 Cusp-form divisor presentation

For a torsion-free modular curve with regular modular cusps, let $f$ be a nonzero meromorphic weight-$k$ form and $C$ the reduced sum of modular cusp points. The local-order divisor of $f$, minus $C$, imposes exactly holomorphy in the interior and vanishing at every modular cusp on the product $f\phi$. Dividing any cusp form by $f$ gives the reverse identification with a Riemann-Roch space. The regular-cusp valence formula on a torsion-free modular curve and Riemann-Roch theorem compute the dimension when the resulting divisor has degree greater than the canonical degree.