Is there a simple correction to this definition of the conductor of a Dirichlet character?

Question

The website LMFDB Dirichlet Characters defines the conductor of a Dirichlet character as follows:

The conductor of a Dirichlet character $\chi$ modulo $q$ is the least positive integer $q_1$ dividing $q$ for which $\chi(n+q_1)=\chi(n)$ for all $n$ coprime to $q$.

This definition does not seem correct since for example $\chi_{5,1}=\{1, 1, 1, 1, 0\}$ has conductor $1$ and yet $\chi _{5,1}(4+1)=0\neq 1=\chi _{5,1}(4)$.

Question: Is there a simple way to correct the definition above?

Answer 1

The definition in Apostol is that the conductor of $\chi$ modulo $q$ is the smallest induced modulus $d$. We have $d\mid q$. And $d>0$ is an induced modulus for $\chi$ if and only if $$ \chi(a)=\chi(b) $$ whenever $(a,q)=(b,q)=1$ and $a\equiv b \bmod d$. And $d=1$ is an induced modulus for $\chi$ if and only if $\chi=\chi_1$, the principal character. This applies to your example $\chi_{5,1}$.

Answer 2

Elsewhere I've seen the conductor of a Dirichlet character $\chi$ defined as the least period of $\chi$ as a function on $\mathbf Z$, which is also incorrect: the quadratic character mod $6$ has values (starting at 1) as $1$, $0$, $0$, $0$, $-1$, $0$, $1$, $0$, $0$, $0$, $-1$, $0$, $\ldots$, which has period $6$, but the conductor of $\chi$ is $3$, not $6$.

A correct (elementary) definition of the conductor of a character $\chi \bmod m$ is the least positive integer $d$ such that (i) $d \mid m$ and (ii) there is a Dirichlet character $\psi \bmod d$ whose lift to modulus $m$ is $\chi$: the composite homomorphism $(\mathbf Z/m\mathbf Z)^\times \stackrel{{\rm red.}}{\rightarrow} (\mathbf Z/d\mathbf Z)^\times \stackrel{\psi}{\rightarrow} S^1$ is $\chi$, where the first map is the standard reduction from modulus $m$ to modulus $d$. Notice this definition involves only units in modular arithmetic, not arbitrary integers in modular arithmetic.

Given a Dirichlet character $\chi \bmod m$ and a factor $d$ of $m$, how can you check in a practical way whether or not $\chi$ is the lift of a Dirichlet character from modulus $d$? A necessary condition is that for all $a \in (\mathbf Z/m\mathbf Z)^\times$ such that $a \equiv 1 \bmod d$ we have $\chi(a \bmod m) = 1$. It turns out this necessary condition is also sufficient (that makes a nice exercise), so you just run through the units mod $m$ that reduce to $1 \bmod d$ and check if $\chi$ has value $1$ at all of them. If it does then there is a Dirichlet character mod $d$ that lifts to $\chi$, and if there is some $a \in (\mathbf Z/m\mathbf Z)^\times$ such that $a \equiv 1 \bmod d$ and $\chi(a \bmod m) \not= 1$ then $\chi$ does not have a lift from modulus $d$. The least $d$ for which $\chi$ is the lift of a Dirichlet character mod $d$ divides all other such $d$, so you could systematically find the least $d$ by first checking whether $\chi$ is the lift of a character modulo one of the maximal proper divisors of $m$ (divide it by one prime factor at a time), and each time you find a success, start over again with that new smaller modulus and character to see if it has a lift from one of its maximal proper divisors. As soon as you get stuck (no lift from any maximal proper divisor), you're done: the modulus you're at is the conductor.