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

A simpler program with no defined functions is

    z1=exp(2*Pi*I/43);z2=r1^6;z3=r1^7;bestappr(prod(n=1,7,\
    m=lift(Mod(3,43)^n);x-2*real(z1^m+z2^m+z3^m)),10^9)