Properties

Label 10.8.a.a.1.1
Level $10$
Weight $8$
Character 10.1
Self dual yes
Analytic conductor $3.124$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,8,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, 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("10.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(3.12385025484\)
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 \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +28.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +224.000 q^{6} +104.000 q^{7} +512.000 q^{8} -1403.00 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +28.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +224.000 q^{6} +104.000 q^{7} +512.000 q^{8} -1403.00 q^{9} +1000.00 q^{10} -5148.00 q^{11} +1792.00 q^{12} -8602.00 q^{13} +832.000 q^{14} +3500.00 q^{15} +4096.00 q^{16} +20274.0 q^{17} -11224.0 q^{18} +45500.0 q^{19} +8000.00 q^{20} +2912.00 q^{21} -41184.0 q^{22} -72072.0 q^{23} +14336.0 q^{24} +15625.0 q^{25} -68816.0 q^{26} -100520. q^{27} +6656.00 q^{28} +231510. q^{29} +28000.0 q^{30} -80128.0 q^{31} +32768.0 q^{32} -144144. q^{33} +162192. q^{34} +13000.0 q^{35} -89792.0 q^{36} +104654. q^{37} +364000. q^{38} -240856. q^{39} +64000.0 q^{40} +584922. q^{41} +23296.0 q^{42} -795532. q^{43} -329472. q^{44} -175375. q^{45} -576576. q^{46} +425664. q^{47} +114688. q^{48} -812727. q^{49} +125000. q^{50} +567672. q^{51} -550528. q^{52} +1.50080e6 q^{53} -804160. q^{54} -643500. q^{55} +53248.0 q^{56} +1.27400e6 q^{57} +1.85208e6 q^{58} +246420. q^{59} +224000. q^{60} +893942. q^{61} -641024. q^{62} -145912. q^{63} +262144. q^{64} -1.07525e6 q^{65} -1.15315e6 q^{66} -2.33684e6 q^{67} +1.29754e6 q^{68} -2.01802e6 q^{69} +104000. q^{70} -203688. q^{71} -718336. q^{72} -3.80570e6 q^{73} +837232. q^{74} +437500. q^{75} +2.91200e6 q^{76} -535392. q^{77} -1.92685e6 q^{78} +5.05304e6 q^{79} +512000. q^{80} +253801. q^{81} +4.67938e6 q^{82} -45492.0 q^{83} +186368. q^{84} +2.53425e6 q^{85} -6.36426e6 q^{86} +6.48228e6 q^{87} -2.63578e6 q^{88} +980010. q^{89} -1.40300e6 q^{90} -894608. q^{91} -4.61261e6 q^{92} -2.24358e6 q^{93} +3.40531e6 q^{94} +5.68750e6 q^{95} +917504. q^{96} -5.24765e6 q^{97} -6.50182e6 q^{98} +7.22264e6 q^{99} +O(q^{100})\)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 28.0000 0.598734 0.299367 0.954138i \(-0.403225\pi\)
0.299367 + 0.954138i \(0.403225\pi\)
\(4\) 64.0000 0.500000
\(5\) 125.000 0.447214
\(6\) 224.000 0.423369
\(7\) 104.000 0.114601 0.0573007 0.998357i \(-0.481751\pi\)
0.0573007 + 0.998357i \(0.481751\pi\)
\(8\) 512.000 0.353553
\(9\) −1403.00 −0.641518
\(10\) 1000.00 0.316228
\(11\) −5148.00 −1.16618 −0.583088 0.812409i \(-0.698156\pi\)
−0.583088 + 0.812409i \(0.698156\pi\)
\(12\) 1792.00 0.299367
\(13\) −8602.00 −1.08592 −0.542960 0.839759i \(-0.682696\pi\)
−0.542960 + 0.839759i \(0.682696\pi\)
\(14\) 832.000 0.0810355
\(15\) 3500.00 0.267762
\(16\) 4096.00 0.250000
\(17\) 20274.0 1.00085 0.500424 0.865780i \(-0.333177\pi\)
0.500424 + 0.865780i \(0.333177\pi\)
\(18\) −11224.0 −0.453622
\(19\) 45500.0 1.52186 0.760928 0.648836i \(-0.224744\pi\)
0.760928 + 0.648836i \(0.224744\pi\)
\(20\) 8000.00 0.223607
\(21\) 2912.00 0.0686158
\(22\) −41184.0 −0.824611
\(23\) −72072.0 −1.23515 −0.617574 0.786513i \(-0.711884\pi\)
−0.617574 + 0.786513i \(0.711884\pi\)
\(24\) 14336.0 0.211684
\(25\) 15625.0 0.200000
\(26\) −68816.0 −0.767861
\(27\) −100520. −0.982832
\(28\) 6656.00 0.0573007
\(29\) 231510. 1.76269 0.881347 0.472470i \(-0.156638\pi\)
0.881347 + 0.472470i \(0.156638\pi\)
\(30\) 28000.0 0.189336
\(31\) −80128.0 −0.483079 −0.241540 0.970391i \(-0.577652\pi\)
−0.241540 + 0.970391i \(0.577652\pi\)
\(32\) 32768.0 0.176777
\(33\) −144144. −0.698229
\(34\) 162192. 0.707707
\(35\) 13000.0 0.0512513
\(36\) −89792.0 −0.320759
\(37\) 104654. 0.339664 0.169832 0.985473i \(-0.445677\pi\)
0.169832 + 0.985473i \(0.445677\pi\)
\(38\) 364000. 1.07612
\(39\) −240856. −0.650177
\(40\) 64000.0 0.158114
\(41\) 584922. 1.32542 0.662711 0.748875i \(-0.269406\pi\)
0.662711 + 0.748875i \(0.269406\pi\)
\(42\) 23296.0 0.0485187
\(43\) −795532. −1.52587 −0.762936 0.646474i \(-0.776243\pi\)
−0.762936 + 0.646474i \(0.776243\pi\)
\(44\) −329472. −0.583088
\(45\) −175375. −0.286896
\(46\) −576576. −0.873382
\(47\) 425664. 0.598032 0.299016 0.954248i \(-0.403342\pi\)
0.299016 + 0.954248i \(0.403342\pi\)
\(48\) 114688. 0.149683
\(49\) −812727. −0.986867
\(50\) 125000. 0.141421
\(51\) 567672. 0.599241
\(52\) −550528. −0.542960
\(53\) 1.50080e6 1.38470 0.692352 0.721560i \(-0.256575\pi\)
0.692352 + 0.721560i \(0.256575\pi\)
\(54\) −804160. −0.694967
\(55\) −643500. −0.521530
\(56\) 53248.0 0.0405177
\(57\) 1.27400e6 0.911187
\(58\) 1.85208e6 1.24641
\(59\) 246420. 0.156205 0.0781023 0.996945i \(-0.475114\pi\)
0.0781023 + 0.996945i \(0.475114\pi\)
\(60\) 224000. 0.133881
\(61\) 893942. 0.504260 0.252130 0.967693i \(-0.418869\pi\)
0.252130 + 0.967693i \(0.418869\pi\)
\(62\) −641024. −0.341589
\(63\) −145912. −0.0735189
\(64\) 262144. 0.125000
\(65\) −1.07525e6 −0.485638
\(66\) −1.15315e6 −0.493722
\(67\) −2.33684e6 −0.949219 −0.474610 0.880196i \(-0.657411\pi\)
−0.474610 + 0.880196i \(0.657411\pi\)
\(68\) 1.29754e6 0.500424
\(69\) −2.01802e6 −0.739525
\(70\) 104000. 0.0362402
\(71\) −203688. −0.0675400 −0.0337700 0.999430i \(-0.510751\pi\)
−0.0337700 + 0.999430i \(0.510751\pi\)
\(72\) −718336. −0.226811
\(73\) −3.80570e6 −1.14500 −0.572499 0.819905i \(-0.694026\pi\)
−0.572499 + 0.819905i \(0.694026\pi\)
\(74\) 837232. 0.240179
\(75\) 437500. 0.119747
\(76\) 2.91200e6 0.760928
\(77\) −535392. −0.133645
\(78\) −1.92685e6 −0.459744
\(79\) 5.05304e6 1.15308 0.576538 0.817070i \(-0.304403\pi\)
0.576538 + 0.817070i \(0.304403\pi\)
\(80\) 512000. 0.111803
\(81\) 253801. 0.0530635
\(82\) 4.67938e6 0.937215
\(83\) −45492.0 −0.00873296 −0.00436648 0.999990i \(-0.501390\pi\)
−0.00436648 + 0.999990i \(0.501390\pi\)
\(84\) 186368. 0.0343079
\(85\) 2.53425e6 0.447593
\(86\) −6.36426e6 −1.07895
\(87\) 6.48228e6 1.05538
\(88\) −2.63578e6 −0.412306
\(89\) 980010. 0.147355 0.0736776 0.997282i \(-0.476526\pi\)
0.0736776 + 0.997282i \(0.476526\pi\)
\(90\) −1.40300e6 −0.202866
\(91\) −894608. −0.124448
\(92\) −4.61261e6 −0.617574
\(93\) −2.24358e6 −0.289236
\(94\) 3.40531e6 0.422872
\(95\) 5.68750e6 0.680595
\(96\) 917504. 0.105842
\(97\) −5.24765e6 −0.583799 −0.291900 0.956449i \(-0.594287\pi\)
−0.291900 + 0.956449i \(0.594287\pi\)
\(98\) −6.50182e6 −0.697820
\(99\) 7.22264e6 0.748123
\(100\) 1.00000e6 0.100000
\(101\) −1.53807e7 −1.48542 −0.742711 0.669612i \(-0.766461\pi\)
−0.742711 + 0.669612i \(0.766461\pi\)
\(102\) 4.54138e6 0.423728
\(103\) 3.25681e6 0.293672 0.146836 0.989161i \(-0.453091\pi\)
0.146836 + 0.989161i \(0.453091\pi\)
\(104\) −4.40422e6 −0.383931
\(105\) 364000. 0.0306859
\(106\) 1.20064e7 0.979133
\(107\) −1.48329e7 −1.17053 −0.585264 0.810842i \(-0.699009\pi\)
−0.585264 + 0.810842i \(0.699009\pi\)
\(108\) −6.43328e6 −0.491416
\(109\) −7.03225e6 −0.520118 −0.260059 0.965593i \(-0.583742\pi\)
−0.260059 + 0.965593i \(0.583742\pi\)
\(110\) −5.14800e6 −0.368777
\(111\) 2.93031e6 0.203368
\(112\) 425984. 0.0286504
\(113\) 1.46074e7 0.952352 0.476176 0.879350i \(-0.342022\pi\)
0.476176 + 0.879350i \(0.342022\pi\)
\(114\) 1.01920e7 0.644306
\(115\) −9.00900e6 −0.552375
\(116\) 1.48166e7 0.881347
\(117\) 1.20686e7 0.696637
\(118\) 1.97136e6 0.110453
\(119\) 2.10850e6 0.114699
\(120\) 1.79200e6 0.0946681
\(121\) 7.01473e6 0.359967
\(122\) 7.15154e6 0.356566
\(123\) 1.63778e7 0.793575
\(124\) −5.12819e6 −0.241540
\(125\) 1.95312e6 0.0894427
\(126\) −1.16730e6 −0.0519857
\(127\) −1.51661e7 −0.656993 −0.328497 0.944505i \(-0.606542\pi\)
−0.328497 + 0.944505i \(0.606542\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −2.22749e7 −0.913591
\(130\) −8.60200e6 −0.343398
\(131\) −2.45985e7 −0.956005 −0.478002 0.878359i \(-0.658639\pi\)
−0.478002 + 0.878359i \(0.658639\pi\)
\(132\) −9.22522e6 −0.349114
\(133\) 4.73200e6 0.174407
\(134\) −1.86947e7 −0.671199
\(135\) −1.25650e7 −0.439536
\(136\) 1.03803e7 0.353853
\(137\) 3.51725e7 1.16864 0.584320 0.811523i \(-0.301361\pi\)
0.584320 + 0.811523i \(0.301361\pi\)
\(138\) −1.61441e7 −0.522923
\(139\) −4.47137e7 −1.41217 −0.706087 0.708125i \(-0.749541\pi\)
−0.706087 + 0.708125i \(0.749541\pi\)
\(140\) 832000. 0.0256257
\(141\) 1.19186e7 0.358062
\(142\) −1.62950e6 −0.0477580
\(143\) 4.42831e7 1.26637
\(144\) −5.74669e6 −0.160380
\(145\) 2.89388e7 0.788300
\(146\) −3.04456e7 −0.809636
\(147\) −2.27564e7 −0.590870
\(148\) 6.69786e6 0.169832
\(149\) 1.73348e7 0.429305 0.214652 0.976691i \(-0.431138\pi\)
0.214652 + 0.976691i \(0.431138\pi\)
\(150\) 3.50000e6 0.0846737
\(151\) 5.92382e7 1.40017 0.700087 0.714057i \(-0.253144\pi\)
0.700087 + 0.714057i \(0.253144\pi\)
\(152\) 2.32960e7 0.538058
\(153\) −2.84444e7 −0.642062
\(154\) −4.28314e6 −0.0945016
\(155\) −1.00160e7 −0.216040
\(156\) −1.54148e7 −0.325088
\(157\) 2.73945e7 0.564955 0.282478 0.959274i \(-0.408844\pi\)
0.282478 + 0.959274i \(0.408844\pi\)
\(158\) 4.04243e7 0.815348
\(159\) 4.20223e7 0.829068
\(160\) 4.09600e6 0.0790569
\(161\) −7.49549e6 −0.141550
\(162\) 2.03041e6 0.0375215
\(163\) −8.85387e7 −1.60131 −0.800657 0.599123i \(-0.795516\pi\)
−0.800657 + 0.599123i \(0.795516\pi\)
\(164\) 3.74350e7 0.662711
\(165\) −1.80180e7 −0.312257
\(166\) −363936. −0.00617514
\(167\) −2.85309e7 −0.474033 −0.237016 0.971506i \(-0.576170\pi\)
−0.237016 + 0.971506i \(0.576170\pi\)
\(168\) 1.49094e6 0.0242593
\(169\) 1.12459e7 0.179222
\(170\) 2.02740e7 0.316496
\(171\) −6.38365e7 −0.976299
\(172\) −5.09140e7 −0.762936
\(173\) −1.04351e8 −1.53227 −0.766134 0.642681i \(-0.777822\pi\)
−0.766134 + 0.642681i \(0.777822\pi\)
\(174\) 5.18582e7 0.746269
\(175\) 1.62500e6 0.0229203
\(176\) −2.10862e7 −0.291544
\(177\) 6.89976e6 0.0935250
\(178\) 7.84008e6 0.104196
\(179\) 1.00761e8 1.31313 0.656565 0.754270i \(-0.272009\pi\)
0.656565 + 0.754270i \(0.272009\pi\)
\(180\) −1.12240e7 −0.143448
\(181\) 3.56842e7 0.447302 0.223651 0.974669i \(-0.428202\pi\)
0.223651 + 0.974669i \(0.428202\pi\)
\(182\) −7.15686e6 −0.0879980
\(183\) 2.50304e7 0.301918
\(184\) −3.69009e7 −0.436691
\(185\) 1.30818e7 0.151902
\(186\) −1.79487e7 −0.204521
\(187\) −1.04371e8 −1.16717
\(188\) 2.72425e7 0.299016
\(189\) −1.04541e7 −0.112634
\(190\) 4.55000e7 0.481253
\(191\) 5.27892e7 0.548186 0.274093 0.961703i \(-0.411622\pi\)
0.274093 + 0.961703i \(0.411622\pi\)
\(192\) 7.34003e6 0.0748417
\(193\) −3.20893e7 −0.321300 −0.160650 0.987011i \(-0.551359\pi\)
−0.160650 + 0.987011i \(0.551359\pi\)
\(194\) −4.19812e7 −0.412808
\(195\) −3.01070e7 −0.290768
\(196\) −5.20145e7 −0.493433
\(197\) 1.67464e8 1.56059 0.780294 0.625412i \(-0.215069\pi\)
0.780294 + 0.625412i \(0.215069\pi\)
\(198\) 5.77812e7 0.529003
\(199\) 1.61770e8 1.45517 0.727583 0.686020i \(-0.240644\pi\)
0.727583 + 0.686020i \(0.240644\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) −6.54314e7 −0.568329
\(202\) −1.23045e8 −1.05035
\(203\) 2.40770e7 0.202007
\(204\) 3.63310e7 0.299621
\(205\) 7.31152e7 0.592747
\(206\) 2.60545e7 0.207657
\(207\) 1.01117e8 0.792370
\(208\) −3.52338e7 −0.271480
\(209\) −2.34234e8 −1.77475
\(210\) 2.91200e6 0.0216982
\(211\) 1.78419e8 1.30753 0.653766 0.756697i \(-0.273188\pi\)
0.653766 + 0.756697i \(0.273188\pi\)
\(212\) 9.60511e7 0.692352
\(213\) −5.70326e6 −0.0404385
\(214\) −1.18663e8 −0.827689
\(215\) −9.94415e7 −0.682391
\(216\) −5.14662e7 −0.347484
\(217\) −8.33331e6 −0.0553616
\(218\) −5.62580e7 −0.367779
\(219\) −1.06560e8 −0.685549
\(220\) −4.11840e7 −0.260765
\(221\) −1.74397e8 −1.08684
\(222\) 2.34425e7 0.143803
\(223\) 1.26953e8 0.766611 0.383305 0.923622i \(-0.374786\pi\)
0.383305 + 0.923622i \(0.374786\pi\)
\(224\) 3.40787e6 0.0202589
\(225\) −2.19219e7 −0.128304
\(226\) 1.16859e8 0.673415
\(227\) −1.57656e8 −0.894582 −0.447291 0.894389i \(-0.647611\pi\)
−0.447291 + 0.894389i \(0.647611\pi\)
\(228\) 8.15360e7 0.455593
\(229\) −3.38259e8 −1.86134 −0.930670 0.365861i \(-0.880775\pi\)
−0.930670 + 0.365861i \(0.880775\pi\)
\(230\) −7.20720e7 −0.390588
\(231\) −1.49910e7 −0.0800181
\(232\) 1.18533e8 0.623206
\(233\) −3.56382e7 −0.184574 −0.0922868 0.995732i \(-0.529418\pi\)
−0.0922868 + 0.995732i \(0.529418\pi\)
\(234\) 9.65488e7 0.492597
\(235\) 5.32080e7 0.267448
\(236\) 1.57709e7 0.0781023
\(237\) 1.41485e8 0.690385
\(238\) 1.68680e7 0.0811042
\(239\) 2.16045e8 1.02365 0.511825 0.859090i \(-0.328970\pi\)
0.511825 + 0.859090i \(0.328970\pi\)
\(240\) 1.43360e7 0.0669405
\(241\) 3.84299e8 1.76852 0.884260 0.466995i \(-0.154663\pi\)
0.884260 + 0.466995i \(0.154663\pi\)
\(242\) 5.61179e7 0.254535
\(243\) 2.26944e8 1.01460
\(244\) 5.72123e7 0.252130
\(245\) −1.01591e8 −0.441340
\(246\) 1.31023e8 0.561142
\(247\) −3.91391e8 −1.65261
\(248\) −4.10255e7 −0.170794
\(249\) −1.27378e6 −0.00522872
\(250\) 1.56250e7 0.0632456
\(251\) −2.36430e8 −0.943724 −0.471862 0.881672i \(-0.656418\pi\)
−0.471862 + 0.881672i \(0.656418\pi\)
\(252\) −9.33837e6 −0.0367595
\(253\) 3.71027e8 1.44040
\(254\) −1.21329e8 −0.464564
\(255\) 7.09590e7 0.267989
\(256\) 1.67772e7 0.0625000
\(257\) −2.37832e7 −0.0873985 −0.0436992 0.999045i \(-0.513914\pi\)
−0.0436992 + 0.999045i \(0.513914\pi\)
\(258\) −1.78199e8 −0.646006
\(259\) 1.08840e7 0.0389260
\(260\) −6.88160e7 −0.242819
\(261\) −3.24809e8 −1.13080
\(262\) −1.96788e8 −0.675997
\(263\) −7.98963e7 −0.270821 −0.135410 0.990790i \(-0.543235\pi\)
−0.135410 + 0.990790i \(0.543235\pi\)
\(264\) −7.38017e7 −0.246861
\(265\) 1.87600e8 0.619258
\(266\) 3.78560e7 0.123324
\(267\) 2.74403e7 0.0882265
\(268\) −1.49558e8 −0.474610
\(269\) 2.33547e8 0.731545 0.365772 0.930704i \(-0.380805\pi\)
0.365772 + 0.930704i \(0.380805\pi\)
\(270\) −1.00520e8 −0.310799
\(271\) −2.79282e8 −0.852414 −0.426207 0.904626i \(-0.640150\pi\)
−0.426207 + 0.904626i \(0.640150\pi\)
\(272\) 8.30423e7 0.250212
\(273\) −2.50490e7 −0.0745112
\(274\) 2.81380e8 0.826354
\(275\) −8.04375e7 −0.233235
\(276\) −1.29153e8 −0.369762
\(277\) 1.63825e8 0.463127 0.231564 0.972820i \(-0.425616\pi\)
0.231564 + 0.972820i \(0.425616\pi\)
\(278\) −3.57709e8 −0.998558
\(279\) 1.12420e8 0.309904
\(280\) 6.65600e6 0.0181201
\(281\) 1.28061e8 0.344307 0.172153 0.985070i \(-0.444928\pi\)
0.172153 + 0.985070i \(0.444928\pi\)
\(282\) 9.53487e7 0.253188
\(283\) −2.69379e8 −0.706498 −0.353249 0.935529i \(-0.614923\pi\)
−0.353249 + 0.935529i \(0.614923\pi\)
\(284\) −1.30360e7 −0.0337700
\(285\) 1.59250e8 0.407495
\(286\) 3.54265e8 0.895461
\(287\) 6.08319e7 0.151895
\(288\) −4.59735e7 −0.113405
\(289\) 696403. 0.00169714
\(290\) 2.31510e8 0.557413
\(291\) −1.46934e8 −0.349540
\(292\) −2.43565e8 −0.572499
\(293\) −1.55781e8 −0.361808 −0.180904 0.983501i \(-0.557902\pi\)
−0.180904 + 0.983501i \(0.557902\pi\)
\(294\) −1.82051e8 −0.417808
\(295\) 3.08025e7 0.0698568
\(296\) 5.35828e7 0.120089
\(297\) 5.17477e8 1.14616
\(298\) 1.38678e8 0.303564
\(299\) 6.19963e8 1.34127
\(300\) 2.80000e7 0.0598734
\(301\) −8.27353e7 −0.174867
\(302\) 4.73905e8 0.990073
\(303\) −4.30658e8 −0.889373
\(304\) 1.86368e8 0.380464
\(305\) 1.11743e8 0.225512
\(306\) −2.27555e8 −0.454007
\(307\) −4.87126e8 −0.960852 −0.480426 0.877035i \(-0.659518\pi\)
−0.480426 + 0.877035i \(0.659518\pi\)
\(308\) −3.42651e7 −0.0668227
\(309\) 9.11906e7 0.175831
\(310\) −8.01280e7 −0.152763
\(311\) 2.88891e8 0.544595 0.272297 0.962213i \(-0.412217\pi\)
0.272297 + 0.962213i \(0.412217\pi\)
\(312\) −1.23318e8 −0.229872
\(313\) −4.09453e8 −0.754743 −0.377371 0.926062i \(-0.623172\pi\)
−0.377371 + 0.926062i \(0.623172\pi\)
\(314\) 2.19156e8 0.399484
\(315\) −1.82390e7 −0.0328787
\(316\) 3.23395e8 0.576538
\(317\) −2.69274e8 −0.474774 −0.237387 0.971415i \(-0.576291\pi\)
−0.237387 + 0.971415i \(0.576291\pi\)
\(318\) 3.36179e8 0.586240
\(319\) −1.19181e9 −2.05561
\(320\) 3.27680e7 0.0559017
\(321\) −4.15321e8 −0.700835
\(322\) −5.99639e7 −0.100091
\(323\) 9.22467e8 1.52315
\(324\) 1.62433e7 0.0265317
\(325\) −1.34406e8 −0.217184
\(326\) −7.08310e8 −1.13230
\(327\) −1.96903e8 −0.311412
\(328\) 2.99480e8 0.468608
\(329\) 4.42691e7 0.0685353
\(330\) −1.44144e8 −0.220799
\(331\) 4.05515e8 0.614624 0.307312 0.951609i \(-0.400570\pi\)
0.307312 + 0.951609i \(0.400570\pi\)
\(332\) −2.91149e6 −0.00436648
\(333\) −1.46830e8 −0.217901
\(334\) −2.28247e8 −0.335192
\(335\) −2.92104e8 −0.424504
\(336\) 1.19276e7 0.0171539
\(337\) −3.83013e8 −0.545141 −0.272570 0.962136i \(-0.587874\pi\)
−0.272570 + 0.962136i \(0.587874\pi\)
\(338\) 8.99671e7 0.126729
\(339\) 4.09007e8 0.570205
\(340\) 1.62192e8 0.223796
\(341\) 4.12499e8 0.563355
\(342\) −5.10692e8 −0.690347
\(343\) −1.70172e8 −0.227698
\(344\) −4.07312e8 −0.539477
\(345\) −2.52252e8 −0.330726
\(346\) −8.34807e8 −1.08348
\(347\) 1.17574e9 1.51063 0.755317 0.655359i \(-0.227483\pi\)
0.755317 + 0.655359i \(0.227483\pi\)
\(348\) 4.14866e8 0.527692
\(349\) 2.88405e8 0.363173 0.181586 0.983375i \(-0.441877\pi\)
0.181586 + 0.983375i \(0.441877\pi\)
\(350\) 1.30000e7 0.0162071
\(351\) 8.64673e8 1.06728
\(352\) −1.68690e8 −0.206153
\(353\) 7.65131e8 0.925816 0.462908 0.886406i \(-0.346806\pi\)
0.462908 + 0.886406i \(0.346806\pi\)
\(354\) 5.51981e7 0.0661321
\(355\) −2.54610e7 −0.0302048
\(356\) 6.27206e7 0.0736776
\(357\) 5.90379e7 0.0686740
\(358\) 8.06089e8 0.928523
\(359\) −5.25904e8 −0.599896 −0.299948 0.953956i \(-0.596969\pi\)
−0.299948 + 0.953956i \(0.596969\pi\)
\(360\) −8.97920e7 −0.101433
\(361\) 1.17638e9 1.31605
\(362\) 2.85474e8 0.316291
\(363\) 1.96413e8 0.215524
\(364\) −5.72549e7 −0.0622240
\(365\) −4.75713e8 −0.512059
\(366\) 2.00243e8 0.213488
\(367\) 8.93571e8 0.943621 0.471811 0.881700i \(-0.343601\pi\)
0.471811 + 0.881700i \(0.343601\pi\)
\(368\) −2.95207e8 −0.308787
\(369\) −8.20646e8 −0.850283
\(370\) 1.04654e8 0.107411
\(371\) 1.56083e8 0.158689
\(372\) −1.43589e8 −0.144618
\(373\) −8.13329e8 −0.811494 −0.405747 0.913985i \(-0.632989\pi\)
−0.405747 + 0.913985i \(0.632989\pi\)
\(374\) −8.34964e8 −0.825310
\(375\) 5.46875e7 0.0535524
\(376\) 2.17940e8 0.211436
\(377\) −1.99145e9 −1.91414
\(378\) −8.36326e7 −0.0796443
\(379\) −1.02095e9 −0.963309 −0.481655 0.876361i \(-0.659964\pi\)
−0.481655 + 0.876361i \(0.659964\pi\)
\(380\) 3.64000e8 0.340298
\(381\) −4.24651e8 −0.393364
\(382\) 4.22313e8 0.387626
\(383\) −9.14343e8 −0.831598 −0.415799 0.909457i \(-0.636498\pi\)
−0.415799 + 0.909457i \(0.636498\pi\)
\(384\) 5.87203e7 0.0529211
\(385\) −6.69240e7 −0.0597681
\(386\) −2.56715e8 −0.227193
\(387\) 1.11613e9 0.978874
\(388\) −3.35849e8 −0.291900
\(389\) 2.23944e8 0.192893 0.0964463 0.995338i \(-0.469252\pi\)
0.0964463 + 0.995338i \(0.469252\pi\)
\(390\) −2.40856e8 −0.205604
\(391\) −1.46119e9 −1.23620
\(392\) −4.16116e8 −0.348910
\(393\) −6.88759e8 −0.572392
\(394\) 1.33971e9 1.10350
\(395\) 6.31630e8 0.515671
\(396\) 4.62249e8 0.374062
\(397\) 1.80757e9 1.44987 0.724934 0.688818i \(-0.241870\pi\)
0.724934 + 0.688818i \(0.241870\pi\)
\(398\) 1.29416e9 1.02896
\(399\) 1.32496e8 0.104423
\(400\) 6.40000e7 0.0500000
\(401\) −1.63569e9 −1.26676 −0.633381 0.773840i \(-0.718333\pi\)
−0.633381 + 0.773840i \(0.718333\pi\)
\(402\) −5.23451e8 −0.401870
\(403\) 6.89261e8 0.524585
\(404\) −9.84362e8 −0.742711
\(405\) 3.17251e7 0.0237307
\(406\) 1.92616e8 0.142841
\(407\) −5.38759e8 −0.396108
\(408\) 2.90648e8 0.211864
\(409\) 2.27381e9 1.64332 0.821661 0.569977i \(-0.193048\pi\)
0.821661 + 0.569977i \(0.193048\pi\)
\(410\) 5.84922e8 0.419136
\(411\) 9.84829e8 0.699704
\(412\) 2.08436e8 0.146836
\(413\) 2.56277e7 0.0179013
\(414\) 8.08936e8 0.560290
\(415\) −5.68650e6 −0.00390550
\(416\) −2.81870e8 −0.191965
\(417\) −1.25198e9 −0.845516
\(418\) −1.87387e9 −1.25494
\(419\) 4.04353e8 0.268542 0.134271 0.990945i \(-0.457131\pi\)
0.134271 + 0.990945i \(0.457131\pi\)
\(420\) 2.32960e7 0.0153429
\(421\) 8.37294e6 0.00546878 0.00273439 0.999996i \(-0.499130\pi\)
0.00273439 + 0.999996i \(0.499130\pi\)
\(422\) 1.42735e9 0.924565
\(423\) −5.97207e8 −0.383648
\(424\) 7.68409e8 0.489567
\(425\) 3.16781e8 0.200170
\(426\) −4.56261e7 −0.0285943
\(427\) 9.29700e7 0.0577890
\(428\) −9.49304e8 −0.585264
\(429\) 1.23993e9 0.758220
\(430\) −7.95532e8 −0.482523
\(431\) −2.77413e8 −0.166900 −0.0834499 0.996512i \(-0.526594\pi\)
−0.0834499 + 0.996512i \(0.526594\pi\)
\(432\) −4.11730e8 −0.245708
\(433\) 8.78768e8 0.520196 0.260098 0.965582i \(-0.416245\pi\)
0.260098 + 0.965582i \(0.416245\pi\)
\(434\) −6.66665e7 −0.0391466
\(435\) 8.10285e8 0.471982
\(436\) −4.50064e8 −0.260059
\(437\) −3.27928e9 −1.87972
\(438\) −8.52477e8 −0.484756
\(439\) 8.96007e8 0.505458 0.252729 0.967537i \(-0.418672\pi\)
0.252729 + 0.967537i \(0.418672\pi\)
\(440\) −3.29472e8 −0.184389
\(441\) 1.14026e9 0.633093
\(442\) −1.39518e9 −0.768512
\(443\) −3.83067e7 −0.0209344 −0.0104672 0.999945i \(-0.503332\pi\)
−0.0104672 + 0.999945i \(0.503332\pi\)
\(444\) 1.87540e8 0.101684
\(445\) 1.22501e8 0.0658993
\(446\) 1.01562e9 0.542076
\(447\) 4.85373e8 0.257039
\(448\) 2.72630e7 0.0143252
\(449\) 7.55714e8 0.393999 0.197000 0.980404i \(-0.436880\pi\)
0.197000 + 0.980404i \(0.436880\pi\)
\(450\) −1.75375e8 −0.0907244
\(451\) −3.01118e9 −1.54568
\(452\) 9.34872e8 0.476176
\(453\) 1.65867e9 0.838332
\(454\) −1.26125e9 −0.632565
\(455\) −1.11826e8 −0.0556548
\(456\) 6.52288e8 0.322153
\(457\) 2.22407e9 1.09004 0.545019 0.838424i \(-0.316522\pi\)
0.545019 + 0.838424i \(0.316522\pi\)
\(458\) −2.70607e9 −1.31617
\(459\) −2.03794e9 −0.983666
\(460\) −5.76576e8 −0.276188
\(461\) 1.19504e9 0.568107 0.284054 0.958808i \(-0.408321\pi\)
0.284054 + 0.958808i \(0.408321\pi\)
\(462\) −1.19928e8 −0.0565813
\(463\) −3.83431e9 −1.79537 −0.897686 0.440636i \(-0.854753\pi\)
−0.897686 + 0.440636i \(0.854753\pi\)
\(464\) 9.48265e8 0.440673
\(465\) −2.80448e8 −0.129350
\(466\) −2.85105e8 −0.130513
\(467\) −1.64554e9 −0.747653 −0.373826 0.927499i \(-0.621954\pi\)
−0.373826 + 0.927499i \(0.621954\pi\)
\(468\) 7.72391e8 0.348319
\(469\) −2.43031e8 −0.108782
\(470\) 4.25664e8 0.189114
\(471\) 7.67045e8 0.338258
\(472\) 1.26167e8 0.0552267
\(473\) 4.09540e9 1.77944
\(474\) 1.13188e9 0.488176
\(475\) 7.10937e8 0.304371
\(476\) 1.34944e8 0.0573493
\(477\) −2.10562e9 −0.888312
\(478\) 1.72836e9 0.723830
\(479\) −7.80831e8 −0.324626 −0.162313 0.986739i \(-0.551895\pi\)
−0.162313 + 0.986739i \(0.551895\pi\)
\(480\) 1.14688e8 0.0473340
\(481\) −9.00234e8 −0.368848
\(482\) 3.07439e9 1.25053
\(483\) −2.09874e8 −0.0847506
\(484\) 4.48943e8 0.179983
\(485\) −6.55956e8 −0.261083
\(486\) 1.81555e9 0.717433
\(487\) 4.79803e9 1.88240 0.941200 0.337849i \(-0.109699\pi\)
0.941200 + 0.337849i \(0.109699\pi\)
\(488\) 4.57698e8 0.178283
\(489\) −2.47908e9 −0.958760
\(490\) −8.12727e8 −0.312075
\(491\) −1.41644e9 −0.540022 −0.270011 0.962857i \(-0.587027\pi\)
−0.270011 + 0.962857i \(0.587027\pi\)
\(492\) 1.04818e9 0.396788
\(493\) 4.69363e9 1.76419
\(494\) −3.13113e9 −1.16857
\(495\) 9.02830e8 0.334571
\(496\) −3.28204e8 −0.120770
\(497\) −2.11836e7 −0.00774019
\(498\) −1.01902e7 −0.00369726
\(499\) −4.18128e9 −1.50646 −0.753230 0.657758i \(-0.771505\pi\)
−0.753230 + 0.657758i \(0.771505\pi\)
\(500\) 1.25000e8 0.0447214
\(501\) −7.98866e8 −0.283819
\(502\) −1.89144e9 −0.667314
\(503\) 2.85222e9 0.999298 0.499649 0.866228i \(-0.333462\pi\)
0.499649 + 0.866228i \(0.333462\pi\)
\(504\) −7.47069e7 −0.0259929
\(505\) −1.92258e9 −0.664301
\(506\) 2.96821e9 1.01852
\(507\) 3.14885e8 0.107306
\(508\) −9.70630e8 −0.328497
\(509\) 8.51999e8 0.286369 0.143185 0.989696i \(-0.454266\pi\)
0.143185 + 0.989696i \(0.454266\pi\)
\(510\) 5.67672e8 0.189497
\(511\) −3.95793e8 −0.131218
\(512\) 1.34218e8 0.0441942
\(513\) −4.57366e9 −1.49573
\(514\) −1.90265e8 −0.0618000
\(515\) 4.07101e8 0.131334
\(516\) −1.42559e9 −0.456795
\(517\) −2.19132e9 −0.697411
\(518\) 8.70721e7 0.0275249
\(519\) −2.92182e9 −0.917420
\(520\) −5.50528e8 −0.171699
\(521\) 2.86477e9 0.887479 0.443740 0.896156i \(-0.353652\pi\)
0.443740 + 0.896156i \(0.353652\pi\)
\(522\) −2.59847e9 −0.799596
\(523\) 2.86816e9 0.876693 0.438347 0.898806i \(-0.355564\pi\)
0.438347 + 0.898806i \(0.355564\pi\)
\(524\) −1.57431e9 −0.478002
\(525\) 4.55000e7 0.0137232
\(526\) −6.39170e8 −0.191499
\(527\) −1.62452e9 −0.483489
\(528\) −5.90414e8 −0.174557
\(529\) 1.78955e9 0.525592
\(530\) 1.50080e9 0.437882
\(531\) −3.45727e8 −0.100208
\(532\) 3.02848e8 0.0872035
\(533\) −5.03150e9 −1.43930
\(534\) 2.19522e8 0.0623856
\(535\) −1.85411e9 −0.523476
\(536\) −1.19646e9 −0.335600
\(537\) 2.82131e9 0.786215
\(538\) 1.86837e9 0.517280
\(539\) 4.18392e9 1.15086
\(540\) −8.04160e8 −0.219768
\(541\) 2.71353e8 0.0736792 0.0368396 0.999321i \(-0.488271\pi\)
0.0368396 + 0.999321i \(0.488271\pi\)
\(542\) −2.23426e9 −0.602747
\(543\) 9.99158e8 0.267815
\(544\) 6.64338e8 0.176927
\(545\) −8.79031e8 −0.232604
\(546\) −2.00392e8 −0.0526874
\(547\) 3.76596e9 0.983829 0.491915 0.870643i \(-0.336297\pi\)
0.491915 + 0.870643i \(0.336297\pi\)
\(548\) 2.25104e9 0.584320
\(549\) −1.25420e9 −0.323492
\(550\) −6.43500e8 −0.164922
\(551\) 1.05337e10 2.68257
\(552\) −1.03322e9 −0.261462
\(553\) 5.25516e8 0.132144
\(554\) 1.31060e9 0.327480
\(555\) 3.66289e8 0.0909491
\(556\) −2.86167e9 −0.706087
\(557\) −1.23789e9 −0.303520 −0.151760 0.988417i \(-0.548494\pi\)
−0.151760 + 0.988417i \(0.548494\pi\)
\(558\) 8.99357e8 0.219135
\(559\) 6.84317e9 1.65697
\(560\) 5.32480e7 0.0128128
\(561\) −2.92238e9 −0.698821
\(562\) 1.02449e9 0.243462
\(563\) −3.72192e9 −0.878998 −0.439499 0.898243i \(-0.644844\pi\)
−0.439499 + 0.898243i \(0.644844\pi\)
\(564\) 7.62790e8 0.179031
\(565\) 1.82592e9 0.425905
\(566\) −2.15503e9 −0.499569
\(567\) 2.63953e7 0.00608115
\(568\) −1.04288e8 −0.0238790
\(569\) 3.89118e9 0.885499 0.442749 0.896645i \(-0.354003\pi\)
0.442749 + 0.896645i \(0.354003\pi\)
\(570\) 1.27400e9 0.288143
\(571\) −7.43685e9 −1.67172 −0.835858 0.548946i \(-0.815029\pi\)
−0.835858 + 0.548946i \(0.815029\pi\)
\(572\) 2.83412e9 0.633187
\(573\) 1.47810e9 0.328217
\(574\) 4.86655e8 0.107406
\(575\) −1.12612e9 −0.247030
\(576\) −3.67788e8 −0.0801898
\(577\) −5.38224e9 −1.16640 −0.583200 0.812329i \(-0.698200\pi\)
−0.583200 + 0.812329i \(0.698200\pi\)
\(578\) 5.57122e6 0.00120006
\(579\) −8.98502e8 −0.192373
\(580\) 1.85208e9 0.394150
\(581\) −4.73117e6 −0.00100081
\(582\) −1.17547e9 −0.247162
\(583\) −7.72611e9 −1.61481
\(584\) −1.94852e9 −0.404818
\(585\) 1.50858e9 0.311546
\(586\) −1.24625e9 −0.255837
\(587\) −4.92134e9 −1.00427 −0.502135 0.864790i \(-0.667452\pi\)
−0.502135 + 0.864790i \(0.667452\pi\)
\(588\) −1.45641e9 −0.295435
\(589\) −3.64582e9 −0.735177
\(590\) 2.46420e8 0.0493962
\(591\) 4.68898e9 0.934377
\(592\) 4.28663e8 0.0849161
\(593\) −7.30205e9 −1.43798 −0.718991 0.695019i \(-0.755396\pi\)
−0.718991 + 0.695019i \(0.755396\pi\)
\(594\) 4.13982e9 0.810454
\(595\) 2.63562e8 0.0512948
\(596\) 1.10942e9 0.214652
\(597\) 4.52956e9 0.871256
\(598\) 4.95971e9 0.948423
\(599\) −2.56041e8 −0.0486761 −0.0243381 0.999704i \(-0.507748\pi\)
−0.0243381 + 0.999704i \(0.507748\pi\)
\(600\) 2.24000e8 0.0423369
\(601\) −3.57507e9 −0.671775 −0.335888 0.941902i \(-0.609036\pi\)
−0.335888 + 0.941902i \(0.609036\pi\)
\(602\) −6.61883e8 −0.123650
\(603\) 3.27858e9 0.608941
\(604\) 3.79124e9 0.700087
\(605\) 8.76842e8 0.160982
\(606\) −3.44527e9 −0.628881
\(607\) 6.27324e9 1.13850 0.569248 0.822166i \(-0.307234\pi\)
0.569248 + 0.822166i \(0.307234\pi\)
\(608\) 1.49094e9 0.269029
\(609\) 6.74157e8 0.120949
\(610\) 8.93942e8 0.159461
\(611\) −3.66156e9 −0.649415
\(612\) −1.82044e9 −0.321031
\(613\) −8.70805e9 −1.52690 −0.763448 0.645870i \(-0.776495\pi\)
−0.763448 + 0.645870i \(0.776495\pi\)
\(614\) −3.89700e9 −0.679425
\(615\) 2.04723e9 0.354898
\(616\) −2.74121e8 −0.0472508
\(617\) 3.92214e8 0.0672241 0.0336121 0.999435i \(-0.489299\pi\)
0.0336121 + 0.999435i \(0.489299\pi\)
\(618\) 7.29525e8 0.124331
\(619\) −6.39170e7 −0.0108318 −0.00541588 0.999985i \(-0.501724\pi\)
−0.00541588 + 0.999985i \(0.501724\pi\)
\(620\) −6.41024e8 −0.108020
\(621\) 7.24468e9 1.21394
\(622\) 2.31113e9 0.385087
\(623\) 1.01921e8 0.0168871
\(624\) −9.86546e8 −0.162544
\(625\) 2.44141e8 0.0400000
\(626\) −3.27562e9 −0.533684
\(627\) −6.55855e9 −1.06260
\(628\) 1.75325e9 0.282478
\(629\) 2.12176e9 0.339952
\(630\) −1.45912e8 −0.0232487
\(631\) 5.26547e9 0.834324 0.417162 0.908832i \(-0.363025\pi\)
0.417162 + 0.908832i \(0.363025\pi\)
\(632\) 2.58716e9 0.407674
\(633\) 4.99573e9 0.782864
\(634\) −2.15419e9 −0.335716
\(635\) −1.89576e9 −0.293816
\(636\) 2.68943e9 0.414534
\(637\) 6.99108e9 1.07166
\(638\) −9.53451e9 −1.45354
\(639\) 2.85774e8 0.0433281
\(640\) 2.62144e8 0.0395285
\(641\) −2.37074e9 −0.355534 −0.177767 0.984073i \(-0.556887\pi\)
−0.177767 + 0.984073i \(0.556887\pi\)
\(642\) −3.32256e9 −0.495565
\(643\) 9.35246e9 1.38735 0.693677 0.720286i \(-0.255989\pi\)
0.693677 + 0.720286i \(0.255989\pi\)
\(644\) −4.79711e8 −0.0707749
\(645\) −2.78436e9 −0.408570
\(646\) 7.37974e9 1.07703
\(647\) 9.02598e9 1.31017 0.655087 0.755553i \(-0.272632\pi\)
0.655087 + 0.755553i \(0.272632\pi\)
\(648\) 1.29946e8 0.0187608
\(649\) −1.26857e9 −0.182162
\(650\) −1.07525e9 −0.153572
\(651\) −2.33333e8 −0.0331468
\(652\) −5.66648e9 −0.800657
\(653\) 6.74203e9 0.947533 0.473767 0.880650i \(-0.342894\pi\)
0.473767 + 0.880650i \(0.342894\pi\)
\(654\) −1.57522e9 −0.220201
\(655\) −3.07482e9 −0.427538
\(656\) 2.39584e9 0.331356
\(657\) 5.33940e9 0.734537
\(658\) 3.54152e8 0.0484618
\(659\) −5.50026e9 −0.748660 −0.374330 0.927296i \(-0.622127\pi\)
−0.374330 + 0.927296i \(0.622127\pi\)
\(660\) −1.15315e9 −0.156129
\(661\) −3.20844e9 −0.432104 −0.216052 0.976382i \(-0.569318\pi\)
−0.216052 + 0.976382i \(0.569318\pi\)
\(662\) 3.24412e9 0.434605
\(663\) −4.88311e9 −0.650728
\(664\) −2.32919e7 −0.00308757
\(665\) 5.91500e8 0.0779972
\(666\) −1.17464e9 −0.154079
\(667\) −1.66854e10 −2.17719
\(668\) −1.82598e9 −0.237016
\(669\) 3.55468e9 0.458996
\(670\) −2.33684e9 −0.300169
\(671\) −4.60201e9 −0.588057
\(672\) 9.54204e7 0.0121297
\(673\) 6.24821e9 0.790138 0.395069 0.918651i \(-0.370721\pi\)
0.395069 + 0.918651i \(0.370721\pi\)
\(674\) −3.06410e9 −0.385473
\(675\) −1.57062e9 −0.196566
\(676\) 7.19737e8 0.0896108
\(677\) −8.32469e9 −1.03112 −0.515558 0.856855i \(-0.672415\pi\)
−0.515558 + 0.856855i \(0.672415\pi\)
\(678\) 3.27205e9 0.403196
\(679\) −5.45755e8 −0.0669042
\(680\) 1.29754e9 0.158248
\(681\) −4.41437e9 −0.535616
\(682\) 3.29999e9 0.398352
\(683\) 9.99323e9 1.20014 0.600072 0.799946i \(-0.295139\pi\)
0.600072 + 0.799946i \(0.295139\pi\)
\(684\) −4.08554e9 −0.488149
\(685\) 4.39656e9 0.522632
\(686\) −1.36138e9 −0.161007
\(687\) −9.47126e9 −1.11445
\(688\) −3.25850e9 −0.381468
\(689\) −1.29099e10 −1.50368
\(690\) −2.01802e9 −0.233858
\(691\) 1.54902e9 0.178601 0.0893004 0.996005i \(-0.471537\pi\)
0.0893004 + 0.996005i \(0.471537\pi\)
\(692\) −6.67845e9 −0.766134
\(693\) 7.51155e8 0.0857360
\(694\) 9.40595e9 1.06818
\(695\) −5.58921e9 −0.631544
\(696\) 3.31893e9 0.373135
\(697\) 1.18587e10 1.32655
\(698\) 2.30724e9 0.256802
\(699\) −9.97869e8 −0.110510
\(700\) 1.04000e8 0.0114601
\(701\) 1.27712e10 1.40029 0.700143 0.714002i \(-0.253119\pi\)
0.700143 + 0.714002i \(0.253119\pi\)
\(702\) 6.91738e9 0.754679
\(703\) 4.76176e9 0.516920
\(704\) −1.34952e9 −0.145772
\(705\) 1.48982e9 0.160130
\(706\) 6.12105e9 0.654650
\(707\) −1.59959e9 −0.170232
\(708\) 4.41585e8 0.0467625
\(709\) −1.34411e10 −1.41636 −0.708178 0.706034i \(-0.750482\pi\)
−0.708178 + 0.706034i \(0.750482\pi\)
\(710\) −2.03688e8 −0.0213580
\(711\) −7.08942e9 −0.739719
\(712\) 5.01765e8 0.0520979
\(713\) 5.77499e9 0.596675
\(714\) 4.72303e8 0.0485598
\(715\) 5.53539e9 0.566339
\(716\) 6.44872e9 0.656565
\(717\) 6.04926e9 0.612893
\(718\) −4.20723e9 −0.424190
\(719\) 8.53708e9 0.856561 0.428281 0.903646i \(-0.359120\pi\)
0.428281 + 0.903646i \(0.359120\pi\)
\(720\) −7.18336e8 −0.0717239
\(721\) 3.38708e8 0.0336552
\(722\) 9.41103e9 0.930587
\(723\) 1.07604e10 1.05887
\(724\) 2.28379e9 0.223651
\(725\) 3.61734e9 0.352539
\(726\) 1.57130e9 0.152399
\(727\) −1.07165e10 −1.03439 −0.517194 0.855868i \(-0.673023\pi\)
−0.517194 + 0.855868i \(0.673023\pi\)
\(728\) −4.58039e8 −0.0439990
\(729\) 5.79936e9 0.554413
\(730\) −3.80570e9 −0.362080
\(731\) −1.61286e10 −1.52717
\(732\) 1.60194e9 0.150959
\(733\) −1.44465e10 −1.35487 −0.677435 0.735583i \(-0.736908\pi\)
−0.677435 + 0.735583i \(0.736908\pi\)
\(734\) 7.14856e9 0.667241
\(735\) −2.84454e9 −0.264245
\(736\) −2.36166e9 −0.218345
\(737\) 1.20300e10 1.10696
\(738\) −6.56516e9 −0.601241
\(739\) 4.70532e9 0.428878 0.214439 0.976737i \(-0.431208\pi\)
0.214439 + 0.976737i \(0.431208\pi\)
\(740\) 8.37232e8 0.0759512
\(741\) −1.09589e10 −0.989476
\(742\) 1.24866e9 0.112210
\(743\) −1.70883e10 −1.52840 −0.764201 0.644978i \(-0.776867\pi\)
−0.764201 + 0.644978i \(0.776867\pi\)
\(744\) −1.14872e9 −0.102260
\(745\) 2.16684e9 0.191991
\(746\) −6.50663e9 −0.573813
\(747\) 6.38253e7 0.00560235
\(748\) −6.67972e9 −0.583583
\(749\) −1.54262e9 −0.134144
\(750\) 4.37500e8 0.0378672
\(751\) 2.08836e10 1.79914 0.899569 0.436778i \(-0.143881\pi\)
0.899569 + 0.436778i \(0.143881\pi\)
\(752\) 1.74352e9 0.149508
\(753\) −6.62005e9 −0.565040
\(754\) −1.59316e10 −1.35350
\(755\) 7.40477e9 0.626177
\(756\) −6.69061e8 −0.0563170
\(757\) −5.69959e9 −0.477538 −0.238769 0.971076i \(-0.576744\pi\)
−0.238769 + 0.971076i \(0.576744\pi\)
\(758\) −8.16757e9 −0.681163
\(759\) 1.03887e10 0.862416
\(760\) 2.91200e9 0.240627
\(761\) −3.00610e9 −0.247262 −0.123631 0.992328i \(-0.539454\pi\)
−0.123631 + 0.992328i \(0.539454\pi\)
\(762\) −3.39721e9 −0.278150
\(763\) −7.31354e8 −0.0596062
\(764\) 3.37851e9 0.274093
\(765\) −3.55555e9 −0.287139
\(766\) −7.31474e9 −0.588028
\(767\) −2.11970e9 −0.169626
\(768\) 4.69762e8 0.0374209
\(769\) −1.11669e10 −0.885500 −0.442750 0.896645i \(-0.645997\pi\)
−0.442750 + 0.896645i \(0.645997\pi\)
\(770\) −5.35392e8 −0.0422624
\(771\) −6.65929e8 −0.0523284
\(772\) −2.05372e9 −0.160650
\(773\) 1.67216e10 1.30212 0.651058 0.759028i \(-0.274325\pi\)
0.651058 + 0.759028i \(0.274325\pi\)
\(774\) 8.92905e9 0.692169
\(775\) −1.25200e9 −0.0966158
\(776\) −2.68679e9 −0.206404
\(777\) 3.04752e8 0.0233063
\(778\) 1.79155e9 0.136396
\(779\) 2.66140e10 2.01710
\(780\) −1.92685e9 −0.145384
\(781\) 1.04859e9 0.0787636
\(782\) −1.16895e10 −0.874123
\(783\) −2.32714e10 −1.73243
\(784\) −3.32893e9 −0.246717
\(785\) 3.42431e9 0.252656
\(786\) −5.51007e9 −0.404742
\(787\) 4.20103e9 0.307216 0.153608 0.988132i \(-0.450911\pi\)
0.153608 + 0.988132i \(0.450911\pi\)
\(788\) 1.07177e10 0.780294
\(789\) −2.23710e9 −0.162149
\(790\) 5.05304e9 0.364635
\(791\) 1.51917e9 0.109141
\(792\) 3.69799e9 0.264501
\(793\) −7.68969e9 −0.547586
\(794\) 1.44606e10 1.02521
\(795\) 5.25279e9 0.370771
\(796\) 1.03533e10 0.727583
\(797\) 8.80492e9 0.616058 0.308029 0.951377i \(-0.400331\pi\)
0.308029 + 0.951377i \(0.400331\pi\)
\(798\) 1.05997e9 0.0738385
\(799\) 8.62991e9 0.598539
\(800\) 5.12000e8 0.0353553
\(801\) −1.37495e9 −0.0945310
\(802\) −1.30855e10 −0.895736
\(803\) 1.95918e10 1.33527
\(804\) −4.18761e9 −0.284165
\(805\) −9.36936e8 −0.0633030
\(806\) 5.51409e9 0.370938
\(807\) 6.53931e9 0.438000
\(808\) −7.87490e9 −0.525176
\(809\) 7.27057e9 0.482779 0.241390 0.970428i \(-0.422397\pi\)
0.241390 + 0.970428i \(0.422397\pi\)
\(810\) 2.53801e8 0.0167801
\(811\) 3.53384e9 0.232634 0.116317 0.993212i \(-0.462891\pi\)
0.116317 + 0.993212i \(0.462891\pi\)
\(812\) 1.54093e9 0.101004
\(813\) −7.81990e9 −0.510369
\(814\) −4.31007e9 −0.280091
\(815\) −1.10673e10 −0.716129
\(816\) 2.32518e9 0.149810
\(817\) −3.61967e10 −2.32216
\(818\) 1.81905e10 1.16200
\(819\) 1.25514e9 0.0798356
\(820\) 4.67938e9 0.296374
\(821\) −2.76875e10 −1.74615 −0.873077 0.487583i \(-0.837879\pi\)
−0.873077 + 0.487583i \(0.837879\pi\)
\(822\) 7.87863e9 0.494766
\(823\) 1.16831e10 0.730562 0.365281 0.930897i \(-0.380973\pi\)
0.365281 + 0.930897i \(0.380973\pi\)
\(824\) 1.66749e9 0.103829
\(825\) −2.25225e9 −0.139646
\(826\) 2.05021e8 0.0126581
\(827\) 2.45526e9 0.150948 0.0754741 0.997148i \(-0.475953\pi\)
0.0754741 + 0.997148i \(0.475953\pi\)
\(828\) 6.47149e9 0.396185
\(829\) 4.71294e9 0.287310 0.143655 0.989628i \(-0.454114\pi\)
0.143655 + 0.989628i \(0.454114\pi\)
\(830\) −4.54920e7 −0.00276161
\(831\) 4.58709e9 0.277290
\(832\) −2.25496e9 −0.135740
\(833\) −1.64772e10 −0.987704
\(834\) −1.00159e10 −0.597870
\(835\) −3.56637e9 −0.211994
\(836\) −1.49910e10 −0.887377
\(837\) 8.05447e9 0.474786
\(838\) 3.23483e9 0.189888
\(839\) −2.77058e10 −1.61958 −0.809792 0.586717i \(-0.800420\pi\)
−0.809792 + 0.586717i \(0.800420\pi\)
\(840\) 1.86368e8 0.0108491
\(841\) 3.63470e10 2.10709
\(842\) 6.69835e7 0.00386701
\(843\) 3.58571e9 0.206148
\(844\) 1.14188e10 0.653766
\(845\) 1.40574e9 0.0801503
\(846\) −4.77765e9 −0.271280
\(847\) 7.29532e8 0.0412527
\(848\) 6.14727e9 0.346176
\(849\) −7.54261e9 −0.423004
\(850\) 2.53425e9 0.141541
\(851\) −7.54262e9 −0.419536
\(852\) −3.65009e8 −0.0202192
\(853\) −9.59318e9 −0.529226 −0.264613 0.964355i \(-0.585244\pi\)
−0.264613 + 0.964355i \(0.585244\pi\)
\(854\) 7.43760e8 0.0408630
\(855\) −7.97956e9 −0.436614
\(856\) −7.59443e9 −0.413844
\(857\) 2.00524e10 1.08826 0.544131 0.839000i \(-0.316859\pi\)
0.544131 + 0.839000i \(0.316859\pi\)
\(858\) 9.91941e9 0.536143
\(859\) −1.40320e9 −0.0755344 −0.0377672 0.999287i \(-0.512025\pi\)
−0.0377672 + 0.999287i \(0.512025\pi\)
\(860\) −6.36426e9 −0.341195
\(861\) 1.70329e9 0.0909449
\(862\) −2.21930e9 −0.118016
\(863\) −3.50518e10 −1.85640 −0.928201 0.372079i \(-0.878645\pi\)
−0.928201 + 0.372079i \(0.878645\pi\)
\(864\) −3.29384e9 −0.173742
\(865\) −1.30439e10 −0.685251
\(866\) 7.03014e9 0.367834
\(867\) 1.94993e7 0.00101614
\(868\) −5.33332e8 −0.0276808
\(869\) −2.60130e10 −1.34469
\(870\) 6.48228e9 0.333742
\(871\) 2.01015e10 1.03078
\(872\) −3.60051e9 −0.183889
\(873\) 7.36245e9 0.374518
\(874\) −2.62342e10 −1.32916
\(875\) 2.03125e8 0.0102503
\(876\) −6.81982e9 −0.342774
\(877\) 1.87881e10 0.940556 0.470278 0.882518i \(-0.344154\pi\)
0.470278 + 0.882518i \(0.344154\pi\)
\(878\) 7.16805e9 0.357413
\(879\) −4.36187e9 −0.216626
\(880\) −2.63578e9 −0.130382
\(881\) 7.29040e9 0.359199 0.179600 0.983740i \(-0.442520\pi\)
0.179600 + 0.983740i \(0.442520\pi\)
\(882\) 9.12205e9 0.447664
\(883\) 3.83994e10 1.87699 0.938496 0.345291i \(-0.112220\pi\)
0.938496 + 0.345291i \(0.112220\pi\)
\(884\) −1.11614e10 −0.543420
\(885\) 8.62470e8 0.0418256
\(886\) −3.06453e8 −0.0148029
\(887\) −1.17256e10 −0.564159 −0.282079 0.959391i \(-0.591024\pi\)
−0.282079 + 0.959391i \(0.591024\pi\)
\(888\) 1.50032e9 0.0719016
\(889\) −1.57727e9 −0.0752924
\(890\) 9.80010e8 0.0465978
\(891\) −1.30657e9 −0.0618814
\(892\) 8.12497e9 0.383305
\(893\) 1.93677e10 0.910119
\(894\) 3.88298e9 0.181754
\(895\) 1.25951e10 0.587249
\(896\) 2.18104e8 0.0101294
\(897\) 1.73590e10 0.803065
\(898\) 6.04571e9 0.278600
\(899\) −1.85504e10 −0.851521
\(900\) −1.40300e9 −0.0641518
\(901\) 3.04272e10 1.38588
\(902\) −2.40894e10 −1.09296
\(903\) −2.31659e9 −0.104699
\(904\) 7.47898e9 0.336707
\(905\) 4.46053e9 0.200040
\(906\) 1.32693e10 0.592790
\(907\) 5.31183e9 0.236384 0.118192 0.992991i \(-0.462290\pi\)
0.118192 + 0.992991i \(0.462290\pi\)
\(908\) −1.00900e10 −0.447291
\(909\) 2.15791e10 0.952926
\(910\) −8.94608e8 −0.0393539
\(911\) 1.84585e10 0.808877 0.404439 0.914565i \(-0.367467\pi\)
0.404439 + 0.914565i \(0.367467\pi\)
\(912\) 5.21830e9 0.227797
\(913\) 2.34193e8 0.0101842
\(914\) 1.77926e10 0.770773
\(915\) 3.12880e9 0.135022
\(916\) −2.16486e10 −0.930670
\(917\) −2.55825e9 −0.109560
\(918\) −1.63035e10 −0.695557
\(919\) −2.59035e9 −0.110092 −0.0550459 0.998484i \(-0.517531\pi\)
−0.0550459 + 0.998484i \(0.517531\pi\)
\(920\) −4.61261e9 −0.195294
\(921\) −1.36395e10 −0.575295
\(922\) 9.56035e9 0.401713
\(923\) 1.75212e9 0.0733430
\(924\) −9.59422e8 −0.0400090
\(925\) 1.63522e9 0.0679328
\(926\) −3.06745e10 −1.26952
\(927\) −4.56930e9 −0.188396
\(928\) 7.58612e9 0.311603
\(929\) 1.64437e9 0.0672893 0.0336446 0.999434i \(-0.489289\pi\)
0.0336446 + 0.999434i \(0.489289\pi\)
\(930\) −2.24358e9 −0.0914644
\(931\) −3.69791e10 −1.50187
\(932\) −2.28084e9 −0.0922868
\(933\) 8.08896e9 0.326067
\(934\) −1.31643e10 −0.528670
\(935\) −1.30463e10 −0.521972
\(936\) 6.17913e9 0.246298
\(937\) −4.12858e10 −1.63950 −0.819751 0.572720i \(-0.805888\pi\)
−0.819751 + 0.572720i \(0.805888\pi\)
\(938\) −1.94425e9 −0.0769204
\(939\) −1.14647e10 −0.451890
\(940\) 3.40531e9 0.133724
\(941\) −2.22631e10 −0.871009 −0.435504 0.900187i \(-0.643430\pi\)
−0.435504 + 0.900187i \(0.643430\pi\)
\(942\) 6.13636e9 0.239184
\(943\) −4.21565e10 −1.63709
\(944\) 1.00934e9 0.0390512
\(945\) −1.30676e9 −0.0503715
\(946\) 3.27632e10 1.25825
\(947\) 5.66138e9 0.216619 0.108310 0.994117i \(-0.465456\pi\)
0.108310 + 0.994117i \(0.465456\pi\)
\(948\) 9.05505e9 0.345193
\(949\) 3.27366e10 1.24338
\(950\) 5.68750e9 0.215223
\(951\) −7.53968e9 −0.284263
\(952\) 1.07955e9 0.0405521
\(953\) 4.12640e9 0.154435 0.0772176 0.997014i \(-0.475396\pi\)
0.0772176 + 0.997014i \(0.475396\pi\)
\(954\) −1.68450e10 −0.628132
\(955\) 6.59864e9 0.245156
\(956\) 1.38269e10 0.511825
\(957\) −3.33708e10 −1.23076
\(958\) −6.24665e9 −0.229545
\(959\) 3.65794e9 0.133928
\(960\) 9.17504e8 0.0334702
\(961\) −2.10921e10 −0.766634
\(962\) −7.20187e9 −0.260815
\(963\) 2.08105e10 0.750915
\(964\) 2.45952e10 0.884260
\(965\) −4.01117e9 −0.143690
\(966\) −1.67899e9 −0.0599278
\(967\) 1.68951e10 0.600853 0.300426 0.953805i \(-0.402871\pi\)
0.300426 + 0.953805i \(0.402871\pi\)
\(968\) 3.59154e9 0.127267
\(969\) 2.58291e10 0.911960
\(970\) −5.24765e9 −0.184614
\(971\) −2.00638e9 −0.0703309 −0.0351655 0.999382i \(-0.511196\pi\)
−0.0351655 + 0.999382i \(0.511196\pi\)
\(972\) 1.45244e10 0.507301
\(973\) −4.65022e9 −0.161837
\(974\) 3.83843e10 1.33106
\(975\) −3.76337e9 −0.130035
\(976\) 3.66159e9 0.126065
\(977\) −3.75233e9 −0.128727 −0.0643636 0.997927i \(-0.520502\pi\)
−0.0643636 + 0.997927i \(0.520502\pi\)
\(978\) −1.98327e10 −0.677946
\(979\) −5.04509e9 −0.171842
\(980\) −6.50182e9 −0.220670
\(981\) 9.86625e9 0.333665
\(982\) −1.13315e10 −0.381853
\(983\) −2.82796e10 −0.949591 −0.474795 0.880096i \(-0.657478\pi\)
−0.474795 + 0.880096i \(0.657478\pi\)
\(984\) 8.38544e9 0.280571
\(985\) 2.09329e10 0.697916
\(986\) 3.75491e10 1.24747
\(987\) 1.23953e9 0.0410344
\(988\) −2.50490e10 −0.826307
\(989\) 5.73356e10 1.88468
\(990\) 7.22264e9 0.236577
\(991\) −1.43828e10 −0.469447 −0.234723 0.972062i \(-0.575418\pi\)
−0.234723 + 0.972062i \(0.575418\pi\)
\(992\) −2.62563e9 −0.0853972
\(993\) 1.13544e10 0.367996
\(994\) −1.69468e8 −0.00547314
\(995\) 2.02213e10 0.650770
\(996\) −8.15217e7 −0.00261436
\(997\) −1.74539e10 −0.557775 −0.278887 0.960324i \(-0.589966\pi\)
−0.278887 + 0.960324i \(0.589966\pi\)
\(998\) −3.34502e10 −1.06523
\(999\) −1.05198e10 −0.333833
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 10.8.a.a.1.1 1
3.2 odd 2 90.8.a.a.1.1 1
4.3 odd 2 80.8.a.a.1.1 1
5.2 odd 4 50.8.b.d.49.2 2
5.3 odd 4 50.8.b.d.49.1 2
5.4 even 2 50.8.a.b.1.1 1
7.6 odd 2 490.8.a.b.1.1 1
8.3 odd 2 320.8.a.f.1.1 1
8.5 even 2 320.8.a.c.1.1 1
15.2 even 4 450.8.c.p.199.1 2
15.8 even 4 450.8.c.p.199.2 2
15.14 odd 2 450.8.a.t.1.1 1
20.3 even 4 400.8.c.i.49.1 2
20.7 even 4 400.8.c.i.49.2 2
20.19 odd 2 400.8.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.8.a.a.1.1 1 1.1 even 1 trivial
50.8.a.b.1.1 1 5.4 even 2
50.8.b.d.49.1 2 5.3 odd 4
50.8.b.d.49.2 2 5.2 odd 4
80.8.a.a.1.1 1 4.3 odd 2
90.8.a.a.1.1 1 3.2 odd 2
320.8.a.c.1.1 1 8.5 even 2
320.8.a.f.1.1 1 8.3 odd 2
400.8.a.m.1.1 1 20.19 odd 2
400.8.c.i.49.1 2 20.3 even 4
400.8.c.i.49.2 2 20.7 even 4
450.8.a.t.1.1 1 15.14 odd 2
450.8.c.p.199.1 2 15.2 even 4
450.8.c.p.199.2 2 15.8 even 4
490.8.a.b.1.1 1 7.6 odd 2