We thoroughly analyze the number of independent zero modes and their zero points on the toroidal orbifold T 2 =Z N ðN ¼ 2; 3; 4; 6Þ with magnetic flux background, inspired by the Atiyah-Singer index theorem. We first show a complete list for the number n η of orbifold zero modes belonging to Z N eigenvalue η. Since it turns out that n η quite complicatedly depends on the flux quanta M, the Scherk-Schwarz twist phase ðα 1 ; α 2 Þ, and the Z N eigenvalue η, it seems hard that n η can be universally explained in a simple formula. We, however, succeed in finding a single zero-mode counting formula n η ¼ ðM − V η Þ=N þ 1, where V η denotes the sum of winding numbers at the fixed points on the orbifold T 2 =Z N. The formula is shown to hold for any pattern.

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