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K.<a> = CyclotomicField(220)
omega = a^20
zeta = a^44
zeta10 = -zeta^3
alpha = [sum([zeta10^(i*j)*omega^(2^i) for i in range(10)]) for j in range(10)]
powtab = [NN(10/gcd(i,10)) for i in range(10)]
atab = [alpha[i]^powtab[i] for i in range(10)]
atab_on_zeta_basis = [(QQ^4)((zeta.coordinates_in_terms_of_powers())(x)) for x in atab]
sqrt5 = 1+2*(zeta+zeta^-1)
sqrtm10m2sqrt5 = 2*(zeta-zeta^-1)
sqrtm10p2sqrt5 = 2*(zeta^2-zeta^-2)
nice_basis = [1, sqrt5, sqrtm10m2sqrt5, sqrtm10p2sqrt5]
m = Matrix(QQ, 4, 4, [(QQ^4)((zeta.coordinates_in_terms_of_powers())(x)) for x in nice_basis])
atab_on_nice_basis = [v * m.inverse() for v in atab_on_zeta_basis]
zetab = [ZZ(floor(arg(CC(N(alpha[i])/N(atab[i]^(1/powtab[i]))))/arg(zeta10)+0.5)) for i in range(10)]
btab = [zeta10^zetab[i] for i in range(10)]
btab_on_zeta_basis = [(QQ^4)((zeta.coordinates_in_terms_of_powers())(x)) for x in btab]
btab_on_nice_basis = [v * m.inverse() for v in btab_on_zeta_basis] 
symbolic_basis = [1, sqrt(5), sqrt(-10-2*sqrt(5)), sqrt(-10+2*sqrt(5))]
symbolic_omega = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)]))^(1/powtab[i]) for i in range(10)])/10
symbolic_cos = sum([sum([btab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)])*(sum([atab_on_nice_basis[i][j]*symbolic_basis[j] for j in range(4)]))^(1/powtab[i]) for i in range(0,10,2)])/10