We know, a character $\chi$ mod $q$ is induced by a primitive character $\chi^{*}$ mod $q^{*}$. I have the following questions.
If $\chi$ is non principal, can $\chi^{*}$ be principal?
Let $\chi$ be a character mod $q$. Can we say $\chi$ is principal iff conductor of $\chi=1?$
If $\chi$ be the principal character mod $q$. What is the primitive character mod $q^{*}$ which induces it?
Any help or hint would be appreciated.