Meromorphic n-differential ($\omega \in {\Gamma }_{\mathrm{mer}}(K_X^{\otimes n})$)

On a Riemann surface, a meromorphic n-differential is a meromorphic section of the $n$th power of the canonical bundle. In a coordinate it is $F(z)(dz{)}^n$, with coefficients transforming by the $n$th power of the derivative. At a ramification coordinate $w=t^e$, its pullback has order $e{\mathrm{ord}}_w\omega +n(e-1)$. At a modular cusp, the exponential coordinate gives $dz=(h/(2\pi i))dq/q$, so a weight-$2n$ form of cusp order $v$ produces a differential of order $v-n$.