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\begin{document}
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\begin{center}
 {\bf\Large Tables for Representations of $\GSp(4)$}

 \vspace{2ex}
 {\sc Brooks Roberts, Ralf Schmidt}
 
 \vspace{2ex}
 {\tiny 1/2007}
\end{center}
\tableofcontents
\nl
\section{Explanations}
The following pages contain several tables with information on the
irreducible, admissible, non--supercuspidal representations of
$\GSp(4,F)$, where $F$ is a non-archimedean local field.
The basis for these tables is the paper
\begin{quote}
 \hspace{-5.1ex}[ST] {\sc Sally, P.; Tadi\'c, M.:} {\em Induced
  representations and classifications for $\GSp(2,F)$ and $\SSp(2,F)$}.
  Soci\'et\'e Math\'ematique de France, M\'emoire 52 (1993), 75--133,
\end{quote}
where a complete list of all such representations is given.
Moreover, [ST] classifies all the unitary, tempered and
square--integrable representations. Our notation is the same
as in [ST], except that we write $\GSp(2n)$ instead of $\GSp(n)$.
\nl
{\bf\ref{maintable}:} This table lists all the
irreducible, admissible, non-supercuspidal representations of
$\GSp(4,F)$. Representations in the same group I -- XI are
constituents of the same induced representation.
The ``tempered'' column gives the precise condition
for a representation to be tempered. The ``$L^2$'' column
indicates which of the tempered representations are
square--integrable. The ``g'' column indicates the
generic representations; in each case, these are subrepresentations
of the full induced representation given in the third column.
The ``$P$'', ``$B$'' and ``$Q$'' columns indicate those
representations occurring in global CAP representations.
Here $P$ stands for the Siegel parabolic subgroup, $B$ stands
for the minimal parabolic subgroup, and $Q$ stands for the
Klingen parabolic subgroup (whose unipotent radical is a
three-dimensional Heisenberg group). A $\bullet$ in the $P$--column means that the representation appears as a
local component in cusp forms that are CAP with respect to $P$
(these are the Saito--Kurokawa representations). A $\circ$
in the $P$--column means that the representation appears as a local
component in non-cuspidal liftings from $\PGL(2)\times\PGL(2)$. Similarly
for the $B$ and $Q$ columns (this will be explained in more detail elsewhere).
\nl
Representations in groups I -- VI are supported in the minimal
parabolic subgroup, i.e., they are constituents of representations
induced from a character of $B(F)$. Representations in groups
VII -- IX are supported in $Q$, and representations in groups
X and XI are supported in $P$; here $\pi$ stands for a
supercuspidal representation of $\GL(2,F)$, and $\omega_\pi$
is the central character of $\pi$. Two types of supercuspidal
representations are also listed in this table, since they
share (conjectural) local Langlands parameters with some of
the other representations: $\rm Va^*$ is in the same $L$-packet
as Va, and $\rm XIa^*$ is in the same $L$-packet as XIa.
The only other cases of $L$-indistinguishability in this
table are the representations VIa and VIb, which constitute
a two-element $L$-packet, and VIIIa, VIIIb, which do also
constitute a two-element $L$-packet.
\nl
{\bf\ref{decompositions}} has more precise information
on how the representations in groups IV, V and VI decompose.
\nl
{\bf\ref{unitarytable}:} This table lists the exact conditions for a representation
to be unitary.
\nl
{\bf\ref{locallanglandstable}:} This table shows an explicit form of
the conjectural Langlands parameters from the Weil--Deligne
group $W'_F$ into the $L$-group $\GSp(4,\C)$ for the non-supercuspidal
representations. The parameters are given as a pair $\varphi=(\rho,N)$, where $\rho$ is a
homomorphism from the Weil group $W_F$ to the $L$-group $\GSp(4,\C)$
and $N$ is a unipotent element
in the Lie algebra of $\GSp(4)$ such that $\rho(x)N\rho(x)^{-1}=|x|N$.
For groups I -- VI, an entry $\tau_1$, \ldots, $\tau_4$ in the ``$\rho$'' column stands
for the map $W_F\ni x\mapsto{\rm diag}(\tau_1(x),\ldots,\tau_4(x))$.
For groups VII -- XI, the ``$\rho$'' column has to be read as block diagonal entries.
The character $\nu$ denotes the normalized absolute value on $F^\times$,
identified with a character of $W_F$.
Note that the representations VIa,b constitute an $L$-packet.
The same is true for the representations VIIIa,b.
The representation Va shares its $L$-parameter with a supercuspidal
representation $\rm Va^*$ of $\theta_{10}$ type, and the representation XIa
shares its $L$-parameter with a supercuspidal representation $\rm XIa^*$.
All other representations in the table constitute singleton $L$-packets.
Finally, the last column lists the number of elements of
$$
 \mathcal{C}(\varphi)={\rm Cent}(\varphi)/{\rm Cent}(\varphi)^0\,\C^\times,
$$
where ${\rm Cent}(\varphi)$ denotes the centralizer of the image of $\varphi$, where
${\rm Cent}(\varphi)^0$ denotes its identity component,
and where $\C^\times$ stands for the center of $\GSp(4,\C)$.
\nl
{\bf\ref{L4table} and \ref{L5table}:} These are the degree $4$ and degree $5$
$L$-functions resulting from the $L$-parameters listed in Table \ref{locallanglandstable}.
\nl
{\bf\ref{epsilon4table} and \ref{epsilon5table}:} These are the degree $4$ and degree $5$
$\varepsilon$-factors resulting from the $L$-parameters listed in Table 
\ref{locallanglandstable}, assuming that the central character of each representation
is trivial.
\nl
{\bf\ref{JacquetSiegeltable} and \ref{JacquetKlingentable}:} These tables
show the semisimplifications of the Jacquet modules with respect to the
unipotent radical of the Siegel resp.\ Klingen parabolic. The Jacquet
modules were computed with the formulas in Sect.\ 2 of [ST].
\nl
{\bf\ref{parahoritable}:} This table contains, for the
Iwahori-spherical representations, the dimensions of the spaces of
invariant vectors under each parahoric subgroup. All the characters
in this table are assumed to be unramified. Under the assumption
that the representation has trivial central character, the signs
under the dimensions indicate how the space splits into $\pm1$
eigenspaces under the Atkin--Lehner involution.
The $\varepsilon$--factor shown is computed from the local parameter
given in the previous table; we give the value at $1/2$ of the
$\varepsilon$--factor under the assumption of trivial central character.
\nl
{\bf\ref{iwahoriparamodulartable}:} This table contains, for the same
representations as the previous table, the dimensions of the spaces
of invariant vectors under the paramodular group (of any level).

%
%
%
\newpage
\section{General tables}
\subsection{Irreducible non-supercuspidal representations of $\GSp(4,F)$}\label{maintable}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}
  \hline&&\mbox{constituent of}&\mbox{representation}
   &{\rm tempered}&L^2\!&\,{\rm g}\,&P&B&Q\\\hline\hline
% I
  {\rm I}&&\multicolumn{2}{|c|}{\chi_1\times\chi_2\rtimes\sigma\quad
   \mbox{(irreducible)}}&\mbox{$\chi_i,\sigma$ unit.}
   &&\bullet&&&\\\hline
% II
  &\mbox{a}&\nu^{1/2}\chi\times\nu^{-1/2}\chi\rtimes\sigma
   &\chi\St_{\GL(2)}\rtimes\sigma
   &\mbox{$\chi,\sigma$ unit.}&&\bullet&\circ&&\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{II}
   &\mbox{b}&(\chi^2\neq\nu^{\pm1},\chi\neq\nu^{\pm 3/2})\!
   &\chi\triv_{\GL(2)}\rtimes\sigma
   &&&&\bullet&&\\\hline
% III
  &\mbox{a}&\chi\times\nu\rtimes\nu^{-1/2}\sigma
   &\chi\rtimes\sigma\St_{\GSp(2)}&\mbox{$\pi,\sigma$ unit.}&
   &\bullet&&&\circ\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{III}
  &\mbox{b}&(\chi\notin\{1,\nu^{\pm2}\})
   &\chi\rtimes\sigma\triv_{\GSp(2)}
   &&&&&&\bullet\\\hline
% IV
  &\mbox{a}&&\sigma\St_{\GSp(4)}&\mbox{$\sigma$ unit.}&\bullet&\bullet&&&\\
   \cline{4-10}
  &\mbox{b}&&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})&&&&&&\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{IV}&
   \mbox{c}&\raisebox{2.5ex}[-2.5ex]{$\nu^2\times\nu\rtimes\nu^{-3/2}\sigma$}
   &L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)&&&&&&\\\cline{4-10}
  &\mbox{d}&&\sigma\triv_{\GSp(4)}&&&&&&\\\hline
% V
  &\mbox{a}&&\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)
   &\mbox{$\sigma$ unit.}&\bullet&\bullet&\circ&\circ&\circ\\\cline{4-10}
  &\mbox{b}&\nu\xi\times\xi\rtimes\nu^{-1/2}\sigma&
   L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2}\sigma)
   &&&&\bullet&\circ&\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{V}&\mbox{c}&(\xi^2=1,\:\xi\neq1)
   &L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2}\sigma)\!
   &&&&\bullet&\circ&\\\cline{4-10}
  &\mbox{d}&&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)
   &&&&&\bullet&\bullet\\\hline
% VI
  &\mbox{a}&&\tau(S,\nu^{-1/2}\sigma)&\mbox{$\sigma$ unit.}&&\bullet&\circ&\circ
   &\circ\\\cline{4-10}
  &\mbox{b}&&\tau(T,\nu^{-1/2}\sigma)&\mbox{$\sigma$ unit.}&&&\bullet&\circ
   &\circ\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{VI}&\mbox{c}
   &\raisebox{2.5ex}[-2.5ex]{$\nu\times1_{F^\times}\rtimes\nu^{-1/2}\sigma$}
   &L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
   &&&&\bullet&\bullet&\\\cline{4-10}
  &\mbox{d}&&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
   &&&&&\bullet&\bullet\\\hline\hline
% VII
  {\rm VII}&&\multicolumn{2}{|c|}{\chi\rtimes\pi\quad
   \mbox{(irreducible)}}&\mbox{$\chi,\pi$ unit.}
   &&\bullet&&&\\\hline
% VIII
  &\mbox{a}&
   &\tau(S,\pi)&\mbox{$\pi$ unit.}&
   &\bullet&&&\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{VIII}
  &\mbox{b}&\raisebox{2.5ex}[-2.5ex]{$1_{F^\times}\rtimes\pi$}
   &\tau(T,\pi)&\mbox{$\pi$ unit.}&&&&&\\\hline
% IX
  &\mbox{a}&\nu\xi\rtimes\nu^{-1/2}\pi
   &\delta(\nu\xi,\nu^{-1/2}\pi)&\mbox{$\pi$ unit.}&\bullet
   &\bullet&&&\circ\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{IX}
  &\mbox{b}&(\xi\neq1,\:\xi\pi=\pi)
   &L(\nu\xi,\nu^{-1/2}\pi)
   &&&&&&\bullet\\\hline\hline
% X
  {\rm X}&&\multicolumn{2}{|c|}{\pi\rtimes\sigma\quad
   \mbox{(irreducible)}}&\mbox{$\pi,\sigma$ unit.}
   &&\bullet&&&\\\hline
% XI
  &\mbox{a}&\nu^{1/2}\pi\rtimes\nu^{-1/2}\sigma
   &\delta(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &\mbox{$\pi,\sigma$ unit.}&\bullet&\bullet&\circ&&\\\cline{4-10}
  \raisebox{2.5ex}[-2.5ex]{XI}
   &\mbox{b}&(\omega_\pi=1)
   &L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &&&&\bullet&&\\\hline\hline
% Va*
  \multicolumn{2}{|c|}{\rm Va^*}
  &\mbox{(supercuspidal)}&\delta^*([\xi,\nu\xi],\nu^{-1/2}\sigma)
   &\mbox{$\sigma$ unit.}&\bullet&&\bullet&\bullet&\bullet\\\hline
% XIa*
  \multicolumn{2}{|c|}{\rm XIa^*}&\mbox{(supercuspidal)}
   &\delta^*(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &\mbox{$\pi,\sigma$ unit.}&\bullet&&\bullet&&\\\hline
 \end{array}
$$
The induced representation
$\chi_1\times\chi_2\rtimes\sigma$ is irreducible if and only if
$\chi_1\neq\nu^{\pm1}$, $\chi_2\neq\nu^{\pm1}$ and
$\chi_1\neq\nu^{\pm1}\chi_2^{\pm1}$.

\vspace{2ex}
\begin{minipage}{21ex}
$L$--Packets:
\end{minipage}
\begin{minipage}{85ex}
 $
  \begin{array}{|c|c|}\hline{\rm VIa}&{\rm VIb}\\\hline\end{array}\qquad
  \begin{array}{|c|c|}\hline{\rm VIIIa}&{\rm VIIIb}\\\hline\end{array}\qquad
  \begin{array}{|c|c|}\hline{\rm Va}&{\rm Va}^*\\\hline\end{array}\qquad
  \begin{array}{|c|c|}\hline{\rm XIa}&{\rm XIa}^*\\\hline\end{array}
 $
\end{minipage}

\vspace{1ex}
\begin{minipage}{20ex}
\raisebox{1ex}[-1ex]{Aubert involution:}
\end{minipage}
\begin{minipage}{85ex}
$
 \begin{array}{cccccc}
 {\rm IIa}\leftrightarrow{\rm IIb}&{\rm IVa}\leftrightarrow{\rm IVd}
  &{\rm Va}\leftrightarrow{\rm Vd}&{\rm VIa}\leftrightarrow{\rm VId}
  &{\rm VIIIa}\leftrightarrow{\rm VIIIb}&{\rm XIa}\leftrightarrow{\rm XIb}\\
 {\rm IIIa}\leftrightarrow{\rm IIIb}&{\rm IVb}\leftrightarrow{\rm IVc}
  &{\rm Vb}\leftrightarrow{\rm Vc}&{\rm VIb}\leftrightarrow{\rm VIc}
  &{\rm IXa}\leftrightarrow{\rm IXb}&
 \end{array}
$
\end{minipage}
%
%
%
\newpage
\subsection{Decompositions for types IV, V and VI}\label{decompositions}
\underline{Group IV}: Constituents of $\nu^2\times\nu\rtimes
\nu^{-3/2}\sigma$.
\begin{align*}
 \nu^2\times\nu\rtimes\nu^{-3/2}\sigma&=\underbrace{\nu^{3/2}\St_{\GL(2)}\rtimes
  \nu^{-3/2}\sigma}_{\rm sub}
  +\underbrace{\nu^{3/2}\triv_{\GL(2)}\rtimes\nu^{-3/2}\sigma}_{\rm quot}\\
 &=\underbrace{\nu^2\rtimes\nu^{-1}\sigma\St_{\GSp(2)}}_{\rm sub}
  +\underbrace{\nu^2\rtimes\nu^{-1}\sigma\triv_{\GSp(2)}}_{\rm quot}.
\end{align*}
Each of the four representations on the right is reducible and
has two irreducible constituents as shown in the following table.
The quotients are on the bottom resp.\ on the right.
\begin{equation}\label{IVcomptable}\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c||c|c|}\hline
 &\nu^{3/2}\St_{\GL(2)}\rtimes\nu^{-3/2}\sigma&
  \nu^{3/2}\triv_{\GL(2)}\rtimes\nu^{-3/2}\sigma\\\hline\hline
 \nu^2\rtimes\nu^{-1}\sigma\St_{\GSp(2)}&\sigma\St_{\GSp(4)}&
  L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})\\\hline
 \nu^2\rtimes\nu^{-1}\sigma\triv_{\GSp(2)}&L(\nu^{3/2}\St_{\GL(2)},
  \nu^{-3/2}\sigma)&\sigma\triv_{\GSp(4)}\\\hline
 \end{array}
\end{equation}
\nl
\underline{Group V}: Constituents of $\nu\xi\times\xi\rtimes
\nu^{-1/2}\sigma$, where $\xi$ is a non-trivial quadratic character.
\begin{align*}
 \nu\xi\times\xi\rtimes\nu^{-1/2}\sigma
  &=\underbrace{\nu^{1/2}\xi\,\St_{\GL(2)}\rtimes\nu^{-1/2}\sigma}_{\rm sub}
  +\underbrace{\nu^{1/2}\xi\,\triv_{\GL(2)}
  \rtimes\nu^{-1/2}\sigma}_{\rm quot}\\
 &=\underbrace{\nu^{1/2}\xi\,\St_{\GL(2)}
  \rtimes\xi\nu^{-1/2}\sigma}_{\rm sub}
  +\underbrace{\nu^{1/2}\xi\,\triv_{\GL(2)}
  \rtimes\xi\nu^{-1/2}\sigma}_{\rm quot}.
\end{align*}
Each of the representations on the right side has two constituents
as indicated in the following table.
The quotients appear on the bottom resp.\ on the right.
\begin{equation}\label{Vcomptable}\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c||c|c|}\hline
 &\nu^{1/2}\xi\,\St_{\GL(2)}\rtimes\xi\nu^{-1/2}\sigma
  &\nu^{1/2}\xi\,\triv_{\GL(2)}\rtimes\xi\nu^{-1/2}\sigma\\\hline\hline
 \nu^{1/2}\xi\,\St_{\GL(2)}\rtimes\nu^{-1/2}\sigma
  &\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)
  &L(\nu^{1/2}\xi\,\St_{\GL(2)},\nu^{-1/2}\sigma)\\\hline
 \nu^{1/2}\xi\,\triv_{\GL(2)}\rtimes\nu^{-1/2}\sigma
  &L(\nu^{1/2}\xi\,\St_{\GL(2)},\xi\nu^{-1/2}\sigma)
  &L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)\\\hline
 \end{array}
\end{equation}
Here $\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)$ is a square integrable
representation.
\nl
\underline{Group VI}: Constituents of $\nu\times1_{F^\times}\rtimes
\nu^{-1/2}\sigma$.
\begin{align*}
 \nu\times1_{F^\times}\rtimes\nu^{-1/2}\sigma
  &=\underbrace{\nu^{1/2}\,\St_{\GL(2)}
  \rtimes\nu^{-1/2}\sigma}_{\rm sub}+\underbrace{\nu^{1/2}\,\triv_{\GL(2)}
  \rtimes\nu^{-1/2}\sigma}_{\rm quot}\nonumber\\
 &=\underbrace{1_{F^\times}\rtimes\sigma\St_{\GSp(2)}}_{\rm sub}+
  \underbrace{1_{F^\times}\rtimes\sigma\triv_{\GSp(2)}}_{\rm quot},
\end{align*}
and each representation on the right side is again reducible.
Their constituents are summarized in the following table, with
the quotients appearing on the bottom resp.\ on the right.
\begin{equation}\label{VIcomptable}\renewcommand{\arraystretch}{1.5}
  \begin{array}{|c||c|c|}\hline
  &\nu^{1/2}\,\St_{\GL(2)}\rtimes\nu^{-1/2}\sigma
   &\nu^{1/2}\,\triv_{\GL(2)}\rtimes\nu^{-1/2}\sigma\\\hline\hline
  1_{F^\times}\rtimes\sigma\St_{\GSp(2)}&\tau(S,\nu^{-1/2}\sigma)&
   \tau(T,\nu^{-1/2}\sigma)\\\hline
  1_{F^\times}\rtimes\sigma\triv_{\GSp(2)}&L(\nu^{1/2}\,\St_{\GL(2)},\nu^{-1/2}\sigma)&
   L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)\\\hline
 \end{array}
\end{equation}
The representations $\tau(S,\nu^{-1/2}\sigma)$ and
$\tau(T,\nu^{-1/2}\sigma)$ are tempered but not
square integrable.
%
%
%
\newpage
\subsection{Unitary representations of $\GSp(4,F)$}\label{unitarytable}
$$
\renewcommand{\arraystretch}{1.3}
 \begin{array}{|c|c|c|c|}
  \hline&&\text{representation}&\text{conditions for unitarity}\\\hline\hline
% I
  &&&e(\chi_1)=e(\chi_2)=e(\sigma)=0\\\cline{4-4}
  &&&\chi_1=\nu^\beta\chi,\:\chi_2=\nu^\beta\chi^{-1},\:e(\sigma)=-\beta,\\
  &&&e(\chi)=0,\:\chi^2\neq1,\:0<\beta<1/2\\\cline{4-4}
  \raisebox{0.5ex}[-0.5ex]{I}&&\raisebox{0.5ex}[-0.5ex]{$\chi_1\times\chi_2\rtimes\sigma
   \quad\text{(irreducible)}$}
   &\chi_1=\nu^\beta,\:e(\chi_2)=0,\:e(\sigma)=-\beta/2,\\
   &&&\chi_2\neq1,\:0<\beta<1\\\cline{4-4}
  &&&\chi_1=\nu^{\beta_1}\chi,\:\chi_2=\nu^{\beta_2}\chi,\:
   e(\sigma)=(-\beta_1-\beta_2)/2,\\
  &&&\chi^2=1,\:0\leq\beta_2\leq\beta_1,\:0<\beta_1<1,\:\beta_1+\beta_2<1\\\hline
% II
  &&&e(\sigma)=e(\chi)=0\\\cline{4-4}
  &\raisebox{2.5ex}[-2.5ex]{a}&\raisebox{2.5ex}[-2.5ex]{$\chi\St_{\GL(2)}\rtimes\sigma$}
   &\chi=\xi\nu^\beta,\:e(\sigma)=-\beta,\:\xi^2=1,\:0<\beta<1/2\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{II}&&&e(\sigma)=e(\chi)=0\\\cline{4-4}
  &\raisebox{2.5ex}[-2.5ex]{b}&\raisebox{2.5ex}[-2.5ex]{$\chi\triv_{\GL(2)}\rtimes\sigma$}
   &\chi=\xi\nu^\beta,\:e(\sigma)=-\beta,\:\xi^2=1,\:0<\beta<1/2\\\hline
% III
  &\text{a}&\chi\rtimes\sigma\St_{\GSp(2)}&e(\sigma)=e(\chi)=0\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{III}
   &\text{b}&\chi\rtimes\sigma\triv_{\GSp(2)}&e(\sigma)=e(\chi)=0\\\hline
% IV
  &\text{a}&\sigma\St_{\GSp(4)}&e(\sigma)=0\\\cline{2-4}
  &\text{b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})&\text{never unitary}\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{IV}
   &\text{c}&L(\nu^{3/2}\St_{\GSp(2)},\nu^{-3/2}\sigma)&\text{never unitary}\\\cline{2-4}
  &\text{d}&\sigma\triv_{\GSp(4)}&e(\sigma)=0\\\hline
% V
  &\text{a}&\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)&e(\sigma)=0\\\cline{2-4}
  &\text{b}&L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2}\sigma)&e(\sigma)=0\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{V}
   &\text{c}&L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2}\sigma)&e(\sigma)=0\\\cline{2-4}
  &\text{d}&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)&e(\sigma)=0\\\hline
% VI
  &\text{a}&\tau(S,\nu^{-1/2}\sigma)&e(\sigma)=0\\\cline{2-4}
  &\text{b}&\tau(T,\nu^{-1/2}\sigma)&e(\sigma)=0\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{VI}
   &\text{c}&L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)&e(\sigma)=0\\\cline{2-4}
  &\text{d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)&e(\sigma)=0\\\hline\hline
% VII
  &&&e(\chi)=e(\pi)=0\\\cline{4-4}
  \text{VII}&&\chi\rtimes\pi\quad\text{(irreducible)}
   &\chi=\nu^\beta\xi,\:\pi=\nu^{-\beta/2}\rho,\:0<\beta<1,\\
  &&&\xi^2=1,\:\xi\neq1,\:e(\rho)=0,\:\xi\rho=\rho\\\hline
% VIII
  &\text{a}&\tau(S,\pi)&e(\pi)=0\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{VIII}&\text{b}&\tau(T,\pi)&e(\pi)=0\\\hline
% IX
  &\text{a}&\delta(\nu\xi,\nu^{-1/2}\pi)&e(\pi)=0\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{IX}&\text{b}&L(\nu\xi,\nu^{-1/2}\pi)&e(\pi)=0\\\hline\hline
% X
  &&&e(\sigma)=e(\pi)=0\\\cline{4-4}
  \raisebox{2.5ex}[-2.5ex]{X}&
   &\raisebox{2.5ex}[-2.5ex]{$\pi\rtimes\sigma\quad\text{(irreducible)}$}
   &\pi=\nu^\beta\rho,\:e(\sigma)=-\beta,\:0<\beta<1/2,\:\omega_\rho=1\\\hline
% XI
  &\text{a}&\delta(\nu^{1/2}\pi,\nu^{-1/2}\sigma)&e(\sigma)=e(\pi)=0\\\cline{2-4}
  \raisebox{2.5ex}[-2.5ex]{XI}
   &\text{b}&L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)&e(\sigma)=e(\pi)=0\\\hline\hline
  \multicolumn{3}{|c|}{\pi\text{ supercuspidal}}&e(\omega_\pi)=0\\\hline
 \end{array}
$$
%
%
%
\section{$L$-parameters and local factors}
\subsection{Local Langlands parameters}\label{locallanglandstable}
$$\renewcommand{\arraystretch}{1.3}
 \begin{array}{|c|c|c|c|c|c|}
  \hline&&\text{representation}&\rho&N&\#\mathcal{C}\\\hline\hline
% I
  {\rm I}&& \chi_1 \times \chi_2 \rtimes \sigma\ \mathrm{(irreducible)}&\chi_1\chi_2\sigma,\:\chi_1\sigma,\:
   \chi_2\sigma,\:\sigma&0&1\\\hline
% II
  &\mbox{a}&\chi \St_{\GL(2)} \rtimes \sigma&&N_1&1\\\cline{2-3}\cline{5-6}
  \raisebox{2.3ex}[-2.3ex]{II}&\mbox{b}&\chi \triv_{\GL(2)} \rtimes \sigma
   &\raisebox{2.3ex}[-2.3ex]{$\chi^2\sigma,\:\nu^{1/2}\chi\sigma,\:\nu^{-1/2}\chi\sigma,\:\sigma$}&0&1\\\hline
% III
  &\mbox{a}&\chi \rtimes \sigma \St_{\GSp(2)}&
   &N_4&1\\\cline{2-3}\cline{5-6}
  \raisebox{2.3ex}[-2.3ex]{III}&\mbox{b}&\chi \rtimes \sigma \triv_{\GSp(2)}
   &\raisebox{2.3ex}[-2.3ex]{$\;\nu^{1/2}\chi\sigma,\:\nu^{-1/2}\chi\sigma,\:\nu^{1/2}\sigma,\:\nu^{-1/2}\sigma\;$}
   &0&1\\\hline
% IV
  &\mbox{a}&\sigma\St_{\GSp(4)}&&N_5&1\\\cline{2-3}\cline{5-6}
  &\mbox{b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})&&N_4
   &1\\\cline{2-3}\cline{5-6}
  \raisebox{2.3ex}[-2.3ex]{IV}&\mbox{c}&L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)
   &\raisebox{2.3ex}[-2.3ex]{$\nu^{3/2}\sigma,\:\nu^{1/2}\sigma,\:\nu^{-1/2}\sigma,\:\nu^{-3/2}\sigma$}
   &N_1&1\\\cline{2-3}\cline{5-6}
  &\mbox{d}&\sigma\triv_{\GSp(4)}&&0&1\\\hline
% V
  &\mbox{a}&\delta([\xi,\nu \xi], \nu^{-1/2} \sigma)&&N_3&2\\\cline{2-3}\cline{5-6}
  &\mbox{b}&L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2} \sigma)&
   &N_1&1\\\cline{2-3}\cline{5-6}
  \raisebox{2.5ex}[-2.5ex]{V}&\mbox{c}&L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2} \sigma)
   &\raisebox{2.5ex}[-2.5ex]{$\nu^{1/2}\sigma,\:\nu^{1/2}\xi\sigma,\: \nu^{-1/2}\xi\sigma,\:\nu^{-1/2}\sigma$}
   &N_2&1\\\cline{2-3}\cline{5-6}
  &\mbox{d}&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)&&0&1\\\hline
% VI
  &\mbox{a}&\tau(S, \nu^{-1/2}\sigma)&&&\\\cline{2-3}
  &\mbox{b}&\tau(T, \nu^{-1/2}\sigma)&&\raisebox{2.3ex}[-2.3ex]{$N_3$}
   &\raisebox{2.3ex}[-2.3ex]{$2$}\\\cline{2-3}\cline{5-6}
  \raisebox{2.3ex}[-2.3ex]{VI}&\mbox{c}&L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
   &\raisebox{2.3ex}[-2.3ex]{$\nu^{1/2}\sigma,\:\nu^{1/2}\sigma,\:\nu^{-1/2}\sigma,\:\nu^{-1/2}\sigma$}&N_1
   &1\\\cline{2-3}\cline{5-6}
  &\mbox{d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)&&0&1\\\hline\hline
% VII
  \mbox{VII}&&\chi \rtimes \pi&\chi\omega_\pi\varphi_\pi',\:\varphi_\pi&0
   &1\\\hline
% VIII
  &\mbox{a}&\tau(S, \pi)&&&\\\cline{2-3}
  \raisebox{2.3ex}[-2.3ex]{VIII}&\mbox{b}&\tau(T, \pi)
   &\raisebox{2.3ex}[-2.3ex]{$\omega_\pi\varphi_\pi',\:\varphi_\pi$}
   &\raisebox{2.3ex}[-2.3ex]{$0$}
   &\raisebox{2.3ex}[-2.3ex]{$2$}\\\hline
% IX
  &\mbox{a}&\delta(\nu\xi,\nu^{-1/2}\pi)&&N_6&1\\\cline{2-3}\cline{5-6}
  \raisebox{2.3ex}[-2.3ex]{IX}&\mbox{b}&L(\nu\xi,\nu^{-1/2}\pi)
   &\raisebox{2.3ex}[-2.3ex]{$\xi\nu^{1/2}\omega_\pi\varphi_\pi',\:\nu^{-1/2}\varphi_\pi$}
   &0&1\\\hline\hline
% X
  \mbox{X}&&\pi \rtimes \sigma&\sigma\omega_\pi,\:\sigma\varphi_\pi,\:\sigma&0
   &1\\\hline
% XI
  &\mbox{a}&\delta (\nu^{1/2}\pi,\nu^{-1/2}\sigma)&&N_2&2\\\cline{2-3}\cline{5-6}
  \raisebox{2.3ex}[-2.3ex]{XI}&\mbox{b}&L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &\raisebox{2.3ex}[-2.3ex]{$\nu^{1/2}\sigma,\:\sigma\varphi_\pi,\:\nu^{-1/2}\sigma$}
   &0&1\\\hline
 \end{array}
$$
Here, $\varphi_\pi:\:W_F\rightarrow\GL(2,\C)$
is the $L$-parameter of the supercuspidal representation $\pi$
of $\GL(2,F)$, and $\omega_\pi$ is its central character. Further,
\vspace{-1ex}
\begin{alignat*}{3}
 &N_1=\begin{bmatrix}0\\&0&1\\&&0\\&&&0\end{bmatrix},
  \qquad&&N_2=\begin{bmatrix}0&&&1\\&0\\&&0\\&&&0\end{bmatrix},
  \qquad&&N_3=\begin{bmatrix}0&&&1\\&0&1\\&&0\\&&&0\end{bmatrix},\\
 &N_4=\begin{bmatrix}0&1\\&0\\&&0&-1\\&&&0\end{bmatrix},
  \qquad&&N_5=\begin{bmatrix}0&1\\&0&1\\&&0&-1\\&&&0\end{bmatrix},
  &&N_6=\begin{bmatrix}0&&y&z\\&0&x&y\\&&0&\\&&&0\end{bmatrix},
\end{alignat*}
where $S=\mat{x}{y}{y}{z}$ is such that $^t\varphi_\pi(w) S\varphi_\pi(w)=
\xi(w)\det(\varphi_\pi(w))S$ for all $w\in W_F$.
%
%
%
\newpage
\subsection{$L$-factors (degree $4$)}\label{L4table}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|}
 \hline&&\mbox{representation}&L(s,\varphi)\\\hline\hline
%I
\mathrm{I}&& \chi_1 \times \chi_2 \rtimes \sigma\ \mathrm{(irreducible)}& 
L(s,\chi_1 \chi_2 \sigma) L(s, \sigma) L(s,\chi_1 \sigma) L(s, \chi_2\sigma) \\\hline
%IIa
&\mathrm{a}& \chi \St_{\GL(2)} \rtimes \sigma & L(s, \chi^2\sigma )L(s, \sigma) L(s,\nu^{1/2}\chi\sigma ) \\
\cline{2-4}
%IIb
\raisebox{2.3ex}[-2.3ex]{II}&\mathrm{b}& \chi \triv_{\GL(2)} \rtimes \sigma & L(s, \chi^2\sigma )L(s, \sigma) L(s,\nu^{1/2}\chi\sigma )L(s,\nu^{-1/2}\chi\sigma ) \\
\hline
%IIIa
&\mathrm{a}& \chi \rtimes \sigma \St_{\GSp(2)} & L(s,\nu^{1/2}\chi\sigma )L(s, \nu^{1/2} \sigma) \\
\cline{2-4}
%IIIb
\raisebox{2.3ex}[-2.3ex]{III}&\mathrm{b}& \chi \rtimes \sigma \triv_{\GSp(2)} 
& L(s,\nu^{1/2}\chi\sigma )L(s, \nu^{1/2} \sigma)L(s,\nu^{-1/2}\chi\sigma )
L(s, \nu^{-1/2} \sigma)\\\hline
%IVa
&\mathrm{a}& \sigma\St_{\GSp(4)} & L(s, \nu^{3/2} \sigma) \\\cline{2-4}
%IVb
&\mathrm{b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)}) 
& L(s, \nu^{3/2} \sigma)L(s, \nu^{-1/2} \sigma) \\\cline{2-4}
%IVc
\raisebox{2.3ex}[-2.3ex]{IV}&\mathrm{c}&L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma) 
& L(s, \nu^{3/2} \sigma)L(s, \nu^{1/2} \sigma)L(s, \nu^{-3/2} \sigma) \\\cline{2-4}
%IVd
&\mathrm{d}&\sigma\triv_{\GSp(4)}& L(s, \nu^{3/2} \sigma)L(s, \nu^{1/2} \sigma)
L(s, \nu^{-1/2} \sigma)L(s, \nu^{-3/2} \sigma) \\\hline
%Va
& \mathrm{a} &\delta([\xi,\nu \xi], \nu^{-1/2} \sigma) &
L(s, \nu^{1/2}\sigma)L(s, \nu^{1/2}\xi\sigma) \\\cline{2-4}
%Vb
& \mathrm{b} &L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2} \sigma) &
L(s, \nu^{1/2}\sigma)L(s, \nu^{1/2}\xi\sigma)L(s, \nu^{-1/2}\sigma) \\\cline{2-4}
%Vc
\raisebox{2.3ex}[-2.3ex]{V}& \mathrm{c} &L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2} \sigma)&
L(s, \nu^{1/2}\sigma)L(s, \nu^{1/2}\xi\sigma)L(s, \nu^{-1/2}\xi\sigma) \\\cline{2-4}
%Vd
& \mathrm{d} &L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)&
L(s, \nu^{1/2}\sigma)L(s, \nu^{1/2}\xi\sigma)
L(s, \nu^{-1/2}\sigma)L(s, \nu^{-1/2}\xi\sigma)\\\hline
%VIa
&\mathrm{a}& \tau(S, \nu^{-1/2}\sigma) & L(s, \nu^{1/2}\sigma)^2 \\\cline{2-4}
%VIb
&\mathrm{b}& \tau(T, \nu^{-1/2}\sigma) & L(s, \nu^{1/2}\sigma)^2 \\\cline{2-4}
%VIc
\raisebox{2.3ex}[-2.3ex]{VI}&\mathrm{c}&L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
& L(s, \nu^{1/2}\sigma)^2L(s, \nu^{-1/2}\sigma) \\\cline{2-4}
%VId
&\mathrm{d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
& L(s, \nu^{1/2}\sigma)^2L(s, \nu^{-1/2}\sigma)^2 \\\hline\hline
%VII
\mathrm{VII}&&\chi \rtimes \pi &1 \\\hline
%VIIIa
&\mathrm{a} &\tau(S, \pi) &1\\\cline{2-4}
%VIIIb
\raisebox{2.3ex}[-2.3ex]{VIII}&\mathrm{b} &\tau(T, \pi) &1\\
\hline
%IXa
&\mathrm{a}& \delta(\nu\xi,\nu^{-1/2}\pi)&1 \\\cline{2-4}
%IXb
\raisebox{2.3ex}[-2.3ex]{IX}&\mathrm{b}&L(\nu\xi,\nu^{-1/2}\pi)&1 \\
\hline\hline
%X
\mathrm{X}&&\pi \rtimes \sigma & L(s,\sigma)L(s,\omega_\pi\sigma)\\
\hline
%XIa
&\mathrm{a}&\delta (\nu^{1/2}\pi,\nu^{-1/2}\sigma)& L(s, \nu^{1/2}\sigma)\\
\cline{2-4}
%XIb
\raisebox{2.3ex}[-2.3ex]{XI}&\mathrm{b}&L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
& L(s, \nu^{1/2}\sigma)L(s, \nu^{-1/2}\sigma)\\\hline
%\multicolumn{3}{|c|}{\mathrm{supercuspidal}}&1 \\\hline
\end{array}
$$

\vspace{3ex}
$\varphi=(\rho,N)$ is the $L$-parameter of the representation, as listed in
Table \ref{locallanglandstable}.
%
%
%
\newpage
\subsection{$\varepsilon$-factors (degree $4$)}\label{epsilon4table}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|c|}
 \hline&&\text{inducing data}&a(\varphi)&\varepsilon(1/2,\varphi)\\\hline\hline
 %I
 \mathrm{I}&&&a(\chi_1\sigma)+a(\chi_2\sigma)+2a(\sigma)
   &\chi_1(-1)\quad(=\chi_2(-1)) \\\hline
 %IIa
 &&\sigma\chi\text{ unr.}&2a(\sigma)+1
  &-\sigma(-1)(\sigma\chi)(\varpi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{a}
  &\sigma\chi\text{ ram.}&2a(\chi\sigma)+2a(\sigma)
  &\chi(-1)\\\cline{2-5}
 %IIb
 \raisebox{2.3ex}[-2.3ex]{II}&&\sigma\chi\text{ unr.}&2a(\sigma)
  &\chi(-1)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{b}
  &\sigma\chi\text{ ram.}&2a(\chi\sigma)+2a(\sigma)&\chi(-1)\\\hline
 %IIIa
 &&\sigma\text{ unr.}&2&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{a}&\sigma\text{ ram.}&4a(\sigma)
  &1\\\cline{2-5}
 %IIIb
 \raisebox{2.3ex}[-2.3ex]{III}&&\sigma\;\text{unr.}&0&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{b}
  &\sigma\;\text{ram.}&4a(\sigma)&1\\\hline
 %IVa
 &&\sigma\;\text{unr.}&3&-\sigma(\varpi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{a}&\sigma\;\text{ram.}
  &4a(\sigma)&1\\\cline{2-5}
 %IVb
 &&\sigma\;\text{unr.}&2&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{b}
  &\sigma\;\text{ram.}&4a(\sigma)&1\\\cline{2-5}
 %IVc
 \raisebox{2.3ex}[-2.3ex]{IV}&&\sigma\;\text{unr.}&1&-\sigma(\varpi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{c}
  &\sigma\;\text{ram.}&4a(\sigma)&1\\\cline{2-5}
 %IVd
 &&\sigma\;\text{unr.}&0&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{d}
  &\sigma\;\text{ram.}&4a(\sigma)&1\\\hline
 %Va
 &&\text{$\sigma$, $\xi$ unr.}&2&-1\\\cline{3-5}
 &&\text{$\sigma$ unr., $\xi$ ram.}&2a(\xi)+1
  &-\sigma(\varpi)\xi(-1)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{a}
  &\text{$\sigma$ ram., $\sigma\xi$ unr.}&2a(\sigma)+1
  &-\sigma(-1)(\sigma\xi)(\varpi)\\\cline{3-5}
 &&\text{$\sigma$, $\sigma\xi$ ram.}&2a(\xi\sigma)+2a(\sigma)&\xi(-1)\\\cline{2-5}
 %Vb
 \raisebox{2.3ex}[-2.3ex]{V}&
  &\text{$\sigma$, $\xi$ unr.}&1&\sigma(\varpi)\\\cline{3-5}
 &&\text{$\sigma$ unr., $\xi$ ram.}&2a(\xi)&\xi(-1)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{b}
  &\text{$\sigma$ ram., $\sigma\xi$ unr.}&2a(\sigma)+1&
  -\xi(-1)(\sigma\xi)(\varpi)\\\cline{3-5}
 &&\text{$\sigma$, $\sigma\xi$ ram.}
  &2a(\xi\sigma)+2a(\sigma)&\xi(-1)\\\hline
 \end{array}
$$

\newpage
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|}
 \hline&&\text{inducing data}&a(\varphi)&\varepsilon(1/2,\varphi)\\\hline\hline
 %Vc
 &&\text{$\sigma$, $\xi$ unr.}&1&-\sigma(\varpi)\\\cline{3-5}
 &&\text{$\sigma$ unr., $\xi$ ram.}&2a(\xi)+1&-\sigma(\varpi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{c}
  &\text{$\sigma$ ram., $\sigma\xi$ unr.}&2a(\sigma)&\xi(-1)\\\cline{3-5}
 \raisebox{2.3ex}[-2.3ex]{V}&
  &\text{$\sigma$, $\sigma\xi$ ram.}&2a(\xi\sigma)+2a(\sigma)&\xi(-1)\\\cline{2-5}
 %Vd
 &&\sigma,\xi\;\text{unr.}&0&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{d}
  &\text{$\sigma$ or $\xi$ ram.}&2a(\xi\sigma)+2a(\sigma)&\xi(-1)\\\hline
 %VIa
 &&\text{$\sigma$ unr.}&2&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{a}&\text{$\sigma$ ram.}&4a(\sigma)
  &1\\\cline{2-5}
 %VIb
 &&\text{$\sigma$ unr.}&2&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{b}
  &\text{$\sigma$ ram.}&4a(\sigma)&1\\\cline{2-5}
 %VIc
 \raisebox{2.3ex}[-2.3ex]{VI}&&\sigma\;\text{unr.}&1
  &-\sigma(\varpi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{c}&\sigma\;\text{ram.}&4a(\sigma)&1\\\cline{2-5}
 %VId
 &&\sigma\;\text{unr.}&0&1\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{d}
  &\sigma\;\text{ram.}&4a(\sigma)&1\\\hline\hline
 %VII
 \mathrm{VII}&&&2a(\pi)&\chi(-1)\quad(=\omega_\pi(-1))\\\hline
 %VIIIa
 &\mathrm{a}&&2a(\pi)&1\\\cline{2-5}
 %VIIIb
 \raisebox{2.3ex}[-2.3ex]{VIII}&\mathrm{b}&&2a(\pi)&1\\\hline
 %IXa
 &\mathrm{a}&&2a(\pi)&\xi(-1)\\\cline{2-5}
 %IXb
 \raisebox{2.3ex}[-2.3ex]{IX}&\mathrm{b}&&2a(\pi)&\xi(-1)\\\hline\hline
 %X
 \mathrm{X}&&&a(\sigma\pi)+2a(\sigma)
  &\sigma(-1)\varepsilon(1/2,\sigma\pi)\\\hline
 %XIa
 &&\sigma\;\text{unr.}&a(\sigma\pi)+1
  &-\sigma(\varpi)\varepsilon(1/2,\sigma\pi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{a}&\sigma\;\text{ram.}
  &a(\sigma\pi)+2a(\sigma)
  &\sigma(-1)\varepsilon(1/2,\sigma\pi)\\\cline{2-5}
 %XIb
 \raisebox{2.3ex}[-2.3ex]{XI}&&\sigma\text{ unr.}&a(\sigma\pi)
  &\varepsilon(1/2,\sigma\pi)\\\cline{3-5}
 &\raisebox{2.3ex}[-2.3ex]{b}
  &\quad\sigma\text{ ram.}\quad&\quad a(\sigma\pi)+2a(\sigma)\quad&
  \sigma(-1)\varepsilon(1/2,\sigma\pi)\\\hline
 \end{array}
$$

\vspace{3ex}
$\varphi=(\rho,N)$ is the $L$-parameter of the representation, as listed in
Table \ref{locallanglandstable}. In this table we assume that all the
representations have trivial central character. Under this assumption the
$\varepsilon$-factors do not depend on the choice of an additive character,
as long as this character has conductor $\OF$. We have
$$
 \varepsilon(s,\varphi)=\varepsilon(1/2,\varphi) q^{-a(\varphi)(s-1/2)}
$$
where $\varepsilon(1/2,\varphi)\in\{\pm1\}$.
%
%
%
\newpage
\subsection{$L$-factors (degree $5$)}\label{L5table}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|}
 \hline&&\mbox{representation}
   &L(s,\rho_5\circ\varphi)\\\hline\hline
%I
\mathrm{I}&& \chi_1 \times \chi_2 \rtimes \sigma\ \mathrm{(irreducible)}& 
L(s,\chi_1)L(s,\chi_1^{-1})L(s,\chi_2)L(s,\chi_2^{-1})L(s,1_{F^\times})\\\hline
%IIa
&\mathrm{a}& \chi \St_{\GL(2)} \rtimes \sigma 
&L(s,\nu^{1/2}\chi)L(s,\nu^{1/2}\chi^{-1})L(s,1_{F^\times})\\
\cline{2-4}
%IIb
\mathrm{II}&&&L(s,\nu^{1/2}\chi)L(s,\nu^{-1/2}\chi)\\
&\raisebox{2.3ex}[-2.3ex]{b}&\raisebox{2.3ex}[-2.3ex]{$\chi\triv_{\GL(2)}\rtimes\sigma$}
&L(s,\nu^{1/2}\chi^{-1})L(s,\nu^{-1/2}\chi^{-1})L(s,1_{F^\times})\\
\hline
%IIIa
&\mathrm{a}& \chi \rtimes \sigma \St_{\GSp(2)}
&L(s,\chi)L(s,\chi^{-1})L(s,\nu)\\
\cline{2-4}
%IIIb
\raisebox{2.3ex}[-2.3ex]{III}&\mathrm{b}& \chi \rtimes \sigma \triv_{\GSp(2)} 
&L(s,\chi)L(s,\chi^{-1})L(s,\nu)L(s,\nu^{-1})L(s,1_{F^\times})\\\hline
%IVa
&\mathrm{a}& \sigma\St_{\GSp(4)} & L(s, \nu^2) \\\cline{2-4}
%IVb
&\mathrm{b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)}) 
&L(s,\nu^2)L(s,\nu)L(s,\nu^{-2})\\\cline{2-4}
%IVc
\raisebox{2.3ex}[-2.3ex]{IV}&\mathrm{c}&L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma) 
&L(s,\nu^2)L(s,1_{F^\times})L(s,\nu^{-1})\\\cline{2-4}
%IVd
&\mathrm{d}&\sigma\triv_{\GSp(4)}
&L(s,\nu^2)L(s,\nu)L(s,1_{F^\times})L(s,\nu^{-1})L(s,\nu^{-2})\\\hline
%Va
& \mathrm{a} &\delta([\xi,\nu \xi], \nu^{-1/2} \sigma) &
L(s,\nu\xi)L(s,\xi)L(s,1_{F^\times})\\\cline{2-4}
%Vb
& \mathrm{b} &L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2} \sigma) &
L(s,\nu\xi)L(s,\xi)L(s,1_{F^\times})\\\cline{2-4}
%Vc
\raisebox{2.3ex}[-2.3ex]{V}& \mathrm{c} &L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2} \sigma)&
L(s,\nu\xi)L(s,\xi)L(s,1_{F^\times})\\\cline{2-4}
%Vd
& \mathrm{d} &L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)&
L(s,\nu\xi)L(s,\nu^{-1}\xi)L(s,\xi)^2L(s,1_{F^\times})\\\hline
%VIa
&\mathrm{a}& \tau(S, \nu^{-1/2}\sigma) &\\\cline{2-3}
%VIb
&\mathrm{b}& \tau(T, \nu^{-1/2}\sigma) &
\raisebox{2.3ex}[-2.3ex]{$L(s,\nu)L(s,1_{F^\times})^2$}\\\cline{2-4}
%VIc
\raisebox{2.3ex}[-2.3ex]{VI}&\mathrm{c}&L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
&L(s,\nu)L(s,1_{F^\times})^2\\\cline{2-4}
%VId
&\mathrm{d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
&L(s,\nu)L(s,\nu^{-1})L(s,1_{F^\times})^3\\\hline\hline
%VII
\mathrm{VII}&&\chi \rtimes \pi &L(s,\chi)L(s,\chi^{-1})L(s,{\rm Sym}^2\circ\mu)\\\hline
%VIIIa
 &\mathrm{a} &\tau(S, \pi) &\\\cline{2-3}
%VIIIb
 \raisebox{2.3ex}[-2.3ex]{VIII}&\mathrm{b} &\tau(T, \pi)
 &\raisebox{2.3ex}[-2.3ex]{$L(s,1_{F^\times})^2L(s,{\rm Sym}^2\circ\mu)$}\\
\hline
%IXa
&\mathrm{a}& \delta(\nu\xi,\nu^{-1/2}\pi)&
L(s,\nu\xi)L(s,{\rm Sym}^2\circ\mu)L(s,\xi)^{-1}\\\cline{2-4}
%IXb
\raisebox{2.3ex}[-2.3ex]{IX}&\mathrm{b}
&L(\nu\xi,\nu^{-1/2}\pi)&L(s,\nu\xi)L(s,\nu^{-1}\xi)L(s,{\rm Sym}^2\circ\mu)\\\hline\hline
%X
\mathrm{X}&&\pi\rtimes\sigma&L(s,\mu)L(s,\det(\mu)^{-1}\mu)L(s,1_{F^\times})\\
\hline
%XIa
&\mathrm{a}&\delta(\nu^{1/2}\pi,\nu^{-1/2}\sigma)&L(s,\nu^{1/2}\mu)L(s,1_{F^\times})\\
\cline{2-4}
%XIb
\raisebox{2.3ex}[-2.3ex]{XI}&\mathrm{b}&L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
&L(s,\nu^{1/2}\mu)L(s,\nu^{-1/2}\mu)L(s,1_{F^\times})\\\hline
\end{array}
$$

\vspace{3ex}
$\varphi=(\rho,N)$ is the $L$-parameter of the representation, as listed in
Table \ref{locallanglandstable}. The homomorphism $\rho_5:\:\GSp(4,\C)\rightarrow\SO(5,\C)$
is a five-dimensional representation of the $L$-group that induces an isomorphism
$\PGSp(4,\C)\cong\SO(5,\C)$.
%
%
%
\newpage
\subsection{$\varepsilon$-factors (degree $5$)}\label{epsilon5table}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|}
 \hline&&\text{representation}&a(\rho_5\circ\varphi)&\varepsilon(1/2,\rho_5\circ\varphi)\\\hline\hline
%I
 \mathrm{I}&& \chi_1 \times \chi_2 \rtimes \sigma\ \mathrm{(irreducible)}
 &2a(\chi_1)+2a(\chi_2)&\chi_1(-1)\chi_2(-1)\\\hline
%IIa
 &\mathrm{a}&\chi \St_{\GL(2)} \rtimes \sigma 
 &\chi\text{ unr.}:\,2,\:\chi\text{ ram.}:\,4a(\chi)&1\\\cline{2-5}
%IIb
 \raisebox{2.3ex}[-2.3ex]{II}&\mathrm{b}&\chi\triv_{\GL(2)}\rtimes\sigma&4a(\chi)&1\\\hline
%IIIa
 &\mathrm{a}&\chi \rtimes \sigma \St_{\GSp(2)}
 &2a(\chi)+2&\chi(-1)\\\cline{2-5}
%IIIb
 \raisebox{2.3ex}[-2.3ex]{III}&\mathrm{b}
 &\chi \rtimes \sigma \triv_{\GSp(2)} &2a(\chi)&\chi(-1)\\\hline
%IVa
&\mathrm{a}& \sigma\St_{\GSp(4)}&4&1 \\\cline{2-5}
%IVb
&\mathrm{b}
&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)}) &2&1\\\cline{2-5}
%IVc
\raisebox{2.3ex}[-2.3ex]{IV}&\mathrm{c}
&L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)
&2&1\\\cline{2-5}
%IVd
 &\mathrm{d}&\sigma\triv_{\GSp(4)}&0&1\\\hline
%Va
& \mathrm{a} &
\delta([\xi,\nu \xi], \nu^{-1/2} \sigma)
&\xi\text{ unr.}:\,2,\:\xi\text{ ram.}:\,4a(\xi)&1\\\cline{2-5}
%Vb
& \mathrm{b} &
L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2} \sigma)
&\xi\text{ unr.}:\,2,\:\xi\text{ ram.}:\,4a(\xi)&1\\\cline{2-5}
%Vc
\raisebox{2.3ex}[-2.3ex]{V}& \mathrm{c} &
L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2} \sigma)
&\xi\text{ unr.}:\,2,\:\xi\text{ ram.}:\,4a(\xi)&1\\\cline{2-5}
%Vd
& \mathrm{d} &
L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)
&4a(\xi)&1\\\hline
%VIa
&\mathrm{a}&\tau(S, \nu^{-1/2}\sigma)&&\\\cline{2-3}
%VIb
&\mathrm{b}&\tau(T, \nu^{-1/2}\sigma)&\raisebox{2.3ex}[-2.3ex]{$2$}
&\raisebox{2.3ex}[-2.3ex]{$1$}\\\cline{2-5}
%VIc
\raisebox{2.3ex}[-2.3ex]{VI}&\mathrm{c}
&L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)&2&1\\\cline{2-5}
%VId
&\mathrm{d}
&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)&0&1\\\hline\hline
%VII
\mathrm{VII}&&\chi \rtimes \pi
&2a(\chi)+a({\rm Sym}^2\circ\mu)&\chi(-1)\varepsilon(\frac12,{\rm Sym}^2\circ\mu)\\\hline
%VIIIa
&\mathrm{a} &\tau(S, \pi)
&&\\\cline{2-3}
%VIIIb
\raisebox{2.3ex}[-2.3ex]{VIII}&\mathrm{b} &\tau(T, \pi)
&\raisebox{2.3ex}[-2.3ex]{$a({\rm Sym}^2\circ\mu)$}
&\raisebox{2.3ex}[-2.3ex]{$\varepsilon(\frac12,{\rm Sym}^2\circ\mu)$}\\
\hline
%IXa
&&&
\xi\text{ unr.}:a({\rm Sym}^2\circ\mu)+2&\\\cline{4-4}
{\rm IX}&\raisebox{2.3ex}[-2.3ex]{a}&\raisebox{2.3ex}[-2.3ex]{$\delta(\nu\xi,\nu^{-1/2}\pi)$}&
\xi\text{ ram.}:2a(\xi)+a({\rm Sym}^2\circ\mu)&
\raisebox{2.3ex}[-2.3ex]{$\xi(-1)\varepsilon(\frac12,{\rm Sym}^2\circ\mu)$}\\\cline{2-5}
%IXb
&\mathrm{b}&L(\nu\xi,\nu^{-1/2}\pi)
&2a(\xi)+a({\rm Sym}^2\circ\mu)&\xi(-1)\varepsilon(\frac12,{\rm Sym}^2\circ\mu)
\\\hline\hline
%X
\mathrm{X}&&\pi\rtimes\sigma
&2a(\mu)&\det(\mu)(-1)\\\hline
%XI
&\mathrm{a}&\delta(\nu^{1/2}\pi,\nu^{-1/2}\sigma)&2a(\mu)&1\\\cline{2-5}
%XI
\raisebox{2.3ex}[-2.3ex]{XI}&\mathrm{b}
&L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
&2a(\mu)&1\\\hline
\end{array}
$$

\vspace{3ex}
$\varphi=(\rho,N)$ is the $L$-parameter of the representation, as listed in
Table \ref{locallanglandstable}. The homomorphism $\rho_5:\:\GSp(4,\C)\rightarrow\SO(5,\C)$
is a five-dimensional representation of the $L$-group that induces an isomorphism
$\PGSp(4,\C)\cong\SO(5,\C)$. In this table we do not require that the
representations have trivial central character.
%
%
%
\newpage
\section{Jacquet modules}
\subsection{Jacquet modules: The Siegel parabolic}\label{JacquetSiegeltable}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|}
 \hline&&\mbox{representation}
   &{\rm s.s.}(s_2(\pi))&\#\\\hline\hline
% I
  &&&(\chi_1\times\chi_2)\otimes\sigma+
   (\chi_1^{-1}\times\chi_2^{-1})\otimes\chi_1\chi_2\sigma&\\
   \raisebox{2.5ex}[-2.5ex]{I}&&\raisebox{2.5ex}[-2.5ex]
   {$\chi_1\times\chi_2\rtimes\sigma\quad\mbox{(irreducible)}$}
   &+(\chi_1\times\chi_2^{-1})\otimes\chi_2\sigma
    +(\chi_2\times\chi_1^{-1})\otimes\chi_1\sigma&\raisebox{2.5ex}[-2.5ex]{$4$}\\\hline
% II
  &&&\chi\St_{\GL(2)}\otimes\sigma
   +\chi^{-1}\St_{\GL(2)}\otimes\chi^2\sigma&\\
   &\raisebox{2.5ex}[-2.5ex]{a}&\raisebox{2.5ex}[-2.5ex]{$\chi\St_{\GL(2)}\rtimes\sigma$}
   &+(\chi\nu^{1/2}\times\chi^{-1}\nu^{1/2})\otimes\chi\nu^{-1/2}\sigma
   &\raisebox{2.5ex}[-2.5ex]{$3$}\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{II}&
   &&\chi\triv_{\GL(2)}\otimes\sigma
   +\chi^{-1}\triv_{\GL(2)}\otimes\chi^2\sigma&\\
   &\raisebox{2.5ex}[-2.5ex]{b}&\raisebox{2.5ex}[-2.5ex]{$\chi\triv_{\GL(2)}\rtimes\sigma$}
   &+(\chi\nu^{-1/2}\times\chi^{-1}\nu^{-1/2})\otimes\chi\nu^{1/2}\sigma
   &\raisebox{2.5ex}[-2.5ex]{$3$}\\\hline
% III
  &{\rm a}&\chi\rtimes\sigma\St_{\GSp(2)}
   &(\chi\times\nu)\otimes\sigma\nu^{-1/2}+(\nu\times\chi^{-1})\otimes\chi\sigma\nu^{-1/2}
   &2\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{III}&{\rm b}
   &\chi\rtimes\sigma\triv_{\GSp(2)}
   &(\chi\times\nu^{-1})\otimes\sigma\nu^{1/2}
   +(\nu^{-1}\times\chi^{-1})\otimes\chi\sigma\nu^{1/2}&2\\\hline
% IV
  &{\rm a}&\sigma\St_{\GSp(4)}&\nu^{3/2}\St_{\GL(2)}\otimes\nu^{-3/2}\sigma
  &1\\\cline{2-5}
  &{\rm b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})
   &\nu^{3/2}\triv_{\GL(2)}\otimes\nu^{-3/2}\sigma
   +(\nu\times\nu^{-2})\otimes\nu^{1/2}\sigma&2\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{IV}&{\rm c}
   &L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)&\nu^{-3/2}\St_{\GL(2)}\otimes\nu^{3/2}\sigma
   +(\nu^2\times\nu^{-1})\otimes\nu^{-1/2}\sigma&2\\\cline{2-5}
  &{\rm d}&\sigma\triv_{\GSp(4)}&\nu^{-3/2}\triv_{\GL(2)}\otimes\nu^{3/2}\sigma&1\\\hline
% V
  &{\rm a}&\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)
   &\nu^{1/2}\xi\St_{\GL(2)}\otimes\nu^{-1/2}\sigma
   +\nu^{1/2}\xi\St_{\GL(2)}\otimes\xi\nu^{-1/2}\sigma&2\\\cline{2-5}
  &{\rm b}&L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2}\sigma)
   &\nu^{-1/2}\xi\St_{\GL(2)}\otimes\nu^{1/2}\sigma
   +\nu^{1/2}\xi\triv_{\GL(2)}\otimes\xi\nu^{-1/2}\sigma&2\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{V}&{\rm c}
   &L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2}\sigma)
   &\nu^{-1/2}\xi\St_{\GL(2)}\otimes\xi\nu^{1/2}\sigma
   +\nu^{1/2}\xi\triv_{\GL(2)}\otimes\nu^{-1/2}\sigma&2\\\cline{2-5}
  &{\rm d}&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)
   &\nu^{-1/2}\xi\triv_{\GL(2)}\otimes\xi\nu^{1/2}\sigma
   +\nu^{-1/2}\xi\triv_{\GL(2)}\otimes\nu^{1/2}\sigma&2\\\hline
% VI
  &{\rm a}&\tau(S,\nu^{-1/2}\sigma)&
  2\cdot(\nu^{1/2}\St_{\GL(2)}\otimes\nu^{-1/2}\sigma)
  +\nu^{1/2}\triv_{\GL(2)}\otimes\nu^{-1/2}\sigma&3\\\cline{2-5}
  &{\rm b}&\tau(T,\nu^{-1/2}\sigma)
   &\nu^{1/2}\triv_{\GL(2)}\otimes\nu^{-1/2}\sigma&1\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{VI}&{\rm c}
   &L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
   &\nu^{-1/2}\St_{\GL(2)}\otimes\nu^{1/2}\sigma&1\\\cline{2-5}
  &{\rm d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
   &2\cdot(\nu^{-1/2}\triv_{\GL(2)}\otimes\nu^{1/2}\sigma)
   +\nu^{-1/2}\St_{\GL(2)}\otimes\nu^{1/2}\sigma&3\\\hline\hline
% VII
  {\rm VII}&&\chi\rtimes\pi
   &0&0\\\hline
% VIII
  &{\rm a}&\tau(S,\pi)&0&0\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{VIII}&{\rm b}
   &\tau(T,\pi)&0&0\\\hline
% IX
  &{\rm a}&\delta(\nu\xi,\nu^{-1/2}\pi)
   &0&0\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{IX}&{\rm b}
   &L(\nu\xi,\nu^{-1/2}\pi)
   &0&0\\\hline\hline
% X
  {\rm X}&&\pi\rtimes\sigma
   &\pi\otimes\sigma+\tilde\pi\otimes\omega_\pi\sigma&2\\\hline
% XI
  &{\rm a}&\delta(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &\nu^{1/2}\pi\otimes\nu^{-1/2}\sigma&1\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{XI}&{\rm b}
   &L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &\nu^{-1/2}\pi\otimes\nu^{1/2}\sigma&1\\\hline
 \end{array}
$$

\vspace{3ex}
Listed are the semisimplifications of the Jacquet modules of all non-supercuspidal
representations with respect to the unipotent radical of the Siegel parabolic
subgroup. The Jacquet modules are representations of $\GL(2,F)\times F^\times$.
The last column lists the number of irreducible constituents.
%
%
%
\newpage
\subsection{Jacquet modules: The Klingen parabolic}\label{JacquetKlingentable}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|}
 \hline&&\mbox{representation}
   &{\rm s.s.}(s_1(\pi))&\#\\\hline\hline
% I
  &&&\chi_1\otimes(\chi_2\rtimes\sigma)+\chi_2\otimes(\chi_1\rtimes\sigma)&\\
   \raisebox{2.5ex}[-2.5ex]{I}&&\raisebox{2.5ex}[-2.5ex]
   {$\chi_1\times\chi_2\rtimes\sigma\quad\mbox{(irreducible)}$}
   &+\chi_2^{-1}\otimes(\chi_1\rtimes\chi_2\sigma)+
   \chi_1^{-1}\otimes(\chi_2\rtimes\chi_1\sigma)&\raisebox{2.5ex}[-2.5ex]{$4$}\\\hline
% II
  &{\rm a}&\chi\St_{\GL(2)}\rtimes\sigma
   &\chi\nu^{1/2}\otimes(\chi\nu^{-1/2}\!\rtimes\sigma)
   +\chi^{-1}\nu^{1/2}\otimes(\chi\nu^{1/2}\!\rtimes\chi\nu^{-1/2}\sigma)&2\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{II}&{\rm b}
   &\chi\triv_{\GL(2)}\rtimes\sigma
   &\!\!\chi\nu^{-1/2}\otimes(\chi\nu^{1/2}\!\rtimes\!\sigma)
   +\chi^{-1}\nu^{-1/2}\otimes(\chi\nu^{-1/2}\!\rtimes\!\chi\nu^{1/2}\sigma)\!\!&2\\\hline
% III
  &&&\chi\otimes\sigma\St_{\GSp(2)}+\chi^{-1}\otimes\chi\sigma\St_{\GSp(2)}&\\
   &\raisebox{2.5ex}[-2.5ex]{a}&\raisebox{2.5ex}[-2.5ex]{$\chi\rtimes\sigma\St_{\GSp(2)}$}
   &+\nu\otimes(\chi\rtimes\sigma\nu^{-1/2})
   &\raisebox{2.5ex}[-2.5ex]{$3$}\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{III}&
   &&\chi\otimes\sigma\triv_{\GSp(2)}+\chi^{-1}\otimes\chi\sigma\triv_{\GSp(2)}&\\
   &\raisebox{2.5ex}[-2.5ex]{b}&\raisebox{2.5ex}[-2.5ex]{$\chi\rtimes\sigma\triv_{\GSp(2)}$}
   &+\nu^{-1}\otimes(\chi\rtimes\sigma\nu^{1/2})
   &\raisebox{2.5ex}[-2.5ex]{$3$}\\\hline
% IV
  &{\rm a}&\sigma\St_{\GSp(4)}&\nu^2\otimes\nu^{-1}\sigma\St_{\GSp(2)}&1\\\cline{2-5}
  &{\rm b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})&\nu^{-2}\otimes\nu\sigma\St_{\GSp(2)}
   +\nu\otimes(\nu^2\rtimes\nu^{-3/2}\sigma)&2\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{IV}&{\rm c}
   &L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)&\nu^2\otimes\nu^{-1}\sigma\triv_{\GSp(2)}
   +\nu^{-1}\otimes(\nu^2\rtimes\nu^{-1/2}\sigma)&2\\\cline{2-5}
  &{\rm d}&\sigma\triv_{\GSp(4)}&\nu^{-2}\otimes\nu\sigma\triv_{\GSp(2)}&1\\\hline
% V
  &{\rm a}&\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)
   &\nu\xi\otimes(\xi\rtimes\nu^{-1/2}\sigma)&1\\\cline{2-5}
  &{\rm b}&L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2}\sigma)
   &\xi\otimes(\nu\xi\rtimes\xi\nu^{-1/2}\sigma)&1\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{V}&{\rm c}
   &L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2}\sigma)
   &\xi\otimes(\nu\xi\rtimes\nu^{-1/2}\sigma)&1\\\cline{2-5}
  &{\rm d}&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)
   &\nu^{-1/2}\xi\otimes(\xi\rtimes\nu^{1/2}\sigma)&1\\\hline
% VI
  &{\rm a}&\tau(S,\nu^{-1/2}\sigma)&\nu\otimes(1_{F^\times}\rtimes\nu^{-1/2}\sigma)
  +1_{F^\times}\otimes\sigma\St_{\GSp(2)}&2\\\cline{2-5}
  &{\rm b}&\tau(T,\nu^{-1/2}\sigma)&1_{F^\times}\otimes\sigma\St_{\GSp(2)}&1\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{VI}&{\rm c}
   &L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
   &1_{F^\times}\otimes\sigma\triv_{\GSp(2)}&1\\\cline{2-5}
  &{\rm d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
   &1_{F^\times}\otimes\sigma\triv_{\GSp(2)}
   +\nu^{-1}\otimes(1_{F^\times}\rtimes\nu^{1/2}\sigma)&2\\\hline\hline
% VII
  {\rm VII}&&\chi\rtimes\pi
   &\chi\otimes\pi+\chi^{-1}\otimes\chi\pi&2\\\hline
% VIII
  &{\rm a}&\tau(S,\pi)
   &1_{F^\times}\otimes\pi&1\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{VIII}&{\rm b}
   &\tau(T,\pi)
   &1_{F^\times}\otimes\pi&1\\\hline
% IX
  &{\rm a}&\delta(\nu\xi,\nu^{-1/2}\pi)
   &\nu\xi\otimes\nu^{-1/2}\pi&1\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{IX}&{\rm b}
   &L(\nu\xi,\nu^{-1/2}\pi)
   &\nu^{-1}\xi\otimes\nu^{1/2}\pi&1\\\hline\hline
% X
  {\rm X}&&\pi\rtimes\sigma
   &0&0\\\hline
% XI
  &{\rm a}&\delta(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &0&0\\\cline{2-5}
  \raisebox{2.5ex}[-2.5ex]{XI}&{\rm b}
   &L(\nu^{1/2}\pi,\nu^{-1/2}\sigma)
   &0&0\\\hline
 \end{array}
$$

\vspace{3ex}
Listed are the semisimplifications of the Jacquet modules of all non-supercuspidal
representations with respect to the unipotent radical of the Klingen parabolic
subgroup. The Jacquet modules are representations of $F^\times\times\GSp(2,F)$.
Note that $\GSp(2,F)=\GL(2,F)$; to translate into standard $\GL(2)$ notation,
use the formula $\chi\rtimes\sigma=\chi\sigma\times\sigma$.
The last column lists the number of irreducible constituents.
%
%
%
\newpage
\section{Invariant vectors}
\subsection{Iwahori-spherical representations: Dimensions of spaces of parahori-invariant vectors}\label{parahoritable}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|c|c|c|}
  \hline&&\mbox{representation}&a&\varepsilon(1/2,\pi)
   &\begin{minipage}{5.6ex}\begin{center}$K$\end{center}\end{minipage}
   &\begin{minipage}{5.6ex}\begin{center}$\K{}$\end{center}\end{minipage}
   &\begin{minipage}{5.6ex}\begin{center}$\Kl{}$\end{center}\end{minipage}
   &\begin{minipage}{5.6ex}\begin{center}$\Si{}$\end{center}\end{minipage}
   &\begin{minipage}{5.6ex}\begin{center}$I$\end{center}\end{minipage}
   \\\hline\hline
% I
  {\rm I}&&\chi_1\times\chi_2\rtimes\sigma\quad
   \mbox{(irreducible)}&0&1&1&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array} &4&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 4 \\ {\scriptscriptstyle ++}\\ {\scriptscriptstyle --}  \end{array} 
    &\!\!\!\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 8 \\ {\scriptscriptstyle ++++}\\ {\scriptscriptstyle ----}\end{array}\!\!\! \\\hline
% II
  &{\rm a}&\chi\St_{\GL(2)}\rtimes\sigma&1&-(\sigma\chi)(\varpi)&0&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -}\end{array}&2&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}
   &\!\!\!\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 4 \\ {\scriptscriptstyle +---} \end{array}\!\!\!
   \\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{II}&{\rm b}
   &\chi\triv_{\GL(2)}\rtimes\sigma&0&1&1
  &\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}&2&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 3 \\ {\scriptscriptstyle ++-} \end{array}
  &\!\!\!\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 4 \\ {\scriptscriptstyle +++-} \end{array}\!\!\!\\\hline
% III
  &{\rm a}&\chi\rtimes\sigma\St_{\GSp(2)}&2&1
   &0&0&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}
   &\!\!\!\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 4 \\ {\scriptscriptstyle ++--} \end{array}\!\!\!\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{III}&{\rm b}
   &\chi\rtimes\sigma\triv_{\GSp(2)}
   &0&1&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}&3&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}
   &\!\!\!\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 4 \\ {\scriptscriptstyle ++--} \end{array}\!\!\!\\\hline
% IV
  &{\rm a}&\sigma\St_{\GSp(4)}&3&-\sigma(\varpi)&0&0&0&0&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}\\\cline{2-10}
  &{\rm b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})&2&1&0&0&1&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 3 \\ {\scriptscriptstyle ++-} \end{array}\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{IV}&{\rm c}
   &L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)&1&-\sigma(\varpi)&0&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}&2&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 3 \\ {\scriptscriptstyle +--} \end{array}
   \\\cline{2-10}
  &{\rm d}&\sigma\triv_{\GSp(4)}&0&1&1&
  \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}\\\hline
% V
  &{\rm a}&\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)&2&-1
   &0&0&1&0&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}\\\cline{2-10}
  &{\rm b}&L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2}\sigma)
   &1&\sigma(\varpi)&0&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle ++} \end{array}\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{V}&{\rm c}
   &\!\!L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2}\sigma)\!\!
   &1&-\sigma(\varpi)&0&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle --} \end{array}
   \\\cline{2-10}
  &{\rm d}&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)&0&1
   &1&0&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}\\\hline
% VI
  &{\rm a}&\tau(S,\nu^{-1/2}\sigma)&2&1&0&0&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 3 \\ {\scriptscriptstyle +--} \end{array}
   \\\cline{2-10}
  &{\rm b}&\tau(T,\nu^{-1/2}\sigma)&2&1&0&0&0&
   \renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}
  &\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}
  \\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{VI}&{\rm c}
   &L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
   &1&-\sigma(\varpi)&0&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}&1&0&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle -} \end{array}
  \\\cline{2-10}
  &{\rm d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
   &0&1&1&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 1 \\ {\scriptscriptstyle +} \end{array}&2&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 2 \\ {\scriptscriptstyle +-} \end{array}&\renewcommand{\arraystretch}{0.5}\begin{array}[t]{c} 3 \\ {\scriptscriptstyle ++-} \end{array}\\\hline
 \end{array}
$$

\vspace{3ex}
\begin{minipage}{80ex}
\begin{tabbing}
$\Kl{}$:\;\=\kill
$I$:\>The Iwahori subgroup.\\
$\Si{}$:\>The Siegel congruence subgroup\quad($=P_1$)\\
$\Kl{}$:\>The Klingen congruence subgroup\quad($=P_2$)\\
$\K{}$:\>The paramodular group\quad($=P_{02}$)\\
$K$:\>$\GSp(4,\OF)$\quad($=P_{12}$)\\
$a$:\>Exponent of the conductor of the $L$-parameter\\
$\varpi$:\>Uniformizer
\end{tabbing}
\end{minipage}
\begin{minipage}{50ex}
\setlength{\unitlength}{4ex}
\hspace{10ex}\begin{picture}(10,2)
 \put(1.1,1.1){\line(1,0){4}}
 \put(1.1,1.2){\line(1,0){4}}
 \multiput(1,1)(2,0){3}{$\bullet$}
 \put(0.9,0.6){$s_0$}
 \put(2.9,0.6){$s_1$}
 \put(4.9,0.6){$s_2$}
\end{picture}
\end{minipage}
\nl
The Atkin--Lehner eigenvalues
listed in this table are correct if one assumes that
\begin{itemize}
 \item in group II, where the central character is $\chi^2\sigma^2$,
  the character $\chi\sigma$ is trivial.
 \item in groups IV, V and VI, where the central character is $\sigma^2$,
  the character $\sigma$ itself is trivial.
\end{itemize}
If these assumptions are not met, then one has to interchange
all plus and minus signs.
%
%
%
\subsection{Iwahori-spherical representations: Dimensions of spaces of paramodular vectors}\label{iwahoriparamodulartable}
$$\renewcommand{\arraystretch}{1.5}
 \begin{array}{|c|c|c|c|c|c|c|c|c|c|}
  \hline&&\mbox{representation}
   &a&\varepsilon(1/2,\pi)&V(0)&V(1)&V(2)&V(3)&V(n)\\\hline\hline
% I
  {\rm I}&&\chi_1\times\chi_2\rtimes\sigma\quad
   \mbox{(irreducible)}&0&1&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&
   \begin{minipage}[t]{2.4ex}$\:2$\\[-1.2ex]$\scriptscriptstyle+-$
   \end{minipage}&\begin{minipage}[t]{5.0ex}$\;\;\,4$\\[-1.2ex]
   $\scriptscriptstyle+++-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{3.8ex}
   $\;\:6$\\[-1.2ex]\rule{0.09ex}{0ex}$\scriptscriptstyle+++$
   \\[-1.5ex]$\scriptscriptstyle---$\\[-2.0ex]\rule{0ex}{0ex}\end{minipage}
   &\big[\frac{(n+2)^2}4\big]\\\hline
% II
  &{\rm a}&\chi\St_{\GL(2)}\rtimes\sigma&1&-(\sigma\chi)(\varpi)&0&
   \begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle-$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle+-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{5.0ex}$\;\;\,4$\\[-1.2ex]
   $\scriptscriptstyle++--$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{(n+1)^2}4\big]\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{II}&{\rm b}
   &\chi\triv_{\GL(2)}\rtimes\sigma&0&1&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&
   \begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle++$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle++$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n+2}2\big]\\\hline
% III
  &{\rm a}&\chi\rtimes\sigma\St_{\GSp(2)}
   &2&1&0&0&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle+-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n^2}4\big]\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{III}&{\rm b}
   &\chi\rtimes\sigma\triv_{\GSp(2)}
   &0&1&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle+-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{3.7ex}$\;\:3$\\[-1.2ex]
   $\scriptscriptstyle++-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{5.0ex}$\;\;\,4$\\[-1.2ex]
   $\scriptscriptstyle++--$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&n+1\\\hline
% IV
  &{\rm a}&\sigma\St_{\GSp(4)}&3&-\sigma(\varpi)&0&0&0
   &\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle-$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\big[\frac{(n-1)^2}4\big]\\\cline{2-10}
  &{\rm b}&L(\nu^2,\nu^{-1}\sigma\St_{\GSp(2)})&2&1&0&0
   &\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\big[\frac n2\big]\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{IV}&{\rm c}
   &L(\nu^{3/2}\St_{\GL(2)},\nu^{-3/2}\sigma)&1&-\sigma(\varpi)&0&
   \begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle-$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle+-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{3.8ex}$\;\:3$\\[-1.2ex]
   $\scriptscriptstyle+--$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&n\\\cline{2-10}
  &{\rm d}&\sigma\triv_{\GSp(4)}&0&1&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&
   \begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&1\\\hline
% V
  &{\rm a}&\delta([\xi,\nu\xi],\nu^{-1/2}\sigma)
   &2&-1&0&0&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle-$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle+-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n^2}4\big]\\\cline{2-10}
  &{\rm b}&L(\nu^{1/2}\xi\St_{\GL(2)},\nu^{-1/2}\sigma)
   &1&\sigma(\varpi)&0&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle++$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n+1}2\big]\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{V}&{\rm c}
   &L(\nu^{1/2}\xi\St_{\GL(2)},\xi\nu^{-1/2}\sigma)
   &1&-\sigma(\varpi)&0&\begin{minipage}[t]{1.3ex}
   $1$\\[-1.2ex]$\scriptscriptstyle-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.5ex}
   $\;2$\\[-1.2ex]$\scriptscriptstyle--$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n+1}2\big]\\\cline{2-10}
  &{\rm d}&L(\nu\xi,\xi\rtimes\nu^{-1/2}\sigma)
   &0&1&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &0&\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&0&\frac{1+(-1)^n}2\\\hline
% VI
  &{\rm a}&\tau(S,\nu^{-1/2}\sigma)&2&1&0&0
   &\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]$\scriptscriptstyle+$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle+-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n^2}4\big]\\\cline{2-10}
  &{\rm b}&\tau(T,\nu^{-1/2}\sigma)&2&1&0&0&0&0&0\\\cline{2-10}
  \raisebox{2.5ex}[-2.5ex]{VI}&{\rm c}
   &L(\nu^{1/2}\St_{\GL(2)},\nu^{-1/2}\sigma)
   &1&-\sigma(\varpi)&0&\begin{minipage}[t]{1.3ex}$1$\\
   [-1.2ex]$\scriptscriptstyle-$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\begin{minipage}[t]{1.2ex}$1$\\[-1.2ex]$\scriptscriptstyle-$\\[-2ex]
   \rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.5ex}
   $\;2$\\[-1.2ex]$\scriptscriptstyle--$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n+1}2\big]\\\cline{2-10}
  &{\rm d}&L(\nu,1_{F^\times}\rtimes\nu^{-1/2}\sigma)
   &0&1&\begin{minipage}[t]{1.3ex}$1$\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{1.2ex}1\\[-1.2ex]
   $\scriptscriptstyle+$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle++$\\[-2ex]\rule{0pt}{0pt}\end{minipage}&\begin{minipage}[t]{2.4ex}
   $\:2$\\[-1.2ex]$\scriptscriptstyle++$\\[-2ex]\rule{0pt}{0pt}\end{minipage}
   &\big[\frac{n+2}2\big]\\\hline
 \end{array}
$$

\vspace{3ex}
Here $V(n)$ is the subspace of vectors invariant under the
paramodular group of level $n$, defined as the
set of all $g\in\GSp(4,F)$ such that
$$
 g\in\left[\begin{matrix}\OF&\OF&\OF&\p^{-n}\\\p^n&\OF&\OF&\OF\\
 \p^n&\OF&\OF&\OF\\\p^n&\p^n&\p^n&\OF\end{matrix}\right]
 \qquad\mbox{and}\qquad\det(g)\in\OF^\times.
$$
The Atkin--Lehner eigenvalues
listed in this table are correct if one assumes that
\begin{itemize}
 \item in group II, where the central character is $\chi^2\sigma^2$,
  the character $\chi\sigma$ is trivial.
 \item in groups IV, V and VI, where the central character is $\sigma^2$,
  the character $\sigma$ itself is trivial.
\end{itemize}
If these assumptions are not met, then one has to interchange
the plus and minus signs in the $V(1)$ and the $V(3)$ column.

\end{document}
 
