LMFDB - Embedded newform 8.8.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8,8,Mod(1,8)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	8.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.49908020387$
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	$\chi$	$=$	8.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+44.0000 q^{3} +430.000 q^{5} -1224.00 q^{7} -251.000 q^{9} -3164.00 q^{11} +6118.00 q^{13} +18920.0 q^{15} -16270.0 q^{17} -5476.00 q^{19} -53856.0 q^{21} +1576.00 q^{23} +106775. q^{25} -107272. q^{27} +122838. q^{29} +251360. q^{31} -139216. q^{33} -526320. q^{35} -52338.0 q^{37} +269192. q^{39} -319398. q^{41} +710788. q^{43} -107930. q^{45} +284112. q^{47} +674633. q^{49} -715880. q^{51} +296062. q^{53} -1.36052e6 q^{55} -240944. q^{57} -897548. q^{59} -884810. q^{61} +307224. q^{63} +2.63074e6 q^{65} +4.65969e6 q^{67} +69344.0 q^{69} -2.71079e6 q^{71} -5.67085e6 q^{73} +4.69810e6 q^{75} +3.87274e6 q^{77} -5.12418e6 q^{79} -4.17103e6 q^{81} -1.56356e6 q^{83} -6.99610e6 q^{85} +5.40487e6 q^{87} +1.16057e7 q^{89} -7.48843e6 q^{91} +1.10598e7 q^{93} -2.35468e6 q^{95} +1.09316e7 q^{97} +794164. q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	44.0000	0.940867	0.470434	−	0.882435i	$-0.344098\pi$
$3$	44.0000	0.940867	0.470434	+	0.882435i	$0.344098\pi$
$4$	0	0
$5$	430.000	1.53841	0.769207	−	0.638999i	$-0.220651\pi$
$5$	430.000	1.53841	0.769207	+	0.638999i	$0.220651\pi$
$6$	0	0
$7$	−1224.00	−1.34877	−0.674386	−	0.738379i	$-0.735591\pi$
$7$	−1224.00	−1.34877	−0.674386	+	0.738379i	$0.735591\pi$
$8$	0	0
$9$	−251.000	−0.114769
$10$	0	0
$11$	−3164.00	−0.716741	−0.358370	−	0.933580i	$-0.616667\pi$
$11$	−3164.00	−0.716741	−0.358370	+	0.933580i	$0.616667\pi$
$12$	0	0
$13$	6118.00	0.772339	0.386169	−	0.922428i	$-0.373798\pi$
$13$	6118.00	0.772339	0.386169	+	0.922428i	$0.373798\pi$
$14$	0	0
$15$	18920.0	1.44744
$16$	0	0
$17$	−16270.0	−0.803186	−0.401593	−	0.915818i	$-0.631543\pi$
$17$	−16270.0	−0.803186	−0.401593	+	0.915818i	$0.631543\pi$
$18$	0	0
$19$	−5476.00	−0.183158	−0.0915790	−	0.995798i	$-0.529191\pi$
$19$	−5476.00	−0.183158	−0.0915790	+	0.995798i	$0.529191\pi$
$20$	0	0
$21$	−53856.0	−1.26901
$22$	0	0
$23$	1576.00	0.0270090	0.0135045	−	0.999909i	$-0.495701\pi$
$23$	1576.00	0.0270090	0.0135045	+	0.999909i	$0.495701\pi$
$24$	0	0
$25$	106775.	1.36672
$26$	0	0
$27$	−107272.	−1.04885
$28$	0	0
$29$	122838.	0.935276	0.467638	−	0.883920i	$-0.345105\pi$
$29$	122838.	0.935276	0.467638	+	0.883920i	$0.345105\pi$
$30$	0	0
$31$	251360.	1.51541	0.757705	−	0.652597i	$-0.226321\pi$
$31$	251360.	1.51541	0.757705	+	0.652597i	$0.226321\pi$
$32$	0	0
$33$	−139216.	−0.674358
$34$	0	0
$35$	−526320.	−2.07497
$36$	0	0
$37$	−52338.0	−0.169868	−0.0849339	−	0.996387i	$-0.527068\pi$
$37$	−52338.0	−0.169868	−0.0849339	+	0.996387i	$0.527068\pi$
$38$	0	0
$39$	269192.	0.726668
$40$	0	0
$41$	−319398.	−0.723750	−0.361875	−	0.932227i	$-0.617863\pi$
$41$	−319398.	−0.723750	−0.361875	+	0.932227i	$0.617863\pi$
$42$	0	0
$43$	710788.	1.36333	0.681664	−	0.731665i	$-0.261257\pi$
$43$	710788.	1.36333	0.681664	+	0.731665i	$0.261257\pi$
$44$	0	0
$45$	−107930.	−0.176562
$46$	0	0
$47$	284112.	0.399160	0.199580	−	0.979882i	$-0.436042\pi$
$47$	284112.	0.399160	0.199580	+	0.979882i	$0.436042\pi$
$48$	0	0
$49$	674633.	0.819184
$50$	0	0
$51$	−715880.	−0.755692
$52$	0	0
$53$	296062.	0.273160	0.136580	−	0.990629i	$-0.456389\pi$
$53$	296062.	0.273160	0.136580	+	0.990629i	$0.456389\pi$
$54$	0	0
$55$	−1.36052e6	−1.10264
$56$	0	0
$57$	−240944.	−0.172327
$58$	0	0
$59$	−897548.	−0.568952	−0.284476	−	0.958683i	$-0.591820\pi$
$59$	−897548.	−0.568952	−0.284476	+	0.958683i	$0.591820\pi$
$60$	0	0
$61$	−884810.	−0.499109	−0.249555	−	0.968361i	$-0.580284\pi$
$61$	−884810.	−0.499109	−0.249555	+	0.968361i	$0.580284\pi$
$62$	0	0
$63$	307224.	0.154797
$64$	0	0
$65$	2.63074e6	1.18818
$66$	0	0
$67$	4.65969e6	1.89276	0.946380	−	0.323057i	$-0.104710\pi$
$67$	4.65969e6	1.89276	0.946380	+	0.323057i	$0.104710\pi$
$68$	0	0
$69$	69344.0	0.0254119
$70$	0	0
$71$	−2.71079e6	−0.898860	−0.449430	−	0.893316i	$-0.648373\pi$
$71$	−2.71079e6	−0.898860	−0.449430	+	0.893316i	$0.648373\pi$
$72$	0	0
$73$	−5.67085e6	−1.70615	−0.853077	−	0.521784i	$-0.825267\pi$
$73$	−5.67085e6	−1.70615	−0.853077	+	0.521784i	$0.825267\pi$
$74$	0	0
$75$	4.69810e6	1.28590
$76$	0	0
$77$	3.87274e6	0.966719
$78$	0	0
$79$	−5.12418e6	−1.16931	−0.584654	−	0.811282i	$-0.698770\pi$
$79$	−5.12418e6	−1.16931	−0.584654	+	0.811282i	$0.698770\pi$
$80$	0	0
$81$	−4.17103e6	−0.872059
$82$	0	0
$83$	−1.56356e6	−0.300151	−0.150076	−	0.988675i	$-0.547952\pi$
$83$	−1.56356e6	−0.300151	−0.150076	+	0.988675i	$0.547952\pi$
$84$	0	0
$85$	−6.99610e6	−1.23563
$86$	0	0
$87$	5.40487e6	0.879970
$88$	0	0
$89$	1.16057e7	1.74504	0.872520	−	0.488579i	$-0.162484\pi$
$89$	1.16057e7	1.74504	0.872520	+	0.488579i	$0.162484\pi$
$90$	0	0
$91$	−7.48843e6	−1.04171
$92$	0	0
$93$	1.10598e7	1.42580
$94$	0	0
$95$	−2.35468e6	−0.281773
$96$	0	0
$97$	1.09316e7	1.21614	0.608070	−	0.793884i	$-0.291944\pi$
$97$	1.09316e7	1.21614	0.608070	+	0.793884i	$0.291944\pi$
$98$	0	0
$99$	794164.	0.0822597
$100$	0	0
$101$	−1.47908e7	−1.42846	−0.714230	−	0.699911i	$-0.753223\pi$
$101$	−1.47908e7	−1.42846	−0.714230	+	0.699911i	$0.753223\pi$
$102$	0	0
$103$	8.76105e6	0.789998	0.394999	−	0.918682i	$-0.370745\pi$
$103$	8.76105e6	0.789998	0.394999	+	0.918682i	$0.370745\pi$
$104$	0	0
$105$	−2.31581e7	−1.95227
$106$	0	0
$107$	−4.29510e6	−0.338946	−0.169473	−	0.985535i	$-0.554206\pi$
$107$	−4.29510e6	−0.338946	−0.169473	+	0.985535i	$0.554206\pi$
$108$	0	0
$109$	3.74465e6	0.276961	0.138480	−	0.990365i	$-0.455778\pi$
$109$	3.74465e6	0.276961	0.138480	+	0.990365i	$0.455778\pi$
$110$	0	0
$111$	−2.30287e6	−0.159823
$112$	0	0
$113$	3.90760e6	0.254763	0.127381	−	0.991854i	$-0.459343\pi$
$113$	3.90760e6	0.254763	0.127381	+	0.991854i	$0.459343\pi$
$114$	0	0
$115$	677680.	0.0415511
$116$	0	0
$117$	−1.53562e6	−0.0886406
$118$	0	0
$119$	1.99145e7	1.08331
$120$	0	0
$121$	−9.47628e6	−0.486283
$122$	0	0
$123$	−1.40535e7	−0.680953
$124$	0	0
$125$	1.23195e7	0.564167
$126$	0	0
$127$	4.23320e7	1.83382	0.916909	−	0.399097i	$-0.130676\pi$
$127$	4.23320e7	1.83382	0.916909	+	0.399097i	$0.130676\pi$
$128$	0	0
$129$	3.12747e7	1.28271
$130$	0	0
$131$	−2.72098e7	−1.05749	−0.528745	−	0.848781i	$-0.677337\pi$
$131$	−2.72098e7	−1.05749	−0.528745	+	0.848781i	$0.677337\pi$
$132$	0	0
$133$	6.70262e6	0.247038
$134$	0	0
$135$	−4.61270e7	−1.61357
$136$	0	0
$137$	−5.01235e7	−1.66540	−0.832702	−	0.553721i	$-0.813207\pi$
$137$	−5.01235e7	−1.66540	−0.832702	+	0.553721i	$0.813207\pi$
$138$	0	0
$139$	−1.22073e7	−0.385539	−0.192770	−	0.981244i	$-0.561747\pi$
$139$	−1.22073e7	−0.385539	−0.192770	+	0.981244i	$0.561747\pi$
$140$	0	0
$141$	1.25009e7	0.375557
$142$	0	0
$143$	−1.93574e7	−0.553567
$144$	0	0
$145$	5.28203e7	1.43884
$146$	0	0
$147$	2.96839e7	0.770743
$148$	0	0
$149$	−4.66478e7	−1.15526	−0.577629	−	0.816299i	$-0.696022\pi$
$149$	−4.66478e7	−1.15526	−0.577629	+	0.816299i	$0.696022\pi$
$150$	0	0
$151$	−3.04359e7	−0.719395	−0.359697	−	0.933069i	$-0.617120\pi$
$151$	−3.04359e7	−0.719395	−0.359697	+	0.933069i	$0.617120\pi$
$152$	0	0
$153$	4.08377e6	0.0921810
$154$	0	0
$155$	1.08085e8	2.33133
$156$	0	0
$157$	6.24060e6	0.128700	0.0643498	−	0.997927i	$-0.479503\pi$
$157$	6.24060e6	0.128700	0.0643498	+	0.997927i	$0.479503\pi$
$158$	0	0
$159$	1.30267e7	0.257007
$160$	0	0
$161$	−1.92902e6	−0.0364290
$162$	0	0
$163$	−2.74060e6	−0.0495665	−0.0247833	−	0.999693i	$-0.507890\pi$
$163$	−2.74060e6	−0.0495665	−0.0247833	+	0.999693i	$0.507890\pi$
$164$	0	0
$165$	−5.98629e7	−1.03744
$166$	0	0
$167$	4.27680e7	0.710578	0.355289	−	0.934757i	$-0.384383\pi$
$167$	4.27680e7	0.710578	0.355289	+	0.934757i	$0.384383\pi$
$168$	0	0
$169$	−2.53186e7	−0.403493
$170$	0	0
$171$	1.37448e6	0.0210209
$172$	0	0
$173$	7.21411e7	1.05931	0.529653	−	0.848214i	$-0.322322\pi$
$173$	7.21411e7	1.05931	0.529653	+	0.848214i	$0.322322\pi$
$174$	0	0
$175$	−1.30693e8	−1.84339
$176$	0	0
$177$	−3.94921e7	−0.535308
$178$	0	0
$179$	−7.21119e7	−0.939769	−0.469885	−	0.882728i	$-0.655704\pi$
$179$	−7.21119e7	−0.939769	−0.469885	+	0.882728i	$0.655704\pi$
$180$	0	0
$181$	1.44881e7	0.181608	0.0908041	−	0.995869i	$-0.471056\pi$
$181$	1.44881e7	0.181608	0.0908041	+	0.995869i	$0.471056\pi$
$182$	0	0
$183$	−3.89316e7	−0.469595
$184$	0	0
$185$	−2.25053e7	−0.261327
$186$	0	0
$187$	5.14783e7	0.575676
$188$	0	0
$189$	1.31301e8	1.41466
$190$	0	0
$191$	1.10503e8	1.14751	0.573755	−	0.819027i	$-0.305486\pi$
$191$	1.10503e8	1.14751	0.573755	+	0.819027i	$0.305486\pi$
$192$	0	0
$193$	3.43354e6	0.0343788	0.0171894	−	0.999852i	$-0.494528\pi$
$193$	3.43354e6	0.0343788	0.0171894	+	0.999852i	$0.494528\pi$
$194$	0	0
$195$	1.15753e8	1.11792
$196$	0	0
$197$	−3.77557e7	−0.351844	−0.175922	−	0.984404i	$-0.556291\pi$
$197$	−3.77557e7	−0.351844	−0.175922	+	0.984404i	$0.556291\pi$
$198$	0	0
$199$	−1.37979e8	−1.24116	−0.620580	−	0.784143i	$-0.713103\pi$
$199$	−1.37979e8	−1.24116	−0.620580	+	0.784143i	$0.713103\pi$
$200$	0	0
$201$	2.05026e8	1.78084
$202$	0	0
$203$	−1.50354e8	−1.26147
$204$	0	0
$205$	−1.37341e8	−1.11343
$206$	0	0
$207$	−395576.	−0.00309980
$208$	0	0
$209$	1.73261e7	0.131277
$210$	0	0
$211$	1.05441e8	0.772720	0.386360	−	0.922348i	$-0.373732\pi$
$211$	1.05441e8	0.772720	0.386360	+	0.922348i	$0.373732\pi$
$212$	0	0
$213$	−1.19275e8	−0.845708
$214$	0	0
$215$	3.05639e8	2.09736
$216$	0	0
$217$	−3.07665e8	−2.04394
$218$	0	0
$219$	−2.49518e8	−1.60527
$220$	0	0
$221$	−9.95399e7	−0.620332
$222$	0	0
$223$	1.32420e8	0.799625	0.399812	−	0.916597i	$-0.369075\pi$
$223$	1.32420e8	0.799625	0.399812	+	0.916597i	$0.369075\pi$
$224$	0	0
$225$	−2.68005e7	−0.156857
$226$	0	0
$227$	−2.23501e8	−1.26820	−0.634102	−	0.773249i	$-0.718630\pi$
$227$	−2.23501e8	−1.26820	−0.634102	+	0.773249i	$0.718630\pi$
$228$	0	0
$229$	3.58523e8	1.97284	0.986422	−	0.164228i	$-0.0525132\pi$
$229$	3.58523e8	1.97284	0.986422	+	0.164228i	$0.0525132\pi$
$230$	0	0
$231$	1.70400e8	0.909554
$232$	0	0
$233$	2.05337e8	1.06346	0.531731	−	0.846913i	$-0.321542\pi$
$233$	2.05337e8	1.06346	0.531731	+	0.846913i	$0.321542\pi$
$234$	0	0
$235$	1.22168e8	0.614074
$236$	0	0
$237$	−2.25464e8	−1.10016
$238$	0	0
$239$	−1.27881e8	−0.605917	−0.302959	−	0.953004i	$-0.597974\pi$
$239$	−1.27881e8	−0.605917	−0.302959	+	0.953004i	$0.597974\pi$
$240$	0	0
$241$	2.13448e8	0.982275	0.491138	−	0.871082i	$-0.336581\pi$
$241$	2.13448e8	0.982275	0.491138	+	0.871082i	$0.336581\pi$
$242$	0	0
$243$	5.10785e7	0.228358
$244$	0	0
$245$	2.90092e8	1.26024
$246$	0	0
$247$	−3.35022e7	−0.141460
$248$	0	0
$249$	−6.87965e7	−0.282402
$250$	0	0
$251$	−2.81046e8	−1.12181	−0.560906	−	0.827880i	$-0.689547\pi$
$251$	−2.81046e8	−1.12181	−0.560906	+	0.827880i	$0.689547\pi$
$252$	0	0
$253$	−4.98646e6	−0.0193585
$254$	0	0
$255$	−3.07828e8	−1.16257
$256$	0	0
$257$	−4.11663e8	−1.51278	−0.756389	−	0.654122i	$-0.773038\pi$
$257$	−4.11663e8	−1.51278	−0.756389	+	0.654122i	$0.773038\pi$
$258$	0	0
$259$	6.40617e7	0.229113
$260$	0	0
$261$	−3.08323e7	−0.107341
$262$	0	0
$263$	−1.01818e8	−0.345127	−0.172563	−	0.984998i	$-0.555205\pi$
$263$	−1.01818e8	−0.345127	−0.172563	+	0.984998i	$0.555205\pi$
$264$	0	0
$265$	1.27307e8	0.420233
$266$	0	0
$267$	5.10650e8	1.64185
$268$	0	0
$269$	−5.30511e6	−0.0166173	−0.00830867	−	0.999965i	$-0.502645\pi$
$269$	−5.30511e6	−0.0166173	−0.00830867	+	0.999965i	$0.502645\pi$
$270$	0	0
$271$	2.51186e8	0.766660	0.383330	−	0.923611i	$-0.374777\pi$
$271$	2.51186e8	0.766660	0.383330	+	0.923611i	$0.374777\pi$
$272$	0	0
$273$	−3.29491e8	−0.980109
$274$	0	0
$275$	−3.37836e8	−0.979584
$276$	0	0
$277$	−1.19651e8	−0.338251	−0.169125	−	0.985595i	$-0.554094\pi$
$277$	−1.19651e8	−0.338251	−0.169125	+	0.985595i	$0.554094\pi$
$278$	0	0
$279$	−6.30914e7	−0.173922
$280$	0	0
$281$	6.21696e8	1.67150	0.835749	−	0.549111i	$-0.185034\pi$
$281$	6.21696e8	1.67150	0.835749	+	0.549111i	$0.185034\pi$
$282$	0	0
$283$	3.41726e8	0.896243	0.448122	−	0.893973i	$-0.352093\pi$
$283$	3.41726e8	0.896243	0.448122	+	0.893973i	$0.352093\pi$
$284$	0	0
$285$	−1.03606e8	−0.265111
$286$	0	0
$287$	3.90943e8	0.976173
$288$	0	0
$289$	−1.45626e8	−0.354892
$290$	0	0
$291$	4.80991e8	1.14423
$292$	0	0
$293$	3.71317e6	0.00862398	0.00431199	−	0.999991i	$-0.498627\pi$
$293$	3.71317e6	0.00862398	0.00431199	+	0.999991i	$0.498627\pi$
$294$	0	0
$295$	−3.85946e8	−0.875284
$296$	0	0
$297$	3.39409e8	0.751753
$298$	0	0
$299$	9.64197e6	0.0208601
$300$	0	0
$301$	−8.70005e8	−1.83882
$302$	0	0
$303$	−6.50797e8	−1.34399
$304$	0	0
$305$	−3.80468e8	−0.767837
$306$	0	0
$307$	−3.23276e7	−0.0637659	−0.0318830	−	0.999492i	$-0.510150\pi$
$307$	−3.23276e7	−0.0637659	−0.0318830	+	0.999492i	$0.510150\pi$
$308$	0	0
$309$	3.85486e8	0.743283
$310$	0	0
$311$	−8.26997e8	−1.55899	−0.779494	−	0.626410i	$-0.784524\pi$
$311$	−8.26997e8	−1.55899	−0.779494	+	0.626410i	$0.784524\pi$
$312$	0	0
$313$	3.53027e8	0.650732	0.325366	−	0.945588i	$-0.394513\pi$
$313$	3.53027e8	0.650732	0.325366	+	0.945588i	$0.394513\pi$
$314$	0	0
$315$	1.32106e8	0.238142
$316$	0	0
$317$	9.30187e8	1.64007	0.820036	−	0.572312i	$-0.193953\pi$
$317$	9.30187e8	1.64007	0.820036	+	0.572312i	$0.193953\pi$
$318$	0	0
$319$	−3.88659e8	−0.670350
$320$	0	0
$321$	−1.88984e8	−0.318903
$322$	0	0
$323$	8.90945e7	0.147110
$324$	0	0
$325$	6.53249e8	1.05557
$326$	0	0
$327$	1.64764e8	0.260583
$328$	0	0
$329$	−3.47753e8	−0.538376
$330$	0	0
$331$	−3.21754e8	−0.487670	−0.243835	−	0.969817i	$-0.578406\pi$
$331$	−3.21754e8	−0.487670	−0.243835	+	0.969817i	$0.578406\pi$
$332$	0	0
$333$	1.31368e7	0.0194956
$334$	0	0
$335$	2.00367e9	2.91185
$336$	0	0
$337$	−2.47705e8	−0.352558	−0.176279	−	0.984340i	$-0.556406\pi$
$337$	−2.47705e8	−0.352558	−0.176279	+	0.984340i	$0.556406\pi$
$338$	0	0
$339$	1.71934e8	0.239698
$340$	0	0
$341$	−7.95303e8	−1.08616
$342$	0	0
$343$	1.82266e8	0.243880
$344$	0	0
$345$	2.98179e7	0.0390940
$346$	0	0
$347$	−2.23076e8	−0.286615	−0.143308	−	0.989678i	$-0.545774\pi$
$347$	−2.23076e8	−0.286615	−0.143308	+	0.989678i	$0.545774\pi$
$348$	0	0
$349$	−5.57087e8	−0.701510	−0.350755	−	0.936467i	$-0.614075\pi$
$349$	−5.57087e8	−0.701510	−0.350755	+	0.936467i	$0.614075\pi$
$350$	0	0
$351$	−6.56290e8	−0.810067
$352$	0	0
$353$	5.64509e8	0.683061	0.341530	−	0.939871i	$-0.389055\pi$
$353$	5.64509e8	0.683061	0.341530	+	0.939871i	$0.389055\pi$
$354$	0	0
$355$	−1.16564e9	−1.38282
$356$	0	0
$357$	8.76237e8	1.01926
$358$	0	0
$359$	1.66005e9	1.89361	0.946803	−	0.321813i	$-0.104292\pi$
$359$	1.66005e9	1.89361	0.946803	+	0.321813i	$0.104292\pi$
$360$	0	0
$361$	−8.63885e8	−0.966453
$362$	0	0
$363$	−4.16956e8	−0.457527
$364$	0	0
$365$	−2.43847e9	−2.62477
$366$	0	0
$367$	−7.00412e8	−0.739643	−0.369822	−	0.929103i	$-0.620581\pi$
$367$	−7.00412e8	−0.739643	−0.369822	+	0.929103i	$0.620581\pi$
$368$	0	0
$369$	8.01689e7	0.0830642
$370$	0	0
$371$	−3.62380e8	−0.368430
$372$	0	0
$373$	4.33445e8	0.432467	0.216233	−	0.976342i	$-0.430623\pi$
$373$	4.33445e8	0.432467	0.216233	+	0.976342i	$0.430623\pi$
$374$	0	0
$375$	5.42058e8	0.530807
$376$	0	0
$377$	7.51523e8	0.722350
$378$	0	0
$379$	1.38421e9	1.30606	0.653032	−	0.757330i	$-0.273496\pi$
$379$	1.38421e9	1.30606	0.653032	+	0.757330i	$0.273496\pi$
$380$	0	0
$381$	1.86261e9	1.72538
$382$	0	0
$383$	−4.70821e8	−0.428213	−0.214107	−	0.976810i	$-0.568684\pi$
$383$	−4.70821e8	−0.428213	−0.214107	+	0.976810i	$0.568684\pi$
$384$	0	0
$385$	1.66528e9	1.48722
$386$	0	0
$387$	−1.78408e8	−0.156468
$388$	0	0
$389$	1.52817e7	0.0131628	0.00658140	−	0.999978i	$-0.497905\pi$
$389$	1.52817e7	0.0131628	0.00658140	+	0.999978i	$0.497905\pi$
$390$	0	0
$391$	−2.56415e7	−0.0216933
$392$	0	0
$393$	−1.19723e9	−0.994957
$394$	0	0
$395$	−2.20340e9	−1.79888
$396$	0	0
$397$	1.02601e9	0.822969	0.411485	−	0.911417i	$-0.365010\pi$
$397$	1.02601e9	0.822969	0.411485	+	0.911417i	$0.365010\pi$
$398$	0	0
$399$	2.94915e8	0.232430
$400$	0	0
$401$	−2.18349e8	−0.169101	−0.0845504	−	0.996419i	$-0.526945\pi$
$401$	−2.18349e8	−0.169101	−0.0845504	+	0.996419i	$0.526945\pi$
$402$	0	0
$403$	1.53782e9	1.17041
$404$	0	0
$405$	−1.79354e9	−1.34159
$406$	0	0
$407$	1.65597e8	0.121751
$408$	0	0
$409$	7.50779e8	0.542601	0.271300	−	0.962495i	$-0.412546\pi$
$409$	7.50779e8	0.542601	0.271300	+	0.962495i	$0.412546\pi$
$410$	0	0
$411$	−2.20544e9	−1.56692
$412$	0	0
$413$	1.09860e9	0.767386
$414$	0	0
$415$	−6.72329e8	−0.461757
$416$	0	0
$417$	−5.37122e8	−0.362741
$418$	0	0
$419$	2.38283e8	0.158250	0.0791251	−	0.996865i	$-0.474787\pi$
$419$	2.38283e8	0.158250	0.0791251	+	0.996865i	$0.474787\pi$
$420$	0	0
$421$	−1.23640e9	−0.807552	−0.403776	−	0.914858i	$-0.632303\pi$
$421$	−1.23640e9	−0.807552	−0.403776	+	0.914858i	$0.632303\pi$
$422$	0	0
$423$	−7.13121e7	−0.0458112
$424$	0	0
$425$	−1.73723e9	−1.09773
$426$	0	0
$427$	1.08301e9	0.673184
$428$	0	0
$429$	−8.51723e8	−0.520833
$430$	0	0
$431$	2.24030e9	1.34783	0.673915	−	0.738809i	$-0.264611\pi$
$431$	2.24030e9	1.34783	0.673915	+	0.738809i	$0.264611\pi$
$432$	0	0
$433$	−1.19040e9	−0.704670	−0.352335	−	0.935874i	$-0.614612\pi$
$433$	−1.19040e9	−0.704670	−0.352335	+	0.935874i	$0.614612\pi$
$434$	0	0
$435$	2.32409e9	1.35376
$436$	0	0
$437$	−8.63018e6	−0.00494692
$438$	0	0
$439$	−1.38822e9	−0.783127	−0.391564	−	0.920151i	$-0.628066\pi$
$439$	−1.38822e9	−0.783127	−0.391564	+	0.920151i	$0.628066\pi$
$440$	0	0
$441$	−1.69333e8	−0.0940170
$442$	0	0
$443$	2.85788e9	1.56182	0.780911	−	0.624642i	$-0.214755\pi$
$443$	2.85788e9	1.56182	0.780911	+	0.624642i	$0.214755\pi$
$444$	0	0
$445$	4.99044e9	2.68460
$446$	0	0
$447$	−2.05250e9	−1.08694
$448$	0	0
$449$	2.96866e7	0.0154774	0.00773870	−	0.999970i	$-0.497537\pi$
$449$	2.96866e7	0.0154774	0.00773870	+	0.999970i	$0.497537\pi$
$450$	0	0
$451$	1.01058e9	0.518741
$452$	0	0
$453$	−1.33918e9	−0.676855
$454$	0	0
$455$	−3.22003e9	−1.60258
$456$	0	0
$457$	1.83015e9	0.896974	0.448487	−	0.893789i	$-0.351963\pi$
$457$	1.83015e9	0.896974	0.448487	+	0.893789i	$0.351963\pi$
$458$	0	0
$459$	1.74532e9	0.842422
$460$	0	0
$461$	−8.13408e8	−0.386683	−0.193342	−	0.981132i	$-0.561933\pi$
$461$	−8.13408e8	−0.386683	−0.193342	+	0.981132i	$0.561933\pi$
$462$	0	0
$463$	−1.01303e9	−0.474337	−0.237168	−	0.971469i	$-0.576219\pi$
$463$	−1.01303e9	−0.474337	−0.237168	+	0.971469i	$0.576219\pi$
$464$	0	0
$465$	4.75573e9	2.19347
$466$	0	0
$467$	−3.80251e9	−1.72767	−0.863837	−	0.503771i	$-0.831945\pi$
$467$	−3.80251e9	−1.72767	−0.863837	+	0.503771i	$0.831945\pi$
$468$	0	0
$469$	−5.70346e9	−2.55290
$470$	0	0
$471$	2.74586e8	0.121089
$472$	0	0
$473$	−2.24893e9	−0.977153
$474$	0	0
$475$	−5.84700e8	−0.250326
$476$	0	0
$477$	−7.43116e7	−0.0313503
$478$	0	0
$479$	−8.67385e8	−0.360610	−0.180305	−	0.983611i	$-0.557708\pi$
$479$	−8.67385e8	−0.360610	−0.180305	+	0.983611i	$0.557708\pi$
$480$	0	0
$481$	−3.20204e8	−0.131195
$482$	0	0
$483$	−8.48771e7	−0.0342748
$484$	0	0
$485$	4.70060e9	1.87093
$486$	0	0
$487$	1.30043e9	0.510195	0.255098	−	0.966915i	$-0.417892\pi$
$487$	1.30043e9	0.510195	0.255098	+	0.966915i	$0.417892\pi$
$488$	0	0
$489$	−1.20586e8	−0.0466355
$490$	0	0
$491$	1.88652e9	0.719244	0.359622	−	0.933098i	$-0.382906\pi$
$491$	1.88652e9	0.719244	0.359622	+	0.933098i	$0.382906\pi$
$492$	0	0
$493$	−1.99857e9	−0.751201
$494$	0	0
$495$	3.41491e8	0.126550
$496$	0	0
$497$	3.31801e9	1.21236
$498$	0	0
$499$	−3.81787e9	−1.37553	−0.687763	−	0.725935i	$-0.741407\pi$
$499$	−3.81787e9	−1.37553	−0.687763	+	0.725935i	$0.741407\pi$
$500$	0	0
$501$	1.88179e9	0.668559
$502$	0	0
$503$	−4.47974e8	−0.156951	−0.0784757	−	0.996916i	$-0.525005\pi$
$503$	−4.47974e8	−0.156951	−0.0784757	+	0.996916i	$0.525005\pi$
$504$	0	0
$505$	−6.36006e9	−2.19756
$506$	0	0
$507$	−1.11402e9	−0.379633
$508$	0	0
$509$	4.50519e9	1.51426	0.757131	−	0.653263i	$-0.226600\pi$
$509$	4.50519e9	1.51426	0.757131	+	0.653263i	$0.226600\pi$
$510$	0	0
$511$	6.94113e9	2.30121
$512$	0	0
$513$	5.87421e8	0.192105
$514$	0	0
$515$	3.76725e9	1.21534
$516$	0	0
$517$	−8.98930e8	−0.286094
$518$	0	0
$519$	3.17421e9	0.996666
$520$	0	0
$521$	4.27301e8	0.132374	0.0661868	−	0.997807i	$-0.478917\pi$
$521$	4.27301e8	0.132374	0.0661868	+	0.997807i	$0.478917\pi$
$522$	0	0
$523$	8.14705e8	0.249026	0.124513	−	0.992218i	$-0.460263\pi$
$523$	8.14705e8	0.249026	0.124513	+	0.992218i	$0.460263\pi$
$524$	0	0
$525$	−5.75047e9	−1.73439
$526$	0	0
$527$	−4.08963e9	−1.21716
$528$	0	0
$529$	−3.40234e9	−0.999271
$530$	0	0
$531$	2.25285e8	0.0652981
$532$	0	0
$533$	−1.95408e9	−0.558980
$534$	0	0
$535$	−1.84689e9	−0.521439
$536$	0	0
$537$	−3.17292e9	−0.884198
$538$	0	0
$539$	−2.13454e9	−0.587142
$540$	0	0
$541$	5.17489e9	1.40511	0.702556	−	0.711629i	$-0.252042\pi$
$541$	5.17489e9	1.40511	0.702556	+	0.711629i	$0.252042\pi$
$542$	0	0
$543$	6.37475e8	0.170869
$544$	0	0
$545$	1.61020e9	0.426080
$546$	0	0
$547$	−6.45830e9	−1.68718	−0.843592	−	0.536984i	$-0.819564\pi$
$547$	−6.45830e9	−1.68718	−0.843592	+	0.536984i	$0.819564\pi$
$548$	0	0
$549$	2.22087e8	0.0572823
$550$	0	0
$551$	−6.72661e8	−0.171303
$552$	0	0
$553$	6.27199e9	1.57713
$554$	0	0
$555$	−9.90235e8	−0.245874
$556$	0	0
$557$	−4.83562e9	−1.18566	−0.592829	−	0.805328i	$-0.701989\pi$
$557$	−4.83562e9	−1.18566	−0.592829	+	0.805328i	$0.701989\pi$
$558$	0	0
$559$	4.34860e9	1.05295
$560$	0	0
$561$	2.26504e9	0.541635
$562$	0	0
$563$	1.16508e9	0.275154	0.137577	−	0.990491i	$-0.456069\pi$
$563$	1.16508e9	0.275154	0.137577	+	0.990491i	$0.456069\pi$
$564$	0	0
$565$	1.68027e9	0.391931
$566$	0	0
$567$	5.10534e9	1.17621
$568$	0	0
$569$	2.79201e9	0.635367	0.317683	−	0.948197i	$-0.397095\pi$
$569$	2.79201e9	0.635367	0.317683	+	0.948197i	$0.397095\pi$
$570$	0	0
$571$	−7.14069e9	−1.60514	−0.802572	−	0.596555i	$-0.796536\pi$
$571$	−7.14069e9	−1.60514	−0.802572	+	0.596555i	$0.796536\pi$
$572$	0	0
$573$	4.86212e9	1.07965
$574$	0	0
$575$	1.68277e8	0.0369138
$576$	0	0
$577$	4.44897e7	0.00964148	0.00482074	−	0.999988i	$-0.498466\pi$
$577$	4.44897e7	0.00964148	0.00482074	+	0.999988i	$0.498466\pi$
$578$	0	0
$579$	1.51076e8	0.0323459
$580$	0	0
$581$	1.91379e9	0.404835
$582$	0	0
$583$	−9.36740e8	−0.195785
$584$	0	0
$585$	−6.60316e8	−0.136366
$586$	0	0
$587$	−4.21422e8	−0.0859972	−0.0429986	−	0.999075i	$-0.513691\pi$
$587$	−4.21422e8	−0.0859972	−0.0429986	+	0.999075i	$0.513691\pi$
$588$	0	0
$589$	−1.37645e9	−0.277559
$590$	0	0
$591$	−1.66125e9	−0.331039
$592$	0	0
$593$	−4.60547e9	−0.906948	−0.453474	−	0.891270i	$-0.649815\pi$
$593$	−4.60547e9	−0.906948	−0.453474	+	0.891270i	$0.649815\pi$
$594$	0	0
$595$	8.56323e9	1.66659
$596$	0	0
$597$	−6.07109e9	−1.16777
$598$	0	0
$599$	−5.96438e8	−0.113389	−0.0566946	−	0.998392i	$-0.518056\pi$
$599$	−5.96438e8	−0.113389	−0.0566946	+	0.998392i	$0.518056\pi$
$600$	0	0
$601$	2.18925e9	0.411372	0.205686	−	0.978618i	$-0.434058\pi$
$601$	2.18925e9	0.411372	0.205686	+	0.978618i	$0.434058\pi$
$602$	0	0
$603$	−1.16958e9	−0.217230
$604$	0	0
$605$	−4.07480e9	−0.748105
$606$	0	0
$607$	9.88184e8	0.179340	0.0896701	−	0.995972i	$-0.471419\pi$
$607$	9.88184e8	0.179340	0.0896701	+	0.995972i	$0.471419\pi$
$608$	0	0
$609$	−6.61556e9	−1.18688
$610$	0	0
$611$	1.73820e9	0.308287
$612$	0	0
$613$	3.18993e8	0.0559332	0.0279666	−	0.999609i	$-0.491097\pi$
$613$	3.18993e8	0.0559332	0.0279666	+	0.999609i	$0.491097\pi$
$614$	0	0
$615$	−6.04301e9	−1.04759
$616$	0	0
$617$	3.83798e9	0.657816	0.328908	−	0.944362i	$-0.393319\pi$
$617$	3.83798e9	0.657816	0.328908	+	0.944362i	$0.393319\pi$
$618$	0	0
$619$	2.96834e9	0.503032	0.251516	−	0.967853i	$-0.419071\pi$
$619$	2.96834e9	0.503032	0.251516	+	0.967853i	$0.419071\pi$
$620$	0	0
$621$	−1.69061e8	−0.0283284
$622$	0	0
$623$	−1.42053e10	−2.35366
$624$	0	0
$625$	−3.04441e9	−0.498796
$626$	0	0
$627$	7.62347e8	0.123514
$628$	0	0
$629$	8.51539e8	0.136436
$630$	0	0
$631$	8.10788e9	1.28471	0.642354	−	0.766408i	$-0.277958\pi$
$631$	8.10788e9	1.28471	0.642354	+	0.766408i	$0.277958\pi$
$632$	0	0
$633$	4.63941e9	0.727026
$634$	0	0
$635$	1.82028e10	2.82117
$636$	0	0
$637$	4.12740e9	0.632687
$638$	0	0
$639$	6.80409e8	0.103161
$640$	0	0
$641$	−1.10357e10	−1.65499	−0.827495	−	0.561473i	$-0.810235\pi$
$641$	−1.10357e10	−1.65499	−0.827495	+	0.561473i	$0.810235\pi$
$642$	0	0
$643$	6.76532e9	1.00358	0.501788	−	0.864991i	$-0.332676\pi$
$643$	6.76532e9	1.00358	0.501788	+	0.864991i	$0.332676\pi$
$644$	0	0
$645$	1.34481e10	1.97334
$646$	0	0
$647$	6.76242e9	0.981606	0.490803	−	0.871271i	$-0.336703\pi$
$647$	6.76242e9	0.981606	0.490803	+	0.871271i	$0.336703\pi$
$648$	0	0
$649$	2.83984e9	0.407791
$650$	0	0
$651$	−1.35372e10	−1.92308
$652$	0	0
$653$	−9.17211e9	−1.28906	−0.644530	−	0.764579i	$-0.722947\pi$
$653$	−9.17211e9	−1.28906	−0.644530	+	0.764579i	$0.722947\pi$
$654$	0	0
$655$	−1.17002e10	−1.62686
$656$	0	0
$657$	1.42338e9	0.195814
$658$	0	0
$659$	−5.62299e9	−0.765364	−0.382682	−	0.923880i	$-0.625000\pi$
$659$	−5.62299e9	−0.765364	−0.382682	+	0.923880i	$0.625000\pi$
$660$	0	0
$661$	3.39966e9	0.457857	0.228928	−	0.973443i	$-0.426478\pi$
$661$	3.39966e9	0.457857	0.228928	+	0.973443i	$0.426478\pi$
$662$	0	0
$663$	−4.37975e9	−0.583650
$664$	0	0
$665$	2.88213e9	0.380047
$666$	0	0
$667$	1.93593e8	0.0252609
$668$	0	0
$669$	5.82647e9	0.752340
$670$	0	0
$671$	2.79954e9	0.357732
$672$	0	0
$673$	1.14073e10	1.44254	0.721271	−	0.692653i	$-0.243558\pi$
$673$	1.14073e10	1.44254	0.721271	+	0.692653i	$0.243558\pi$
$674$	0	0
$675$	−1.14540e10	−1.43348
$676$	0	0
$677$	3.10086e9	0.384080	0.192040	−	0.981387i	$-0.438490\pi$
$677$	3.10086e9	0.384080	0.192040	+	0.981387i	$0.438490\pi$
$678$	0	0
$679$	−1.33803e10	−1.64029
$680$	0	0
$681$	−9.83405e9	−1.19321
$682$	0	0
$683$	6.43067e9	0.772296	0.386148	−	0.922437i	$-0.373805\pi$
$683$	6.43067e9	0.772296	0.386148	+	0.922437i	$0.373805\pi$
$684$	0	0
$685$	−2.15531e10	−2.56208
$686$	0	0
$687$	1.57750e10	1.85618
$688$	0	0
$689$	1.81131e9	0.210972
$690$	0	0
$691$	−4.53885e9	−0.523326	−0.261663	−	0.965159i	$-0.584271\pi$
$691$	−4.53885e9	−0.523326	−0.261663	+	0.965159i	$0.584271\pi$
$692$	0	0
$693$	−9.72057e8	−0.110949
$694$	0	0
$695$	−5.24915e9	−0.593119
$696$	0	0
$697$	5.19661e9	0.581306
$698$	0	0
$699$	9.03484e9	1.00058
$700$	0	0
$701$	−7.84396e9	−0.860047	−0.430024	−	0.902818i	$-0.641495\pi$
$701$	−7.84396e9	−0.860047	−0.430024	+	0.902818i	$0.641495\pi$
$702$	0	0
$703$	2.86603e8	0.0311126
$704$	0	0
$705$	5.37540e9	0.577762
$706$	0	0
$707$	1.81040e10	1.92666
$708$	0	0
$709$	4.05387e9	0.427177	0.213588	−	0.976924i	$-0.431485\pi$
$709$	4.05387e9	0.427177	0.213588	+	0.976924i	$0.431485\pi$
$710$	0	0
$711$	1.28617e9	0.134201
$712$	0	0
$713$	3.96143e8	0.0409297
$714$	0	0
$715$	−8.32366e9	−0.851615
$716$	0	0
$717$	−5.62676e9	−0.570087
$718$	0	0
$719$	7.84938e9	0.787561	0.393780	−	0.919205i	$-0.371167\pi$
$719$	7.84938e9	0.787561	0.393780	+	0.919205i	$0.371167\pi$
$720$	0	0
$721$	−1.07235e10	−1.06553
$722$	0	0
$723$	9.39173e9	0.924190
$724$	0	0
$725$	1.31160e10	1.27826
$726$	0	0
$727$	6.44318e9	0.621913	0.310957	−	0.950424i	$-0.399351\pi$
$727$	6.44318e9	0.621913	0.310957	+	0.950424i	$0.399351\pi$
$728$	0	0
$729$	1.13695e10	1.08691
$730$	0	0
$731$	−1.15645e10	−1.09501
$732$	0	0
$733$	9.48714e9	0.889757	0.444879	−	0.895591i	$-0.353247\pi$
$733$	9.48714e9	0.889757	0.444879	+	0.895591i	$0.353247\pi$
$734$	0	0
$735$	1.27641e10	1.18572
$736$	0	0
$737$	−1.47433e10	−1.35662
$738$	0	0
$739$	−2.07907e10	−1.89502	−0.947508	−	0.319733i	$-0.896407\pi$
$739$	−2.07907e10	−1.89502	−0.947508	+	0.319733i	$0.896407\pi$
$740$	0	0
$741$	−1.47410e9	−0.133095
$742$	0	0
$743$	−5.87068e9	−0.525083	−0.262541	−	0.964921i	$-0.584561\pi$
$743$	−5.87068e9	−0.525083	−0.262541	+	0.964921i	$0.584561\pi$
$744$	0	0
$745$	−2.00585e10	−1.77727
$746$	0	0
$747$	3.92453e8	0.0344481
$748$	0	0
$749$	5.25720e9	0.457160
$750$	0	0
$751$	7.08413e9	0.610305	0.305152	−	0.952304i	$-0.401293\pi$
$751$	7.08413e9	0.610305	0.305152	+	0.952304i	$0.401293\pi$
$752$	0	0
$753$	−1.23660e10	−1.05548
$754$	0	0
$755$	−1.30874e10	−1.10673
$756$	0	0
$757$	−1.12028e10	−0.938622	−0.469311	−	0.883033i	$-0.655498\pi$
$757$	−1.12028e10	−0.938622	−0.469311	+	0.883033i	$0.655498\pi$
$758$	0	0
$759$	−2.19404e8	−0.0182137
$760$	0	0
$761$	2.08007e10	1.71093	0.855465	−	0.517861i	$-0.173272\pi$
$761$	2.08007e10	1.71093	0.855465	+	0.517861i	$0.173272\pi$
$762$	0	0
$763$	−4.58345e9	−0.373557
$764$	0	0
$765$	1.75602e9	0.141813
$766$	0	0
$767$	−5.49120e9	−0.439424
$768$	0	0
$769$	1.94604e10	1.54315	0.771576	−	0.636137i	$-0.219469\pi$
$769$	1.94604e10	1.54315	0.771576	+	0.636137i	$0.219469\pi$
$770$	0	0
$771$	−1.81132e10	−1.42332
$772$	0	0
$773$	1.81342e7	0.00141211	0.000706057	−	1.00000i	$-0.499775\pi$
$773$	1.81342e7	0.00141211	0.000706057		1.00000i	$0.499775\pi$
$774$	0	0
$775$	2.68390e10	2.07114
$776$	0	0
$777$	2.81872e9	0.215565
$778$	0	0
$779$	1.74902e9	0.132561
$780$	0	0
$781$	8.57695e9	0.644249
$782$	0	0
$783$	−1.31771e10	−0.980964
$784$	0	0
$785$	2.68346e9	0.197993
$786$	0	0
$787$	3.48980e9	0.255205	0.127603	−	0.991825i	$-0.459272\pi$
$787$	3.48980e9	0.255205	0.127603	+	0.991825i	$0.459272\pi$
$788$	0	0
$789$	−4.47998e9	−0.324718
$790$	0	0
$791$	−4.78290e9	−0.343617
$792$	0	0
$793$	−5.41327e9	−0.385481
$794$	0	0
$795$	5.60149e9	0.395384
$796$	0	0
$797$	9.60101e9	0.671758	0.335879	−	0.941905i	$-0.390967\pi$
$797$	9.60101e9	0.671758	0.335879	+	0.941905i	$0.390967\pi$
$798$	0	0
$799$	−4.62250e9	−0.320600
$800$	0	0
$801$	−2.91302e9	−0.200277
$802$	0	0
$803$	1.79426e10	1.22287
$804$	0	0
$805$	−8.29480e8	−0.0560429
$806$	0	0
$807$	−2.33425e8	−0.0156347
$808$	0	0
$809$	1.28054e10	0.850305	0.425152	−	0.905122i	$-0.360221\pi$
$809$	1.28054e10	0.850305	0.425152	+	0.905122i	$0.360221\pi$
$810$	0	0
$811$	1.41860e10	0.933871	0.466936	−	0.884291i	$-0.345358\pi$
$811$	1.41860e10	0.933871	0.466936	+	0.884291i	$0.345358\pi$
$812$	0	0
$813$	1.10522e10	0.721325
$814$	0	0
$815$	−1.17846e9	−0.0762539
$816$	0	0
$817$	−3.89228e9	−0.249704
$818$	0	0
$819$	1.87960e9	0.119556
$820$	0	0
$821$	−1.88610e10	−1.18950	−0.594749	−	0.803912i	$-0.702749\pi$
$821$	−1.88610e10	−1.18950	−0.594749	+	0.803912i	$0.702749\pi$
$822$	0	0
$823$	−7.98216e9	−0.499138	−0.249569	−	0.968357i	$-0.580289\pi$
$823$	−7.98216e9	−0.499138	−0.249569	+	0.968357i	$0.580289\pi$
$824$	0	0
$825$	−1.48648e10	−0.921658
$826$	0	0
$827$	−1.69379e10	−1.04134	−0.520668	−	0.853759i	$-0.674317\pi$
$827$	−1.69379e10	−1.04134	−0.520668	+	0.853759i	$0.674317\pi$
$828$	0	0
$829$	−1.93636e10	−1.18044	−0.590221	−	0.807242i	$-0.700960\pi$
$829$	−1.93636e10	−1.18044	−0.590221	+	0.807242i	$0.700960\pi$
$830$	0	0
$831$	−5.26466e9	−0.318249
$832$	0	0
$833$	−1.09763e10	−0.657957
$834$	0	0
$835$	1.83903e10	1.09316
$836$	0	0
$837$	−2.69639e10	−1.58944
$838$	0	0
$839$	−9.16289e9	−0.535631	−0.267815	−	0.963470i	$-0.586302\pi$
$839$	−9.16289e9	−0.535631	−0.267815	+	0.963470i	$0.586302\pi$
$840$	0	0
$841$	−2.16070e9	−0.125259
$842$	0	0
$843$	2.73546e10	1.57266
$844$	0	0
$845$	−1.08870e10	−0.620740
$846$	0	0
$847$	1.15990e10	0.655884
$848$	0	0
$849$	1.50360e10	0.843246
$850$	0	0
$851$	−8.24847e7	−0.00458796
$852$	0	0
$853$	2.16335e10	1.19345	0.596727	−	0.802445i	$-0.296468\pi$
$853$	2.16335e10	1.19345	0.596727	+	0.802445i	$0.296468\pi$
$854$	0	0
$855$	5.91025e8	0.0323388
$856$	0	0
$857$	−1.34574e10	−0.730345	−0.365172	−	0.930940i	$-0.618990\pi$
$857$	−1.34574e10	−0.730345	−0.365172	+	0.930940i	$0.618990\pi$
$858$	0	0
$859$	−2.10347e10	−1.13230	−0.566148	−	0.824304i	$-0.691567\pi$
$859$	−2.10347e10	−1.13230	−0.566148	+	0.824304i	$0.691567\pi$
$860$	0	0
$861$	1.72015e10	0.918449
$862$	0	0
$863$	−3.15666e10	−1.67182	−0.835912	−	0.548863i	$-0.815061\pi$
$863$	−3.15666e10	−1.67182	−0.835912	+	0.548863i	$0.815061\pi$
$864$	0	0
$865$	3.10207e10	1.62965
$866$	0	0
$867$	−6.40753e9	−0.333906
$868$	0	0
$869$	1.62129e10	0.838091
$870$	0	0
$871$	2.85080e10	1.46185
$872$	0	0
$873$	−2.74384e9	−0.139575
$874$	0	0
$875$	−1.50791e10	−0.760933
$876$	0	0
$877$	3.09323e10	1.54851	0.774254	−	0.632875i	$-0.218125\pi$
$877$	3.09323e10	1.54851	0.774254	+	0.632875i	$0.218125\pi$
$878$	0	0
$879$	1.63379e8	0.00811402
$880$	0	0
$881$	2.59947e10	1.28077	0.640383	−	0.768056i	$-0.278776\pi$
$881$	2.59947e10	1.28077	0.640383	+	0.768056i	$0.278776\pi$
$882$	0	0
$883$	2.77295e10	1.35544	0.677718	−	0.735322i	$-0.262969\pi$
$883$	2.77295e10	1.35544	0.677718	+	0.735322i	$0.262969\pi$
$884$	0	0
$885$	−1.69816e10	−0.823526
$886$	0	0
$887$	−6.18068e9	−0.297374	−0.148687	−	0.988884i	$-0.547505\pi$
$887$	−6.18068e9	−0.297374	−0.148687	+	0.988884i	$0.547505\pi$
$888$	0	0
$889$	−5.18144e10	−2.47340
$890$	0	0
$891$	1.31971e10	0.625040
$892$	0	0
$893$	−1.55580e9	−0.0731093
$894$	0	0
$895$	−3.10081e10	−1.44575
$896$	0	0
$897$	4.24247e8	0.0196266
$898$	0	0
$899$	3.08766e10	1.41733
$900$	0	0
$901$	−4.81693e9	−0.219398
$902$	0	0
$903$	−3.82802e10	−1.73008
$904$	0	0
$905$	6.22987e9	0.279389
$906$	0	0
$907$	−9.47637e9	−0.421713	−0.210856	−	0.977517i	$-0.567625\pi$
$907$	−9.47637e9	−0.421713	−0.210856	+	0.977517i	$0.567625\pi$
$908$	0	0
$909$	3.71250e9	0.163943
$910$	0	0
$911$	2.63833e10	1.15615	0.578075	−	0.815983i	$-0.303804\pi$
$911$	2.63833e10	1.15615	0.578075	+	0.815983i	$0.303804\pi$
$912$	0	0
$913$	4.94709e9	0.215131
$914$	0	0
$915$	−1.67406e10	−0.722433
$916$	0	0
$917$	3.33048e10	1.42631
$918$	0	0
$919$	−2.34537e10	−0.996797	−0.498398	−	0.866948i	$-0.666078\pi$
$919$	−2.34537e10	−0.996797	−0.498398	+	0.866948i	$0.666078\pi$
$920$	0	0
$921$	−1.42241e9	−0.0599952
$922$	0	0
$923$	−1.65846e10	−0.694224
$924$	0	0
$925$	−5.58839e9	−0.232162
$926$	0	0
$927$	−2.19902e9	−0.0906673
$928$	0	0
$929$	−4.04287e10	−1.65438	−0.827189	−	0.561924i	$-0.810061\pi$
$929$	−4.04287e10	−1.65438	−0.827189	+	0.561924i	$0.810061\pi$
$930$	0	0
$931$	−3.69429e9	−0.150040
$932$	0	0
$933$	−3.63879e10	−1.46680
$934$	0	0
$935$	2.21357e10	0.885629
$936$	0	0
$937$	−1.09610e10	−0.435273	−0.217636	−	0.976030i	$-0.569835\pi$
$937$	−1.09610e10	−0.435273	−0.217636	+	0.976030i	$0.569835\pi$
$938$	0	0
$939$	1.55332e10	0.612252
$940$	0	0
$941$	−1.87938e9	−0.0735276	−0.0367638	−	0.999324i	$-0.511705\pi$
$941$	−1.87938e9	−0.0735276	−0.0367638	+	0.999324i	$0.511705\pi$
$942$	0	0
$943$	−5.03371e8	−0.0195478
$944$	0	0
$945$	5.64594e10	2.17633
$946$	0	0
$947$	−1.97695e10	−0.756435	−0.378218	−	0.925717i	$-0.623463\pi$
$947$	−1.97695e10	−0.756435	−0.378218	+	0.925717i	$0.623463\pi$
$948$	0	0
$949$	−3.46943e10	−1.31773
$950$	0	0
$951$	4.09282e10	1.54309
$952$	0	0
$953$	2.81049e9	0.105186	0.0525929	−	0.998616i	$-0.483251\pi$
$953$	2.81049e9	0.105186	0.0525929	+	0.998616i	$0.483251\pi$
$954$	0	0
$955$	4.75162e10	1.76535
$956$	0	0
$957$	−1.71010e10	−0.630711
$958$	0	0
$959$	6.13512e10	2.24625
$960$	0	0
$961$	3.56692e10	1.29647
$962$	0	0
$963$	1.07807e9	0.0389005
$964$	0	0
$965$	1.47642e9	0.0528889
$966$	0	0
$967$	−2.06586e10	−0.734697	−0.367348	−	0.930083i	$-0.619734\pi$
$967$	−2.06586e10	−0.734697	−0.367348	+	0.930083i	$0.619734\pi$
$968$	0	0
$969$	3.92016e9	0.138411
$970$	0	0
$971$	−4.53900e10	−1.59108	−0.795542	−	0.605899i	$-0.792814\pi$
$971$	−4.53900e10	−1.59108	−0.795542	+	0.605899i	$0.792814\pi$
$972$	0	0
$973$	1.49418e10	0.520004
$974$	0	0
$975$	2.87430e10	0.993152
$976$	0	0
$977$	−2.14529e10	−0.735960	−0.367980	−	0.929834i	$-0.619950\pi$
$977$	−2.14529e10	−0.735960	−0.367980	+	0.929834i	$0.619950\pi$
$978$	0	0
$979$	−3.67204e10	−1.25074
$980$	0	0
$981$	−9.39906e8	−0.0317865
$982$	0	0
$983$	3.89213e10	1.30692	0.653462	−	0.756960i	$-0.273316\pi$
$983$	3.89213e10	1.30692	0.653462	+	0.756960i	$0.273316\pi$
$984$	0	0
$985$	−1.62349e10	−0.541282
$986$	0	0
$987$	−1.53011e10	−0.506540
$988$	0	0
$989$	1.12020e9	0.0368222
$990$	0	0
$991$	−3.27057e10	−1.06749	−0.533747	−	0.845644i	$-0.679216\pi$
$991$	−3.27057e10	−1.06749	−0.533747	+	0.845644i	$0.679216\pi$
$992$	0	0
$993$	−1.41572e10	−0.458833
$994$	0	0
$995$	−5.93311e10	−1.90942
$996$	0	0
$997$	−9.21065e9	−0.294345	−0.147173	−	0.989111i	$-0.547017\pi$
$997$	−9.21065e9	−0.294345	−0.147173	+	0.989111i	$0.547017\pi$
$998$	0	0
$999$	5.61440e9	0.178166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.8.a.b.1.1	✓	1
3.2	odd	2		72.8.a.a.1.1		1
4.3	odd	2		16.8.a.a.1.1		1
5.2	odd	4		200.8.c.c.49.1		2
5.3	odd	4		200.8.c.c.49.2		2
5.4	even	2		200.8.a.b.1.1		1
7.6	odd	2		392.8.a.b.1.1		1
8.3	odd	2		64.8.a.f.1.1		1
8.5	even	2		64.8.a.b.1.1		1
12.11	even	2		144.8.a.a.1.1		1
16.3	odd	4		256.8.b.a.129.1		2
16.5	even	4		256.8.b.g.129.1		2
16.11	odd	4		256.8.b.a.129.2		2
16.13	even	4		256.8.b.g.129.2		2
20.3	even	4		400.8.c.f.49.1		2
20.7	even	4		400.8.c.f.49.2		2
20.19	odd	2		400.8.a.p.1.1		1
24.5	odd	2		576.8.a.y.1.1		1
24.11	even	2		576.8.a.z.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.a.b.1.1	✓	1	1.1	even	1	trivial
16.8.a.a.1.1		1	4.3	odd	2
64.8.a.b.1.1		1	8.5	even	2
64.8.a.f.1.1		1	8.3	odd	2
72.8.a.a.1.1		1	3.2	odd	2
144.8.a.a.1.1		1	12.11	even	2
200.8.a.b.1.1		1	5.4	even	2
200.8.c.c.49.1		2	5.2	odd	4
200.8.c.c.49.2		2	5.3	odd	4
256.8.b.a.129.1		2	16.3	odd	4
256.8.b.a.129.2		2	16.11	odd	4
256.8.b.g.129.1		2	16.5	even	4
256.8.b.g.129.2		2	16.13	even	4
392.8.a.b.1.1		1	7.6	odd	2
400.8.a.p.1.1		1	20.19	odd	2
400.8.c.f.49.1		2	20.3	even	4
400.8.c.f.49.2		2	20.7	even	4
576.8.a.y.1.1		1	24.5	odd	2
576.8.a.z.1.1		1	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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