Consider characters of $PGL(2)$. The conductor of a local component $\pi\_p$ of an automorphic representation is defined as the smallest index such that there is nontrivial fixed vector by the congruence subgroup
$$K\_0(p^r) = \left\{
M \in GL(2, \mathcal{O}\_{F\_p}) \ : \ M \equiv \left(
\begin{array}{cc}
\star & \star \\
& \star
\end{array}
\right)
\mod p^r
\right\}$$


In the case of a character of $GL(2)$ of the form $\chi\_0 \circ \det$, where $\chi\_0$ is a character on the base field $F\_p$, is there a natural relation between this notion and the conductor of $\chi\_0$, defined as the smallest index $r$ such that $\chi\_0$ is trivial on $1 + p^r$?



Conductor of characters
The lower-right corner is supposed to be a $1$, not a star (unless the central character is trivial, which it isn't in your case unless $\chi\_0$ is quadratic). 


Regardless, as LSpice notes, these representations do not have a conductor in that sense, unless the conductor of $\chi\_0$ is $1$, because they are not invariant under any of these subgroups.


This is not so surprising as the theorem only says that generic representations have a conductor.


Maybe a better thing to try is to tensor your representation with the Steinberg representation, obtaining an infinitesimally equivalent representation which is generic, whose conductor is equal to $\min(c(\chi\_0)^2, p)$.


