reducing the modulus of a Dirichlet character

Question

Let $\chi$ be a Dirichlet character modulo $N$. Let $M$ be a positive divisor of $N$ such that $$\text{radical}(N)=\text{radical}(M).$$ Is $\chi$ be a character modulo $M$?

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Answer 1

The answer is "no". Consider the following example. There is a Dirichlet character modulo 4 defined by $$ \chi_4(n) = \begin{cases} 1 &\text{if $n = 1 \bmod 4$} \\ -1 & \text{if $n = -1 \bmod 4$}\\ 0 & \text{if $n$ is even}.\end{cases}$$ Then $\chi_4$ is a Dirichlet character modulo $N = 4$; $M = 2$ is a positive divisor of $N$ which has the same radical as $N$; but $\chi_4$ is not a Dirichlet character modulo 2 (since evidently $\chi_4(1) \ne \chi_4(3)$).