  We thoroughly analyze the number of independent zero modes and their zero
points on the toroidal orbifold $T^2/\mathbb{Z}_N$ ($N = 2, 3, 4, 6$) with
magnetic flux background, inspired by the Atiyah-Singer index theorem. We first
show a complete list for the number $n_{\eta}$ of orbifold zero modes belonging
to $\mathbb{Z}_{N}$ eigenvalue $\eta$. Since it turns out that $n_{\eta}$ quite
complicatedly depends on the flux quanta $M$, the Scherk-Schwarz twist phase
$(\alpha_1, \alpha_2)$, and the $\mathbb{Z}_{N}$ eigenvalue $\eta$, it seems
hard that $n_{\eta}$ can be universally explained in a simple formula. We,
however, succeed in finding a single zero-mode counting formula $n_{\eta} =
(M-V_{\eta})/N + 1$, where $V_{\eta}$ denotes the sum of winding numbers at the
fixed points on the orbifold $T^2/\mathbb{Z}_N$. The formula is shown to hold
for any pattern.
