I came across through simulation that multiplying and adding certain modulo sequences yield equal results. Consider the following two sequences \begin{align} g_0[k] &= \sum_{n=0}^{N-1} \left< a n \right>_N \left< k n \right>_N \\ g_1[k] &= \sum_{n=0}^{N-1} \left< b n \right>_N \left< k n \right>_N \end{align} where $a, b \in \mathbb{Z}$, $\left_N = a \pmod N$and $\text{GCD}(a,N) = \text{GCD}(b,N) = 1$. It appears that there exist $k_0, k_1 \in \mathbb{Z}$ such that \begin{align} g_0[k_1 k] &= g_1[k] \\ g_1[k_0 k] &= g_0[k] \end{align}
Outside of running several computer simulations that seem to indicate that such integers might always exist for arbitrary $a$ and $b$ with $\text{GCD}(a,N) = \text{GCD}(b,N)$, I cannot find out how to 1) determine under what conditions this holds and 2) figure out the value of $k_0$ and $k_1$ directly.