It's a mistake everyone makes every now and then - seeing pi/n and thinking it's 2pi/n.
What? Like who, 6th graders?
No, like anyone who isn't watching out for traps caused by bad notation. It's much easier to copy down numbers than it is to alter them appropriately. If you see "e^(pi i/3)", what stands out is the 3 in the denominator. Except oops, pi actually only means half a circle, so this is a sixth root of unity, not a third one. Part of why I like to just write zeta_n instead of e^(2pi i/n). Sure, this can be avoided with a bit of thought, but thought shouldn't be required here; notation that forces you to think about something so trivial, is not good notation.
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