Fricke involution
= Fricke involution
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{title2=$W_N$}
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The Fricke matrix $W_N=\begin{pmatrix}0&-1\\N&0\end{pmatrix}$ normalizes $\Gamma_1(N)$ and $\Gamma_0(N)$ under the appropriate determinant-normalized slash action. It exchanges the cusps zero and infinity.