  An elliptic orbifold is the quotient of an elliptic curve by a finite group.
In 2001, Eskin and Okounkov proved that generating functions for the number of
branched covers of an elliptic curve with specified ramification are
quasimodular forms for $SL_2(\mathbb{Z}).$ In 2006, they generalized this
theorem to the enumeration of branched covers of the quotient of an elliptic
curve by $\pm 1$, proving quasi-modularity for $\Gamma_1(2)$. In 2017, the
author generalized their work to the quotient of an elliptic curve by $\langle
\zeta_N\rangle$ for $N=3, 4, 6$, proving quasimodularity for $\Gamma_1(N)$.
  In these works, both Eskin-Okounkov and the author had to assume that there
was at least one orbifold point of order $N$ over which there was no
ramification. Here we remove that assumption, with the caveat that the
generating functions are only quasimodular for $\Gamma(N)$. We deduce the
following corollary: Let $h_6(\vec{\kappa},q)$ be the generating function whose
$q^n$ coefficient is the number of surface triangulations with $2n$ triangles,
such that the set of non-zero curvatures is $\kappa_i$. Here the curvature of a
vertex is six minus its valence. Then under the substitution $q=e^{2\pi i
\tau}$, the function $h_6(\vec{\kappa},q)$ is a quasimodular form for
$\Gamma_1(6)$ with weight bounded in terms of $\vec{\kappa}$. This statement in
turn implies that the Masur-Veech volume of any stratum of sextic differentials
is polynomial in $\pi$.
