Conductor of a Dirichlet character from its Conrey label (reviewed)

Let $$\chi_q(n,\cdot)=\prod_{p|q}\chi_{p^e}(n,\cdot)$$ be the unique factorization of the Dirichlet character $\chi_q(n,\cdot)$ into characters of prime power modulus $p^e$ under the Conrey labeling system. The conductor of $\chi_q(n,\cdot)$ is the product of the conductors of the $\chi_{p^e}(n,\cdot)$.

For odd prime $p$ and $f\leq e$, the isomorphism between $(\Z/p^f\Z)^\times$ and the kernel of $x\mapsto x^{p^{e-f}}$ in $(\Z/p^e\Z)^\times$ is compatible with the Conrey labeling system. This implies that $\chi_{p^e}(n,\cdot)$ is induced by a Dirichlet character of modulus $p^{f}$ whenever $p^{e-f}$ divides the order of $n$ modulo $p^e$.

Thus for odd primes $p$, the conductor of $\chi_{p^e}(n,\cdot)$ is

• $1$ if $n\equiv 1\bmod p^e$;
• $p^f$ if $n^{p-1}$ has order $p^{f-1}$ modulo $p^e$.

For $p=2$ the conductor of $\chi_{p^e}(n,\cdot)$ is

• 1 if $n\equiv 1\bmod 2^e$;
• $4$ if $e=2$ and $n\equiv 3\bmod 4$;
• $2^{f+2}$ if $e\geq3$ and $\varepsilon n$ has order $2^f$ modulo $2^e$, where $\varepsilon\in\{-1,1\}$ satisfies $n\equiv \epsilon\bmod 4$.