Properties

Label 177.12.a.a.1.8
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-43.6565 q^{2} -243.000 q^{3} -142.110 q^{4} +7124.43 q^{5} +10608.5 q^{6} +50312.1 q^{7} +95612.5 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-43.6565 q^{2} -243.000 q^{3} -142.110 q^{4} +7124.43 q^{5} +10608.5 q^{6} +50312.1 q^{7} +95612.5 q^{8} +59049.0 q^{9} -311028. q^{10} +365340. q^{11} +34532.8 q^{12} +675470. q^{13} -2.19645e6 q^{14} -1.73124e6 q^{15} -3.88307e6 q^{16} +5.75558e6 q^{17} -2.57787e6 q^{18} +6.02412e6 q^{19} -1.01246e6 q^{20} -1.22259e7 q^{21} -1.59495e7 q^{22} -3.76836e7 q^{23} -2.32338e7 q^{24} +1.92941e6 q^{25} -2.94886e7 q^{26} -1.43489e7 q^{27} -7.14987e6 q^{28} -1.23657e8 q^{29} +7.55797e7 q^{30} -3.05132e8 q^{31} -2.62934e7 q^{32} -8.87776e7 q^{33} -2.51269e8 q^{34} +3.58445e8 q^{35} -8.39147e6 q^{36} +4.58803e8 q^{37} -2.62992e8 q^{38} -1.64139e8 q^{39} +6.81185e8 q^{40} -1.12154e9 q^{41} +5.33738e8 q^{42} -1.10767e9 q^{43} -5.19186e7 q^{44} +4.20691e8 q^{45} +1.64513e9 q^{46} +1.34464e9 q^{47} +9.43585e8 q^{48} +5.53985e8 q^{49} -8.42311e7 q^{50} -1.39861e9 q^{51} -9.59912e7 q^{52} -5.46355e9 q^{53} +6.26423e8 q^{54} +2.60284e9 q^{55} +4.81047e9 q^{56} -1.46386e9 q^{57} +5.39841e9 q^{58} +7.14924e8 q^{59} +2.46027e8 q^{60} +2.87070e8 q^{61} +1.33210e10 q^{62} +2.97088e9 q^{63} +9.10040e9 q^{64} +4.81234e9 q^{65} +3.87572e9 q^{66} +6.29348e9 q^{67} -8.17927e8 q^{68} +9.15711e9 q^{69} -1.56485e10 q^{70} -1.39937e10 q^{71} +5.64583e9 q^{72} -2.18805e10 q^{73} -2.00297e10 q^{74} -4.68846e8 q^{75} -8.56090e8 q^{76} +1.83810e10 q^{77} +7.16574e9 q^{78} -1.00071e9 q^{79} -2.76646e10 q^{80} +3.48678e9 q^{81} +4.89627e10 q^{82} -8.58714e9 q^{83} +1.73742e9 q^{84} +4.10052e10 q^{85} +4.83571e10 q^{86} +3.00485e10 q^{87} +3.49311e10 q^{88} -9.54335e10 q^{89} -1.83659e10 q^{90} +3.39843e10 q^{91} +5.35522e9 q^{92} +7.41471e10 q^{93} -5.87023e10 q^{94} +4.29185e10 q^{95} +6.38929e9 q^{96} +7.20283e10 q^{97} -2.41851e10 q^{98} +2.15730e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −43.6565 −0.964681 −0.482341 0.875984i \(-0.660213\pi\)
−0.482341 + 0.875984i \(0.660213\pi\)
\(3\) −243.000 −0.577350
\(4\) −142.110 −0.0693898
\(5\) 7124.43 1.01957 0.509783 0.860303i \(-0.329726\pi\)
0.509783 + 0.860303i \(0.329726\pi\)
\(6\) 10608.5 0.556959
\(7\) 50312.1 1.13145 0.565723 0.824596i \(-0.308597\pi\)
0.565723 + 0.824596i \(0.308597\pi\)
\(8\) 95612.5 1.03162
\(9\) 59049.0 0.333333
\(10\) −311028. −0.983556
\(11\) 365340. 0.683971 0.341985 0.939705i \(-0.388901\pi\)
0.341985 + 0.939705i \(0.388901\pi\)
\(12\) 34532.8 0.0400622
\(13\) 675470. 0.504565 0.252283 0.967654i \(-0.418819\pi\)
0.252283 + 0.967654i \(0.418819\pi\)
\(14\) −2.19645e6 −1.09148
\(15\) −1.73124e6 −0.588647
\(16\) −3.88307e6 −0.925795
\(17\) 5.75558e6 0.983151 0.491575 0.870835i \(-0.336421\pi\)
0.491575 + 0.870835i \(0.336421\pi\)
\(18\) −2.57787e6 −0.321560
\(19\) 6.02412e6 0.558148 0.279074 0.960270i \(-0.409973\pi\)
0.279074 + 0.960270i \(0.409973\pi\)
\(20\) −1.01246e6 −0.0707474
\(21\) −1.22259e7 −0.653240
\(22\) −1.59495e7 −0.659814
\(23\) −3.76836e7 −1.22081 −0.610406 0.792089i \(-0.708994\pi\)
−0.610406 + 0.792089i \(0.708994\pi\)
\(24\) −2.32338e7 −0.595606
\(25\) 1.92941e6 0.0395142
\(26\) −2.94886e7 −0.486745
\(27\) −1.43489e7 −0.192450
\(28\) −7.14987e6 −0.0785108
\(29\) −1.23657e8 −1.11951 −0.559755 0.828658i \(-0.689105\pi\)
−0.559755 + 0.828658i \(0.689105\pi\)
\(30\) 7.55797e7 0.567856
\(31\) −3.05132e8 −1.91425 −0.957125 0.289674i \(-0.906453\pi\)
−0.957125 + 0.289674i \(0.906453\pi\)
\(32\) −2.62934e7 −0.138523
\(33\) −8.87776e7 −0.394891
\(34\) −2.51269e8 −0.948427
\(35\) 3.58445e8 1.15358
\(36\) −8.39147e6 −0.0231299
\(37\) 4.58803e8 1.08772 0.543860 0.839176i \(-0.316962\pi\)
0.543860 + 0.839176i \(0.316962\pi\)
\(38\) −2.62992e8 −0.538435
\(39\) −1.64139e8 −0.291311
\(40\) 6.81185e8 1.05180
\(41\) −1.12154e9 −1.51184 −0.755919 0.654665i \(-0.772810\pi\)
−0.755919 + 0.654665i \(0.772810\pi\)
\(42\) 5.33738e8 0.630169
\(43\) −1.10767e9 −1.14904 −0.574519 0.818491i \(-0.694811\pi\)
−0.574519 + 0.818491i \(0.694811\pi\)
\(44\) −5.19186e7 −0.0474606
\(45\) 4.20691e8 0.339855
\(46\) 1.64513e9 1.17769
\(47\) 1.34464e9 0.855200 0.427600 0.903968i \(-0.359359\pi\)
0.427600 + 0.903968i \(0.359359\pi\)
\(48\) 9.43585e8 0.534508
\(49\) 5.53985e8 0.280169
\(50\) −8.42311e7 −0.0381186
\(51\) −1.39861e9 −0.567622
\(52\) −9.59912e7 −0.0350117
\(53\) −5.46355e9 −1.79456 −0.897279 0.441463i \(-0.854460\pi\)
−0.897279 + 0.441463i \(0.854460\pi\)
\(54\) 6.26423e8 0.185653
\(55\) 2.60284e9 0.697353
\(56\) 4.81047e9 1.16722
\(57\) −1.46386e9 −0.322247
\(58\) 5.39841e9 1.07997
\(59\) 7.14924e8 0.130189
\(60\) 2.46027e8 0.0408461
\(61\) 2.87070e8 0.0435184 0.0217592 0.999763i \(-0.493073\pi\)
0.0217592 + 0.999763i \(0.493073\pi\)
\(62\) 1.33210e10 1.84664
\(63\) 2.97088e9 0.377148
\(64\) 9.10040e9 1.05943
\(65\) 4.81234e9 0.514438
\(66\) 3.87572e9 0.380944
\(67\) 6.29348e9 0.569481 0.284740 0.958605i \(-0.408093\pi\)
0.284740 + 0.958605i \(0.408093\pi\)
\(68\) −8.17927e8 −0.0682206
\(69\) 9.15711e9 0.704836
\(70\) −1.56485e10 −1.11284
\(71\) −1.39937e10 −0.920474 −0.460237 0.887796i \(-0.652235\pi\)
−0.460237 + 0.887796i \(0.652235\pi\)
\(72\) 5.64583e9 0.343873
\(73\) −2.18805e10 −1.23533 −0.617663 0.786443i \(-0.711920\pi\)
−0.617663 + 0.786443i \(0.711920\pi\)
\(74\) −2.00297e10 −1.04930
\(75\) −4.68846e8 −0.0228136
\(76\) −8.56090e8 −0.0387297
\(77\) 1.83810e10 0.773876
\(78\) 7.16574e9 0.281022
\(79\) −1.00071e9 −0.0365896 −0.0182948 0.999833i \(-0.505824\pi\)
−0.0182948 + 0.999833i \(0.505824\pi\)
\(80\) −2.76646e10 −0.943909
\(81\) 3.48678e9 0.111111
\(82\) 4.89627e10 1.45844
\(83\) −8.58714e9 −0.239287 −0.119643 0.992817i \(-0.538175\pi\)
−0.119643 + 0.992817i \(0.538175\pi\)
\(84\) 1.73742e9 0.0453282
\(85\) 4.10052e10 1.00239
\(86\) 4.83571e10 1.10846
\(87\) 3.00485e10 0.646350
\(88\) 3.49311e10 0.705598
\(89\) −9.54335e10 −1.81157 −0.905787 0.423734i \(-0.860719\pi\)
−0.905787 + 0.423734i \(0.860719\pi\)
\(90\) −1.83659e10 −0.327852
\(91\) 3.39843e10 0.570888
\(92\) 5.35522e9 0.0847119
\(93\) 7.41471e10 1.10519
\(94\) −5.87023e10 −0.824996
\(95\) 4.29185e10 0.569068
\(96\) 6.38929e9 0.0799763
\(97\) 7.20283e10 0.851645 0.425823 0.904807i \(-0.359985\pi\)
0.425823 + 0.904807i \(0.359985\pi\)
\(98\) −2.41851e10 −0.270274
\(99\) 2.15730e10 0.227990
\(100\) −2.74188e8 −0.00274188
\(101\) 1.68819e11 1.59828 0.799141 0.601144i \(-0.205288\pi\)
0.799141 + 0.601144i \(0.205288\pi\)
\(102\) 6.10583e10 0.547575
\(103\) −1.88225e11 −1.59982 −0.799912 0.600117i \(-0.795121\pi\)
−0.799912 + 0.600117i \(0.795121\pi\)
\(104\) 6.45834e10 0.520520
\(105\) −8.71022e10 −0.666021
\(106\) 2.38519e11 1.73118
\(107\) −1.29332e11 −0.891447 −0.445723 0.895171i \(-0.647053\pi\)
−0.445723 + 0.895171i \(0.647053\pi\)
\(108\) 2.03913e9 0.0133541
\(109\) −2.33095e11 −1.45107 −0.725535 0.688186i \(-0.758407\pi\)
−0.725535 + 0.688186i \(0.758407\pi\)
\(110\) −1.13631e11 −0.672724
\(111\) −1.11489e11 −0.627995
\(112\) −1.95365e11 −1.04749
\(113\) 1.34951e11 0.689041 0.344520 0.938779i \(-0.388042\pi\)
0.344520 + 0.938779i \(0.388042\pi\)
\(114\) 6.39071e10 0.310865
\(115\) −2.68474e11 −1.24470
\(116\) 1.75729e10 0.0776826
\(117\) 3.98858e10 0.168188
\(118\) −3.12111e10 −0.125591
\(119\) 2.89576e11 1.11238
\(120\) −1.65528e11 −0.607260
\(121\) −1.51838e11 −0.532184
\(122\) −1.25325e10 −0.0419814
\(123\) 2.72535e11 0.872860
\(124\) 4.33624e10 0.132829
\(125\) −3.34127e11 −0.979278
\(126\) −1.29698e11 −0.363828
\(127\) 3.29222e11 0.884236 0.442118 0.896957i \(-0.354227\pi\)
0.442118 + 0.896957i \(0.354227\pi\)
\(128\) −3.43443e11 −0.883485
\(129\) 2.69164e11 0.663398
\(130\) −2.10090e11 −0.496268
\(131\) −7.93659e11 −1.79739 −0.898694 0.438577i \(-0.855483\pi\)
−0.898694 + 0.438577i \(0.855483\pi\)
\(132\) 1.26162e10 0.0274014
\(133\) 3.03087e11 0.631514
\(134\) −2.74751e11 −0.549368
\(135\) −1.02228e11 −0.196216
\(136\) 5.50306e11 1.01424
\(137\) 1.10881e12 1.96289 0.981443 0.191755i \(-0.0614181\pi\)
0.981443 + 0.191755i \(0.0614181\pi\)
\(138\) −3.99767e11 −0.679942
\(139\) 9.06147e10 0.148121 0.0740606 0.997254i \(-0.476404\pi\)
0.0740606 + 0.997254i \(0.476404\pi\)
\(140\) −5.09388e10 −0.0800469
\(141\) −3.26747e11 −0.493750
\(142\) 6.10915e11 0.887964
\(143\) 2.46776e11 0.345108
\(144\) −2.29291e11 −0.308598
\(145\) −8.80983e11 −1.14141
\(146\) 9.55226e11 1.19170
\(147\) −1.34618e11 −0.161756
\(148\) −6.52006e10 −0.0754766
\(149\) −1.10067e12 −1.22781 −0.613905 0.789380i \(-0.710402\pi\)
−0.613905 + 0.789380i \(0.710402\pi\)
\(150\) 2.04682e10 0.0220078
\(151\) −3.52927e11 −0.365857 −0.182928 0.983126i \(-0.558558\pi\)
−0.182928 + 0.983126i \(0.558558\pi\)
\(152\) 5.75982e11 0.575797
\(153\) 3.39861e11 0.327717
\(154\) −8.02452e11 −0.746543
\(155\) −2.17389e12 −1.95170
\(156\) 2.33259e10 0.0202140
\(157\) 1.37272e12 1.14851 0.574254 0.818677i \(-0.305292\pi\)
0.574254 + 0.818677i \(0.305292\pi\)
\(158\) 4.36873e10 0.0352973
\(159\) 1.32764e12 1.03609
\(160\) −1.87325e11 −0.141233
\(161\) −1.89594e12 −1.38128
\(162\) −1.52221e11 −0.107187
\(163\) −4.44642e11 −0.302677 −0.151338 0.988482i \(-0.548358\pi\)
−0.151338 + 0.988482i \(0.548358\pi\)
\(164\) 1.59383e11 0.104906
\(165\) −6.32490e11 −0.402617
\(166\) 3.74884e11 0.230836
\(167\) −8.71475e11 −0.519176 −0.259588 0.965719i \(-0.583587\pi\)
−0.259588 + 0.965719i \(0.583587\pi\)
\(168\) −1.16894e12 −0.673896
\(169\) −1.33590e12 −0.745414
\(170\) −1.79015e12 −0.966984
\(171\) 3.55719e11 0.186049
\(172\) 1.57412e11 0.0797315
\(173\) −6.43858e11 −0.315891 −0.157945 0.987448i \(-0.550487\pi\)
−0.157945 + 0.987448i \(0.550487\pi\)
\(174\) −1.31181e12 −0.623522
\(175\) 9.70725e10 0.0447082
\(176\) −1.41864e12 −0.633217
\(177\) −1.73727e11 −0.0751646
\(178\) 4.16629e12 1.74759
\(179\) −4.41573e12 −1.79602 −0.898009 0.439978i \(-0.854986\pi\)
−0.898009 + 0.439978i \(0.854986\pi\)
\(180\) −5.97845e10 −0.0235825
\(181\) −3.14280e12 −1.20250 −0.601250 0.799061i \(-0.705330\pi\)
−0.601250 + 0.799061i \(0.705330\pi\)
\(182\) −1.48364e12 −0.550725
\(183\) −6.97580e10 −0.0251254
\(184\) −3.60302e12 −1.25941
\(185\) 3.26871e12 1.10900
\(186\) −3.23700e12 −1.06616
\(187\) 2.10274e12 0.672446
\(188\) −1.91087e11 −0.0593422
\(189\) −7.21924e11 −0.217747
\(190\) −1.87367e12 −0.548970
\(191\) 6.11106e12 1.73953 0.869767 0.493462i \(-0.164269\pi\)
0.869767 + 0.493462i \(0.164269\pi\)
\(192\) −2.21140e12 −0.611660
\(193\) 1.44042e11 0.0387191 0.0193595 0.999813i \(-0.493837\pi\)
0.0193595 + 0.999813i \(0.493837\pi\)
\(194\) −3.14450e12 −0.821566
\(195\) −1.16940e12 −0.297011
\(196\) −7.87270e10 −0.0194409
\(197\) 5.66244e12 1.35969 0.679844 0.733357i \(-0.262047\pi\)
0.679844 + 0.733357i \(0.262047\pi\)
\(198\) −9.41800e11 −0.219938
\(199\) −5.90518e12 −1.34135 −0.670673 0.741753i \(-0.733995\pi\)
−0.670673 + 0.741753i \(0.733995\pi\)
\(200\) 1.84475e11 0.0407637
\(201\) −1.52931e12 −0.328790
\(202\) −7.37004e12 −1.54183
\(203\) −6.22143e12 −1.26667
\(204\) 1.98756e11 0.0393872
\(205\) −7.99037e12 −1.54142
\(206\) 8.21724e12 1.54332
\(207\) −2.22518e12 −0.406937
\(208\) −2.62289e12 −0.467124
\(209\) 2.20085e12 0.381757
\(210\) 3.80258e12 0.642499
\(211\) −4.07921e12 −0.671464 −0.335732 0.941958i \(-0.608984\pi\)
−0.335732 + 0.941958i \(0.608984\pi\)
\(212\) 7.76427e11 0.124524
\(213\) 3.40047e12 0.531436
\(214\) 5.64618e12 0.859962
\(215\) −7.89153e12 −1.17152
\(216\) −1.37194e12 −0.198535
\(217\) −1.53519e13 −2.16587
\(218\) 1.01761e13 1.39982
\(219\) 5.31696e12 0.713216
\(220\) −3.69890e11 −0.0483892
\(221\) 3.88772e12 0.496064
\(222\) 4.86723e12 0.605815
\(223\) −4.49033e12 −0.545258 −0.272629 0.962119i \(-0.587893\pi\)
−0.272629 + 0.962119i \(0.587893\pi\)
\(224\) −1.32288e12 −0.156731
\(225\) 1.13929e11 0.0131714
\(226\) −5.89149e12 −0.664705
\(227\) −1.31337e13 −1.44625 −0.723126 0.690716i \(-0.757295\pi\)
−0.723126 + 0.690716i \(0.757295\pi\)
\(228\) 2.08030e11 0.0223606
\(229\) 4.35598e12 0.457078 0.228539 0.973535i \(-0.426605\pi\)
0.228539 + 0.973535i \(0.426605\pi\)
\(230\) 1.17206e13 1.20074
\(231\) −4.46659e12 −0.446797
\(232\) −1.18231e13 −1.15491
\(233\) 6.64973e12 0.634376 0.317188 0.948363i \(-0.397261\pi\)
0.317188 + 0.948363i \(0.397261\pi\)
\(234\) −1.74127e12 −0.162248
\(235\) 9.57979e12 0.871933
\(236\) −1.01598e11 −0.00903378
\(237\) 2.43172e11 0.0211250
\(238\) −1.26419e13 −1.07309
\(239\) 7.80698e11 0.0647582 0.0323791 0.999476i \(-0.489692\pi\)
0.0323791 + 0.999476i \(0.489692\pi\)
\(240\) 6.72251e12 0.544966
\(241\) −4.00251e12 −0.317131 −0.158566 0.987348i \(-0.550687\pi\)
−0.158566 + 0.987348i \(0.550687\pi\)
\(242\) 6.62873e12 0.513388
\(243\) −8.47289e11 −0.0641500
\(244\) −4.07956e10 −0.00301974
\(245\) 3.94683e12 0.285651
\(246\) −1.18979e13 −0.842032
\(247\) 4.06911e12 0.281622
\(248\) −2.91745e13 −1.97478
\(249\) 2.08667e12 0.138152
\(250\) 1.45868e13 0.944692
\(251\) 9.31420e12 0.590120 0.295060 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(252\) −4.22193e11 −0.0261703
\(253\) −1.37673e13 −0.835000
\(254\) −1.43727e13 −0.853006
\(255\) −9.96428e12 −0.578728
\(256\) −3.64411e12 −0.207144
\(257\) 2.27295e13 1.26462 0.632308 0.774717i \(-0.282108\pi\)
0.632308 + 0.774717i \(0.282108\pi\)
\(258\) −1.17508e13 −0.639967
\(259\) 2.30834e13 1.23070
\(260\) −6.83883e11 −0.0356967
\(261\) −7.30180e12 −0.373170
\(262\) 3.46484e13 1.73391
\(263\) 3.42919e13 1.68049 0.840244 0.542208i \(-0.182412\pi\)
0.840244 + 0.542208i \(0.182412\pi\)
\(264\) −8.48825e12 −0.407377
\(265\) −3.89247e13 −1.82967
\(266\) −1.32317e13 −0.609209
\(267\) 2.31904e13 1.04591
\(268\) −8.94368e11 −0.0395162
\(269\) 3.52108e13 1.52419 0.762094 0.647467i \(-0.224172\pi\)
0.762094 + 0.647467i \(0.224172\pi\)
\(270\) 4.46291e12 0.189285
\(271\) 1.33516e13 0.554885 0.277442 0.960742i \(-0.410513\pi\)
0.277442 + 0.960742i \(0.410513\pi\)
\(272\) −2.23493e13 −0.910196
\(273\) −8.25819e12 −0.329602
\(274\) −4.84069e13 −1.89356
\(275\) 7.04889e11 0.0270266
\(276\) −1.30132e12 −0.0489084
\(277\) −3.73820e12 −0.137728 −0.0688642 0.997626i \(-0.521938\pi\)
−0.0688642 + 0.997626i \(0.521938\pi\)
\(278\) −3.95592e12 −0.142890
\(279\) −1.80178e13 −0.638084
\(280\) 3.42719e13 1.19006
\(281\) 3.84494e13 1.30920 0.654598 0.755978i \(-0.272838\pi\)
0.654598 + 0.755978i \(0.272838\pi\)
\(282\) 1.42646e13 0.476311
\(283\) 4.47772e13 1.46633 0.733165 0.680051i \(-0.238042\pi\)
0.733165 + 0.680051i \(0.238042\pi\)
\(284\) 1.98865e12 0.0638715
\(285\) −1.04292e13 −0.328552
\(286\) −1.07734e13 −0.332919
\(287\) −5.64273e13 −1.71056
\(288\) −1.55260e12 −0.0461743
\(289\) −1.14518e12 −0.0334145
\(290\) 3.84606e13 1.10110
\(291\) −1.75029e13 −0.491698
\(292\) 3.10944e12 0.0857190
\(293\) 2.23241e13 0.603951 0.301976 0.953316i \(-0.402354\pi\)
0.301976 + 0.953316i \(0.402354\pi\)
\(294\) 5.87697e12 0.156043
\(295\) 5.09343e12 0.132736
\(296\) 4.38673e13 1.12211
\(297\) −5.24223e12 −0.131630
\(298\) 4.80512e13 1.18445
\(299\) −2.54541e13 −0.615980
\(300\) 6.66278e10 0.00158303
\(301\) −5.57293e13 −1.30007
\(302\) 1.54075e13 0.352935
\(303\) −4.10230e13 −0.922768
\(304\) −2.33921e13 −0.516730
\(305\) 2.04521e12 0.0443699
\(306\) −1.48372e13 −0.316142
\(307\) −4.79932e13 −1.00443 −0.502213 0.864744i \(-0.667481\pi\)
−0.502213 + 0.864744i \(0.667481\pi\)
\(308\) −2.61213e12 −0.0536991
\(309\) 4.57386e13 0.923659
\(310\) 9.49046e13 1.88277
\(311\) 7.55495e13 1.47248 0.736240 0.676720i \(-0.236599\pi\)
0.736240 + 0.676720i \(0.236599\pi\)
\(312\) −1.56938e13 −0.300522
\(313\) −4.55788e13 −0.857568 −0.428784 0.903407i \(-0.641058\pi\)
−0.428784 + 0.903407i \(0.641058\pi\)
\(314\) −5.99281e13 −1.10794
\(315\) 2.11658e13 0.384528
\(316\) 1.42211e11 0.00253894
\(317\) −1.03923e13 −0.182341 −0.0911705 0.995835i \(-0.529061\pi\)
−0.0911705 + 0.995835i \(0.529061\pi\)
\(318\) −5.79602e13 −0.999496
\(319\) −4.51767e13 −0.765712
\(320\) 6.48352e13 1.08015
\(321\) 3.14277e13 0.514677
\(322\) 8.27701e13 1.33250
\(323\) 3.46723e13 0.548743
\(324\) −4.95508e11 −0.00770998
\(325\) 1.30326e12 0.0199375
\(326\) 1.94115e13 0.291986
\(327\) 5.66422e13 0.837775
\(328\) −1.07234e14 −1.55964
\(329\) 6.76517e13 0.967612
\(330\) 2.76123e13 0.388397
\(331\) −1.01393e13 −0.140266 −0.0701331 0.997538i \(-0.522342\pi\)
−0.0701331 + 0.997538i \(0.522342\pi\)
\(332\) 1.22032e12 0.0166041
\(333\) 2.70919e13 0.362573
\(334\) 3.80456e13 0.500839
\(335\) 4.48374e13 0.580623
\(336\) 4.74738e13 0.604767
\(337\) 7.30955e13 0.916065 0.458033 0.888935i \(-0.348554\pi\)
0.458033 + 0.888935i \(0.348554\pi\)
\(338\) 5.83208e13 0.719087
\(339\) −3.27931e13 −0.397818
\(340\) −5.82727e12 −0.0695554
\(341\) −1.11477e14 −1.30929
\(342\) −1.55294e13 −0.179478
\(343\) −7.16114e13 −0.814450
\(344\) −1.05907e14 −1.18537
\(345\) 6.52392e13 0.718627
\(346\) 2.81086e13 0.304734
\(347\) 1.43624e14 1.53255 0.766275 0.642513i \(-0.222108\pi\)
0.766275 + 0.642513i \(0.222108\pi\)
\(348\) −4.27021e12 −0.0448501
\(349\) 7.75416e13 0.801669 0.400834 0.916150i \(-0.368720\pi\)
0.400834 + 0.916150i \(0.368720\pi\)
\(350\) −4.23785e12 −0.0431292
\(351\) −9.69225e12 −0.0971037
\(352\) −9.60603e12 −0.0947456
\(353\) −6.17172e13 −0.599301 −0.299651 0.954049i \(-0.596870\pi\)
−0.299651 + 0.954049i \(0.596870\pi\)
\(354\) 7.58429e12 0.0725099
\(355\) −9.96971e13 −0.938483
\(356\) 1.35621e13 0.125705
\(357\) −7.03669e13 −0.642234
\(358\) 1.92775e14 1.73258
\(359\) 7.25911e13 0.642486 0.321243 0.946997i \(-0.395899\pi\)
0.321243 + 0.946997i \(0.395899\pi\)
\(360\) 4.02233e13 0.350602
\(361\) −8.02002e13 −0.688471
\(362\) 1.37204e14 1.16003
\(363\) 3.68967e13 0.307257
\(364\) −4.82952e12 −0.0396138
\(365\) −1.55886e14 −1.25950
\(366\) 3.04539e12 0.0242380
\(367\) 2.32677e14 1.82427 0.912137 0.409886i \(-0.134432\pi\)
0.912137 + 0.409886i \(0.134432\pi\)
\(368\) 1.46328e14 1.13022
\(369\) −6.62261e13 −0.503946
\(370\) −1.42701e14 −1.06983
\(371\) −2.74883e14 −2.03045
\(372\) −1.05371e13 −0.0766891
\(373\) −1.34371e14 −0.963622 −0.481811 0.876275i \(-0.660021\pi\)
−0.481811 + 0.876275i \(0.660021\pi\)
\(374\) −9.17985e13 −0.648696
\(375\) 8.11928e13 0.565387
\(376\) 1.28564e14 0.882242
\(377\) −8.35263e13 −0.564866
\(378\) 3.15167e13 0.210056
\(379\) −2.47448e13 −0.162543 −0.0812714 0.996692i \(-0.525898\pi\)
−0.0812714 + 0.996692i \(0.525898\pi\)
\(380\) −6.09915e12 −0.0394875
\(381\) −8.00010e13 −0.510514
\(382\) −2.66787e14 −1.67810
\(383\) 1.39855e14 0.867134 0.433567 0.901121i \(-0.357255\pi\)
0.433567 + 0.901121i \(0.357255\pi\)
\(384\) 8.34566e13 0.510081
\(385\) 1.30954e14 0.789017
\(386\) −6.28838e12 −0.0373516
\(387\) −6.54069e13 −0.383013
\(388\) −1.02360e13 −0.0590955
\(389\) −1.11895e14 −0.636924 −0.318462 0.947936i \(-0.603166\pi\)
−0.318462 + 0.947936i \(0.603166\pi\)
\(390\) 5.10518e13 0.286521
\(391\) −2.16891e14 −1.20024
\(392\) 5.29679e13 0.289028
\(393\) 1.92859e14 1.03772
\(394\) −2.47202e14 −1.31167
\(395\) −7.12946e12 −0.0373055
\(396\) −3.06574e12 −0.0158202
\(397\) −1.57558e14 −0.801847 −0.400924 0.916111i \(-0.631311\pi\)
−0.400924 + 0.916111i \(0.631311\pi\)
\(398\) 2.57799e14 1.29397
\(399\) −7.36500e13 −0.364605
\(400\) −7.49201e12 −0.0365821
\(401\) −1.29783e14 −0.625063 −0.312531 0.949907i \(-0.601177\pi\)
−0.312531 + 0.949907i \(0.601177\pi\)
\(402\) 6.67645e13 0.317178
\(403\) −2.06108e14 −0.965865
\(404\) −2.39909e13 −0.110904
\(405\) 2.48414e13 0.113285
\(406\) 2.71606e14 1.22193
\(407\) 1.67619e14 0.743968
\(408\) −1.33724e14 −0.585571
\(409\) 2.02205e14 0.873604 0.436802 0.899558i \(-0.356111\pi\)
0.436802 + 0.899558i \(0.356111\pi\)
\(410\) 3.48831e14 1.48698
\(411\) −2.69441e14 −1.13327
\(412\) 2.67487e13 0.111011
\(413\) 3.59694e13 0.147302
\(414\) 9.71434e13 0.392565
\(415\) −6.11785e13 −0.243969
\(416\) −1.77604e13 −0.0698939
\(417\) −2.20194e13 −0.0855178
\(418\) −9.60816e13 −0.368274
\(419\) 3.51636e14 1.33020 0.665099 0.746755i \(-0.268389\pi\)
0.665099 + 0.746755i \(0.268389\pi\)
\(420\) 1.23781e13 0.0462151
\(421\) −2.61825e14 −0.964849 −0.482425 0.875937i \(-0.660244\pi\)
−0.482425 + 0.875937i \(0.660244\pi\)
\(422\) 1.78084e14 0.647748
\(423\) 7.93996e13 0.285067
\(424\) −5.22384e14 −1.85130
\(425\) 1.11049e13 0.0388484
\(426\) −1.48452e14 −0.512666
\(427\) 1.44431e13 0.0492387
\(428\) 1.83794e13 0.0618573
\(429\) −5.99666e13 −0.199248
\(430\) 3.44517e14 1.13014
\(431\) 7.24476e13 0.234638 0.117319 0.993094i \(-0.462570\pi\)
0.117319 + 0.993094i \(0.462570\pi\)
\(432\) 5.57178e13 0.178169
\(433\) 3.43810e14 1.08551 0.542756 0.839890i \(-0.317381\pi\)
0.542756 + 0.839890i \(0.317381\pi\)
\(434\) 6.70208e14 2.08937
\(435\) 2.14079e14 0.658996
\(436\) 3.31253e13 0.100689
\(437\) −2.27010e14 −0.681393
\(438\) −2.32120e14 −0.688026
\(439\) −1.03638e14 −0.303364 −0.151682 0.988429i \(-0.548469\pi\)
−0.151682 + 0.988429i \(0.548469\pi\)
\(440\) 2.48864e14 0.719404
\(441\) 3.27123e13 0.0933896
\(442\) −1.69724e14 −0.478544
\(443\) 3.73050e13 0.103883 0.0519417 0.998650i \(-0.483459\pi\)
0.0519417 + 0.998650i \(0.483459\pi\)
\(444\) 1.58438e13 0.0435764
\(445\) −6.79910e14 −1.84702
\(446\) 1.96032e14 0.526000
\(447\) 2.67462e14 0.708876
\(448\) 4.57861e14 1.19868
\(449\) 2.04923e14 0.529952 0.264976 0.964255i \(-0.414636\pi\)
0.264976 + 0.964255i \(0.414636\pi\)
\(450\) −4.97376e12 −0.0127062
\(451\) −4.09745e14 −1.03405
\(452\) −1.91779e13 −0.0478124
\(453\) 8.57612e13 0.211228
\(454\) 5.73370e14 1.39517
\(455\) 2.42119e14 0.582058
\(456\) −1.39964e14 −0.332436
\(457\) −2.24389e14 −0.526577 −0.263289 0.964717i \(-0.584807\pi\)
−0.263289 + 0.964717i \(0.584807\pi\)
\(458\) −1.90167e14 −0.440935
\(459\) −8.25863e13 −0.189207
\(460\) 3.81529e13 0.0863693
\(461\) −4.58977e14 −1.02668 −0.513340 0.858185i \(-0.671592\pi\)
−0.513340 + 0.858185i \(0.671592\pi\)
\(462\) 1.94996e14 0.431017
\(463\) 2.09824e14 0.458310 0.229155 0.973390i \(-0.426404\pi\)
0.229155 + 0.973390i \(0.426404\pi\)
\(464\) 4.80167e14 1.03644
\(465\) 5.28256e14 1.12682
\(466\) −2.90304e14 −0.611970
\(467\) 7.64430e14 1.59256 0.796278 0.604931i \(-0.206799\pi\)
0.796278 + 0.604931i \(0.206799\pi\)
\(468\) −5.66818e12 −0.0116706
\(469\) 3.16638e14 0.644337
\(470\) −4.18220e14 −0.841137
\(471\) −3.33571e14 −0.663091
\(472\) 6.83557e13 0.134306
\(473\) −4.04677e14 −0.785909
\(474\) −1.06160e13 −0.0203789
\(475\) 1.16230e13 0.0220548
\(476\) −4.11517e13 −0.0771879
\(477\) −3.22617e14 −0.598186
\(478\) −3.40825e13 −0.0624710
\(479\) −5.73863e14 −1.03983 −0.519915 0.854218i \(-0.674036\pi\)
−0.519915 + 0.854218i \(0.674036\pi\)
\(480\) 4.55201e13 0.0815411
\(481\) 3.09908e14 0.548826
\(482\) 1.74736e14 0.305931
\(483\) 4.60714e14 0.797484
\(484\) 2.15778e13 0.0369281
\(485\) 5.13161e14 0.868308
\(486\) 3.69897e13 0.0618843
\(487\) −1.01994e15 −1.68719 −0.843597 0.536976i \(-0.819566\pi\)
−0.843597 + 0.536976i \(0.819566\pi\)
\(488\) 2.74475e13 0.0448945
\(489\) 1.08048e14 0.174750
\(490\) −1.72305e14 −0.275562
\(491\) −6.75363e14 −1.06804 −0.534022 0.845471i \(-0.679320\pi\)
−0.534022 + 0.845471i \(0.679320\pi\)
\(492\) −3.87301e13 −0.0605676
\(493\) −7.11715e14 −1.10065
\(494\) −1.77643e14 −0.271675
\(495\) 1.53695e14 0.232451
\(496\) 1.18485e15 1.77220
\(497\) −7.04053e14 −1.04147
\(498\) −9.10969e13 −0.133273
\(499\) 5.97022e14 0.863848 0.431924 0.901910i \(-0.357835\pi\)
0.431924 + 0.901910i \(0.357835\pi\)
\(500\) 4.74828e13 0.0679519
\(501\) 2.11768e14 0.299746
\(502\) −4.06625e14 −0.569278
\(503\) −1.01245e15 −1.40201 −0.701005 0.713156i \(-0.747265\pi\)
−0.701005 + 0.713156i \(0.747265\pi\)
\(504\) 2.84054e14 0.389074
\(505\) 1.20274e15 1.62955
\(506\) 6.01033e14 0.805509
\(507\) 3.24624e14 0.430365
\(508\) −4.67858e13 −0.0613570
\(509\) −1.31283e15 −1.70318 −0.851591 0.524207i \(-0.824362\pi\)
−0.851591 + 0.524207i \(0.824362\pi\)
\(510\) 4.35005e14 0.558288
\(511\) −1.10086e15 −1.39770
\(512\) 8.62460e14 1.08331
\(513\) −8.64396e13 −0.107416
\(514\) −9.92292e14 −1.21995
\(515\) −1.34099e15 −1.63113
\(516\) −3.82510e13 −0.0460330
\(517\) 4.91251e14 0.584932
\(518\) −1.00774e15 −1.18723
\(519\) 1.56458e14 0.182380
\(520\) 4.60120e14 0.530704
\(521\) 1.32049e15 1.50704 0.753522 0.657423i \(-0.228353\pi\)
0.753522 + 0.657423i \(0.228353\pi\)
\(522\) 3.18771e14 0.359990
\(523\) 1.46506e15 1.63718 0.818591 0.574377i \(-0.194756\pi\)
0.818591 + 0.574377i \(0.194756\pi\)
\(524\) 1.12787e14 0.124720
\(525\) −2.35886e13 −0.0258123
\(526\) −1.49707e15 −1.62114
\(527\) −1.75621e15 −1.88200
\(528\) 3.44729e14 0.365588
\(529\) 4.67241e14 0.490382
\(530\) 1.69932e15 1.76505
\(531\) 4.22156e13 0.0433963
\(532\) −4.30717e13 −0.0438206
\(533\) −7.57569e14 −0.762821
\(534\) −1.01241e15 −1.00897
\(535\) −9.21417e14 −0.908888
\(536\) 6.01735e14 0.587488
\(537\) 1.07302e15 1.03693
\(538\) −1.53718e15 −1.47036
\(539\) 2.02393e14 0.191627
\(540\) 1.45276e13 0.0136154
\(541\) −1.07000e15 −0.992654 −0.496327 0.868136i \(-0.665318\pi\)
−0.496327 + 0.868136i \(0.665318\pi\)
\(542\) −5.82885e14 −0.535287
\(543\) 7.63701e14 0.694263
\(544\) −1.51334e14 −0.136189
\(545\) −1.66067e15 −1.47946
\(546\) 3.60524e14 0.317961
\(547\) −1.56936e15 −1.37022 −0.685111 0.728439i \(-0.740246\pi\)
−0.685111 + 0.728439i \(0.740246\pi\)
\(548\) −1.57574e14 −0.136204
\(549\) 1.69512e13 0.0145061
\(550\) −3.07730e13 −0.0260720
\(551\) −7.44923e14 −0.624852
\(552\) 8.75534e14 0.727123
\(553\) −5.03477e13 −0.0413991
\(554\) 1.63197e14 0.132864
\(555\) −7.94297e14 −0.640282
\(556\) −1.28773e13 −0.0102781
\(557\) −1.18643e15 −0.937647 −0.468823 0.883292i \(-0.655322\pi\)
−0.468823 + 0.883292i \(0.655322\pi\)
\(558\) 7.86592e14 0.615547
\(559\) −7.48199e14 −0.579765
\(560\) −1.39187e15 −1.06798
\(561\) −5.10967e14 −0.388237
\(562\) −1.67856e15 −1.26296
\(563\) 1.77729e15 1.32422 0.662112 0.749405i \(-0.269660\pi\)
0.662112 + 0.749405i \(0.269660\pi\)
\(564\) 4.64342e13 0.0342612
\(565\) 9.61449e14 0.702522
\(566\) −1.95482e15 −1.41454
\(567\) 1.75428e14 0.125716
\(568\) −1.33797e15 −0.949579
\(569\) −4.24631e14 −0.298466 −0.149233 0.988802i \(-0.547680\pi\)
−0.149233 + 0.988802i \(0.547680\pi\)
\(570\) 4.55302e14 0.316948
\(571\) −2.59400e15 −1.78843 −0.894214 0.447641i \(-0.852264\pi\)
−0.894214 + 0.447641i \(0.852264\pi\)
\(572\) −3.50694e13 −0.0239470
\(573\) −1.48499e15 −1.00432
\(574\) 2.46342e15 1.65015
\(575\) −7.27069e13 −0.0482394
\(576\) 5.37369e14 0.353142
\(577\) −1.60915e15 −1.04744 −0.523721 0.851890i \(-0.675457\pi\)
−0.523721 + 0.851890i \(0.675457\pi\)
\(578\) 4.99945e13 0.0322343
\(579\) −3.50023e13 −0.0223545
\(580\) 1.25197e14 0.0792025
\(581\) −4.32037e14 −0.270740
\(582\) 7.64114e14 0.474332
\(583\) −1.99605e15 −1.22743
\(584\) −2.09205e15 −1.27439
\(585\) 2.84164e14 0.171479
\(586\) −9.74592e14 −0.582620
\(587\) −5.57425e14 −0.330124 −0.165062 0.986283i \(-0.552782\pi\)
−0.165062 + 0.986283i \(0.552782\pi\)
\(588\) 1.91307e13 0.0112242
\(589\) −1.83815e15 −1.06843
\(590\) −2.22361e14 −0.128048
\(591\) −1.37597e15 −0.785016
\(592\) −1.78156e15 −1.00701
\(593\) 7.68689e14 0.430477 0.215238 0.976562i \(-0.430947\pi\)
0.215238 + 0.976562i \(0.430947\pi\)
\(594\) 2.28857e14 0.126981
\(595\) 2.06306e15 1.13415
\(596\) 1.56416e14 0.0851975
\(597\) 1.43496e15 0.774427
\(598\) 1.11124e15 0.594224
\(599\) −9.18042e14 −0.486424 −0.243212 0.969973i \(-0.578201\pi\)
−0.243212 + 0.969973i \(0.578201\pi\)
\(600\) −4.48275e13 −0.0235349
\(601\) 2.37400e15 1.23501 0.617505 0.786567i \(-0.288143\pi\)
0.617505 + 0.786567i \(0.288143\pi\)
\(602\) 2.43295e15 1.25416
\(603\) 3.71624e14 0.189827
\(604\) 5.01545e13 0.0253867
\(605\) −1.08176e15 −0.542597
\(606\) 1.79092e15 0.890177
\(607\) 2.78400e15 1.37130 0.685648 0.727933i \(-0.259519\pi\)
0.685648 + 0.727933i \(0.259519\pi\)
\(608\) −1.58395e14 −0.0773163
\(609\) 1.51181e15 0.731310
\(610\) −8.92867e13 −0.0428028
\(611\) 9.08263e14 0.431504
\(612\) −4.82978e13 −0.0227402
\(613\) 2.31529e15 1.08037 0.540185 0.841546i \(-0.318354\pi\)
0.540185 + 0.841546i \(0.318354\pi\)
\(614\) 2.09521e15 0.968952
\(615\) 1.94166e15 0.889938
\(616\) 1.75746e15 0.798346
\(617\) 7.90650e13 0.0355972 0.0177986 0.999842i \(-0.494334\pi\)
0.0177986 + 0.999842i \(0.494334\pi\)
\(618\) −1.99679e15 −0.891036
\(619\) 1.54696e15 0.684197 0.342099 0.939664i \(-0.388862\pi\)
0.342099 + 0.939664i \(0.388862\pi\)
\(620\) 3.08933e14 0.135428
\(621\) 5.40718e14 0.234945
\(622\) −3.29823e15 −1.42047
\(623\) −4.80147e15 −2.04970
\(624\) 6.37363e14 0.269694
\(625\) −2.47467e15 −1.03795
\(626\) 1.98981e15 0.827280
\(627\) −5.34807e14 −0.220407
\(628\) −1.95078e14 −0.0796947
\(629\) 2.64068e15 1.06939
\(630\) −9.24027e14 −0.370947
\(631\) −3.12319e15 −1.24290 −0.621452 0.783452i \(-0.713457\pi\)
−0.621452 + 0.783452i \(0.713457\pi\)
\(632\) −9.56801e13 −0.0377466
\(633\) 9.91248e14 0.387670
\(634\) 4.53690e14 0.175901
\(635\) 2.34552e15 0.901537
\(636\) −1.88672e14 −0.0718940
\(637\) 3.74200e14 0.141363
\(638\) 1.97226e15 0.738669
\(639\) −8.26313e14 −0.306825
\(640\) −2.44683e15 −0.900771
\(641\) 3.29251e15 1.20173 0.600867 0.799349i \(-0.294822\pi\)
0.600867 + 0.799349i \(0.294822\pi\)
\(642\) −1.37202e15 −0.496499
\(643\) −4.86563e15 −1.74574 −0.872869 0.487955i \(-0.837743\pi\)
−0.872869 + 0.487955i \(0.837743\pi\)
\(644\) 2.69433e14 0.0958469
\(645\) 1.91764e15 0.676378
\(646\) −1.51367e15 −0.529362
\(647\) 2.31233e15 0.801817 0.400909 0.916118i \(-0.368694\pi\)
0.400909 + 0.916118i \(0.368694\pi\)
\(648\) 3.33380e14 0.114624
\(649\) 2.61190e14 0.0890454
\(650\) −5.68956e13 −0.0192333
\(651\) 3.73050e15 1.25047
\(652\) 6.31882e13 0.0210027
\(653\) −3.09718e15 −1.02081 −0.510404 0.859935i \(-0.670504\pi\)
−0.510404 + 0.859935i \(0.670504\pi\)
\(654\) −2.47280e15 −0.808186
\(655\) −5.65437e15 −1.83255
\(656\) 4.35503e15 1.39965
\(657\) −1.29202e15 −0.411775
\(658\) −2.95344e15 −0.933438
\(659\) −8.36963e14 −0.262323 −0.131162 0.991361i \(-0.541871\pi\)
−0.131162 + 0.991361i \(0.541871\pi\)
\(660\) 8.98834e13 0.0279375
\(661\) −6.02877e14 −0.185832 −0.0929160 0.995674i \(-0.529619\pi\)
−0.0929160 + 0.995674i \(0.529619\pi\)
\(662\) 4.42645e14 0.135312
\(663\) −9.44716e14 −0.286403
\(664\) −8.21038e14 −0.246853
\(665\) 2.15932e15 0.643870
\(666\) −1.18274e15 −0.349768
\(667\) 4.65982e15 1.36671
\(668\) 1.23846e14 0.0360255
\(669\) 1.09115e15 0.314805
\(670\) −1.95745e15 −0.560116
\(671\) 1.04878e14 0.0297653
\(672\) 3.21459e14 0.0904888
\(673\) 2.95976e15 0.826369 0.413185 0.910647i \(-0.364416\pi\)
0.413185 + 0.910647i \(0.364416\pi\)
\(674\) −3.19109e15 −0.883711
\(675\) −2.76849e13 −0.00760452
\(676\) 1.89845e14 0.0517241
\(677\) 1.61475e15 0.436382 0.218191 0.975906i \(-0.429984\pi\)
0.218191 + 0.975906i \(0.429984\pi\)
\(678\) 1.43163e15 0.383767
\(679\) 3.62390e15 0.963590
\(680\) 3.92062e15 1.03408
\(681\) 3.19148e15 0.834994
\(682\) 4.86670e15 1.26305
\(683\) 4.94165e15 1.27221 0.636104 0.771603i \(-0.280545\pi\)
0.636104 + 0.771603i \(0.280545\pi\)
\(684\) −5.05513e13 −0.0129099
\(685\) 7.89966e15 2.00129
\(686\) 3.12630e15 0.785684
\(687\) −1.05850e15 −0.263894
\(688\) 4.30116e15 1.06377
\(689\) −3.69046e15 −0.905472
\(690\) −2.84811e15 −0.693246
\(691\) −3.06757e15 −0.740738 −0.370369 0.928885i \(-0.620769\pi\)
−0.370369 + 0.928885i \(0.620769\pi\)
\(692\) 9.14989e13 0.0219196
\(693\) 1.08538e15 0.257959
\(694\) −6.27012e15 −1.47842
\(695\) 6.45578e14 0.151019
\(696\) 2.87302e15 0.666788
\(697\) −6.45514e15 −1.48636
\(698\) −3.38520e15 −0.773355
\(699\) −1.61588e15 −0.366257
\(700\) −1.37950e13 −0.00310229
\(701\) 5.24474e15 1.17024 0.585120 0.810947i \(-0.301047\pi\)
0.585120 + 0.810947i \(0.301047\pi\)
\(702\) 4.23130e14 0.0936741
\(703\) 2.76389e15 0.607108
\(704\) 3.32474e15 0.724616
\(705\) −2.32789e15 −0.503411
\(706\) 2.69435e15 0.578135
\(707\) 8.49364e15 1.80837
\(708\) 2.46883e13 0.00521566
\(709\) 8.25422e15 1.73030 0.865151 0.501512i \(-0.167223\pi\)
0.865151 + 0.501512i \(0.167223\pi\)
\(710\) 4.35243e15 0.905337
\(711\) −5.90907e13 −0.0121965
\(712\) −9.12464e15 −1.86886
\(713\) 1.14985e16 2.33694
\(714\) 3.07197e15 0.619551
\(715\) 1.75814e15 0.351860
\(716\) 6.27520e14 0.124625
\(717\) −1.89710e14 −0.0373882
\(718\) −3.16907e15 −0.619794
\(719\) −4.62022e15 −0.896715 −0.448357 0.893854i \(-0.647991\pi\)
−0.448357 + 0.893854i \(0.647991\pi\)
\(720\) −1.63357e15 −0.314636
\(721\) −9.47000e15 −1.81011
\(722\) 3.50126e15 0.664155
\(723\) 9.72610e14 0.183096
\(724\) 4.46624e14 0.0834412
\(725\) −2.38584e14 −0.0442366
\(726\) −1.61078e15 −0.296405
\(727\) 3.01603e14 0.0550803 0.0275402 0.999621i \(-0.491233\pi\)
0.0275402 + 0.999621i \(0.491233\pi\)
\(728\) 3.24933e15 0.588940
\(729\) 2.05891e14 0.0370370
\(730\) 6.80544e15 1.21501
\(731\) −6.37530e15 −1.12968
\(732\) 9.91332e12 0.00174345
\(733\) 5.27949e15 0.921552 0.460776 0.887517i \(-0.347571\pi\)
0.460776 + 0.887517i \(0.347571\pi\)
\(734\) −1.01579e16 −1.75984
\(735\) −9.59080e14 −0.164920
\(736\) 9.90829e14 0.169111
\(737\) 2.29926e15 0.389508
\(738\) 2.89120e15 0.486147
\(739\) 3.18645e15 0.531818 0.265909 0.963998i \(-0.414328\pi\)
0.265909 + 0.963998i \(0.414328\pi\)
\(740\) −4.64518e14 −0.0769534
\(741\) −9.88795e14 −0.162595
\(742\) 1.20004e16 1.95873
\(743\) −1.56705e15 −0.253889 −0.126944 0.991910i \(-0.540517\pi\)
−0.126944 + 0.991910i \(0.540517\pi\)
\(744\) 7.08940e15 1.14014
\(745\) −7.84162e15 −1.25183
\(746\) 5.86616e15 0.929588
\(747\) −5.07062e14 −0.0797623
\(748\) −2.98822e14 −0.0466609
\(749\) −6.50697e15 −1.00862
\(750\) −3.54459e15 −0.545418
\(751\) 2.59659e15 0.396629 0.198314 0.980138i \(-0.436453\pi\)
0.198314 + 0.980138i \(0.436453\pi\)
\(752\) −5.22133e15 −0.791740
\(753\) −2.26335e15 −0.340706
\(754\) 3.64646e15 0.544916
\(755\) −2.51440e15 −0.373015
\(756\) 1.02593e14 0.0151094
\(757\) −5.34562e14 −0.0781575 −0.0390788 0.999236i \(-0.512442\pi\)
−0.0390788 + 0.999236i \(0.512442\pi\)
\(758\) 1.08027e15 0.156802
\(759\) 3.34546e15 0.482087
\(760\) 4.10354e15 0.587062
\(761\) 9.47799e15 1.34617 0.673086 0.739564i \(-0.264968\pi\)
0.673086 + 0.739564i \(0.264968\pi\)
\(762\) 3.49256e15 0.492484
\(763\) −1.17275e16 −1.64181
\(764\) −8.68445e14 −0.120706
\(765\) 2.42132e15 0.334129
\(766\) −6.10560e15 −0.836508
\(767\) 4.82910e14 0.0656888
\(768\) 8.85519e14 0.119595
\(769\) −6.04843e15 −0.811050 −0.405525 0.914084i \(-0.632911\pi\)
−0.405525 + 0.914084i \(0.632911\pi\)
\(770\) −5.71701e15 −0.761150
\(771\) −5.52328e15 −0.730126
\(772\) −2.04699e13 −0.00268671
\(773\) −1.02296e15 −0.133313 −0.0666564 0.997776i \(-0.521233\pi\)
−0.0666564 + 0.997776i \(0.521233\pi\)
\(774\) 2.85544e15 0.369485
\(775\) −5.88724e14 −0.0756401
\(776\) 6.88681e15 0.878575
\(777\) −5.60926e15 −0.710542
\(778\) 4.88494e15 0.614428
\(779\) −6.75632e15 −0.843829
\(780\) 1.66183e14 0.0206095
\(781\) −5.11246e15 −0.629577
\(782\) 9.46869e15 1.15785
\(783\) 1.77434e15 0.215450
\(784\) −2.15116e15 −0.259379
\(785\) 9.77985e15 1.17098
\(786\) −8.41955e15 −1.00107
\(787\) 7.90190e15 0.932976 0.466488 0.884528i \(-0.345519\pi\)
0.466488 + 0.884528i \(0.345519\pi\)
\(788\) −8.04691e14 −0.0943485
\(789\) −8.33294e15 −0.970230
\(790\) 3.11247e14 0.0359879
\(791\) 6.78968e15 0.779612
\(792\) 2.06265e15 0.235199
\(793\) 1.93907e14 0.0219579
\(794\) 6.87841e15 0.773527
\(795\) 9.45870e15 1.05636
\(796\) 8.39186e14 0.0930757
\(797\) 1.68582e15 0.185690 0.0928452 0.995681i \(-0.470404\pi\)
0.0928452 + 0.995681i \(0.470404\pi\)
\(798\) 3.21530e15 0.351727
\(799\) 7.73918e15 0.840791
\(800\) −5.07306e13 −0.00547363
\(801\) −5.63526e15 −0.603858
\(802\) 5.66587e15 0.602986
\(803\) −7.99382e15 −0.844927
\(804\) 2.17331e14 0.0228147
\(805\) −1.35075e16 −1.40831
\(806\) 8.99794e15 0.931752
\(807\) −8.55622e15 −0.879990
\(808\) 1.61412e16 1.64882
\(809\) 2.09869e15 0.212927 0.106463 0.994317i \(-0.466047\pi\)
0.106463 + 0.994317i \(0.466047\pi\)
\(810\) −1.08449e15 −0.109284
\(811\) −1.42566e16 −1.42692 −0.713461 0.700695i \(-0.752873\pi\)
−0.713461 + 0.700695i \(0.752873\pi\)
\(812\) 8.84129e14 0.0878936
\(813\) −3.24444e15 −0.320363
\(814\) −7.31767e15 −0.717692
\(815\) −3.16782e15 −0.308599
\(816\) 5.43088e15 0.525502
\(817\) −6.67275e15 −0.641333
\(818\) −8.82758e15 −0.842749
\(819\) 2.00674e15 0.190296
\(820\) 1.13551e15 0.106959
\(821\) 1.05580e16 0.987856 0.493928 0.869503i \(-0.335561\pi\)
0.493928 + 0.869503i \(0.335561\pi\)
\(822\) 1.17629e16 1.09325
\(823\) 1.15070e16 1.06234 0.531169 0.847266i \(-0.321753\pi\)
0.531169 + 0.847266i \(0.321753\pi\)
\(824\) −1.79967e16 −1.65041
\(825\) −1.71288e14 −0.0156038
\(826\) −1.57030e15 −0.142099
\(827\) −4.73492e15 −0.425630 −0.212815 0.977093i \(-0.568263\pi\)
−0.212815 + 0.977093i \(0.568263\pi\)
\(828\) 3.16220e14 0.0282373
\(829\) −1.24379e16 −1.10331 −0.551654 0.834073i \(-0.686003\pi\)
−0.551654 + 0.834073i \(0.686003\pi\)
\(830\) 2.67084e15 0.235352
\(831\) 9.08382e14 0.0795175
\(832\) 6.14704e15 0.534550
\(833\) 3.18851e15 0.275448
\(834\) 9.61289e14 0.0824975
\(835\) −6.20877e15 −0.529334
\(836\) −3.12764e14 −0.0264900
\(837\) 4.37831e15 0.368398
\(838\) −1.53512e16 −1.28322
\(839\) −7.25098e15 −0.602152 −0.301076 0.953600i \(-0.597346\pi\)
−0.301076 + 0.953600i \(0.597346\pi\)
\(840\) −8.32807e15 −0.687081
\(841\) 3.09044e15 0.253304
\(842\) 1.14304e16 0.930772
\(843\) −9.34319e15 −0.755864
\(844\) 5.79698e14 0.0465927
\(845\) −9.51754e15 −0.759998
\(846\) −3.46631e15 −0.274999
\(847\) −7.63931e15 −0.602137
\(848\) 2.12153e16 1.66139
\(849\) −1.08809e16 −0.846586
\(850\) −4.84799e14 −0.0374764
\(851\) −1.72893e16 −1.32790
\(852\) −4.83241e14 −0.0368762
\(853\) −4.88298e15 −0.370225 −0.185112 0.982717i \(-0.559265\pi\)
−0.185112 + 0.982717i \(0.559265\pi\)
\(854\) −6.30535e14 −0.0474997
\(855\) 2.53429e15 0.189689
\(856\) −1.23658e16 −0.919635
\(857\) 1.27260e16 0.940366 0.470183 0.882569i \(-0.344188\pi\)
0.470183 + 0.882569i \(0.344188\pi\)
\(858\) 2.61793e15 0.192211
\(859\) −7.72016e15 −0.563202 −0.281601 0.959532i \(-0.590865\pi\)
−0.281601 + 0.959532i \(0.590865\pi\)
\(860\) 1.12147e15 0.0812915
\(861\) 1.37118e16 0.987593
\(862\) −3.16281e15 −0.226351
\(863\) −2.00729e16 −1.42742 −0.713708 0.700443i \(-0.752986\pi\)
−0.713708 + 0.700443i \(0.752986\pi\)
\(864\) 3.77281e14 0.0266588
\(865\) −4.58713e15 −0.322071
\(866\) −1.50095e16 −1.04717
\(867\) 2.78278e14 0.0192919
\(868\) 2.18166e15 0.150289
\(869\) −3.65598e14 −0.0250262
\(870\) −9.34593e15 −0.635721
\(871\) 4.25105e15 0.287340
\(872\) −2.22868e16 −1.49695
\(873\) 4.25320e15 0.283882
\(874\) 9.91048e15 0.657328
\(875\) −1.68106e16 −1.10800
\(876\) −7.55595e14 −0.0494899
\(877\) 2.56450e16 1.66919 0.834594 0.550866i \(-0.185703\pi\)
0.834594 + 0.550866i \(0.185703\pi\)
\(878\) 4.52447e15 0.292649
\(879\) −5.42475e15 −0.348691
\(880\) −1.01070e16 −0.645606
\(881\) 1.81453e16 1.15185 0.575925 0.817502i \(-0.304642\pi\)
0.575925 + 0.817502i \(0.304642\pi\)
\(882\) −1.42810e15 −0.0900912
\(883\) −1.47084e16 −0.922107 −0.461054 0.887372i \(-0.652528\pi\)
−0.461054 + 0.887372i \(0.652528\pi\)
\(884\) −5.52485e14 −0.0344218
\(885\) −1.23770e15 −0.0766353
\(886\) −1.62861e15 −0.100214
\(887\) 6.28753e15 0.384503 0.192252 0.981346i \(-0.438421\pi\)
0.192252 + 0.981346i \(0.438421\pi\)
\(888\) −1.06598e16 −0.647853
\(889\) 1.65639e16 1.00047
\(890\) 2.96825e16 1.78178
\(891\) 1.27386e15 0.0759967
\(892\) 6.38123e14 0.0378353
\(893\) 8.10028e15 0.477328
\(894\) −1.16764e16 −0.683840
\(895\) −3.14595e16 −1.83116
\(896\) −1.72793e16 −0.999616
\(897\) 6.18535e15 0.355636
\(898\) −8.94624e15 −0.511235
\(899\) 3.77316e16 2.14302
\(900\) −1.61905e13 −0.000913961 0
\(901\) −3.14459e16 −1.76432
\(902\) 1.78880e16 0.997531
\(903\) 1.35422e16 0.750598
\(904\) 1.29030e16 0.710828
\(905\) −2.23907e16 −1.22603
\(906\) −3.74403e15 −0.203767
\(907\) −2.27848e16 −1.23255 −0.616275 0.787531i \(-0.711359\pi\)
−0.616275 + 0.787531i \(0.711359\pi\)
\(908\) 1.86643e15 0.100355
\(909\) 9.96858e15 0.532760
\(910\) −1.05701e16 −0.561501
\(911\) −1.34480e16 −0.710078 −0.355039 0.934852i \(-0.615532\pi\)
−0.355039 + 0.934852i \(0.615532\pi\)
\(912\) 5.68427e15 0.298334
\(913\) −3.13723e15 −0.163665
\(914\) 9.79603e15 0.507980
\(915\) −4.96986e14 −0.0256170
\(916\) −6.19029e14 −0.0317165
\(917\) −3.99307e16 −2.03365
\(918\) 3.60543e15 0.182525
\(919\) −1.66171e16 −0.836217 −0.418109 0.908397i \(-0.637307\pi\)
−0.418109 + 0.908397i \(0.637307\pi\)
\(920\) −2.56695e16 −1.28406
\(921\) 1.16623e16 0.579906
\(922\) 2.00373e16 0.990420
\(923\) −9.45231e15 −0.464439
\(924\) 6.34749e14 0.0310032
\(925\) 8.85217e14 0.0429804
\(926\) −9.16017e15 −0.442123
\(927\) −1.11145e16 −0.533275
\(928\) 3.25135e15 0.155078
\(929\) −1.95972e16 −0.929197 −0.464598 0.885522i \(-0.653801\pi\)
−0.464598 + 0.885522i \(0.653801\pi\)
\(930\) −2.30618e16 −1.08702
\(931\) 3.33728e15 0.156376
\(932\) −9.44995e14 −0.0440192
\(933\) −1.83585e16 −0.850137
\(934\) −3.33723e16 −1.53631
\(935\) 1.49809e16 0.685603
\(936\) 3.81358e15 0.173507
\(937\) 3.08038e16 1.39327 0.696637 0.717423i \(-0.254679\pi\)
0.696637 + 0.717423i \(0.254679\pi\)
\(938\) −1.38233e16 −0.621580
\(939\) 1.10756e16 0.495117
\(940\) −1.36139e15 −0.0605032
\(941\) 8.75412e15 0.386785 0.193392 0.981121i \(-0.438051\pi\)
0.193392 + 0.981121i \(0.438051\pi\)
\(942\) 1.45625e16 0.639671
\(943\) 4.22638e16 1.84567
\(944\) −2.77610e15 −0.120528
\(945\) −5.14330e15 −0.222007
\(946\) 1.76668e16 0.758152
\(947\) 4.88945e15 0.208610 0.104305 0.994545i \(-0.466738\pi\)
0.104305 + 0.994545i \(0.466738\pi\)
\(948\) −3.45572e13 −0.00146586
\(949\) −1.47796e16 −0.623303
\(950\) −5.07419e14 −0.0212758
\(951\) 2.52532e15 0.105275
\(952\) 2.76871e16 1.14756
\(953\) 4.11751e15 0.169677 0.0848386 0.996395i \(-0.472963\pi\)
0.0848386 + 0.996395i \(0.472963\pi\)
\(954\) 1.40843e16 0.577059
\(955\) 4.35378e16 1.77357
\(956\) −1.10945e14 −0.00449356
\(957\) 1.09779e16 0.442084
\(958\) 2.50528e16 1.00311
\(959\) 5.57867e16 2.22090
\(960\) −1.57549e16 −0.623627
\(961\) 6.76972e16 2.66436
\(962\) −1.35295e16 −0.529442
\(963\) −7.63693e15 −0.297149
\(964\) 5.68798e14 0.0220057
\(965\) 1.02622e15 0.0394766
\(966\) −2.01131e16 −0.769318
\(967\) 3.95380e15 0.150373 0.0751864 0.997169i \(-0.476045\pi\)
0.0751864 + 0.997169i \(0.476045\pi\)
\(968\) −1.45176e16 −0.549012
\(969\) −8.42538e15 −0.316817
\(970\) −2.24028e16 −0.837641
\(971\) −5.07440e16 −1.88660 −0.943299 0.331945i \(-0.892295\pi\)
−0.943299 + 0.331945i \(0.892295\pi\)
\(972\) 1.20408e14 0.00445136
\(973\) 4.55902e15 0.167591
\(974\) 4.45270e16 1.62761
\(975\) −3.16691e14 −0.0115109
\(976\) −1.11471e15 −0.0402892
\(977\) −1.65424e16 −0.594536 −0.297268 0.954794i \(-0.596075\pi\)
−0.297268 + 0.954794i \(0.596075\pi\)
\(978\) −4.71700e15 −0.168578
\(979\) −3.48657e16 −1.23906
\(980\) −5.60885e14 −0.0198212
\(981\) −1.37641e16 −0.483690
\(982\) 2.94840e16 1.03032
\(983\) −4.79624e16 −1.66670 −0.833348 0.552749i \(-0.813579\pi\)
−0.833348 + 0.552749i \(0.813579\pi\)
\(984\) 2.60578e16 0.900460
\(985\) 4.03417e16 1.38629
\(986\) 3.10710e16 1.06177
\(987\) −1.64394e16 −0.558651
\(988\) −5.78263e14 −0.0195417
\(989\) 4.17410e16 1.40276
\(990\) −6.70979e15 −0.224241
\(991\) −4.09919e16 −1.36236 −0.681182 0.732114i \(-0.738534\pi\)
−0.681182 + 0.732114i \(0.738534\pi\)
\(992\) 8.02296e15 0.265168
\(993\) 2.46384e15 0.0809827
\(994\) 3.07365e16 1.00468
\(995\) −4.20710e16 −1.36759
\(996\) −2.96538e14 −0.00958636
\(997\) −6.55021e15 −0.210587 −0.105294 0.994441i \(-0.533578\pi\)
−0.105294 + 0.994441i \(0.533578\pi\)
\(998\) −2.60639e16 −0.833338
\(999\) −6.58332e15 −0.209332
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 177.12.a.a.1.8 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.8 26 1.1 even 1 trivial