I am looking for a seventh degree polynomial with integer coefficients, which has the following roots. $$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$ $$x_2=2\left(\cos\frac{6\pi}{43}+\cos\frac{36\pi}{43}+\cos\frac{42\pi}{43}\right),$$ $$x_3=2\left(\cos\frac{18\pi}{43}+\cos\frac{22\pi}{43}+\cos\frac{40\pi}{43}\right)$$ $$x_4=2\left(\cos\frac{20\pi}{43}+\cos\frac{32\pi}{43}+\cos\frac{34\pi}{43}\right),$$ $$x_5=2\left(\cos\frac{10\pi}{43}+\cos\frac{16\pi}{43}+\cos\frac{26\pi}{43}\right),$$ $$x_6=2\left(\cos\frac{8\pi}{43}+\cos\frac{30\pi}{43}+\cos\frac{38\pi}{43}\right)$$ and $$x_7=2\left(\cos\frac{4\pi}{43}+\cos\frac{24\pi}{43}+\cos\frac{28\pi}{43}\right).$$ I see only that $\sum\limits_{k=1}^7x_k=-1$, but the computations for $\sum\limits_{1\leq i<j\leq7}x_ix_j$ and the similar are very complicated by hand and I have no any software besides WA, which does not help.

Thank you for your help!

Update.

I got $$\sum\limits_{1\leq i<j\leq7}x_ix_j=-18.$$