Eigenvalues and eigenvectors of certain diagonal constant matrices

Question

Suppose I have an infinite complex diagonal constant (Toeplitz) matrix, which is also Hermitian. This is given by finite number of complex parameters $z_1, z_2, \cdots, z_k$. If, $z_1$ is the parameter of principal diagonal, then it is real; and if $z_s$ is the parameter for $j$-th diagonal then $\bar{z}_s$ is the parameters of $-j$-th diagonal. All this is to make the matrix Hermitian.

Question: What are eigenvalues and eigenvectors of any $n \times n$ principal block matrix. This is a Toeplitz matrix. I expect (like circulant matrices) there is a nice periodicity in eigenvalues and eigenvectors in terms of the dimension $n$, but I can not explicitly write it.

Notes:

1. I know the asymptotic behaviour of eigenvalues (Kac-Murdock-Szego theorem). But here, I am interested in each eigenvalue and eigenvector (and like to write a spectral decomposition of this matrix).

2. To make life simpler, I have assumed that there exists only a finite number of nonzero parameters.

Advanced thanks for any help, suggestions, and etc.

As a test case, I may like to know these cases, when the matrix contains only one or two parameter(s).

1. When $j$ the diagonal (as well as $-j$th diagonal) is $1$.

2. When $j$ the diagonal is $\imath =\sqrt{-1}$., and hence $-j$ th is $-\imath$.

3. when $j_1$-th diagonal is $1$, $j_2$-th is $\imath$. ($-j_1$ and $-j_2$-th diagonals are parametrized by $1$ ans $-\imath$ respectively.)