Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}\_2(F)$, where $F$ is local non-archimedean. The local conductor associated to $\pi$ can be defined in two usual manners:


**By its associated L-function**


Godement and Jacquet associate to it the automorphic L-function $L(s, \pi)$. This L-function satisfies a functional equation of the form
$$L(1-s, \tilde{\pi}) = \varepsilon(s, \pi) L(s, \pi)$$


This $\varepsilon$-factor is essentially of the form $c^{s - \frac{1}{2}}$, and we define the appearing real number $c(\pi)=c$ to be the conductor of $\pi$.


**By some "depth" property**


The other way to define the conductor is to follow the work of Casselman for $\mathrm{GL}\_2$, and more generaly JPSS for $\mathrm{GL}\_n$. We consider the decreasing sequence of compact open congruence subgroups, for $r \geqslant 0$:
$$
K\_{0, \mathfrak{p}}\left(\mathfrak{p}^r\right) = 
\left\{
g \in \mathrm{GL}\_2\left(\mathcal{O}\_{\mathfrak{p}}\right) \ : \ 
g \equiv
\left(
\begin{array}{cc}
\star & \star \\
0 & \star
\end{array}
\right)
\mod \mathfrak{p}^r
\right\} \subseteq \mathrm{GL}\_2(F).
$$


The conductor of an irreducible admissible infinite-dimensional representation $\pi\_\mathfrak{p}$ of $\tilde{G\_\mathfrak{p}}$ with trivial central character (to ease notations) is then defined by:
$$
c(\pi) = N\mathfrak{p}^{f(\pi\_{\mathfrak{p}})} ,
$$


where:
$$
 f\left(\pi\_{\mathfrak{p}}\right) = \min \left\{r \geqslant 0 \ : \ \pi\_{\mathfrak{p}}^{K\_{0, \mathfrak{p}}\left(\mathfrak{p}^r\right)} \neq 0\right\}
$$


**Are they consistent?**


Here is my question: is it obvious that those two definitions are the same?



On the consistency of the definition of the conductor for automorphic forms
These definitions are consistent, though it's not immediate.


The conductor quantifies the extent to which $\pi$ is ramified. As an aside, I prefer to write $c(\pi)$ for the conductor exponent of $\pi$, which is a nonnegative integer, so that $\mathfrak{p}^{c(\pi)}$ is the conductor of $\pi$, and $q^{c(\pi)}$ is the absolute conductor of $\pi$, where $q = N(\mathfrak{p}) = \# \mathcal{O}\_F / \mathfrak{p}$. This isn't standard terminology though, as these are all unfortunately called the same thing by different people.


The conductor exponent is tied to a distinguished vector in $\pi$, called the newform of $\pi$. (It is also called the Whittaker newform, the essential Whittaker function, the newvector, or the essential vector; the terminology here hasn't reached a consensus, but newform is best in my opinion because for $\mathrm{GL}\_2$, it is the local component of a classical newform.) There are two definitions of the newform of $\pi$, which are equivalent. I'll state them for $\mathrm{GL}\_n$ instead of $\mathrm{GL}\_2$. Throughout, $n \geq 2$ and $F$ is a nonarchimedean local field.


1. Recall that for a spherical (that is, unramified principal series) representation $\pi'$ of $\mathrm{GL}\_{n-1}(F)$, there exists a spherical vector $W'^{\circ}$ in the Whittaker model $\mathcal{W}(\pi',\overline{\psi})$ of $\pi'$, so that $W'^{\circ}(1\_{n-1}) = 1$ and $W'^{\circ}(gk) = W(g)$ for all $g \in \mathrm{GL}\_{n-1}(F)$ and $k \in K\_{n-1} = \mathrm{GL}\_{n-1}(\mathcal{O}\_F)$; that such a vector exists and is unique is classical. Now let $\pi$ be a (possibly ramified) generic irreducible admissible representation of $\mathrm{GL}\_n(F)$. Then there exists a unique Whittaker function $W^{\circ}$ in the Whittaker model $\mathcal{W}(\pi,\psi)$ of $\pi$ satisfying
$$W^{\circ} \left(g \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}\right) = W^{\circ}(g), \qquad W^{\circ}(1\_n) = 1$$
for every $g \in \mathrm{GL}\_n(F)$ and $k \in K\_{n-1}$ such that for every spherical representation $\pi'$ of $\mathrm{GL}\_{n-1}(F)$ with associated spherical vector $W'^{\circ}$, the Eulerian integral
$$\Psi(s,W^{\circ},W'^{\circ}) = \int\limits\_{N\_{n-1}(F) \backslash \mathrm{GL}\_{n-1}(F)} W^{\circ} \begin{pmatrix} g & 0 \\ 0 & 1 \end{pmatrix} W'^{\circ}(g) |\det g|^{s - 1/2} \, dg$$
is equal to the Rankin-Selberg $L$-function $L(s, \pi \times \pi')$. $W^{\circ}$ is called the newform of $\pi$. (In general, $\Psi(s,W,W')/L(s,\pi \times \pi')$ is a polynomial in $q^{-s}$; the point is that there exists a distinguished choice of $W \in \mathcal{W}(\pi,\psi)$, $W' \in \mathcal{W}(\pi',\overline{\psi})$ for which this polynomial is equal to $1$.)
2. For a nonnegative integer $m$, let $K\_1(\mathfrak{p}^m)$ denote the congruence subgroup of $K\_n = \mathrm{GL}\_n(\mathcal{O}\_F)$ given by
$$\left\{\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in K\_n : c \in \mathrm{Mat}\_{1 \times (n-1)}(\mathfrak{p}^m), \\ d - 1 \in \mathfrak{p}^m \right\}.$$
(Note that necessarily $a \in K\_{n-1}$ and $b \in \mathrm{Mat}\_{n-1 \times 1}(\mathcal{O}\_F)$.) Let $\pi$ be a (possibly ramified) generic irreducible admissible representation of $\mathrm{GL}\_n(F)$. Then there exists a minimal $m$ for which the vector subspace
$$\pi^{K\_1(\mathfrak{p}^m)} = \left\{v \in \pi : \pi(k) \cdot v = v \quad \text{for all $k \in K\_1(\mathfrak{p}^m)$}\right\}$$
of $\pi$ is not equal to $\{0\}$. We denote by $c(\pi)$ this minimal $m$. Then $\pi^{K\_1(\mathfrak{p}^{c(\pi)})}$ is one dimensional, so that there exists a unique Whittaker function $W^{\circ}$ in the Whittaker model $\mathcal{W}(\pi,\psi)$ of $\pi$ satisfying
$$W^{\circ} (gk) = W^{\circ}(g), \qquad W^{\circ}(1\_n) = 1$$
for every $g \in \mathrm{GL}\_n(F)$ and $k \in K\_1(\mathfrak{p}^{c(\pi)})$.


The original proof of the existence and uniqueness of the newform $W^{\circ}$ of $\pi$ is via (1), in the paper [Conducteur des répresentations du group linéaire](https://eudml.org/doc/163506) by Jacquet, Piatetski-Shapiro, and Shalika. However, it was noticed by Matringe several years ago that the proof was in fact incomplete. He gave a [correct proof](http://www.math.uiuc.edu/documenta/vol-18/37.pdf), as did [Jacquet](http://msp.org/pjm/2012/260-2/pjm-v260-n2-p09-p.pdf).


In the last section of Jacquet, Piatetski-Shapiro, and Shalika's paper, they show that (1) implies (2) via the functional equation for $L(s,\pi \times \pi')$. They make use of the fact that the epsilon factor $\epsilon(s,\pi \times \pi',\psi)$ is equal to $\epsilon(1/2, \pi \times \pi',\psi) q^{-(n - 1) c(\pi) (s - 1/2)}$ for some nonnegative integer $c(\pi)$, and it is precisely this nonnegative integer $c(\pi)$ that satisfies $\pi^{K\_1(\mathfrak{p}^{c(\pi)})} \ni W^{\circ}$ and $\pi^{K\_1(\mathfrak{p}^m)} = \{0\}$ for all $m < c(\pi)$.


I don't know if the fact that (2) implies (1) has appeared anywhere in print, but I do know how to prove it. There is a [paper](https://projecteuclid.org/euclid.jmsj/1390600834) by Miyauchi that shows that the (Whittaker) newform $W^{\circ}$ given by (2) is such that
$$\Psi(s,W^{\circ}) = \int\_{F^{\times}} W^{\circ} \begin{pmatrix} x & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & & \cdots & 1 \end{pmatrix} |x|^{s - \frac{n - 1}{2}} \, d^{\times} x$$
is equal to the $L$-function $L(s,\pi)$, and the same method of proof (using Hecke operators) shows that $\Psi(s,W^{\circ},W'^{\circ}) = L(s, \pi \times \pi')$ for all spherical representations $\pi'$ of $\mathrm{GL}\_{n-1}(F)$. Now you can work backwards to show that $\epsilon(s,\pi \times \pi',\psi)$ is equal to $\epsilon(1/2, \pi \times \pi',\psi) q^{-(n - 1) c(\pi) (s - 1/2)}$, as in Martin Dickson's answer.


