Eigenvectors of modular multiplication

Let $N\ge 2$ and $a$ be coprime to $N$. Then $U_a = \sum_{x=0}^{N - 1} \ket{ax\bmod N}\!\bra{x}$ is an $N$-dimensional unitary that, for every $k\in\{0,\ldots,r-1\}$, satisfies $$U_a \ket{\psi_k}=e^{2\pi i k/r}\ket{\psi_k}.$$ Here $r$ is the least positive integer such that $a^r\equiv 1\pmod N$ and $\ket{\psi_k} = \sum_{j=0}^{r-1}e^{-2\pi i k j/r}\ket{a^j\bmod N} / \sqrt{r}$. Moreover, $\ket{1}=\sum_{k=0}^{r-1}\ket{\psi_k} / \sqrt{r}$.