linear algebra - How to find eigenvalues of the matrix

This is a question from our end-semester exam:

How to find the eigenvalues of the given matrix:

M=\begin{bmatrix}
5,1,1,1,1,1\\
1,5,1,1,1,1\\
1,1,5,1,1,1\\

1,1,1,5,1,1\\
1,1,1,1,4,0\\
1,1,1,1,0,4\\
\end{bmatrix}

I know that $4$ is an eigenvalue of $M$ with multiplicity atleast $3$ since $M-4I$ has $4$ identical rows.

Is there any way to find all eigenvalues of this matrix? I could find only $3$ out of $6$.