Atkin–Lehner involutions

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.