Valence formula for the modular group

For a nonzero meromorphic modular form $f$ of weight $k$,where the sum takes one representative from each modular-group orbit.

A nonzero weight-fourteen cusp form would have order at infinity at least one and order at $i$ at least one, since $i^{14}=-1$. The valence formula for the modular group would then give $1+1/2\le 14/12$, which is impossible.

A weight-two modular form obeys $f(i)=i^{2}f(i)=-f(i)$, so it vanishes at $i$. The valence formula for the modular group would then have left side at least $1/2$, but right side $2/12=1/6$. All orders are nonnegative by holomorphy, giving a contradiction for nonzero $f$.

For nonnegative integer $k$ and a finite-index subgroup $\Gamma$, a nonzero modular form has order at infinity, measured in a genuine periodic cusp parameter, at most $k[S{L}_2(\mathbb{Z}):\Gamma ]/12$. The product of its slash translates over the left cosets is a nonzero level-one form; the translates along the infinity-cusp orbit contribute its full local order to this product. The valence formula for the modular group gives the bound. Initial coefficients through that bound therefore give an injective linear map.