Pre-calc complex roots $7$-th root of unity help

Question

Let $\omega$ be a complex number such that $\omega^7 = 1$ and $\omega \neq 1$. Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. Then $\alpha$ and $\beta$ are roots of the quadratic [$x^2 + px + q = 0$]for some integers $p$ and $q$. Find the ordered pair $(p,q)$. I have already found that p = 1, correct me if I'm wrong. How do I find q?

Answer 1

Hint: $\,\alpha\beta=\omega^4(1+\omega+\omega^3)(1+\omega^2+\omega^3)=\omega^4(\omega^6 + \omega^5 + \omega^4 + 3 \omega^3 + \omega^2 + \omega + 1)\,$, then use the first step.