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A weight-$2n$ meromorphic modular form satisfies $f(\gamma z)=(cz+d{)}^{2n}f(z)$, while $d(\gamma z)=(cz+d{)}^{-2}dz$. The factors cancel in $f(z)(dz{)}^n$, giving a $\Gamma$-invariant meromorphic n-differential on the complex upper half-plane.

Away from elliptic points and cusps, descend this differential through local inverse branches of the quotient map. At an elliptic point of effective stabilizer order $e$, choose a coordinate $t$ in which the stabilizer acts by $t\mapsto \zeta t$, and use $w=t^e$ downstairs. If the lifted coefficient has order $r$, invariance forces $r+n$ to be divisible by $e$. The descended differential has integer orderIndeed $F(w)(dw{)}^n$ pulls back to $e^n{t}^{n(e-1)}F(t^e)(dt{)}^n$. At a modular cusp of width $h$, the coordinate $q_c=e^{2\pi iz/h}$ gives $dz=(h/(2\pi i))d{q}_c/q_c$. ThereforeBoth formulas provide meromorphic extensions over the omitted points, producing the unique differential with the prescribed pullback on the compactified modular curve.

For the character-valued question, write $X=X_0(N)$ and let $f$ be the chosen nonzero meromorphic form. The quotient of two forms with the same weight and Dirichlet character is a meromorphic function on $X$. At an interior point $P$ with effective elliptic order $e_P$ (one at ordinary points), let $r_P$ be the vanishing order of the lifted $f$. At a cusp let $v_c$ be its order in the width coordinate; this can be fractional because of the character multiplier. Multiplying by a meromorphic function $\phi$ changes these orders by $e_P{\mathrm{ord}}_P\phi$ and ${\mathrm{ord}}_c\phi$, respectively. Interior holomorphy requires a nonnegative order, while cusp vanishing requires a strictly positive order. The exact divisor is consequentlyOnly finitely many coefficients are nonzero. The order inequalities are equivalent to $\mathrm{div}(\phi )+D(f)\ge 0$. Division by $f$ proves the reverse inclusion, so the Riemann-Roch space givesThis extends the cusp-form divisor presentation to elliptic points and character multipliers.

Choose a nonzero meromorphic one-differential on $X$, and let $g$ be its weight-two pullback. For $f_t=f{g}^t$, the interior order is $r_P(f)+t{r}_P(g)$ and the cusp order is $v_c(f)+t{v}_c(g)$. The elliptic orders are one, two or three, so adding six to $t$ changes each interior floor by $6r_P(g)/e_P$, and each cusp ceiling by $6v_c(g)$. The valence formula for the modular group for $g$ says ${\sum }_P{r}_P(g)/e_P+{\sum }_c{v}_c(g)=d/6$, where $d=[S{L}_2(\mathbb{Z}):{\Gamma }_0(N)]$. ThusThe degrees tend to infinity. For sufficiently large $t$, the Riemann-Roch theorem therefore givesThe periodic formula obtained this way is an eventual formula. For small $t$, the exact answer includes the Riemann-Roch correction $\ell (K_X-D(f_t))$; it should not silently be discarded without the large-degree hypothesis.

Let $\rho =e^{2\pi i/3}$. For a nonzero meromorphic modular form $f$ of weight $k$, the valence formula for the modular group isHere $v_z(f)$ is positive for a zero and negative for a pole, and the factors $1/2$ and $1/3$ account for the stabilizers of the two elliptic points. Equivalently, the weighted number of zeros minus poles in $\Gamma (1)\backslash \mathfrak{h}$ is $k/12-v_{\infty }(f)$.