What are the eigenvalues of the 8 point DFT matrix?
The eigenvalues belong to the same set of quartic roots of unity verifying $\lambda^4=1$, whatever the order of the DFT. 

For more details on their multiplicity, you can read: 
Eigenvectors and Functions of the Discrete Fourier Transform, 1982, Dickinson and Steiglitz (online).