On the consistency of the definition of the conductor for automorphic forms 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?
 A: 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.

*

*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 $1$.)

*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 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, as did Jacquet.
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 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.
A: Yes this is true, but I don't think it's completely obvious.  The following argument is taken from Roberts--Schmidt ``Local newforms for $GSp_4$'', which uses the same argument as the $GL_2$ case.
Jacquet--Langlands proves the local functional equation
$$
Z(1-s, \pi(w_0) W) = \gamma(s, W) Z(s, W)
$$
where $W$ is in the Whittaker model, $Z(s, W)$ is the usual zeta integral, and $w_0 = \begin{bmatrix} & -1 \\ 1 \end{bmatrix}$ is the long Weyl element.  For any positive integer $n$, define $w_n = \begin{bmatrix} & -1 \\ \varpi^n \end{bmatrix}$; we can write the above as
$$
Z(1-s, \pi(w_n) W) = q^{n(s-1/2)} \gamma(s, W) Z(s, W).
$$
Now take $\pi$ of conductor $\mathfrak{p}^n$ (in the depth sense), and $W$ a generator for the space of $K(\mathfrak{p}^n)$-fixed vectors.  It is known (presumably implicit in J--L) that one can choose $W$ s.t. $Z(s, W) = L(s, \pi)$.  Also, one easily sees that $\pi(w_n)$ normalises $K(\mathfrak{p}^n)$ and hence acts on its fixed vectors, and so the LHS is also the corresponding local $L$-function, times the eigenvalue of $w_n$.  Now dividing by the local $L$-functions to get the definition of the $\epsilon$-factor, one now sees that the conductor in the $\epsilon$-factor is what it should be.
