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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.

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• Last edited by John Voight on 2020-10-29 11:17:58
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