Say I have a Dirichlet character $\chi$ mod $N$ and I know that $\chi$ is induced by a Dirichlet character $\chi'$ mod $M$ with $M|N$. I want to then say that the conductor of $\chi$ divides $M$, which intuitively seems to be the case. Does anyone know a simple proof (or disproof) of this?
For instance, it is not clear to me why $\chi$ can't also induced by a character mod $Q$ with $\gcd(M,Q)=1$.
Best Answer
A Dirichlet character $\chi$ modulo $N$ is induced by a character $\chi'$ modulo $M$ exactly if it is constant on the residue classes modulo $M$ (see Lemma $17.20$ in these lecture notes). If it is constant on the residue classes modulo $M$ and $Q$, then it is also constant on the residue classes modulo $\gcd(M,Q)$, and thus it is also induced by a character modulo $\gcd(M,Q)$. It follows that the conductor of $\chi$ divides the modulus of every character that induces $\chi$, as otherwise their greatest common divisor would be less than the conductor and yet would be the modulus of a character that induces $\chi$.
Regarding your second paragraph, it follows from the above that $\gcd(M,Q)=1$ for $M$ and $Q$ moduli of characters that induce $\chi$ only if $\chi$ has conductor $1$, i.e. is a principal character.