Exponents in 2015

Algebra Level 5

Let \(a_1,\) \(a_2,\) \(a_3,\) \(a_4\) be complex numbers such that

\[ \begin{align*} a_1 + a_2 + a_3 + a_4 &= 2 \\ a_1^2 + a_2^2 + a_3^2 + a_4^2 &= 0 \\ a_1^3 + a_2^3 + a_3^3 + a_4^3 &= 1 \\ a_1^4 + a_2^4 + a_3^4 + a_4^4 &= 5. \end{align*} \]

If \(a_1^6 + a_2^6 + a_3^6 + a_4^6 = \frac{m}{n}\) for some relatively prime positive integers \(m\) and \(n,\) find \(m + n.\)

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