LMFDB - Embedded newform 21.6.a.c.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)

$f(q)$

$=$

$q+5.00000 q^{2} +9.00000 q^{3} -7.00000 q^{4} +94.0000 q^{5} +45.0000 q^{6} -49.0000 q^{7} -195.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+5.00000 q^{2} +9.00000 q^{3} -7.00000 q^{4} +94.0000 q^{5} +45.0000 q^{6} -49.0000 q^{7} -195.000 q^{8} +81.0000 q^{9} +470.000 q^{10} +52.0000 q^{11} -63.0000 q^{12} -770.000 q^{13} -245.000 q^{14} +846.000 q^{15} -751.000 q^{16} -2022.00 q^{17} +405.000 q^{18} +1732.00 q^{19} -658.000 q^{20} -441.000 q^{21} +260.000 q^{22} -576.000 q^{23} -1755.00 q^{24} +5711.00 q^{25} -3850.00 q^{26} +729.000 q^{27} +343.000 q^{28} +5518.00 q^{29} +4230.00 q^{30} +6336.00 q^{31} +2485.00 q^{32} +468.000 q^{33} -10110.0 q^{34} -4606.00 q^{35} -567.000 q^{36} -7338.00 q^{37} +8660.00 q^{38} -6930.00 q^{39} -18330.0 q^{40} -3262.00 q^{41} -2205.00 q^{42} +5420.00 q^{43} -364.000 q^{44} +7614.00 q^{45} -2880.00 q^{46} +864.000 q^{47} -6759.00 q^{48} +2401.00 q^{49} +28555.0 q^{50} -18198.0 q^{51} +5390.00 q^{52} +4182.00 q^{53} +3645.00 q^{54} +4888.00 q^{55} +9555.00 q^{56} +15588.0 q^{57} +27590.0 q^{58} -11220.0 q^{59} -5922.00 q^{60} -45602.0 q^{61} +31680.0 q^{62} -3969.00 q^{63} +36457.0 q^{64} -72380.0 q^{65} +2340.00 q^{66} +1396.00 q^{67} +14154.0 q^{68} -5184.00 q^{69} -23030.0 q^{70} +18720.0 q^{71} -15795.0 q^{72} +46362.0 q^{73} -36690.0 q^{74} +51399.0 q^{75} -12124.0 q^{76} -2548.00 q^{77} -34650.0 q^{78} +97424.0 q^{79} -70594.0 q^{80} +6561.00 q^{81} -16310.0 q^{82} -81228.0 q^{83} +3087.00 q^{84} -190068. q^{85} +27100.0 q^{86} +49662.0 q^{87} -10140.0 q^{88} -3182.00 q^{89} +38070.0 q^{90} +37730.0 q^{91} +4032.00 q^{92} +57024.0 q^{93} +4320.00 q^{94} +162808. q^{95} +22365.0 q^{96} +4914.00 q^{97} +12005.0 q^{98} +4212.00 q^{99} +O(q^{100})\)

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 $

$2$	5.00000	0.883883	0.441942	−	0.897044i	$-0.354290\pi$
$2$	5.00000	0.883883	0.441942	+	0.897044i	$0.354290\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−7.00000	−0.218750
$5$	94.0000	1.68152	0.840762	−	0.541406i	$-0.182108\pi$
$5$	94.0000	1.68152	0.840762	+	0.541406i	$0.182108\pi$
$6$	45.0000	0.510310
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−195.000	−1.07723
$9$	81.0000	0.333333
$10$	470.000	1.48627
$11$	52.0000	0.129575	0.0647876	−	0.997899i	$-0.479363\pi$
$11$	52.0000	0.129575	0.0647876	+	0.997899i	$0.479363\pi$
$12$	−63.0000	−0.126295
$13$	−770.000	−1.26367	−0.631833	−	0.775104i	$-0.717697\pi$
$13$	−770.000	−1.26367	−0.631833	+	0.775104i	$0.717697\pi$
$14$	−245.000	−0.334077
$15$	846.000	0.970828
$16$	−751.000	−0.733398
$17$	−2022.00	−1.69691	−0.848455	−	0.529267i	$-0.822467\pi$
$17$	−2022.00	−1.69691	−0.848455	+	0.529267i	$0.822467\pi$
$18$	405.000	0.294628
$19$	1732.00	1.10069	0.550344	−	0.834938i	$-0.314497\pi$
$19$	1732.00	1.10069	0.550344	+	0.834938i	$0.314497\pi$
$20$	−658.000	−0.367833
$21$	−441.000	−0.218218
$22$	260.000	0.114529
$23$	−576.000	−0.227040	−0.113520	−	0.993536i	$-0.536213\pi$
$23$	−576.000	−0.227040	−0.113520	+	0.993536i	$0.536213\pi$
$24$	−1755.00	−0.621941
$25$	5711.00	1.82752
$26$	−3850.00	−1.11693
$27$	729.000	0.192450
$28$	343.000	0.0826797
$29$	5518.00	1.21839	0.609196	−	0.793020i	$-0.291492\pi$
$29$	5518.00	1.21839	0.609196	+	0.793020i	$0.291492\pi$
$30$	4230.00	0.858099
$31$	6336.00	1.18416	0.592081	−	0.805879i	$-0.298307\pi$
$31$	6336.00	1.18416	0.592081	+	0.805879i	$0.298307\pi$
$32$	2485.00	0.428994
$33$	468.000	0.0748102
$34$	−10110.0	−1.49987
$35$	−4606.00	−0.635556
$36$	−567.000	−0.0729167
$37$	−7338.00	−0.881198	−0.440599	−	0.897704i	$-0.645234\pi$
$37$	−7338.00	−0.881198	−0.440599	+	0.897704i	$0.645234\pi$
$38$	8660.00	0.972879
$39$	−6930.00	−0.729578
$40$	−18330.0	−1.81139
$41$	−3262.00	−0.303057	−0.151528	−	0.988453i	$-0.548420\pi$
$41$	−3262.00	−0.303057	−0.151528	+	0.988453i	$0.548420\pi$
$42$	−2205.00	−0.192879
$43$	5420.00	0.447021	0.223511	−	0.974701i	$-0.428248\pi$
$43$	5420.00	0.447021	0.223511	+	0.974701i	$0.428248\pi$
$44$	−364.000	−0.0283446
$45$	7614.00	0.560508
$46$	−2880.00	−0.200677
$47$	864.000	0.0570518	0.0285259	−	0.999593i	$-0.490919\pi$
$47$	864.000	0.0570518	0.0285259	+	0.999593i	$0.490919\pi$
$48$	−6759.00	−0.423428
$49$	2401.00	0.142857
$50$	28555.0	1.61531
$51$	−18198.0	−0.979712
$52$	5390.00	0.276427
$53$	4182.00	0.204500	0.102250	−	0.994759i	$-0.467396\pi$
$53$	4182.00	0.204500	0.102250	+	0.994759i	$0.467396\pi$
$54$	3645.00	0.170103
$55$	4888.00	0.217884
$56$	9555.00	0.407156
$57$	15588.0	0.635482
$58$	27590.0	1.07692
$59$	−11220.0	−0.419626	−0.209813	−	0.977741i	$-0.567286\pi$
$59$	−11220.0	−0.419626	−0.209813	+	0.977741i	$0.567286\pi$
$60$	−5922.00	−0.212369
$61$	−45602.0	−1.56913	−0.784566	−	0.620046i	$-0.787114\pi$
$61$	−45602.0	−1.56913	−0.784566	+	0.620046i	$0.787114\pi$
$62$	31680.0	1.04666
$63$	−3969.00	−0.125988
$64$	36457.0	1.11258
$65$	−72380.0	−2.12488
$66$	2340.00	0.0661235
$67$	1396.00	0.0379925	0.0189963	−	0.999820i	$-0.493953\pi$
$67$	1396.00	0.0379925	0.0189963	+	0.999820i	$0.493953\pi$
$68$	14154.0	0.371199
$69$	−5184.00	−0.131082
$70$	−23030.0	−0.561757
$71$	18720.0	0.440717	0.220359	−	0.975419i	$-0.429277\pi$
$71$	18720.0	0.440717	0.220359	+	0.975419i	$0.429277\pi$
$72$	−15795.0	−0.359078
$73$	46362.0	1.01825	0.509126	−	0.860692i	$-0.329969\pi$
$73$	46362.0	1.01825	0.509126	+	0.860692i	$0.329969\pi$
$74$	−36690.0	−0.778876
$75$	51399.0	1.05512
$76$	−12124.0	−0.240775
$77$	−2548.00	−0.0489748
$78$	−34650.0	−0.644862
$79$	97424.0	1.75630	0.878149	−	0.478387i	$-0.158778\pi$
$79$	97424.0	1.75630	0.878149	+	0.478387i	$0.158778\pi$
$80$	−70594.0	−1.23323
$81$	6561.00	0.111111
$82$	−16310.0	−0.267867
$83$	−81228.0	−1.29423	−0.647114	−	0.762394i	$-0.724024\pi$
$83$	−81228.0	−1.29423	−0.647114	+	0.762394i	$0.724024\pi$
$84$	3087.00	0.0477352
$85$	−190068.	−2.85339
$86$	27100.0	0.395115
$87$	49662.0	0.703438
$88$	−10140.0	−0.139583
$89$	−3182.00	−0.0425819	−0.0212910	−	0.999773i	$-0.506778\pi$
$89$	−3182.00	−0.0425819	−0.0212910	+	0.999773i	$0.506778\pi$
$90$	38070.0	0.495424
$91$	37730.0	0.477621
$92$	4032.00	0.0496651
$93$	57024.0	0.683676
$94$	4320.00	0.0504271
$95$	162808.	1.85083
$96$	22365.0	0.247680
$97$	4914.00	0.0530281	0.0265140	−	0.999648i	$-0.491559\pi$
$97$	4914.00	0.0530281	0.0265140	+	0.999648i	$0.491559\pi$
$98$	12005.0	0.126269
$99$	4212.00	0.0431917
$100$	−39977.0	−0.399770
$101$	−166354.	−1.62267	−0.811334	−	0.584583i	$-0.801258\pi$
$101$	−166354.	−1.62267	−0.811334	+	0.584583i	$0.801258\pi$
$102$	−90990.0	−0.865951
$103$	157160.	1.45965	0.729825	−	0.683634i	$-0.239601\pi$
$103$	157160.	1.45965	0.729825	+	0.683634i	$0.239601\pi$
$104$	150150.	1.36126
$105$	−41454.0	−0.366938
$106$	20910.0	0.180755
$107$	−6764.00	−0.0571142	−0.0285571	−	0.999592i	$-0.509091\pi$
$107$	−6764.00	−0.0571142	−0.0285571	+	0.999592i	$0.509091\pi$
$108$	−5103.00	−0.0420985
$109$	178398.	1.43821	0.719107	−	0.694899i	$-0.244551\pi$
$109$	178398.	1.43821	0.719107	+	0.694899i	$0.244551\pi$
$110$	24440.0	0.192584
$111$	−66042.0	−0.508760
$112$	36799.0	0.277199
$113$	−45134.0	−0.332512	−0.166256	−	0.986083i	$-0.553168\pi$
$113$	−45134.0	−0.332512	−0.166256	+	0.986083i	$0.553168\pi$
$114$	77940.0	0.561692
$115$	−54144.0	−0.381773
$116$	−38626.0	−0.266523
$117$	−62370.0	−0.421222
$118$	−56100.0	−0.370901
$119$	99078.0	0.641372
$120$	−164970.	−1.04581
$121$	−158347.	−0.983210
$122$	−228010.	−1.38693
$123$	−29358.0	−0.174970
$124$	−44352.0	−0.259035
$125$	243084.	1.39149
$126$	−19845.0	−0.111359
$127$	−205056.	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$127$	−205056.	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$128$	102765.	0.554396
$129$	48780.0	0.258088
$130$	−361900.	−1.87815
$131$	72964.0	0.371476	0.185738	−	0.982599i	$-0.440532\pi$
$131$	72964.0	0.371476	0.185738	+	0.982599i	$0.440532\pi$
$132$	−3276.00	−0.0163647
$133$	−84868.0	−0.416021
$134$	6980.00	0.0335810
$135$	68526.0	0.323609
$136$	394290.	1.82797
$137$	−94182.0	−0.428713	−0.214356	−	0.976756i	$-0.568765\pi$
$137$	−94182.0	−0.428713	−0.214356	+	0.976756i	$0.568765\pi$
$138$	−25920.0	−0.115861
$139$	−47796.0	−0.209824	−0.104912	−	0.994482i	$-0.533456\pi$
$139$	−47796.0	−0.209824	−0.104912	+	0.994482i	$0.533456\pi$
$140$	32242.0	0.139028
$141$	7776.00	0.0329389
$142$	93600.0	0.389543
$143$	−40040.0	−0.163740
$144$	−60831.0	−0.244466
$145$	518692.	2.04875
$146$	231810.	0.900016
$147$	21609.0	0.0824786
$148$	51366.0	0.192762
$149$	−124266.	−0.458550	−0.229275	−	0.973362i	$-0.573636\pi$
$149$	−124266.	−0.458550	−0.229275	+	0.973362i	$0.573636\pi$
$150$	256995.	0.932602
$151$	−446296.	−1.59287	−0.796436	−	0.604723i	$-0.793284\pi$
$151$	−446296.	−1.59287	−0.796436	+	0.604723i	$0.793284\pi$
$152$	−337740.	−1.18570
$153$	−163782.	−0.565637
$154$	−12740.0	−0.0432880
$155$	595584.	1.99119
$156$	48510.0	0.159595
$157$	−159746.	−0.517227	−0.258613	−	0.965981i	$-0.583266\pi$
$157$	−159746.	−0.517227	−0.258613	+	0.965981i	$0.583266\pi$
$158$	487120.	1.55236
$159$	37638.0	0.118068
$160$	233590.	0.721364
$161$	28224.0	0.0858132
$162$	32805.0	0.0982093
$163$	247252.	0.728905	0.364452	−	0.931222i	$-0.381256\pi$
$163$	247252.	0.728905	0.364452	+	0.931222i	$0.381256\pi$
$164$	22834.0	0.0662937
$165$	43992.0	0.125795
$166$	−406140.	−1.14395
$167$	−684488.	−1.89922	−0.949609	−	0.313438i	$-0.898519\pi$
$167$	−684488.	−1.89922	−0.949609	+	0.313438i	$0.898519\pi$
$168$	85995.0	0.235072
$169$	221607.	0.596852
$170$	−950340.	−2.52207
$171$	140292.	0.366896
$172$	−37940.0	−0.0977859
$173$	−610474.	−1.55079	−0.775393	−	0.631479i	$-0.782448\pi$
$173$	−610474.	−1.55079	−0.775393	+	0.631479i	$0.782448\pi$
$174$	248310.	0.621758
$175$	−279839.	−0.690738
$176$	−39052.0	−0.0950302
$177$	−100980.	−0.242271
$178$	−15910.0	−0.0376374
$179$	662252.	1.54487	0.772433	−	0.635097i	$-0.219040\pi$
$179$	662252.	1.54487	0.772433	+	0.635097i	$0.219040\pi$
$180$	−53298.0	−0.122611
$181$	154630.	0.350830	0.175415	−	0.984495i	$-0.443873\pi$
$181$	154630.	0.350830	0.175415	+	0.984495i	$0.443873\pi$
$182$	188650.	0.422161
$183$	−410418.	−0.905938
$184$	112320.	0.244575
$185$	−689772.	−1.48175
$186$	285120.	0.604290
$187$	−105144.	−0.219877
$188$	−6048.00	−0.0124801
$189$	−35721.0	−0.0727393
$190$	814040.	1.63592
$191$	486904.	0.965739	0.482870	−	0.875692i	$-0.339594\pi$
$191$	486904.	0.965739	0.482870	+	0.875692i	$0.339594\pi$
$192$	328113.	0.642348
$193$	620546.	1.19917	0.599585	−	0.800311i	$-0.295332\pi$
$193$	620546.	1.19917	0.599585	+	0.800311i	$0.295332\pi$
$194$	24570.0	0.0468706
$195$	−651420.	−1.22680
$196$	−16807.0	−0.0312500
$197$	−236570.	−0.434304	−0.217152	−	0.976138i	$-0.569677\pi$
$197$	−236570.	−0.434304	−0.217152	+	0.976138i	$0.569677\pi$
$198$	21060.0	0.0381764
$199$	82104.0	0.146971	0.0734855	−	0.997296i	$-0.476588\pi$
$199$	82104.0	0.146971	0.0734855	+	0.997296i	$0.476588\pi$
$200$	−1.11364e6	−1.96866
$201$	12564.0	0.0219350
$202$	−831770.	−1.43425
$203$	−270382.	−0.460509
$204$	127386.	0.214312
$205$	−306628.	−0.509597
$206$	785800.	1.29016
$207$	−46656.0	−0.0756801
$208$	578270.	0.926771
$209$	90064.0	0.142622
$210$	−207270.	−0.324331
$211$	99892.0	0.154463	0.0772315	−	0.997013i	$-0.475392\pi$
$211$	99892.0	0.154463	0.0772315	+	0.997013i	$0.475392\pi$
$212$	−29274.0	−0.0447345
$213$	168480.	0.254448
$214$	−33820.0	−0.0504823
$215$	509480.	0.751677
$216$	−142155.	−0.207314
$217$	−310464.	−0.447571
$218$	891990.	1.27121
$219$	417258.	0.587888
$220$	−34216.0	−0.0476620
$221$	1.55694e6	2.14433
$222$	−330210.	−0.449684
$223$	−186704.	−0.251415	−0.125708	−	0.992067i	$-0.540120\pi$
$223$	−186704.	−0.251415	−0.125708	+	0.992067i	$0.540120\pi$
$224$	−121765.	−0.162145
$225$	462591.	0.609173
$226$	−225670.	−0.293902
$227$	336372.	0.433267	0.216633	−	0.976253i	$-0.430492\pi$
$227$	336372.	0.433267	0.216633	+	0.976253i	$0.430492\pi$
$228$	−109116.	−0.139012
$229$	−926314.	−1.16727	−0.583633	−	0.812018i	$-0.698369\pi$
$229$	−926314.	−1.16727	−0.583633	+	0.812018i	$0.698369\pi$
$230$	−270720.	−0.337443
$231$	−22932.0	−0.0282756
$232$	−1.07601e6	−1.31249
$233$	1.25711e6	1.51700	0.758499	−	0.651675i	$-0.225933\pi$
$233$	1.25711e6	1.51700	0.758499	+	0.651675i	$0.225933\pi$
$234$	−311850.	−0.372311
$235$	81216.0	0.0959339
$236$	78540.0	0.0917933
$237$	876816.	1.01400
$238$	495390.	0.566898
$239$	−347016.	−0.392966	−0.196483	−	0.980507i	$-0.562952\pi$
$239$	−347016.	−0.392966	−0.196483	+	0.980507i	$0.562952\pi$
$240$	−635346.	−0.712004
$241$	99170.0	0.109986	0.0549930	−	0.998487i	$-0.482486\pi$
$241$	99170.0	0.109986	0.0549930	+	0.998487i	$0.482486\pi$
$242$	−791735.	−0.869043
$243$	59049.0	0.0641500
$244$	319214.	0.343247
$245$	225694.	0.240218
$246$	−146790.	−0.154653
$247$	−1.33364e6	−1.39090
$248$	−1.23552e6	−1.27562
$249$	−731052.	−0.747222
$250$	1.21542e6	1.22992
$251$	344428.	0.345076	0.172538	−	0.985003i	$-0.444803\pi$
$251$	344428.	0.345076	0.172538	+	0.985003i	$0.444803\pi$
$252$	27783.0	0.0275599
$253$	−29952.0	−0.0294188
$254$	−1.02528e6	−0.997145
$255$	−1.71061e6	−1.64741
$256$	−652799.	−0.622558
$257$	295130.	0.278728	0.139364	−	0.990241i	$-0.455494\pi$
$257$	295130.	0.278728	0.139364	+	0.990241i	$0.455494\pi$
$258$	243900.	0.228120
$259$	359562.	0.333061
$260$	506660.	0.464818
$261$	446958.	0.406130
$262$	364820.	0.328341
$263$	−1.27246e6	−1.13437	−0.567187	−	0.823589i	$-0.691968\pi$
$263$	−1.27246e6	−1.13437	−0.567187	+	0.823589i	$0.691968\pi$
$264$	−91260.0	−0.0805881
$265$	393108.	0.343872
$266$	−424340.	−0.367714
$267$	−28638.0	−0.0245847
$268$	−9772.00	−0.00831087
$269$	276774.	0.233209	0.116604	−	0.993178i	$-0.462799\pi$
$269$	276774.	0.233209	0.116604	+	0.993178i	$0.462799\pi$
$270$	342630.	0.286033
$271$	−1.28994e6	−1.06695	−0.533476	−	0.845815i	$-0.679115\pi$
$271$	−1.28994e6	−1.06695	−0.533476	+	0.845815i	$0.679115\pi$
$272$	1.51852e6	1.24451
$273$	339570.	0.275755
$274$	−470910.	−0.378932
$275$	296972.	0.236801
$276$	36288.0	0.0286741
$277$	1.71655e6	1.34418	0.672089	−	0.740470i	$-0.265397\pi$
$277$	1.71655e6	1.34418	0.672089	+	0.740470i	$0.265397\pi$
$278$	−238980.	−0.185460
$279$	513216.	0.394720
$280$	898170.	0.684642
$281$	−1.47218e6	−1.11223	−0.556116	−	0.831104i	$-0.687709\pi$
$281$	−1.47218e6	−1.11223	−0.556116	+	0.831104i	$0.687709\pi$
$282$	38880.0	0.0291141
$283$	1.02881e6	0.763607	0.381804	−	0.924244i	$-0.375303\pi$
$283$	1.02881e6	0.763607	0.381804	+	0.924244i	$0.375303\pi$
$284$	−131040.	−0.0964069
$285$	1.46527e6	1.06858
$286$	−200200.	−0.144727
$287$	159838.	0.114545
$288$	201285.	0.142998
$289$	2.66863e6	1.87950
$290$	2.59346e6	1.81086
$291$	44226.0	0.0306158
$292$	−324534.	−0.222742
$293$	−1.18607e6	−0.807123	−0.403562	−	0.914952i	$-0.632228\pi$
$293$	−1.18607e6	−0.807123	−0.403562	+	0.914952i	$0.632228\pi$
$294$	108045.	0.0729015
$295$	−1.05468e6	−0.705612
$296$	1.43091e6	0.949255
$297$	37908.0	0.0249367
$298$	−621330.	−0.405305
$299$	443520.	0.286903
$300$	−359793.	−0.230807
$301$	−265580.	−0.168958
$302$	−2.23148e6	−1.40791
$303$	−1.49719e6	−0.936848
$304$	−1.30073e6	−0.807242
$305$	−4.28659e6	−2.63853
$306$	−818910.	−0.499957
$307$	1.51892e6	0.919788	0.459894	−	0.887974i	$-0.347887\pi$
$307$	1.51892e6	0.919788	0.459894	+	0.887974i	$0.347887\pi$
$308$	17836.0	0.0107132
$309$	1.41444e6	0.842730
$310$	2.97792e6	1.75998
$311$	212808.	0.124763	0.0623817	−	0.998052i	$-0.480130\pi$
$311$	212808.	0.124763	0.0623817	+	0.998052i	$0.480130\pi$
$312$	1.35135e6	0.785925
$313$	−1894.00	−0.00109275	−0.000546373	−	1.00000i	$-0.500174\pi$
$313$	−1894.00	−0.00109275	−0.000546373		1.00000i	$0.500174\pi$
$314$	−798730.	−0.457168
$315$	−373086.	−0.211852
$316$	−681968.	−0.384190
$317$	−1.57898e6	−0.882527	−0.441263	−	0.897378i	$-0.645470\pi$
$317$	−1.57898e6	−0.882527	−0.441263	+	0.897378i	$0.645470\pi$
$318$	188190.	0.104359
$319$	286936.	0.157873
$320$	3.42696e6	1.87083
$321$	−60876.0	−0.0329749
$322$	141120.	0.0758488
$323$	−3.50210e6	−1.86777
$324$	−45927.0	−0.0243056
$325$	−4.39747e6	−2.30938
$326$	1.23626e6	0.644267
$327$	1.60558e6	0.830354
$328$	636090.	0.326463
$329$	−42336.0	−0.0215635
$330$	219960.	0.111188
$331$	−3.39471e6	−1.70307	−0.851535	−	0.524298i	$-0.824328\pi$
$331$	−3.39471e6	−1.70307	−0.851535	+	0.524298i	$0.824328\pi$
$332$	568596.	0.283112
$333$	−594378.	−0.293733
$334$	−3.42244e6	−1.67869
$335$	131224.	0.0638853
$336$	331191.	0.160041
$337$	2.02731e6	0.972403	0.486201	−	0.873847i	$-0.338382\pi$
$337$	2.02731e6	0.972403	0.486201	+	0.873847i	$0.338382\pi$
$338$	1.10804e6	0.527548
$339$	−406206.	−0.191976
$340$	1.33048e6	0.624180
$341$	329472.	0.153438
$342$	701460.	0.324293
$343$	−117649.	−0.0539949
$344$	−1.05690e6	−0.481546
$345$	−487296.	−0.220417
$346$	−3.05237e6	−1.37071
$347$	3.48885e6	1.55546	0.777730	−	0.628598i	$-0.216371\pi$
$347$	3.48885e6	1.55546	0.777730	+	0.628598i	$0.216371\pi$
$348$	−347634.	−0.153877
$349$	965566.	0.424344	0.212172	−	0.977232i	$-0.431946\pi$
$349$	965566.	0.424344	0.212172	+	0.977232i	$0.431946\pi$
$350$	−1.39919e6	−0.610532
$351$	−561330.	−0.243193
$352$	129220.	0.0555870
$353$	1.15393e6	0.492882	0.246441	−	0.969158i	$-0.420739\pi$
$353$	1.15393e6	0.492882	0.246441	+	0.969158i	$0.420739\pi$
$354$	−504900.	−0.214140
$355$	1.75968e6	0.741076
$356$	22274.0	0.00931479
$357$	891702.	0.370296
$358$	3.31126e6	1.36548
$359$	1.61110e6	0.659762	0.329881	−	0.944022i	$-0.392991\pi$
$359$	1.61110e6	0.659762	0.329881	+	0.944022i	$0.392991\pi$
$360$	−1.48473e6	−0.603797
$361$	523725.	0.211512
$362$	773150.	0.310093
$363$	−1.42512e6	−0.567657
$364$	−264110.	−0.104480
$365$	4.35803e6	1.71221
$366$	−2.05209e6	−0.800744
$367$	3.67747e6	1.42523	0.712614	−	0.701557i	$-0.247511\pi$
$367$	3.67747e6	1.42523	0.712614	+	0.701557i	$0.247511\pi$
$368$	432576.	0.166511
$369$	−264222.	−0.101019
$370$	−3.44886e6	−1.30970
$371$	−204918.	−0.0772939
$372$	−399168.	−0.149554
$373$	649766.	0.241816	0.120908	−	0.992664i	$-0.461419\pi$
$373$	649766.	0.241816	0.120908	+	0.992664i	$0.461419\pi$
$374$	−525720.	−0.194346
$375$	2.18776e6	0.803379
$376$	−168480.	−0.0614580
$377$	−4.24886e6	−1.53964
$378$	−178605.	−0.0642931
$379$	320700.	0.114683	0.0573417	−	0.998355i	$-0.481738\pi$
$379$	320700.	0.114683	0.0573417	+	0.998355i	$0.481738\pi$
$380$	−1.13966e6	−0.404869
$381$	−1.84550e6	−0.651332
$382$	2.43452e6	0.853601
$383$	−2.36189e6	−0.822740	−0.411370	−	0.911469i	$-0.634950\pi$
$383$	−2.36189e6	−0.822740	−0.411370	+	0.911469i	$0.634950\pi$
$384$	924885.	0.320081
$385$	−239512.	−0.0823522
$386$	3.10273e6	1.05993
$387$	439020.	0.149007
$388$	−34398.0	−0.0115999
$389$	−3.53390e6	−1.18408	−0.592039	−	0.805910i	$-0.701677\pi$
$389$	−3.53390e6	−1.18408	−0.592039	+	0.805910i	$0.701677\pi$
$390$	−3.25710e6	−1.08435
$391$	1.16467e6	0.385267
$392$	−468195.	−0.153890
$393$	656676.	0.214472
$394$	−1.18285e6	−0.383874
$395$	9.15786e6	2.95326
$396$	−29484.0	−0.00944819
$397$	4.04811e6	1.28907	0.644534	−	0.764575i	$-0.277051\pi$
$397$	4.04811e6	1.28907	0.644534	+	0.764575i	$0.277051\pi$
$398$	410520.	0.129905
$399$	−763812.	−0.240190
$400$	−4.28896e6	−1.34030
$401$	2.07645e6	0.644853	0.322426	−	0.946595i	$-0.395502\pi$
$401$	2.07645e6	0.644853	0.322426	+	0.946595i	$0.395502\pi$
$402$	62820.0	0.0193880
$403$	−4.87872e6	−1.49638
$404$	1.16448e6	0.354959
$405$	616734.	0.186836
$406$	−1.35191e6	−0.407036
$407$	−381576.	−0.114181
$408$	3.54861e6	1.05538
$409$	2.57431e6	0.760945	0.380472	−	0.924792i	$-0.375761\pi$
$409$	2.57431e6	0.760945	0.380472	+	0.924792i	$0.375761\pi$
$410$	−1.53314e6	−0.450425
$411$	−847638.	−0.247517
$412$	−1.10012e6	−0.319299
$413$	549780.	0.158604
$414$	−233280.	−0.0668924
$415$	−7.63543e6	−2.17627
$416$	−1.91345e6	−0.542105
$417$	−430164.	−0.121142
$418$	450320.	0.126061
$419$	848148.	0.236013	0.118007	−	0.993013i	$-0.462350\pi$
$419$	848148.	0.236013	0.118007	+	0.993013i	$0.462350\pi$
$420$	290178.	0.0802678
$421$	1.43682e6	0.395092	0.197546	−	0.980294i	$-0.436703\pi$
$421$	1.43682e6	0.395092	0.197546	+	0.980294i	$0.436703\pi$
$422$	499460.	0.136527
$423$	69984.0	0.0190173
$424$	−815490.	−0.220295
$425$	−1.15476e7	−3.10114
$426$	842400.	0.224903
$427$	2.23450e6	0.593076
$428$	47348.0	0.0124937
$429$	−360360.	−0.0945352
$430$	2.54740e6	0.664394
$431$	2.35438e6	0.610496	0.305248	−	0.952273i	$-0.401261\pi$
$431$	2.35438e6	0.610496	0.305248	+	0.952273i	$0.401261\pi$
$432$	−547479.	−0.141143
$433$	−3.78808e6	−0.970955	−0.485478	−	0.874249i	$-0.661354\pi$
$433$	−3.78808e6	−0.970955	−0.485478	+	0.874249i	$0.661354\pi$
$434$	−1.55232e6	−0.395601
$435$	4.66823e6	1.18285
$436$	−1.24879e6	−0.314609
$437$	−997632.	−0.249900
$438$	2.08629e6	0.519624
$439$	−3.64322e6	−0.902245	−0.451123	−	0.892462i	$-0.648976\pi$
$439$	−3.64322e6	−0.902245	−0.451123	+	0.892462i	$0.648976\pi$
$440$	−953160.	−0.234711
$441$	194481.	0.0476190
$442$	7.78470e6	1.89534
$443$	2.48389e6	0.601345	0.300672	−	0.953728i	$-0.402789\pi$
$443$	2.48389e6	0.601345	0.300672	+	0.953728i	$0.402789\pi$
$444$	462294.	0.111291
$445$	−299108.	−0.0716025
$446$	−933520.	−0.222222
$447$	−1.11839e6	−0.264744
$448$	−1.78639e6	−0.420515
$449$	−2.63177e6	−0.616074	−0.308037	−	0.951374i	$-0.599672\pi$
$449$	−2.63177e6	−0.616074	−0.308037	+	0.951374i	$0.599672\pi$
$450$	2.31295e6	0.538438
$451$	−169624.	−0.0392686
$452$	315938.	0.0727371
$453$	−4.01666e6	−0.919645
$454$	1.68186e6	0.382957
$455$	3.54662e6	0.803131
$456$	−3.03966e6	−0.684562
$457$	−1.16130e6	−0.260109	−0.130054	−	0.991507i	$-0.541515\pi$
$457$	−1.16130e6	−0.260109	−0.130054	+	0.991507i	$0.541515\pi$
$458$	−4.63157e6	−1.03173
$459$	−1.47404e6	−0.326571
$460$	379008.	0.0835129
$461$	2.81385e6	0.616663	0.308332	−	0.951279i	$-0.400229\pi$
$461$	2.81385e6	0.616663	0.308332	+	0.951279i	$0.400229\pi$
$462$	−114660.	−0.0249923
$463$	6.84299e6	1.48352	0.741760	−	0.670665i	$-0.233991\pi$
$463$	6.84299e6	1.48352	0.741760	+	0.670665i	$0.233991\pi$
$464$	−4.14402e6	−0.893566
$465$	5.36026e6	1.14962
$466$	6.28557e6	1.34085
$467$	3.34314e6	0.709353	0.354676	−	0.934989i	$-0.384591\pi$
$467$	3.34314e6	0.709353	0.354676	+	0.934989i	$0.384591\pi$
$468$	436590.	0.0921423
$469$	−68404.0	−0.0143598
$470$	406080.	0.0847944
$471$	−1.43771e6	−0.298621
$472$	2.18790e6	0.452035
$473$	281840.	0.0579228
$474$	4.38408e6	0.896257
$475$	9.89145e6	2.01153
$476$	−693546.	−0.140300
$477$	338742.	0.0681668
$478$	−1.73508e6	−0.347336
$479$	−4.28248e6	−0.852818	−0.426409	−	0.904530i	$-0.640222\pi$
$479$	−4.28248e6	−0.852818	−0.426409	+	0.904530i	$0.640222\pi$
$480$	2.10231e6	0.416480
$481$	5.65026e6	1.11354
$482$	495850.	0.0972149
$483$	254016.	0.0495442
$484$	1.10843e6	0.215077
$485$	461916.	0.0891679
$486$	295245.	0.0567012
$487$	−8.93175e6	−1.70653	−0.853266	−	0.521477i	$-0.825381\pi$
$487$	−8.93175e6	−1.70653	−0.853266	+	0.521477i	$0.825381\pi$
$488$	8.89239e6	1.69032
$489$	2.22527e6	0.420833
$490$	1.12847e6	0.212324
$491$	2.75306e6	0.515361	0.257681	−	0.966230i	$-0.417042\pi$
$491$	2.75306e6	0.515361	0.257681	+	0.966230i	$0.417042\pi$
$492$	205506.	0.0382747
$493$	−1.11574e7	−2.06750
$494$	−6.66820e6	−1.22939
$495$	395928.	0.0726279
$496$	−4.75834e6	−0.868462
$497$	−917280.	−0.166575
$498$	−3.65526e6	−0.660458
$499$	4.80408e6	0.863693	0.431846	−	0.901947i	$-0.357862\pi$
$499$	4.80408e6	0.863693	0.431846	+	0.901947i	$0.357862\pi$
$500$	−1.70159e6	−0.304389
$501$	−6.16039e6	−1.09651
$502$	1.72214e6	0.305007
$503$	−6.02465e6	−1.06172	−0.530862	−	0.847458i	$-0.678132\pi$
$503$	−6.02465e6	−1.06172	−0.530862	+	0.847458i	$0.678132\pi$
$504$	773955.	0.135719
$505$	−1.56373e7	−2.72855
$506$	−149760.	−0.0260028
$507$	1.99446e6	0.344593
$508$	1.43539e6	0.246781
$509$	−8.42987e6	−1.44220	−0.721101	−	0.692830i	$-0.756364\pi$
$509$	−8.42987e6	−1.44220	−0.721101	+	0.692830i	$0.756364\pi$
$510$	−8.55306e6	−1.45612
$511$	−2.27174e6	−0.384863
$512$	−6.55248e6	−1.10466
$513$	1.26263e6	0.211827
$514$	1.47565e6	0.246363
$515$	1.47730e7	2.45444
$516$	−341460.	−0.0564567
$517$	44928.0	0.00739249
$518$	1.79781e6	0.294388
$519$	−5.49427e6	−0.895347
$520$	1.41141e7	2.28899
$521$	9.25058e6	1.49305	0.746525	−	0.665357i	$-0.231721\pi$
$521$	9.25058e6	1.49305	0.746525	+	0.665357i	$0.231721\pi$
$522$	2.23479e6	0.358972
$523$	5.84494e6	0.934385	0.467192	−	0.884156i	$-0.345266\pi$
$523$	5.84494e6	0.934385	0.467192	+	0.884156i	$0.345266\pi$
$524$	−510748.	−0.0812603
$525$	−2.51855e6	−0.398798
$526$	−6.36232e6	−1.00265
$527$	−1.28114e7	−2.00942
$528$	−351468.	−0.0548657
$529$	−6.10457e6	−0.948453
$530$	1.96554e6	0.303943
$531$	−908820.	−0.139875
$532$	594076.	0.0910045
$533$	2.51174e6	0.382963
$534$	−143190.	−0.0217300
$535$	−635816.	−0.0960389
$536$	−272220.	−0.0409268
$537$	5.96027e6	0.891929
$538$	1.38387e6	0.206129
$539$	124852.	0.0185107
$540$	−479682.	−0.0707895
$541$	9.22533e6	1.35515	0.677577	−	0.735452i	$-0.263030\pi$
$541$	9.22533e6	1.35515	0.677577	+	0.735452i	$0.263030\pi$
$542$	−6.44968e6	−0.943061
$543$	1.39167e6	0.202552
$544$	−5.02467e6	−0.727965
$545$	1.67694e7	2.41839
$546$	1.69785e6	0.243735
$547$	−6.44337e6	−0.920757	−0.460378	−	0.887723i	$-0.652286\pi$
$547$	−6.44337e6	−0.920757	−0.460378	+	0.887723i	$0.652286\pi$
$548$	659274.	0.0937809
$549$	−3.69376e6	−0.523044
$550$	1.48486e6	0.209305
$551$	9.55718e6	1.34107
$552$	1.01088e6	0.141206
$553$	−4.77378e6	−0.663818
$554$	8.58275e6	1.18810
$555$	−6.20795e6	−0.855491
$556$	334572.	0.0458989
$557$	3.74213e6	0.511070	0.255535	−	0.966800i	$-0.417748\pi$
$557$	3.74213e6	0.511070	0.255535	+	0.966800i	$0.417748\pi$
$558$	2.56608e6	0.348887
$559$	−4.17340e6	−0.564886
$560$	3.45911e6	0.466116
$561$	−946296.	−0.126946
$562$	−7.36091e6	−0.983084
$563$	−1.46384e7	−1.94635	−0.973176	−	0.230060i	$-0.926108\pi$
$563$	−1.46384e7	−1.94635	−0.973176	+	0.230060i	$0.926108\pi$
$564$	−54432.0	−0.00720537
$565$	−4.24260e6	−0.559127
$566$	5.14406e6	0.674940
$567$	−321489.	−0.0419961
$568$	−3.65040e6	−0.474755
$569$	1.41805e7	1.83616	0.918078	−	0.396400i	$-0.129741\pi$
$569$	1.41805e7	1.83616	0.918078	+	0.396400i	$0.129741\pi$
$570$	7.32636e6	0.944498
$571$	−1.25160e6	−0.160648	−0.0803242	−	0.996769i	$-0.525596\pi$
$571$	−1.25160e6	−0.160648	−0.0803242	+	0.996769i	$0.525596\pi$
$572$	280280.	0.0358181
$573$	4.38214e6	0.557570
$574$	799190.	0.101244
$575$	−3.28954e6	−0.414921
$576$	2.95302e6	0.370860
$577$	5.94378e6	0.743230	0.371615	−	0.928387i	$-0.378804\pi$
$577$	5.94378e6	0.743230	0.371615	+	0.928387i	$0.378804\pi$
$578$	1.33431e7	1.66126
$579$	5.58491e6	0.692341
$580$	−3.63084e6	−0.448165
$581$	3.98017e6	0.489172
$582$	221130.	0.0270608
$583$	217464.	0.0264982
$584$	−9.04059e6	−1.09689
$585$	−5.86278e6	−0.708295
$586$	−5.93033e6	−0.713403
$587$	−6.46192e6	−0.774046	−0.387023	−	0.922070i	$-0.626497\pi$
$587$	−6.46192e6	−0.774046	−0.387023	+	0.922070i	$0.626497\pi$
$588$	−151263.	−0.0180422
$589$	1.09740e7	1.30339
$590$	−5.27340e6	−0.623678
$591$	−2.12913e6	−0.250746
$592$	5.51084e6	0.646269
$593$	−2.34605e6	−0.273969	−0.136984	−	0.990573i	$-0.543741\pi$
$593$	−2.34605e6	−0.273969	−0.136984	+	0.990573i	$0.543741\pi$
$594$	189540.	0.0220412
$595$	9.31333e6	1.07848
$596$	869862.	0.100308
$597$	738936.	0.0848537
$598$	2.21760e6	0.253589
$599$	−1.34959e7	−1.53686	−0.768432	−	0.639931i	$-0.778963\pi$
$599$	−1.34959e7	−1.53686	−0.768432	+	0.639931i	$0.778963\pi$
$600$	−1.00228e7	−1.13661
$601$	3.87849e6	0.438002	0.219001	−	0.975725i	$-0.429720\pi$
$601$	3.87849e6	0.438002	0.219001	+	0.975725i	$0.429720\pi$
$602$	−1.32790e6	−0.149339
$603$	113076.	0.0126642
$604$	3.12407e6	0.348441
$605$	−1.48846e7	−1.65329
$606$	−7.48593e6	−0.828065
$607$	−533488.	−0.0587696	−0.0293848	−	0.999568i	$-0.509355\pi$
$607$	−533488.	−0.0587696	−0.0293848	+	0.999568i	$0.509355\pi$
$608$	4.30402e6	0.472188
$609$	−2.43344e6	−0.265875
$610$	−2.14329e7	−2.33215
$611$	−665280.	−0.0720944
$612$	1.14647e6	0.123733
$613$	5.14610e6	0.553130	0.276565	−	0.960995i	$-0.410804\pi$
$613$	5.14610e6	0.553130	0.276565	+	0.960995i	$0.410804\pi$
$614$	7.59458e6	0.812986
$615$	−2.75965e6	−0.294216
$616$	496860.	0.0527573
$617$	−2.37860e6	−0.251541	−0.125770	−	0.992059i	$-0.540140\pi$
$617$	−2.37860e6	−0.251541	−0.125770	+	0.992059i	$0.540140\pi$
$618$	7.07220e6	0.744875
$619$	1.60023e7	1.67863	0.839317	−	0.543642i	$-0.182955\pi$
$619$	1.60023e7	1.67863	0.839317	+	0.543642i	$0.182955\pi$
$620$	−4.16909e6	−0.435574
$621$	−419904.	−0.0436939
$622$	1.06404e6	0.110276
$623$	155918.	0.0160944
$624$	5.20443e6	0.535071
$625$	5.00302e6	0.512309
$626$	−9470.00	−0.000965860 0
$627$	810576.	0.0823427
$628$	1.11822e6	0.113143
$629$	1.48374e7	1.49531
$630$	−1.86543e6	−0.187252
$631$	1.23459e7	1.23439	0.617193	−	0.786812i	$-0.288270\pi$
$631$	1.23459e7	1.23439	0.617193	+	0.786812i	$0.288270\pi$
$632$	−1.89977e7	−1.89194
$633$	899028.	0.0891793
$634$	−7.89489e6	−0.780051
$635$	−1.92753e7	−1.89699
$636$	−263466.	−0.0258275
$637$	−1.84877e6	−0.180524
$638$	1.43468e6	0.139541
$639$	1.51632e6	0.146906
$640$	9.65991e6	0.932230
$641$	−3.43755e6	−0.330449	−0.165224	−	0.986256i	$-0.552835\pi$
$641$	−3.43755e6	−0.330449	−0.165224	+	0.986256i	$0.552835\pi$
$642$	−304380.	−0.0291460
$643$	−1.62191e7	−1.54703	−0.773515	−	0.633778i	$-0.781503\pi$
$643$	−1.62191e7	−1.54703	−0.773515	+	0.633778i	$0.781503\pi$
$644$	−197568.	−0.0187716
$645$	4.58532e6	0.433981
$646$	−1.75105e7	−1.65089
$647$	1.19929e7	1.12632	0.563160	−	0.826348i	$-0.309585\pi$
$647$	1.19929e7	1.12632	0.563160	+	0.826348i	$0.309585\pi$
$648$	−1.27940e6	−0.119693
$649$	−583440.	−0.0543731
$650$	−2.19873e7	−2.04122
$651$	−2.79418e6	−0.258405
$652$	−1.73076e6	−0.159448
$653$	1.58009e6	0.145011	0.0725053	−	0.997368i	$-0.476901\pi$
$653$	1.58009e6	0.145011	0.0725053	+	0.997368i	$0.476901\pi$
$654$	8.02791e6	0.733936
$655$	6.85862e6	0.624645
$656$	2.44976e6	0.222262
$657$	3.75532e6	0.339417
$658$	−211680.	−0.0190597
$659$	6.98358e6	0.626419	0.313209	−	0.949684i	$-0.398596\pi$
$659$	6.98358e6	0.626419	0.313209	+	0.949684i	$0.398596\pi$
$660$	−307944.	−0.0275177
$661$	3.69602e6	0.329027	0.164513	−	0.986375i	$-0.447395\pi$
$661$	3.69602e6	0.329027	0.164513	+	0.986375i	$0.447395\pi$
$662$	−1.69735e7	−1.50532
$663$	1.40125e7	1.23803
$664$	1.58395e7	1.39418
$665$	−7.97759e6	−0.699548
$666$	−2.97189e6	−0.259625
$667$	−3.17837e6	−0.276624
$668$	4.79142e6	0.415454
$669$	−1.68034e6	−0.145155
$670$	656120.	0.0564672
$671$	−2.37130e6	−0.203320
$672$	−1.09588e6	−0.0936142
$673$	1.84688e6	0.157182	0.0785908	−	0.996907i	$-0.474958\pi$
$673$	1.84688e6	0.157182	0.0785908	+	0.996907i	$0.474958\pi$
$674$	1.01366e7	0.859491
$675$	4.16332e6	0.351706
$676$	−1.55125e6	−0.130561
$677$	−7.68501e6	−0.644426	−0.322213	−	0.946667i	$-0.604427\pi$
$677$	−7.68501e6	−0.644426	−0.322213	+	0.946667i	$0.604427\pi$
$678$	−2.03103e6	−0.169684
$679$	−240786.	−0.0200427
$680$	3.70633e7	3.07377
$681$	3.02735e6	0.250147
$682$	1.64736e6	0.135621
$683$	7.12180e6	0.584168	0.292084	−	0.956393i	$-0.405651\pi$
$683$	7.12180e6	0.584168	0.292084	+	0.956393i	$0.405651\pi$
$684$	−982044.	−0.0802584
$685$	−8.85311e6	−0.720891
$686$	−588245.	−0.0477252
$687$	−8.33683e6	−0.673921
$688$	−4.07042e6	−0.327845
$689$	−3.22014e6	−0.258420
$690$	−2.43648e6	−0.194823
$691$	−3.23787e6	−0.257967	−0.128983	−	0.991647i	$-0.541171\pi$
$691$	−3.23787e6	−0.257967	−0.128983	+	0.991647i	$0.541171\pi$
$692$	4.27332e6	0.339234
$693$	−206388.	−0.0163249
$694$	1.74443e7	1.37485
$695$	−4.49282e6	−0.352823
$696$	−9.68409e6	−0.757767
$697$	6.59576e6	0.514260
$698$	4.82783e6	0.375071
$699$	1.13140e7	0.875839
$700$	1.95887e6	0.151099
$701$	−7.39163e6	−0.568127	−0.284063	−	0.958805i	$-0.591683\pi$
$701$	−7.39163e6	−0.568127	−0.284063	+	0.958805i	$0.591683\pi$
$702$	−2.80665e6	−0.214954
$703$	−1.27094e7	−0.969923
$704$	1.89576e6	0.144163
$705$	730944.	0.0553874
$706$	5.76965e6	0.435650
$707$	8.15135e6	0.613311
$708$	706860.	0.0529969
$709$	−5.33361e6	−0.398479	−0.199240	−	0.979951i	$-0.563847\pi$
$709$	−5.33361e6	−0.398479	−0.199240	+	0.979951i	$0.563847\pi$
$710$	8.79840e6	0.655025
$711$	7.89134e6	0.585433
$712$	620490.	0.0458706
$713$	−3.64954e6	−0.268852
$714$	4.45851e6	0.327299
$715$	−3.76376e6	−0.275332
$716$	−4.63576e6	−0.337939
$717$	−3.12314e6	−0.226879
$718$	8.05552e6	0.583153
$719$	−1.14564e7	−0.826468	−0.413234	−	0.910625i	$-0.635601\pi$
$719$	−1.14564e7	−0.826468	−0.413234	+	0.910625i	$0.635601\pi$
$720$	−5.71811e6	−0.411075
$721$	−7.70084e6	−0.551696
$722$	2.61862e6	0.186952
$723$	892530.	0.0635005
$724$	−1.08241e6	−0.0767442
$725$	3.15133e7	2.22663
$726$	−7.12562e6	−0.501742
$727$	−2.49540e7	−1.75107	−0.875536	−	0.483153i	$-0.839492\pi$
$727$	−2.49540e7	−1.75107	−0.875536	+	0.483153i	$0.839492\pi$
$728$	−7.35735e6	−0.514509
$729$	531441.	0.0370370
$730$	2.17901e7	1.51340
$731$	−1.09592e7	−0.758555
$732$	2.87293e6	0.198174
$733$	−1.43398e7	−0.985789	−0.492894	−	0.870089i	$-0.664061\pi$
$733$	−1.43398e7	−0.985789	−0.492894	+	0.870089i	$0.664061\pi$
$734$	1.83874e7	1.25974
$735$	2.03125e6	0.138690
$736$	−1.43136e6	−0.0973990
$737$	72592.0	0.00492289
$738$	−1.32111e6	−0.0892890
$739$	922932.	0.0621668	0.0310834	−	0.999517i	$-0.490104\pi$
$739$	922932.	0.0621668	0.0310834	+	0.999517i	$0.490104\pi$
$740$	4.82840e6	0.324134
$741$	−1.20028e7	−0.803037
$742$	−1.02459e6	−0.0683188
$743$	−9.38995e6	−0.624010	−0.312005	−	0.950081i	$-0.601001\pi$
$743$	−9.38995e6	−0.624010	−0.312005	+	0.950081i	$0.601001\pi$
$744$	−1.11197e7	−0.736478
$745$	−1.16810e7	−0.771062
$746$	3.24883e6	0.213737
$747$	−6.57947e6	−0.431409
$748$	736008.	0.0480982
$749$	331436.	0.0215871
$750$	1.09388e7	0.710094
$751$	−408032.	−0.0263994	−0.0131997	−	0.999913i	$-0.504202\pi$
$751$	−408032.	−0.0263994	−0.0131997	+	0.999913i	$0.504202\pi$
$752$	−648864.	−0.0418417
$753$	3.09985e6	0.199229
$754$	−2.12443e7	−1.36086
$755$	−4.19518e7	−2.67845
$756$	250047.	0.0159117
$757$	2.59605e7	1.64654	0.823271	−	0.567649i	$-0.192147\pi$
$757$	2.59605e7	1.64654	0.823271	+	0.567649i	$0.192147\pi$
$758$	1.60350e6	0.101367
$759$	−269568.	−0.0169849
$760$	−3.17476e7	−1.99378
$761$	1.83554e7	1.14895	0.574477	−	0.818521i	$-0.305206\pi$
$761$	1.83554e7	1.14895	0.574477	+	0.818521i	$0.305206\pi$
$762$	−9.22752e6	−0.575702
$763$	−8.74150e6	−0.543594
$764$	−3.40833e6	−0.211255
$765$	−1.53955e7	−0.951131
$766$	−1.18094e7	−0.727206
$767$	8.63940e6	0.530268
$768$	−5.87519e6	−0.359434
$769$	−747166.	−0.0455618	−0.0227809	−	0.999740i	$-0.507252\pi$
$769$	−747166.	−0.0455618	−0.0227809	+	0.999740i	$0.507252\pi$
$770$	−1.19756e6	−0.0727898
$771$	2.65617e6	0.160924
$772$	−4.34382e6	−0.262318
$773$	2.02692e7	1.22008	0.610038	−	0.792372i	$-0.291154\pi$
$773$	2.02692e7	1.22008	0.610038	+	0.792372i	$0.291154\pi$
$774$	2.19510e6	0.131705
$775$	3.61849e7	2.16408
$776$	−958230.	−0.0571236
$777$	3.23606e6	0.192293
$778$	−1.76695e7	−1.04659
$779$	−5.64978e6	−0.333571
$780$	4.55994e6	0.268363
$781$	973440.	0.0571060
$782$	5.82336e6	0.340531
$783$	4.02262e6	0.234479
$784$	−1.80315e6	−0.104771
$785$	−1.50161e7	−0.869729
$786$	3.28338e6	0.189568
$787$	−4.69982e6	−0.270486	−0.135243	−	0.990812i	$-0.543181\pi$
$787$	−4.69982e6	−0.270486	−0.135243	+	0.990812i	$0.543181\pi$
$788$	1.65599e6	0.0950041
$789$	−1.14522e7	−0.654931
$790$	4.57893e7	2.61033
$791$	2.21157e6	0.125678
$792$	−821340.	−0.0465275
$793$	3.51135e7	1.98286
$794$	2.02406e7	1.13939
$795$	3.53797e6	0.198535
$796$	−574728.	−0.0321499
$797$	584710.	0.0326058	0.0163029	−	0.999867i	$-0.494810\pi$
$797$	584710.	0.0326058	0.0163029	+	0.999867i	$0.494810\pi$
$798$	−3.81906e6	−0.212300
$799$	−1.74701e6	−0.0968117
$800$	1.41918e7	0.783996
$801$	−257742.	−0.0141940
$802$	1.03822e7	0.569975
$803$	2.41082e6	0.131940
$804$	−87948.0	−0.00479828
$805$	2.65306e6	0.144297
$806$	−2.43936e7	−1.32263
$807$	2.49097e6	0.134643
$808$	3.24390e7	1.74799
$809$	−1.64013e7	−0.881061	−0.440531	−	0.897738i	$-0.645210\pi$
$809$	−1.64013e7	−0.881061	−0.440531	+	0.897738i	$0.645210\pi$
$810$	3.08367e6	0.165141
$811$	−304948.	−0.0162807	−0.00814036	−	0.999967i	$-0.502591\pi$
$811$	−304948.	−0.0162807	−0.00814036	+	0.999967i	$0.502591\pi$
$812$	1.89267e6	0.100736
$813$	−1.16094e7	−0.616005
$814$	−1.90788e6	−0.100923
$815$	2.32417e7	1.22567
$816$	1.36667e7	0.718519
$817$	9.38744e6	0.492031
$818$	1.28716e7	0.672587
$819$	3.05613e6	0.159207
$820$	2.14640e6	0.111474
$821$	3.43428e7	1.77819	0.889095	−	0.457722i	$-0.151335\pi$
$821$	3.43428e7	1.77819	0.889095	+	0.457722i	$0.151335\pi$
$822$	−4.23819e6	−0.218777
$823$	1.56684e7	0.806351	0.403176	−	0.915123i	$-0.367906\pi$
$823$	1.56684e7	0.806351	0.403176	+	0.915123i	$0.367906\pi$
$824$	−3.06462e7	−1.57238
$825$	2.67275e6	0.136717
$826$	2.74890e6	0.140187
$827$	−2.96886e7	−1.50948	−0.754738	−	0.656026i	$-0.772236\pi$
$827$	−2.96886e7	−1.50948	−0.754738	+	0.656026i	$0.772236\pi$
$828$	326592.	0.0165550
$829$	−2.30708e7	−1.16594	−0.582970	−	0.812494i	$-0.698110\pi$
$829$	−2.30708e7	−1.16594	−0.582970	+	0.812494i	$0.698110\pi$
$830$	−3.81772e7	−1.92357
$831$	1.54490e7	0.776062
$832$	−2.80719e7	−1.40593
$833$	−4.85482e6	−0.242416
$834$	−2.15082e6	−0.107075
$835$	−6.43419e7	−3.19358
$836$	−630448.	−0.0311985
$837$	4.61894e6	0.227892
$838$	4.24074e6	0.208608
$839$	2.32642e7	1.14100	0.570498	−	0.821299i	$-0.306750\pi$
$839$	2.32642e7	1.14100	0.570498	+	0.821299i	$0.306750\pi$
$840$	8.08353e6	0.395278
$841$	9.93718e6	0.484477
$842$	7.18411e6	0.349215
$843$	−1.32496e7	−0.642148
$844$	−699244.	−0.0337888
$845$	2.08311e7	1.00362
$846$	349920.	0.0168090
$847$	7.75900e6	0.371619
$848$	−3.14068e6	−0.149980
$849$	9.25931e6	0.440869
$850$	−5.77382e7	−2.74104
$851$	4.22669e6	0.200067
$852$	−1.17936e6	−0.0556605
$853$	1.91515e7	0.901219	0.450610	−	0.892721i	$-0.351207\pi$
$853$	1.91515e7	0.901219	0.450610	+	0.892721i	$0.351207\pi$
$854$	1.11725e7	0.524210
$855$	1.31874e7	0.616944
$856$	1.31898e6	0.0615253
$857$	−5.34683e6	−0.248682	−0.124341	−	0.992240i	$-0.539682\pi$
$857$	−5.34683e6	−0.248682	−0.124341	+	0.992240i	$0.539682\pi$
$858$	−1.80180e6	−0.0835581
$859$	3.95858e7	1.83045	0.915223	−	0.402948i	$-0.132014\pi$
$859$	3.95858e7	1.83045	0.915223	+	0.402948i	$0.132014\pi$
$860$	−3.56636e6	−0.164429
$861$	1.43854e6	0.0661325
$862$	1.17719e7	0.539607
$863$	−2.50284e7	−1.14395	−0.571973	−	0.820272i	$-0.693822\pi$
$863$	−2.50284e7	−1.14395	−0.571973	+	0.820272i	$0.693822\pi$
$864$	1.81156e6	0.0825600
$865$	−5.73846e7	−2.60768
$866$	−1.89404e7	−0.858211
$867$	2.40176e7	1.08513
$868$	2.17325e6	0.0979062
$869$	5.06605e6	0.227573
$870$	2.33411e7	1.04550
$871$	−1.07492e6	−0.0480099
$872$	−3.47876e7	−1.54929
$873$	398034.	0.0176760
$874$	−4.98816e6	−0.220883
$875$	−1.19111e7	−0.525935
$876$	−2.92081e6	−0.128600
$877$	−5.02589e6	−0.220655	−0.110328	−	0.993895i	$-0.535190\pi$
$877$	−5.02589e6	−0.220655	−0.110328	+	0.993895i	$0.535190\pi$
$878$	−1.82161e7	−0.797480
$879$	−1.06746e7	−0.465993
$880$	−3.67089e6	−0.159795
$881$	−2.60490e7	−1.13071	−0.565356	−	0.824847i	$-0.691261\pi$
$881$	−2.60490e7	−1.13071	−0.565356	+	0.824847i	$0.691261\pi$
$882$	972405.	0.0420897
$883$	−6.82462e6	−0.294562	−0.147281	−	0.989095i	$-0.547052\pi$
$883$	−6.82462e6	−0.294562	−0.147281	+	0.989095i	$0.547052\pi$
$884$	−1.08986e7	−0.469072
$885$	−9.49212e6	−0.407385
$886$	1.24195e7	0.531519
$887$	2.33835e7	0.997931	0.498965	−	0.866622i	$-0.333713\pi$
$887$	2.33835e7	0.997931	0.498965	+	0.866622i	$0.333713\pi$
$888$	1.28782e7	0.548053
$889$	1.00477e7	0.426397
$890$	−1.49554e6	−0.0632882
$891$	341172.	0.0143972
$892$	1.30693e6	0.0549971
$893$	1.49645e6	0.0627961
$894$	−5.59197e6	−0.234003
$895$	6.22517e7	2.59773
$896$	−5.03548e6	−0.209542
$897$	3.99168e6	0.165644
$898$	−1.31589e7	−0.544537
$899$	3.49620e7	1.44277
$900$	−3.23814e6	−0.133257
$901$	−8.45600e6	−0.347019
$902$	−848120.	−0.0347089
$903$	−2.39022e6	−0.0975480
$904$	8.80113e6	0.358193
$905$	1.45352e7	0.589930
$906$	−2.00833e7	−0.812859
$907$	3.95959e7	1.59820	0.799102	−	0.601196i	$-0.205309\pi$
$907$	3.95959e7	1.59820	0.799102	+	0.601196i	$0.205309\pi$
$908$	−2.35460e6	−0.0947771
$909$	−1.34747e7	−0.540890
$910$	1.77331e7	0.709874
$911$	−4.67570e6	−0.186660	−0.0933300	−	0.995635i	$-0.529751\pi$
$911$	−4.67570e6	−0.186660	−0.0933300	+	0.995635i	$0.529751\pi$
$912$	−1.17066e7	−0.466062
$913$	−4.22386e6	−0.167700
$914$	−5.80651e6	−0.229906
$915$	−3.85793e7	−1.52336
$916$	6.48420e6	0.255339
$917$	−3.57524e6	−0.140405
$918$	−7.37019e6	−0.288650
$919$	−4.92594e6	−0.192398	−0.0961990	−	0.995362i	$-0.530669\pi$
$919$	−4.92594e6	−0.192398	−0.0961990	+	0.995362i	$0.530669\pi$
$920$	1.05581e7	0.411259
$921$	1.36702e7	0.531040
$922$	1.40692e7	0.545058
$923$	−1.44144e7	−0.556919
$924$	160524.	0.00618529
$925$	−4.19073e7	−1.61041
$926$	3.42150e7	1.31126
$927$	1.27300e7	0.486550
$928$	1.37122e7	0.522683
$929$	−3.23688e7	−1.23052	−0.615258	−	0.788326i	$-0.710948\pi$
$929$	−3.23688e7	−1.23052	−0.615258	+	0.788326i	$0.710948\pi$
$930$	2.68013e7	1.01613
$931$	4.15853e6	0.157241
$932$	−8.79980e6	−0.331843
$933$	1.91527e6	0.0720321
$934$	1.67157e7	0.626985
$935$	−9.88354e6	−0.369729
$936$	1.21622e7	0.453754
$937$	−3.32337e7	−1.23660	−0.618301	−	0.785941i	$-0.712179\pi$
$937$	−3.32337e7	−1.23660	−0.618301	+	0.785941i	$0.712179\pi$
$938$	−342020.	−0.0126924
$939$	−17046.0	−0.000630897 0
$940$	−568512.	−0.0209855
$941$	−2.66426e7	−0.980852	−0.490426	−	0.871483i	$-0.663159\pi$
$941$	−2.66426e7	−0.980852	−0.490426	+	0.871483i	$0.663159\pi$
$942$	−7.18857e6	−0.263946
$943$	1.87891e6	0.0688061
$944$	8.42622e6	0.307753
$945$	−3.35777e6	−0.122313
$946$	1.40920e6	0.0511970
$947$	3.14663e7	1.14017	0.570086	−	0.821585i	$-0.306910\pi$
$947$	3.14663e7	1.14017	0.570086	+	0.821585i	$0.306910\pi$
$948$	−6.13771e6	−0.221812
$949$	−3.56987e7	−1.28673
$950$	4.94573e7	1.77796
$951$	−1.42108e7	−0.509527
$952$	−1.93202e7	−0.690907
$953$	−1.34516e7	−0.479779	−0.239890	−	0.970800i	$-0.577111\pi$
$953$	−1.34516e7	−0.479779	−0.239890	+	0.970800i	$0.577111\pi$
$954$	1.69371e6	0.0602515
$955$	4.57690e7	1.62391
$956$	2.42911e6	0.0859613
$957$	2.58242e6	0.0911481
$958$	−2.14124e7	−0.753792
$959$	4.61492e6	0.162038
$960$	3.08426e7	1.08012
$961$	1.15157e7	0.402238
$962$	2.82513e7	0.984239
$963$	−547884.	−0.0190381
$964$	−694190.	−0.0240595
$965$	5.83313e7	2.01643
$966$	1.27008e6	0.0437913
$967$	−2.84963e7	−0.979992	−0.489996	−	0.871725i	$-0.663002\pi$
$967$	−2.84963e7	−0.979992	−0.489996	+	0.871725i	$0.663002\pi$
$968$	3.08777e7	1.05915
$969$	−3.15189e7	−1.07836
$970$	2.30958e6	0.0788141
$971$	1.81858e7	0.618990	0.309495	−	0.950901i	$-0.399840\pi$
$971$	1.81858e7	0.618990	0.309495	+	0.950901i	$0.399840\pi$
$972$	−413343.	−0.0140328
$973$	2.34200e6	0.0793059
$974$	−4.46588e7	−1.50837
$975$	−3.95772e7	−1.33332
$976$	3.42471e7	1.15080
$977$	3.20941e7	1.07569	0.537847	−	0.843042i	$-0.319238\pi$
$977$	3.20941e7	1.07569	0.537847	+	0.843042i	$0.319238\pi$
$978$	1.11263e7	0.371968
$979$	−165464.	−0.00551756
$980$	−1.57986e6	−0.0525476
$981$	1.44502e7	0.479405
$982$	1.37653e7	0.455519
$983$	1.56154e7	0.515429	0.257715	−	0.966221i	$-0.417031\pi$
$983$	1.56154e7	0.515429	0.257715	+	0.966221i	$0.417031\pi$
$984$	5.72481e6	0.188483
$985$	−2.22376e7	−0.730293
$986$	−5.57870e7	−1.82743
$987$	−381024.	−0.0124497
$988$	9.33548e6	0.304260
$989$	−3.12192e6	−0.101492
$990$	1.97964e6	0.0641946
$991$	4.84499e7	1.56714	0.783572	−	0.621301i	$-0.213395\pi$
$991$	4.84499e7	1.56714	0.783572	+	0.621301i	$0.213395\pi$
$992$	1.57450e7	0.507998
$993$	−3.05524e7	−0.983268
$994$	−4.58640e6	−0.147233
$995$	7.71778e6	0.247135
$996$	5.11736e6	0.163455
$997$	−4.54336e7	−1.44757	−0.723784	−	0.690027i	$-0.757599\pi$
$997$	−4.54336e7	−1.44757	−0.723784	+	0.690027i	$0.757599\pi$
$998$	2.40204e7	0.763404
$999$	−5.34940e6	−0.169587

Twists

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

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