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I also used PARI/GP with the following program:

c(n) = 2*cos(2*n*Pi/43);
x(n) = c(1*n) + c(6*n) + c(7*n);
print(bestappr(prod(n=1,7, z-x(n+3*(n>5))),10^9))

with the resulting output

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