= Solution
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 order
$$
\boxed{\operatorname{ord}_{w}\omega(f)=\frac{r+n}{e}-n.}
$$
Indeed $F(w)(dw)^n$ pulls back to $e^nt^{n(e-1)}F(t^e)(dt)^n$. At a <modular cusp> of width $h$, the coordinate $q_c=e^{2\pi iz/h}$ gives $dz=(h/(2\pi i))dq_c/q_c$. Therefore
$$
\boxed{\operatorname{ord}_{c}\omega(f)=v_c(f)-n.}
$$
Both 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 $\varphi$ changes these orders by $e_P\operatorname{ord}_P\varphi$ and $\operatorname{ord}_c\varphi$, respectively. Interior holomorphy requires a nonnegative order, while cusp vanishing requires a strictly positive order. The exact <divisor> is consequently
$$
D(f)=\sum_{P\text{ interior}}\left\lfloor\frac{r_P}{e_P}\right\rfloor P
+\sum_{c\text{ cusp}}(\lceil v_c\rceil-1)c.
$$
Only finitely many coefficients are nonzero. The order inequalities are equivalent to $\operatorname{div}(\varphi)+D(f)\ge0$. Division by $f$ proves the reverse inclusion, so the <Riemann-Roch space> gives
$$
\boxed{S_k(\Gamma_1(N),\chi)=fL(D(f)).}
$$
This 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 $f_t=fg^t$, 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 $d=[SL_2(\mathbb Z):\Gamma_0(N)]$. Thus
$$
\deg D(f_{t+6})-\deg D(f_t)=d,\qquad
\deg D(f_t)=dt/6+B(t\bmod6).
$$
The degrees tend to infinity. For sufficiently large $t$, the <Riemann-Roch theorem> therefore gives
$$
\boxed{\dim S_{k+2t}(\Gamma_1(N),\chi)=\frac{dt}{6}+A(t\bmod6)>0,\qquad A=B+1-g(X).}
$$
The 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.
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