LMFDB - Embedded newform 21.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,6,Mod(1,21)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	21.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.36806021607$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	21.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +78.0000 q^{5} +54.0000 q^{6} +49.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +78.0000 q^{5} +54.0000 q^{6} +49.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} -468.000 q^{10} +444.000 q^{11} -36.0000 q^{12} -442.000 q^{13} -294.000 q^{14} -702.000 q^{15} -1136.00 q^{16} -126.000 q^{17} -486.000 q^{18} +2684.00 q^{19} +312.000 q^{20} -441.000 q^{21} -2664.00 q^{22} +4200.00 q^{23} -1512.00 q^{24} +2959.00 q^{25} +2652.00 q^{26} -729.000 q^{27} +196.000 q^{28} -5442.00 q^{29} +4212.00 q^{30} +80.0000 q^{31} +1440.00 q^{32} -3996.00 q^{33} +756.000 q^{34} +3822.00 q^{35} +324.000 q^{36} -5434.00 q^{37} -16104.0 q^{38} +3978.00 q^{39} +13104.0 q^{40} +7962.00 q^{41} +2646.00 q^{42} -11524.0 q^{43} +1776.00 q^{44} +6318.00 q^{45} -25200.0 q^{46} -13920.0 q^{47} +10224.0 q^{48} +2401.00 q^{49} -17754.0 q^{50} +1134.00 q^{51} -1768.00 q^{52} -9594.00 q^{53} +4374.00 q^{54} +34632.0 q^{55} +8232.00 q^{56} -24156.0 q^{57} +32652.0 q^{58} +27492.0 q^{59} -2808.00 q^{60} +49478.0 q^{61} -480.000 q^{62} +3969.00 q^{63} +27712.0 q^{64} -34476.0 q^{65} +23976.0 q^{66} -59356.0 q^{67} -504.000 q^{68} -37800.0 q^{69} -22932.0 q^{70} +32040.0 q^{71} +13608.0 q^{72} -61846.0 q^{73} +32604.0 q^{74} -26631.0 q^{75} +10736.0 q^{76} +21756.0 q^{77} -23868.0 q^{78} -65776.0 q^{79} -88608.0 q^{80} +6561.00 q^{81} -47772.0 q^{82} +40188.0 q^{83} -1764.00 q^{84} -9828.00 q^{85} +69144.0 q^{86} +48978.0 q^{87} +74592.0 q^{88} -7974.00 q^{89} -37908.0 q^{90} -21658.0 q^{91} +16800.0 q^{92} -720.000 q^{93} +83520.0 q^{94} +209352. q^{95} -12960.0 q^{96} -143662. q^{97} -14406.0 q^{98} +35964.0 q^{99} +O(q^{100})\)

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 $

$2$	−6.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−6.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	4.00000	0.125000
$5$	78.0000	1.39531	0.697653	−	0.716436i	$-0.254228\pi$
$5$	78.0000	1.39531	0.697653	+	0.716436i	$0.254228\pi$
$6$	54.0000	0.612372
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	168.000	0.928078
$9$	81.0000	0.333333
$10$	−468.000	−1.47995
$11$	444.000	1.10637	0.553186	−	0.833058i	$-0.313412\pi$
$11$	444.000	1.10637	0.553186	+	0.833058i	$0.313412\pi$
$12$	−36.0000	−0.0721688
$13$	−442.000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−442.000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	−294.000	−0.400892
$15$	−702.000	−0.805581
$16$	−1136.00	−1.10938
$17$	−126.000	−0.105742	−0.0528711	−	0.998601i	$-0.516837\pi$
$17$	−126.000	−0.105742	−0.0528711	+	0.998601i	$0.516837\pi$
$18$	−486.000	−0.353553
$19$	2684.00	1.70568	0.852842	−	0.522169i	$-0.174877\pi$
$19$	2684.00	1.70568	0.852842	+	0.522169i	$0.174877\pi$
$20$	312.000	0.174413
$21$	−441.000	−0.218218
$22$	−2664.00	−1.17348
$23$	4200.00	1.65550	0.827751	−	0.561096i	$-0.189620\pi$
$23$	4200.00	1.65550	0.827751	+	0.561096i	$0.189620\pi$
$24$	−1512.00	−0.535826
$25$	2959.00	0.946880
$26$	2652.00	0.769379
$27$	−729.000	−0.192450
$28$	196.000	0.0472456
$29$	−5442.00	−1.20161	−0.600805	−	0.799396i	$-0.705153\pi$
$29$	−5442.00	−1.20161	−0.600805	+	0.799396i	$0.705153\pi$
$30$	4212.00	0.854447
$31$	80.0000	0.0149515	0.00747577	−	0.999972i	$-0.497620\pi$
$31$	80.0000	0.0149515	0.00747577	+	0.999972i	$0.497620\pi$
$32$	1440.00	0.248592
$33$	−3996.00	−0.638764
$34$	756.000	0.112157
$35$	3822.00	0.527376
$36$	324.000	0.0416667
$37$	−5434.00	−0.652552	−0.326276	−	0.945274i	$-0.605794\pi$
$37$	−5434.00	−0.652552	−0.326276	+	0.945274i	$0.605794\pi$
$38$	−16104.0	−1.80915
$39$	3978.00	0.418797
$40$	13104.0	1.29495
$41$	7962.00	0.739712	0.369856	−	0.929089i	$-0.379407\pi$
$41$	7962.00	0.739712	0.369856	+	0.929089i	$0.379407\pi$
$42$	2646.00	0.231455
$43$	−11524.0	−0.950456	−0.475228	−	0.879863i	$-0.657634\pi$
$43$	−11524.0	−0.950456	−0.475228	+	0.879863i	$0.657634\pi$
$44$	1776.00	0.138297
$45$	6318.00	0.465102
$46$	−25200.0	−1.75592
$47$	−13920.0	−0.919167	−0.459584	−	0.888134i	$-0.652001\pi$
$47$	−13920.0	−0.919167	−0.459584	+	0.888134i	$0.652001\pi$
$48$	10224.0	0.640498
$49$	2401.00	0.142857
$50$	−17754.0	−1.00432
$51$	1134.00	0.0610503
$52$	−1768.00	−0.0906721
$53$	−9594.00	−0.469148	−0.234574	−	0.972098i	$-0.575370\pi$
$53$	−9594.00	−0.469148	−0.234574	+	0.972098i	$0.575370\pi$
$54$	4374.00	0.204124
$55$	34632.0	1.54373
$56$	8232.00	0.350780
$57$	−24156.0	−0.984777
$58$	32652.0	1.27450
$59$	27492.0	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$59$	27492.0	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$60$	−2808.00	−0.100698
$61$	49478.0	1.70250	0.851251	−	0.524759i	$-0.175845\pi$
$61$	49478.0	1.70250	0.851251	+	0.524759i	$0.175845\pi$
$62$	−480.000	−0.0158585
$63$	3969.00	0.125988
$64$	27712.0	0.845703
$65$	−34476.0	−1.01212
$66$	23976.0	0.677512
$67$	−59356.0	−1.61539	−0.807695	−	0.589600i	$-0.799285\pi$
$67$	−59356.0	−1.61539	−0.807695	+	0.589600i	$0.799285\pi$
$68$	−504.000	−0.0132178
$69$	−37800.0	−0.955805
$70$	−22932.0	−0.559367
$71$	32040.0	0.754304	0.377152	−	0.926151i	$-0.376903\pi$
$71$	32040.0	0.754304	0.377152	+	0.926151i	$0.376903\pi$
$72$	13608.0	0.309359
$73$	−61846.0	−1.35833	−0.679164	−	0.733987i	$-0.737657\pi$
$73$	−61846.0	−1.35833	−0.679164	+	0.733987i	$0.737657\pi$
$74$	32604.0	0.692136
$75$	−26631.0	−0.546681
$76$	10736.0	0.213210
$77$	21756.0	0.418169
$78$	−23868.0	−0.444201
$79$	−65776.0	−1.18577	−0.592884	−	0.805288i	$-0.702011\pi$
$79$	−65776.0	−1.18577	−0.592884	+	0.805288i	$0.702011\pi$
$80$	−88608.0	−1.54792
$81$	6561.00	0.111111
$82$	−47772.0	−0.784583
$83$	40188.0	0.640326	0.320163	−	0.947362i	$-0.396262\pi$
$83$	40188.0	0.640326	0.320163	+	0.947362i	$0.396262\pi$
$84$	−1764.00	−0.0272772
$85$	−9828.00	−0.147543
$86$	69144.0	1.00811
$87$	48978.0	0.693750
$88$	74592.0	1.02680
$89$	−7974.00	−0.106709	−0.0533545	−	0.998576i	$-0.516991\pi$
$89$	−7974.00	−0.106709	−0.0533545	+	0.998576i	$0.516991\pi$
$90$	−37908.0	−0.493315
$91$	−21658.0	−0.274167
$92$	16800.0	0.206938
$93$	−720.000	−0.00863227
$94$	83520.0	0.974924
$95$	209352.	2.37995
$96$	−12960.0	−0.143525
$97$	−143662.	−1.55029	−0.775144	−	0.631784i	$-0.782323\pi$
$97$	−143662.	−1.55029	−0.775144	+	0.631784i	$0.782323\pi$
$98$	−14406.0	−0.151523
$99$	35964.0	0.368791
$100$	11836.0	0.118360
$101$	−2706.00	−0.0263952	−0.0131976	−	0.999913i	$-0.504201\pi$
$101$	−2706.00	−0.0263952	−0.0131976	+	0.999913i	$0.504201\pi$
$102$	−6804.00	−0.0647536
$103$	131768.	1.22382	0.611909	−	0.790928i	$-0.290402\pi$
$103$	131768.	1.22382	0.611909	+	0.790928i	$0.290402\pi$
$104$	−74256.0	−0.673206
$105$	−34398.0	−0.304481
$106$	57564.0	0.497607
$107$	−128916.	−1.08855	−0.544274	−	0.838908i	$-0.683195\pi$
$107$	−128916.	−1.08855	−0.544274	+	0.838908i	$0.683195\pi$
$108$	−2916.00	−0.0240563
$109$	−100978.	−0.814068	−0.407034	−	0.913413i	$-0.633437\pi$
$109$	−100978.	−0.814068	−0.407034	+	0.913413i	$0.633437\pi$
$110$	−207792.	−1.63737
$111$	48906.0	0.376751
$112$	−55664.0	−0.419304
$113$	220146.	1.62186	0.810932	−	0.585140i	$-0.198960\pi$
$113$	220146.	1.62186	0.810932	+	0.585140i	$0.198960\pi$
$114$	144936.	1.04451
$115$	327600.	2.30993
$116$	−21768.0	−0.150201
$117$	−35802.0	−0.241792
$118$	−164952.	−1.09057
$119$	−6174.00	−0.0399668
$120$	−117936.	−0.747641
$121$	36085.0	0.224059
$122$	−296868.	−1.80578
$123$	−71658.0	−0.427073
$124$	320.000	0.00186894
$125$	−12948.0	−0.0741187
$126$	−23814.0	−0.133631
$127$	−74320.0	−0.408880	−0.204440	−	0.978879i	$-0.565537\pi$
$127$	−74320.0	−0.408880	−0.204440	+	0.978879i	$0.565537\pi$
$128$	−212352.	−1.14560
$129$	103716.	0.548746
$130$	206856.	1.07352
$131$	−155316.	−0.790748	−0.395374	−	0.918520i	$-0.629385\pi$
$131$	−155316.	−0.790748	−0.395374	+	0.918520i	$0.629385\pi$
$132$	−15984.0	−0.0798455
$133$	131516.	0.644688
$134$	356136.	1.71338
$135$	−56862.0	−0.268527
$136$	−21168.0	−0.0981369
$137$	−264246.	−1.20284	−0.601419	−	0.798934i	$-0.705398\pi$
$137$	−264246.	−1.20284	−0.601419	+	0.798934i	$0.705398\pi$
$138$	226800.	1.01378
$139$	224612.	0.986043	0.493022	−	0.870017i	$-0.335892\pi$
$139$	224612.	0.986043	0.493022	+	0.870017i	$0.335892\pi$
$140$	15288.0	0.0659220
$141$	125280.	0.530682
$142$	−192240.	−0.800061
$143$	−196248.	−0.802537
$144$	−92016.0	−0.369792
$145$	−424476.	−1.67661
$146$	371076.	1.44072
$147$	−21609.0	−0.0824786
$148$	−21736.0	−0.0815690
$149$	−82074.0	−0.302859	−0.151429	−	0.988468i	$-0.548388\pi$
$149$	−82074.0	−0.302859	−0.151429	+	0.988468i	$0.548388\pi$
$150$	159786.	0.579843
$151$	−287032.	−1.02444	−0.512222	−	0.858853i	$-0.671177\pi$
$151$	−287032.	−1.02444	−0.512222	+	0.858853i	$0.671177\pi$
$152$	450912.	1.58301
$153$	−10206.0	−0.0352474
$154$	−130536.	−0.443536
$155$	6240.00	0.0208620
$156$	15912.0	0.0523496
$157$	129878.	0.420520	0.210260	−	0.977646i	$-0.432569\pi$
$157$	129878.	0.420520	0.210260	+	0.977646i	$0.432569\pi$
$158$	394656.	1.25770
$159$	86346.0	0.270863
$160$	112320.	0.346862
$161$	205800.	0.625721
$162$	−39366.0	−0.117851
$163$	555284.	1.63699	0.818495	−	0.574513i	$-0.194809\pi$
$163$	555284.	1.63699	0.818495	+	0.574513i	$0.194809\pi$
$164$	31848.0	0.0924640
$165$	−311688.	−0.891272
$166$	−241128.	−0.679168
$167$	43512.0	0.120731	0.0603654	−	0.998176i	$-0.480773\pi$
$167$	43512.0	0.120731	0.0603654	+	0.998176i	$0.480773\pi$
$168$	−74088.0	−0.202523
$169$	−175929.	−0.473828
$170$	58968.0	0.156493
$171$	217404.	0.568561
$172$	−46096.0	−0.118807
$173$	−18330.0	−0.0465637	−0.0232818	−	0.999729i	$-0.507412\pi$
$173$	−18330.0	−0.0465637	−0.0232818	+	0.999729i	$0.507412\pi$
$174$	−293868.	−0.735833
$175$	144991.	0.357887
$176$	−504384.	−1.22738
$177$	−247428.	−0.593630
$178$	47844.0	0.113182
$179$	−153324.	−0.357666	−0.178833	−	0.983879i	$-0.557232\pi$
$179$	−153324.	−0.357666	−0.178833	+	0.983879i	$0.557232\pi$
$180$	25272.0	0.0581378
$181$	−382066.	−0.866846	−0.433423	−	0.901191i	$-0.642694\pi$
$181$	−382066.	−0.866846	−0.433423	+	0.901191i	$0.642694\pi$
$182$	129948.	0.290798
$183$	−445302.	−0.982940
$184$	705600.	1.53643
$185$	−423852.	−0.910510
$186$	4320.00	0.00915591
$187$	−55944.0	−0.116990
$188$	−55680.0	−0.114896
$189$	−35721.0	−0.0727393
$190$	−1.25611e6	−2.52432
$191$	−273408.	−0.542285	−0.271143	−	0.962539i	$-0.587402\pi$
$191$	−273408.	−0.542285	−0.271143	+	0.962539i	$0.587402\pi$
$192$	−249408.	−0.488267
$193$	153602.	0.296827	0.148414	−	0.988925i	$-0.452583\pi$
$193$	153602.	0.296827	0.148414	+	0.988925i	$0.452583\pi$
$194$	861972.	1.64433
$195$	310284.	0.584350
$196$	9604.00	0.0178571
$197$	154422.	0.283494	0.141747	−	0.989903i	$-0.454728\pi$
$197$	154422.	0.283494	0.141747	+	0.989903i	$0.454728\pi$
$198$	−215784.	−0.391162
$199$	−366856.	−0.656694	−0.328347	−	0.944557i	$-0.606492\pi$
$199$	−366856.	−0.656694	−0.328347	+	0.944557i	$0.606492\pi$
$200$	497112.	0.878778
$201$	534204.	0.932646
$202$	16236.0	0.0279963
$203$	−266658.	−0.454166
$204$	4536.00	0.00763128
$205$	621036.	1.03212
$206$	−790608.	−1.29806
$207$	340200.	0.551834
$208$	502112.	0.804715
$209$	1.19170e6	1.88712
$210$	206388.	0.322951
$211$	520244.	0.804453	0.402227	−	0.915540i	$-0.368236\pi$
$211$	520244.	0.804453	0.402227	+	0.915540i	$0.368236\pi$
$212$	−38376.0	−0.0586435
$213$	−288360.	−0.435498
$214$	773496.	1.15458
$215$	−898872.	−1.32618
$216$	−122472.	−0.178609
$217$	3920.00	0.00565115
$218$	605868.	0.863449
$219$	556614.	0.784231
$220$	138528.	0.192966
$221$	55692.0	0.0767030
$222$	−293436.	−0.399605
$223$	304736.	0.410357	0.205178	−	0.978725i	$-0.434223\pi$
$223$	304736.	0.410357	0.205178	+	0.978725i	$0.434223\pi$
$224$	70560.0	0.0939590
$225$	239679.	0.315627
$226$	−1.32088e6	−1.72025
$227$	288588.	0.371718	0.185859	−	0.982576i	$-0.440493\pi$
$227$	288588.	0.371718	0.185859	+	0.982576i	$0.440493\pi$
$228$	−96624.0	−0.123097
$229$	772190.	0.973051	0.486525	−	0.873666i	$-0.338264\pi$
$229$	772190.	0.973051	0.486525	+	0.873666i	$0.338264\pi$
$230$	−1.96560e6	−2.45005
$231$	−195804.	−0.241430
$232$	−914256.	−1.11519
$233$	252234.	0.304378	0.152189	−	0.988351i	$-0.451368\pi$
$233$	252234.	0.304378	0.152189	+	0.988351i	$0.451368\pi$
$234$	214812.	0.256460
$235$	−1.08576e6	−1.28252
$236$	109968.	0.128525
$237$	591984.	0.684603
$238$	37044.0	0.0423912
$239$	−1.45114e6	−1.64329	−0.821643	−	0.570002i	$-0.806942\pi$
$239$	−1.45114e6	−1.64329	−0.821643	+	0.570002i	$0.806942\pi$
$240$	797472.	0.893691
$241$	−146398.	−0.162365	−0.0811825	−	0.996699i	$-0.525870\pi$
$241$	−146398.	−0.162365	−0.0811825	+	0.996699i	$0.525870\pi$
$242$	−216510.	−0.237651
$243$	−59049.0	−0.0641500
$244$	197912.	0.212813
$245$	187278.	0.199329
$246$	429948.	0.452979
$247$	−1.18633e6	−1.23726
$248$	13440.0	0.0138762
$249$	−361692.	−0.369692
$250$	77688.0	0.0786147
$251$	607860.	0.609003	0.304501	−	0.952512i	$-0.401510\pi$
$251$	607860.	0.609003	0.304501	+	0.952512i	$0.401510\pi$
$252$	15876.0	0.0157485
$253$	1.86480e6	1.83160
$254$	445920.	0.433683
$255$	88452.0	0.0851838
$256$	387328.	0.369385
$257$	95586.0	0.0902737	0.0451369	−	0.998981i	$-0.485628\pi$
$257$	95586.0	0.0902737	0.0451369	+	0.998981i	$0.485628\pi$
$258$	−622296.	−0.582033
$259$	−266266.	−0.246642
$260$	−137904.	−0.126515
$261$	−440802.	−0.400537
$262$	931896.	0.838715
$263$	−2.20034e6	−1.96156	−0.980779	−	0.195121i	$-0.937490\pi$
$263$	−2.20034e6	−1.96156	−0.980779	+	0.195121i	$0.937490\pi$
$264$	−671328.	−0.592823
$265$	−748332.	−0.654605
$266$	−789096.	−0.683795
$267$	71766.0	0.0616085
$268$	−237424.	−0.201924
$269$	1.77025e6	1.49160	0.745801	−	0.666169i	$-0.232067\pi$
$269$	1.77025e6	1.49160	0.745801	+	0.666169i	$0.232067\pi$
$270$	341172.	0.284816
$271$	−223504.	−0.184868	−0.0924341	−	0.995719i	$-0.529465\pi$
$271$	−223504.	−0.184868	−0.0924341	+	0.995719i	$0.529465\pi$
$272$	143136.	0.117308
$273$	194922.	0.158290
$274$	1.58548e6	1.27580
$275$	1.31380e6	1.04760
$276$	−151200.	−0.119476
$277$	−342778.	−0.268419	−0.134210	−	0.990953i	$-0.542850\pi$
$277$	−342778.	−0.268419	−0.134210	+	0.990953i	$0.542850\pi$
$278$	−1.34767e6	−1.04586
$279$	6480.00	0.00498384
$280$	642096.	0.489446
$281$	480378.	0.362925	0.181463	−	0.983398i	$-0.441917\pi$
$281$	480378.	0.362925	0.181463	+	0.983398i	$0.441917\pi$
$282$	−751680.	−0.562873
$283$	−29980.0	−0.0222518	−0.0111259	−	0.999938i	$-0.503542\pi$
$283$	−29980.0	−0.0222518	−0.0111259	+	0.999938i	$0.503542\pi$
$284$	128160.	0.0942880
$285$	−1.88417e6	−1.37407
$286$	1.17749e6	0.851219
$287$	390138.	0.279585
$288$	116640.	0.0828641
$289$	−1.40398e6	−0.988819
$290$	2.54686e6	1.77832
$291$	1.29296e6	0.895060
$292$	−247384.	−0.169791
$293$	−198066.	−0.134785	−0.0673924	−	0.997727i	$-0.521468\pi$
$293$	−198066.	−0.134785	−0.0673924	+	0.997727i	$0.521468\pi$
$294$	129654.	0.0874818
$295$	2.14438e6	1.43465
$296$	−912912.	−0.605619
$297$	−323676.	−0.212921
$298$	492444.	0.321230
$299$	−1.85640e6	−1.20086
$300$	−106524.	−0.0683352
$301$	−564676.	−0.359239
$302$	1.72219e6	1.08659
$303$	24354.0	0.0152393
$304$	−3.04902e6	−1.89224
$305$	3.85928e6	2.37551
$306$	61236.0	0.0373855
$307$	−1.04564e6	−0.633191	−0.316595	−	0.948561i	$-0.602540\pi$
$307$	−1.04564e6	−0.633191	−0.316595	+	0.948561i	$0.602540\pi$
$308$	87024.0	0.0522712
$309$	−1.18591e6	−0.706572
$310$	−37440.0	−0.0221275
$311$	1.83718e6	1.07708	0.538542	−	0.842598i	$-0.318975\pi$
$311$	1.83718e6	1.07708	0.538542	+	0.842598i	$0.318975\pi$
$312$	668304.	0.388676
$313$	−365494.	−0.210872	−0.105436	−	0.994426i	$-0.533624\pi$
$313$	−365494.	−0.210872	−0.105436	+	0.994426i	$0.533624\pi$
$314$	−779268.	−0.446029
$315$	309582.	0.175792
$316$	−263104.	−0.148221
$317$	−28338.0	−0.0158388	−0.00791938	−	0.999969i	$-0.502521\pi$
$317$	−28338.0	−0.0158388	−0.00791938	+	0.999969i	$0.502521\pi$
$318$	−518076.	−0.287293
$319$	−2.41625e6	−1.32943
$320$	2.16154e6	1.18001
$321$	1.16024e6	0.628473
$322$	−1.23480e6	−0.663677
$323$	−338184.	−0.180363
$324$	26244.0	0.0138889
$325$	−1.30788e6	−0.686845
$326$	−3.33170e6	−1.73629
$327$	908802.	0.470002
$328$	1.33762e6	0.686510
$329$	−682080.	−0.347413
$330$	1.87013e6	0.945337
$331$	1.93392e6	0.970214	0.485107	−	0.874455i	$-0.338781\pi$
$331$	1.93392e6	0.970214	0.485107	+	0.874455i	$0.338781\pi$
$332$	160752.	0.0800408
$333$	−440154.	−0.217517
$334$	−261072.	−0.128054
$335$	−4.62977e6	−2.25397
$336$	500976.	0.242085
$337$	−1.88817e6	−0.905664	−0.452832	−	0.891596i	$-0.649586\pi$
$337$	−1.88817e6	−0.905664	−0.452832	+	0.891596i	$0.649586\pi$
$338$	1.05557e6	0.502570
$339$	−1.98131e6	−0.936384
$340$	−39312.0	−0.0184428
$341$	35520.0	0.0165420
$342$	−1.30442e6	−0.603050
$343$	117649.	0.0539949
$344$	−1.93603e6	−0.882097
$345$	−2.94840e6	−1.33364
$346$	109980.	0.0493882
$347$	2.91937e6	1.30156	0.650782	−	0.759264i	$-0.274441\pi$
$347$	2.91937e6	1.30156	0.650782	+	0.759264i	$0.274441\pi$
$348$	195912.	0.0867187
$349$	−780682.	−0.343092	−0.171546	−	0.985176i	$-0.554876\pi$
$349$	−780682.	−0.343092	−0.171546	+	0.985176i	$0.554876\pi$
$350$	−869946.	−0.379596
$351$	322218.	0.139599
$352$	639360.	0.275036
$353$	1.33437e6	0.569954	0.284977	−	0.958534i	$-0.408014\pi$
$353$	1.33437e6	0.569954	0.284977	+	0.958534i	$0.408014\pi$
$354$	1.48457e6	0.629639
$355$	2.49912e6	1.05249
$356$	−31896.0	−0.0133386
$357$	55566.0	0.0230748
$358$	919944.	0.379362
$359$	1.01743e6	0.416648	0.208324	−	0.978060i	$-0.433199\pi$
$359$	1.01743e6	0.416648	0.208324	+	0.978060i	$0.433199\pi$
$360$	1.06142e6	0.431651
$361$	4.72776e6	1.90936
$362$	2.29240e6	0.919429
$363$	−324765.	−0.129361
$364$	−86632.0	−0.0342709
$365$	−4.82399e6	−1.89528
$366$	2.67181e6	1.04257
$367$	837680.	0.324648	0.162324	−	0.986737i	$-0.448101\pi$
$367$	837680.	0.324648	0.162324	+	0.986737i	$0.448101\pi$
$368$	−4.77120e6	−1.83657
$369$	644922.	0.246571
$370$	2.54311e6	0.965742
$371$	−470106.	−0.177321
$372$	−2880.00	−0.00107903
$373$	−1.51993e6	−0.565655	−0.282827	−	0.959171i	$-0.591272\pi$
$373$	−1.51993e6	−0.565655	−0.282827	+	0.959171i	$0.591272\pi$
$374$	335664.	0.124087
$375$	116532.	0.0427924
$376$	−2.33856e6	−0.853059
$377$	2.40536e6	0.871620
$378$	214326.	0.0771517
$379$	2.64465e6	0.945737	0.472869	−	0.881133i	$-0.343219\pi$
$379$	2.64465e6	0.945737	0.472869	+	0.881133i	$0.343219\pi$
$380$	837408.	0.297494
$381$	668880.	0.236067
$382$	1.64045e6	0.575180
$383$	2.01336e6	0.701333	0.350667	−	0.936500i	$-0.385955\pi$
$383$	2.01336e6	0.701333	0.350667	+	0.936500i	$0.385955\pi$
$384$	1.91117e6	0.661410
$385$	1.69697e6	0.583474
$386$	−921612.	−0.314833
$387$	−933444.	−0.316819
$388$	−574648.	−0.193786
$389$	−726234.	−0.243334	−0.121667	−	0.992571i	$-0.538824\pi$
$389$	−726234.	−0.243334	−0.121667	+	0.992571i	$0.538824\pi$
$390$	−1.86170e6	−0.619796
$391$	−529200.	−0.175056
$392$	403368.	0.132583
$393$	1.39784e6	0.456538
$394$	−926532.	−0.300691
$395$	−5.13053e6	−1.65451
$396$	143856.	0.0460988
$397$	4.57578e6	1.45710	0.728549	−	0.684993i	$-0.240195\pi$
$397$	4.57578e6	1.45710	0.728549	+	0.684993i	$0.240195\pi$
$398$	2.20114e6	0.696529
$399$	−1.18364e6	−0.372211
$400$	−3.36142e6	−1.05045
$401$	−33870.0	−0.0105185	−0.00525926	−	0.999986i	$-0.501674\pi$
$401$	−33870.0	−0.0105185	−0.00525926	+	0.999986i	$0.501674\pi$
$402$	−3.20522e6	−0.989221
$403$	−35360.0	−0.0108455
$404$	−10824.0	−0.00329940
$405$	511758.	0.155034
$406$	1.59995e6	0.481716
$407$	−2.41270e6	−0.721966
$408$	190512.	0.0566594
$409$	−5.86178e6	−1.73269	−0.866346	−	0.499444i	$-0.833538\pi$
$409$	−5.86178e6	−1.73269	−0.866346	+	0.499444i	$0.833538\pi$
$410$	−3.72622e6	−1.09473
$411$	2.37821e6	0.694459
$412$	527072.	0.152977
$413$	1.34711e6	0.388622
$414$	−2.04120e6	−0.585308
$415$	3.13466e6	0.893451
$416$	−636480.	−0.180323
$417$	−2.02151e6	−0.569292
$418$	−7.15018e6	−2.00159
$419$	302748.	0.0842454	0.0421227	−	0.999112i	$-0.486588\pi$
$419$	302748.	0.0842454	0.0421227	+	0.999112i	$0.486588\pi$
$420$	−137592.	−0.0380601
$421$	−5.36708e6	−1.47582	−0.737909	−	0.674900i	$-0.764187\pi$
$421$	−5.36708e6	−1.47582	−0.737909	+	0.674900i	$0.764187\pi$
$422$	−3.12146e6	−0.853252
$423$	−1.12752e6	−0.306389
$424$	−1.61179e6	−0.435406
$425$	−372834.	−0.100125
$426$	1.73016e6	0.461915
$427$	2.42442e6	0.643485
$428$	−515664.	−0.136068
$429$	1.76623e6	0.463345
$430$	5.39323e6	1.40662
$431$	1.17706e6	0.305214	0.152607	−	0.988287i	$-0.451233\pi$
$431$	1.17706e6	0.305214	0.152607	+	0.988287i	$0.451233\pi$
$432$	828144.	0.213499
$433$	−3.66249e6	−0.938766	−0.469383	−	0.882995i	$-0.655524\pi$
$433$	−3.66249e6	−0.938766	−0.469383	+	0.882995i	$0.655524\pi$
$434$	−23520.0	−0.00599395
$435$	3.82028e6	0.967994
$436$	−403912.	−0.101758
$437$	1.12728e7	2.82376
$438$	−3.33968e6	−0.831802
$439$	−2.53674e6	−0.628225	−0.314113	−	0.949386i	$-0.601707\pi$
$439$	−2.53674e6	−0.628225	−0.314113	+	0.949386i	$0.601707\pi$
$440$	5.81818e6	1.43270
$441$	194481.	0.0476190
$442$	−334152.	−0.0813558
$443$	6.01504e6	1.45623	0.728113	−	0.685457i	$-0.240397\pi$
$443$	6.01504e6	1.45623	0.728113	+	0.685457i	$0.240397\pi$
$444$	195624.	0.0470939
$445$	−621972.	−0.148892
$446$	−1.82842e6	−0.435249
$447$	738666.	0.174856
$448$	1.35789e6	0.319646
$449$	5.65965e6	1.32487	0.662436	−	0.749119i	$-0.269523\pi$
$449$	5.65965e6	1.32487	0.662436	+	0.749119i	$0.269523\pi$
$450$	−1.43807e6	−0.334773
$451$	3.53513e6	0.818397
$452$	880584.	0.202733
$453$	2.58329e6	0.591463
$454$	−1.73153e6	−0.394267
$455$	−1.68932e6	−0.382547
$456$	−4.05821e6	−0.913949
$457$	−6.46159e6	−1.44727	−0.723634	−	0.690184i	$-0.757530\pi$
$457$	−6.46159e6	−1.44727	−0.723634	+	0.690184i	$0.757530\pi$
$458$	−4.63314e6	−1.03208
$459$	91854.0	0.0203501
$460$	1.31040e6	0.288742
$461$	−3.37353e6	−0.739320	−0.369660	−	0.929167i	$-0.620526\pi$
$461$	−3.37353e6	−0.739320	−0.369660	+	0.929167i	$0.620526\pi$
$462$	1.17482e6	0.256075
$463$	−4.54974e6	−0.986358	−0.493179	−	0.869928i	$-0.664165\pi$
$463$	−4.54974e6	−0.986358	−0.493179	+	0.869928i	$0.664165\pi$
$464$	6.18211e6	1.33304
$465$	−56160.0	−0.0120447
$466$	−1.51340e6	−0.322842
$467$	2.01136e6	0.426773	0.213386	−	0.976968i	$-0.431551\pi$
$467$	2.01136e6	0.426773	0.213386	+	0.976968i	$0.431551\pi$
$468$	−143208.	−0.0302240
$469$	−2.90844e6	−0.610560
$470$	6.51456e6	1.36032
$471$	−1.16890e6	−0.242787
$472$	4.61866e6	0.954247
$473$	−5.11666e6	−1.05156
$474$	−3.55190e6	−0.726132
$475$	7.94196e6	1.61508
$476$	−24696.0	−0.00499585
$477$	−777114.	−0.156383
$478$	8.70682e6	1.74297
$479$	−7.60402e6	−1.51427	−0.757137	−	0.653257i	$-0.773402\pi$
$479$	−7.60402e6	−1.51427	−0.757137	+	0.653257i	$0.773402\pi$
$480$	−1.01088e6	−0.200261
$481$	2.40183e6	0.473347
$482$	878388.	0.172214
$483$	−1.85220e6	−0.361260
$484$	144340.	0.0280074
$485$	−1.12056e7	−2.16313
$486$	354294.	0.0680414
$487$	673112.	0.128607	0.0643035	−	0.997930i	$-0.479517\pi$
$487$	673112.	0.128607	0.0643035	+	0.997930i	$0.479517\pi$
$488$	8.31230e6	1.58005
$489$	−4.99756e6	−0.945117
$490$	−1.12367e6	−0.211421
$491$	−2.47170e6	−0.462692	−0.231346	−	0.972872i	$-0.574313\pi$
$491$	−2.47170e6	−0.462692	−0.231346	+	0.972872i	$0.574313\pi$
$492$	−286632.	−0.0533841
$493$	685692.	0.127061
$494$	7.11797e6	1.31232
$495$	2.80519e6	0.514576
$496$	−90880.0	−0.0165869
$497$	1.56996e6	0.285100
$498$	2.17015e6	0.392118
$499$	6.08152e6	1.09335	0.546677	−	0.837343i	$-0.315892\pi$
$499$	6.08152e6	1.09335	0.546677	+	0.837343i	$0.315892\pi$
$500$	−51792.0	−0.00926483
$501$	−391608.	−0.0697039
$502$	−3.64716e6	−0.645945
$503$	−846216.	−0.149129	−0.0745644	−	0.997216i	$-0.523757\pi$
$503$	−846216.	−0.149129	−0.0745644	+	0.997216i	$0.523757\pi$
$504$	666792.	0.116927
$505$	−211068.	−0.0368293
$506$	−1.11888e7	−1.94271
$507$	1.58336e6	0.273565
$508$	−297280.	−0.0511101
$509$	−7.66785e6	−1.31183	−0.655917	−	0.754833i	$-0.727718\pi$
$509$	−7.66785e6	−1.31183	−0.655917	+	0.754833i	$0.727718\pi$
$510$	−530712.	−0.0903511
$511$	−3.03045e6	−0.513400
$512$	4.47130e6	0.753804
$513$	−1.95664e6	−0.328259
$514$	−573516.	−0.0957498
$515$	1.02779e7	1.70760
$516$	414864.	0.0685933
$517$	−6.18048e6	−1.01694
$518$	1.59760e6	0.261603
$519$	164970.	0.0268835
$520$	−5.79197e6	−0.939329
$521$	−9.68938e6	−1.56387	−0.781937	−	0.623357i	$-0.785768\pi$
$521$	−9.68938e6	−1.56387	−0.781937	+	0.623357i	$0.785768\pi$
$522$	2.64481e6	0.424833
$523$	−7.51678e6	−1.20165	−0.600824	−	0.799381i	$-0.705161\pi$
$523$	−7.51678e6	−1.20165	−0.600824	+	0.799381i	$0.705161\pi$
$524$	−621264.	−0.0988435
$525$	−1.30492e6	−0.206626
$526$	1.32021e7	2.08055
$527$	−10080.0	−0.00158101
$528$	4.53946e6	0.708629
$529$	1.12037e7	1.74069
$530$	4.48999e6	0.694314
$531$	2.22685e6	0.342732
$532$	526064.	0.0805860
$533$	−3.51920e6	−0.536570
$534$	−430596.	−0.0653457
$535$	−1.00554e7	−1.51886
$536$	−9.97181e6	−1.49921
$537$	1.37992e6	0.206499
$538$	−1.06215e7	−1.58208
$539$	1.06604e6	0.158053
$540$	−227448.	−0.0335659
$541$	7.34325e6	1.07869	0.539343	−	0.842086i	$-0.318673\pi$
$541$	7.34325e6	1.07869	0.539343	+	0.842086i	$0.318673\pi$
$542$	1.34102e6	0.196082
$543$	3.43859e6	0.500474
$544$	−181440.	−0.0262867
$545$	−7.87628e6	−1.13587
$546$	−1.16953e6	−0.167892
$547$	2.18296e6	0.311945	0.155973	−	0.987761i	$-0.450149\pi$
$547$	2.18296e6	0.311945	0.155973	+	0.987761i	$0.450149\pi$
$548$	−1.05698e6	−0.150355
$549$	4.00772e6	0.567501
$550$	−7.88278e6	−1.11115
$551$	−1.46063e7	−2.04957
$552$	−6.35040e6	−0.887061
$553$	−3.22302e6	−0.448178
$554$	2.05667e6	0.284702
$555$	3.81467e6	0.525683
$556$	898448.	0.123255
$557$	1.25466e7	1.71351	0.856755	−	0.515724i	$-0.172477\pi$
$557$	1.25466e7	1.71351	0.856755	+	0.515724i	$0.172477\pi$
$558$	−38880.0	−0.00528617
$559$	5.09361e6	0.689439
$560$	−4.34179e6	−0.585058
$561$	503496.	0.0675443
$562$	−2.88227e6	−0.384940
$563$	5.15972e6	0.686050	0.343025	−	0.939326i	$-0.388549\pi$
$563$	5.15972e6	0.686050	0.343025	+	0.939326i	$0.388549\pi$
$564$	501120.	0.0663352
$565$	1.71714e7	2.26300
$566$	179880.	0.0236016
$567$	321489.	0.0419961
$568$	5.38272e6	0.700053
$569$	1.17452e7	1.52083	0.760414	−	0.649439i	$-0.224996\pi$
$569$	1.17452e7	1.52083	0.760414	+	0.649439i	$0.224996\pi$
$570$	1.13050e7	1.45742
$571$	−7.54728e6	−0.968725	−0.484362	−	0.874867i	$-0.660948\pi$
$571$	−7.54728e6	−0.968725	−0.484362	+	0.874867i	$0.660948\pi$
$572$	−784992.	−0.100317
$573$	2.46067e6	0.313089
$574$	−2.34083e6	−0.296544
$575$	1.24278e7	1.56756
$576$	2.24467e6	0.281901
$577$	9.28483e6	1.16101	0.580503	−	0.814258i	$-0.302856\pi$
$577$	9.28483e6	1.16101	0.580503	+	0.814258i	$0.302856\pi$
$578$	8.42389e6	1.04880
$579$	−1.38242e6	−0.171373
$580$	−1.69790e6	−0.209577
$581$	1.96921e6	0.242020
$582$	−7.75775e6	−0.949354
$583$	−4.25974e6	−0.519053
$584$	−1.03901e7	−1.26063
$585$	−2.79256e6	−0.337374
$586$	1.18840e6	0.142961
$587$	1.47623e6	0.176831	0.0884155	−	0.996084i	$-0.471820\pi$
$587$	1.47623e6	0.176831	0.0884155	+	0.996084i	$0.471820\pi$
$588$	−86436.0	−0.0103098
$589$	214720.	0.0255026
$590$	−1.28663e7	−1.52168
$591$	−1.38980e6	−0.163675
$592$	6.17302e6	0.723925
$593$	−1.24007e7	−1.44813	−0.724067	−	0.689729i	$-0.757730\pi$
$593$	−1.24007e7	−1.44813	−0.724067	+	0.689729i	$0.757730\pi$
$594$	1.94206e6	0.225837
$595$	−481572.	−0.0557659
$596$	−328296.	−0.0378573
$597$	3.30170e6	0.379142
$598$	1.11384e7	1.27371
$599$	−3.69127e6	−0.420348	−0.210174	−	0.977664i	$-0.567403\pi$
$599$	−3.69127e6	−0.420348	−0.210174	+	0.977664i	$0.567403\pi$
$600$	−4.47401e6	−0.507363
$601$	9.12223e6	1.03018	0.515092	−	0.857135i	$-0.327758\pi$
$601$	9.12223e6	1.03018	0.515092	+	0.857135i	$0.327758\pi$
$602$	3.38806e6	0.381030
$603$	−4.80784e6	−0.538464
$604$	−1.14813e6	−0.128055
$605$	2.81463e6	0.312632
$606$	−146124.	−0.0161637
$607$	−5.67914e6	−0.625620	−0.312810	−	0.949816i	$-0.601270\pi$
$607$	−5.67914e6	−0.625620	−0.312810	+	0.949816i	$0.601270\pi$
$608$	3.86496e6	0.424020
$609$	2.39992e6	0.262213
$610$	−2.31557e7	−2.51961
$611$	6.15264e6	0.666743
$612$	−40824.0	−0.00440592
$613$	−1.40106e7	−1.50593	−0.752966	−	0.658060i	$-0.771377\pi$
$613$	−1.40106e7	−1.50593	−0.752966	+	0.658060i	$0.771377\pi$
$614$	6.27382e6	0.671600
$615$	−5.58932e6	−0.595897
$616$	3.65501e6	0.388094
$617$	−253686.	−0.0268277	−0.0134139	−	0.999910i	$-0.504270\pi$
$617$	−253686.	−0.0268277	−0.0134139	+	0.999910i	$0.504270\pi$
$618$	7.11547e6	0.749433
$619$	4.30034e6	0.451103	0.225552	−	0.974231i	$-0.427582\pi$
$619$	4.30034e6	0.451103	0.225552	+	0.974231i	$0.427582\pi$
$620$	24960.0	0.00260775
$621$	−3.06180e6	−0.318602
$622$	−1.10231e7	−1.14242
$623$	−390726.	−0.0403322
$624$	−4.51901e6	−0.464603
$625$	−1.02568e7	−1.05030
$626$	2.19296e6	0.223664
$627$	−1.07253e7	−1.08953
$628$	519512.	0.0525650
$629$	684684.	0.0690023
$630$	−1.85749e6	−0.186456
$631$	1.04150e7	1.04132	0.520662	−	0.853763i	$-0.325685\pi$
$631$	1.04150e7	1.04132	0.520662	+	0.853763i	$0.325685\pi$
$632$	−1.10504e7	−1.10048
$633$	−4.68220e6	−0.464451
$634$	170028.	0.0167995
$635$	−5.79696e6	−0.570514
$636$	345384.	0.0338579
$637$	−1.06124e6	−0.103625
$638$	1.44975e7	1.41007
$639$	2.59524e6	0.251435
$640$	−1.65635e7	−1.59846
$641$	4.52714e6	0.435190	0.217595	−	0.976039i	$-0.430179\pi$
$641$	4.52714e6	0.435190	0.217595	+	0.976039i	$0.430179\pi$
$642$	−6.96146e6	−0.666596
$643$	1.49687e7	1.42776	0.713882	−	0.700266i	$-0.246935\pi$
$643$	1.49687e7	1.42776	0.713882	+	0.700266i	$0.246935\pi$
$644$	823200.	0.0782151
$645$	8.08985e6	0.765669
$646$	2.02910e6	0.191304
$647$	−1.73020e7	−1.62493	−0.812465	−	0.583010i	$-0.801875\pi$
$647$	−1.73020e7	−1.62493	−0.812465	+	0.583010i	$0.801875\pi$
$648$	1.10225e6	0.103120
$649$	1.22064e7	1.13757
$650$	7.84727e6	0.728509
$651$	−35280.0	−0.00326269
$652$	2.22114e6	0.204624
$653$	4.07470e6	0.373949	0.186975	−	0.982365i	$-0.440132\pi$
$653$	4.07470e6	0.373949	0.186975	+	0.982365i	$0.440132\pi$
$654$	−5.45281e6	−0.498513
$655$	−1.21146e7	−1.10334
$656$	−9.04483e6	−0.820618
$657$	−5.00953e6	−0.452776
$658$	4.09248e6	0.368487
$659$	−3.79475e6	−0.340384	−0.170192	−	0.985411i	$-0.554439\pi$
$659$	−3.79475e6	−0.340384	−0.170192	+	0.985411i	$0.554439\pi$
$660$	−1.24675e6	−0.111409
$661$	1.64261e7	1.46228	0.731142	−	0.682225i	$-0.238988\pi$
$661$	1.64261e7	1.46228	0.731142	+	0.682225i	$0.238988\pi$
$662$	−1.16035e7	−1.02907
$663$	−501228.	−0.0442845
$664$	6.75158e6	0.594272
$665$	1.02582e7	0.899537
$666$	2.64092e6	0.230712
$667$	−2.28564e7	−1.98927
$668$	174048.	0.0150913
$669$	−2.74262e6	−0.236920
$670$	2.77786e7	2.39069
$671$	2.19682e7	1.88360
$672$	−635040.	−0.0542473
$673$	5.50675e6	0.468660	0.234330	−	0.972157i	$-0.424710\pi$
$673$	5.50675e6	0.468660	0.234330	+	0.972157i	$0.424710\pi$
$674$	1.13290e7	0.960602
$675$	−2.15711e6	−0.182227
$676$	−703716.	−0.0592285
$677$	1.83957e7	1.54257	0.771286	−	0.636488i	$-0.219614\pi$
$677$	1.83957e7	1.54257	0.771286	+	0.636488i	$0.219614\pi$
$678$	1.18879e7	0.993185
$679$	−7.03944e6	−0.585954
$680$	−1.65110e6	−0.136931
$681$	−2.59729e6	−0.214612
$682$	−213120.	−0.0175454
$683$	1.75835e6	0.144229	0.0721146	−	0.997396i	$-0.477025\pi$
$683$	1.75835e6	0.144229	0.0721146	+	0.997396i	$0.477025\pi$
$684$	869616.	0.0710702
$685$	−2.06112e7	−1.67833
$686$	−705894.	−0.0572703
$687$	−6.94971e6	−0.561791
$688$	1.30913e7	1.05441
$689$	4.24055e6	0.340309
$690$	1.76904e7	1.41454
$691$	−5.36314e6	−0.427291	−0.213646	−	0.976911i	$-0.568534\pi$
$691$	−5.36314e6	−0.427291	−0.213646	+	0.976911i	$0.568534\pi$
$692$	−73320.0	−0.00582046
$693$	1.76224e6	0.139390
$694$	−1.75162e7	−1.38052
$695$	1.75197e7	1.37583
$696$	8.22830e6	0.643854
$697$	−1.00321e6	−0.0782187
$698$	4.68409e6	0.363904
$699$	−2.27011e6	−0.175733
$700$	579964.	0.0447359
$701$	−2.12606e7	−1.63411	−0.817054	−	0.576561i	$-0.804394\pi$
$701$	−2.12606e7	−1.63411	−0.817054	+	0.576561i	$0.804394\pi$
$702$	−1.93331e6	−0.148067
$703$	−1.45849e7	−1.11305
$704$	1.23041e7	0.935662
$705$	9.77184e6	0.740463
$706$	−8.00622e6	−0.604527
$707$	−132594.	−0.00997643
$708$	−989712.	−0.0742037
$709$	2.07729e6	0.155196	0.0775980	−	0.996985i	$-0.475275\pi$
$709$	2.07729e6	0.155196	0.0775980	+	0.996985i	$0.475275\pi$
$710$	−1.49947e7	−1.11633
$711$	−5.32786e6	−0.395256
$712$	−1.33963e6	−0.0990343
$713$	336000.	0.0247523
$714$	−333396.	−0.0244746
$715$	−1.53073e7	−1.11979
$716$	−613296.	−0.0447082
$717$	1.30602e7	0.948752
$718$	−6.10459e6	−0.441922
$719$	4.23619e6	0.305600	0.152800	−	0.988257i	$-0.451171\pi$
$719$	4.23619e6	0.305600	0.152800	+	0.988257i	$0.451171\pi$
$720$	−7.17725e6	−0.515973
$721$	6.45663e6	0.462560
$722$	−2.83665e7	−2.02518
$723$	1.31758e6	0.0937415
$724$	−1.52826e6	−0.108356
$725$	−1.61029e7	−1.13778
$726$	1.94859e6	0.137208
$727$	2.14524e7	1.50536	0.752678	−	0.658389i	$-0.228762\pi$
$727$	2.14524e7	1.50536	0.752678	+	0.658389i	$0.228762\pi$
$728$	−3.63854e6	−0.254448
$729$	531441.	0.0370370
$730$	2.89439e7	2.01025
$731$	1.45202e6	0.100503
$732$	−1.78121e6	−0.122867
$733$	−1.48892e7	−1.02355	−0.511777	−	0.859118i	$-0.671013\pi$
$733$	−1.48892e7	−1.02355	−0.511777	+	0.859118i	$0.671013\pi$
$734$	−5.02608e6	−0.344341
$735$	−1.68550e6	−0.115083
$736$	6.04800e6	0.411545
$737$	−2.63541e7	−1.78722
$738$	−3.86953e6	−0.261528
$739$	6.99324e6	0.471050	0.235525	−	0.971868i	$-0.424319\pi$
$739$	6.99324e6	0.471050	0.235525	+	0.971868i	$0.424319\pi$
$740$	−1.69541e6	−0.113814
$741$	1.06770e7	0.714335
$742$	2.82064e6	0.188078
$743$	1.90428e6	0.126549	0.0632745	−	0.997996i	$-0.479846\pi$
$743$	1.90428e6	0.126549	0.0632745	+	0.997996i	$0.479846\pi$
$744$	−120960.	−0.00801142
$745$	−6.40177e6	−0.422581
$746$	9.11958e6	0.599968
$747$	3.25523e6	0.213442
$748$	−223776.	−0.0146238
$749$	−6.31688e6	−0.411432
$750$	−699192.	−0.0453882
$751$	1.95361e7	1.26398	0.631988	−	0.774978i	$-0.282239\pi$
$751$	1.95361e7	1.26398	0.631988	+	0.774978i	$0.282239\pi$
$752$	1.58131e7	1.01970
$753$	−5.47074e6	−0.351608
$754$	−1.44322e7	−0.924493
$755$	−2.23885e7	−1.42941
$756$	−142884.	−0.00909241
$757$	1.25183e6	0.0793973	0.0396986	−	0.999212i	$-0.487360\pi$
$757$	1.25183e6	0.0793973	0.0396986	+	0.999212i	$0.487360\pi$
$758$	−1.58679e7	−1.00311
$759$	−1.67832e7	−1.05748
$760$	3.51711e7	2.20878
$761$	2.04472e7	1.27989	0.639944	−	0.768422i	$-0.278958\pi$
$761$	2.04472e7	1.27989	0.639944	+	0.768422i	$0.278958\pi$
$762$	−4.01328e6	−0.250387
$763$	−4.94792e6	−0.307689
$764$	−1.09363e6	−0.0677857
$765$	−796068.	−0.0491809
$766$	−1.20802e7	−0.743876
$767$	−1.21515e7	−0.745831
$768$	−3.48595e6	−0.213264
$769$	2.21064e6	0.134804	0.0674020	−	0.997726i	$-0.478529\pi$
$769$	2.21064e6	0.134804	0.0674020	+	0.997726i	$0.478529\pi$
$770$	−1.01818e7	−0.618868
$771$	−860274.	−0.0521196
$772$	614408.	0.0371034
$773$	1.29151e7	0.777405	0.388703	−	0.921363i	$-0.372923\pi$
$773$	1.29151e7	0.777405	0.388703	+	0.921363i	$0.372923\pi$
$774$	5.60066e6	0.336037
$775$	236720.	0.0141573
$776$	−2.41352e7	−1.43879
$777$	2.39639e6	0.142399
$778$	4.35740e6	0.258095
$779$	2.13700e7	1.26171
$780$	1.24114e6	0.0730437
$781$	1.42258e7	0.834541
$782$	3.17520e6	0.185675
$783$	3.96722e6	0.231250
$784$	−2.72754e6	−0.158482
$785$	1.01305e7	0.586754
$786$	−8.38706e6	−0.484232
$787$	−1.35499e7	−0.779830	−0.389915	−	0.920851i	$-0.627496\pi$
$787$	−1.35499e7	−0.779830	−0.389915	+	0.920851i	$0.627496\pi$
$788$	617688.	0.0354367
$789$	1.98031e7	1.13251
$790$	3.07832e7	1.75487
$791$	1.07872e7	0.613007
$792$	6.04195e6	0.342266
$793$	−2.18693e7	−1.23496
$794$	−2.74547e7	−1.54549
$795$	6.73499e6	0.377937
$796$	−1.46742e6	−0.0820867
$797$	−2.45956e7	−1.37155	−0.685776	−	0.727813i	$-0.740537\pi$
$797$	−2.45956e7	−1.37155	−0.685776	+	0.727813i	$0.740537\pi$
$798$	7.10186e6	0.394789
$799$	1.75392e6	0.0971948
$800$	4.26096e6	0.235387
$801$	−645894.	−0.0355697
$802$	203220.	0.0111566
$803$	−2.74596e7	−1.50282
$804$	2.13682e6	0.116581
$805$	1.60524e7	0.873072
$806$	212160.	0.0115034
$807$	−1.59322e7	−0.861177
$808$	−454608.	−0.0244968
$809$	1.55237e7	0.833920	0.416960	−	0.908925i	$-0.363095\pi$
$809$	1.55237e7	0.833920	0.416960	+	0.908925i	$0.363095\pi$
$810$	−3.07055e6	−0.164438
$811$	−2.66262e7	−1.42153	−0.710766	−	0.703429i	$-0.751651\pi$
$811$	−2.66262e7	−1.42153	−0.710766	+	0.703429i	$0.751651\pi$
$812$	−1.06663e6	−0.0567707
$813$	2.01154e6	0.106734
$814$	1.44762e7	0.765760
$815$	4.33122e7	2.28410
$816$	−1.28822e6	−0.0677276
$817$	−3.09304e7	−1.62118
$818$	3.51707e7	1.83780
$819$	−1.75430e6	−0.0913889
$820$	2.48414e6	0.129016
$821$	−1.23891e7	−0.641477	−0.320739	−	0.947168i	$-0.603931\pi$
$821$	−1.23891e7	−0.641477	−0.320739	+	0.947168i	$0.603931\pi$
$822$	−1.42693e7	−0.736585
$823$	−3.65630e6	−0.188166	−0.0940831	−	0.995564i	$-0.529992\pi$
$823$	−3.65630e6	−0.188166	−0.0940831	+	0.995564i	$0.529992\pi$
$824$	2.21370e7	1.13580
$825$	−1.18242e7	−0.604833
$826$	−8.08265e6	−0.412196
$827$	2.80463e7	1.42597	0.712987	−	0.701178i	$-0.247342\pi$
$827$	2.80463e7	1.42597	0.712987	+	0.701178i	$0.247342\pi$
$828$	1.36080e6	0.0689792
$829$	2.11153e7	1.06712	0.533558	−	0.845763i	$-0.320855\pi$
$829$	2.11153e7	1.06712	0.533558	+	0.845763i	$0.320855\pi$
$830$	−1.88080e7	−0.947648
$831$	3.08500e6	0.154972
$832$	−1.22487e7	−0.613454
$833$	−302526.	−0.0151060
$834$	1.21290e7	0.603826
$835$	3.39394e6	0.168456
$836$	4.76678e6	0.235890
$837$	−58320.0	−0.00287742
$838$	−1.81649e6	−0.0893557
$839$	1.33947e7	0.656944	0.328472	−	0.944514i	$-0.393466\pi$
$839$	1.33947e7	0.656944	0.328472	+	0.944514i	$0.393466\pi$
$840$	−5.77886e6	−0.282582
$841$	9.10422e6	0.443867
$842$	3.22025e7	1.56534
$843$	−4.32340e6	−0.209535
$844$	2.08098e6	0.100557
$845$	−1.37225e7	−0.661135
$846$	6.76512e6	0.324975
$847$	1.76816e6	0.0846865
$848$	1.08988e7	0.520461
$849$	269820.	0.0128471
$850$	2.23700e6	0.106199
$851$	−2.28228e7	−1.08030
$852$	−1.15344e6	−0.0544372
$853$	3.01513e7	1.41884	0.709420	−	0.704786i	$-0.248957\pi$
$853$	3.01513e7	1.41884	0.709420	+	0.704786i	$0.248957\pi$
$854$	−1.45465e7	−0.682519
$855$	1.69575e7	0.793317
$856$	−2.16579e7	−1.01026
$857$	2.39894e7	1.11575	0.557875	−	0.829925i	$-0.311617\pi$
$857$	2.39894e7	1.11575	0.557875	+	0.829925i	$0.311617\pi$
$858$	−1.05974e7	−0.491452
$859$	−8.87576e6	−0.410414	−0.205207	−	0.978719i	$-0.565787\pi$
$859$	−8.87576e6	−0.410414	−0.205207	+	0.978719i	$0.565787\pi$
$860$	−3.59549e6	−0.165772
$861$	−3.51124e6	−0.161418
$862$	−7.06234e6	−0.323728
$863$	−8.71286e6	−0.398230	−0.199115	−	0.979976i	$-0.563807\pi$
$863$	−8.71286e6	−0.398230	−0.199115	+	0.979976i	$0.563807\pi$
$864$	−1.04976e6	−0.0478416
$865$	−1.42974e6	−0.0649706
$866$	2.19750e7	0.995711
$867$	1.26358e7	0.570895
$868$	15680.0	0.000706394 0
$869$	−2.92045e7	−1.31190
$870$	−2.29217e7	−1.02671
$871$	2.62354e7	1.17177
$872$	−1.69643e7	−0.755518
$873$	−1.16366e7	−0.516763
$874$	−6.76368e7	−2.99505
$875$	−634452.	−0.0280142
$876$	2.22646e6	0.0980288
$877$	−2.95788e7	−1.29862	−0.649310	−	0.760524i	$-0.724942\pi$
$877$	−2.95788e7	−1.29862	−0.649310	+	0.760524i	$0.724942\pi$
$878$	1.52205e7	0.666333
$879$	1.78259e6	0.0778180
$880$	−3.93420e7	−1.71257
$881$	2.45670e7	1.06638	0.533190	−	0.845995i	$-0.320993\pi$
$881$	2.45670e7	1.06638	0.533190	+	0.845995i	$0.320993\pi$
$882$	−1.16689e6	−0.0505076
$883$	1.45682e7	0.628788	0.314394	−	0.949293i	$-0.398199\pi$
$883$	1.45682e7	0.628788	0.314394	+	0.949293i	$0.398199\pi$
$884$	222768.	0.00958787
$885$	−1.92994e7	−0.828295
$886$	−3.60902e7	−1.54456
$887$	1.61714e7	0.690141	0.345070	−	0.938577i	$-0.387855\pi$
$887$	1.61714e7	0.690141	0.345070	+	0.938577i	$0.387855\pi$
$888$	8.21621e6	0.349654
$889$	−3.64168e6	−0.154542
$890$	3.73183e6	0.157924
$891$	2.91308e6	0.122930
$892$	1.21894e6	0.0512946
$893$	−3.73613e7	−1.56781
$894$	−4.43200e6	−0.185462
$895$	−1.19593e7	−0.499054
$896$	−1.04052e7	−0.432995
$897$	1.67076e7	0.693319
$898$	−3.39579e7	−1.40524
$899$	−435360.	−0.0179659
$900$	958716.	0.0394533
$901$	1.20884e6	0.0496087
$902$	−2.12108e7	−0.868041
$903$	5.08208e6	0.207407
$904$	3.69845e7	1.50522
$905$	−2.98011e7	−1.20952
$906$	−1.54997e7	−0.627341
$907$	3.14446e7	1.26919	0.634596	−	0.772844i	$-0.281167\pi$
$907$	3.14446e7	1.26919	0.634596	+	0.772844i	$0.281167\pi$
$908$	1.15435e6	0.0464648
$909$	−219186.	−0.00879839
$910$	1.01359e7	0.405752
$911$	1.51427e7	0.604514	0.302257	−	0.953227i	$-0.402260\pi$
$911$	1.51427e7	0.604514	0.302257	+	0.953227i	$0.402260\pi$
$912$	2.74412e7	1.09249
$913$	1.78435e7	0.708439
$914$	3.87695e7	1.53506
$915$	−3.47336e7	−1.37150
$916$	3.08876e6	0.121631
$917$	−7.61048e6	−0.298875
$918$	−551124.	−0.0215845
$919$	4.14876e7	1.62043	0.810214	−	0.586134i	$-0.199351\pi$
$919$	4.14876e7	1.62043	0.810214	+	0.586134i	$0.199351\pi$
$920$	5.50368e7	2.14380
$921$	9.41072e6	0.365573
$922$	2.02412e7	0.784167
$923$	−1.41617e7	−0.547155
$924$	−783216.	−0.0301788
$925$	−1.60792e7	−0.617889
$926$	2.72985e7	1.04619
$927$	1.06732e7	0.407939
$928$	−7.83648e6	−0.298711
$929$	−1.78495e7	−0.678556	−0.339278	−	0.940686i	$-0.610183\pi$
$929$	−1.78495e7	−0.678556	−0.339278	+	0.940686i	$0.610183\pi$
$930$	336960.	0.0127753
$931$	6.44428e6	0.243669
$932$	1.00894e6	0.0380473
$933$	−1.65346e7	−0.621855
$934$	−1.20681e7	−0.452661
$935$	−4.36363e6	−0.163237
$936$	−6.01474e6	−0.224402
$937$	2.96399e7	1.10288	0.551439	−	0.834215i	$-0.314079\pi$
$937$	2.96399e7	1.10288	0.551439	+	0.834215i	$0.314079\pi$
$938$	1.74507e7	0.647597
$939$	3.28945e6	0.121747
$940$	−4.34304e6	−0.160315
$941$	−3.22282e7	−1.18648	−0.593242	−	0.805024i	$-0.702152\pi$
$941$	−3.22282e7	−1.18648	−0.593242	+	0.805024i	$0.702152\pi$
$942$	7.01341e6	0.257515
$943$	3.34404e7	1.22459
$944$	−3.12309e7	−1.14066
$945$	−2.78624e6	−0.101494
$946$	3.06999e7	1.11535
$947$	4.84885e7	1.75697	0.878484	−	0.477772i	$-0.158556\pi$
$947$	4.84885e7	1.75697	0.878484	+	0.477772i	$0.158556\pi$
$948$	2.36794e6	0.0855754
$949$	2.73359e7	0.985300
$950$	−4.76517e7	−1.71305
$951$	255042.	0.00914451
$952$	−1.03723e6	−0.0370923
$953$	−2.03264e7	−0.724983	−0.362491	−	0.931987i	$-0.618074\pi$
$953$	−2.03264e7	−0.724983	−0.362491	+	0.931987i	$0.618074\pi$
$954$	4.66268e6	0.165869
$955$	−2.13258e7	−0.756654
$956$	−5.80454e6	−0.205411
$957$	2.17462e7	0.767546
$958$	4.56241e7	1.60613
$959$	−1.29481e7	−0.454630
$960$	−1.94538e7	−0.681282
$961$	−2.86228e7	−0.999776
$962$	−1.44110e7	−0.502060
$963$	−1.04422e7	−0.362849
$964$	−585592.	−0.0202956
$965$	1.19810e7	0.414165
$966$	1.11132e7	0.383174
$967$	−3.66292e6	−0.125968	−0.0629841	−	0.998015i	$-0.520062\pi$
$967$	−3.66292e6	−0.125968	−0.0629841	+	0.998015i	$0.520062\pi$
$968$	6.06228e6	0.207945
$969$	3.04366e6	0.104132
$970$	6.72338e7	2.29434
$971$	1.48741e6	0.0506271	0.0253136	−	0.999680i	$-0.491942\pi$
$971$	1.48741e6	0.0506271	0.0253136	+	0.999680i	$0.491942\pi$
$972$	−236196.	−0.00801875
$973$	1.10060e7	0.372689
$974$	−4.03867e6	−0.136408
$975$	1.17709e7	0.396550
$976$	−5.62070e7	−1.88871
$977$	4.07930e7	1.36725	0.683627	−	0.729831i	$-0.260401\pi$
$977$	4.07930e7	1.36725	0.683627	+	0.729831i	$0.260401\pi$
$978$	2.99853e7	1.00245
$979$	−3.54046e6	−0.118060
$980$	749112.	0.0249162
$981$	−8.17922e6	−0.271356
$982$	1.48302e7	0.490759
$983$	−9.26326e6	−0.305759	−0.152880	−	0.988245i	$-0.548855\pi$
$983$	−9.26326e6	−0.305759	−0.152880	+	0.988245i	$0.548855\pi$
$984$	−1.20385e7	−0.396357
$985$	1.20449e7	0.395561
$986$	−4.11415e6	−0.134768
$987$	6.13872e6	0.200579
$988$	−4.74531e6	−0.154658
$989$	−4.84008e7	−1.57348
$990$	−1.68312e7	−0.545790
$991$	−5.22051e7	−1.68861	−0.844303	−	0.535866i	$-0.819985\pi$
$991$	−5.22051e7	−1.68861	−0.844303	+	0.535866i	$0.819985\pi$
$992$	115200.	0.00371684
$993$	−1.74052e7	−0.560153
$994$	−9.41976e6	−0.302394
$995$	−2.86148e7	−0.916289
$996$	−1.44677e6	−0.0462116
$997$	−1.86609e7	−0.594560	−0.297280	−	0.954790i	$-0.596079\pi$
$997$	−1.86609e7	−0.594560	−0.297280	+	0.954790i	$0.596079\pi$
$998$	−3.64891e7	−1.15968
$999$	3.96139e6	0.125584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.6.a.a.1.1	✓	1
3.2	odd	2		63.6.a.d.1.1		1
4.3	odd	2		336.6.a.r.1.1		1
5.2	odd	4		525.6.d.b.274.1		2
5.3	odd	4		525.6.d.b.274.2		2
5.4	even	2		525.6.a.d.1.1		1
7.2	even	3		147.6.e.j.67.1		2
7.3	odd	6		147.6.e.i.79.1		2
7.4	even	3		147.6.e.j.79.1		2
7.5	odd	6		147.6.e.i.67.1		2
7.6	odd	2		147.6.a.b.1.1		1
12.11	even	2		1008.6.a.c.1.1		1
21.20	even	2		441.6.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.a.1.1	✓	1	1.1	even	1	trivial
63.6.a.d.1.1		1	3.2	odd	2
147.6.a.b.1.1		1	7.6	odd	2
147.6.e.i.67.1		2	7.5	odd	6
147.6.e.i.79.1		2	7.3	odd	6
147.6.e.j.67.1		2	7.2	even	3
147.6.e.j.79.1		2	7.4	even	3
336.6.a.r.1.1		1	4.3	odd	2
441.6.a.j.1.1		1	21.20	even	2
525.6.a.d.1.1		1	5.4	even	2
525.6.d.b.274.1		2	5.2	odd	4
525.6.d.b.274.2		2	5.3	odd	4
1008.6.a.c.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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