2c.5547.b.16641.1.top
- lmfdb/classical_modular_forms/main.py (line 873)
- lmfdb/classical_modular_forms/templates/cmf_newform.html (line 190)
- lmfdb/classical_modular_forms/templates/cmf_space.html (line 136)

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

- 2021-08-04 12:13:07 by John Voight (Reviewed)
- 2021-08-04 10:19:20 by Edgar Costa
- 2021-08-03 15:11:28 by Andrew Sutherland
- 2021-08-03 14:48:40 by Andrew Sutherland
- 2021-08-03 10:44:07 by Andrew Sutherland
- 2020-10-29 17:16:45 by John Voight (Reviewed)
- 2020-10-29 11:29:35 by John Voight
- 2020-10-29 11:28:58 by John Voight
- 2020-10-29 11:28:48 by John Voight
- 2020-10-29 11:23:10 by John Voight
- 2020-10-22 14:45:58 by Andrew Sutherland (Reviewed)
- 2019-04-28 22:17:49 by David Farmer (Reviewed)
- 2018-10-03 15:27:16 by David Roe (Reviewed)

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