LMFDB - Sato-Tate groups: statistics

The database currently contains 10465 Sato-Tate groups. The statistics below omit the infinite family $\mu(n)$ with trivial identity component since they are generated dynamically in search results.

Distribution of component groups and identity components

		identity component
		$\operatorname{SO}(1)$	$\operatorname{SU}(2)$	$\operatorname{SU}(2)^3$	$\operatorname{SU}(2)_2$	$\operatorname{SU}(2)_3$	$\operatorname{SU}(2)\times\operatorname{SU}(2)$	$\operatorname{SU}(2)\times\operatorname{SU}(2)_2$	$\operatorname{SU}(2)\times\operatorname{U}(1)_2$	$\operatorname{SU}(2)\times\operatorname{USp}(4)$	$\operatorname{U}(1)$	$\operatorname{U}(1)^2\times\operatorname{SU}(2)$	$\operatorname{U}(1)^3$	$\operatorname{U}(1)_2$	$\operatorname{U}(1)_3$	$\operatorname{U}(1)\times\operatorname{SU}(2)$	$\operatorname{U}(1)\times\operatorname{SU}(2)^2$	$\operatorname{U}(1)\times\operatorname{SU}(2)_2$	$\operatorname{U}(1)\times\operatorname{U}(1)$	$\operatorname{U}(1)\times\operatorname{U}(1)_2$	$\operatorname{U}(1)\times\operatorname{USp}(4)$	$\operatorname{U}(3)$	$\operatorname{USp}(4)$	$\operatorname{USp}(6)$	Total
component group	$C_1$	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	23
	$C_1$	50.00%	100.00%	25.00%	10.00%	9.09%	50.00%	10.00%	3.12%	100.00%	50.00%	20.00%	7.69%	3.12%	0.58%	50.00%	20.00%	3.23%	20.00%	0.82%	50.00%	50.00%	100.00%	100.00%	4.93%
	$C_2$	1		1	2	1	1	2	3		1	2	3	3	2	1	3	5	2	7	1	1			42
	$C_2$	50.00%		25.00%	20.00%	9.09%	50.00%	20.00%	9.38%		50.00%	40.00%	23.08%	9.38%	1.17%	50.00%	60.00%	16.13%	40.00%	5.74%	50.00%	50.00%			8.99%
	$C_3$			1	1	1		1	1				1	1	2			1		1					11
	$C_3$			25.00%	10.00%	9.09%		10.00%	3.12%				7.69%	3.12%	1.17%			3.23%		0.82%					2.36%
	$C_4$				1	1		1	2			1	2	2	3			2	1	4					20
	$C_4$				10.00%	9.09%		10.00%	6.25%			20.00%	15.38%	6.25%	1.75%			6.45%	20.00%	3.28%					4.28%
	$C_2^2$				1	1		1	3			1	3	3	2		1	5	1	12					34
	$C_2^2$				10.00%	9.09%		10.00%	9.38%			20.00%	23.08%	9.38%	1.17%		20.00%	16.13%	20.00%	9.84%					7.28%
	$S_3$			1	1	1		1	2					2	3			2		4					17
	$S_3$			25.00%	10.00%	9.09%		10.00%	6.25%					6.25%	1.75%			6.45%		3.28%					3.64%
	$C_6$				1	1		1	3				1	3	4			3		7					24
	$C_6$				10.00%	9.09%		10.00%	9.38%				7.69%	9.38%	2.34%			9.68%		5.74%					5.14%
	$C_7$														1										1
	$C_7$														0.58%										0.21%
	$C_8$														3										3
	$C_8$														1.75%										0.64%
	$C_2\times C_4$								1				1	1	2			1		6					12
	$C_2\times C_4$								3.12%				7.69%	3.12%	1.17%			3.23%		4.92%					2.57%
	$D_4$				1	1		1	3					3	4			3		10					26
	$D_4$				10.00%	9.09%		10.00%	9.38%					9.38%	2.34%			9.68%		8.20%					5.57%
	$Q_8$														2										2
	$Q_8$														1.17%										0.43%
	$C_2^3$								1				1	1	1			1		7					12
	$C_2^3$								3.12%				7.69%	3.12%	0.58%			3.23%		5.74%					2.57%
	$C_3^2$														2										2
	$C_3^2$														1.17%										0.43%
	$C_3:C_4$														2										2
	$C_3:C_4$														1.17%										0.43%
	$C_{12}$														3										3
	$C_{12}$														1.75%										0.64%
	$A_4$					1			1					1	1					1					5
	$A_4$					9.09%			3.12%					3.12%	0.58%					0.82%					1.07%
	$D_6$				1	1		1	4					4	5			4		16					36
	$D_6$				10.00%	9.09%		10.00%	12.50%					12.50%	2.92%			12.90%		13.11%					7.71%
	$C_2\times C_6$								1					1	3			1		7					13
	$C_2\times C_6$								3.12%					3.12%	1.75%			3.23%		5.74%					2.78%
	$D_7$														1										1
	$D_7$														0.58%										0.21%
	$C_4^2$														1										1
	$C_4^2$														0.58%										0.21%
	$\OD_{16}$														3										3
	$\OD_{16}$														1.75%										0.64%
	$D_8$														3										3
	$D_8$														1.75%										0.64%
	$\SD_{16}$														3										3
	$\SD_{16}$														1.75%										0.64%
	$C_2^2\times C_4$																			1					1
	$C_2^2\times C_4$																			0.82%					0.21%
	$C_2\times D_4$								1					1	2			1		9					14
	$C_2\times D_4$								3.12%					3.12%	1.17%			3.23%		7.38%					3.00%
	$D_4:C_2$														3										3
	$D_4:C_2$														1.75%										0.64%
	$C_2^4$																			1					1
	$C_2^4$																			0.82%					0.21%
	$C_3\times S_3$														3										3
	$C_3\times S_3$														1.75%										0.64%
	$C_3:S_3$														2										2
	$C_3:S_3$														1.17%										0.43%
	$C_3\times C_6$														1										1
	$C_3\times C_6$														0.58%										0.21%
	$C_7:C_3$														1										1
	$C_7:C_3$														0.58%										0.21%
$C_3:C_8$														2										2
$C_3:C_8$														1.17%										0.43%
$\SL(2,3)$														2										2
$\SL(2,3)$														1.17%										0.43%
$C_4\times S_3$														3										3
$C_4\times S_3$														1.75%										0.64%
$D_{12}$														3										3
$D_{12}$														1.75%										0.64%
$C_3:D_4$														3										3
$C_3:D_4$														1.75%										0.64%
$C_3\times D_4$														3										3
$C_3\times D_4$														1.75%										0.64%
$C_3\times Q_8$														2										2
$C_3\times Q_8$														1.17%										0.43%
$S_4$					1			2					2	2					4					11
$S_4$					9.09%			6.25%					6.25%	1.17%					3.28%					2.36%
$C_2\times A_4$								1					1	1					3					6
$C_2\times A_4$								3.12%					3.12%	0.58%					2.46%					1.28%
$C_2\times D_6$								1					1	3			1		10					16
$C_2\times D_6$								3.12%					3.12%	1.75%			3.23%		8.20%					3.43%
$C_2^2\times C_6$																			1					1
$C_2^2\times C_6$																			0.82%					0.21%
$\He_3$														1										1
$\He_3$														0.58%										0.21%
$\OD_{16}:C_2$														1										1
$\OD_{16}:C_2$														0.58%										0.21%
$C_4\wr C_2$														2										2
$C_4\wr C_2$														1.17%										0.43%
$C_4:D_4$														1										1
$C_4:D_4$														0.58%										0.21%
$D_8:C_2$														3										3
$D_8:C_2$														1.75%										0.64%
$C_2^2\times D_4$																			1					1
$C_2^2\times D_4$																			0.82%					0.21%
$D_4:C_2^2$														1										1
$D_4:C_2^2$														0.58%										0.21%
$C_3:C_{12}$														2										2
$C_3:C_{12}$														1.17%										0.43%
$C_3^2:C_4$														1										1
$C_3^2:C_4$														0.58%										0.21%
$S_3^2$														2										2
$S_3^2$														1.17%										0.43%
$C_3\times A_4$														1										1
$C_3\times A_4$														0.58%										0.21%
$C_6\times S_3$														2										2
$C_6\times S_3$														1.17%										0.43%
$C_6:S_3$														1										1
$C_6:S_3$														0.58%										0.21%
$C_6^2$														1										1
$C_6^2$														0.58%										0.21%
$F_7$														1										1
$F_7$														0.58%										0.21%
$C_4^2:C_3$														1										1
$C_4^2:C_3$														0.58%										0.21%
$C_3:D_8$														2										2
$C_3:D_8$														1.17%										0.43%
$Q_8:S_3$														2										2
$Q_8:S_3$														1.17%										0.43%
$\GL(2,3)$														3										3
$\GL(2,3)$														1.75%										0.64%
$\SL(2,3):C_2$														2										2
$\SL(2,3):C_2$														1.17%										0.43%
$S_3\times D_4$														3										3
$S_3\times D_4$														1.75%										0.64%
$D_{12}:C_2$														2										2
$D_{12}:C_2$														1.17%										0.43%
$C_2\times S_4$								1					1	1					6					9
$C_2\times S_4$								3.12%					3.12%	0.58%					4.92%					1.93%
$C_2^2\times A_4$																			1					1
$C_2^2\times A_4$																			0.82%					0.21%
$C_2^2\times D_6$																			1					1
$C_2^2\times D_6$																			0.82%					0.21%
$C_3^2:C_6$														2										2
$C_3^2:C_6$														1.17%										0.43%
$C_3^2:S_3$														1										1
$C_3^2:S_3$														0.58%										0.21%
$D_4:D_4$														1										1
$D_4:D_4$														0.58%										0.21%
$C_6.D_6$														1										1
$C_6.D_6$														0.58%										0.21%
$C_3:D_{12}$														2										2
$C_3:D_{12}$														1.17%										0.43%
$C_3\times \SL(2,3)$														1										1
$C_3\times \SL(2,3)$														0.58%										0.21%
$C_6\wr C_2$														2										2
$C_6\wr C_2$														1.17%										0.43%
$F_9$														1										1
$F_9$														0.58%										0.21%
$\SOPlus(4,2)$														1										1
$\SOPlus(4,2)$														0.58%										0.21%
$\PSU(3,2)$														1										1
$\PSU(3,2)$														0.58%										0.21%
$C_3\times S_4$														1										1
$C_3\times S_4$														0.58%										0.21%
$C_3:S_4$														1										1
$C_3:S_4$														0.58%										0.21%
$S_3\times A_4$														1										1
$S_3\times A_4$														0.58%										0.21%
$S_3\times D_6$														1										1
$S_3\times D_6$														0.58%										0.21%
$C_6:D_6$														1										1
$C_6:D_6$														0.58%										0.21%
$C_4^2:S_3$														2										2
$C_4^2:S_3$														1.17%										0.43%
$\GU(2,3)$														1										1
$\GU(2,3)$														0.58%										0.21%
$C_4^2:C_6$														1										1
$C_4^2:C_6$														0.58%										0.21%
$\GL(2,3):C_2$														2										2
$\GL(2,3):C_2$														1.17%										0.43%
$\SL(2,3):C_2^2$														1										1
$\SL(2,3):C_2^2$														0.58%										0.21%
$C_2^2\times S_4$																			1					1
$C_2^2\times S_4$																			0.82%					0.21%
$C_3^2:D_6$														1										1
$C_3^2:D_6$														0.58%										0.21%
$C_3^2:A_4$														1										1
$C_3^2:A_4$														0.58%										0.21%
$C_3:\GL(2,3)$														1										1
$C_3:\GL(2,3)$														0.58%										0.21%
$\SL(2,3):S_3$														1										1
$\SL(2,3):S_3$														0.58%										0.21%
$D_6:D_6$														1										1
$D_6:D_6$														0.58%										0.21%
$F_9:C_2$														1										1
$F_9:C_2$														0.58%										0.21%
$S_3\times S_4$														1										1
$S_3\times S_4$														0.58%										0.21%
$\GL(3,2)$														1										1
$\GL(3,2)$														0.58%										0.21%
$C_4^2:D_6$														1										1
$C_4^2:D_6$														0.58%										0.21%
$\SL(2,3):D_4$														1										1
$\SL(2,3):D_4$														0.58%										0.21%
$C_3^2:S_4$														1										1
$C_3^2:S_4$														0.58%										0.21%
$C_3^2:S_4$														1										1
$C_3^2:S_4$														0.58%										0.21%
$C_6^2:C_6$														1										1
$C_6^2:C_6$														0.58%										0.21%
$\PGU(3,2)$														1										1
$\PGU(3,2)$														0.58%										0.21%
$\PGL(2,7)$														1										1
$\PGL(2,7)$														0.58%										0.21%
$C_6^2:D_6$														1										1
$C_6^2:D_6$														0.58%										0.21%
$C_3^2:\GL(2,3)$														1										1
$C_3^2:\GL(2,3)$														0.58%										0.21%
Total	2	1	4	10	11	2	10	32	1	2	5	13	32	171	2	5	31	5	122	2	2	1	1	467
Total	0.43%	0.21%	0.86%	2.14%	2.36%	0.43%	2.14%	6.85%	0.21%	0.43%	1.07%	2.78%	6.85%	36.62%	0.43%	1.07%	6.64%	1.07%	26.12%	0.43%	0.43%	0.21%	0.21%

Distribution of maximal subgroups per identity component

identity component	$\operatorname{SO}(1)$	$\operatorname{SU}(2)$	$\operatorname{SU}(2)^3$	$\operatorname{SU}(2)_2$	$\operatorname{SU}(2)_3$	$\operatorname{SU}(2)\times\operatorname{SU}(2)$	$\operatorname{SU}(2)\times\operatorname{SU}(2)_2$	$\operatorname{SU}(2)\times\operatorname{U}(1)_2$	$\operatorname{SU}(2)\times\operatorname{USp}(4)$	$\operatorname{U}(1)$
count	1	1	1	2	2	1	2	2	1	1
proportion	2.17%	2.17%	2.17%	4.35%	4.35%	2.17%	4.35%	4.35%	2.17%	2.17%
identity component	$\operatorname{U}(1)^2\times\operatorname{SU}(2)$	$\operatorname{U}(1)^3$	$\operatorname{U}(1)_2$	$\operatorname{U}(1)_3$	$\operatorname{U}(1)\times\operatorname{SU}(2)$	$\operatorname{U}(1)\times\operatorname{SU}(2)^2$	$\operatorname{U}(1)\times\operatorname{SU}(2)_2$	$\operatorname{U}(1)\times\operatorname{U}(1)$	$\operatorname{U}(1)\times\operatorname{U}(1)_2$	$\operatorname{U}(1)\times\operatorname{USp}(4)$
count	2	3	2	12	1	1	2	3	2	1
proportion	4.35%	6.52%	4.35%	26.09%	2.17%	2.17%	4.35%	6.52%	4.35%	2.17%
identity component	$\operatorname{U}(3)$	$\operatorname{USp}(4)$	$\operatorname{USp}(6)$
count	1	1	1
proportion	2.17%	2.17%	2.17%

Distribution of trace zero densities and identity components

		identity component
		$\operatorname{SO}(1)$	$\operatorname{SU}(2)$	$\operatorname{SU}(2)^3$	$\operatorname{SU}(2)_2$	$\operatorname{SU}(2)_3$	$\operatorname{SU}(2)\times\operatorname{SU}(2)$	$\operatorname{SU}(2)\times\operatorname{SU}(2)_2$	$\operatorname{SU}(2)\times\operatorname{U}(1)_2$	$\operatorname{SU}(2)\times\operatorname{USp}(4)$	$\operatorname{U}(1)$	$\operatorname{U}(1)^2\times\operatorname{SU}(2)$	$\operatorname{U}(1)^3$	$\operatorname{U}(1)_2$	$\operatorname{U}(1)_3$	$\operatorname{U}(1)\times\operatorname{SU}(2)$	$\operatorname{U}(1)\times\operatorname{SU}(2)^2$	$\operatorname{U}(1)\times\operatorname{SU}(2)_2$	$\operatorname{U}(1)\times\operatorname{U}(1)$	$\operatorname{U}(1)\times\operatorname{U}(1)_2$	$\operatorname{U}(1)\times\operatorname{USp}(4)$	$\operatorname{U}(3)$	$\operatorname{USp}(4)$	$\operatorname{USp}(6)$	Total
trace zero density	0	2	1	2	2	5	1	10	32	1	1	5	6	2	24	2	3	13	2	35	2	1	1	1	154
	0	100.00%	100.00%	50.00%	20.00%	45.45%	50.00%	100.00%	100.00%	100.00%	50.00%	100.00%	46.15%	6.25%	14.04%	100.00%	60.00%	41.94%	40.00%	28.69%	100.00%	50.00%	100.00%	100.00%	32.98%
	1/36														1										1
	1/36														0.58%										0.21%
	1/18														3										3
	1/18														1.75%										0.64%
	1/12														4			1		1					6
	1/12														2.34%			3.23%		0.82%					1.28%
	1/9														4										4
	1/9														2.34%										0.86%
	1/8												1					1		2					4
	1/8												7.69%					3.23%		1.64%					0.86%
	1/6				1	1								1	6			1		1					11
	1/6				10.00%	9.09%								3.12%	3.51%			3.23%		0.82%					2.36%
	3/16																			1					1
	3/16																			0.82%					0.21%
	2/9														2										2
	2/9														1.17%										0.43%
	1/4				1								1	2			1	4	1	12					22
	1/4				10.00%								7.69%	6.25%			20.00%	12.90%	20.00%	9.84%					4.71%
	7/27														1										1
	7/27														0.58%										0.21%
	7/24																	1		3					4
	7/24																	3.23%		2.46%					0.86%
	5/16																	1		5					6
	5/16																	3.23%		4.10%					1.28%
	1/3			1		3									6			1		4					15
	1/3			25.00%		27.27%									3.51%			3.23%		3.28%					3.21%
	37/108														1										1
	37/108														0.58%										0.21%
	11/32																			1					1
	11/32																			0.82%					0.21%
	13/36														1										1
	13/36														0.58%										0.21%
	10/27														1										1
	10/27														0.58%										0.21%
	3/8												1	1				1		10					13
	3/8												7.69%	3.12%				3.23%		8.20%					2.78%
	19/48																			1					1
	19/48																			0.82%					0.21%
	13/32																			1					1
	13/32																			0.82%					0.21%
	5/12																			2					2
	5/12																			1.64%					0.43%
	7/16																			2					2
	7/16																			1.64%					0.43%
	4/9														1										1
	4/9														0.58%										0.21%
	1/2				3		1				1		2	7	44		1	7	1	41		1			109
	1/2				30.00%		50.00%				50.00%		15.38%	21.88%	25.73%		20.00%	22.58%	20.00%	33.61%		50.00%			23.34%
	37/72														1										1
	37/72														0.58%										0.21%
	19/36														6										6
	19/36														3.51%										1.28%
	13/24														6										6
	13/24														3.51%										1.28%
	5/9														7										7
	5/9														4.09%										1.50%
	7/12				1									3	12										16
	7/12				10.00%									9.38%	7.02%										3.43%
	11/18														4										4
	11/18														2.34%										0.86%
	5/8				1									5											6
	5/8				10.00%									15.62%											1.28%
17/27														1										1
17/27														0.58%										0.21%
2/3			1		2							1		13										17
2/3			25.00%		18.18%							7.69%		7.60%										3.64%
145/216														1										1
145/216														0.58%										0.21%
49/72														1										1
49/72														0.58%										0.21%
37/54														2										2
37/54														1.17%										0.43%
11/16													1											1
11/16													3.12%											0.21%
13/18														2										2
13/18														1.17%										0.43%
20/27														1										1
20/27														0.58%										0.21%
3/4				1									6					1						8
3/4				10.00%									18.75%					20.00%						1.71%
19/24													1											1
19/24													3.12%											0.21%
13/16													1											1
13/16													3.12%											0.21%
5/6												1		7										8
5/6												7.69%		4.09%										1.71%
91/108														2										2
91/108														1.17%										0.43%
31/36														2										2
31/36														1.17%										0.43%
47/54														2										2
47/54														1.17%										0.43%
7/8													2											2
7/8													6.25%											0.43%
8/9														1										1
8/9														0.58%										0.21%
17/18														1										1
17/18														0.58%										0.21%
Total	2	1	4	10	11	2	10	32	1	2	5	13	32	171	2	5	31	5	122	2	2	1	1	467
Total	0.43%	0.21%	0.86%	2.14%	2.36%	0.43%	2.14%	6.85%	0.21%	0.43%	1.07%	2.78%	6.85%	36.62%	0.43%	1.07%	6.64%	1.07%	26.12%	0.43%	0.43%	0.21%	0.21%

Distribution of second trace moments and identity components

		identity component
		$\operatorname{SO}(1)$	$\operatorname{SU}(2)$	$\operatorname{SU}(2)^3$	$\operatorname{SU}(2)_2$	$\operatorname{SU}(2)_3$	$\operatorname{SU}(2)\times\operatorname{SU}(2)$	$\operatorname{SU}(2)\times\operatorname{SU}(2)_2$	$\operatorname{SU}(2)\times\operatorname{U}(1)_2$	$\operatorname{SU}(2)\times\operatorname{USp}(4)$	$\operatorname{U}(1)$	$\operatorname{U}(1)^2\times\operatorname{SU}(2)$	$\operatorname{U}(1)^3$	$\operatorname{U}(1)_2$	$\operatorname{U}(1)_3$	$\operatorname{U}(1)\times\operatorname{SU}(2)$	$\operatorname{U}(1)\times\operatorname{SU}(2)^2$	$\operatorname{U}(1)\times\operatorname{SU}(2)_2$	$\operatorname{U}(1)\times\operatorname{U}(1)$	$\operatorname{U}(1)\times\operatorname{U}(1)_2$	$\operatorname{U}(1)\times\operatorname{USp}(4)$	$\operatorname{U}(3)$	$\operatorname{USp}(4)$	$\operatorname{USp}(6)$	Total
second trace moment	1	2	1	2	4	2	1				1		1	9	23				1			1	1	1	50
	1	100.00%	100.00%	50.00%	40.00%	18.18%	50.00%				50.00%		7.69%	28.12%	13.45%				20.00%			50.00%	100.00%	100.00%	10.71%
	2			1	5	3	1	4	9	1	1	1	3	16	56	1	2	11	2	36	1	1			155
	2			25.00%	50.00%	27.27%	50.00%	40.00%	28.12%	100.00%	50.00%	20.00%	23.08%	50.00%	32.75%	50.00%	40.00%	35.48%	40.00%	29.51%	50.00%	50.00%			33.19%
	3			1		4		5	16			2	5		38	1	2	13	1	51	1				140
	3			25.00%		36.36%		50.00%	50.00%			40.00%	38.46%		22.22%	50.00%	40.00%	41.94%	20.00%	41.80%	50.00%				29.98%
	4				1							1	2	6	24		1	5	1	16					57
	4				10.00%							20.00%	15.38%	18.75%	14.04%		20.00%	16.13%	20.00%	13.11%					12.21%
	5					1		1	6			1	1		7			1		11					29
	5					9.09%		10.00%	18.75%			20.00%	7.69%		4.09%			3.23%		9.02%					6.21%
	6												1		17			1		6					25
	6												7.69%		9.94%			3.23%		4.92%					5.35%
	8													1											1
	8													3.12%											0.21%
	9					1			1						1					1					4
	9					9.09%			3.12%						0.58%					0.82%					0.86%
	10														4					1					5
	10														2.34%					0.82%					1.07%
	18														1										1
	18														0.58%										0.21%
	Total	2	1	4	10	11	2	10	32	1	2	5	13	32	171	2	5	31	5	122	2	2	1	1	467
	Total	0.43%	0.21%	0.86%	2.14%	2.36%	0.43%	2.14%	6.85%	0.21%	0.43%	1.07%	2.78%	6.85%	36.62%	0.43%	1.07%	6.64%	1.07%	26.12%	0.43%	0.43%	0.21%	0.21%

Distribution of fourth trace moments and identity components

		identity component
		$\operatorname{SO}(1)$	$\operatorname{SU}(2)$	$\operatorname{SU}(2)^3$	$\operatorname{SU}(2)_2$	$\operatorname{SU}(2)_3$	$\operatorname{SU}(2)\times\operatorname{SU}(2)$	$\operatorname{SU}(2)\times\operatorname{SU}(2)_2$	$\operatorname{SU}(2)\times\operatorname{U}(1)_2$	$\operatorname{SU}(2)\times\operatorname{USp}(4)$	$\operatorname{U}(1)$	$\operatorname{U}(1)^2\times\operatorname{SU}(2)$	$\operatorname{U}(1)^3$	$\operatorname{U}(1)_2$	$\operatorname{U}(1)_3$	$\operatorname{U}(1)\times\operatorname{SU}(2)$	$\operatorname{U}(1)\times\operatorname{SU}(2)^2$	$\operatorname{U}(1)\times\operatorname{SU}(2)_2$	$\operatorname{U}(1)\times\operatorname{U}(1)$	$\operatorname{U}(1)\times\operatorname{U}(1)_2$	$\operatorname{U}(1)\times\operatorname{USp}(4)$	$\operatorname{U}(3)$	$\operatorname{USp}(4)$	$\operatorname{USp}(6)$	Total
fourth trace moment	1	2																							2
	1	100.00%																							0.43%
	2		1																						1
	2		100.00%																						0.21%
	3										1												1	1	3
	3										50.00%												100.00%	100.00%	0.64%
	5			1			1																		2
	5			25.00%			50.00%																		0.43%
	6				3						1			3	3							1			11
	6				30.00%						50.00%			9.38%	1.75%							50.00%			2.36%
	8			1	1	1																			3
	8			25.00%	10.00%	9.09%																			0.64%
	9													4	6				1						11
	9													12.50%	3.51%				20.00%						2.36%
	10						1																		1
	10						50.00%																		0.21%
	11									1						1									2
	11									100.00%						50.00%									0.43%
	12				3									4	4				1		1	1			14
	12				30.00%									12.50%	2.34%				20.00%		50.00%	50.00%			3.00%
	13			1																					1
	13			25.00%																					0.21%
	14					1		3	3								1								8
	14					9.09%		30.00%	9.38%								20.00%								1.71%
	15												1		10			5		5					21
	15												7.69%		5.85%			16.13%		4.10%					4.50%
	16				2			1																	3
	16				20.00%			10.00%																	0.64%
	17								4			1						1							6
	17								12.50%			20.00%						3.23%							1.28%
	18												1	10	5				1	8					25
	18												7.69%	31.25%	2.92%				20.00%	6.56%					5.35%
	20					1			2							1	1								5
	20					9.09%			6.25%							50.00%	20.00%								1.07%
	21														9			3	1	5	1				19
	21														5.26%			9.68%	20.00%	4.10%	50.00%				4.07%
	22					1																			1
	22					9.09%																			0.21%
	23																1	2							3
	23																20.00%	6.45%							0.64%
	24			1									1	4	14			3		15					38
	24			25.00%									7.69%	12.50%	8.19%			9.68%		12.30%					8.14%
	25																1								1
	25																20.00%								0.21%
	26							3	2			1						1							7
	26							30.00%	6.25%			20.00%						3.23%							1.50%
	27												3		11			5		13					32
	27												23.08%		6.43%			16.13%		10.66%					6.85%
	28					1																			1
	28					9.09%																			0.21%
	30							2					1		10					2					15
	30							20.00%					7.69%		5.85%					1.64%					3.21%
	31																	2							2
	31																	6.45%							0.43%
	32				1				10			1													12
	32				10.00%				31.25%			20.00%													2.57%
	33												1		4					18					23
	33												7.69%		2.34%					14.75%					4.93%
	36													3					1						4
	36													9.38%					20.00%						0.86%
	38					1			4																5
	38					9.09%			12.50%																1.07%
	39																			4					4
	39																			3.28%					0.86%
40																1								1
40																20.00%								0.21%
41											1													1
41											20.00%													0.21%
42					2							1		7			3		2					15
42					18.18%							7.69%		4.09%			9.68%		1.64%					3.21%
43																	2							2
43																	6.45%							0.43%
45												1		17					10					28
45												7.69%		9.94%					8.20%					6.00%
46																	2							2
46																	6.45%							0.43%
48												1	3	9					10					23
48												7.69%	9.38%	5.26%					8.20%					4.93%
51														11					7					18
51														6.43%					5.74%					3.85%
54					1									6					4					11
54					9.09%									3.51%					3.28%					2.36%
57														3										3
57														1.75%										0.64%
58							1																	1
58							10.00%																	0.21%
59																	1							1
59																	3.23%							0.21%
60														2										2
60														1.17%										0.43%
62								3			1													4
62								9.38%			20.00%													0.86%
63												1		5					5					11
63												7.69%		2.92%					4.10%					2.36%
66														2										2
66														1.17%										0.43%
74								3																3
74								9.38%																0.64%
75																			3					3
75																			2.46%					0.64%
81														2										2
81														1.17%										0.43%
82					1																			1
82					9.09%																			0.21%
84														1										1
84														0.58%										0.21%
86																	1							1
86																	3.23%							0.21%
90												1		8					3					12
90												7.69%		4.68%					2.46%					2.57%
96													1											1
96													3.12%											0.21%
99														5					3					8
99														2.92%					2.46%					1.71%
102														5					3					8
102														2.92%					2.46%					1.71%
114														1										1
114														0.58%										0.21%
123														2										2
123														1.17%										0.43%
126														2										2
126														1.17%										0.43%
146								1																1
146								3.12%																0.21%
147																			1					1
147																			0.82%					0.21%
162					1									1										2
162					9.09%									0.58%										0.43%
198														3					1					4
198														1.75%					0.82%					0.86%
243														1										1
243														0.58%										0.21%
246														1										1
246														0.58%										0.21%
486														1										1
486														0.58%										0.21%
Total	2	1	4	10	11	2	10	32	1	2	5	13	32	171	2	5	31	5	122	2	2	1	1	467
Total	0.43%	0.21%	0.86%	2.14%	2.36%	0.43%	2.14%	6.85%	0.21%	0.43%	1.07%	2.78%	6.85%	36.62%	0.43%	1.07%	6.64%	1.07%	26.12%	0.43%	0.43%	0.21%	0.21%

Distribution of first $a_2$ moment and identity components

		identity component
		$\operatorname{SU}(2)^3$	$\operatorname{SU}(2)_2$	$\operatorname{SU}(2)_3$	$\operatorname{SU}(2)\times\operatorname{SU}(2)$	$\operatorname{SU}(2)\times\operatorname{SU}(2)_2$	$\operatorname{SU}(2)\times\operatorname{U}(1)_2$	$\operatorname{SU}(2)\times\operatorname{USp}(4)$	$\operatorname{U}(1)^2\times\operatorname{SU}(2)$	$\operatorname{U}(1)^3$	$\operatorname{U}(1)_2$	$\operatorname{U}(1)_3$	$\operatorname{U}(1)\times\operatorname{SU}(2)$	$\operatorname{U}(1)\times\operatorname{SU}(2)^2$	$\operatorname{U}(1)\times\operatorname{SU}(2)_2$	$\operatorname{U}(1)\times\operatorname{U}(1)$	$\operatorname{U}(1)\times\operatorname{U}(1)_2$	$\operatorname{U}(1)\times\operatorname{USp}(4)$	$\operatorname{U}(3)$	$\operatorname{USp}(4)$	$\operatorname{USp}(6)$	Total
first a2 moment	1	2	8	2	1					2	22	40				1			2	1	1	82
	1	50.00%	80.00%	18.18%	50.00%					15.38%	68.75%	23.39%				20.00%			100.00%	100.00%	100.00%	17.75%
	2	1	1	6	1	8	22	1	1	3	8	87	2	3	26	4	91	2				267
	2	25.00%	10.00%	54.55%	50.00%	80.00%	68.75%	100.00%	20.00%	23.08%	25.00%	50.88%	100.00%	60.00%	83.87%	80.00%	74.59%	100.00%				57.79%
	3	1	1	1		1	8		4	8	1	34		2	3		26					90
	3	25.00%	10.00%	9.09%		10.00%	25.00%		80.00%	61.54%	3.12%	19.88%		40.00%	9.68%		21.31%					19.48%
	4			1		1	1				1	4			2		3					13
	4			9.09%		10.00%	3.12%				3.12%	2.34%			6.45%		2.46%					2.81%
	5						1					4					2					7
	5						3.12%					2.34%					1.64%					1.52%
	6			1								1										2
	6			9.09%								0.58%										0.43%
	9											1										1
	9											0.58%										0.22%
	Total	4	10	11	2	10	32	1	5	13	32	171	2	5	31	5	122	2	2	1	1	462
	Total	0.87%	2.16%	2.38%	0.43%	2.16%	6.93%	0.22%	1.08%	2.81%	6.93%	37.01%	0.43%	1.08%	6.71%	1.08%	26.41%	0.43%	0.43%	0.22%	0.22%

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

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Varieties

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Distribution of component groups and identity components

Distribution of maximal subgroups per identity component

Distribution of trace zero densities and identity components

Distribution of second trace moments and identity components

Distribution of fourth trace moments and identity components

Distribution of first $a_2$ moment and identity components

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

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Distribution of component groups and identity components

Distribution of maximal subgroups per identity component

Distribution of trace zero densities and identity components

Distribution of second trace moments and identity components

Distribution of fourth trace moments and identity components

Distribution of first a2a_2a2​ moment and identity components

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Distribution of first $a_2$ moment and identity components