Properties

Label 5.12.a.b.1.2
Level 5
Weight 12
Character 5.1
Self dual yes
Analytic conductor 3.842
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.84171590280\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{151}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(12.2882\)
Character \(\chi\) = 5.1

$q$-expansion

\(f(q)\) \(=\) \(q+63.7292 q^{2} +283.223 q^{3} +2013.42 q^{4} -3125.00 q^{5} +18049.6 q^{6} +41926.3 q^{7} -2204.06 q^{8} -96932.0 q^{9} +O(q^{10})\) \(q+63.7292 q^{2} +283.223 q^{3} +2013.42 q^{4} -3125.00 q^{5} +18049.6 q^{6} +41926.3 q^{7} -2204.06 q^{8} -96932.0 q^{9} -199154. q^{10} -957905. q^{11} +570245. q^{12} +1.39098e6 q^{13} +2.67193e6 q^{14} -885071. q^{15} -4.26394e6 q^{16} +3.76857e6 q^{17} -6.17740e6 q^{18} +9.41036e6 q^{19} -6.29192e6 q^{20} +1.18745e7 q^{21} -6.10466e7 q^{22} +3.02942e7 q^{23} -624238. q^{24} +9.76562e6 q^{25} +8.86460e7 q^{26} -7.76254e7 q^{27} +8.44151e7 q^{28} +1.03553e8 q^{29} -5.64049e7 q^{30} -5.48554e7 q^{31} -2.67224e8 q^{32} -2.71300e8 q^{33} +2.40168e8 q^{34} -1.31020e8 q^{35} -1.95164e8 q^{36} +4.78282e8 q^{37} +5.99715e8 q^{38} +3.93957e8 q^{39} +6.88767e6 q^{40} -9.29557e8 q^{41} +7.56752e8 q^{42} -2.68608e7 q^{43} -1.92866e9 q^{44} +3.02912e8 q^{45} +1.93063e9 q^{46} -1.20497e9 q^{47} -1.20764e9 q^{48} -2.19508e8 q^{49} +6.22356e8 q^{50} +1.06735e9 q^{51} +2.80062e9 q^{52} -4.02058e9 q^{53} -4.94700e9 q^{54} +2.99345e9 q^{55} -9.24080e7 q^{56} +2.66523e9 q^{57} +6.59933e9 q^{58} +7.97972e9 q^{59} -1.78201e9 q^{60} -2.07472e9 q^{61} -3.49589e9 q^{62} -4.06400e9 q^{63} -8.29741e9 q^{64} -4.34681e9 q^{65} -1.72898e10 q^{66} +5.61370e9 q^{67} +7.58770e9 q^{68} +8.58000e9 q^{69} -8.34979e9 q^{70} +1.51224e10 q^{71} +2.13643e8 q^{72} -6.64484e9 q^{73} +3.04806e10 q^{74} +2.76585e9 q^{75} +1.89470e10 q^{76} -4.01615e10 q^{77} +2.51066e10 q^{78} -1.57985e10 q^{79} +1.33248e10 q^{80} -4.81405e9 q^{81} -5.92399e10 q^{82} -2.04046e10 q^{83} +2.39083e10 q^{84} -1.17768e10 q^{85} -1.71182e9 q^{86} +2.93285e10 q^{87} +2.11128e9 q^{88} -4.21030e10 q^{89} +1.93044e10 q^{90} +5.83187e10 q^{91} +6.09948e10 q^{92} -1.55363e10 q^{93} -7.67919e10 q^{94} -2.94074e10 q^{95} -7.56837e10 q^{96} +1.10181e11 q^{97} -1.39891e10 q^{98} +9.28516e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 20q^{2} - 220q^{3} + 6976q^{4} - 6250q^{5} + 60184q^{6} + 57900q^{7} - 246240q^{8} - 20846q^{9} + O(q^{10}) \) \( 2q - 20q^{2} - 220q^{3} + 6976q^{4} - 6250q^{5} + 60184q^{6} + 57900q^{7} - 246240q^{8} - 20846q^{9} + 62500q^{10} - 618176q^{11} - 1927040q^{12} + 3414260q^{13} + 1334472q^{14} + 687500q^{15} + 6005632q^{16} + 1317940q^{17} - 12548020q^{18} + 5325320q^{19} - 21800000q^{20} + 3836184q^{21} - 89491840q^{22} + 58943940q^{23} + 122180160q^{24} + 19531250q^{25} - 80761736q^{26} - 26769160q^{27} + 163685760q^{28} + 94140380q^{29} - 188075000q^{30} + 244543464q^{31} - 627301120q^{32} - 442259840q^{33} + 445358072q^{34} - 180937500q^{35} + 182418752q^{36} + 21003220q^{37} + 941752240q^{38} - 624203992q^{39} + 769500000q^{40} - 745743316q^{41} + 1429793040q^{42} + 629950100q^{43} - 242725888q^{44} + 65143750q^{45} - 468194856q^{46} - 1402061540q^{47} - 6375522560q^{48} - 1941677414q^{49} - 195312500q^{50} + 2300559784q^{51} + 12841321600q^{52} + 1138320580q^{53} - 9205154480q^{54} + 1931800000q^{55} - 3990553920q^{56} + 4720910480q^{57} + 7387417960q^{58} + 7317515560q^{59} + 6022000000q^{60} - 1516425676q^{61} - 28564327440q^{62} - 2848632180q^{63} + 819531776q^{64} - 10669562500q^{65} - 2975464192q^{66} + 15734290140q^{67} - 4573774720q^{68} - 5837195832q^{69} - 4170225000q^{70} + 32938471544q^{71} - 18354067680q^{72} - 29982848860q^{73} + 68768198072q^{74} - 2148437500q^{75} - 1325392640q^{76} - 34734748800q^{77} + 110356370800q^{78} - 3302823120q^{79} - 18767600000q^{80} - 43884431798q^{81} - 74630515640q^{82} + 13299102420q^{83} - 15982487808q^{84} - 4118562500q^{85} - 56706093896q^{86} + 34064940920q^{87} - 80794874880q^{88} - 12674770860q^{89} + 39212562500q^{90} + 90637859064q^{91} + 203171571840q^{92} - 166200542640q^{93} - 60289765528q^{94} - 16641625000q^{95} + 105515416064q^{96} - 3080703740q^{97} + 130206802940q^{98} + 118700272448q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 63.7292 1.40823 0.704115 0.710086i \(-0.251344\pi\)
0.704115 + 0.710086i \(0.251344\pi\)
\(3\) 283.223 0.672916 0.336458 0.941698i \(-0.390771\pi\)
0.336458 + 0.941698i \(0.390771\pi\)
\(4\) 2013.42 0.983113
\(5\) −3125.00 −0.447214
\(6\) 18049.6 0.947621
\(7\) 41926.3 0.942861 0.471431 0.881903i \(-0.343738\pi\)
0.471431 + 0.881903i \(0.343738\pi\)
\(8\) −2204.06 −0.0237809
\(9\) −96932.0 −0.547184
\(10\) −199154. −0.629780
\(11\) −957905. −1.79334 −0.896670 0.442699i \(-0.854021\pi\)
−0.896670 + 0.442699i \(0.854021\pi\)
\(12\) 570245. 0.661553
\(13\) 1.39098e6 1.03904 0.519520 0.854458i \(-0.326111\pi\)
0.519520 + 0.854458i \(0.326111\pi\)
\(14\) 2.67193e6 1.32777
\(15\) −885071. −0.300937
\(16\) −4.26394e6 −1.01660
\(17\) 3.76857e6 0.643736 0.321868 0.946785i \(-0.395689\pi\)
0.321868 + 0.946785i \(0.395689\pi\)
\(18\) −6.17740e6 −0.770561
\(19\) 9.41036e6 0.871889 0.435945 0.899973i \(-0.356414\pi\)
0.435945 + 0.899973i \(0.356414\pi\)
\(20\) −6.29192e6 −0.439661
\(21\) 1.18745e7 0.634467
\(22\) −6.10466e7 −2.52544
\(23\) 3.02942e7 0.981423 0.490712 0.871322i \(-0.336737\pi\)
0.490712 + 0.871322i \(0.336737\pi\)
\(24\) −624238. −0.0160025
\(25\) 9.76562e6 0.200000
\(26\) 8.86460e7 1.46321
\(27\) −7.76254e7 −1.04113
\(28\) 8.44151e7 0.926939
\(29\) 1.03553e8 0.937502 0.468751 0.883330i \(-0.344704\pi\)
0.468751 + 0.883330i \(0.344704\pi\)
\(30\) −5.64049e7 −0.423789
\(31\) −5.48554e7 −0.344136 −0.172068 0.985085i \(-0.555045\pi\)
−0.172068 + 0.985085i \(0.555045\pi\)
\(32\) −2.67224e8 −1.40783
\(33\) −2.71300e8 −1.20677
\(34\) 2.40168e8 0.906529
\(35\) −1.31020e8 −0.421660
\(36\) −1.95164e8 −0.537943
\(37\) 4.78282e8 1.13390 0.566950 0.823752i \(-0.308123\pi\)
0.566950 + 0.823752i \(0.308123\pi\)
\(38\) 5.99715e8 1.22782
\(39\) 3.93957e8 0.699187
\(40\) 6.88767e6 0.0106351
\(41\) −9.29557e8 −1.25304 −0.626520 0.779406i \(-0.715521\pi\)
−0.626520 + 0.779406i \(0.715521\pi\)
\(42\) 7.56752e8 0.893475
\(43\) −2.68608e7 −0.0278639 −0.0139320 0.999903i \(-0.504435\pi\)
−0.0139320 + 0.999903i \(0.504435\pi\)
\(44\) −1.92866e9 −1.76306
\(45\) 3.02912e8 0.244708
\(46\) 1.93063e9 1.38207
\(47\) −1.20497e9 −0.766370 −0.383185 0.923672i \(-0.625173\pi\)
−0.383185 + 0.923672i \(0.625173\pi\)
\(48\) −1.20764e9 −0.684088
\(49\) −2.19508e8 −0.111013
\(50\) 6.22356e8 0.281646
\(51\) 1.06735e9 0.433181
\(52\) 2.80062e9 1.02149
\(53\) −4.02058e9 −1.32060 −0.660301 0.751001i \(-0.729571\pi\)
−0.660301 + 0.751001i \(0.729571\pi\)
\(54\) −4.94700e9 −1.46614
\(55\) 2.99345e9 0.802006
\(56\) −9.24080e7 −0.0224221
\(57\) 2.66523e9 0.586709
\(58\) 6.59933e9 1.32022
\(59\) 7.97972e9 1.45312 0.726560 0.687103i \(-0.241118\pi\)
0.726560 + 0.687103i \(0.241118\pi\)
\(60\) −1.78201e9 −0.295855
\(61\) −2.07472e9 −0.314518 −0.157259 0.987557i \(-0.550266\pi\)
−0.157259 + 0.987557i \(0.550266\pi\)
\(62\) −3.49589e9 −0.484623
\(63\) −4.06400e9 −0.515918
\(64\) −8.29741e9 −0.965946
\(65\) −4.34681e9 −0.464673
\(66\) −1.72898e10 −1.69941
\(67\) 5.61370e9 0.507969 0.253985 0.967208i \(-0.418259\pi\)
0.253985 + 0.967208i \(0.418259\pi\)
\(68\) 7.58770e9 0.632865
\(69\) 8.58000e9 0.660416
\(70\) −8.34979e9 −0.593795
\(71\) 1.51224e10 0.994714 0.497357 0.867546i \(-0.334304\pi\)
0.497357 + 0.867546i \(0.334304\pi\)
\(72\) 2.13643e8 0.0130125
\(73\) −6.64484e9 −0.375153 −0.187577 0.982250i \(-0.560063\pi\)
−0.187577 + 0.982250i \(0.560063\pi\)
\(74\) 3.04806e10 1.59679
\(75\) 2.76585e9 0.134583
\(76\) 1.89470e10 0.857166
\(77\) −4.01615e10 −1.69087
\(78\) 2.51066e10 0.984616
\(79\) −1.57985e10 −0.577652 −0.288826 0.957382i \(-0.593265\pi\)
−0.288826 + 0.957382i \(0.593265\pi\)
\(80\) 1.33248e10 0.454638
\(81\) −4.81405e9 −0.153406
\(82\) −5.92399e10 −1.76457
\(83\) −2.04046e10 −0.568589 −0.284295 0.958737i \(-0.591759\pi\)
−0.284295 + 0.958737i \(0.591759\pi\)
\(84\) 2.39083e10 0.623752
\(85\) −1.17768e10 −0.287888
\(86\) −1.71182e9 −0.0392389
\(87\) 2.93285e10 0.630861
\(88\) 2.11128e9 0.0426472
\(89\) −4.21030e10 −0.799223 −0.399611 0.916685i \(-0.630855\pi\)
−0.399611 + 0.916685i \(0.630855\pi\)
\(90\) 1.93044e10 0.344605
\(91\) 5.83187e10 0.979670
\(92\) 6.09948e10 0.964850
\(93\) −1.55363e10 −0.231575
\(94\) −7.67919e10 −1.07923
\(95\) −2.94074e10 −0.389921
\(96\) −7.56837e10 −0.947351
\(97\) 1.10181e11 1.30275 0.651376 0.758755i \(-0.274192\pi\)
0.651376 + 0.758755i \(0.274192\pi\)
\(98\) −1.39891e10 −0.156331
\(99\) 9.28516e10 0.981287
\(100\) 1.96623e10 0.196623
\(101\) −3.33271e10 −0.315522 −0.157761 0.987477i \(-0.550428\pi\)
−0.157761 + 0.987477i \(0.550428\pi\)
\(102\) 6.80211e10 0.610018
\(103\) 8.49419e10 0.721966 0.360983 0.932572i \(-0.382441\pi\)
0.360983 + 0.932572i \(0.382441\pi\)
\(104\) −3.06580e9 −0.0247093
\(105\) −3.71078e10 −0.283742
\(106\) −2.56229e11 −1.85971
\(107\) 1.43774e11 0.990989 0.495494 0.868611i \(-0.334987\pi\)
0.495494 + 0.868611i \(0.334987\pi\)
\(108\) −1.56292e11 −1.02354
\(109\) −3.01296e11 −1.87563 −0.937816 0.347134i \(-0.887155\pi\)
−0.937816 + 0.347134i \(0.887155\pi\)
\(110\) 1.90771e11 1.12941
\(111\) 1.35460e11 0.763020
\(112\) −1.78771e11 −0.958515
\(113\) −2.22891e11 −1.13805 −0.569026 0.822320i \(-0.692680\pi\)
−0.569026 + 0.822320i \(0.692680\pi\)
\(114\) 1.69853e11 0.826221
\(115\) −9.46694e10 −0.438906
\(116\) 2.08495e11 0.921671
\(117\) −1.34830e11 −0.568546
\(118\) 5.08541e11 2.04633
\(119\) 1.58003e11 0.606954
\(120\) 1.95074e9 0.00715655
\(121\) 6.32271e11 2.21607
\(122\) −1.32220e11 −0.442914
\(123\) −2.63271e11 −0.843191
\(124\) −1.10447e11 −0.338324
\(125\) −3.05176e10 −0.0894427
\(126\) −2.58996e11 −0.726532
\(127\) −1.79939e11 −0.483287 −0.241644 0.970365i \(-0.577687\pi\)
−0.241644 + 0.970365i \(0.577687\pi\)
\(128\) 1.84863e10 0.0475549
\(129\) −7.60759e9 −0.0187501
\(130\) −2.77019e11 −0.654366
\(131\) −5.65153e11 −1.27989 −0.639946 0.768420i \(-0.721043\pi\)
−0.639946 + 0.768420i \(0.721043\pi\)
\(132\) −5.46240e11 −1.18639
\(133\) 3.94542e11 0.822071
\(134\) 3.57757e11 0.715338
\(135\) 2.42579e11 0.465605
\(136\) −8.30615e9 −0.0153086
\(137\) 1.24499e9 0.00220396 0.00110198 0.999999i \(-0.499649\pi\)
0.00110198 + 0.999999i \(0.499649\pi\)
\(138\) 5.46797e11 0.930017
\(139\) 4.68084e11 0.765142 0.382571 0.923926i \(-0.375039\pi\)
0.382571 + 0.923926i \(0.375039\pi\)
\(140\) −2.63797e11 −0.414540
\(141\) −3.41275e11 −0.515703
\(142\) 9.63736e11 1.40079
\(143\) −1.33243e12 −1.86335
\(144\) 4.13312e11 0.556268
\(145\) −3.23602e11 −0.419264
\(146\) −4.23470e11 −0.528302
\(147\) −6.21697e10 −0.0747022
\(148\) 9.62981e11 1.11475
\(149\) 1.05439e12 1.17619 0.588096 0.808791i \(-0.299878\pi\)
0.588096 + 0.808791i \(0.299878\pi\)
\(150\) 1.76265e11 0.189524
\(151\) 4.91301e11 0.509301 0.254651 0.967033i \(-0.418040\pi\)
0.254651 + 0.967033i \(0.418040\pi\)
\(152\) −2.07410e10 −0.0207343
\(153\) −3.65295e11 −0.352242
\(154\) −2.55946e12 −2.38114
\(155\) 1.71423e11 0.153902
\(156\) 7.93198e11 0.687379
\(157\) 6.17375e11 0.516536 0.258268 0.966073i \(-0.416848\pi\)
0.258268 + 0.966073i \(0.416848\pi\)
\(158\) −1.00683e12 −0.813467
\(159\) −1.13872e12 −0.888654
\(160\) 8.35074e11 0.629600
\(161\) 1.27013e12 0.925346
\(162\) −3.06795e11 −0.216031
\(163\) 2.97234e11 0.202333 0.101166 0.994870i \(-0.467743\pi\)
0.101166 + 0.994870i \(0.467743\pi\)
\(164\) −1.87158e12 −1.23188
\(165\) 8.47814e11 0.539683
\(166\) −1.30037e12 −0.800705
\(167\) −2.53524e11 −0.151035 −0.0755177 0.997144i \(-0.524061\pi\)
−0.0755177 + 0.997144i \(0.524061\pi\)
\(168\) −2.61720e10 −0.0150882
\(169\) 1.42662e11 0.0796035
\(170\) −7.50526e11 −0.405412
\(171\) −9.12165e11 −0.477084
\(172\) −5.40820e10 −0.0273934
\(173\) 1.80555e12 0.885843 0.442922 0.896560i \(-0.353942\pi\)
0.442922 + 0.896560i \(0.353942\pi\)
\(174\) 1.86908e12 0.888397
\(175\) 4.09437e11 0.188572
\(176\) 4.08445e12 1.82311
\(177\) 2.26004e12 0.977828
\(178\) −2.68319e12 −1.12549
\(179\) 1.63020e12 0.663054 0.331527 0.943446i \(-0.392436\pi\)
0.331527 + 0.943446i \(0.392436\pi\)
\(180\) 6.09888e11 0.240576
\(181\) −4.16028e12 −1.59181 −0.795904 0.605423i \(-0.793004\pi\)
−0.795904 + 0.605423i \(0.793004\pi\)
\(182\) 3.71660e12 1.37960
\(183\) −5.87608e11 −0.211644
\(184\) −6.67701e10 −0.0233391
\(185\) −1.49463e12 −0.507096
\(186\) −9.90115e11 −0.326110
\(187\) −3.60994e12 −1.15444
\(188\) −2.42611e12 −0.753429
\(189\) −3.25455e12 −0.981636
\(190\) −1.87411e12 −0.549098
\(191\) −3.03600e12 −0.864208 −0.432104 0.901824i \(-0.642229\pi\)
−0.432104 + 0.901824i \(0.642229\pi\)
\(192\) −2.35001e12 −0.650000
\(193\) 3.77397e12 1.01445 0.507227 0.861812i \(-0.330670\pi\)
0.507227 + 0.861812i \(0.330670\pi\)
\(194\) 7.02174e12 1.83457
\(195\) −1.23111e12 −0.312686
\(196\) −4.41961e11 −0.109138
\(197\) −1.03079e12 −0.247518 −0.123759 0.992312i \(-0.539495\pi\)
−0.123759 + 0.992312i \(0.539495\pi\)
\(198\) 5.91736e12 1.38188
\(199\) 1.14293e12 0.259614 0.129807 0.991539i \(-0.458564\pi\)
0.129807 + 0.991539i \(0.458564\pi\)
\(200\) −2.15240e10 −0.00475617
\(201\) 1.58993e12 0.341821
\(202\) −2.12391e12 −0.444328
\(203\) 4.34159e12 0.883935
\(204\) 2.14901e12 0.425865
\(205\) 2.90487e12 0.560376
\(206\) 5.41328e12 1.01670
\(207\) −2.93648e12 −0.537019
\(208\) −5.93105e12 −1.05629
\(209\) −9.01423e12 −1.56359
\(210\) −2.36485e12 −0.399574
\(211\) 6.96492e12 1.14647 0.573235 0.819391i \(-0.305688\pi\)
0.573235 + 0.819391i \(0.305688\pi\)
\(212\) −8.09510e12 −1.29830
\(213\) 4.28299e12 0.669359
\(214\) 9.16259e12 1.39554
\(215\) 8.39400e10 0.0124611
\(216\) 1.71091e11 0.0247589
\(217\) −2.29989e12 −0.324472
\(218\) −1.92014e13 −2.64132
\(219\) −1.88197e12 −0.252447
\(220\) 6.02707e12 0.788463
\(221\) 5.24201e12 0.668868
\(222\) 8.63279e12 1.07451
\(223\) 6.80403e12 0.826208 0.413104 0.910684i \(-0.364445\pi\)
0.413104 + 0.910684i \(0.364445\pi\)
\(224\) −1.12037e13 −1.32739
\(225\) −9.46601e11 −0.109437
\(226\) −1.42047e13 −1.60264
\(227\) 8.00368e12 0.881348 0.440674 0.897667i \(-0.354739\pi\)
0.440674 + 0.897667i \(0.354739\pi\)
\(228\) 5.36621e12 0.576801
\(229\) 1.20624e13 1.26572 0.632862 0.774265i \(-0.281880\pi\)
0.632862 + 0.774265i \(0.281880\pi\)
\(230\) −6.03321e12 −0.618081
\(231\) −1.13746e13 −1.13781
\(232\) −2.28236e11 −0.0222946
\(233\) −1.05744e13 −1.00878 −0.504390 0.863476i \(-0.668283\pi\)
−0.504390 + 0.863476i \(0.668283\pi\)
\(234\) −8.59263e12 −0.800643
\(235\) 3.76554e12 0.342731
\(236\) 1.60665e13 1.42858
\(237\) −4.47449e12 −0.388711
\(238\) 1.00694e13 0.854731
\(239\) −4.81329e12 −0.399258 −0.199629 0.979872i \(-0.563974\pi\)
−0.199629 + 0.979872i \(0.563974\pi\)
\(240\) 3.77389e12 0.305933
\(241\) −1.29674e13 −1.02745 −0.513724 0.857956i \(-0.671734\pi\)
−0.513724 + 0.857956i \(0.671734\pi\)
\(242\) 4.02941e13 3.12074
\(243\) 1.23877e13 0.937896
\(244\) −4.17728e12 −0.309207
\(245\) 6.85963e11 0.0496464
\(246\) −1.67781e13 −1.18741
\(247\) 1.30896e13 0.905928
\(248\) 1.20904e11 0.00818385
\(249\) −5.77905e12 −0.382613
\(250\) −1.94486e12 −0.125956
\(251\) 5.10147e12 0.323214 0.161607 0.986855i \(-0.448332\pi\)
0.161607 + 0.986855i \(0.448332\pi\)
\(252\) −8.18253e12 −0.507206
\(253\) −2.90190e13 −1.76003
\(254\) −1.14674e13 −0.680580
\(255\) −3.33545e12 −0.193724
\(256\) 1.81712e13 1.03291
\(257\) −1.20976e13 −0.673079 −0.336539 0.941669i \(-0.609256\pi\)
−0.336539 + 0.941669i \(0.609256\pi\)
\(258\) −4.84826e11 −0.0264045
\(259\) 2.00526e13 1.06911
\(260\) −8.75193e12 −0.456826
\(261\) −1.00376e13 −0.512986
\(262\) −3.60167e13 −1.80238
\(263\) −1.09431e13 −0.536271 −0.268135 0.963381i \(-0.586407\pi\)
−0.268135 + 0.963381i \(0.586407\pi\)
\(264\) 5.97961e11 0.0286980
\(265\) 1.25643e13 0.590591
\(266\) 2.51439e13 1.15767
\(267\) −1.19245e13 −0.537810
\(268\) 1.13027e13 0.499391
\(269\) 1.23621e13 0.535125 0.267562 0.963541i \(-0.413782\pi\)
0.267562 + 0.963541i \(0.413782\pi\)
\(270\) 1.54594e13 0.655680
\(271\) 1.34683e13 0.559734 0.279867 0.960039i \(-0.409710\pi\)
0.279867 + 0.960039i \(0.409710\pi\)
\(272\) −1.60690e13 −0.654424
\(273\) 1.65172e13 0.659236
\(274\) 7.93424e10 0.00310368
\(275\) −9.35454e12 −0.358668
\(276\) 1.72751e13 0.649263
\(277\) 2.20976e13 0.814153 0.407077 0.913394i \(-0.366548\pi\)
0.407077 + 0.913394i \(0.366548\pi\)
\(278\) 2.98306e13 1.07750
\(279\) 5.31724e12 0.188306
\(280\) 2.88775e11 0.0100274
\(281\) 7.15247e12 0.243541 0.121770 0.992558i \(-0.461143\pi\)
0.121770 + 0.992558i \(0.461143\pi\)
\(282\) −2.17492e13 −0.726229
\(283\) −9.90996e12 −0.324524 −0.162262 0.986748i \(-0.551879\pi\)
−0.162262 + 0.986748i \(0.551879\pi\)
\(284\) 3.04476e13 0.977917
\(285\) −8.32883e12 −0.262384
\(286\) −8.49145e13 −2.62403
\(287\) −3.89729e13 −1.18144
\(288\) 2.59025e13 0.770341
\(289\) −2.00697e13 −0.585604
\(290\) −2.06229e13 −0.590420
\(291\) 3.12057e13 0.876642
\(292\) −1.33788e13 −0.368818
\(293\) −6.76585e13 −1.83042 −0.915209 0.402980i \(-0.867974\pi\)
−0.915209 + 0.402980i \(0.867974\pi\)
\(294\) −3.96203e12 −0.105198
\(295\) −2.49366e13 −0.649855
\(296\) −1.05416e12 −0.0269651
\(297\) 7.43577e13 1.86709
\(298\) 6.71957e13 1.65635
\(299\) 4.21386e13 1.01974
\(300\) 5.56880e12 0.132311
\(301\) −1.12618e12 −0.0262718
\(302\) 3.13103e13 0.717213
\(303\) −9.43898e12 −0.212320
\(304\) −4.01252e13 −0.886364
\(305\) 6.48351e12 0.140657
\(306\) −2.32800e13 −0.496038
\(307\) −2.76335e13 −0.578328 −0.289164 0.957280i \(-0.593377\pi\)
−0.289164 + 0.957280i \(0.593377\pi\)
\(308\) −8.08617e13 −1.66232
\(309\) 2.40575e13 0.485823
\(310\) 1.09247e13 0.216730
\(311\) −1.21728e13 −0.237251 −0.118626 0.992939i \(-0.537849\pi\)
−0.118626 + 0.992939i \(0.537849\pi\)
\(312\) −8.68302e11 −0.0166273
\(313\) 8.17271e13 1.53770 0.768851 0.639428i \(-0.220829\pi\)
0.768851 + 0.639428i \(0.220829\pi\)
\(314\) 3.93448e13 0.727402
\(315\) 1.27000e13 0.230726
\(316\) −3.18089e13 −0.567897
\(317\) 7.61714e13 1.33649 0.668246 0.743941i \(-0.267045\pi\)
0.668246 + 0.743941i \(0.267045\pi\)
\(318\) −7.25697e13 −1.25143
\(319\) −9.91937e13 −1.68126
\(320\) 2.59294e13 0.431984
\(321\) 4.07200e13 0.666852
\(322\) 8.09441e13 1.30310
\(323\) 3.54636e13 0.561267
\(324\) −9.69267e12 −0.150816
\(325\) 1.35838e13 0.207808
\(326\) 1.89425e13 0.284931
\(327\) −8.53338e13 −1.26214
\(328\) 2.04880e12 0.0297984
\(329\) −5.05201e13 −0.722581
\(330\) 5.40305e13 0.759998
\(331\) 8.05093e13 1.11376 0.556881 0.830592i \(-0.311998\pi\)
0.556881 + 0.830592i \(0.311998\pi\)
\(332\) −4.10830e13 −0.558988
\(333\) −4.63608e13 −0.620452
\(334\) −1.61569e13 −0.212693
\(335\) −1.75428e13 −0.227171
\(336\) −5.06321e13 −0.645000
\(337\) −7.90669e13 −0.990901 −0.495451 0.868636i \(-0.664997\pi\)
−0.495451 + 0.868636i \(0.664997\pi\)
\(338\) 9.09176e12 0.112100
\(339\) −6.31279e13 −0.765813
\(340\) −2.37116e13 −0.283026
\(341\) 5.25463e13 0.617153
\(342\) −5.81316e13 −0.671844
\(343\) −9.21053e13 −1.04753
\(344\) 5.92027e10 0.000662629 0
\(345\) −2.68125e13 −0.295347
\(346\) 1.15067e14 1.24747
\(347\) 1.40156e14 1.49554 0.747772 0.663956i \(-0.231124\pi\)
0.747772 + 0.663956i \(0.231124\pi\)
\(348\) 5.90504e13 0.620207
\(349\) −9.88347e13 −1.02181 −0.510905 0.859637i \(-0.670689\pi\)
−0.510905 + 0.859637i \(0.670689\pi\)
\(350\) 2.60931e13 0.265553
\(351\) −1.07975e14 −1.08177
\(352\) 2.55975e14 2.52472
\(353\) −2.74254e13 −0.266313 −0.133156 0.991095i \(-0.542511\pi\)
−0.133156 + 0.991095i \(0.542511\pi\)
\(354\) 1.44030e14 1.37701
\(355\) −4.72573e13 −0.444850
\(356\) −8.47708e13 −0.785726
\(357\) 4.47499e13 0.408429
\(358\) 1.03891e14 0.933733
\(359\) 1.57856e14 1.39715 0.698574 0.715538i \(-0.253818\pi\)
0.698574 + 0.715538i \(0.253818\pi\)
\(360\) −6.67636e11 −0.00581937
\(361\) −2.79354e13 −0.239809
\(362\) −2.65132e14 −2.24163
\(363\) 1.79073e14 1.49123
\(364\) 1.17420e14 0.963127
\(365\) 2.07651e13 0.167774
\(366\) −3.74478e13 −0.298044
\(367\) −1.95534e14 −1.53306 −0.766529 0.642210i \(-0.778018\pi\)
−0.766529 + 0.642210i \(0.778018\pi\)
\(368\) −1.29173e14 −0.997717
\(369\) 9.01038e13 0.685643
\(370\) −9.52518e13 −0.714107
\(371\) −1.68568e14 −1.24514
\(372\) −3.12810e13 −0.227664
\(373\) 8.35940e13 0.599482 0.299741 0.954021i \(-0.403100\pi\)
0.299741 + 0.954021i \(0.403100\pi\)
\(374\) −2.30059e14 −1.62572
\(375\) −8.64327e12 −0.0601875
\(376\) 2.65583e12 0.0182249
\(377\) 1.44040e14 0.974102
\(378\) −2.07410e14 −1.38237
\(379\) 2.81905e14 1.85177 0.925887 0.377802i \(-0.123320\pi\)
0.925887 + 0.377802i \(0.123320\pi\)
\(380\) −5.92093e13 −0.383336
\(381\) −5.09629e13 −0.325212
\(382\) −1.93482e14 −1.21700
\(383\) 2.51689e13 0.156053 0.0780264 0.996951i \(-0.475138\pi\)
0.0780264 + 0.996951i \(0.475138\pi\)
\(384\) 5.23574e12 0.0320005
\(385\) 1.25505e14 0.756181
\(386\) 2.40512e14 1.42859
\(387\) 2.60367e12 0.0152467
\(388\) 2.21840e14 1.28075
\(389\) −1.44036e14 −0.819879 −0.409939 0.912113i \(-0.634450\pi\)
−0.409939 + 0.912113i \(0.634450\pi\)
\(390\) −7.84580e13 −0.440334
\(391\) 1.14166e14 0.631778
\(392\) 4.83809e11 0.00263998
\(393\) −1.60064e14 −0.861261
\(394\) −6.56916e13 −0.348563
\(395\) 4.93703e13 0.258334
\(396\) 1.86949e14 0.964716
\(397\) 3.78848e13 0.192804 0.0964021 0.995342i \(-0.469267\pi\)
0.0964021 + 0.995342i \(0.469267\pi\)
\(398\) 7.28380e13 0.365596
\(399\) 1.11743e14 0.553185
\(400\) −4.16400e13 −0.203320
\(401\) 1.24111e13 0.0597747 0.0298874 0.999553i \(-0.490485\pi\)
0.0298874 + 0.999553i \(0.490485\pi\)
\(402\) 1.01325e14 0.481363
\(403\) −7.63027e13 −0.357571
\(404\) −6.71012e13 −0.310194
\(405\) 1.50439e13 0.0686053
\(406\) 2.76686e14 1.24478
\(407\) −4.58149e14 −2.03347
\(408\) −2.35249e12 −0.0103014
\(409\) −4.37138e14 −1.88860 −0.944301 0.329082i \(-0.893261\pi\)
−0.944301 + 0.329082i \(0.893261\pi\)
\(410\) 1.85125e14 0.789139
\(411\) 3.52610e11 0.00148308
\(412\) 1.71023e14 0.709775
\(413\) 3.34560e14 1.37009
\(414\) −1.87139e14 −0.756246
\(415\) 6.37644e13 0.254281
\(416\) −3.71702e14 −1.46279
\(417\) 1.32572e14 0.514877
\(418\) −5.74470e14 −2.20190
\(419\) 5.14822e14 1.94751 0.973755 0.227598i \(-0.0730871\pi\)
0.973755 + 0.227598i \(0.0730871\pi\)
\(420\) −7.47134e13 −0.278951
\(421\) −3.90682e14 −1.43970 −0.719850 0.694130i \(-0.755789\pi\)
−0.719850 + 0.694130i \(0.755789\pi\)
\(422\) 4.43869e14 1.61449
\(423\) 1.16800e14 0.419345
\(424\) 8.86159e12 0.0314050
\(425\) 3.68025e13 0.128747
\(426\) 2.72952e14 0.942612
\(427\) −8.69855e13 −0.296547
\(428\) 2.89476e14 0.974254
\(429\) −3.77373e14 −1.25388
\(430\) 5.34943e12 0.0175482
\(431\) −3.29050e14 −1.06571 −0.532853 0.846208i \(-0.678880\pi\)
−0.532853 + 0.846208i \(0.678880\pi\)
\(432\) 3.30990e14 1.05841
\(433\) −5.59793e14 −1.76744 −0.883718 0.468019i \(-0.844968\pi\)
−0.883718 + 0.468019i \(0.844968\pi\)
\(434\) −1.46570e14 −0.456932
\(435\) −9.16514e13 −0.282129
\(436\) −6.06634e14 −1.84396
\(437\) 2.85079e14 0.855693
\(438\) −1.19936e14 −0.355503
\(439\) 1.25752e14 0.368095 0.184047 0.982917i \(-0.441080\pi\)
0.184047 + 0.982917i \(0.441080\pi\)
\(440\) −6.59774e12 −0.0190724
\(441\) 2.12774e13 0.0607443
\(442\) 3.34069e14 0.941920
\(443\) 3.33211e14 0.927894 0.463947 0.885863i \(-0.346433\pi\)
0.463947 + 0.885863i \(0.346433\pi\)
\(444\) 2.72738e14 0.750135
\(445\) 1.31572e14 0.357423
\(446\) 4.33616e14 1.16349
\(447\) 2.98628e14 0.791479
\(448\) −3.47880e14 −0.910753
\(449\) −1.08196e14 −0.279804 −0.139902 0.990165i \(-0.544679\pi\)
−0.139902 + 0.990165i \(0.544679\pi\)
\(450\) −6.03262e13 −0.154112
\(451\) 8.90427e14 2.24713
\(452\) −4.48773e14 −1.11883
\(453\) 1.39148e14 0.342717
\(454\) 5.10068e14 1.24114
\(455\) −1.82246e14 −0.438122
\(456\) −5.87431e12 −0.0139524
\(457\) −1.81057e14 −0.424890 −0.212445 0.977173i \(-0.568143\pi\)
−0.212445 + 0.977173i \(0.568143\pi\)
\(458\) 7.68728e14 1.78243
\(459\) −2.92537e14 −0.670210
\(460\) −1.90609e14 −0.431494
\(461\) −3.74610e14 −0.837962 −0.418981 0.907995i \(-0.637613\pi\)
−0.418981 + 0.907995i \(0.637613\pi\)
\(462\) −7.24897e14 −1.60231
\(463\) 2.33341e13 0.0509678 0.0254839 0.999675i \(-0.491887\pi\)
0.0254839 + 0.999675i \(0.491887\pi\)
\(464\) −4.41542e14 −0.953067
\(465\) 4.85509e13 0.103563
\(466\) −6.73896e14 −1.42060
\(467\) −7.37382e14 −1.53621 −0.768103 0.640326i \(-0.778799\pi\)
−0.768103 + 0.640326i \(0.778799\pi\)
\(468\) −2.71469e14 −0.558945
\(469\) 2.35362e14 0.478945
\(470\) 2.39975e14 0.482645
\(471\) 1.74854e14 0.347586
\(472\) −1.75877e13 −0.0345564
\(473\) 2.57301e13 0.0499695
\(474\) −2.85156e14 −0.547395
\(475\) 9.18980e13 0.174378
\(476\) 3.18125e14 0.596704
\(477\) 3.89723e14 0.722612
\(478\) −3.06747e14 −0.562247
\(479\) −3.39424e14 −0.615030 −0.307515 0.951543i \(-0.599497\pi\)
−0.307515 + 0.951543i \(0.599497\pi\)
\(480\) 2.36512e14 0.423668
\(481\) 6.65281e14 1.17817
\(482\) −8.26404e14 −1.44688
\(483\) 3.59728e14 0.622680
\(484\) 1.27302e15 2.17865
\(485\) −3.44315e14 −0.582608
\(486\) 7.89456e14 1.32077
\(487\) −3.27468e14 −0.541700 −0.270850 0.962622i \(-0.587305\pi\)
−0.270850 + 0.962622i \(0.587305\pi\)
\(488\) 4.57280e12 0.00747952
\(489\) 8.41833e13 0.136153
\(490\) 4.37159e13 0.0699135
\(491\) −3.28187e14 −0.519006 −0.259503 0.965742i \(-0.583559\pi\)
−0.259503 + 0.965742i \(0.583559\pi\)
\(492\) −5.30075e14 −0.828952
\(493\) 3.90246e14 0.603504
\(494\) 8.34191e14 1.27576
\(495\) −2.90161e14 −0.438845
\(496\) 2.33900e14 0.349849
\(497\) 6.34025e14 0.937878
\(498\) −3.68294e14 −0.538807
\(499\) 2.84812e14 0.412103 0.206052 0.978541i \(-0.433939\pi\)
0.206052 + 0.978541i \(0.433939\pi\)
\(500\) −6.14446e13 −0.0879323
\(501\) −7.18038e13 −0.101634
\(502\) 3.25112e14 0.455159
\(503\) −2.01563e13 −0.0279117 −0.0139559 0.999903i \(-0.504442\pi\)
−0.0139559 + 0.999903i \(0.504442\pi\)
\(504\) 8.95729e12 0.0122690
\(505\) 1.04147e14 0.141106
\(506\) −1.84936e15 −2.47852
\(507\) 4.04052e13 0.0535665
\(508\) −3.62292e14 −0.475126
\(509\) −6.91697e14 −0.897362 −0.448681 0.893692i \(-0.648106\pi\)
−0.448681 + 0.893692i \(0.648106\pi\)
\(510\) −2.12566e14 −0.272808
\(511\) −2.78594e14 −0.353717
\(512\) 1.12018e15 1.40703
\(513\) −7.30482e14 −0.907746
\(514\) −7.70969e14 −0.947850
\(515\) −2.65443e14 −0.322873
\(516\) −1.53172e13 −0.0184335
\(517\) 1.15425e15 1.37436
\(518\) 1.27794e15 1.50555
\(519\) 5.11374e14 0.596098
\(520\) 9.58061e12 0.0110503
\(521\) 5.00013e14 0.570656 0.285328 0.958430i \(-0.407898\pi\)
0.285328 + 0.958430i \(0.407898\pi\)
\(522\) −6.39686e14 −0.722403
\(523\) −2.89272e14 −0.323256 −0.161628 0.986852i \(-0.551674\pi\)
−0.161628 + 0.986852i \(0.551674\pi\)
\(524\) −1.13789e15 −1.25828
\(525\) 1.15962e14 0.126893
\(526\) −6.97396e14 −0.755193
\(527\) −2.06727e14 −0.221533
\(528\) 1.15681e15 1.22680
\(529\) −3.50713e13 −0.0368083
\(530\) 8.00715e14 0.831688
\(531\) −7.73490e14 −0.795123
\(532\) 7.94377e14 0.808188
\(533\) −1.29299e15 −1.30196
\(534\) −7.59940e14 −0.757360
\(535\) −4.49293e14 −0.443184
\(536\) −1.23729e13 −0.0120800
\(537\) 4.61709e14 0.446180
\(538\) 7.87828e14 0.753579
\(539\) 2.10268e14 0.199084
\(540\) 4.88413e14 0.457743
\(541\) 1.20798e15 1.12066 0.560331 0.828269i \(-0.310674\pi\)
0.560331 + 0.828269i \(0.310674\pi\)
\(542\) 8.58325e14 0.788235
\(543\) −1.17829e15 −1.07115
\(544\) −1.00705e15 −0.906271
\(545\) 9.41550e14 0.838808
\(546\) 1.05263e15 0.928356
\(547\) −1.74686e15 −1.52521 −0.762603 0.646867i \(-0.776079\pi\)
−0.762603 + 0.646867i \(0.776079\pi\)
\(548\) 2.50668e12 0.00216674
\(549\) 2.01107e14 0.172099
\(550\) −5.96158e14 −0.505087
\(551\) 9.74468e14 0.817398
\(552\) −1.89108e13 −0.0157053
\(553\) −6.62373e14 −0.544646
\(554\) 1.40826e15 1.14652
\(555\) −4.23314e14 −0.341233
\(556\) 9.42447e14 0.752221
\(557\) 8.14516e14 0.643719 0.321859 0.946787i \(-0.395692\pi\)
0.321859 + 0.946787i \(0.395692\pi\)
\(558\) 3.38864e14 0.265178
\(559\) −3.73628e13 −0.0289517
\(560\) 5.58660e14 0.428661
\(561\) −1.02242e15 −0.776840
\(562\) 4.55821e14 0.342961
\(563\) −8.94179e14 −0.666237 −0.333118 0.942885i \(-0.608101\pi\)
−0.333118 + 0.942885i \(0.608101\pi\)
\(564\) −6.87129e14 −0.506994
\(565\) 6.96536e14 0.508952
\(566\) −6.31554e14 −0.457004
\(567\) −2.01835e14 −0.144641
\(568\) −3.33305e13 −0.0236552
\(569\) 5.58478e14 0.392544 0.196272 0.980549i \(-0.437116\pi\)
0.196272 + 0.980549i \(0.437116\pi\)
\(570\) −5.30790e14 −0.369497
\(571\) 1.19826e15 0.826136 0.413068 0.910700i \(-0.364457\pi\)
0.413068 + 0.910700i \(0.364457\pi\)
\(572\) −2.68273e15 −1.83189
\(573\) −8.59863e14 −0.581539
\(574\) −2.48371e15 −1.66374
\(575\) 2.95842e14 0.196285
\(576\) 8.04284e14 0.528550
\(577\) −1.70453e15 −1.10953 −0.554764 0.832008i \(-0.687192\pi\)
−0.554764 + 0.832008i \(0.687192\pi\)
\(578\) −1.27903e15 −0.824665
\(579\) 1.06887e15 0.682643
\(580\) −6.51545e14 −0.412184
\(581\) −8.55491e14 −0.536101
\(582\) 1.98872e15 1.23451
\(583\) 3.85134e15 2.36829
\(584\) 1.46456e13 0.00892147
\(585\) 4.21345e14 0.254261
\(586\) −4.31182e15 −2.57765
\(587\) 8.77553e14 0.519713 0.259857 0.965647i \(-0.416325\pi\)
0.259857 + 0.965647i \(0.416325\pi\)
\(588\) −1.25173e14 −0.0734407
\(589\) −5.16209e14 −0.300048
\(590\) −1.58919e15 −0.915145
\(591\) −2.91944e14 −0.166559
\(592\) −2.03937e15 −1.15273
\(593\) 3.16216e15 1.77086 0.885428 0.464776i \(-0.153865\pi\)
0.885428 + 0.464776i \(0.153865\pi\)
\(594\) 4.73876e15 2.62930
\(595\) −4.93758e14 −0.271438
\(596\) 2.12293e15 1.15633
\(597\) 3.23704e14 0.174698
\(598\) 2.68546e15 1.43603
\(599\) 1.86633e15 0.988873 0.494437 0.869214i \(-0.335374\pi\)
0.494437 + 0.869214i \(0.335374\pi\)
\(600\) −6.09608e12 −0.00320051
\(601\) 8.57728e14 0.446211 0.223105 0.974794i \(-0.428381\pi\)
0.223105 + 0.974794i \(0.428381\pi\)
\(602\) −7.17703e13 −0.0369968
\(603\) −5.44147e14 −0.277953
\(604\) 9.89193e14 0.500701
\(605\) −1.97585e15 −0.991057
\(606\) −6.01539e14 −0.298995
\(607\) 2.90938e15 1.43306 0.716528 0.697558i \(-0.245730\pi\)
0.716528 + 0.697558i \(0.245730\pi\)
\(608\) −2.51467e15 −1.22747
\(609\) 1.22964e15 0.594814
\(610\) 4.13189e14 0.198077
\(611\) −1.67609e15 −0.796289
\(612\) −7.35491e14 −0.346294
\(613\) −3.95415e15 −1.84510 −0.922551 0.385875i \(-0.873900\pi\)
−0.922551 + 0.385875i \(0.873900\pi\)
\(614\) −1.76106e15 −0.814419
\(615\) 8.22723e14 0.377086
\(616\) 8.85181e13 0.0402104
\(617\) −1.44545e15 −0.650778 −0.325389 0.945580i \(-0.605495\pi\)
−0.325389 + 0.945580i \(0.605495\pi\)
\(618\) 1.53316e15 0.684151
\(619\) −4.27132e15 −1.88914 −0.944569 0.328314i \(-0.893519\pi\)
−0.944569 + 0.328314i \(0.893519\pi\)
\(620\) 3.45146e14 0.151303
\(621\) −2.35160e15 −1.02178
\(622\) −7.75764e14 −0.334105
\(623\) −1.76522e15 −0.753556
\(624\) −1.67981e15 −0.710794
\(625\) 9.53674e13 0.0400000
\(626\) 5.20840e15 2.16544
\(627\) −2.55303e15 −1.05217
\(628\) 1.24303e15 0.507813
\(629\) 1.80244e15 0.729933
\(630\) 8.09362e14 0.324915
\(631\) 2.19799e15 0.874711 0.437355 0.899289i \(-0.355915\pi\)
0.437355 + 0.899289i \(0.355915\pi\)
\(632\) 3.48207e13 0.0137371
\(633\) 1.97262e15 0.771478
\(634\) 4.85435e15 1.88209
\(635\) 5.62310e14 0.216133
\(636\) −2.29272e15 −0.873647
\(637\) −3.05331e14 −0.115347
\(638\) −6.32154e15 −2.36760
\(639\) −1.46584e15 −0.544292
\(640\) −5.77698e13 −0.0212672
\(641\) −6.69744e14 −0.244450 −0.122225 0.992502i \(-0.539003\pi\)
−0.122225 + 0.992502i \(0.539003\pi\)
\(642\) 2.59505e15 0.939082
\(643\) 3.43086e15 1.23096 0.615478 0.788154i \(-0.288963\pi\)
0.615478 + 0.788154i \(0.288963\pi\)
\(644\) 2.55729e15 0.909720
\(645\) 2.37737e13 0.00838530
\(646\) 2.26007e15 0.790393
\(647\) −2.12502e15 −0.736869 −0.368435 0.929654i \(-0.620106\pi\)
−0.368435 + 0.929654i \(0.620106\pi\)
\(648\) 1.06104e13 0.00364813
\(649\) −7.64381e15 −2.60594
\(650\) 8.65684e14 0.292641
\(651\) −6.51380e14 −0.218343
\(652\) 5.98455e14 0.198916
\(653\) 4.76025e15 1.56894 0.784471 0.620166i \(-0.212935\pi\)
0.784471 + 0.620166i \(0.212935\pi\)
\(654\) −5.43826e15 −1.77739
\(655\) 1.76610e15 0.572385
\(656\) 3.96357e15 1.27384
\(657\) 6.44097e14 0.205278
\(658\) −3.21961e15 −1.01756
\(659\) 1.31734e15 0.412886 0.206443 0.978459i \(-0.433811\pi\)
0.206443 + 0.978459i \(0.433811\pi\)
\(660\) 1.70700e15 0.530569
\(661\) −2.45528e15 −0.756822 −0.378411 0.925638i \(-0.623529\pi\)
−0.378411 + 0.925638i \(0.623529\pi\)
\(662\) 5.13080e15 1.56843
\(663\) 1.48465e15 0.450092
\(664\) 4.49729e13 0.0135215
\(665\) −1.23294e15 −0.367641
\(666\) −2.95454e15 −0.873739
\(667\) 3.13705e15 0.920087
\(668\) −5.10449e14 −0.148485
\(669\) 1.92706e15 0.555969
\(670\) −1.11799e15 −0.319909
\(671\) 1.98739e15 0.564038
\(672\) −3.17314e15 −0.893220
\(673\) −8.60705e14 −0.240310 −0.120155 0.992755i \(-0.538339\pi\)
−0.120155 + 0.992755i \(0.538339\pi\)
\(674\) −5.03888e15 −1.39542
\(675\) −7.58060e14 −0.208225
\(676\) 2.87238e14 0.0782593
\(677\) 1.90003e15 0.513479 0.256739 0.966481i \(-0.417352\pi\)
0.256739 + 0.966481i \(0.417352\pi\)
\(678\) −4.02309e15 −1.07844
\(679\) 4.61948e15 1.22831
\(680\) 2.59567e13 0.00684622
\(681\) 2.26682e15 0.593073
\(682\) 3.34873e15 0.869093
\(683\) −2.35084e15 −0.605214 −0.302607 0.953115i \(-0.597857\pi\)
−0.302607 + 0.953115i \(0.597857\pi\)
\(684\) −1.83657e15 −0.469027
\(685\) −3.89060e12 −0.000985640 0
\(686\) −5.86980e15 −1.47516
\(687\) 3.41635e15 0.851726
\(688\) 1.14533e14 0.0283265
\(689\) −5.59255e15 −1.37216
\(690\) −1.70874e15 −0.415916
\(691\) −6.60668e15 −1.59534 −0.797672 0.603092i \(-0.793935\pi\)
−0.797672 + 0.603092i \(0.793935\pi\)
\(692\) 3.63533e15 0.870884
\(693\) 3.89293e15 0.925217
\(694\) 8.93203e15 2.10607
\(695\) −1.46276e15 −0.342182
\(696\) −6.46416e13 −0.0150024
\(697\) −3.50310e15 −0.806627
\(698\) −6.29866e15 −1.43894
\(699\) −2.99490e15 −0.678825
\(700\) 8.24367e14 0.185388
\(701\) −8.01300e14 −0.178791 −0.0893956 0.995996i \(-0.528494\pi\)
−0.0893956 + 0.995996i \(0.528494\pi\)
\(702\) −6.88118e15 −1.52338
\(703\) 4.50081e15 0.988636
\(704\) 7.94813e15 1.73227
\(705\) 1.06649e15 0.230629
\(706\) −1.74780e15 −0.375030
\(707\) −1.39728e15 −0.297493
\(708\) 4.55039e15 0.961315
\(709\) 4.10789e15 0.861121 0.430560 0.902562i \(-0.358316\pi\)
0.430560 + 0.902562i \(0.358316\pi\)
\(710\) −3.01167e15 −0.626451
\(711\) 1.53138e15 0.316082
\(712\) 9.27973e13 0.0190062
\(713\) −1.66180e15 −0.337743
\(714\) 2.85188e15 0.575162
\(715\) 4.16383e15 0.833316
\(716\) 3.28227e15 0.651857
\(717\) −1.36323e15 −0.268667
\(718\) 1.00601e16 1.96751
\(719\) 2.12875e15 0.413157 0.206578 0.978430i \(-0.433767\pi\)
0.206578 + 0.978430i \(0.433767\pi\)
\(720\) −1.29160e15 −0.248771
\(721\) 3.56130e15 0.680714
\(722\) −1.78030e15 −0.337706
\(723\) −3.67267e15 −0.691386
\(724\) −8.37637e15 −1.56493
\(725\) 1.01126e15 0.187500
\(726\) 1.14122e16 2.10000
\(727\) −4.60450e15 −0.840898 −0.420449 0.907316i \(-0.638127\pi\)
−0.420449 + 0.907316i \(0.638127\pi\)
\(728\) −1.28538e14 −0.0232974
\(729\) 4.36126e15 0.784531
\(730\) 1.32334e15 0.236264
\(731\) −1.01227e14 −0.0179370
\(732\) −1.18310e15 −0.208070
\(733\) −3.94830e15 −0.689190 −0.344595 0.938752i \(-0.611984\pi\)
−0.344595 + 0.938752i \(0.611984\pi\)
\(734\) −1.24612e16 −2.15890
\(735\) 1.94280e14 0.0334078
\(736\) −8.09532e15 −1.38168
\(737\) −5.37739e15 −0.910962
\(738\) 5.74224e15 0.965543
\(739\) 2.01640e15 0.336537 0.168269 0.985741i \(-0.446182\pi\)
0.168269 + 0.985741i \(0.446182\pi\)
\(740\) −3.00932e15 −0.498532
\(741\) 3.70727e15 0.609613
\(742\) −1.07427e16 −1.75345
\(743\) −7.15627e15 −1.15944 −0.579720 0.814816i \(-0.696838\pi\)
−0.579720 + 0.814816i \(0.696838\pi\)
\(744\) 3.42428e13 0.00550704
\(745\) −3.29498e15 −0.526009
\(746\) 5.32738e15 0.844209
\(747\) 1.97786e15 0.311123
\(748\) −7.26830e15 −1.13494
\(749\) 6.02791e15 0.934365
\(750\) −5.50829e14 −0.0847578
\(751\) −2.55257e14 −0.0389904 −0.0194952 0.999810i \(-0.506206\pi\)
−0.0194952 + 0.999810i \(0.506206\pi\)
\(752\) 5.13792e15 0.779094
\(753\) 1.44485e15 0.217496
\(754\) 9.17953e15 1.37176
\(755\) −1.53532e15 −0.227766
\(756\) −6.55276e15 −0.965059
\(757\) −5.67357e15 −0.829524 −0.414762 0.909930i \(-0.636135\pi\)
−0.414762 + 0.909930i \(0.636135\pi\)
\(758\) 1.79656e16 2.60772
\(759\) −8.21883e15 −1.18435
\(760\) 6.48155e13 0.00927265
\(761\) 8.59220e15 1.22036 0.610181 0.792262i \(-0.291097\pi\)
0.610181 + 0.792262i \(0.291097\pi\)
\(762\) −3.24782e15 −0.457973
\(763\) −1.26322e16 −1.76846
\(764\) −6.11273e15 −0.849614
\(765\) 1.14155e15 0.157527
\(766\) 1.60400e15 0.219758
\(767\) 1.10996e16 1.50985
\(768\) 5.14650e15 0.695064
\(769\) 1.23682e16 1.65849 0.829243 0.558888i \(-0.188772\pi\)
0.829243 + 0.558888i \(0.188772\pi\)
\(770\) 7.99831e15 1.06488
\(771\) −3.42630e15 −0.452925
\(772\) 7.59856e15 0.997324
\(773\) 8.62674e15 1.12424 0.562120 0.827055i \(-0.309986\pi\)
0.562120 + 0.827055i \(0.309986\pi\)
\(774\) 1.65930e14 0.0214709
\(775\) −5.35697e14 −0.0688272
\(776\) −2.42845e14 −0.0309806
\(777\) 5.67936e15 0.719422
\(778\) −9.17933e15 −1.15458
\(779\) −8.74746e15 −1.09251
\(780\) −2.47875e15 −0.307405
\(781\) −1.44858e16 −1.78386
\(782\) 7.27571e15 0.889689
\(783\) −8.03831e15 −0.976057
\(784\) 9.35970e14 0.112856
\(785\) −1.92930e15 −0.231002
\(786\) −1.02008e16 −1.21285
\(787\) 1.18931e16 1.40421 0.702106 0.712073i \(-0.252243\pi\)
0.702106 + 0.712073i \(0.252243\pi\)
\(788\) −2.07541e15 −0.243338
\(789\) −3.09933e15 −0.360865
\(790\) 3.14633e15 0.363794
\(791\) −9.34502e15 −1.07302
\(792\) −2.04650e14 −0.0233359
\(793\) −2.88589e15 −0.326797
\(794\) 2.41437e15 0.271513
\(795\) 3.55850e15 0.397418
\(796\) 2.30119e15 0.255230
\(797\) −6.33100e15 −0.697351 −0.348676 0.937243i \(-0.613369\pi\)
−0.348676 + 0.937243i \(0.613369\pi\)
\(798\) 7.12131e15 0.779012
\(799\) −4.54103e15 −0.493340
\(800\) −2.60961e15 −0.281566
\(801\) 4.08112e15 0.437322
\(802\) 7.90952e14 0.0841766
\(803\) 6.36512e15 0.672777
\(804\) 3.20118e15 0.336049
\(805\) −3.96914e15 −0.413827
\(806\) −4.86271e15 −0.503542
\(807\) 3.50123e15 0.360094
\(808\) 7.34547e13 0.00750339
\(809\) 1.06496e16 1.08048 0.540238 0.841512i \(-0.318334\pi\)
0.540238 + 0.841512i \(0.318334\pi\)
\(810\) 9.58736e14 0.0966121
\(811\) 6.79444e15 0.680047 0.340024 0.940417i \(-0.389565\pi\)
0.340024 + 0.940417i \(0.389565\pi\)
\(812\) 8.74141e15 0.869008
\(813\) 3.81453e15 0.376654
\(814\) −2.91975e16 −2.86359
\(815\) −9.28855e14 −0.0904860
\(816\) −4.55109e15 −0.440372
\(817\) −2.52770e14 −0.0242943
\(818\) −2.78585e16 −2.65959
\(819\) −5.65294e15 −0.536060
\(820\) 5.84870e15 0.550913
\(821\) 2.03099e16 1.90029 0.950145 0.311807i \(-0.100934\pi\)
0.950145 + 0.311807i \(0.100934\pi\)
\(822\) 2.24715e13 0.00208852
\(823\) −1.88821e15 −0.174322 −0.0871609 0.996194i \(-0.527779\pi\)
−0.0871609 + 0.996194i \(0.527779\pi\)
\(824\) −1.87217e14 −0.0171690
\(825\) −2.64942e15 −0.241354
\(826\) 2.13213e16 1.92940
\(827\) −1.10204e16 −0.990644 −0.495322 0.868710i \(-0.664950\pi\)
−0.495322 + 0.868710i \(0.664950\pi\)
\(828\) −5.91235e15 −0.527950
\(829\) −8.45180e15 −0.749720 −0.374860 0.927081i \(-0.622309\pi\)
−0.374860 + 0.927081i \(0.622309\pi\)
\(830\) 4.06366e15 0.358086
\(831\) 6.25854e15 0.547857
\(832\) −1.15415e16 −1.00366
\(833\) −8.27233e14 −0.0714629
\(834\) 8.44870e15 0.725065
\(835\) 7.92263e14 0.0675451
\(836\) −1.81494e16 −1.53719
\(837\) 4.25817e15 0.358288
\(838\) 3.28092e16 2.74254
\(839\) 5.30367e15 0.440439 0.220219 0.975450i \(-0.429323\pi\)
0.220219 + 0.975450i \(0.429323\pi\)
\(840\) 8.17876e13 0.00674763
\(841\) −1.47735e15 −0.121089
\(842\) −2.48979e16 −2.02743
\(843\) 2.02574e15 0.163882
\(844\) 1.40233e16 1.12711
\(845\) −4.45820e14 −0.0355998
\(846\) 7.44359e15 0.590535
\(847\) 2.65088e16 2.08945
\(848\) 1.71435e16 1.34253
\(849\) −2.80672e15 −0.218377
\(850\) 2.34539e15 0.181306
\(851\) 1.44892e16 1.11284
\(852\) 8.62344e15 0.658056
\(853\) −4.43767e15 −0.336462 −0.168231 0.985748i \(-0.553805\pi\)
−0.168231 + 0.985748i \(0.553805\pi\)
\(854\) −5.54352e15 −0.417607
\(855\) 2.85051e15 0.213358
\(856\) −3.16885e14 −0.0235666
\(857\) −2.12803e16 −1.57247 −0.786236 0.617926i \(-0.787973\pi\)
−0.786236 + 0.617926i \(0.787973\pi\)
\(858\) −2.40497e16 −1.76575
\(859\) −4.61602e15 −0.336748 −0.168374 0.985723i \(-0.553852\pi\)
−0.168374 + 0.985723i \(0.553852\pi\)
\(860\) 1.69006e14 0.0122507
\(861\) −1.10380e16 −0.795012
\(862\) −2.09701e16 −1.50076
\(863\) 6.10081e15 0.433839 0.216919 0.976190i \(-0.430399\pi\)
0.216919 + 0.976190i \(0.430399\pi\)
\(864\) 2.07433e16 1.46573
\(865\) −5.64235e15 −0.396161
\(866\) −3.56752e16 −2.48896
\(867\) −5.68420e15 −0.394062
\(868\) −4.63062e15 −0.318993
\(869\) 1.51334e16 1.03593
\(870\) −5.84088e15 −0.397303
\(871\) 7.80854e15 0.527800
\(872\) 6.64073e14 0.0446041
\(873\) −1.06800e16 −0.712844
\(874\) 1.81679e16 1.20501
\(875\) −1.27949e15 −0.0843321
\(876\) −3.78918e15 −0.248184
\(877\) 9.32501e14 0.0606948 0.0303474 0.999539i \(-0.490339\pi\)
0.0303474 + 0.999539i \(0.490339\pi\)
\(878\) 8.01407e15 0.518362
\(879\) −1.91624e16 −1.23172
\(880\) −1.27639e16 −0.815321
\(881\) −1.26049e16 −0.800150 −0.400075 0.916482i \(-0.631016\pi\)
−0.400075 + 0.916482i \(0.631016\pi\)
\(882\) 1.35599e15 0.0855420
\(883\) 2.18236e16 1.36818 0.684090 0.729397i \(-0.260199\pi\)
0.684090 + 0.729397i \(0.260199\pi\)
\(884\) 1.05543e16 0.657572
\(885\) −7.06261e15 −0.437298
\(886\) 2.12353e16 1.30669
\(887\) −2.71175e16 −1.65832 −0.829162 0.559008i \(-0.811182\pi\)
−0.829162 + 0.559008i \(0.811182\pi\)
\(888\) −2.98562e14 −0.0181453
\(889\) −7.54420e15 −0.455673
\(890\) 8.38497e15 0.503334
\(891\) 4.61140e15 0.275109
\(892\) 1.36993e16 0.812256
\(893\) −1.13392e16 −0.668190
\(894\) 1.90313e16 1.11458
\(895\) −5.09437e15 −0.296527
\(896\) 7.75064e14 0.0448377
\(897\) 1.19346e16 0.686198
\(898\) −6.89522e15 −0.394029
\(899\) −5.68042e15 −0.322628
\(900\) −1.90590e15 −0.107589
\(901\) −1.51519e16 −0.850119
\(902\) 5.67463e16 3.16447
\(903\) −3.18958e14 −0.0176787
\(904\) 4.91265e14 0.0270638
\(905\) 1.30009e16 0.711878
\(906\) 8.86777e15 0.482624
\(907\) 2.41689e16 1.30742 0.653712 0.756744i \(-0.273211\pi\)
0.653712 + 0.756744i \(0.273211\pi\)
\(908\) 1.61147e16 0.866465
\(909\) 3.23046e15 0.172649
\(910\) −1.16144e16 −0.616977
\(911\) −2.08024e16 −1.09841 −0.549203 0.835689i \(-0.685069\pi\)
−0.549203 + 0.835689i \(0.685069\pi\)
\(912\) −1.13644e16 −0.596449
\(913\) 1.95457e16 1.01967
\(914\) −1.15386e16 −0.598342
\(915\) 1.83628e15 0.0946503
\(916\) 2.42866e16 1.24435
\(917\) −2.36948e16 −1.20676
\(918\) −1.86432e16 −0.943810
\(919\) −1.27606e16 −0.642151 −0.321076 0.947054i \(-0.604044\pi\)
−0.321076 + 0.947054i \(0.604044\pi\)
\(920\) 2.08657e14 0.0104376
\(921\) −7.82642e15 −0.389166
\(922\) −2.38736e16 −1.18004
\(923\) 2.10349e16 1.03355
\(924\) −2.29019e16 −1.11860
\(925\) 4.67073e15 0.226780
\(926\) 1.48707e15 0.0717744
\(927\) −8.23358e15 −0.395048
\(928\) −2.76717e16 −1.31984
\(929\) 2.66057e16 1.26150 0.630751 0.775985i \(-0.282747\pi\)
0.630751 + 0.775985i \(0.282747\pi\)
\(930\) 3.09411e15 0.145841
\(931\) −2.06565e15 −0.0967908
\(932\) −2.12906e16 −0.991745
\(933\) −3.44761e15 −0.159650
\(934\) −4.69928e16 −2.16333
\(935\) 1.12811e16 0.516281
\(936\) 2.97174e14 0.0135205
\(937\) −1.35312e16 −0.612023 −0.306012 0.952028i \(-0.598995\pi\)
−0.306012 + 0.952028i \(0.598995\pi\)
\(938\) 1.49994e16 0.674465
\(939\) 2.31470e16 1.03474
\(940\) 7.58159e15 0.336944
\(941\) 2.75735e16 1.21829 0.609143 0.793061i \(-0.291514\pi\)
0.609143 + 0.793061i \(0.291514\pi\)
\(942\) 1.11433e16 0.489481
\(943\) −2.81602e16 −1.22976
\(944\) −3.40250e16 −1.47724
\(945\) 1.01705e16 0.439001
\(946\) 1.63976e15 0.0703686
\(947\) 4.64018e16 1.97975 0.989873 0.141955i \(-0.0453390\pi\)
0.989873 + 0.141955i \(0.0453390\pi\)
\(948\) −9.00900e15 −0.382147
\(949\) −9.24283e15 −0.389799
\(950\) 5.85659e15 0.245564
\(951\) 2.15735e16 0.899347
\(952\) −3.48246e14 −0.0144339
\(953\) 5.27189e15 0.217248 0.108624 0.994083i \(-0.465356\pi\)
0.108624 + 0.994083i \(0.465356\pi\)
\(954\) 2.48367e16 1.01760
\(955\) 9.48750e15 0.386485
\(956\) −9.69115e15 −0.392516
\(957\) −2.80939e16 −1.13135
\(958\) −2.16312e16 −0.866105
\(959\) 5.21979e13 0.00207803
\(960\) 7.34379e15 0.290689
\(961\) −2.23994e16 −0.881571
\(962\) 4.23978e16 1.65913
\(963\) −1.39363e16 −0.542253
\(964\) −2.61088e16 −1.01010
\(965\) −1.17936e16 −0.453678
\(966\) 2.29252e16 0.876877
\(967\) 3.13722e16 1.19316 0.596581 0.802553i \(-0.296526\pi\)
0.596581 + 0.802553i \(0.296526\pi\)
\(968\) −1.39356e15 −0.0527001
\(969\) 1.00441e16 0.377686
\(970\) −2.19429e16 −0.820446
\(971\) −4.02261e16 −1.49555 −0.747776 0.663951i \(-0.768878\pi\)
−0.747776 + 0.663951i \(0.768878\pi\)
\(972\) 2.49415e16 0.922057
\(973\) 1.96250e16 0.721423
\(974\) −2.08693e16 −0.762839
\(975\) 3.84723e15 0.139837
\(976\) 8.84648e15 0.319740
\(977\) −1.61108e16 −0.579025 −0.289513 0.957174i \(-0.593493\pi\)
−0.289513 + 0.957174i \(0.593493\pi\)
\(978\) 5.36494e15 0.191735
\(979\) 4.03307e16 1.43328
\(980\) 1.38113e15 0.0488080
\(981\) 2.92052e16 1.02631
\(982\) −2.09151e16 −0.730880
\(983\) −3.39027e16 −1.17812 −0.589060 0.808089i \(-0.700502\pi\)
−0.589060 + 0.808089i \(0.700502\pi\)
\(984\) 5.80265e14 0.0200518
\(985\) 3.22123e15 0.110694
\(986\) 2.48701e16 0.849873
\(987\) −1.43084e16 −0.486236
\(988\) 2.63548e16 0.890629
\(989\) −8.13727e14 −0.0273463
\(990\) −1.84918e16 −0.617995
\(991\) −4.18340e16 −1.39035 −0.695174 0.718841i \(-0.744673\pi\)
−0.695174 + 0.718841i \(0.744673\pi\)
\(992\) 1.46586e16 0.484484
\(993\) 2.28021e16 0.749468
\(994\) 4.04059e16 1.32075
\(995\) −3.57166e15 −0.116103
\(996\) −1.16356e16 −0.376152
\(997\) 3.94947e16 1.26974 0.634871 0.772618i \(-0.281053\pi\)
0.634871 + 0.772618i \(0.281053\pi\)
\(998\) 1.81509e16 0.580336
\(999\) −3.71268e16 −1.18053
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 5.12.a.b.1.2 2
3.2 odd 2 45.12.a.d.1.1 2
4.3 odd 2 80.12.a.j.1.1 2
5.2 odd 4 25.12.b.c.24.3 4
5.3 odd 4 25.12.b.c.24.2 4
5.4 even 2 25.12.a.c.1.1 2
7.6 odd 2 245.12.a.b.1.2 2
15.2 even 4 225.12.b.f.199.2 4
15.8 even 4 225.12.b.f.199.3 4
15.14 odd 2 225.12.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.12.a.b.1.2 2 1.1 even 1 trivial
25.12.a.c.1.1 2 5.4 even 2
25.12.b.c.24.2 4 5.3 odd 4
25.12.b.c.24.3 4 5.2 odd 4
45.12.a.d.1.1 2 3.2 odd 2
80.12.a.j.1.1 2 4.3 odd 2
225.12.a.h.1.2 2 15.14 odd 2
225.12.b.f.199.2 4 15.2 even 4
225.12.b.f.199.3 4 15.8 even 4
245.12.a.b.1.2 2 7.6 odd 2