I wanted to understand the definition of the conductor for Dirichlet characters and I read https://en.wikipedia.org/wiki/Dirichlet_character: "We can formalize this differently by defining characters $\chi_1$ mod $N_1$ and $\chi_2$ mod $N_2$ to be co-trained if for some modulus $N$ such that $N_1$ and $N_2$ both divide $N$ we have $\chi_1(n) = \chi_2(n)$ for all $n$ coprime to $N$: that is, there is some character $\chi^*$ induced by each of $\chi_1$ and $\chi_2$. This is an equivalence relation on characters. A character with the smallest modulus in an equivalence class is primitive and this smallest modulus is the conductor of the characters in the class."
I am just really confused what they mean by it. Could someone possibly explain me what it means in a way that I can understand?