$ \newcommand{\SL}{{\rm SL}} \newcommand{\SU}{{\rm SU}} \newcommand{\GL}{{\rm GL}} \newcommand{\GSp}{{\rm GSp}} \newcommand{\PGSp}{{\rm PGSp}} \newcommand{\SO}{{\rm SO}} \newcommand{\Sp}{{\rm Sp}} \newcommand{\triv}{1} \newcommand{\p}{\mathfrak{p}} \newcommand{\A}{\mathbb{A}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\St}{\mathrm{St}} \newcommand{\sc}{\mathrm{sc}} $

Automorphic Representations of GSp(4)

Elliptic newforms of level dividing 32 and their Saito-Kurokawa lifts

The following table shows all irreducible, admissible representations $\pi$ of $\GL(2,\Q_2)$ with trivial central character and conductor exponent $a(\pi)\leq7$.
orbittype of representation$\pi$$a(\pi)$$\varepsilon(\frac12,\pi)$
$[\chi\times\chi^{-1}]$unramified principal series$\chi\times\chi^{-1}$01
$(\chi\times\chi^{-1})\otimes\chi_5$01
$(\chi\times\chi^{-1})\otimes\chi_3$4-1
$(\chi\times\chi^{-1})\otimes\chi_7$4-1
$(\chi\times\chi^{-1})\otimes\chi_2$61
$(\chi\times\chi^{-1})\otimes\chi_{2\cdot5}$61
$(\chi\times\chi^{-1})\otimes\chi_{2\cdot3}$6-1
$(\chi\times\chi^{-1})\otimes\chi_{2\cdot7}$6-1
$[\St_{\GL(2)}]$Steinberg$\St_{\GL(2)}$1-1
$\St_{\GL(2)}\otimes\chi_5$11
$\St_{\GL(2)}\otimes\chi_3$4-1
$\St_{\GL(2)}\otimes\chi_7$4-1
$\St_{\GL(2)}\otimes\chi_2$61
$\St_{\GL(2)}\otimes\chi_{2\cdot5}$61
$\St_{\GL(2)}\otimes\chi_{2\cdot3}$6-1
$\St_{\GL(2)}\otimes\chi_{2\cdot7}$6-1
$[\tau_2]$monomial supercuspidal from $\Q_2(\sqrt{5})$$\tau_2$2-1
$\tau_2\otimes\chi_3$4-1
$\tau_2\otimes\chi_2$61
$\tau_2\otimes\chi_{2\cdot3}$6-1
$[\tau_3]$exceptional supercuspidal$\tau_3$31
$\tau_3\otimes\chi_5$3-1
$\tau_3\otimes\chi_3$41
$\tau_3\otimes\chi_7$41
$\tau_3\otimes\chi_2$61
$\tau_3\otimes\chi_{2\cdot5}$61
$\tau_3\otimes\chi_{2\cdot3}$6-1
$\tau_3\otimes\chi_{2\cdot7}$6-1
$[\tau_{5a}]$monomial supercuspidal from $\Q_2(\sqrt{3})$$\tau_{5a}$51
$\tau_{5a}\otimes\chi_5$5-1
$\tau_{5a}\otimes\chi_2$61
$\tau_{5a}\otimes\chi_{2\cdot5}$61
$[\tau_{5b}]$monomial supercuspidal from $\Q_2(\sqrt{7})$$\tau_{5b}$51
$\tau_{5b}\otimes\chi_5$5-1
$\tau_{5b}\otimes\chi_2$6-1
$\tau_{5b}\otimes\chi_{2\cdot5}$6-1
$[\tau_7]$exceptional supercuspidal$\tau_7$71
$\tau_7\otimes\chi_5$7-1
$\tau_7\otimes\chi_3$7-1
$\tau_7\otimes\chi_7$71
$\tau_7\otimes\chi_2$7-1
$\tau_7\otimes\chi_{2\cdot5}$71
$\tau_7\otimes\chi_{2\cdot3}$7-1
$\tau_7\otimes\chi_{2\cdot7}$71
The table below lists the cuspidal automorphic representations of $\GL(2,\A)$ with trivial central character whose conductor divides 32 and whose archimedean component is a discrete series representation of weight $\leq8$. To each $\pi$ and choice of quadratic Hecke character $\chi$ there is associated a finite packet of automorphic representations $\Pi$ of $\GSp(4,\A)$ with trivial central character such that, up to finitely many Euler factors, $L(s,\Pi)=L(s,\pi)L(s-\frac12,\chi)L(s+\frac12,\chi)$. These are the packets of Arthur type $\mathrm{\bf(P)}$. The elements of the packet are parametrized by subsets $S$, satisfying a parity condition, of the set of places where $\pi$ is square-integrable. The table below lists the possibilities for which $S$ contains the archimedean place, in which case $S$ is uniquely determined. The archimedean component $\Pi_\infty$ is then a (limit of) holomorphic discrete series representation. Tables like the one below can be constructed for any weight in a mechanical way from the data given in the LMFDB, because this data allows for the determination of the 2-component of the underlying automorphic representation for any eigenform in $S_k(\Gamma_0(N))$, where $N\mid32$.
$\GL(2)$ $\GSp(4)$
weightlevel $\pi$ size $\pi_2$ $a(\pi_2)$ $\varepsilon(\frac12,\pi_2)$ $\varepsilon(\frac12,\pi)$ $\chi$$S$weightconductor $\Pi=\Pi(\pi,\chi,S)$ $\Pi_2$$a(\Pi_2)$$\varepsilon(\frac12,\Pi_2)$$\varepsilon(\frac12,\Pi)$para
232 32.2.a.a 1 $\tau_{5b}\otimes\chi_5$ 5 -1 1 1$\{\infty,2\}$264 P.64.2.0.a XIa*611
$\xi$$\{\infty,2\}$512 P.512.2.0.a XIa*911
48 8.4.a.a 1 $\tau_3$ 3 1 1 1$\{\infty,2\}$316 P.16.3.0.a XIa*4-11
$\xi$$\{\infty,2\}$128 P.128.3.0.a XIa*7-11
416 16.4.a.a 1 $\tau_3\otimes\chi_7$ 4 1 1 1$\{\infty,2\}$332 P.32.3.0.a XIa*5-11
$\xi$$\{\infty,2\}$256 P.256.3.0.a XIa*8-11
432 32.4.a.a 1 $\tau_{5a}\otimes\chi_5$ 5 -1 -1 1$\{\infty\}$332 P.32.3.0.b XIb5-1132
$\xi$$\{\infty,2\}$512 P.512.3.0.a XIa*9-11
432 32.4.a.b 1 $\tau_{5b}$ 5 1 1 1$\{\infty,2\}$364 P.64.3.0.a XIa*6-11
$\xi$$\{\infty,2\}$512 P.512.3.0.b XIa*9-11
432 32.4.a.c 1 $\tau_{5a}$ 5 1 1 1$\{\infty,2\}$364 P.64.3.0.b XIa*6-11
$\xi$$\{\infty\}$512 P.512.3.0.c XIb9-11
64 4.6.a.a 1 $\tau_2$ 2 -1 1 1$\{\infty,2\}$48 P.8.4.0.a XIa*311
$\xi$$\{\infty,2\}$64 P.64.4.0.a XIa*611
68 8.6.a.a 1 $\tau_3\otimes\chi_5$ 3 -1 1 1$\{\infty,2\}$416 P.16.4.0.a XIa*411
$\xi$$\{\infty\}$128 P.128.4.0.a XIb711
616 16.6.a.a 1 $\tau_3\otimes\chi_3$ 4 1 -1 1$\{\infty\}$416 P.16.4.0.b XIb41116
$\xi$$\{\infty,2\}$256 P.256.4.0.a XIa*8-1-1
616 16.6.a.b 1 $\tau_2\otimes\chi_3$ 4 -1 1 1$\{\infty,2\}$432 P.32.4.0.a XIa*511
$\xi$$\{\infty,2\}$256 P.256.4.0.b XIa*811
632 32.6.a.a 1 $\tau_{5a}$ 5 1 -1 1$\{\infty\}$432 P.32.4.0.b XIb51132
$\xi$$\{\infty,2\}$512 P.512.4.0.a XIa*9-1-1
632 32.6.a.b 1 $\tau_{5b}$ 5 1 -1 1$\{\infty\}$432 P.32.4.0.c XIb51132
$\xi$$\{\infty\}$256 P.256.4.0.c XIb8-1-1
632 32.6.a.c 1 $\tau_{5a}\otimes\chi_5$ 5 -1 1 1$\{\infty,2\}$464 P.64.4.0.b XIa*611
$\xi$$\{\infty\}$512 P.512.4.0.b XIb911
632 32.6.a.d 2 $\tau_{5b}\otimes\chi_5$ 5 -1 1 1$\{\infty,2\}$464 P.64.4.0.c XIa*611
$\xi$$\{\infty,2\}$512 P.512.4.0.c XIa*911
82 2.8.a.a 1 $\St_{\GL(2)}\otimes\chi_5$ 1 1 1 1$\{\infty,2\}$54 P.4.5.0.a Va*2-11
$\xi$$\{\infty\}$32 P.32.5.0.a Vb5-11
88 8.8.a.a 1 $\tau_3\otimes\chi_5$ 3 -1 -1 1$\{\infty\}$58 P.8.5.0.a XIb3-118
$\xi$$\{\infty,2\}$128 P.128.5.0.a XIa*71-1
88 8.8.a.b 1 $\tau_3$ 3 1 1 1$\{\infty,2\}$516 P.16.5.0.a XIa*4-11
$\xi$$\{\infty,2\}$128 P.128.5.0.b XIa*7-11
816 16.8.a.a 1 $\tau_3\otimes\chi_7$ 4 1 1 1$\{\infty,2\}$532 P.32.5.0.b XIa*5-11
$\xi$$\{\infty,2\}$256 P.256.5.0.a XIa*8-11
816 16.8.a.b 1 $\St_{\GL(2)}\otimes\chi_3$ 4 -1 -1 1$\{\infty\}$516 P.16.5.0.b Vb4-1116
$\xi$$\{\infty,2\}$256 P.256.5.0.b Va*81-1
816 16.8.a.c 1 $\tau_3\otimes\chi_3$ 4 1 1 1$\{\infty,2\}$532 P.32.5.0.c XIa*5-11
$\xi$$\{\infty\}$256 P.256.5.0.c XIb8-11
832 32.8.a.a 1 $\tau_{5b}\otimes\chi_5$ 5 -1 -1 1$\{\infty\}$532 P.32.5.0.d XIb5-1132
$\xi$$\{\infty\}$512 P.512.5.0.a XIb91-1
832 32.8.a.b 2 $\tau_{5a}$ 5 1 1 1$\{\infty,2\}$564 P.64.5.0.b XIa*6-11
$\xi$$\{\infty\}$512 P.512.5.0.b XIb9-11
832 32.8.a.c 2 $\tau_{5b}$ 5 1 1 1$\{\infty,2\}$564 P.64.5.0.c XIa*6-11
$\xi$$\{\infty,2\}$512 P.512.5.0.c XIa*9-11
832 32.8.a.d 2 $\tau_{5a}\otimes\chi_5$ 5 -1 -1 1$\{\infty\}$532 P.32.5.0.e XIb5-1132
$\xi$$\{\infty,2\}$512 P.512.5.0.d XIa*91-1