$ \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}} $

Self-twist

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 $a(\pi)\leq7$.

• The representations are organized by twisting orbits. The twisting characters $\chi_y$ are given by the Hilbert symbol $\chi_y(x)=(x,y)$, where $y\in\{5,3,7,2,2\cdot5,2\cdot3,2\cdot7\}$. Note:
  • $\chi_5$ is the unique non-trivial quadratic character of $\Q_2^\times$. We have $\chi_5(2)=-1$.
  • $\chi_3$ and $\chi_7$ are trivial on $1+4\Z_2$, but not on $1+2\Z_2$ ($=\Z_2^\times$). We have $\chi_3(2)=-1$ and $\chi_7(2)=1$.
  • $\chi_{2\cdot y}$ for $y\in\{1,3,5,7\}$ are trivial on $1+8\Z_2$ ($=\Z_2^{\times2}$), but not on $1+4\Z_2$. We have $\chi_2(2)=1$.
• A supercuspidal representation $\pi$ is monomial from a quadratic extension $E/\Q_2$ if it is obtained via automorphic induction from a character of $E^\times$, i.e., its Langlands parameter $W_\Q\to\GL(2,\C)$ is induced from the index-$2$ subgroup $W_E$. This happens if and only if $\pi\otimes\chi\cong\pi$, where $\chi$ is the quadratic character of $\Q_2^\times$ corresponding to $E$.
• Supercuspidal representations that are not monomial are called exceptional. Their Langlands parameters are primitive (not induced).
• $\tau_3$ is the $2$-component of the automorphic representation corresponding to this elliptic curve.
• $\tau_7$ is the $2$-component of the automorphic representation corresponding to this elliptic curve.

orbit	type of representation	$\pi$	$a(\pi)$	$\varepsilon(\frac12,\pi)$
$[\chi\times\chi^{-1}]$	unramified principal series	$\chi\times\chi^{-1}$	0	1
		$(\chi\times\chi^{-1})\otimes\chi_5$	0	1
		$(\chi\times\chi^{-1})\otimes\chi_3$	4	-1
		$(\chi\times\chi^{-1})\otimes\chi_7$	4	-1
		$(\chi\times\chi^{-1})\otimes\chi_2$	6	1
		$(\chi\times\chi^{-1})\otimes\chi_{2\cdot5}$	6	1
		$(\chi\times\chi^{-1})\otimes\chi_{2\cdot3}$	6	-1
		$(\chi\times\chi^{-1})\otimes\chi_{2\cdot7}$	6	-1
$[\St_{\GL(2)}]$	Steinberg	$\St_{\GL(2)}$	1	-1
		$\St_{\GL(2)}\otimes\chi_5$	1	1
		$\St_{\GL(2)}\otimes\chi_3$	4	-1
		$\St_{\GL(2)}\otimes\chi_7$	4	-1
		$\St_{\GL(2)}\otimes\chi_2$	6	1
		$\St_{\GL(2)}\otimes\chi_{2\cdot5}$	6	1
		$\St_{\GL(2)}\otimes\chi_{2\cdot3}$	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
		$\tau_2\otimes\chi_2$	6	1
		$\tau_2\otimes\chi_{2\cdot3}$	6	-1
$[\tau_3]$	exceptional supercuspidal	$\tau_3$	3	1
		$\tau_3\otimes\chi_5$	3	-1
		$\tau_3\otimes\chi_3$	4	1
		$\tau_3\otimes\chi_7$	4	1
		$\tau_3\otimes\chi_2$	6	1
		$\tau_3\otimes\chi_{2\cdot5}$	6	1
		$\tau_3\otimes\chi_{2\cdot3}$	6	-1
		$\tau_3\otimes\chi_{2\cdot7}$	6	-1
$[\tau_{5a}]$	monomial supercuspidal from $\Q_2(\sqrt{3})$	$\tau_{5a}$	5	1
		$\tau_{5a}\otimes\chi_5$	5	-1
		$\tau_{5a}\otimes\chi_2$	6	1
		$\tau_{5a}\otimes\chi_{2\cdot5}$	6	1
$[\tau_{5b}]$	monomial supercuspidal from $\Q_2(\sqrt{7})$	$\tau_{5b}$	5	1
		$\tau_{5b}\otimes\chi_5$	5	-1
		$\tau_{5b}\otimes\chi_2$	6	-1
		$\tau_{5b}\otimes\chi_{2\cdot5}$	6	-1
$[\tau_7]$	exceptional supercuspidal	$\tau_7$	7	1
		$\tau_7\otimes\chi_5$	7	-1
		$\tau_7\otimes\chi_3$	7	-1
		$\tau_7\otimes\chi_7$	7	1
		$\tau_7\otimes\chi_2$	7	-1
		$\tau_7\otimes\chi_{2\cdot5}$	7	1
		$\tau_7\otimes\chi_{2\cdot3}$	7	-1
		$\tau_7\otimes\chi_{2\cdot7}$	7	1

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.

• $\xi=\otimes\xi_p$ denotes the character of $\Q^\times\backslash\A^\times$ corresponding to the non-trivial Dirichlet character mod $4$. We have $\xi_2=\chi_7$, with the local character $\chi_7$ as above. If $\pi=\otimes\pi_p$ is an automorphic representation of $\GL(2,\A)$ or of $\GSp(4,\A)$ ramified only at $2$, then $\pi\otimes\xi$ is also such a representation. Hovering over an entry in the $\pi$ or $\Pi$ column highlights this entry and its $\xi$-twist.
• $S$ is a finite set of places subject to the following conditions:
  • $S$ is a subset of the set of places $v$ for which $\pi_v$ is square-integrable.
  • $\infty\in S$ (since we want to produce holomorphic lifts).
  • $(-1)^{\#S}=\varepsilon(\frac12,\pi\otimes\chi)$.

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 $N\mid32$.

$\GL(2)$										$\GSp(4)$
weight	level	$\pi$	size	$\pi_2$	$a(\pi_2)$	$\varepsilon(\frac12,\pi_2)$	$\varepsilon(\frac12,\pi)$	$\chi$	$S$	weight	conductor	$\Pi=\Pi(\pi,\chi,S)$	$\Pi_2$	$a(\Pi_2)$	$\varepsilon(\frac12,\Pi_2)$	$\varepsilon(\frac12,\Pi)$	para
2	32	32.2.a.a	1	$\tau_{5b}\otimes\chi_5$	5	-1	1	1	$\{\infty,2\}$	2	64	P.64.2.0.a	XIa*	6	1	1
								$\xi$	$\{\infty,2\}$		512	P.512.2.0.a	XIa*	9	1	1
4	8	8.4.a.a	1	$\tau_3$	3	1	1	1	$\{\infty,2\}$	3	16	P.16.3.0.a	XIa*	4	-1	1
								$\xi$	$\{\infty,2\}$		128	P.128.3.0.a	XIa*	7	-1	1
4	16	16.4.a.a	1	$\tau_3\otimes\chi_7$	4	1	1	1	$\{\infty,2\}$	3	32	P.32.3.0.a	XIa*	5	-1	1
								$\xi$	$\{\infty,2\}$		256	P.256.3.0.a	XIa*	8	-1	1
4	32	32.4.a.a	1	$\tau_{5a}\otimes\chi_5$	5	-1	-1	1	$\{\infty\}$	3	32	P.32.3.0.b	XIb	5	-1	1	32
								$\xi$	$\{\infty,2\}$		512	P.512.3.0.a	XIa*	9	-1	1
4	32	32.4.a.b	1	$\tau_{5b}$	5	1	1	1	$\{\infty,2\}$	3	64	P.64.3.0.a	XIa*	6	-1	1
								$\xi$	$\{\infty,2\}$		512	P.512.3.0.b	XIa*	9	-1	1
4	32	32.4.a.c	1	$\tau_{5a}$	5	1	1	1	$\{\infty,2\}$	3	64	P.64.3.0.b	XIa*	6	-1	1
								$\xi$	$\{\infty\}$		512	P.512.3.0.c	XIb	9	-1	1
6	4	4.6.a.a	1	$\tau_2$	2	-1	1	1	$\{\infty,2\}$	4	8	P.8.4.0.a	XIa*	3	1	1
								$\xi$	$\{\infty,2\}$		64	P.64.4.0.a	XIa*	6	1	1
6	8	8.6.a.a	1	$\tau_3\otimes\chi_5$	3	-1	1	1	$\{\infty,2\}$	4	16	P.16.4.0.a	XIa*	4	1	1
								$\xi$	$\{\infty\}$		128	P.128.4.0.a	XIb	7	1	1
6	16	16.6.a.a	1	$\tau_3\otimes\chi_3$	4	1	-1	1	$\{\infty\}$	4	16	P.16.4.0.b	XIb	4	1	1	16
								$\xi$	$\{\infty,2\}$		256	P.256.4.0.a	XIa*	8	-1	-1
6	16	16.6.a.b	1	$\tau_2\otimes\chi_3$	4	-1	1	1	$\{\infty,2\}$	4	32	P.32.4.0.a	XIa*	5	1	1
								$\xi$	$\{\infty,2\}$		256	P.256.4.0.b	XIa*	8	1	1
6	32	32.6.a.a	1	$\tau_{5a}$	5	1	-1	1	$\{\infty\}$	4	32	P.32.4.0.b	XIb	5	1	1	32
								$\xi$	$\{\infty,2\}$		512	P.512.4.0.a	XIa*	9	-1	-1
6	32	32.6.a.b	1	$\tau_{5b}$	5	1	-1	1	$\{\infty\}$	4	32	P.32.4.0.c	XIb	5	1	1	32
								$\xi$	$\{\infty\}$		256	P.256.4.0.c	XIb	8	-1	-1
6	32	32.6.a.c	1	$\tau_{5a}\otimes\chi_5$	5	-1	1	1	$\{\infty,2\}$	4	64	P.64.4.0.b	XIa*	6	1	1
								$\xi$	$\{\infty\}$		512	P.512.4.0.b	XIb	9	1	1
6	32	32.6.a.d	2	$\tau_{5b}\otimes\chi_5$	5	-1	1	1	$\{\infty,2\}$	4	64	P.64.4.0.c	XIa*	6	1	1
								$\xi$	$\{\infty,2\}$		512	P.512.4.0.c	XIa*	9	1	1
8	2	2.8.a.a	1	$\St_{\GL(2)}\otimes\chi_5$	1	1	1	1	$\{\infty,2\}$	5	4	P.4.5.0.a	Va*	2	-1	1
								$\xi$	$\{\infty\}$		32	P.32.5.0.a	Vb	5	-1	1
8	8	8.8.a.a	1	$\tau_3\otimes\chi_5$	3	-1	-1	1	$\{\infty\}$	5	8	P.8.5.0.a	XIb	3	-1	1	8
								$\xi$	$\{\infty,2\}$		128	P.128.5.0.a	XIa*	7	1	-1
8	8	8.8.a.b	1	$\tau_3$	3	1	1	1	$\{\infty,2\}$	5	16	P.16.5.0.a	XIa*	4	-1	1
								$\xi$	$\{\infty,2\}$		128	P.128.5.0.b	XIa*	7	-1	1
8	16	16.8.a.a	1	$\tau_3\otimes\chi_7$	4	1	1	1	$\{\infty,2\}$	5	32	P.32.5.0.b	XIa*	5	-1	1
								$\xi$	$\{\infty,2\}$		256	P.256.5.0.a	XIa*	8	-1	1
8	16	16.8.a.b	1	$\St_{\GL(2)}\otimes\chi_3$	4	-1	-1	1	$\{\infty\}$	5	16	P.16.5.0.b	Vb	4	-1	1	16
								$\xi$	$\{\infty,2\}$		256	P.256.5.0.b	Va*	8	1	-1
8	16	16.8.a.c	1	$\tau_3\otimes\chi_3$	4	1	1	1	$\{\infty,2\}$	5	32	P.32.5.0.c	XIa*	5	-1	1
								$\xi$	$\{\infty\}$		256	P.256.5.0.c	XIb	8	-1	1
8	32	32.8.a.a	1	$\tau_{5b}\otimes\chi_5$	5	-1	-1	1	$\{\infty\}$	5	32	P.32.5.0.d	XIb	5	-1	1	32
								$\xi$	$\{\infty\}$		512	P.512.5.0.a	XIb	9	1	-1
8	32	32.8.a.b	2	$\tau_{5a}$	5	1	1	1	$\{\infty,2\}$	5	64	P.64.5.0.b	XIa*	6	-1	1
								$\xi$	$\{\infty\}$		512	P.512.5.0.b	XIb	9	-1	1
8	32	32.8.a.c	2	$\tau_{5b}$	5	1	1	1	$\{\infty,2\}$	5	64	P.64.5.0.c	XIa*	6	-1	1
								$\xi$	$\{\infty,2\}$		512	P.512.5.0.c	XIa*	9	-1	1
8	32	32.8.a.d	2	$\tau_{5a}\otimes\chi_5$	5	-1	-1	1	$\{\infty\}$	5	32	P.32.5.0.e	XIb	5	-1	1	32
								$\xi$	$\{\infty,2\}$		512	P.512.5.0.d	XIa*	9	1	-1