Force Eigenvalues of a Matrix to be Rational

Question

I am working on a problem for which I need to force the eigenvalues of matrices of the following format to be rational: $$\left[\begin{array}{ccccc} q_1 & 1 & 1 &1&0\\ 1 & q_2 & 0 & 0 &1\\ 1 & 0 & q_3 & 0 &1 \\ 1 & 0 & 0& q_4 &1\\ 0&1&1 & 1 &q_5\\ \end{array}\right]$$ The $q_i$'s need to be symmetric about the middle $q_{floor(n/2)+1}$, and the maximum $q$ must be the middle $q$. $5x5$ is the smallest matrix I can use, but any odd x odd square matrix bigger than 5x5 will work as well. I've already built a brute force algorithm to generate random symmetric $q$'s and check if the eigenvalues of the resulting matrix are rational and have gotten some results, but it is exceptionally slow for any matrix larger than 10x10 and only uses integer value $q$'s. Does anybody know a more elegant way to tackle this?