Fricke involution for Gamma_0 (reviewed)

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.