Suppose $\beta = e^{2\pi i/7} = \cos(2\pi/7) + i \sin(2\pi/7)$. What is $w^2 = (\beta + \beta^2 + \beta^4)^2$?

Question

Suppose $\beta = e^{2\pi i/7} = \cos(2\pi/7) + i \sin(2\pi/7)$. What is $w^2 = (\beta + \beta^2 + \beta^4)^2$?

We have $\sum_{k=0}^6\beta^k = 0.$

So far I have figured out that $w^2 = -\beta^4 - \beta^2 - \beta - 2$. We must find a quadratic equation with integer coefficients that $w$ satisfies. How do I complete this?