Let $\alpha = \omega + \omega^2 + \omega^4$ and $\beta = \omega^3 + \omega^5 + \omega^6$. 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 got that $p=-1$ but does not know how to go on to finding $q$.  All help is appreciated!

method is due to Gauss. This book 1875.  $x^2 + x + 2.$
gp-pari to check; note a^8 = a
? x = a + a^2 + a^4
%1 = a^4 + a^2 + a
? q = x^2 + x + 2
%2 = a^8 + 2*a^6 + 2*a^5 + 2*a^4 + 2*a^3 + 2*a^2 + a + 2
? 

If we switched to one of the real numbers $$ t = \omega + \omega^6, $$ we would have a root of
$$ t^3 + t^2 - 2 t - 1  $$

