 Solution

For matrices of positive determinant, use the determinant-normalized slash operator for modular formsIt satisfies the composition law $(f{|}_k[\alpha ]){|}_k[\beta ]=f{|}_k[\alpha \beta ]$. IfthenThus the Fricke involution normalizes the group, and the composition law proves that $f{|}_k[{\alpha }_N]$ has the required transformation law.

The matrix ${\alpha }_N$ permutes the rational cusps. Applying it to a local Fourier expansion merely transports that expansion to the image cusp, with a nonzero change of local parameter. It therefore preserves holomorphy and vanishing at every cusp. Hence