<s>[INST] Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots 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 $$\sum\limits_{1\leq i<j\leq7}x_ix_j=-18.$$
 [/INST] I also used PARI/GP with the following program:
z1 = Mod(z, (z^43-1)/(z-1));
e(n) = lift(Mod(3,43)^n);
c(n) = z1^n + z1^-n;
r(n) = c(1*n) + c(6*n) + c(7*n);
p = prod(n=1,7, x - r(e(n)));
lift(p)

with the resulting output
z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49

A simpler program with complex numbers is
z1=exp(2*Pi*I/43); z2=z1^6; z3=z1^7;
bestappr(prod(n=1,7, m=lift(Mod(3,43)^n);\
x - 2*real(z1^m + z2^m + z3^m)), 10^9)

</s>