Multiplication by the modular discriminant ($M_k\stackrel{f\mapsto \Delta f}{\to }{S}_{k+12}$)

The modular discriminant has a simple zero at infinity and no zero in the upper half-plane, by the valence formula for the modular group. Multiplication by it identifies weight-$k$ modular forms with weight-$(k+12)$ cusp forms. Conversely dividing a cusp form by it preserves interior holomorphy and removes one order of vanishing at infinity.