The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix
$$A_k(\chi)=\left[G_N(\chi^{ki+ki})\right]_{0\le i,j\le \varphi(N)/k-1},$$
where $N=p^m$ is a prime power, $\varphi(\cdot)$ is the Euler totient function, $k$ is a divisor of $\varphi(N)$, $\chi$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and
$$G_N(\chi^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}\chi^{ki+kj}(x)e^{2\pi ix/N}$$
is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.