LMFDB - Bianchi modular forms: statistics

The database currently contains 215,630 Bianchi modular forms of weight 2 over 32 imaginary quadratic fields, and 400,198 spaces of cusp forms over 32 imaginary quadratic fields, including all with class number one.

Distribution of Bianchi modular forms by base field and level norm

		level norm
		1-100	101-1000	1001-10000	10001-50000	50001-100000	100001-150000	Total
base field	$\Q(\sqrt{-3}) $	3	127	2080	11072	14445	14616	42343
	$\Q(\sqrt{-3}) $	0.01%	0.30%	4.91%	26.15%	34.11%	34.52%	19.64%
	$\Q(\sqrt{-1}) $	5	188	2967	15982	20888		40030
	$\Q(\sqrt{-1}) $	0.01%	0.47%	7.41%	39.93%	52.18%		18.56%
	$\Q(\sqrt{-7}) $	16	301	5606	29901			35824
	$\Q(\sqrt{-7}) $	0.04%	0.84%	15.65%	83.47%			16.61%
	$\Q(\sqrt{-2}) $	10	345	5884	30604			36843
	$\Q(\sqrt{-2}) $	0.03%	0.94%	15.97%	83.07%			17.09%
	$\Q(\sqrt{-11}) $	12	314	4879	25336			30541
	$\Q(\sqrt{-11}) $	0.04%	1.03%	15.98%	82.96%			14.16%
	$\Q(\sqrt{-15}) $	33	658					691
	$\Q(\sqrt{-15}) $	4.78%	95.22%					0.32%
	$\Q(\sqrt{-19}) $	11	200	2386	1386			3983
	$\Q(\sqrt{-19}) $	0.28%	5.02%	59.90%	34.80%			1.85%
	$\Q(\sqrt{-5}) $	25	635					660
	$\Q(\sqrt{-5}) $	3.79%	96.21%					0.31%
	$\Q(\sqrt{-23}) $	41	889	2813				3743
	$\Q(\sqrt{-23}) $	1.10%	23.75%	75.15%				1.74%
	$\Q(\sqrt{-6}) $	22	492					514
	$\Q(\sqrt{-6}) $	4.28%	95.72%					0.24%
	$\Q(\sqrt{-31}) $	52	759	4352				5163
	$\Q(\sqrt{-31}) $	1.01%	14.70%	84.29%				2.39%
	$\Q(\sqrt{-35}) $	31	400					431
	$\Q(\sqrt{-35}) $	7.19%	92.81%					0.20%
	$\Q(\sqrt{-39}) $	59	1123					1182
	$\Q(\sqrt{-39}) $	4.99%	95.01%					0.55%
	$\Q(\sqrt{-10}) $	26	312					338
	$\Q(\sqrt{-10}) $	7.69%	92.31%					0.16%
	$\Q(\sqrt{-43}) $	1	65	653	345			1064
	$\Q(\sqrt{-43}) $	0.09%	6.11%	61.37%	32.42%			0.49%
	$\Q(\sqrt{-47}) $	64	1224					1288
	$\Q(\sqrt{-47}) $	4.97%	95.03%					0.60%
	$\Q(\sqrt{-51}) $	37	267					304
	$\Q(\sqrt{-51}) $	12.17%	87.83%					0.14%
	$\Q(\sqrt{-13}) $	21	269					290
	$\Q(\sqrt{-13}) $	7.24%	92.76%					0.13%
	$\Q(\sqrt{-55}) $	56	807					863
	$\Q(\sqrt{-55}) $	6.49%	93.51%					0.40%
	$\Q(\sqrt{-14}) $	53	809					862
	$\Q(\sqrt{-14}) $	6.15%	93.85%					0.40%
	$\Q(\sqrt{-59}) $	38	526					564
	$\Q(\sqrt{-59}) $	6.74%	93.26%					0.26%
	$\Q(\sqrt{-67}) $	1	31	419				451
	$\Q(\sqrt{-67}) $	0.22%	6.87%	92.90%				0.21%
	$\Q(\sqrt{-17}) $	69	707					776
	$\Q(\sqrt{-17}) $	8.89%	91.11%					0.36%
	$\Q(\sqrt{-71}) $	76	1698					1774
	$\Q(\sqrt{-71}) $	4.28%	95.72%					0.82%
	$\Q(\sqrt{-79}) $	61	741					802
	$\Q(\sqrt{-79}) $	7.61%	92.39%					0.37%
	$\Q(\sqrt{-83}) $	13	289					302
	$\Q(\sqrt{-83}) $	4.30%	95.70%					0.14%
	$\Q(\sqrt{-21}) $	56	632					688
	$\Q(\sqrt{-21}) $	8.14%	91.86%					0.32%
	$\Q(\sqrt{-87}) $	97	921					1018
	$\Q(\sqrt{-87}) $	9.53%	90.47%					0.47%
	$\Q(\sqrt{-22}) $	16	193					209
	$\Q(\sqrt{-22}) $	7.66%	92.34%					0.10%
	$\Q(\sqrt{-91}) $	25	194					219
	$\Q(\sqrt{-91}) $	11.42%	88.58%					0.10%
	$\Q(\sqrt{-95}) $	68	1691					1759
	$\Q(\sqrt{-95}) $	3.87%	96.13%					0.82%
	$\Q(\sqrt{-163}) $		26	85				111
	$\Q(\sqrt{-163}) $		23.42%	76.58%				0.05%
Total	1098	17833	32124	114626	35333	14616	215630
Total	0.51%	8.27%	14.90%	53.16%	16.39%	6.78%

Distribution of computed $\operatorname{GL}_2$ levels by base field and level norm

The set of weights computed for each level varies.

		level norm
		1-100	101-1000	1001-10000	10001-50000	50001-100000	100001-150000	Total
base field	$\Q(\sqrt{-3}) $	3	161	2368	11389	14356	14526	42803
	$\Q(\sqrt{-3}) $	0.01%	0.38%	5.53%	26.61%	33.54%	33.94%	22.68%
	$\Q(\sqrt{-1}) $	5	253	3425	16061	20206		39950
	$\Q(\sqrt{-1}) $	0.01%	0.63%	8.57%	40.20%	50.58%		21.16%
	$\Q(\sqrt{-7}) $	24	477	6053	27474			34028
	$\Q(\sqrt{-7}) $	0.07%	1.40%	17.79%	80.74%			18.03%
	$\Q(\sqrt{-2}) $	13	417	5279	23925			29634
	$\Q(\sqrt{-2}) $	0.04%	1.41%	17.81%	80.73%			15.70%
	$\Q(\sqrt{-11}) $	20	368	4434	20228			25050
	$\Q(\sqrt{-11}) $	0.08%	1.47%	17.70%	80.75%			13.27%
	$\Q(\sqrt{-19}) $	18	252	2683	1590			4543
	$\Q(\sqrt{-19}) $	0.40%	5.55%	59.06%	35.00%			2.41%
	$\Q(\sqrt{-23}) $	198	2130	1212				3540
	$\Q(\sqrt{-23}) $	5.59%	60.17%	34.24%				1.88%
	$\Q(\sqrt{-31}) $	164	1796	4108				6068
	$\Q(\sqrt{-31}) $	2.70%	29.60%	67.70%				3.21%
	$\Q(\sqrt{-43}) $	6	100	1222	713			2041
	$\Q(\sqrt{-43}) $	0.29%	4.90%	59.87%	34.93%			1.08%
	$\Q(\sqrt{-67}) $	6	69	872				947
	$\Q(\sqrt{-67}) $	0.63%	7.29%	92.08%				0.50%
	$\Q(\sqrt{-163}) $	2	34	122				158
	$\Q(\sqrt{-163}) $	1.27%	21.52%	77.22%				0.08%

Distribution of computed $\operatorname{SL}_2$ levels by base field and level norm

The set of weights computed for each level varies.

		level norm
		1-100	101-200	201-400	401-800	801-1600	1601-3200	3201-6400	Total
base field	$\Q(\sqrt{-3}) $	5		1	1				7
	$\Q(\sqrt{-3}) $	71.43%		14.29%	14.29%				0.15%
	$\Q(\sqrt{-1}) $	63	47	88	196	131			525
	$\Q(\sqrt{-1}) $	12.00%	8.95%	16.76%	37.33%	24.95%			11.34%
	$\Q(\sqrt{-7}) $	105	74	142	297	500	142		1260
	$\Q(\sqrt{-7}) $	8.33%	5.87%	11.27%	23.57%	39.68%	11.27%		27.22%
	$\Q(\sqrt{-2}) $	107	71	109	218	175			680
	$\Q(\sqrt{-2}) $	15.74%	10.44%	16.03%	32.06%	25.74%			14.69%
	$\Q(\sqrt{-11}) $	97	91	132	222	402	202		1146
	$\Q(\sqrt{-11}) $	8.46%	7.94%	11.52%	19.37%	35.08%	17.63%		24.76%
	$\Q(\sqrt{-19}) $	74	71	136	99	114			494
	$\Q(\sqrt{-19}) $	14.98%	14.37%	27.53%	20.04%	23.08%			10.67%
	$\Q(\sqrt{-5}) $	3							3
	$\Q(\sqrt{-5}) $	100.00%							0.06%
	$\Q(\sqrt{-43}) $	47	51	91	161		1		351
	$\Q(\sqrt{-43}) $	13.39%	14.53%	25.93%	45.87%		0.28%		7.58%
	$\Q(\sqrt{-67}) $	39	40	55				1	135
	$\Q(\sqrt{-67}) $	28.89%	29.63%	40.74%				0.74%	2.92%
	$\Q(\sqrt{-163}) $	25	3						28
	$\Q(\sqrt{-163}) $	89.29%	10.71%						0.60%

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Distribution of Bianchi modular forms by base field and level norm

Distribution of computed $\operatorname{GL}_2$ levels by base field and level norm

Distribution of computed $\operatorname{SL}_2$ levels by base field and level norm

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