$ \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}} \renewcommand{\sc}{{\sf sc}} $

Automorphic Representations of GSp(4)

Weight 11, 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 $(11,11)$. These correspond to scalar-valued Siegel modular forms of weight $11$.
• 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.4.11.0.a		1	X	-	$1+1344T+2^{19}T^2$	$-13464$	0	0	1	2	4	0	1	0	3	0	15	[4,1,1]+[3,3]
G.8.11.0.a		2	X	-	$1-32(-7\pm\sqrt{55})T+2^{19}T^2$	$-24(781\pm128\sqrt{55})$	0	0	0	2	4	0	0	0	0	0	0
G.16.11.0.a		1	X	+	$1+928T+2^{19}T^2$	$-66096$	0	0	0	0	1	0	0	0	0	0	0
G.16.11.0.b		1	sc(16)	-	$1$	$8040$	0	0	0	0	1	0	0	0	1	0	9	[2,2,1,1]
G.16.11.0.c		2	XIa	-	$1-2^9T$	$24(-1245\pm32\sqrt{21})$	0	0	0	0	2	0	0	0	0	0	0
G.16.11.0.d		2	XIa	-	$1+2^9T$	$120(111\pm8\sqrt{69})$	0	0	0	0	2	0	0	0	0	0	0
G.16.11.0.e		1	IIa or X	-	$1-1152T+2^{19}T^2$	$-73584$	0	0	0	0	1	0	0	0	0	0	0
G.16.11.0.f		1	IIa or X	-	$1-256T+2^{19}T^2$	$18768$	0	0	0	0	1	0	0	0	0	0	0
G.16.11.0.g		1	IIa or X	-	$1+192T+2^{19}T^2$	$35568$	0	0	0	0	1	0	0	0	0	0	0
G.16.11.0.h		2	IIa or X	-	$1-8(-107\pm\sqrt{3961})T+2^{19}T^2$	$48(425\pm2\sqrt{3961})$	0	0	0	0	2	0	0	0	0	0	0
G.16.11.0.i		4	IIa or X	-	$1-256\,$$t_{11}$$T+2^{19}T^2$	$\alpha_{11,16}$ (degree 4)	0	0	0	0	4	0	0	0	0	0	0
P.2.11.0.a	2.20.a.b	1	VIc	-	$(1-2^9T)^2(1-2^{10}T)$	$25704$	0	1	1	2	2	0	0	1	2	1	5	[5,1]
P.4.11.0.a	2.20.a.a	1	Va*	-	$(1+2^9T)(1-2^9T)(1-2^{10}T)$	$65640$	0	0	0	0	0	0	0	0	0	0	1	[1,1,1,1,1,1]
P.4.11.0.b	4.20.a.a	1	XIb	-	$(1-2^9T)(1-2^{10}T)$	$78696$	0	0	1	1	2	0	0	0	1	0	5	[3,3]
P.8.11.0.a	8.20.a.a	2	XIb	-	$(1-2^9T)(1-2^{10}T)$	$24(2699\pm40\sqrt{1453})$	0	0	0	2	2	0	0	0	0	0	0
P.16.11.0.a	8.20.a.b	3	XIa*	-	$1-2^9T$	$86643-$$\beta_1$	0	0	0	0	0	0	0	0	0	0	0
P.16.11.0.b	16.20.a.a	1	IIb	-	$(1-2^9T)(1-2^{10}T)$	$28080$	0	0	0	0	1	0	0	0	0	0	0
P.16.11.0.c	16.20.a.b	1	XIb	-	$(1-2^9T)(1-2^{10}T)$	$78768$	0	0	0	0	1	0	0	0	0	0	0
P.16.11.0.d	16.20.a.c	1	Vb	-	$(1-2^9T)(1-2^{10}T)$	$91824$	0	0	0	0	1	0	0	0	0	0	0
P.16.11.0.e	16.20.a.d	1	Vb	-	$(1-2^9T)(1-2^{10}T)$	$131760$	0	0	0	0	1	0	0	0	0	0	0
$\mathrm{dim}\:S_{11}(\Gamma)$							0	1	3	9	33	0	1	1	7	1	35