Fricke involution ($W_N$)

The Fricke matrix $W_N=\left(\begin{array}{cc}0& -1\\ N& 0\end{array}\right)$ normalizes ${\Gamma }_1(N)$ and ${\Gamma }_0(N)$ under the appropriate determinant-normalized slash action. It exchanges the cusps zero and infinity.

If a one-dimensional modular cusp space is stable under the phase-normalized Fricke involution, its scalar action is determined by evaluation at the fixed point. The phase-normalized prefactor is one there; if the form is nonzero at that point, the eigenvalue is $+1$. A product with strictly positive convergent factors on the imaginary axis supplies such a value and also makes the central completed L-function of a cusp form positive when its central Mellin integral has real positive kernel.