I havn't checked your working so far but to calculate
note that
where k=0,1
z^2-2wz-1=0, where z is a complex number and w the third root of unity. I've tried substituting w=e^(i2pi/3)
z^2-2(e^(i2pi/3))z-1=0
z^2+2*[0.5-(i*root(3)/2)]z-1=0
Using the quadratic equation gives me a square root of i, which I've never encountered before, any help is greatly appreciated, thanks in advance .