Coset norm of a modular form ($\mathrm{Nm}(f)={\prod }_{\alpha \in \Gamma \backslash {\Gamma }^{\prime }}f{|}_k\alpha$)

For a finite-index inclusion of modular groups and a form with trivial multiplier, multiply its slash operator for modular forms translates over left coset representatives. Right multiplication permutes the factors, producing a form of weight $k[{\Gamma }^{\prime }:\Gamma ]$ for the larger group. modular cusp cusp widths determine the orders contributed by the factors; the product is nonzero if the original form is nonzero.