LMFDB - Embedded newform 1849.4.a.i.1.50

Newspace parameters

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.50
Character	$\chi$	$=$	1849.1

$q$-expansion

$f(q)$	$=$	$q+5.33237 q^{2} -5.38049 q^{3} +20.4342 q^{4} -1.13957 q^{5} -28.6907 q^{6} -19.4433 q^{7} +66.3036 q^{8} +1.94964 q^{9} +O(q^{10})$	\(q+5.33237 q^{2} -5.38049 q^{3} +20.4342 q^{4} -1.13957 q^{5} -28.6907 q^{6} -19.4433 q^{7} +66.3036 q^{8} +1.94964 q^{9} -6.07658 q^{10} +11.2409 q^{11} -109.946 q^{12} +82.3500 q^{13} -103.679 q^{14} +6.13142 q^{15} +190.082 q^{16} -132.630 q^{17} +10.3962 q^{18} -41.5526 q^{19} -23.2861 q^{20} +104.614 q^{21} +59.9405 q^{22} +97.0017 q^{23} -356.746 q^{24} -123.701 q^{25} +439.121 q^{26} +134.783 q^{27} -397.308 q^{28} +94.9288 q^{29} +32.6950 q^{30} -265.836 q^{31} +483.159 q^{32} -60.4813 q^{33} -707.235 q^{34} +22.1569 q^{35} +39.8392 q^{36} +70.0555 q^{37} -221.574 q^{38} -443.083 q^{39} -75.5573 q^{40} +30.4482 q^{41} +557.843 q^{42} +229.698 q^{44} -2.22174 q^{45} +517.249 q^{46} -418.189 q^{47} -1022.73 q^{48} +35.0417 q^{49} -659.622 q^{50} +713.617 q^{51} +1682.75 q^{52} -399.944 q^{53} +718.714 q^{54} -12.8097 q^{55} -1289.16 q^{56} +223.573 q^{57} +506.195 q^{58} -57.1424 q^{59} +125.290 q^{60} -331.172 q^{61} -1417.54 q^{62} -37.9074 q^{63} +1055.73 q^{64} -93.8432 q^{65} -322.509 q^{66} +156.263 q^{67} -2710.19 q^{68} -521.916 q^{69} +118.149 q^{70} +564.356 q^{71} +129.268 q^{72} -69.0808 q^{73} +373.562 q^{74} +665.574 q^{75} -849.093 q^{76} -218.559 q^{77} -2362.68 q^{78} -569.629 q^{79} -216.611 q^{80} -777.839 q^{81} +162.361 q^{82} -785.399 q^{83} +2137.71 q^{84} +151.141 q^{85} -510.763 q^{87} +745.310 q^{88} +429.427 q^{89} -11.8471 q^{90} -1601.15 q^{91} +1982.15 q^{92} +1430.33 q^{93} -2229.94 q^{94} +47.3519 q^{95} -2599.63 q^{96} -1161.42 q^{97} +186.855 q^{98} +21.9156 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) $	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.33237	1.88528	0.942639	−	0.333814i	$-0.108336\pi$
$2$	5.33237	1.88528	0.942639	+	0.333814i	$0.108336\pi$
$3$	−5.38049	−1.03548	−0.517738	−	0.855539i	$-0.673226\pi$
$3$	−5.38049	−1.03548	−0.517738	+	0.855539i	$0.673226\pi$
$4$	20.4342	2.55427
$5$	−1.13957	−0.101926	−0.0509629	−	0.998701i	$-0.516229\pi$
$5$	−1.13957	−0.101926	−0.0509629	+	0.998701i	$0.516229\pi$
$6$	−28.6907	−1.95216
$7$	−19.4433	−1.04984	−0.524920	−	0.851152i	$-0.675905\pi$
$7$	−19.4433	−1.04984	−0.524920	+	0.851152i	$0.675905\pi$
$8$	66.3036	2.93023
$9$	1.94964	0.0722088
$10$	−6.07658	−0.192158
$11$	11.2409	0.308113	0.154057	−	0.988062i	$-0.450766\pi$
$11$	11.2409	0.308113	0.154057	+	0.988062i	$0.450766\pi$
$12$	−109.946	−2.64488
$13$	82.3500	1.75691	0.878453	−	0.477829i	$-0.158576\pi$
$13$	82.3500	1.75691	0.878453	+	0.477829i	$0.158576\pi$
$14$	−103.679	−1.97924
$15$	6.13142	0.105542
$16$	190.082	2.97003
$17$	−132.630	−1.89221	−0.946106	−	0.323856i	$-0.895021\pi$
$17$	−132.630	−1.89221	−0.946106	+	0.323856i	$0.895021\pi$
$18$	10.3962	0.136134
$19$	−41.5526	−0.501727	−0.250864	−	0.968022i	$-0.580715\pi$
$19$	−41.5526	−0.501727	−0.250864	+	0.968022i	$0.580715\pi$
$20$	−23.2861	−0.260346
$21$	104.614	1.08708
$22$	59.9405	0.580879
$23$	97.0017	0.879402	0.439701	−	0.898144i	$-0.355084\pi$
$23$	97.0017	0.879402	0.439701	+	0.898144i	$0.355084\pi$
$24$	−356.746	−3.03418
$25$	−123.701	−0.989611
$26$	439.121	3.31225
$27$	134.783	0.960705
$28$	−397.308	−2.68157
$29$	94.9288	0.607856	0.303928	−	0.952695i	$-0.401702\pi$
$29$	94.9288	0.607856	0.303928	+	0.952695i	$0.401702\pi$
$30$	32.6950	0.198975
$31$	−265.836	−1.54018	−0.770091	−	0.637935i	$-0.779789\pi$
$31$	−265.836	−1.54018	−0.770091	+	0.637935i	$0.779789\pi$
$32$	483.159	2.66910
$33$	−60.4813	−0.319044
$34$	−707.235	−3.56735
$35$	22.1569	0.107006
$36$	39.8392	0.184441
$37$	70.0555	0.311272	0.155636	−	0.987814i	$-0.450257\pi$
$37$	70.0555	0.311272	0.155636	+	0.987814i	$0.450257\pi$
$38$	−221.574	−0.945896
$39$	−443.083	−1.81923
$40$	−75.5573	−0.298666
$41$	30.4482	0.115981	0.0579903	−	0.998317i	$-0.481531\pi$
$41$	30.4482	0.115981	0.0579903	+	0.998317i	$0.481531\pi$
$42$	557.843	2.04945
$43$	0	0
$43$	0	0
$44$	229.698	0.787006
$45$	−2.22174	−0.00735994
$46$	517.249	1.65792
$47$	−418.189	−1.29785	−0.648927	−	0.760850i	$-0.724782\pi$
$47$	−418.189	−1.29785	−0.648927	+	0.760850i	$0.724782\pi$
$48$	−1022.73	−3.07539
$49$	35.0417	0.102162
$50$	−659.622	−1.86569
$51$	713.617	1.95934
$52$	1682.75	4.48761
$53$	−399.944	−1.03654	−0.518269	−	0.855218i	$-0.673423\pi$
$53$	−399.944	−1.03654	−0.518269	+	0.855218i	$0.673423\pi$
$54$	718.714	1.81120
$55$	−12.8097	−0.0314047
$56$	−1289.16	−3.07627
$57$	223.573	0.519526
$58$	506.195	1.14598
$59$	−57.1424	−0.126090	−0.0630450	−	0.998011i	$-0.520081\pi$
$59$	−57.1424	−0.126090	−0.0630450	+	0.998011i	$0.520081\pi$
$60$	125.290	0.269582
$61$	−331.172	−0.695118	−0.347559	−	0.937658i	$-0.612989\pi$
$61$	−331.172	−0.695118	−0.347559	+	0.937658i	$0.612989\pi$
$62$	−1417.54	−2.90367
$63$	−37.9074	−0.0758077
$64$	1055.73	2.06196
$65$	−93.8432	−0.179074
$66$	−322.509	−0.601486
$67$	156.263	0.284934	0.142467	−	0.989800i	$-0.454496\pi$
$67$	156.263	0.284934	0.142467	+	0.989800i	$0.454496\pi$
$68$	−2710.19	−4.83322
$69$	−521.916	−0.910599
$70$	118.149	0.201735
$71$	564.356	0.943335	0.471667	−	0.881777i	$-0.343652\pi$
$71$	564.356	0.943335	0.471667	+	0.881777i	$0.343652\pi$
$72$	129.268	0.211589
$73$	−69.0808	−0.110758	−0.0553788	−	0.998465i	$-0.517637\pi$
$73$	−69.0808	−0.110758	−0.0553788	+	0.998465i	$0.517637\pi$
$74$	373.562	0.586834
$75$	665.574	1.02472
$76$	−849.093	−1.28155
$77$	−218.559	−0.323470
$78$	−2362.68	−3.42976
$79$	−569.629	−0.811244	−0.405622	−	0.914041i	$-0.632945\pi$
$79$	−569.629	−0.811244	−0.405622	+	0.914041i	$0.632945\pi$
$80$	−216.611	−0.302723
$81$	−777.839	−1.06699
$82$	162.361	0.218656
$83$	−785.399	−1.03866	−0.519330	−	0.854574i	$-0.673818\pi$
$83$	−785.399	−1.03866	−0.519330	+	0.854574i	$0.673818\pi$
$84$	2137.71	2.77670
$85$	151.141	0.192865
$86$	0	0
$87$	−510.763	−0.629420
$88$	745.310	0.902844
$89$	429.427	0.511451	0.255726	−	0.966749i	$-0.417686\pi$
$89$	429.427	0.511451	0.255726	+	0.966749i	$0.417686\pi$
$90$	−11.8471	−0.0138755
$91$	−1601.15	−1.84447
$92$	1982.15	2.24623
$93$	1430.33	1.59482
$94$	−2229.94	−2.44682
$95$	47.3519	0.0511390
$96$	−2599.63	−2.76379
$97$	−1161.42	−1.21572	−0.607858	−	0.794046i	$-0.707971\pi$
$97$	−1161.42	−1.21572	−0.607858	+	0.794046i	$0.707971\pi$
$98$	186.855	0.192604
$99$	21.9156	0.0222485
$100$	−2527.74	−2.52774
$101$	190.011	0.187196	0.0935978	−	0.995610i	$-0.470163\pi$
$101$	190.011	0.187196	0.0935978	+	0.995610i	$0.470163\pi$
$102$	3805.27	3.69390
$103$	−552.932	−0.528952	−0.264476	−	0.964392i	$-0.585199\pi$
$103$	−552.932	−0.528952	−0.264476	+	0.964392i	$0.585199\pi$
$104$	5460.10	5.14814
$105$	−119.215	−0.110802
$106$	−2132.65	−1.95416
$107$	−1660.97	−1.50067	−0.750336	−	0.661057i	$-0.770108\pi$
$107$	−1660.97	−1.50067	−0.750336	+	0.661057i	$0.770108\pi$
$108$	2754.18	2.45390
$109$	−851.383	−0.748144	−0.374072	−	0.927400i	$-0.622039\pi$
$109$	−851.383	−0.748144	−0.374072	+	0.927400i	$0.622039\pi$
$110$	−68.3061	−0.0592066
$111$	−376.933	−0.322314
$112$	−3695.82	−3.11806
$113$	−336.677	−0.280282	−0.140141	−	0.990132i	$-0.544756\pi$
$113$	−336.677	−0.280282	−0.140141	+	0.990132i	$0.544756\pi$
$114$	1192.18	0.979451
$115$	−110.540	−0.0896337
$116$	1939.79	1.55263
$117$	160.553	0.126864
$118$	−304.705	−0.237715
$119$	2578.77	1.98652
$120$	406.535	0.309262
$121$	−1204.64	−0.905066
$122$	−1765.93	−1.31049
$123$	−163.826	−0.120095
$124$	−5432.15	−3.93404
$125$	283.411	0.202793
$126$	−202.136	−0.142918
$127$	1906.99	1.33242	0.666212	−	0.745762i	$-0.267914\pi$
$127$	1906.99	1.33242	0.666212	+	0.745762i	$0.267914\pi$
$128$	1764.25	1.21827
$129$	0	0
$130$	−500.407	−0.337604
$131$	−1298.17	−0.865815	−0.432908	−	0.901438i	$-0.642512\pi$
$131$	−1298.17	−0.865815	−0.432908	+	0.901438i	$0.642512\pi$
$132$	−1235.89	−0.814925
$133$	807.920	0.526733
$134$	833.253	0.537180
$135$	−153.594	−0.0979206
$136$	−8793.88	−5.54462
$137$	259.404	0.161769	0.0808847	−	0.996723i	$-0.474225\pi$
$137$	259.404	0.161769	0.0808847	+	0.996723i	$0.474225\pi$
$138$	−2783.05	−1.71673
$139$	−52.2455	−0.0318806	−0.0159403	−	0.999873i	$-0.505074\pi$
$139$	−52.2455	−0.0318806	−0.0159403	+	0.999873i	$0.505074\pi$
$140$	452.758	0.273322
$141$	2250.06	1.34390
$142$	3009.36	1.77845
$143$	925.685	0.541326
$144$	370.591	0.214462
$145$	−108.178	−0.0619562
$146$	−368.365	−0.208809
$147$	−188.541	−0.105787
$148$	1431.53	0.795073
$149$	3239.65	1.78123	0.890613	−	0.454762i	$-0.150276\pi$
$149$	3239.65	1.78123	0.890613	+	0.454762i	$0.150276\pi$
$150$	3549.09	1.93188
$151$	−2294.66	−1.23667	−0.618334	−	0.785915i	$-0.712192\pi$
$151$	−2294.66	−1.23667	−0.618334	+	0.785915i	$0.712192\pi$
$152$	−2755.09	−1.47018
$153$	−258.581	−0.136634
$154$	−1165.44	−0.609830
$155$	302.938	0.156984
$156$	−9054.03	−4.64681
$157$	−3333.45	−1.69451	−0.847257	−	0.531184i	$-0.821747\pi$
$157$	−3333.45	−1.69451	−0.847257	+	0.531184i	$0.821747\pi$
$158$	−3037.47	−1.52942
$159$	2151.89	1.07331
$160$	−550.591	−0.272050
$161$	−1886.03	−0.923231
$162$	−4147.73	−2.01158
$163$	1886.07	0.906310	0.453155	−	0.891432i	$-0.350298\pi$
$163$	1886.07	0.906310	0.453155	+	0.891432i	$0.350298\pi$
$164$	622.183	0.296246
$165$	68.9224	0.0325188
$166$	−4188.04	−1.95816
$167$	−628.615	−0.291280	−0.145640	−	0.989338i	$-0.546524\pi$
$167$	−628.615	−0.291280	−0.145640	+	0.989338i	$0.546524\pi$
$168$	6936.31	3.18541
$169$	4584.52	2.08672
$170$	805.940	0.363605
$171$	−81.0126	−0.0362292
$172$	0	0
$173$	2010.27	0.883456	0.441728	−	0.897149i	$-0.354366\pi$
$173$	2010.27	0.883456	0.441728	+	0.897149i	$0.354366\pi$
$174$	−2723.58	−1.18663
$175$	2405.16	1.03893
$176$	2136.69	0.915107
$177$	307.454	0.130563
$178$	2289.86	0.964228
$179$	−1442.47	−0.602320	−0.301160	−	0.953574i	$-0.597374\pi$
$179$	−1442.47	−0.602320	−0.301160	+	0.953574i	$0.597374\pi$
$180$	−45.3994	−0.0187993
$181$	−1243.78	−0.510770	−0.255385	−	0.966839i	$-0.582202\pi$
$181$	−1243.78	−0.510770	−0.255385	+	0.966839i	$0.582202\pi$
$182$	−8537.95	−3.47734
$183$	1781.86	0.719777
$184$	6431.56	2.57685
$185$	−79.8329	−0.0317266
$186$	7627.04	3.00668
$187$	−1490.88	−0.583016
$188$	−8545.35	−3.31507
$189$	−2620.63	−1.00859
$190$	252.498	0.0964112
$191$	−3169.52	−1.20072	−0.600362	−	0.799728i	$-0.704977\pi$
$191$	−3169.52	−1.20072	−0.600362	+	0.799728i	$0.704977\pi$
$192$	−5680.32	−2.13511
$193$	−2034.25	−0.758698	−0.379349	−	0.925254i	$-0.623852\pi$
$193$	−2034.25	−0.758698	−0.379349	+	0.925254i	$0.623852\pi$
$194$	−6193.13	−2.29196
$195$	504.922	0.185427
$196$	716.048	0.260950
$197$	3188.91	1.15330	0.576650	−	0.816991i	$-0.304360\pi$
$197$	3188.91	1.15330	0.576650	+	0.816991i	$0.304360\pi$
$198$	116.862	0.0419446
$199$	1546.27	0.550814	0.275407	−	0.961328i	$-0.411187\pi$
$199$	1546.27	0.550814	0.275407	+	0.961328i	$0.411187\pi$
$200$	−8201.85	−2.89979
$201$	−840.772	−0.295042
$202$	1013.21	0.352916
$203$	−1845.73	−0.638151
$204$	14582.2	5.00468
$205$	−34.6977	−0.0118214
$206$	−2948.44	−0.997221
$207$	189.118	0.0635006
$208$	15653.3	5.21807
$209$	−467.087	−0.154589
$210$	−635.698	−0.208892
$211$	−4947.03	−1.61406	−0.807032	−	0.590508i	$-0.798928\pi$
$211$	−4947.03	−1.61406	−0.807032	+	0.590508i	$0.798928\pi$
$212$	−8172.52	−2.64760
$213$	−3036.51	−0.976800
$214$	−8856.90	−2.82918
$215$	0	0
$216$	8936.61	2.81509
$217$	5168.73	1.61694
$218$	−4539.89	−1.41046
$219$	371.688	0.114687
$220$	−261.756	−0.0802162
$221$	−10922.1	−3.32444
$222$	−2009.95	−0.607652
$223$	1020.68	0.306501	0.153251	−	0.988187i	$-0.451026\pi$
$223$	1020.68	0.306501	0.153251	+	0.988187i	$0.451026\pi$
$224$	−9394.20	−2.80213
$225$	−241.173	−0.0714587
$226$	−1795.29	−0.528410
$227$	−1897.59	−0.554836	−0.277418	−	0.960749i	$-0.589479\pi$
$227$	−1897.59	−0.554836	−0.277418	+	0.960749i	$0.589479\pi$
$228$	4568.53	1.32701
$229$	3784.45	1.09207	0.546034	−	0.837763i	$-0.316137\pi$
$229$	3784.45	1.09207	0.546034	+	0.837763i	$0.316137\pi$
$230$	−589.439	−0.168984
$231$	1175.96	0.334945
$232$	6294.12	1.78116
$233$	5821.37	1.63678	0.818391	−	0.574661i	$-0.194866\pi$
$233$	5821.37	1.63678	0.818391	+	0.574661i	$0.194866\pi$
$234$	856.126	0.239174
$235$	476.554	0.132285
$236$	−1167.66	−0.322068
$237$	3064.88	0.840023
$238$	13751.0	3.74514
$239$	3277.24	0.886975	0.443488	−	0.896280i	$-0.353741\pi$
$239$	3277.24	0.886975	0.443488	+	0.896280i	$0.353741\pi$
$240$	1165.47	0.313462
$241$	3524.53	0.942054	0.471027	−	0.882119i	$-0.343883\pi$
$241$	3524.53	0.942054	0.471027	+	0.882119i	$0.343883\pi$
$242$	−6423.60	−1.70630
$243$	546.008	0.144142
$244$	−6767.22	−1.77552
$245$	−39.9323	−0.0104130
$246$	−873.581	−0.226413
$247$	−3421.86	−0.881488
$248$	−17625.9	−4.51309
$249$	4225.83	1.07551
$250$	1511.25	0.382321
$251$	658.193	0.165517	0.0827585	−	0.996570i	$-0.473627\pi$
$251$	658.193	0.165517	0.0827585	+	0.996570i	$0.473627\pi$
$252$	−774.606	−0.193633
$253$	1090.38	0.270956
$254$	10168.8	2.51199
$255$	−813.213	−0.199707
$256$	961.825	0.234821
$257$	−4247.86	−1.03103	−0.515514	−	0.856881i	$-0.672399\pi$
$257$	−4247.86	−1.03103	−0.515514	+	0.856881i	$0.672399\pi$
$258$	0	0
$259$	−1362.11	−0.326785
$260$	−1917.61	−0.457404
$261$	185.077	0.0438926
$262$	−6922.33	−1.63230
$263$	−2058.13	−0.482547	−0.241274	−	0.970457i	$-0.577565\pi$
$263$	−2058.13	−0.482547	−0.241274	+	0.970457i	$0.577565\pi$
$264$	−4010.13	−0.934873
$265$	455.762	0.105650
$266$	4308.13	0.993038
$267$	−2310.53	−0.529595
$268$	3193.11	0.727799
$269$	4941.92	1.12013	0.560064	−	0.828450i	$-0.310777\pi$
$269$	4941.92	1.12013	0.560064	+	0.828450i	$0.310777\pi$
$270$	−819.021	−0.184608
$271$	7166.71	1.60644	0.803222	−	0.595679i	$-0.203117\pi$
$271$	7166.71	1.60644	0.803222	+	0.595679i	$0.203117\pi$
$272$	−25210.7	−5.61993
$273$	8614.99	1.90990
$274$	1383.24	0.304980
$275$	−1390.51	−0.304913
$276$	−10664.9	−2.32592
$277$	1097.77	0.238118	0.119059	−	0.992887i	$-0.462012\pi$
$277$	1097.77	0.238118	0.119059	+	0.992887i	$0.462012\pi$
$278$	−278.592	−0.0601038
$279$	−518.285	−0.111215
$280$	1469.08	0.313552
$281$	−3052.94	−0.648125	−0.324062	−	0.946036i	$-0.605049\pi$
$281$	−3052.94	−0.648125	−0.324062	+	0.946036i	$0.605049\pi$
$282$	11998.2	2.53362
$283$	−3952.97	−0.830316	−0.415158	−	0.909749i	$-0.636274\pi$
$283$	−3952.97	−0.830316	−0.415158	+	0.909749i	$0.636274\pi$
$284$	11532.2	2.40953
$285$	−254.776	−0.0529531
$286$	4936.10	1.02055
$287$	−592.013	−0.121761
$288$	941.985	0.192733
$289$	12677.8	2.58047
$290$	−576.843	−0.116805
$291$	6249.01	1.25884
$292$	−1411.61	−0.282905
$293$	−229.679	−0.0457953	−0.0228976	−	0.999738i	$-0.507289\pi$
$293$	−229.679	−0.0457953	−0.0228976	+	0.999738i	$0.507289\pi$
$294$	−1005.37	−0.199437
$295$	65.1175	0.0128518
$296$	4644.93	0.912099
$297$	1515.08	0.296006
$298$	17275.0	3.35810
$299$	7988.08	1.54503
$300$	13600.4	2.61741
$301$	0	0
$302$	−12236.0	−2.33146
$303$	−1022.35	−0.193836
$304$	−7898.40	−1.49015
$305$	377.392	0.0708504
$306$	−1378.85	−0.257594
$307$	3264.44	0.606878	0.303439	−	0.952851i	$-0.401865\pi$
$307$	3264.44	0.606878	0.303439	+	0.952851i	$0.401865\pi$
$308$	−4466.08	−0.826229
$309$	2975.05	0.547717
$310$	1615.38	0.295959
$311$	6556.37	1.19543	0.597713	−	0.801710i	$-0.296076\pi$
$311$	6556.37	1.19543	0.597713	+	0.801710i	$0.296076\pi$
$312$	−29378.0	−5.33077
$313$	7030.21	1.26956	0.634778	−	0.772694i	$-0.281091\pi$
$313$	7030.21	1.26956	0.634778	+	0.772694i	$0.281091\pi$
$314$	−17775.2	−3.19463
$315$	43.1979	0.00772676
$316$	−11639.9	−2.07214
$317$	6485.38	1.14907	0.574535	−	0.818480i	$-0.305183\pi$
$317$	6485.38	1.14907	0.574535	+	0.818480i	$0.305183\pi$
$318$	11474.7	2.02349
$319$	1067.08	0.187289
$320$	−1203.07	−0.210167
$321$	8936.82	1.55391
$322$	−10057.0	−1.74055
$323$	5511.14	0.949375
$324$	−15894.5	−2.72539
$325$	−10186.8	−1.73865
$326$	10057.2	1.70865
$327$	4580.85	0.774684
$328$	2018.82	0.339850
$329$	8130.98	1.36254
$330$	367.520	0.0613070
$331$	5097.58	0.846490	0.423245	−	0.906015i	$-0.360891\pi$
$331$	5097.58	0.846490	0.423245	+	0.906015i	$0.360891\pi$
$332$	−16049.0	−2.65302
$333$	136.583	0.0224766
$334$	−3352.01	−0.549143
$335$	−178.072	−0.0290422
$336$	19885.3	3.22867
$337$	−6280.94	−1.01527	−0.507633	−	0.861574i	$-0.669479\pi$
$337$	−6280.94	−1.01527	−0.507633	+	0.861574i	$0.669479\pi$
$338$	24446.4	3.93404
$339$	1811.49	0.290225
$340$	3088.44	0.492630
$341$	−2988.23	−0.474551
$342$	−431.989	−0.0683020
$343$	5987.72	0.942585
$344$	0	0
$345$	594.757	0.0928135
$346$	10719.5	1.66556
$347$	2180.02	0.337261	0.168631	−	0.985679i	$-0.446065\pi$
$347$	2180.02	0.337261	0.168631	+	0.985679i	$0.446065\pi$
$348$	−10437.0	−1.60771
$349$	−9644.24	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$349$	−9644.24	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$350$	12825.2	1.95868
$351$	11099.4	1.68787
$352$	5431.12	0.822386
$353$	−5127.42	−0.773103	−0.386551	−	0.922268i	$-0.626334\pi$
$353$	−5127.42	−0.773103	−0.386551	+	0.922268i	$0.626334\pi$
$354$	1639.46	0.246148
$355$	−643.121	−0.0961502
$356$	8774.98	1.30639
$357$	−13875.1	−2.05699
$358$	−7691.79	−1.13554
$359$	−5232.56	−0.769259	−0.384629	−	0.923071i	$-0.625671\pi$
$359$	−5232.56	−0.769259	−0.384629	+	0.923071i	$0.625671\pi$
$360$	−147.309	−0.0215664
$361$	−5132.38	−0.748270
$362$	−6632.30	−0.962944
$363$	6481.57	0.937173
$364$	−32718.3	−4.71127
$365$	78.7221	0.0112890
$366$	9501.56	1.35698
$367$	5608.15	0.797666	0.398833	−	0.917024i	$-0.369415\pi$
$367$	5608.15	0.797666	0.398833	+	0.917024i	$0.369415\pi$
$368$	18438.3	2.61185
$369$	59.3629	0.00837483
$370$	−425.698	−0.0598135
$371$	7776.23	1.08820
$372$	29227.6	4.07360
$373$	5236.17	0.726860	0.363430	−	0.931621i	$-0.381606\pi$
$373$	5236.17	0.726860	0.363430	+	0.931621i	$0.381606\pi$
$374$	−7949.93	−1.09915
$375$	−1524.89	−0.209987
$376$	−27727.5	−3.80302
$377$	7817.38	1.06795
$378$	−13974.2	−1.90146
$379$	−3323.14	−0.450392	−0.225196	−	0.974314i	$-0.572302\pi$
$379$	−3323.14	−0.450392	−0.225196	+	0.974314i	$0.572302\pi$
$380$	967.597	0.130623
$381$	−10260.5	−1.37969
$382$	−16901.0	−2.26370
$383$	−7934.47	−1.05857	−0.529285	−	0.848444i	$-0.677540\pi$
$383$	−7934.47	−1.05857	−0.529285	+	0.848444i	$0.677540\pi$
$384$	−9492.52	−1.26149
$385$	249.063	0.0329699
$386$	−10847.4	−1.43036
$387$	0	0
$388$	−23732.7	−3.10527
$389$	−2071.19	−0.269958	−0.134979	−	0.990848i	$-0.543097\pi$
$389$	−2071.19	−0.269958	−0.134979	+	0.990848i	$0.543097\pi$
$390$	2692.43	0.349581
$391$	−12865.4	−1.66402
$392$	2323.39	0.299360
$393$	6984.80	0.896530
$394$	17004.4	2.17429
$395$	649.129	0.0826867
$396$	447.828	0.0568287
$397$	7588.62	0.959349	0.479675	−	0.877446i	$-0.340755\pi$
$397$	7588.62	0.959349	0.479675	+	0.877446i	$0.340755\pi$
$398$	8245.27	1.03844
$399$	−4347.00	−0.545419
$400$	−23513.4	−2.93918
$401$	−2081.48	−0.259212	−0.129606	−	0.991566i	$-0.541371\pi$
$401$	−2081.48	−0.259212	−0.129606	+	0.991566i	$0.541371\pi$
$402$	−4483.31	−0.556237
$403$	−21891.6	−2.70595
$404$	3882.71	0.478148
$405$	886.398	0.108754
$406$	−9842.11	−1.20309
$407$	787.485	0.0959071
$408$	47315.4	5.74132
$409$	3217.52	0.388988	0.194494	−	0.980904i	$-0.437694\pi$
$409$	3217.52	0.388988	0.194494	+	0.980904i	$0.437694\pi$
$410$	−185.021	−0.0222867
$411$	−1395.72	−0.167508
$412$	−11298.7	−1.35109
$413$	1111.04	0.132374
$414$	1008.45	0.119716
$415$	895.013	0.105866
$416$	39788.1	4.68936
$417$	281.106	0.0330116
$418$	−2490.68	−0.291443
$419$	7694.73	0.897165	0.448582	−	0.893741i	$-0.351929\pi$
$419$	7694.73	0.897165	0.448582	+	0.893741i	$0.351929\pi$
$420$	−2436.06	−0.283018
$421$	−1838.05	−0.212781	−0.106391	−	0.994324i	$-0.533929\pi$
$421$	−1838.05	−0.212781	−0.106391	+	0.994324i	$0.533929\pi$
$422$	−26379.4	−3.04296
$423$	−815.318	−0.0937166
$424$	−26517.7	−3.03730
$425$	16406.6	1.87255
$426$	−16191.8	−1.84154
$427$	6439.07	0.729762
$428$	−33940.5	−3.83312
$429$	−4980.64	−0.560530
$430$	0	0
$431$	−4840.45	−0.540966	−0.270483	−	0.962725i	$-0.587183\pi$
$431$	−4840.45	−0.540966	−0.270483	+	0.962725i	$0.587183\pi$
$432$	25619.9	2.85332
$433$	3091.82	0.343148	0.171574	−	0.985171i	$-0.445115\pi$
$433$	3091.82	0.343148	0.171574	+	0.985171i	$0.445115\pi$
$434$	27561.6	3.04839
$435$	582.048	0.0641541
$436$	−17397.3	−1.91096
$437$	−4030.67	−0.441220
$438$	1981.98	0.216216
$439$	−8495.03	−0.923566	−0.461783	−	0.886993i	$-0.652790\pi$
$439$	−8495.03	−0.923566	−0.461783	+	0.886993i	$0.652790\pi$
$440$	−849.329	−0.0920231
$441$	68.3186	0.00737702
$442$	−58240.8	−6.26749
$443$	−7752.60	−0.831461	−0.415730	−	0.909488i	$-0.636474\pi$
$443$	−7752.60	−0.831461	−0.415730	+	0.909488i	$0.636474\pi$
$444$	−7702.31	−0.823278
$445$	−489.360	−0.0521301
$446$	5442.64	0.577840
$447$	−17430.9	−1.84441
$448$	−20526.8	−2.16473
$449$	2654.77	0.279034	0.139517	−	0.990220i	$-0.455445\pi$
$449$	2654.77	0.279034	0.139517	+	0.990220i	$0.455445\pi$
$450$	−1286.02	−0.134719
$451$	342.264	0.0357352
$452$	−6879.72	−0.715917
$453$	12346.4	1.28054
$454$	−10118.7	−1.04602
$455$	1824.62	0.187999
$456$	14823.7	1.52233
$457$	−1223.05	−0.125190	−0.0625952	−	0.998039i	$-0.519938\pi$
$457$	−1223.05	−0.125190	−0.0625952	+	0.998039i	$0.519938\pi$
$458$	20180.1	2.05885
$459$	−17876.4	−1.81786
$460$	−2258.79	−0.228949
$461$	−15665.7	−1.58270	−0.791350	−	0.611364i	$-0.790621\pi$
$461$	−15665.7	−1.58270	−0.791350	+	0.611364i	$0.790621\pi$
$462$	6270.63	0.631464
$463$	7084.10	0.711072	0.355536	−	0.934663i	$-0.384298\pi$
$463$	7084.10	0.711072	0.355536	+	0.934663i	$0.384298\pi$
$464$	18044.3	1.80535
$465$	−1629.95	−0.162553
$466$	31041.7	3.08579
$467$	−9122.22	−0.903910	−0.451955	−	0.892041i	$-0.649273\pi$
$467$	−9122.22	−0.903910	−0.451955	+	0.892041i	$0.649273\pi$
$468$	3280.76	0.324045
$469$	−3038.27	−0.299135
$470$	2541.16	0.249394
$471$	17935.6	1.75463
$472$	−3788.75	−0.369473
$473$	0	0
$474$	16343.1	1.58368
$475$	5140.12	0.496515
$476$	52695.1	5.07411
$477$	−779.746	−0.0748472
$478$	17475.5	1.67219
$479$	5530.27	0.527525	0.263762	−	0.964588i	$-0.415037\pi$
$479$	5530.27	0.527525	0.263762	+	0.964588i	$0.415037\pi$
$480$	2962.45	0.281701
$481$	5769.07	0.546875
$482$	18794.1	1.77603
$483$	10147.8	0.955982
$484$	−24615.9	−2.31178
$485$	1323.52	0.123913
$486$	2911.52	0.271747
$487$	8915.86	0.829602	0.414801	−	0.909912i	$-0.363851\pi$
$487$	8915.86	0.829602	0.414801	+	0.909912i	$0.363851\pi$
$488$	−21957.9	−2.03686
$489$	−10148.0	−0.938462
$490$	−212.934	−0.0196314
$491$	10178.5	0.935535	0.467767	−	0.883852i	$-0.345058\pi$
$491$	10178.5	0.935535	0.467767	+	0.883852i	$0.345058\pi$
$492$	−3347.65	−0.306755
$493$	−12590.4	−1.15019
$494$	−18246.6	−1.66185
$495$	−24.9743	−0.00226770
$496$	−50530.7	−4.57439
$497$	−10972.9	−0.990350
$498$	22533.7	2.02763
$499$	2520.11	0.226083	0.113042	−	0.993590i	$-0.463941\pi$
$499$	2520.11	0.226083	0.113042	+	0.993590i	$0.463941\pi$
$500$	5791.28	0.517988
$501$	3382.25	0.301613
$502$	3509.73	0.312045
$503$	−13769.0	−1.22053	−0.610267	−	0.792196i	$-0.708938\pi$
$503$	−13769.0	−1.22053	−0.610267	+	0.792196i	$0.708938\pi$
$504$	−2513.40	−0.222134
$505$	−216.529	−0.0190801
$506$	5814.32	0.510826
$507$	−24666.9	−2.16074
$508$	38967.7	3.40337
$509$	−2487.17	−0.216585	−0.108292	−	0.994119i	$-0.534538\pi$
$509$	−2487.17	−0.216585	−0.108292	+	0.994119i	$0.534538\pi$
$510$	−4336.35	−0.376504
$511$	1343.16	0.116278
$512$	−8985.19	−0.775572
$513$	−5600.59	−0.482012
$514$	−22651.1	−1.94377
$515$	630.103	0.0539139
$516$	0	0
$517$	−4700.81	−0.399887
$518$	−7263.28	−0.616081
$519$	−10816.2	−0.914797
$520$	−6222.14	−0.524729
$521$	−10805.0	−0.908594	−0.454297	−	0.890850i	$-0.650109\pi$
$521$	−10805.0	−0.908594	−0.454297	+	0.890850i	$0.650109\pi$
$522$	986.898	0.0827497
$523$	−15831.2	−1.32361	−0.661805	−	0.749676i	$-0.730209\pi$
$523$	−15831.2	−1.32361	−0.661805	+	0.749676i	$0.730209\pi$
$524$	−26527.1	−2.21153
$525$	−12940.9	−1.07579
$526$	−10974.7	−0.909736
$527$	35258.0	2.91435
$528$	−11496.4	−0.947570
$529$	−2757.68	−0.226652
$530$	2430.29	0.199180
$531$	−111.407	−0.00910481
$532$	16509.2	1.34542
$533$	2507.41	0.203767
$534$	−12320.6	−0.998434
$535$	1892.78	0.152957
$536$	10360.8	0.834924
$537$	7761.20	0.623688
$538$	26352.2	2.11175
$539$	393.899	0.0314776
$540$	−3138.57	−0.250116
$541$	6016.44	0.478128	0.239064	−	0.971004i	$-0.423159\pi$
$541$	6016.44	0.478128	0.239064	+	0.971004i	$0.423159\pi$
$542$	38215.5	3.02859
$543$	6692.14	0.528890
$544$	−64081.6	−5.05051
$545$	970.206	0.0762552
$546$	45938.3	3.60069
$547$	13119.1	1.02547	0.512734	−	0.858547i	$-0.328633\pi$
$547$	13119.1	1.02547	0.512734	+	0.858547i	$0.328633\pi$
$548$	5300.71	0.413203
$549$	−645.665	−0.0501936
$550$	−7414.72	−0.574845
$551$	−3944.54	−0.304978
$552$	−34604.9	−2.66827
$553$	11075.5	0.851676
$554$	5853.73	0.448919
$555$	429.540	0.0328521
$556$	−1067.59	−0.0814317
$557$	−7200.35	−0.547736	−0.273868	−	0.961767i	$-0.588303\pi$
$557$	−7200.35	−0.547736	−0.273868	+	0.961767i	$0.588303\pi$
$558$	−2763.69	−0.209671
$559$	0	0
$560$	4211.63	0.317810
$561$	8021.67	0.603699
$562$	−16279.4	−1.22189
$563$	−22074.0	−1.65241	−0.826205	−	0.563369i	$-0.809505\pi$
$563$	−22074.0	−1.65241	−0.826205	+	0.563369i	$0.809505\pi$
$564$	45978.2	3.43268
$565$	383.665	0.0285680
$566$	−21078.7	−1.56538
$567$	15123.8	1.12017
$568$	37418.9	2.76419
$569$	13386.6	0.986286	0.493143	−	0.869948i	$-0.335848\pi$
$569$	13386.6	0.986286	0.493143	+	0.869948i	$0.335848\pi$
$570$	−1358.56	−0.0998314
$571$	22610.4	1.65712	0.828561	−	0.559899i	$-0.189160\pi$
$571$	22610.4	1.65712	0.828561	+	0.559899i	$0.189160\pi$
$572$	18915.6	1.38269
$573$	17053.6	1.24332
$574$	−3156.83	−0.229553
$575$	−11999.2	−0.870266
$576$	2058.28	0.148892
$577$	−1067.23	−0.0770006	−0.0385003	−	0.999259i	$-0.512258\pi$
$577$	−1067.23	−0.0770006	−0.0385003	+	0.999259i	$0.512258\pi$
$578$	67603.0	4.86490
$579$	10945.3	0.785613
$580$	−2210.52	−0.158253
$581$	15270.7	1.09043
$582$	33322.0	2.37327
$583$	−4495.72	−0.319371
$584$	−4580.31	−0.324545
$585$	−182.960	−0.0129307
$586$	−1224.74	−0.0863368
$587$	−6633.97	−0.466462	−0.233231	−	0.972421i	$-0.574930\pi$
$587$	−6633.97	−0.466462	−0.233231	+	0.972421i	$0.574930\pi$
$588$	−3852.69	−0.270208
$589$	11046.2	0.772751
$590$	347.231	0.0242293
$591$	−17157.9	−1.19421
$592$	13316.3	0.924487
$593$	20524.6	1.42133	0.710663	−	0.703533i	$-0.248395\pi$
$593$	20524.6	1.42133	0.710663	+	0.703533i	$0.248395\pi$
$594$	8078.96	0.558054
$595$	−2938.68	−0.202478
$596$	66199.6	4.54973
$597$	−8319.68	−0.570355
$598$	42595.4	2.91280
$599$	4328.83	0.295277	0.147639	−	0.989041i	$-0.452833\pi$
$599$	4328.83	0.295277	0.147639	+	0.989041i	$0.452833\pi$
$600$	44129.9	3.00266
$601$	22901.3	1.55435	0.777174	−	0.629286i	$-0.216652\pi$
$601$	22901.3	1.55435	0.777174	+	0.629286i	$0.216652\pi$
$602$	0	0
$603$	304.657	0.0205748
$604$	−46889.5	−3.15879
$605$	1372.77	0.0922496
$606$	−5451.54	−0.365435
$607$	3876.30	0.259199	0.129600	−	0.991566i	$-0.458631\pi$
$607$	3876.30	0.259199	0.129600	+	0.991566i	$0.458631\pi$
$608$	−20076.5	−1.33916
$609$	9930.92	0.660790
$610$	2012.39	0.133573
$611$	−34437.9	−2.28021
$612$	−5283.90	−0.349002
$613$	26240.8	1.72897	0.864484	−	0.502660i	$-0.167645\pi$
$613$	26240.8	1.72897	0.864484	+	0.502660i	$0.167645\pi$
$614$	17407.2	1.14413
$615$	186.690	0.0122408
$616$	−14491.3	−0.947841
$617$	29037.2	1.89464	0.947321	−	0.320286i	$-0.103779\pi$
$617$	29037.2	1.89464	0.947321	+	0.320286i	$0.103779\pi$
$618$	15864.0	1.03260
$619$	951.665	0.0617942	0.0308971	−	0.999523i	$-0.490164\pi$
$619$	951.665	0.0617942	0.0308971	+	0.999523i	$0.490164\pi$
$620$	6190.28	0.400980
$621$	13074.2	0.844846
$622$	34961.0	2.25371
$623$	−8349.47	−0.536942
$624$	−84222.1	−5.40318
$625$	15139.7	0.968941
$626$	37487.7	2.39347
$627$	2513.16	0.160073
$628$	−68116.4	−4.32825
$629$	−9291.50	−0.588993
$630$	230.347	0.0145671
$631$	−7974.08	−0.503079	−0.251540	−	0.967847i	$-0.580937\pi$
$631$	−7974.08	−0.503079	−0.251540	+	0.967847i	$0.580937\pi$
$632$	−37768.5	−2.37713
$633$	26617.4	1.67132
$634$	34582.5	2.16632
$635$	−2173.14	−0.135808
$636$	43972.2	2.74152
$637$	2885.68	0.179490
$638$	5690.07	0.353091
$639$	1100.29	0.0681171
$640$	−2010.48	−0.124174
$641$	25568.9	1.57552	0.787761	−	0.615981i	$-0.211240\pi$
$641$	25568.9	1.57552	0.787761	+	0.615981i	$0.211240\pi$
$642$	47654.4	2.92955
$643$	−14114.5	−0.865663	−0.432832	−	0.901475i	$-0.642486\pi$
$643$	−14114.5	−0.865663	−0.432832	+	0.901475i	$0.642486\pi$
$644$	−38539.5	−2.35818
$645$	0	0
$646$	29387.5	1.78984
$647$	−4730.36	−0.287434	−0.143717	−	0.989619i	$-0.545905\pi$
$647$	−4730.36	−0.287434	−0.143717	+	0.989619i	$0.545905\pi$
$648$	−51573.5	−3.12654
$649$	−642.330	−0.0388500
$650$	−54319.8	−3.27784
$651$	−27810.3	−1.67430
$652$	38540.3	2.31496
$653$	1693.66	0.101498	0.0507490	−	0.998711i	$-0.483839\pi$
$653$	1693.66	0.101498	0.0507490	+	0.998711i	$0.483839\pi$
$654$	24426.8	1.46050
$655$	1479.35	0.0882489
$656$	5787.65	0.344466
$657$	−134.683	−0.00799767
$658$	43357.4	2.56876
$659$	−8066.13	−0.476801	−0.238400	−	0.971167i	$-0.576623\pi$
$659$	−8066.13	−0.476801	−0.238400	+	0.971167i	$0.576623\pi$
$660$	1408.37	0.0830619
$661$	−226.554	−0.0133312	−0.00666560	−	0.999978i	$-0.502122\pi$
$661$	−226.554	−0.0133312	−0.00666560	+	0.999978i	$0.502122\pi$
$662$	27182.2	1.59587
$663$	58766.3	3.44237
$664$	−52074.8	−3.04351
$665$	−920.677	−0.0536877
$666$	728.311	0.0423746
$667$	9208.25	0.534550
$668$	−12845.2	−0.744007
$669$	−5491.75	−0.317374
$670$	−949.547	−0.0547525
$671$	−3722.65	−0.214175
$672$	50545.4	2.90153
$673$	7867.98	0.450651	0.225326	−	0.974284i	$-0.427655\pi$
$673$	7867.98	0.450651	0.225326	+	0.974284i	$0.427655\pi$
$674$	−33492.3	−1.91406
$675$	−16672.9	−0.950724
$676$	93680.8	5.33004
$677$	20935.2	1.18848	0.594242	−	0.804287i	$-0.297452\pi$
$677$	20935.2	1.18848	0.594242	+	0.804287i	$0.297452\pi$
$678$	9659.52	0.547156
$679$	22581.9	1.27631
$680$	10021.2	0.565140
$681$	10210.0	0.574519
$682$	−15934.4	−0.894660
$683$	13418.1	0.751726	0.375863	−	0.926675i	$-0.377346\pi$
$683$	13418.1	0.751726	0.375863	+	0.926675i	$0.377346\pi$
$684$	−1655.42	−0.0925391
$685$	−295.608	−0.0164885
$686$	31928.8	1.77703
$687$	−20362.2	−1.13081
$688$	0	0
$689$	−32935.4	−1.82110
$690$	3171.47	0.174979
$691$	−28272.5	−1.55649	−0.778246	−	0.627960i	$-0.783890\pi$
$691$	−28272.5	−1.55649	−0.778246	+	0.627960i	$0.783890\pi$
$692$	41078.2	2.25659
$693$	−426.112	−0.0233574
$694$	11624.7	0.635831
$695$	59.5371	0.00324946
$696$	−33865.4	−1.84435
$697$	−4038.36	−0.219460
$698$	−51426.6	−2.78872
$699$	−31321.8	−1.69485
$700$	49147.5	2.65372
$701$	−8270.14	−0.445590	−0.222795	−	0.974865i	$-0.571518\pi$
$701$	−8270.14	−0.445590	−0.222795	+	0.974865i	$0.571518\pi$
$702$	59186.0	3.18210
$703$	−2910.99	−0.156174
$704$	11867.3	0.635319
$705$	−2564.09	−0.136978
$706$	−27341.3	−1.45751
$707$	−3694.43	−0.196525
$708$	6282.57	0.333493
$709$	−18435.9	−0.976549	−0.488274	−	0.872690i	$-0.662373\pi$
$709$	−18435.9	−0.976549	−0.488274	+	0.872690i	$0.662373\pi$
$710$	−3429.36	−0.181270
$711$	−1110.57	−0.0585790
$712$	28472.6	1.49867
$713$	−25786.6	−1.35444
$714$	−73986.9	−3.87800
$715$	−1054.88	−0.0551751
$716$	−29475.7	−1.53849
$717$	−17633.2	−0.918441
$718$	−27901.9	−1.45027
$719$	807.291	0.0418733	0.0209366	−	0.999781i	$-0.493335\pi$
$719$	807.291	0.0418733	0.0209366	+	0.999781i	$0.493335\pi$
$720$	−422.313	−0.0218593
$721$	10750.8	0.555314
$722$	−27367.8	−1.41070
$723$	−18963.7	−0.975474
$724$	−25415.6	−1.30465
$725$	−11742.8	−0.601541
$726$	34562.1	1.76683
$727$	−6429.96	−0.328025	−0.164012	−	0.986458i	$-0.552444\pi$
$727$	−6429.96	−0.328025	−0.164012	+	0.986458i	$0.552444\pi$
$728$	−106162.	−5.40472
$729$	18063.9	0.917739
$730$	419.775	0.0212830
$731$	0	0
$732$	36410.9	1.83851
$733$	21819.9	1.09951	0.549753	−	0.835327i	$-0.314722\pi$
$733$	21819.9	1.09951	0.549753	+	0.835327i	$0.314722\pi$
$734$	29904.8	1.50382
$735$	214.855	0.0107824
$736$	46867.2	2.34721
$737$	1756.53	0.0877921
$738$	316.545	0.0157889
$739$	−19531.5	−0.972228	−0.486114	−	0.873896i	$-0.661586\pi$
$739$	−19531.5	−0.972228	−0.486114	+	0.873896i	$0.661586\pi$
$740$	−1631.32	−0.0810384
$741$	18411.3	0.912759
$742$	41465.7	2.05156
$743$	−24083.9	−1.18917	−0.594585	−	0.804033i	$-0.702684\pi$
$743$	−24083.9	−1.18917	−0.594585	+	0.804033i	$0.702684\pi$
$744$	94836.0	4.67319
$745$	−3691.79	−0.181553
$746$	27921.2	1.37033
$747$	−1531.24	−0.0750004
$748$	−30464.9	−1.48918
$749$	32294.7	1.57546
$750$	−8131.29	−0.395883
$751$	12480.5	0.606417	0.303209	−	0.952924i	$-0.401942\pi$
$751$	12480.5	0.606417	0.303209	+	0.952924i	$0.401942\pi$
$752$	−79490.3	−3.85467
$753$	−3541.40	−0.171389
$754$	41685.2	2.01337
$755$	2614.92	0.126048
$756$	−53550.4	−2.57620
$757$	37150.9	1.78372	0.891858	−	0.452315i	$-0.149402\pi$
$757$	37150.9	1.78372	0.891858	+	0.452315i	$0.149402\pi$
$758$	−17720.2	−0.849113
$759$	−5866.79	−0.280568
$760$	3139.60	0.149849
$761$	32130.3	1.53052	0.765258	−	0.643724i	$-0.222612\pi$
$761$	32130.3	1.53052	0.765258	+	0.643724i	$0.222612\pi$
$762$	−54712.9	−2.60110
$763$	16553.7	0.785431
$764$	−64766.5	−3.06698
$765$	294.670	0.0139266
$766$	−42309.5	−1.99570
$767$	−4705.68	−0.221528
$768$	−5175.09	−0.243151
$769$	−19725.3	−0.924984	−0.462492	−	0.886624i	$-0.653045\pi$
$769$	−19725.3	−0.924984	−0.462492	+	0.886624i	$0.653045\pi$
$770$	1328.09	0.0621574
$771$	22855.5	1.06760
$772$	−41568.3	−1.93792
$773$	4215.71	0.196156	0.0980780	−	0.995179i	$-0.468731\pi$
$773$	4215.71	0.196156	0.0980780	+	0.995179i	$0.468731\pi$
$774$	0	0
$775$	32884.3	1.52418
$776$	−77006.4	−3.56233
$777$	7328.82	0.338378
$778$	−11044.4	−0.508945
$779$	−1265.20	−0.0581907
$780$	10317.7	0.473630
$781$	6343.85	0.290654
$782$	−68602.9	−3.13713
$783$	12794.8	0.583970
$784$	6660.79	0.303425
$785$	3798.69	0.172715
$786$	37245.5	1.69021
$787$	−35244.1	−1.59634	−0.798169	−	0.602434i	$-0.794198\pi$
$787$	−35244.1	−1.59634	−0.798169	+	0.602434i	$0.794198\pi$
$788$	65162.6	2.94584
$789$	11073.8	0.499666
$790$	3461.40	0.155887
$791$	6546.11	0.294251
$792$	1453.08	0.0651933
$793$	−27272.0	−1.22126
$794$	40465.3	1.80864
$795$	−2452.22	−0.109398
$796$	31596.7	1.40693
$797$	17073.8	0.758828	0.379414	−	0.925227i	$-0.376126\pi$
$797$	17073.8	0.758828	0.379414	+	0.925227i	$0.376126\pi$
$798$	−23179.8	−1.02827
$799$	55464.7	2.45582
$800$	−59767.4	−2.64137
$801$	837.227	0.0369313
$802$	−11099.2	−0.488687
$803$	−776.528	−0.0341259
$804$	−17180.5	−0.753618
$805$	2149.26	0.0941010
$806$	−116734.	−5.10147
$807$	−26589.9	−1.15986
$808$	12598.4	0.548527
$809$	12016.0	0.522202	0.261101	−	0.965312i	$-0.415914\pi$
$809$	12016.0	0.522202	0.261101	+	0.965312i	$0.415914\pi$
$810$	4726.60	0.205032
$811$	−41834.3	−1.81135	−0.905673	−	0.423976i	$-0.860634\pi$
$811$	−41834.3	−1.81135	−0.905673	+	0.423976i	$0.860634\pi$
$812$	−37715.9	−1.63001
$813$	−38560.4	−1.66343
$814$	4199.16	0.180811
$815$	−2149.30	−0.0923764
$816$	135646.	5.81930
$817$	0	0
$818$	17157.0	0.733350
$819$	−3121.67	−0.133187
$820$	−709.018	−0.0301951
$821$	26438.7	1.12389	0.561947	−	0.827173i	$-0.310052\pi$
$821$	26438.7	1.12389	0.561947	+	0.827173i	$0.310052\pi$
$822$	−7442.50	−0.315799
$823$	16854.2	0.713853	0.356926	−	0.934132i	$-0.383825\pi$
$823$	16854.2	0.713853	0.356926	+	0.934132i	$0.383825\pi$
$824$	−36661.4	−1.54995
$825$	7481.62	0.315729
$826$	5924.46	0.249562
$827$	−20262.0	−0.851970	−0.425985	−	0.904730i	$-0.640072\pi$
$827$	−20262.0	−0.851970	−0.425985	+	0.904730i	$0.640072\pi$
$828$	3864.47	0.162198
$829$	15273.0	0.639869	0.319935	−	0.947440i	$-0.396339\pi$
$829$	15273.0	0.639869	0.319935	+	0.947440i	$0.396339\pi$
$830$	4772.54	0.199587
$831$	−5906.55	−0.246566
$832$	86939.0	3.62268
$833$	−4647.60	−0.193313
$834$	1498.96	0.0622360
$835$	716.348	0.0296889
$836$	−9544.54	−0.394862
$837$	−35830.3	−1.47966
$838$	41031.1	1.69140
$839$	−38209.7	−1.57228	−0.786142	−	0.618046i	$-0.787924\pi$
$839$	−38209.7	−1.57228	−0.786142	+	0.618046i	$0.787924\pi$
$840$	−7904.38	−0.324675
$841$	−15377.5	−0.630511
$842$	−9801.15	−0.401152
$843$	16426.3	0.671117
$844$	−101088.	−4.12276
$845$	−5224.36	−0.212690
$846$	−4347.58	−0.176682
$847$	23422.2	0.950174
$848$	−76022.2	−3.07855
$849$	21268.9	0.859772
$850$	87485.9	3.53029
$851$	6795.50	0.273733
$852$	−62048.6	−2.49501
$853$	−37931.9	−1.52258	−0.761292	−	0.648409i	$-0.775435\pi$
$853$	−37931.9	−1.52258	−0.761292	+	0.648409i	$0.775435\pi$
$854$	34335.5	1.37580
$855$	92.3191	0.00369269
$856$	−110128.	−4.39732
$857$	13948.7	0.555984	0.277992	−	0.960583i	$-0.410331\pi$
$857$	13948.7	0.555984	0.277992	+	0.960583i	$0.410331\pi$
$858$	−26558.6	−1.05675
$859$	−15379.3	−0.610869	−0.305434	−	0.952213i	$-0.598802\pi$
$859$	−15379.3	−0.610869	−0.305434	+	0.952213i	$0.598802\pi$
$860$	0	0
$861$	3185.32	0.126081
$862$	−25811.1	−1.01987
$863$	4157.99	0.164009	0.0820043	−	0.996632i	$-0.473868\pi$
$863$	4157.99	0.164009	0.0820043	+	0.996632i	$0.473868\pi$
$864$	65121.7	2.56422
$865$	−2290.83	−0.0900470
$866$	16486.7	0.646930
$867$	−68213.0	−2.67201
$868$	105619.	4.13011
$869$	−6403.12	−0.249955
$870$	3103.69	0.120948
$871$	12868.3	0.500603
$872$	−56449.8	−2.19224
$873$	−2264.35	−0.0877854
$874$	−21493.0	−0.831822
$875$	−5510.45	−0.212900
$876$	7595.15	0.292941
$877$	−38274.3	−1.47370	−0.736848	−	0.676059i	$-0.763687\pi$
$877$	−38274.3	−1.47370	−0.736848	+	0.676059i	$0.763687\pi$
$878$	−45298.6	−1.74118
$879$	1235.79	0.0474199
$880$	−2434.89	−0.0932730
$881$	−16695.5	−0.638462	−0.319231	−	0.947677i	$-0.603425\pi$
$881$	−16695.5	−0.638462	−0.319231	+	0.947677i	$0.603425\pi$
$882$	364.300	0.0139077
$883$	−45928.1	−1.75040	−0.875200	−	0.483762i	$-0.839270\pi$
$883$	−45928.1	−1.75040	−0.875200	+	0.483762i	$0.839270\pi$
$884$	−223184.	−8.49152
$885$	−350.364	−0.0133077
$886$	−41339.7	−1.56753
$887$	14180.1	0.536779	0.268389	−	0.963311i	$-0.413509\pi$
$887$	14180.1	0.536779	0.268389	+	0.963311i	$0.413509\pi$
$888$	−24992.0	−0.944456
$889$	−37078.2	−1.39883
$890$	−2609.45	−0.0982797
$891$	−8743.58	−0.328755
$892$	20856.7	0.782887
$893$	17376.9	0.651170
$894$	−92948.0	−3.47723
$895$	1643.79	0.0613920
$896$	−34302.8	−1.27899
$897$	−42979.8	−1.59984
$898$	14156.2	0.526057
$899$	−25235.5	−0.936209
$900$	−4928.17	−0.182525
$901$	53044.8	1.96135
$902$	1825.08	0.0673708
$903$	0	0
$904$	−22322.9	−0.821293
$905$	1417.37	0.0520607
$906$	65835.5	2.41417
$907$	23302.1	0.853071	0.426535	−	0.904471i	$-0.359734\pi$
$907$	23302.1	0.853071	0.426535	+	0.904471i	$0.359734\pi$
$908$	−38775.8	−1.41720
$909$	370.452	0.0135172
$910$	9729.55	0.354430
$911$	26596.3	0.967260	0.483630	−	0.875273i	$-0.339318\pi$
$911$	26596.3	0.967260	0.483630	+	0.875273i	$0.339318\pi$
$912$	42497.3	1.54301
$913$	−8828.56	−0.320025
$914$	−6521.77	−0.236019
$915$	−2030.55	−0.0733639
$916$	77332.2	2.78944
$917$	25240.7	0.908967
$918$	−95323.3	−3.42717
$919$	40133.3	1.44056	0.720281	−	0.693683i	$-0.244013\pi$
$919$	40133.3	1.44056	0.720281	+	0.693683i	$0.244013\pi$
$920$	−7329.18	−0.262648
$921$	−17564.3	−0.628407
$922$	−83535.3	−2.98383
$923$	46474.7	1.65735
$924$	24029.7	0.855540
$925$	−8665.97	−0.308038
$926$	37775.1	1.34057
$927$	−1078.02	−0.0381950
$928$	45865.7	1.62243
$929$	8697.44	0.307162	0.153581	−	0.988136i	$-0.450919\pi$
$929$	8697.44	0.307162	0.153581	+	0.988136i	$0.450919\pi$
$930$	−8691.51	−0.306458
$931$	−1456.07	−0.0512577
$932$	118955.	4.18079
$933$	−35276.5	−1.23783
$934$	−48643.0	−1.70412
$935$	1698.96	0.0594244
$936$	10645.2	0.371741
$937$	−33520.7	−1.16870	−0.584351	−	0.811501i	$-0.698651\pi$
$937$	−33520.7	−1.16870	−0.584351	+	0.811501i	$0.698651\pi$
$938$	−16201.2	−0.563953
$939$	−37826.0	−1.31459
$940$	9737.99	0.337892
$941$	−24899.8	−0.862604	−0.431302	−	0.902208i	$-0.641946\pi$
$941$	−24899.8	−0.862604	−0.431302	+	0.902208i	$0.641946\pi$
$942$	95639.3	3.30796
$943$	2953.52	0.101994
$944$	−10861.7	−0.374491
$945$	2986.38	0.102801
$946$	0	0
$947$	−22969.3	−0.788175	−0.394087	−	0.919073i	$-0.628939\pi$
$947$	−22969.3	−0.788175	−0.394087	+	0.919073i	$0.628939\pi$
$948$	62628.3	2.14565
$949$	−5688.80	−0.194591
$950$	27409.0	0.936069
$951$	−34894.5	−1.18983
$952$	170982.	5.82096
$953$	23618.2	0.802799	0.401399	−	0.915903i	$-0.368524\pi$
$953$	23618.2	0.802799	0.401399	+	0.915903i	$0.368524\pi$
$954$	−4157.89	−0.141108
$955$	3611.87	0.122385
$956$	66967.7	2.26558
$957$	−5741.42	−0.193933
$958$	29489.4	0.994531
$959$	−5043.67	−0.169832
$960$	6473.09	0.217623
$961$	40878.0	1.37216
$962$	30762.8	1.03101
$963$	−3238.29	−0.108362
$964$	72020.9	2.40626
$965$	2318.17	0.0773310
$966$	54111.7	1.80229
$967$	41439.3	1.37807	0.689036	−	0.724727i	$-0.258034\pi$
$967$	41439.3	1.37807	0.689036	+	0.724727i	$0.258034\pi$
$968$	−79872.2	−2.65205
$969$	−29652.6	−0.983054
$970$	7057.47	0.233610
$971$	−47537.5	−1.57111	−0.785557	−	0.618790i	$-0.787623\pi$
$971$	−47537.5	−1.57111	−0.785557	+	0.618790i	$0.787623\pi$
$972$	11157.2	0.368177
$973$	1015.82	0.0334695
$974$	47542.7	1.56403
$975$	54810.0	1.80033
$976$	−62949.8	−2.06452
$977$	−46979.4	−1.53839	−0.769194	−	0.639016i	$-0.779342\pi$
$977$	−46979.4	−1.53839	−0.769194	+	0.639016i	$0.779342\pi$
$978$	−54112.8	−1.76926
$979$	4827.13	0.157585
$980$	−815.983	−0.0265976
$981$	−1659.89	−0.0540226
$982$	54275.3	1.76374
$983$	6249.99	0.202791	0.101396	−	0.994846i	$-0.467669\pi$
$983$	6249.99	0.202791	0.101396	+	0.994846i	$0.467669\pi$
$984$	−10862.3	−0.351907
$985$	−3633.97	−0.117551
$986$	−67136.9	−2.16843
$987$	−43748.6	−1.41088
$988$	−69922.8	−2.25156
$989$	0	0
$990$	−133.172	−0.00427524
$991$	−9689.00	−0.310576	−0.155288	−	0.987869i	$-0.549631\pi$
$991$	−9689.00	−0.310576	−0.155288	+	0.987869i	$0.549631\pi$
$992$	−128441.	−4.11090
$993$	−27427.5	−0.876519
$994$	−58511.8	−1.86708
$995$	−1762.07	−0.0561422
$996$	86351.3	2.74713
$997$	26955.2	0.856247	0.428124	−	0.903720i	$-0.359175\pi$
$997$	26955.2	0.856247	0.428124	+	0.903720i	$0.359175\pi$
$998$	13438.1	0.426229
$999$	9442.30	0.299040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.50	✓	50
43.42	odd	2		1849.4.a.j.1.1	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.50	✓	50	1.1	even	1	trivial
1849.4.a.j.1.1	yes	50	43.42	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+5.33237 q^{2} -5.38049 q^{3} +20.4342 q^{4} -1.13957 q^{5} -28.6907 q^{6} -19.4433 q^{7} +66.3036 q^{8} +1.94964 q^{9} +O(q^{10})\)	\(q+5.33237 q^{2} -5.38049 q^{3} +20.4342 q^{4} -1.13957 q^{5} -28.6907 q^{6} -19.4433 q^{7} +66.3036 q^{8} +1.94964 q^{9} -6.07658 q^{10} +11.2409 q^{11} -109.946 q^{12} +82.3500 q^{13} -103.679 q^{14} +6.13142 q^{15} +190.082 q^{16} -132.630 q^{17} +10.3962 q^{18} -41.5526 q^{19} -23.2861 q^{20} +104.614 q^{21} +59.9405 q^{22} +97.0017 q^{23} -356.746 q^{24} -123.701 q^{25} +439.121 q^{26} +134.783 q^{27} -397.308 q^{28} +94.9288 q^{29} +32.6950 q^{30} -265.836 q^{31} +483.159 q^{32} -60.4813 q^{33} -707.235 q^{34} +22.1569 q^{35} +39.8392 q^{36} +70.0555 q^{37} -221.574 q^{38} -443.083 q^{39} -75.5573 q^{40} +30.4482 q^{41} +557.843 q^{42} +229.698 q^{44} -2.22174 q^{45} +517.249 q^{46} -418.189 q^{47} -1022.73 q^{48} +35.0417 q^{49} -659.622 q^{50} +713.617 q^{51} +1682.75 q^{52} -399.944 q^{53} +718.714 q^{54} -12.8097 q^{55} -1289.16 q^{56} +223.573 q^{57} +506.195 q^{58} -57.1424 q^{59} +125.290 q^{60} -331.172 q^{61} -1417.54 q^{62} -37.9074 q^{63} +1055.73 q^{64} -93.8432 q^{65} -322.509 q^{66} +156.263 q^{67} -2710.19 q^{68} -521.916 q^{69} +118.149 q^{70} +564.356 q^{71} +129.268 q^{72} -69.0808 q^{73} +373.562 q^{74} +665.574 q^{75} -849.093 q^{76} -218.559 q^{77} -2362.68 q^{78} -569.629 q^{79} -216.611 q^{80} -777.839 q^{81} +162.361 q^{82} -785.399 q^{83} +2137.71 q^{84} +151.141 q^{85} -510.763 q^{87} +745.310 q^{88} +429.427 q^{89} -11.8471 q^{90} -1601.15 q^{91} +1982.15 q^{92} +1430.33 q^{93} -2229.94 q^{94} +47.3519 q^{95} -2599.63 q^{96} -1161.42 q^{97} +186.855 q^{98} +21.9156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} + O(q^{10}) \)	\( 50q - 10q^{2} - 2q^{3} + 186q^{4} + 8q^{5} - 51q^{6} - 6q^{7} - 138q^{8} + 360q^{9} - 137q^{10} - 252q^{11} - 48q^{12} - 192q^{13} - 272q^{14} - 314q^{15} + 542q^{16} - 236q^{17} - 386q^{18} + 12q^{19} + 108q^{20} - 408q^{21} + 1235q^{22} - 630q^{23} - 613q^{24} + 1098q^{25} + 1493q^{26} + 10q^{27} - 242q^{28} + 208q^{29} - 48q^{30} - 932q^{31} - 1124q^{32} + 254q^{33} + 765q^{34} - 1452q^{35} + 747q^{36} - 90q^{37} - 1213q^{38} - 1610q^{39} - 1693q^{40} - 1354q^{41} - 16q^{42} - 2704q^{44} + 4508q^{45} + 233q^{46} - 3484q^{47} - 376q^{48} + 1324q^{49} - 408q^{50} + 4054q^{51} - 2176q^{52} - 726q^{53} - 6497q^{54} - 3288q^{55} - 7097q^{56} - 870q^{57} + 275q^{58} - 4370q^{59} - 3891q^{60} + 1172q^{61} - 1546q^{62} - 3686q^{63} + 606q^{64} + 2610q^{65} - 4697q^{66} - 344q^{67} - 3221q^{68} + 136q^{69} - 1310q^{70} + 162q^{71} - 5814q^{72} + 746q^{73} - 4332q^{74} + 236q^{75} + 1338q^{76} + 2024q^{77} - 2782q^{78} - 2656q^{79} - 5713q^{80} - 86q^{81} - 4168q^{82} - 3514q^{83} - 4269q^{84} - 7558q^{85} - 10278q^{87} + 11692q^{88} + 2640q^{89} - 8286q^{90} - 5946q^{91} - 4271q^{92} - 2q^{93} + 9062q^{94} - 12140q^{95} - 700q^{96} - 3864q^{97} - 2826q^{98} - 8174q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

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