Let $C \in \mathbb R^{n \times n}$ be a circulant matrix. Then its eigenvalues have the form (e.g. here [PDF], page 32) $$\lambda_k = \sum_{j = 0}^{n - 1} c_j \omega_k^j, \tag1$$ where $c = (c_0, c_1, \ldots, c_{n - 1})$ is the first row of $C$ and $$\omega_k = \exp\left(\frac{2\pi i}{n} k\right)$$ If $C$ is also stochastic, its eigenvalues are in the interval $[-1, 1]$, and one of them is $1$.
I'd like to obtain the second greatest eigenvalue, but unfortunately equation $(1)$ does not give them in sorted order. For example, for $$c = \frac{1}{3}(0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1) \in \mathbb R^{12}$$ the second-greatest eigenvalue is $0.6666$ and is found at $k = 2$. On the other hand, for $$c = \frac{1}{2}(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1) \in \mathbb R^{12}$$ the second-greatest eigenvalue is $0.8660$ and is found at $k = 1$.
It's not clear how I can sort these sum of exponentials.