Mathematics Complex Numbers

Let \(z = 1 + ai\) be a complex number, \(a > 0\), such that \(z^{3}\) is a real number. Then the sum \(1 + z + z^{2} + \ldots + z^{11}\) is equal to:

1
\(1365 \sqrt {3} i\)
2
\(-1365 \sqrt {3} i\)
3
\(-1250 \sqrt {3} i\)
4
\(1250 \sqrt {3} i\)

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