Imaginary summation?

Algebra Level 5

If nn is an integer and ω\omega is an imaginary cube root of unity, let AA denote the value of

k=0n(nk)ωk{\displaystyle\sum_{k=0}^{n} \binom{n}{k} \omega^{k}}

Then the number of possible values of eAe^A is?

Note:This question is a part of set KVPY 2014 SB

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