Fricke sign from a nonvanishing fixed-point value (${\tau }_{\ast }=i/\sqrt{N}$)

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.