$ \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}} $

Automorphic Representations of GSp(4)

Weight 12, conductor dividing 16

The following table gives a complete list of the Galois orbits of cuspidal automorphic representations $\pi\cong\otimes\pi_v$ of $\GSp(4,\A_\Q)$ with the following properties:

• The archimedean component $\pi_\infty$ is a holomorphic discrete series representation with scalar minimal $K$-type $(12,12)$. These correspond to scalar-valued Siegel modular forms of weight $12$.
• The non-archimedean component $\pi_2$ has conductor exponent $\leq4$.
• The non-archimedean components $\pi_p$ for $p\geq3$ are unramified.

label	lift from	size	$p=2$			$p=3$	paramodular					Siegel		Klingen		Borel	principal
			type	$\varepsilon$	$L$	$T(3)$	$K(1)$	$K(2)$	$K(4)$	$K(8)$	$K(16)$	$\Gamma_0(2)$	$\Gamma_0(4)$	$\Gamma_0'(2)$	$\Gamma_0'(4)$	$B(2)$	$\Gamma(2)$	$S_6$ types
G.4.12.0.a		1	IIIa	+	$1+7\cdot2^8T+2^{20}T^2$	$-88488$	0	0	1	2	4	2	8	1	5	4	30	[4,2]+[3,2,1]+[2,2,2]
G.8.12.0.a		1	IVa	+	$1+2^9T$	$-14760$	0	0	0	1	2	0	2	0	2	1	16	[3,2,1]
G.8.12.0.b		1	XIa	+	$1-2^{10}T$	$-229032$	0	0	0	1	2	0	1	0	2	0	10	[4,1,1]
G.8.12.0.c		2	X	+	$1+2^6(12\pm5\sqrt{6})T+2^{21}T^2$	$504(65\pm64\sqrt{6})$	0	0	0	2	4	0	0	0	0	0	0
G.16.12.0.a		1	VII	+	$1$	$-12456$	0	0	0	0	1	0	4	0	2	0	15	[3,1,1,1]+[2,1,1,1,1]
G.16.12.0.b		2	IXa	+	$1$	$72(819\pm64\sqrt{85})$	0	0	0	0	2	0	6	0	1	0	20	2[3,1,1,1]
G.16.12.0.c		2	XIa	+	$1-2^{10}T$	$72(-521\pm128\sqrt{5})$	0	0	0	0	2	0	0	0	0	0	0
G.16.12.0.d		2	XIa	+	$1+2^{10}T$	$72(831\pm8\sqrt{85})$	0	0	0	0	2	0	0	0	0	0	0
G.16.12.0.e		5	X	+	$1-2^9$$t_{12,a}$$T+2^{21}T^2$	$\alpha_{12,16,a}$	0	0	0	0	5	0	0	0	0	0	0
G.16.12.0.f		8	X	+	$1-2^9$$t_{12,b}$$T+2^{21}T^2$	$\alpha_{12,16,b}$	0	0	0	0	8	0	0	0	0	0	0
G.16.12.0.g		1	IIa or X	-	$1+21\cdot2^6T+2^{21}T^2$	$-185616$	0	0	0	0	1	0	0	0	0	0	0
P.1.12.0.a	1.22.a.a	1	IIb	+	$(1+288T+2^{21}T^2)(1-2^{10}T)(1-2^{11}T)$	$107352$	1	1	2	2	3	3	7	2	4	4	15	[6]+[4,2]+[2,2,2]
P.2.12.0.a	2.22.a.a	1	Vb	+	$(1+2^{10}T)(1-2^{10}T)(1-2^{11}T)$	$307800$	0	1	1	2	2	1	3	1	2	2	9	[4,2]
P.4.12.0.a	2.22.a.b	1	VIb	+	$(1-2^{10}T)^2$	$295512$	0	0	0	0	0	1	3	0	0	1	5	[2,2,2]
P.8.12.0.a	4.22.a.a	2	XIa*	+	$1-2^{10}T$	$24(11209\pm112\sqrt{2161})$	0	0	0	0	0	0	0	0	0	0	0
P.8.12.0.b	8.22.a.a	2	XIb	+	$(1-2^{10}T)(1-2^{11}T)$	$24(7645\pm8\sqrt{358549}$	0	0	0	2	2	0	0	0	0	0	0
P.16.12.0.a	8.22.a.b	3	XIa*	+	$1-2^{10}T$	$268451-$$\beta_1$	0	0	0	0	0	0	0	0	0	0	0
P.16.12.0.b	16.22.a.e	2	XIb	+	$(1-2^{10}T)(1-2^{11}T)$	$48(6019\pm4\sqrt{358549}$)	0	0	0	0	2	0	0	0	0	0	0
P.16.12.0.c	16.22.a.f	3	XIb	+	$(1-2^{10}T)(1-2^{11}T)$	$203941+$$\beta_1$	0	0	0	0	3	0	0	0	0	0	0
$\mathrm{dim}\:S_{12}(\Gamma)$							1	2	4	12	45	7	34	4	18	12	120