# Is there anyway to find the frequency of DFT eigenvectors (basis) from its eigenvalues?

I've read this document which talks about the DFT. It describes that DFT bases are the eigenvectors of a circulant matrix. I know that every basis has a frequency in it, but I don't know what is it? And it there any relation between this frequency and eigenvalues? For example, one of the bases of DFT with $$n$$ points is as follow:

$$\begin{equation*} F^{(k)} = \begin{pmatrix} w_n^{0k} \\ w_n^{1k} \\ \vdots \\ w_n^{(n-1)k} \end{pmatrix} \end{equation*}$$ $$\begin{equation*} w_n^{jk} = e^{\frac{2\pi i}{n}jk} \end{equation*}$$ and the eigenvalue related to this eigenvector($$F^{(k)}$$) is $$\lambda_k = \sum_{j=0}^{n-1}c_jw_n^{jk}$$, where $$c_i$$ is the $$i$$-th element of the first row of circulant matrix. What is the frequency of $$F^{(k)}$$? Is there any relation between this frequency and $$\lambda_k$$? Can we talk about the frequency based on the $$\lambda_k$$? Since $$\lambda_k$$ is related to the elements of this circulant matrix but the eigenvectors are not related, so can we define special circulant matrix that we can find the frequency from its eigenvalues?