I know that the eigenvectors and eigenvalues of any circulant matrix have a nice general form (See the wikipedia page). The wikipedia page also generalizes the eigenvalues (but not eigenvectors) for real symmetric circulant matrices as well, namely as: $$\lambda_j = c_0 +2c_1\Re w_j + 2c_2\Re w_j^2+...+2c_{n/2-1}\Re w_j^{n/2-1}+c_{n/2} w_j^{n/2}$$ My question is:
Can we also generalize the real eigenvectors for real symmetric circulant matrices given this information, and if so how? I have been trying to work this out and I can't quite get to a sensible answer.