Let $q$ be a natural number and $\chi$ be a Dirichlet character mod $q$. We say that $q_1$ is a period of $\chi$ if $$n \equiv m \; (\mathrm{mod} \; q_1),(n,q)=(m,q)=1 \Rightarrow \chi(n)=\chi(m).$$ My questions are
- If $q_1$ is a period of $\chi$ , then $q_1|q$ ?
- If $q_1$ and $q_2$ are periods of $\chi$ , then is $d=\mathrm{gcd}(q_1,q_2)$ also period of $\chi$ ?
Every Dirichlet character $\chi$ has a minimal period $q$, sometimes called the conductor of $\chi$. All other periods of $\chi$ are multiples of $q$, and any nontrivial positive multiple of $q$ is a period.
For example, there is a unique Dirichlet character $\psi$ with conductor $2$, defined by $\psi(n) = 0$ if $n$ is even and $\psi(n) = 1$ if $n$ is odd. Note that this character is periodic mod $2$, but also mod $4$ or mod $8$ or any multiple of $2$. Thus the answer to your first question is no. As all periods are multiples of the conductor, the answer to your second question is yes.