Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc}
0 & 1 & 0 & \ldots & 0 \\
0 & 0 & 1 & \cdots & 0 \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
\vdots & & \ddots & 0 & 1 \\
1 & 0 & \ldots & 0 & 0
\end{array} \right).$$
Deduce the value of the determinant
$$\left| \begin{array}{ccccc}
a_{0} & a_{1} & \ldots & a_{n-2} & a_{n-1} \\
a_{n-1} & a_{0} & \ddots & & a_{n-2} \\
\vdots & \ddots & \ddots & \ddots & \vdots \\
a_{2} & & \ddots & a_{0} & a_{1} \\
a_{1} & a_{2} & \cdots & a_{n-1} & a_{0}
\end{array} \right|$$
where $a_{0}, \ldots, a_{n-1}$ are arbitrary complex numbers.