LMFDB - Embedded newform 38.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,8,Mod(1,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	38.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:	$11.8706309684$
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	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{2} +77.0000 q^{3} +64.0000 q^{4} +440.000 q^{5} -616.000 q^{6} +951.000 q^{7} -512.000 q^{8} +3742.00 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} +77.0000 q^{3} +64.0000 q^{4} +440.000 q^{5} -616.000 q^{6} +951.000 q^{7} -512.000 q^{8} +3742.00 q^{9} -3520.00 q^{10} -8398.00 q^{11} +4928.00 q^{12} -6223.00 q^{13} -7608.00 q^{14} +33880.0 q^{15} +4096.00 q^{16} +26211.0 q^{17} -29936.0 q^{18} -6859.00 q^{19} +28160.0 q^{20} +73227.0 q^{21} +67184.0 q^{22} -64213.0 q^{23} -39424.0 q^{24} +115475. q^{25} +49784.0 q^{26} +119735. q^{27} +60864.0 q^{28} +65845.0 q^{29} -271040. q^{30} -32708.0 q^{31} -32768.0 q^{32} -646646. q^{33} -209688. q^{34} +418440. q^{35} +239488. q^{36} -436694. q^{37} +54872.0 q^{38} -479171. q^{39} -225280. q^{40} -28808.0 q^{41} -585816. q^{42} +650272. q^{43} -537472. q^{44} +1.64648e6 q^{45} +513704. q^{46} +58736.0 q^{47} +315392. q^{48} +80858.0 q^{49} -923800. q^{50} +2.01825e6 q^{51} -398272. q^{52} -918703. q^{53} -957880. q^{54} -3.69512e6 q^{55} -486912. q^{56} -528143. q^{57} -526760. q^{58} -787635. q^{59} +2.16832e6 q^{60} +3.10686e6 q^{61} +261664. q^{62} +3.55864e6 q^{63} +262144. q^{64} -2.73812e6 q^{65} +5.17317e6 q^{66} +2.72600e6 q^{67} +1.67750e6 q^{68} -4.94440e6 q^{69} -3.34752e6 q^{70} -1.80096e6 q^{71} -1.91590e6 q^{72} -1.43622e6 q^{73} +3.49355e6 q^{74} +8.89158e6 q^{75} -438976. q^{76} -7.98650e6 q^{77} +3.83337e6 q^{78} +3.40211e6 q^{79} +1.80224e6 q^{80} +1.03584e6 q^{81} +230464. q^{82} -9.45404e6 q^{83} +4.68653e6 q^{84} +1.15328e7 q^{85} -5.20218e6 q^{86} +5.07006e6 q^{87} +4.29978e6 q^{88} -40980.0 q^{89} -1.31718e7 q^{90} -5.91807e6 q^{91} -4.10963e6 q^{92} -2.51852e6 q^{93} -469888. q^{94} -3.01796e6 q^{95} -2.52314e6 q^{96} +4.28165e6 q^{97} -646864. q^{98} -3.14253e7 q^{99} +O(q^{100})\)

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 $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	77.0000	1.64652	0.823259	−	0.567666i	$-0.192154\pi$
$3$	77.0000	1.64652	0.823259	+	0.567666i	$0.192154\pi$
$4$	64.0000	0.500000
$5$	440.000	1.57419	0.787096	−	0.616831i	$-0.211584\pi$
$5$	440.000	1.57419	0.787096	+	0.616831i	$0.211584\pi$
$6$	−616.000	−1.16426
$7$	951.000	1.04794	0.523971	−	0.851736i	$-0.324450\pi$
$7$	951.000	1.04794	0.523971	+	0.851736i	$0.324450\pi$
$8$	−512.000	−0.353553
$9$	3742.00	1.71102
$10$	−3520.00	−1.11312
$11$	−8398.00	−1.90240	−0.951199	−	0.308578i	$-0.900147\pi$
$11$	−8398.00	−1.90240	−0.951199	+	0.308578i	$0.900147\pi$
$12$	4928.00	0.823259
$13$	−6223.00	−0.785594	−0.392797	−	0.919625i	$-0.628492\pi$
$13$	−6223.00	−0.785594	−0.392797	+	0.919625i	$0.628492\pi$
$14$	−7608.00	−0.741007
$15$	33880.0	2.59193
$16$	4096.00	0.250000
$17$	26211.0	1.29393	0.646967	−	0.762518i	$-0.276037\pi$
$17$	26211.0	1.29393	0.646967	+	0.762518i	$0.276037\pi$
$18$	−29936.0	−1.20987
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	28160.0	0.787096
$21$	73227.0	1.72546
$22$	67184.0	1.34520
$23$	−64213.0	−1.10046	−0.550232	−	0.835012i	$-0.685461\pi$
$23$	−64213.0	−1.10046	−0.550232	+	0.835012i	$0.685461\pi$
$24$	−39424.0	−0.582132
$25$	115475.	1.47808
$26$	49784.0	0.555499
$27$	119735.	1.17071
$28$	60864.0	0.523971
$29$	65845.0	0.501337	0.250669	−	0.968073i	$-0.419350\pi$
$29$	65845.0	0.501337	0.250669	+	0.968073i	$0.419350\pi$
$30$	−271040.	−1.83277
$31$	−32708.0	−0.197191	−0.0985957	−	0.995128i	$-0.531435\pi$
$31$	−32708.0	−0.197191	−0.0985957	+	0.995128i	$0.531435\pi$
$32$	−32768.0	−0.176777
$33$	−646646.	−3.13233
$34$	−209688.	−0.914950
$35$	418440.	1.64966
$36$	239488.	0.855510
$37$	−436694.	−1.41733	−0.708665	−	0.705545i	$-0.750702\pi$
$37$	−436694.	−1.41733	−0.708665	+	0.705545i	$0.750702\pi$
$38$	54872.0	0.162221
$39$	−479171.	−1.29349
$40$	−225280.	−0.556561
$41$	−28808.0	−0.0652784	−0.0326392	−	0.999467i	$-0.510391\pi$
$41$	−28808.0	−0.0652784	−0.0326392	+	0.999467i	$0.510391\pi$
$42$	−585816.	−1.22008
$43$	650272.	1.24726	0.623628	−	0.781721i	$-0.285658\pi$
$43$	650272.	1.24726	0.623628	+	0.781721i	$0.285658\pi$
$44$	−537472.	−0.951199
$45$	1.64648e6	2.69347
$46$	513704.	0.778145
$47$	58736.0	0.0825205	0.0412603	−	0.999148i	$-0.486863\pi$
$47$	58736.0	0.0825205	0.0412603	+	0.999148i	$0.486863\pi$
$48$	315392.	0.411629
$49$	80858.0	0.0981831
$50$	−923800.	−1.04516
$51$	2.01825e6	2.13049
$52$	−398272.	−0.392797
$53$	−918703.	−0.847636	−0.423818	−	0.905747i	$-0.639310\pi$
$53$	−918703.	−0.847636	−0.423818	+	0.905747i	$0.639310\pi$
$54$	−957880.	−0.827814
$55$	−3.69512e6	−2.99474
$56$	−486912.	−0.370504
$57$	−528143.	−0.377737
$58$	−526760.	−0.354499
$59$	−787635.	−0.499279	−0.249639	−	0.968339i	$-0.580312\pi$
$59$	−787635.	−0.499279	−0.249639	+	0.968339i	$0.580312\pi$
$60$	2.16832e6	1.29597
$61$	3.10686e6	1.75254	0.876269	−	0.481822i	$-0.160025\pi$
$61$	3.10686e6	1.75254	0.876269	+	0.481822i	$0.160025\pi$
$62$	261664.	0.139435
$63$	3.55864e6	1.79305
$64$	262144.	0.125000
$65$	−2.73812e6	−1.23668
$66$	5.17317e6	2.21489
$67$	2.72600e6	1.10730	0.553649	−	0.832750i	$-0.313235\pi$
$67$	2.72600e6	1.10730	0.553649	+	0.832750i	$0.313235\pi$
$68$	1.67750e6	0.646967
$69$	−4.94440e6	−1.81193
$70$	−3.34752e6	−1.16649
$71$	−1.80096e6	−0.597172	−0.298586	−	0.954383i	$-0.596515\pi$
$71$	−1.80096e6	−0.597172	−0.298586	+	0.954383i	$0.596515\pi$
$72$	−1.91590e6	−0.604937
$73$	−1.43622e6	−0.432108	−0.216054	−	0.976381i	$-0.569319\pi$
$73$	−1.43622e6	−0.432108	−0.216054	+	0.976381i	$0.569319\pi$
$74$	3.49355e6	1.00220
$75$	8.89158e6	2.43368
$76$	−438976.	−0.114708
$77$	−7.98650e6	−1.99360
$78$	3.83337e6	0.914638
$79$	3.40211e6	0.776343	0.388171	−	0.921587i	$-0.373107\pi$
$79$	3.40211e6	0.776343	0.388171	+	0.921587i	$0.373107\pi$
$80$	1.80224e6	0.393548
$81$	1.03584e6	0.216569
$82$	230464.	0.0461588
$83$	−9.45404e6	−1.81486	−0.907432	−	0.420199i	$-0.861960\pi$
$83$	−9.45404e6	−1.81486	−0.907432	+	0.420199i	$0.861960\pi$
$84$	4.68653e6	0.862728
$85$	1.15328e7	2.03690
$86$	−5.20218e6	−0.881943
$87$	5.07006e6	0.825460
$88$	4.29978e6	0.672599
$89$	−40980.0	−0.00616179	−0.00308090	−	0.999995i	$-0.500981\pi$
$89$	−40980.0	−0.00616179	−0.00308090	+	0.999995i	$0.500981\pi$
$90$	−1.31718e7	−1.90457
$91$	−5.91807e6	−0.823257
$92$	−4.10963e6	−0.550232
$93$	−2.51852e6	−0.324679
$94$	−469888.	−0.0583508
$95$	−3.01796e6	−0.361144
$96$	−2.52314e6	−0.291066
$97$	4.28165e6	0.476332	0.238166	−	0.971224i	$-0.423454\pi$
$97$	4.28165e6	0.476332	0.238166	+	0.971224i	$0.423454\pi$
$98$	−646864.	−0.0694259
$99$	−3.14253e7	−3.25504
$100$	7.39040e6	0.739040
$101$	−2.48364e6	−0.239863	−0.119932	−	0.992782i	$-0.538268\pi$
$101$	−2.48364e6	−0.239863	−0.119932	+	0.992782i	$0.538268\pi$
$102$	−1.61460e7	−1.50648
$103$	−1.25032e7	−1.12743	−0.563716	−	0.825969i	$-0.690629\pi$
$103$	−1.25032e7	−1.12743	−0.563716	+	0.825969i	$0.690629\pi$
$104$	3.18618e6	0.277749
$105$	3.22199e7	2.71620
$106$	7.34962e6	0.599369
$107$	2.22469e6	0.175560	0.0877802	−	0.996140i	$-0.472023\pi$
$107$	2.22469e6	0.175560	0.0877802	+	0.996140i	$0.472023\pi$
$108$	7.66304e6	0.585353
$109$	1.41411e7	1.04590	0.522949	−	0.852364i	$-0.324832\pi$
$109$	1.41411e7	1.04590	0.522949	+	0.852364i	$0.324832\pi$
$110$	2.95610e7	2.11760
$111$	−3.36254e7	−2.33366
$112$	3.89530e6	0.261986
$113$	−7.38354e6	−0.481382	−0.240691	−	0.970602i	$-0.577374\pi$
$113$	−7.38354e6	−0.481382	−0.240691	+	0.970602i	$0.577374\pi$
$114$	4.22514e6	0.267100
$115$	−2.82537e7	−1.73234
$116$	4.21408e6	0.250669
$117$	−2.32865e7	−1.34417
$118$	6.30108e6	0.353043
$119$	2.49267e7	1.35597
$120$	−1.73466e7	−0.916387
$121$	5.10392e7	2.61912
$122$	−2.48549e7	−1.23923
$123$	−2.21822e6	−0.107482
$124$	−2.09331e6	−0.0985957
$125$	1.64340e7	0.752590
$126$	−2.84691e7	−1.26788
$127$	−2.24906e7	−0.974288	−0.487144	−	0.873322i	$-0.661961\pi$
$127$	−2.24906e7	−0.974288	−0.487144	+	0.873322i	$0.661961\pi$
$128$	−2.09715e6	−0.0883883
$129$	5.00709e7	2.05363
$130$	2.19050e7	0.874462
$131$	−3.90761e6	−0.151866	−0.0759332	−	0.997113i	$-0.524194\pi$
$131$	−3.90761e6	−0.151866	−0.0759332	+	0.997113i	$0.524194\pi$
$132$	−4.13853e7	−1.56617
$133$	−6.52291e6	−0.240414
$134$	−2.18080e7	−0.782977
$135$	5.26834e7	1.84292
$136$	−1.34200e7	−0.457475
$137$	−2.98680e7	−0.992393	−0.496197	−	0.868210i	$-0.665270\pi$
$137$	−2.98680e7	−0.992393	−0.496197	+	0.868210i	$0.665270\pi$
$138$	3.95552e7	1.28123
$139$	9.35214e6	0.295365	0.147683	−	0.989035i	$-0.452819\pi$
$139$	9.35214e6	0.295365	0.147683	+	0.989035i	$0.452819\pi$
$140$	2.67802e7	0.824831
$141$	4.52267e6	0.135871
$142$	1.44077e7	0.422264
$143$	5.22608e7	1.49451
$144$	1.53272e7	0.427755
$145$	2.89718e7	0.789201
$146$	1.14898e7	0.305546
$147$	6.22607e6	0.161660
$148$	−2.79484e7	−0.708665
$149$	2.48262e7	0.614835	0.307417	−	0.951575i	$-0.400535\pi$
$149$	2.48262e7	0.614835	0.307417	+	0.951575i	$0.400535\pi$
$150$	−7.11326e7	−1.72087
$151$	−1.11303e7	−0.263080	−0.131540	−	0.991311i	$-0.541992\pi$
$151$	−1.11303e7	−0.263080	−0.131540	+	0.991311i	$0.541992\pi$
$152$	3.51181e6	0.0811107
$153$	9.80816e7	2.21395
$154$	6.38920e7	1.40969
$155$	−1.43915e7	−0.310417
$156$	−3.06669e7	−0.646747
$157$	4.52922e7	0.934059	0.467030	−	0.884242i	$-0.345324\pi$
$157$	4.52922e7	0.934059	0.467030	+	0.884242i	$0.345324\pi$
$158$	−2.72169e7	−0.548957
$159$	−7.07401e7	−1.39565
$160$	−1.44179e7	−0.278280
$161$	−6.10666e7	−1.15322
$162$	−8.28673e6	−0.153137
$163$	2.33565e7	0.422427	0.211214	−	0.977440i	$-0.432258\pi$
$163$	2.33565e7	0.422427	0.211214	+	0.977440i	$0.432258\pi$
$164$	−1.84371e6	−0.0326392
$165$	−2.84524e8	−4.93089
$166$	7.56323e7	1.28330
$167$	−6.83692e7	−1.13593	−0.567967	−	0.823051i	$-0.692270\pi$
$167$	−6.83692e7	−1.13593	−0.567967	+	0.823051i	$0.692270\pi$
$168$	−3.74922e7	−0.610041
$169$	−2.40228e7	−0.382842
$170$	−9.22627e7	−1.44031
$171$	−2.56664e7	−0.392535
$172$	4.16174e7	0.623628
$173$	4.58676e7	0.673511	0.336755	−	0.941592i	$-0.390670\pi$
$173$	4.58676e7	0.673511	0.336755	+	0.941592i	$0.390670\pi$
$174$	−4.05605e7	−0.583689
$175$	1.09817e8	1.54894
$176$	−3.43982e7	−0.475600
$177$	−6.06479e7	−0.822071
$178$	327840.	0.00435704
$179$	−3.50479e7	−0.456748	−0.228374	−	0.973573i	$-0.573341\pi$
$179$	−3.50479e7	−0.456748	−0.228374	+	0.973573i	$0.573341\pi$
$180$	1.05375e8	1.34674
$181$	1.30380e7	0.163431	0.0817155	−	0.996656i	$-0.473960\pi$
$181$	1.30380e7	0.163431	0.0817155	+	0.996656i	$0.473960\pi$
$182$	4.73446e7	0.582131
$183$	2.39228e8	2.88559
$184$	3.28771e7	0.389073
$185$	−1.92145e8	−2.23115
$186$	2.01481e7	0.229583
$187$	−2.20120e8	−2.46158
$188$	3.75910e6	0.0412603
$189$	1.13868e8	1.22683
$190$	2.41437e7	0.255368
$191$	−7.52311e7	−0.781233	−0.390616	−	0.920554i	$-0.627738\pi$
$191$	−7.52311e7	−0.781233	−0.390616	+	0.920554i	$0.627738\pi$
$192$	2.01851e7	0.205815
$193$	8.77425e7	0.878536	0.439268	−	0.898356i	$-0.355238\pi$
$193$	8.77425e7	0.878536	0.439268	+	0.898356i	$0.355238\pi$
$194$	−3.42532e7	−0.336818
$195$	−2.10835e8	−2.03621
$196$	5.17491e6	0.0490915
$197$	1.69049e8	1.57536	0.787681	−	0.616083i	$-0.211281\pi$
$197$	1.69049e8	1.57536	0.787681	+	0.616083i	$0.211281\pi$
$198$	2.51403e8	2.30166
$199$	1.39490e8	1.25475	0.627377	−	0.778716i	$-0.284128\pi$
$199$	1.39490e8	1.25475	0.627377	+	0.778716i	$0.284128\pi$
$200$	−5.91232e7	−0.522580
$201$	2.09902e8	1.82318
$202$	1.98691e7	0.169609
$203$	6.26186e7	0.525372
$204$	1.29168e8	1.06524
$205$	−1.26755e7	−0.102761
$206$	1.00025e8	0.797214
$207$	−2.40285e8	−1.88291
$208$	−2.54894e7	−0.196398
$209$	5.76019e7	0.436440
$210$	−2.57759e8	−1.92064
$211$	−2.25998e8	−1.65621	−0.828105	−	0.560573i	$-0.810581\pi$
$211$	−2.25998e8	−1.65621	−0.828105	+	0.560573i	$0.810581\pi$
$212$	−5.87970e7	−0.423818
$213$	−1.38674e8	−0.983254
$214$	−1.77975e7	−0.124140
$215$	2.86120e8	1.96342
$216$	−6.13043e7	−0.413907
$217$	−3.11053e7	−0.206645
$218$	−1.13128e8	−0.739561
$219$	−1.10589e8	−0.711473
$220$	−2.36488e8	−1.49737
$221$	−1.63111e8	−1.01651
$222$	2.69004e8	1.65015
$223$	1.59102e8	0.960746	0.480373	−	0.877064i	$-0.340501\pi$
$223$	1.59102e8	0.960746	0.480373	+	0.877064i	$0.340501\pi$
$224$	−3.11624e7	−0.185252
$225$	4.32107e8	2.52902
$226$	5.90683e7	0.340389
$227$	2.73408e8	1.55139	0.775693	−	0.631110i	$-0.217400\pi$
$227$	2.73408e8	1.55139	0.775693	+	0.631110i	$0.217400\pi$
$228$	−3.38012e7	−0.188869
$229$	9.36336e7	0.515238	0.257619	−	0.966247i	$-0.417062\pi$
$229$	9.36336e7	0.515238	0.257619	+	0.966247i	$0.417062\pi$
$230$	2.26030e8	1.22495
$231$	−6.14960e8	−3.28250
$232$	−3.37126e7	−0.177249
$233$	−2.77939e7	−0.143947	−0.0719736	−	0.997407i	$-0.522930\pi$
$233$	−2.77939e7	−0.143947	−0.0719736	+	0.997407i	$0.522930\pi$
$234$	1.86292e8	0.950469
$235$	2.58438e7	0.129903
$236$	−5.04086e7	−0.249639
$237$	2.61962e8	1.27826
$238$	−1.99413e8	−0.958815
$239$	2.65907e8	1.25990	0.629952	−	0.776634i	$-0.283075\pi$
$239$	2.65907e8	1.25990	0.629952	+	0.776634i	$0.283075\pi$
$240$	1.38772e8	0.647984
$241$	−1.15586e8	−0.531917	−0.265959	−	0.963984i	$-0.585688\pi$
$241$	−1.15586e8	−0.531917	−0.265959	+	0.963984i	$0.585688\pi$
$242$	−4.08314e8	−1.85200
$243$	−1.82101e8	−0.814122
$244$	1.98839e8	0.876269
$245$	3.55775e7	0.154559
$246$	1.77457e7	0.0760013
$247$	4.26836e7	0.180228
$248$	1.67465e7	0.0697177
$249$	−7.27961e8	−2.98820
$250$	−1.31472e8	−0.532161
$251$	9.91605e7	0.395804	0.197902	−	0.980222i	$-0.436587\pi$
$251$	9.91605e7	0.395804	0.197902	+	0.980222i	$0.436587\pi$
$252$	2.27753e8	0.896525
$253$	5.39261e8	2.09352
$254$	1.79925e8	0.688926
$255$	8.88029e8	3.35379
$256$	1.67772e7	0.0625000
$257$	−3.72577e8	−1.36915	−0.684573	−	0.728944i	$-0.740011\pi$
$257$	−3.72577e8	−1.36915	−0.684573	+	0.728944i	$0.740011\pi$
$258$	−4.00568e8	−1.45213
$259$	−4.15296e8	−1.48528
$260$	−1.75240e8	−0.618338
$261$	2.46392e8	0.857798
$262$	3.12609e7	0.107386
$263$	−3.58686e8	−1.21582	−0.607911	−	0.794005i	$-0.707992\pi$
$263$	−3.58686e8	−1.21582	−0.607911	+	0.794005i	$0.707992\pi$
$264$	3.31083e8	1.10745
$265$	−4.04229e8	−1.33434
$266$	5.21833e7	0.169999
$267$	−3.15546e6	−0.0101455
$268$	1.74464e8	0.553649
$269$	5.09148e8	1.59482	0.797408	−	0.603440i	$-0.206204\pi$
$269$	5.09148e8	1.59482	0.797408	+	0.603440i	$0.206204\pi$
$270$	−4.21467e8	−1.30314
$271$	3.12525e8	0.953877	0.476938	−	0.878937i	$-0.341746\pi$
$271$	3.12525e8	0.953877	0.476938	+	0.878937i	$0.341746\pi$
$272$	1.07360e8	0.323484
$273$	−4.55692e8	−1.35551
$274$	2.38944e8	0.701728
$275$	−9.69759e8	−2.81190
$276$	−3.16442e8	−0.905966
$277$	3.42007e8	0.966844	0.483422	−	0.875388i	$-0.339394\pi$
$277$	3.42007e8	0.966844	0.483422	+	0.875388i	$0.339394\pi$
$278$	−7.48171e7	−0.208855
$279$	−1.22393e8	−0.337398
$280$	−2.14241e8	−0.583244
$281$	−4.28706e8	−1.15262	−0.576311	−	0.817230i	$-0.695508\pi$
$281$	−4.28706e8	−1.15262	−0.576311	+	0.817230i	$0.695508\pi$
$282$	−3.61814e7	−0.0960756
$283$	4.52256e8	1.18613	0.593064	−	0.805155i	$-0.297918\pi$
$283$	4.52256e8	1.18613	0.593064	+	0.805155i	$0.297918\pi$
$284$	−1.15261e8	−0.298586
$285$	−2.32383e8	−0.594631
$286$	−4.18086e8	−1.05678
$287$	−2.73964e7	−0.0684080
$288$	−1.22618e8	−0.302468
$289$	2.76678e8	0.674267
$290$	−2.31774e8	−0.558049
$291$	3.29687e8	0.784289
$292$	−9.19183e7	−0.216054
$293$	4.17395e8	0.969416	0.484708	−	0.874676i	$-0.338926\pi$
$293$	4.17395e8	0.969416	0.484708	+	0.874676i	$0.338926\pi$
$294$	−4.98085e7	−0.114311
$295$	−3.46559e8	−0.785960
$296$	2.23587e8	0.501102
$297$	−1.00553e9	−2.22715
$298$	−1.98610e8	−0.434754
$299$	3.99597e8	0.864517
$300$	5.69061e8	1.21684
$301$	6.18409e8	1.30705
$302$	8.90424e7	0.186025
$303$	−1.91240e8	−0.394939
$304$	−2.80945e7	−0.0573539
$305$	1.36702e9	2.75883
$306$	−7.84652e8	−1.56550
$307$	3.15746e8	0.622807	0.311404	−	0.950278i	$-0.399201\pi$
$307$	3.15746e8	0.622807	0.311404	+	0.950278i	$0.399201\pi$
$308$	−5.11136e8	−0.996802
$309$	−9.62745e8	−1.85634
$310$	1.15132e8	0.219498
$311$	5.25059e8	0.989798	0.494899	−	0.868950i	$-0.335205\pi$
$311$	5.25059e8	0.989798	0.494899	+	0.868950i	$0.335205\pi$
$312$	2.45336e8	0.457319
$313$	9.13906e8	1.68460	0.842299	−	0.539011i	$-0.181202\pi$
$313$	9.13906e8	1.68460	0.842299	+	0.539011i	$0.181202\pi$
$314$	−3.62337e8	−0.660480
$315$	1.56580e9	2.82260
$316$	2.17735e8	0.388171
$317$	−3.56880e8	−0.629237	−0.314619	−	0.949218i	$-0.601877\pi$
$317$	−3.56880e8	−0.629237	−0.314619	+	0.949218i	$0.601877\pi$
$318$	5.65921e8	0.986872
$319$	−5.52966e8	−0.953743
$320$	1.15343e8	0.196774
$321$	1.71301e8	0.289063
$322$	4.88533e8	0.815451
$323$	−1.79781e8	−0.296849
$324$	6.62938e7	0.108284
$325$	−7.18601e8	−1.16117
$326$	−1.86852e8	−0.298701
$327$	1.08886e9	1.72209
$328$	1.47497e7	0.0230794
$329$	5.58579e7	0.0864767
$330$	2.27619e9	3.48667
$331$	−1.19720e9	−1.81455	−0.907274	−	0.420541i	$-0.861840\pi$
$331$	−1.19720e9	−1.81455	−0.907274	+	0.420541i	$0.861840\pi$
$332$	−6.05058e8	−0.907432
$333$	−1.63411e9	−2.42508
$334$	5.46954e8	0.803227
$335$	1.19944e9	1.74310
$336$	2.99938e8	0.431364
$337$	2.91487e8	0.414872	0.207436	−	0.978249i	$-0.433488\pi$
$337$	2.91487e8	0.414872	0.207436	+	0.978249i	$0.433488\pi$
$338$	1.92182e8	0.270710
$339$	−5.68532e8	−0.792604
$340$	7.38102e8	1.01845
$341$	2.74682e8	0.375137
$342$	2.05331e8	0.277564
$343$	−7.06293e8	−0.945052
$344$	−3.32939e8	−0.440971
$345$	−2.17554e9	−2.85233
$346$	−3.66941e8	−0.476244
$347$	2.16207e8	0.277791	0.138895	−	0.990307i	$-0.455645\pi$
$347$	2.16207e8	0.277791	0.138895	+	0.990307i	$0.455645\pi$
$348$	3.24484e8	0.412730
$349$	−1.03896e9	−1.30831	−0.654154	−	0.756362i	$-0.726975\pi$
$349$	−1.03896e9	−1.30831	−0.654154	+	0.756362i	$0.726975\pi$
$350$	−8.78534e8	−1.09527
$351$	−7.45111e8	−0.919700
$352$	2.75186e8	0.336300
$353$	1.16693e8	0.141200	0.0705999	−	0.997505i	$-0.477509\pi$
$353$	1.16693e8	0.141200	0.0705999	+	0.997505i	$0.477509\pi$
$354$	4.85183e8	0.581292
$355$	−7.92422e8	−0.940063
$356$	−2.62272e6	−0.00308090
$357$	1.91935e9	2.23263
$358$	2.80383e8	0.322970
$359$	−1.28194e9	−1.46230	−0.731150	−	0.682217i	$-0.761016\pi$
$359$	−1.28194e9	−1.46230	−0.731150	+	0.682217i	$0.761016\pi$
$360$	−8.42998e8	−0.952287
$361$	4.70459e7	0.0526316
$362$	−1.04304e8	−0.115563
$363$	3.93002e9	4.31243
$364$	−3.78757e8	−0.411629
$365$	−6.31938e8	−0.680220
$366$	−1.91383e9	−2.04042
$367$	−1.91665e8	−0.202400	−0.101200	−	0.994866i	$-0.532268\pi$
$367$	−1.91665e8	−0.202400	−0.101200	+	0.994866i	$0.532268\pi$
$368$	−2.63016e8	−0.275116
$369$	−1.07800e8	−0.111693
$370$	1.53716e9	1.57766
$371$	−8.73687e8	−0.888274
$372$	−1.61185e8	−0.162340
$373$	1.00128e8	0.0999023	0.0499512	−	0.998752i	$-0.484093\pi$
$373$	1.00128e8	0.0999023	0.0499512	+	0.998752i	$0.484093\pi$
$374$	1.76096e9	1.74060
$375$	1.26542e9	1.23915
$376$	−3.00728e7	−0.0291754
$377$	−4.09753e8	−0.393847
$378$	−9.10944e8	−0.867502
$379$	−1.03285e9	−0.974543	−0.487272	−	0.873251i	$-0.662008\pi$
$379$	−1.03285e9	−0.974543	−0.487272	+	0.873251i	$0.662008\pi$
$380$	−1.93149e8	−0.180572
$381$	−1.73177e9	−1.60418
$382$	6.01848e8	0.552415
$383$	1.41475e9	1.28672	0.643358	−	0.765566i	$-0.277541\pi$
$383$	1.41475e9	1.28672	0.643358	+	0.765566i	$0.277541\pi$
$384$	−1.61481e8	−0.145533
$385$	−3.51406e9	−3.13831
$386$	−7.01940e8	−0.621219
$387$	2.43332e9	2.13408
$388$	2.74025e8	0.238166
$389$	6.84402e8	0.589505	0.294752	−	0.955574i	$-0.404763\pi$
$389$	6.84402e8	0.589505	0.294752	+	0.955574i	$0.404763\pi$
$390$	1.68668e9	1.43982
$391$	−1.68309e9	−1.42393
$392$	−4.13993e7	−0.0347130
$393$	−3.00886e8	−0.250051
$394$	−1.35239e9	−1.11395
$395$	1.49693e9	1.22211
$396$	−2.01122e9	−1.62752
$397$	−3.21394e8	−0.257793	−0.128897	−	0.991658i	$-0.541144\pi$
$397$	−3.21394e8	−0.257793	−0.128897	+	0.991658i	$0.541144\pi$
$398$	−1.11592e9	−0.887245
$399$	−5.02264e8	−0.395847
$400$	4.72986e8	0.369520
$401$	1.49873e9	1.16070	0.580349	−	0.814368i	$-0.302916\pi$
$401$	1.49873e9	1.16070	0.580349	+	0.814368i	$0.302916\pi$
$402$	−1.67922e9	−1.28919
$403$	2.03542e8	0.154912
$404$	−1.58953e8	−0.119932
$405$	4.55770e8	0.340921
$406$	−5.00949e8	−0.371494
$407$	3.66736e9	2.69633
$408$	−1.03334e9	−0.753241
$409$	−1.67043e9	−1.20725	−0.603623	−	0.797270i	$-0.706277\pi$
$409$	−1.67043e9	−1.20725	−0.603623	+	0.797270i	$0.706277\pi$
$410$	1.01404e8	0.0726628
$411$	−2.29983e9	−1.63399
$412$	−8.00203e8	−0.563716
$413$	−7.49041e8	−0.523215
$414$	1.92228e9	1.33142
$415$	−4.15978e9	−2.85694
$416$	2.03915e8	0.138875
$417$	7.20115e8	0.486324
$418$	−4.60815e8	−0.308610
$419$	−8.15512e8	−0.541603	−0.270802	−	0.962635i	$-0.587289\pi$
$419$	−8.15512e8	−0.541603	−0.270802	+	0.962635i	$0.587289\pi$
$420$	2.06207e9	1.35810
$421$	1.94494e8	0.127034	0.0635169	−	0.997981i	$-0.479768\pi$
$421$	1.94494e8	0.127034	0.0635169	+	0.997981i	$0.479768\pi$
$422$	1.80798e9	1.17112
$423$	2.19790e8	0.141194
$424$	4.70376e8	0.299685
$425$	3.02672e9	1.91254
$426$	1.10939e9	0.695266
$427$	2.95463e9	1.83656
$428$	1.42380e8	0.0877802
$429$	4.02408e9	2.46074
$430$	−2.28896e9	−1.38835
$431$	−2.62496e9	−1.57925	−0.789627	−	0.613587i	$-0.789726\pi$
$431$	−2.62496e9	−1.57925	−0.789627	+	0.613587i	$0.789726\pi$
$432$	4.90435e8	0.292677
$433$	1.90404e9	1.12712	0.563559	−	0.826076i	$-0.309432\pi$
$433$	1.90404e9	1.12712	0.563559	+	0.826076i	$0.309432\pi$
$434$	2.48842e8	0.146120
$435$	2.23083e9	1.29943
$436$	9.05028e8	0.522949
$437$	4.40437e8	0.252464
$438$	8.84713e8	0.503087
$439$	−1.51564e9	−0.855007	−0.427503	−	0.904014i	$-0.640607\pi$
$439$	−1.51564e9	−0.855007	−0.427503	+	0.904014i	$0.640607\pi$
$440$	1.89190e9	1.05880
$441$	3.02571e8	0.167993
$442$	1.30489e9	0.718779
$443$	−2.83361e9	−1.54856	−0.774279	−	0.632844i	$-0.781887\pi$
$443$	−2.83361e9	−1.54856	−0.774279	+	0.632844i	$0.781887\pi$
$444$	−2.15203e9	−1.16683
$445$	−1.80312e7	−0.00969984
$446$	−1.27282e9	−0.679350
$447$	1.91162e9	1.01234
$448$	2.49299e8	0.130993
$449$	7.85104e8	0.409322	0.204661	−	0.978833i	$-0.434391\pi$
$449$	7.85104e8	0.409322	0.204661	+	0.978833i	$0.434391\pi$
$450$	−3.45686e9	−1.78829
$451$	2.41930e8	0.124186
$452$	−4.72546e8	−0.240691
$453$	−8.57033e8	−0.433165
$454$	−2.18726e9	−1.09700
$455$	−2.60395e9	−1.29596
$456$	2.70409e8	0.133550
$457$	1.46139e8	0.0716242	0.0358121	−	0.999359i	$-0.488598\pi$
$457$	1.46139e8	0.0716242	0.0358121	+	0.999359i	$0.488598\pi$
$458$	−7.49069e8	−0.364328
$459$	3.13837e9	1.51482
$460$	−1.80824e9	−0.866170
$461$	1.31176e9	0.623593	0.311797	−	0.950149i	$-0.399069\pi$
$461$	1.31176e9	0.623593	0.311797	+	0.950149i	$0.399069\pi$
$462$	4.91968e9	2.32108
$463$	3.88164e9	1.81753	0.908764	−	0.417310i	$-0.137027\pi$
$463$	3.88164e9	1.81753	0.908764	+	0.417310i	$0.137027\pi$
$464$	2.69701e8	0.125334
$465$	−1.10815e9	−0.511107
$466$	2.22351e8	0.101786
$467$	1.16917e9	0.531211	0.265606	−	0.964082i	$-0.414428\pi$
$467$	1.16917e9	0.531211	0.265606	+	0.964082i	$0.414428\pi$
$468$	−1.49033e9	−0.672083
$469$	2.59243e9	1.16038
$470$	−2.06751e8	−0.0918554
$471$	3.48750e9	1.53795
$472$	4.03269e8	0.176522
$473$	−5.46098e9	−2.37278
$474$	−2.09570e9	−0.903868
$475$	−7.92043e8	−0.339095
$476$	1.59531e9	0.677984
$477$	−3.43779e9	−1.45032
$478$	−2.12726e9	−0.890886
$479$	−1.99948e9	−0.831271	−0.415635	−	0.909531i	$-0.636441\pi$
$479$	−1.99948e9	−0.831271	−0.415635	+	0.909531i	$0.636441\pi$
$480$	−1.11018e9	−0.458194
$481$	2.71755e9	1.11345
$482$	9.24685e8	0.376122
$483$	−4.70213e9	−1.89880
$484$	3.26651e9	1.30956
$485$	1.88392e9	0.749838
$486$	1.45681e9	0.575671
$487$	−1.34306e9	−0.526920	−0.263460	−	0.964670i	$-0.584864\pi$
$487$	−1.34306e9	−0.526920	−0.263460	+	0.964670i	$0.584864\pi$
$488$	−1.59071e9	−0.619616
$489$	1.79845e9	0.695534
$490$	−2.84620e8	−0.109290
$491$	4.30613e9	1.64173	0.820865	−	0.571122i	$-0.193492\pi$
$491$	4.30613e9	1.64173	0.820865	+	0.571122i	$0.193492\pi$
$492$	−1.41966e8	−0.0537410
$493$	1.72586e9	0.648697
$494$	−3.41468e8	−0.127440
$495$	−1.38271e10	−5.12406
$496$	−1.33972e8	−0.0492979
$497$	−1.71271e9	−0.625802
$498$	5.82369e9	2.11298
$499$	2.28288e9	0.822492	0.411246	−	0.911524i	$-0.365094\pi$
$499$	2.28288e9	0.822492	0.411246	+	0.911524i	$0.365094\pi$
$500$	1.05178e9	0.376295
$501$	−5.26443e9	−1.87034
$502$	−7.93284e8	−0.279876
$503$	1.75365e9	0.614405	0.307202	−	0.951644i	$-0.400607\pi$
$503$	1.75365e9	0.614405	0.307202	+	0.951644i	$0.400607\pi$
$504$	−1.82202e9	−0.633939
$505$	−1.09280e9	−0.377591
$506$	−4.31409e9	−1.48034
$507$	−1.84975e9	−0.630357
$508$	−1.43940e9	−0.487144
$509$	5.93337e9	1.99429	0.997147	−	0.0754798i	$-0.0240488\pi$
$509$	5.93337e9	1.99429	0.997147	+	0.0754798i	$0.0240488\pi$
$510$	−7.10423e9	−2.37149
$511$	−1.36585e9	−0.452824
$512$	−1.34218e8	−0.0441942
$513$	−8.21262e8	−0.268578
$514$	2.98061e9	0.968132
$515$	−5.50140e9	−1.77479
$516$	3.20454e9	1.02681
$517$	−4.93265e8	−0.156987
$518$	3.32237e9	1.05025
$519$	3.53180e9	1.10895
$520$	1.40192e9	0.437231
$521$	3.43082e9	1.06284	0.531418	−	0.847110i	$-0.321659\pi$
$521$	3.43082e9	1.06284	0.531418	+	0.847110i	$0.321659\pi$
$522$	−1.97114e9	−0.606554
$523$	5.51108e8	0.168454	0.0842269	−	0.996447i	$-0.473158\pi$
$523$	5.51108e8	0.168454	0.0842269	+	0.996447i	$0.473158\pi$
$524$	−2.50087e8	−0.0759332
$525$	8.45589e9	2.55036
$526$	2.86949e9	0.859716
$527$	−8.57309e8	−0.255153
$528$	−2.64866e9	−0.783083
$529$	7.18484e8	0.211019
$530$	3.23383e9	0.943522
$531$	−2.94733e9	−0.854276
$532$	−4.17466e8	−0.120207
$533$	1.79272e8	0.0512823
$534$	2.52437e7	0.00717395
$535$	9.78864e8	0.276366
$536$	−1.39571e9	−0.391489
$537$	−2.69869e9	−0.752044
$538$	−4.07318e9	−1.12771
$539$	−6.79045e8	−0.186783
$540$	3.37174e9	0.921458
$541$	−1.74658e9	−0.474240	−0.237120	−	0.971480i	$-0.576203\pi$
$541$	−1.74658e9	−0.474240	−0.237120	+	0.971480i	$0.576203\pi$
$542$	−2.50020e9	−0.674493
$543$	1.00392e9	0.269092
$544$	−8.58882e8	−0.228738
$545$	6.22206e9	1.64644
$546$	3.64553e9	0.958488
$547$	−5.18146e9	−1.35362	−0.676810	−	0.736158i	$-0.736638\pi$
$547$	−5.18146e9	−1.35362	−0.676810	+	0.736158i	$0.736638\pi$
$548$	−1.91155e9	−0.496197
$549$	1.16259e10	2.99863
$550$	7.75807e9	1.98831
$551$	−4.51631e8	−0.115015
$552$	2.53153e9	0.640615
$553$	3.23541e9	0.813562
$554$	−2.73606e9	−0.683662
$555$	−1.47952e10	−3.67363
$556$	5.98537e8	0.147683
$557$	1.08397e9	0.265780	0.132890	−	0.991131i	$-0.457574\pi$
$557$	1.08397e9	0.265780	0.132890	+	0.991131i	$0.457574\pi$
$558$	9.79147e8	0.238577
$559$	−4.04664e9	−0.979836
$560$	1.71393e9	0.412416
$561$	−1.69492e10	−4.05303
$562$	3.42965e9	0.815027
$563$	4.58547e9	1.08294	0.541471	−	0.840720i	$-0.317868\pi$
$563$	4.58547e9	1.08294	0.541471	+	0.840720i	$0.317868\pi$
$564$	2.89451e8	0.0679357
$565$	−3.24876e9	−0.757788
$566$	−3.61805e9	−0.838720
$567$	9.85085e8	0.226951
$568$	9.22090e8	0.211132
$569$	−6.15694e9	−1.40111	−0.700555	−	0.713598i	$-0.747064\pi$
$569$	−6.15694e9	−1.40111	−0.700555	+	0.713598i	$0.747064\pi$
$570$	1.85906e9	0.420467
$571$	3.85757e9	0.867136	0.433568	−	0.901121i	$-0.357254\pi$
$571$	3.85757e9	0.867136	0.433568	+	0.901121i	$0.357254\pi$
$572$	3.34469e9	0.747256
$573$	−5.79279e9	−1.28631
$574$	2.19171e8	0.0483718
$575$	−7.41500e9	−1.62657
$576$	9.80943e8	0.213877
$577$	−5.15448e9	−1.11704	−0.558521	−	0.829490i	$-0.688631\pi$
$577$	−5.15448e9	−1.11704	−0.558521	+	0.829490i	$0.688631\pi$
$578$	−2.21342e9	−0.476779
$579$	6.75618e9	1.44653
$580$	1.85420e9	0.394600
$581$	−8.99079e9	−1.90187
$582$	−2.63749e9	−0.554576
$583$	7.71527e9	1.61254
$584$	7.35346e8	0.152773
$585$	−1.02460e10	−2.11598
$586$	−3.33916e9	−0.685481
$587$	−1.46585e9	−0.299127	−0.149564	−	0.988752i	$-0.547787\pi$
$587$	−1.46585e9	−0.299127	−0.149564	+	0.988752i	$0.547787\pi$
$588$	3.98468e8	0.0808301
$589$	2.24344e8	0.0452388
$590$	2.77248e9	0.555758
$591$	1.30168e10	2.59386
$592$	−1.78870e9	−0.354333
$593$	4.20253e9	0.827599	0.413799	−	0.910368i	$-0.364202\pi$
$593$	4.20253e9	0.827599	0.413799	+	0.910368i	$0.364202\pi$
$594$	8.04428e9	1.57483
$595$	1.09677e10	2.13456
$596$	1.58888e9	0.307417
$597$	1.07408e10	2.06597
$598$	−3.19678e9	−0.611306
$599$	−1.57872e9	−0.300132	−0.150066	−	0.988676i	$-0.547949\pi$
$599$	−1.57872e9	−0.300132	−0.150066	+	0.988676i	$0.547949\pi$
$600$	−4.55249e9	−0.860437
$601$	9.22428e9	1.73329	0.866647	−	0.498923i	$-0.166271\pi$
$601$	9.22428e9	1.73329	0.866647	+	0.498923i	$0.166271\pi$
$602$	−4.94727e9	−0.924225
$603$	1.02007e10	1.89461
$604$	−7.12339e8	−0.131540
$605$	2.24573e10	4.12300
$606$	1.52992e9	0.279264
$607$	3.36747e9	0.611143	0.305572	−	0.952169i	$-0.401152\pi$
$607$	3.36747e9	0.611143	0.305572	+	0.952169i	$0.401152\pi$
$608$	2.24756e8	0.0405554
$609$	4.82163e9	0.865035
$610$	−1.09362e10	−1.95079
$611$	−3.65514e8	−0.0648276
$612$	6.27722e9	1.10697
$613$	1.66253e9	0.291512	0.145756	−	0.989321i	$-0.453439\pi$
$613$	1.66253e9	0.291512	0.145756	+	0.989321i	$0.453439\pi$
$614$	−2.52597e9	−0.440391
$615$	−9.76015e8	−0.169197
$616$	4.08909e9	0.704845
$617$	−4.02484e9	−0.689843	−0.344922	−	0.938632i	$-0.612094\pi$
$617$	−4.02484e9	−0.689843	−0.344922	+	0.938632i	$0.612094\pi$
$618$	7.70196e9	1.31263
$619$	1.05076e10	1.78068	0.890338	−	0.455300i	$-0.150468\pi$
$619$	1.05076e10	1.78068	0.890338	+	0.455300i	$0.150468\pi$
$620$	−9.21057e8	−0.155209
$621$	−7.68854e9	−1.28832
$622$	−4.20047e9	−0.699893
$623$	−3.89720e7	−0.00645720
$624$	−1.96268e9	−0.323373
$625$	−1.79052e9	−0.293360
$626$	−7.31124e9	−1.19119
$627$	4.43534e9	0.718606
$628$	2.89870e9	0.467030
$629$	−1.14462e10	−1.83393
$630$	−1.25264e10	−1.99588
$631$	−1.43832e9	−0.227905	−0.113952	−	0.993486i	$-0.536351\pi$
$631$	−1.43832e9	−0.227905	−0.113952	+	0.993486i	$0.536351\pi$
$632$	−1.74188e9	−0.274479
$633$	−1.74018e10	−2.72698
$634$	2.85504e9	0.444938
$635$	−9.89585e9	−1.53372
$636$	−4.52737e9	−0.697824
$637$	−5.03179e8	−0.0771320
$638$	4.42373e9	0.674398
$639$	−6.73918e9	−1.02177
$640$	−9.22747e8	−0.139140
$641$	−6.45356e9	−0.967824	−0.483912	−	0.875117i	$-0.660785\pi$
$641$	−6.45356e9	−0.967824	−0.483912	+	0.875117i	$0.660785\pi$
$642$	−1.37041e9	−0.204399
$643$	−5.47199e9	−0.811721	−0.405860	−	0.913935i	$-0.633028\pi$
$643$	−5.47199e9	−0.811721	−0.405860	+	0.913935i	$0.633028\pi$
$644$	−3.90826e9	−0.576611
$645$	2.20312e10	3.23280
$646$	1.43825e9	0.209904
$647$	9.50275e9	1.37938	0.689691	−	0.724104i	$-0.257746\pi$
$647$	9.50275e9	1.37938	0.689691	+	0.724104i	$0.257746\pi$
$648$	−5.30351e8	−0.0765686
$649$	6.61456e9	0.949827
$650$	5.74881e9	0.821072
$651$	−2.39511e9	−0.340245
$652$	1.49482e9	0.211214
$653$	1.02097e10	1.43489	0.717445	−	0.696615i	$-0.245311\pi$
$653$	1.02097e10	1.43489	0.717445	+	0.696615i	$0.245311\pi$
$654$	−8.71089e9	−1.21770
$655$	−1.71935e9	−0.239067
$656$	−1.17998e8	−0.0163196
$657$	−5.37435e9	−0.739345
$658$	−4.46863e8	−0.0611483
$659$	−8.46676e9	−1.15244	−0.576220	−	0.817295i	$-0.695473\pi$
$659$	−8.46676e9	−1.15244	−0.576220	+	0.817295i	$0.695473\pi$
$660$	−1.82096e10	−2.46545
$661$	−7.70863e9	−1.03818	−0.519089	−	0.854720i	$-0.673729\pi$
$661$	−7.70863e9	−1.03818	−0.519089	+	0.854720i	$0.673729\pi$
$662$	9.57759e9	1.28308
$663$	−1.25596e10	−1.67370
$664$	4.84047e9	0.641651
$665$	−2.87008e9	−0.378458
$666$	1.30729e10	1.71479
$667$	−4.22810e9	−0.551703
$668$	−4.37563e9	−0.567967
$669$	1.22509e10	1.58189
$670$	−9.59552e9	−1.23256
$671$	−2.60914e10	−3.33403
$672$	−2.39950e9	−0.305020
$673$	4.39307e9	0.555541	0.277770	−	0.960648i	$-0.410405\pi$
$673$	4.39307e9	0.555541	0.277770	+	0.960648i	$0.410405\pi$
$674$	−2.33189e9	−0.293359
$675$	1.38264e10	1.73040
$676$	−1.53746e9	−0.191421
$677$	−7.42240e9	−0.919357	−0.459678	−	0.888085i	$-0.652035\pi$
$677$	−7.42240e9	−0.919357	−0.459678	+	0.888085i	$0.652035\pi$
$678$	4.54826e9	0.560456
$679$	4.07185e9	0.499168
$680$	−5.90481e9	−0.720153
$681$	2.10524e10	2.55439
$682$	−2.19745e9	−0.265262
$683$	−1.34040e10	−1.60976	−0.804882	−	0.593435i	$-0.797771\pi$
$683$	−1.34040e10	−1.60976	−0.804882	+	0.593435i	$0.797771\pi$
$684$	−1.64265e9	−0.196267
$685$	−1.31419e10	−1.56222
$686$	5.65035e9	0.668253
$687$	7.20979e9	0.848348
$688$	2.66351e9	0.311814
$689$	5.71709e9	0.665898
$690$	1.74043e10	2.01690
$691$	−1.43025e10	−1.64907	−0.824535	−	0.565811i	$-0.808563\pi$
$691$	−1.43025e10	−1.64907	−0.824535	+	0.565811i	$0.808563\pi$
$692$	2.93553e9	0.336755
$693$	−2.98855e10	−3.41110
$694$	−1.72966e9	−0.196428
$695$	4.11494e9	0.464961
$696$	−2.59587e9	−0.291844
$697$	−7.55086e8	−0.0844660
$698$	8.31168e9	0.925113
$699$	−2.14013e9	−0.237012
$700$	7.02827e9	0.774471
$701$	7.95614e9	0.872347	0.436174	−	0.899862i	$-0.356333\pi$
$701$	7.95614e9	0.872347	0.436174	+	0.899862i	$0.356333\pi$
$702$	5.96089e9	0.650326
$703$	2.99528e9	0.325158
$704$	−2.20149e9	−0.237800
$705$	1.98998e9	0.213888
$706$	−9.33546e8	−0.0998434
$707$	−2.36194e9	−0.251363
$708$	−3.88147e9	−0.411035
$709$	1.07008e10	1.12760	0.563798	−	0.825912i	$-0.309339\pi$
$709$	1.07008e10	1.12760	0.563798	+	0.825912i	$0.309339\pi$
$710$	6.33937e9	0.664725
$711$	1.27307e10	1.32834
$712$	2.09818e7	0.00217852
$713$	2.10028e9	0.217002
$714$	−1.53548e10	−1.57871
$715$	2.29947e10	2.35265
$716$	−2.24307e9	−0.228374
$717$	2.04748e10	2.07445
$718$	1.02555e10	1.03400
$719$	1.29294e9	0.129726	0.0648630	−	0.997894i	$-0.479339\pi$
$719$	1.29294e9	0.129726	0.0648630	+	0.997894i	$0.479339\pi$
$720$	6.74398e9	0.673368
$721$	−1.18905e10	−1.18148
$722$	−3.76367e8	−0.0372161
$723$	−8.90009e9	−0.875811
$724$	8.34430e8	0.0817155
$725$	7.60345e9	0.741016
$726$	−3.14402e10	−3.04935
$727$	−4.13504e9	−0.399125	−0.199562	−	0.979885i	$-0.563952\pi$
$727$	−4.13504e9	−0.399125	−0.199562	+	0.979885i	$0.563952\pi$
$728$	3.03005e9	0.291065
$729$	−1.62871e10	−1.55704
$730$	5.05550e9	0.480988
$731$	1.70443e10	1.61387
$732$	1.53106e10	1.44279
$733$	7.76128e9	0.727896	0.363948	−	0.931419i	$-0.381429\pi$
$733$	7.76128e9	0.727896	0.363948	+	0.931419i	$0.381429\pi$
$734$	1.53332e9	0.143119
$735$	2.73947e9	0.254484
$736$	2.10413e9	0.194536
$737$	−2.28930e10	−2.10652
$738$	8.62396e8	0.0789786
$739$	8.13711e9	0.741676	0.370838	−	0.928697i	$-0.379070\pi$
$739$	8.13711e9	0.741676	0.370838	+	0.928697i	$0.379070\pi$
$740$	−1.22973e10	−1.11558
$741$	3.28663e9	0.296748
$742$	6.98949e9	0.628105
$743$	−1.46014e9	−0.130597	−0.0652987	−	0.997866i	$-0.520800\pi$
$743$	−1.46014e9	−0.130597	−0.0652987	+	0.997866i	$0.520800\pi$
$744$	1.28948e9	0.114791
$745$	1.09235e10	0.967868
$746$	−8.01026e8	−0.0706416
$747$	−3.53770e10	−3.10527
$748$	−1.40877e10	−1.23079
$749$	2.11568e9	0.183977
$750$	−1.01233e10	−0.876213
$751$	1.24940e9	0.107637	0.0538187	−	0.998551i	$-0.482861\pi$
$751$	1.24940e9	0.107637	0.0538187	+	0.998551i	$0.482861\pi$
$752$	2.40583e8	0.0206301
$753$	7.63536e9	0.651699
$754$	3.27803e9	0.278492
$755$	−4.89733e9	−0.414138
$756$	7.28755e9	0.613416
$757$	−1.72633e9	−0.144640	−0.0723202	−	0.997381i	$-0.523040\pi$
$757$	−1.72633e9	−0.144640	−0.0723202	+	0.997381i	$0.523040\pi$
$758$	8.26282e9	0.689106
$759$	4.15231e10	3.44702
$760$	1.54520e9	0.127684
$761$	−1.88538e10	−1.55079	−0.775393	−	0.631478i	$-0.782448\pi$
$761$	−1.88538e10	−1.55079	−0.775393	+	0.631478i	$0.782448\pi$
$762$	1.38542e10	1.13433
$763$	1.34481e10	1.09604
$764$	−4.81479e9	−0.390616
$765$	4.31559e10	3.48518
$766$	−1.13180e10	−0.909845
$767$	4.90145e9	0.392230
$768$	1.29185e9	0.102907
$769$	1.02437e10	0.812299	0.406149	−	0.913807i	$-0.366871\pi$
$769$	1.02437e10	0.812299	0.406149	+	0.913807i	$0.366871\pi$
$770$	2.81125e10	2.21912
$771$	−2.86884e10	−2.25432
$772$	5.61552e9	0.439268
$773$	−1.70435e10	−1.32718	−0.663590	−	0.748096i	$-0.730968\pi$
$773$	−1.70435e10	−1.32718	−0.663590	+	0.748096i	$0.730968\pi$
$774$	−1.94665e10	−1.50902
$775$	−3.77696e9	−0.291465
$776$	−2.19220e9	−0.168409
$777$	−3.19778e10	−2.44554
$778$	−5.47521e9	−0.416843
$779$	1.97594e8	0.0149759
$780$	−1.34935e10	−1.01810
$781$	1.51244e10	1.13606
$782$	1.34647e10	1.00687
$783$	7.88395e9	0.586918
$784$	3.31194e8	0.0245458
$785$	1.99286e10	1.47039
$786$	2.40709e9	0.176812
$787$	−2.29012e10	−1.67474	−0.837370	−	0.546637i	$-0.815908\pi$
$787$	−2.29012e10	−1.67474	−0.837370	+	0.546637i	$0.815908\pi$
$788$	1.08191e10	0.787681
$789$	−2.76189e10	−2.00187
$790$	−1.19754e10	−0.864164
$791$	−7.02174e9	−0.504461
$792$	1.60898e10	1.15083
$793$	−1.93340e10	−1.37678
$794$	2.57115e9	0.182287
$795$	−3.11257e10	−2.19702
$796$	8.92739e9	0.627377
$797$	4.91683e9	0.344018	0.172009	−	0.985095i	$-0.444974\pi$
$797$	4.91683e9	0.344018	0.172009	+	0.985095i	$0.444974\pi$
$798$	4.01811e9	0.279906
$799$	1.53953e9	0.106776
$800$	−3.78388e9	−0.261290
$801$	−1.53347e8	−0.0105429
$802$	−1.19899e10	−0.820737
$803$	1.20614e10	0.822041
$804$	1.34337e10	0.911592
$805$	−2.68693e10	−1.81539
$806$	−1.62834e9	−0.109540
$807$	3.92044e10	2.62589
$808$	1.27162e9	0.0848044
$809$	−1.08294e10	−0.719089	−0.359545	−	0.933128i	$-0.617068\pi$
$809$	−1.08294e10	−0.719089	−0.359545	+	0.933128i	$0.617068\pi$
$810$	−3.64616e9	−0.241067
$811$	1.60764e10	1.05832	0.529158	−	0.848523i	$-0.322508\pi$
$811$	1.60764e10	1.05832	0.529158	+	0.848523i	$0.322508\pi$
$812$	4.00759e9	0.262686
$813$	2.40644e10	1.57057
$814$	−2.93388e10	−1.90659
$815$	1.02769e10	0.664981
$816$	8.26674e9	0.532622
$817$	−4.46022e9	−0.286140
$818$	1.33634e10	0.853651
$819$	−2.21454e10	−1.40861
$820$	−8.11233e8	−0.0513804
$821$	1.74122e10	1.09813	0.549064	−	0.835780i	$-0.314984\pi$
$821$	1.74122e10	1.09813	0.549064	+	0.835780i	$0.314984\pi$
$822$	1.83987e10	1.15541
$823$	−1.26645e10	−0.791934	−0.395967	−	0.918265i	$-0.629591\pi$
$823$	−1.26645e10	−0.791934	−0.395967	+	0.918265i	$0.629591\pi$
$824$	6.40163e9	0.398607
$825$	−7.46714e10	−4.62984
$826$	5.99233e9	0.369969
$827$	2.86299e10	1.76015	0.880077	−	0.474830i	$-0.157491\pi$
$827$	2.86299e10	1.76015	0.880077	+	0.474830i	$0.157491\pi$
$828$	−1.53782e10	−0.941457
$829$	−2.75171e10	−1.67749	−0.838747	−	0.544521i	$-0.816712\pi$
$829$	−2.75171e10	−1.67749	−0.838747	+	0.544521i	$0.816712\pi$
$830$	3.32782e10	2.02016
$831$	2.63346e10	1.59192
$832$	−1.63132e9	−0.0981992
$833$	2.11937e9	0.127043
$834$	−5.76092e9	−0.343883
$835$	−3.00825e10	−1.78818
$836$	3.68652e9	0.218220
$837$	−3.91629e9	−0.230853
$838$	6.52410e9	0.382971
$839$	9.77182e9	0.571227	0.285613	−	0.958345i	$-0.407803\pi$
$839$	9.77182e9	0.571227	0.285613	+	0.958345i	$0.407803\pi$
$840$	−1.64966e10	−0.960321
$841$	−1.29143e10	−0.748661
$842$	−1.55595e9	−0.0898265
$843$	−3.30103e10	−1.89781
$844$	−1.44639e10	−0.828105
$845$	−1.05700e10	−0.602667
$846$	−1.75832e9	−0.0998394
$847$	4.85383e10	2.74469
$848$	−3.76301e9	−0.211909
$849$	3.48237e10	1.95298
$850$	−2.42137e10	−1.35237
$851$	2.80414e10	1.55972
$852$	−8.87512e9	−0.491627
$853$	−3.51519e9	−0.193922	−0.0969611	−	0.995288i	$-0.530912\pi$
$853$	−3.51519e9	−0.193922	−0.0969611	+	0.995288i	$0.530912\pi$
$854$	−2.36370e10	−1.29864
$855$	−1.12932e10	−0.617925
$856$	−1.13904e9	−0.0620700
$857$	−1.61455e10	−0.876233	−0.438117	−	0.898918i	$-0.644354\pi$
$857$	−1.61455e10	−0.876233	−0.438117	+	0.898918i	$0.644354\pi$
$858$	−3.21926e10	−1.74001
$859$	1.97382e10	1.06250	0.531252	−	0.847214i	$-0.321722\pi$
$859$	1.97382e10	1.06250	0.531252	+	0.847214i	$0.321722\pi$
$860$	1.83117e10	0.981710
$861$	−2.10952e9	−0.112635
$862$	2.09997e10	1.11670
$863$	8.05160e8	0.0426427	0.0213213	−	0.999773i	$-0.493213\pi$
$863$	8.05160e8	0.0426427	0.0213213	+	0.999773i	$0.493213\pi$
$864$	−3.92348e9	−0.206954
$865$	2.01817e10	1.06024
$866$	−1.52324e10	−0.796993
$867$	2.13042e10	1.11019
$868$	−1.99074e9	−0.103323
$869$	−2.85709e10	−1.47691
$870$	−1.78466e10	−0.918838
$871$	−1.69639e10	−0.869886
$872$	−7.24022e9	−0.369781
$873$	1.60219e10	0.815013
$874$	−3.52350e9	−0.178519
$875$	1.56287e10	0.788671
$876$	−7.07771e9	−0.355736
$877$	6.23056e9	0.311909	0.155955	−	0.987764i	$-0.450155\pi$
$877$	6.23056e9	0.311909	0.155955	+	0.987764i	$0.450155\pi$
$878$	1.21251e10	0.604581
$879$	3.21394e10	1.59616
$880$	−1.51352e10	−0.748685
$881$	9.81947e9	0.483807	0.241904	−	0.970300i	$-0.422228\pi$
$881$	9.81947e9	0.483807	0.241904	+	0.970300i	$0.422228\pi$
$882$	−2.42057e9	−0.118789
$883$	−9.86220e9	−0.482071	−0.241036	−	0.970516i	$-0.577487\pi$
$883$	−9.86220e9	−0.482071	−0.241036	+	0.970516i	$0.577487\pi$
$884$	−1.04391e10	−0.508254
$885$	−2.66851e10	−1.29410
$886$	2.26689e10	1.09500
$887$	−1.78336e10	−0.858037	−0.429019	−	0.903296i	$-0.641141\pi$
$887$	−1.78336e10	−0.858037	−0.429019	+	0.903296i	$0.641141\pi$
$888$	1.72162e10	0.825073
$889$	−2.13885e10	−1.02100
$890$	1.44250e8	0.00685882
$891$	−8.69899e9	−0.412000
$892$	1.01825e10	0.480373
$893$	−4.02870e8	−0.0189315
$894$	−1.52929e10	−0.715830
$895$	−1.54211e10	−0.719009
$896$	−1.99439e9	−0.0926259
$897$	3.07690e10	1.42344
$898$	−6.28083e9	−0.289434
$899$	−2.15366e9	−0.0988594
$900$	2.76549e10	1.26451
$901$	−2.40801e10	−1.09679
$902$	−1.93544e9	−0.0878124
$903$	4.76175e10	2.15208
$904$	3.78037e9	0.170194
$905$	5.73670e9	0.257272
$906$	6.85626e9	0.306294
$907$	9.62665e8	0.0428400	0.0214200	−	0.999771i	$-0.493181\pi$
$907$	9.62665e8	0.0428400	0.0214200	+	0.999771i	$0.493181\pi$
$908$	1.74981e10	0.775693
$909$	−9.29377e9	−0.410411
$910$	2.08316e10	0.916385
$911$	3.09938e8	0.0135819	0.00679096	−	0.999977i	$-0.497838\pi$
$911$	3.09938e8	0.0135819	0.00679096	+	0.999977i	$0.497838\pi$
$912$	−2.16327e9	−0.0944343
$913$	7.93950e10	3.45259
$914$	−1.16911e9	−0.0506460
$915$	1.05260e11	4.54246
$916$	5.99255e9	0.257619
$917$	−3.71614e9	−0.159147
$918$	−2.51070e10	−1.07114
$919$	−4.47873e9	−0.190349	−0.0951744	−	0.995461i	$-0.530341\pi$
$919$	−4.47873e9	−0.190349	−0.0951744	+	0.995461i	$0.530341\pi$
$920$	1.44659e10	0.612475
$921$	2.43125e10	1.02546
$922$	−1.04941e10	−0.440947
$923$	1.12074e10	0.469135
$924$	−3.93575e10	−1.64125
$925$	−5.04272e10	−2.09493
$926$	−3.10531e10	−1.28519
$927$	−4.67869e10	−1.92906
$928$	−2.15761e9	−0.0886247
$929$	−2.62394e10	−1.07374	−0.536870	−	0.843665i	$-0.680394\pi$
$929$	−2.62394e10	−1.07374	−0.536870	+	0.843665i	$0.680394\pi$
$930$	8.86518e9	0.361407
$931$	−5.54605e8	−0.0225247
$932$	−1.77881e9	−0.0719736
$933$	4.04295e10	1.62972
$934$	−9.35333e9	−0.375623
$935$	−9.68528e10	−3.87500
$936$	1.19227e10	0.475235
$937$	−5.22218e9	−0.207378	−0.103689	−	0.994610i	$-0.533065\pi$
$937$	−5.22218e9	−0.207378	−0.103689	+	0.994610i	$0.533065\pi$
$938$	−2.07394e10	−0.820515
$939$	7.03707e10	2.77372
$940$	1.65401e9	0.0649516
$941$	3.43325e10	1.34320	0.671602	−	0.740913i	$-0.265607\pi$
$941$	3.43325e10	1.34320	0.671602	+	0.740913i	$0.265607\pi$
$942$	−2.79000e10	−1.08749
$943$	1.84985e9	0.0718365
$944$	−3.22615e9	−0.124820
$945$	5.01019e10	1.93127
$946$	4.36879e10	1.67781
$947$	−2.00808e10	−0.768346	−0.384173	−	0.923261i	$-0.625513\pi$
$947$	−2.00808e10	−0.768346	−0.384173	+	0.923261i	$0.625513\pi$
$948$	1.67656e10	0.639131
$949$	8.93762e9	0.339461
$950$	6.33634e9	0.239776
$951$	−2.74797e10	−1.03605
$952$	−1.27625e10	−0.479407
$953$	4.99401e10	1.86906	0.934532	−	0.355880i	$-0.115819\pi$
$953$	4.99401e10	1.86906	0.934532	+	0.355880i	$0.115819\pi$
$954$	2.75023e10	1.02553
$955$	−3.31017e10	−1.22981
$956$	1.70181e10	0.629952
$957$	−4.25784e10	−1.57035
$958$	1.59958e10	0.587797
$959$	−2.84044e10	−1.03997
$960$	8.88144e9	0.323992
$961$	−2.64428e10	−0.961116
$962$	−2.17404e10	−0.787325
$963$	8.32479e9	0.300387
$964$	−7.39748e9	−0.265959
$965$	3.86067e10	1.38298
$966$	3.76170e10	1.34265
$967$	−3.94773e10	−1.40396	−0.701981	−	0.712196i	$-0.747701\pi$
$967$	−3.94773e10	−1.40396	−0.701981	+	0.712196i	$0.747701\pi$
$968$	−2.61321e10	−0.925999
$969$	−1.38432e10	−0.488767
$970$	−1.50714e10	−0.530215
$971$	−1.83646e10	−0.643746	−0.321873	−	0.946783i	$-0.604312\pi$
$971$	−1.83646e10	−0.643746	−0.321873	+	0.946783i	$0.604312\pi$
$972$	−1.16544e10	−0.407061
$973$	8.89389e9	0.309526
$974$	1.07445e10	0.372589
$975$	−5.53323e10	−1.91189
$976$	1.27257e10	0.438135
$977$	−4.72867e10	−1.62221	−0.811107	−	0.584897i	$-0.801135\pi$
$977$	−4.72867e10	−1.62221	−0.811107	+	0.584897i	$0.801135\pi$
$978$	−1.43876e10	−0.491816
$979$	3.44150e8	0.0117222
$980$	2.27696e9	0.0772795
$981$	5.29158e10	1.78955
$982$	−3.44490e10	−1.16088
$983$	−1.03196e10	−0.346517	−0.173258	−	0.984876i	$-0.555430\pi$
$983$	−1.03196e10	−0.346517	−0.173258	+	0.984876i	$0.555430\pi$
$984$	1.13573e9	0.0380006
$985$	7.43815e10	2.47992
$986$	−1.38069e10	−0.458698
$987$	4.30106e9	0.142385
$988$	2.73175e9	0.0901138
$989$	−4.17559e10	−1.37256
$990$	1.10617e11	3.62326
$991$	−2.32803e10	−0.759856	−0.379928	−	0.925016i	$-0.624051\pi$
$991$	−2.32803e10	−0.759856	−0.379928	+	0.925016i	$0.624051\pi$
$992$	1.07178e9	0.0348589
$993$	−9.21843e10	−2.98768
$994$	1.37017e10	0.442509
$995$	6.13758e10	1.97522
$996$	−4.65895e10	−1.49410
$997$	−5.08802e9	−0.162598	−0.0812991	−	0.996690i	$-0.525907\pi$
$997$	−5.08802e9	−0.162598	−0.0812991	+	0.996690i	$0.525907\pi$
$998$	−1.82631e10	−0.581590
$999$	−5.22876e10	−1.65928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.a.1.1	✓	1
3.2	odd	2		342.8.a.e.1.1		1
4.3	odd	2		304.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.a.1.1	✓	1	1.1	even	1	trivial
304.8.a.a.1.1		1	4.3	odd	2
342.8.a.e.1.1		1	3.2	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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