Properties

Label 43.8.a.b.1.12
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(17.8435\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+18.8435 q^{2} -90.3374 q^{3} +227.079 q^{4} +70.1157 q^{5} -1702.28 q^{6} +1065.82 q^{7} +1867.00 q^{8} +5973.84 q^{9} +O(q^{10})\) \(q+18.8435 q^{2} -90.3374 q^{3} +227.079 q^{4} +70.1157 q^{5} -1702.28 q^{6} +1065.82 q^{7} +1867.00 q^{8} +5973.84 q^{9} +1321.23 q^{10} -1455.37 q^{11} -20513.7 q^{12} +9741.20 q^{13} +20083.8 q^{14} -6334.07 q^{15} +6114.77 q^{16} +21354.5 q^{17} +112568. q^{18} +45230.6 q^{19} +15921.8 q^{20} -96283.4 q^{21} -27424.2 q^{22} -39774.0 q^{23} -168660. q^{24} -73208.8 q^{25} +183559. q^{26} -342093. q^{27} +242025. q^{28} +106837. q^{29} -119356. q^{30} -195235. q^{31} -123752. q^{32} +131474. q^{33} +402394. q^{34} +74730.8 q^{35} +1.35653e6 q^{36} -34774.3 q^{37} +852305. q^{38} -879994. q^{39} +130906. q^{40} +217638. q^{41} -1.81432e6 q^{42} -79507.0 q^{43} -330483. q^{44} +418860. q^{45} -749483. q^{46} +914679. q^{47} -552392. q^{48} +312430. q^{49} -1.37951e6 q^{50} -1.92911e6 q^{51} +2.21202e6 q^{52} +1.04811e6 q^{53} -6.44624e6 q^{54} -102044. q^{55} +1.98989e6 q^{56} -4.08602e6 q^{57} +2.01318e6 q^{58} +2.62505e6 q^{59} -1.43833e6 q^{60} +305720. q^{61} -3.67892e6 q^{62} +6.36704e6 q^{63} -3.11462e6 q^{64} +683011. q^{65} +2.47743e6 q^{66} -1.51055e6 q^{67} +4.84916e6 q^{68} +3.59308e6 q^{69} +1.40819e6 q^{70} -3.03204e6 q^{71} +1.11532e7 q^{72} -4.88668e6 q^{73} -655271. q^{74} +6.61349e6 q^{75} +1.02709e7 q^{76} -1.55116e6 q^{77} -1.65822e7 q^{78} -2.20917e6 q^{79} +428742. q^{80} +1.78390e7 q^{81} +4.10107e6 q^{82} -8.61841e6 q^{83} -2.18639e7 q^{84} +1.49729e6 q^{85} -1.49819e6 q^{86} -9.65134e6 q^{87} -2.71717e6 q^{88} -4.49534e6 q^{89} +7.89280e6 q^{90} +1.03824e7 q^{91} -9.03184e6 q^{92} +1.76370e7 q^{93} +1.72358e7 q^{94} +3.17138e6 q^{95} +1.11794e7 q^{96} +1.11102e7 q^{97} +5.88729e6 q^{98} -8.69412e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 18.8435 1.66555 0.832775 0.553612i \(-0.186751\pi\)
0.832775 + 0.553612i \(0.186751\pi\)
\(3\) −90.3374 −1.93171 −0.965857 0.259075i \(-0.916582\pi\)
−0.965857 + 0.259075i \(0.916582\pi\)
\(4\) 227.079 1.77405
\(5\) 70.1157 0.250854 0.125427 0.992103i \(-0.459970\pi\)
0.125427 + 0.992103i \(0.459970\pi\)
\(6\) −1702.28 −3.21737
\(7\) 1065.82 1.17447 0.587234 0.809418i \(-0.300217\pi\)
0.587234 + 0.809418i \(0.300217\pi\)
\(8\) 1867.00 1.28923
\(9\) 5973.84 2.73152
\(10\) 1321.23 0.417809
\(11\) −1455.37 −0.329684 −0.164842 0.986320i \(-0.552711\pi\)
−0.164842 + 0.986320i \(0.552711\pi\)
\(12\) −20513.7 −3.42697
\(13\) 9741.20 1.22973 0.614866 0.788631i \(-0.289210\pi\)
0.614866 + 0.788631i \(0.289210\pi\)
\(14\) 20083.8 1.95613
\(15\) −6334.07 −0.484578
\(16\) 6114.77 0.373216
\(17\) 21354.5 1.05419 0.527094 0.849807i \(-0.323281\pi\)
0.527094 + 0.849807i \(0.323281\pi\)
\(18\) 112568. 4.54948
\(19\) 45230.6 1.51285 0.756424 0.654082i \(-0.226945\pi\)
0.756424 + 0.654082i \(0.226945\pi\)
\(20\) 15921.8 0.445028
\(21\) −96283.4 −2.26874
\(22\) −27424.2 −0.549105
\(23\) −39774.0 −0.681635 −0.340817 0.940130i \(-0.610704\pi\)
−0.340817 + 0.940130i \(0.610704\pi\)
\(24\) −168660. −2.49042
\(25\) −73208.8 −0.937072
\(26\) 183559. 2.04818
\(27\) −342093. −3.34481
\(28\) 242025. 2.08357
\(29\) 106837. 0.813443 0.406721 0.913552i \(-0.366672\pi\)
0.406721 + 0.913552i \(0.366672\pi\)
\(30\) −119356. −0.807088
\(31\) −195235. −1.17704 −0.588521 0.808482i \(-0.700290\pi\)
−0.588521 + 0.808482i \(0.700290\pi\)
\(32\) −123752. −0.667617
\(33\) 131474. 0.636855
\(34\) 402394. 1.75580
\(35\) 74730.8 0.294619
\(36\) 1.35653e6 4.84587
\(37\) −34774.3 −0.112863 −0.0564316 0.998406i \(-0.517972\pi\)
−0.0564316 + 0.998406i \(0.517972\pi\)
\(38\) 852305. 2.51972
\(39\) −879994. −2.37549
\(40\) 130906. 0.323407
\(41\) 217638. 0.493164 0.246582 0.969122i \(-0.420692\pi\)
0.246582 + 0.969122i \(0.420692\pi\)
\(42\) −1.81432e6 −3.77869
\(43\) −79507.0 −0.152499
\(44\) −330483. −0.584877
\(45\) 418860. 0.685212
\(46\) −749483. −1.13530
\(47\) 914679. 1.28507 0.642534 0.766257i \(-0.277883\pi\)
0.642534 + 0.766257i \(0.277883\pi\)
\(48\) −552392. −0.720947
\(49\) 312430. 0.379373
\(50\) −1.37951e6 −1.56074
\(51\) −1.92911e6 −2.03639
\(52\) 2.21202e6 2.18161
\(53\) 1.04811e6 0.967031 0.483515 0.875336i \(-0.339360\pi\)
0.483515 + 0.875336i \(0.339360\pi\)
\(54\) −6.44624e6 −5.57094
\(55\) −102044. −0.0827024
\(56\) 1.98989e6 1.51415
\(57\) −4.08602e6 −2.92239
\(58\) 2.01318e6 1.35483
\(59\) 2.62505e6 1.66401 0.832005 0.554768i \(-0.187193\pi\)
0.832005 + 0.554768i \(0.187193\pi\)
\(60\) −1.43833e6 −0.859667
\(61\) 305720. 0.172452 0.0862262 0.996276i \(-0.472519\pi\)
0.0862262 + 0.996276i \(0.472519\pi\)
\(62\) −3.67892e6 −1.96042
\(63\) 6.36704e6 3.20808
\(64\) −3.11462e6 −1.48517
\(65\) 683011. 0.308483
\(66\) 2.47743e6 1.06071
\(67\) −1.51055e6 −0.613583 −0.306791 0.951777i \(-0.599255\pi\)
−0.306791 + 0.951777i \(0.599255\pi\)
\(68\) 4.84916e6 1.87019
\(69\) 3.59308e6 1.31672
\(70\) 1.40819e6 0.490703
\(71\) −3.03204e6 −1.00538 −0.502690 0.864467i \(-0.667656\pi\)
−0.502690 + 0.864467i \(0.667656\pi\)
\(72\) 1.11532e7 3.52155
\(73\) −4.88668e6 −1.47023 −0.735113 0.677945i \(-0.762871\pi\)
−0.735113 + 0.677945i \(0.762871\pi\)
\(74\) −655271. −0.187979
\(75\) 6.61349e6 1.81016
\(76\) 1.02709e7 2.68387
\(77\) −1.55116e6 −0.387203
\(78\) −1.65822e7 −3.95650
\(79\) −2.20917e6 −0.504120 −0.252060 0.967712i \(-0.581108\pi\)
−0.252060 + 0.967712i \(0.581108\pi\)
\(80\) 428742. 0.0936226
\(81\) 1.78390e7 3.72969
\(82\) 4.10107e6 0.821389
\(83\) −8.61841e6 −1.65445 −0.827225 0.561871i \(-0.810082\pi\)
−0.827225 + 0.561871i \(0.810082\pi\)
\(84\) −2.18639e7 −4.02486
\(85\) 1.49729e6 0.264447
\(86\) −1.49819e6 −0.253994
\(87\) −9.65134e6 −1.57134
\(88\) −2.71717e6 −0.425037
\(89\) −4.49534e6 −0.675923 −0.337962 0.941160i \(-0.609737\pi\)
−0.337962 + 0.941160i \(0.609737\pi\)
\(90\) 7.89280e6 1.14125
\(91\) 1.03824e7 1.44428
\(92\) −9.03184e6 −1.20926
\(93\) 1.76370e7 2.27371
\(94\) 1.72358e7 2.14034
\(95\) 3.17138e6 0.379503
\(96\) 1.11794e7 1.28965
\(97\) 1.11102e7 1.23601 0.618003 0.786175i \(-0.287942\pi\)
0.618003 + 0.786175i \(0.287942\pi\)
\(98\) 5.88729e6 0.631865
\(99\) −8.69412e6 −0.900539
\(100\) −1.66242e7 −1.66242
\(101\) −4.64538e6 −0.448638 −0.224319 0.974516i \(-0.572016\pi\)
−0.224319 + 0.974516i \(0.572016\pi\)
\(102\) −3.63513e7 −3.39171
\(103\) −1.71624e6 −0.154756 −0.0773782 0.997002i \(-0.524655\pi\)
−0.0773782 + 0.997002i \(0.524655\pi\)
\(104\) 1.81868e7 1.58540
\(105\) −6.75098e6 −0.569120
\(106\) 1.97501e7 1.61064
\(107\) −7.01098e6 −0.553268 −0.276634 0.960975i \(-0.589219\pi\)
−0.276634 + 0.960975i \(0.589219\pi\)
\(108\) −7.76821e7 −5.93387
\(109\) 9.93632e6 0.734908 0.367454 0.930042i \(-0.380230\pi\)
0.367454 + 0.930042i \(0.380230\pi\)
\(110\) −1.92287e6 −0.137745
\(111\) 3.14142e6 0.218020
\(112\) 6.51725e6 0.438330
\(113\) −1.02922e7 −0.671017 −0.335508 0.942037i \(-0.608908\pi\)
−0.335508 + 0.942037i \(0.608908\pi\)
\(114\) −7.69950e7 −4.86738
\(115\) −2.78878e6 −0.170990
\(116\) 2.42604e7 1.44309
\(117\) 5.81923e7 3.35904
\(118\) 4.94653e7 2.77149
\(119\) 2.27601e7 1.23811
\(120\) −1.18257e7 −0.624730
\(121\) −1.73691e7 −0.891308
\(122\) 5.76085e6 0.287228
\(123\) −1.96609e7 −0.952653
\(124\) −4.43338e7 −2.08814
\(125\) −1.06109e7 −0.485922
\(126\) 1.19978e8 5.34322
\(127\) −2.57945e7 −1.11741 −0.558706 0.829365i \(-0.688702\pi\)
−0.558706 + 0.829365i \(0.688702\pi\)
\(128\) −4.28502e7 −1.80600
\(129\) 7.18245e6 0.294584
\(130\) 1.28703e7 0.513793
\(131\) −218805. −0.00850371 −0.00425185 0.999991i \(-0.501353\pi\)
−0.00425185 + 0.999991i \(0.501353\pi\)
\(132\) 2.98550e7 1.12982
\(133\) 4.82077e7 1.77679
\(134\) −2.84641e7 −1.02195
\(135\) −2.39861e7 −0.839057
\(136\) 3.98689e7 1.35909
\(137\) 1.69620e7 0.563578 0.281789 0.959476i \(-0.409072\pi\)
0.281789 + 0.959476i \(0.409072\pi\)
\(138\) 6.77063e7 2.19307
\(139\) −4.70595e7 −1.48626 −0.743131 0.669146i \(-0.766660\pi\)
−0.743131 + 0.669146i \(0.766660\pi\)
\(140\) 1.69698e7 0.522671
\(141\) −8.26297e7 −2.48238
\(142\) −5.71343e7 −1.67451
\(143\) −1.41770e7 −0.405423
\(144\) 3.65287e7 1.01945
\(145\) 7.49092e6 0.204055
\(146\) −9.20824e7 −2.44873
\(147\) −2.82241e7 −0.732841
\(148\) −7.89652e6 −0.200226
\(149\) 3.38648e7 0.838681 0.419341 0.907829i \(-0.362261\pi\)
0.419341 + 0.907829i \(0.362261\pi\)
\(150\) 1.24622e8 3.01491
\(151\) −5.68824e7 −1.34449 −0.672247 0.740327i \(-0.734671\pi\)
−0.672247 + 0.740327i \(0.734671\pi\)
\(152\) 8.44456e7 1.95040
\(153\) 1.27568e8 2.87954
\(154\) −2.92293e7 −0.644906
\(155\) −1.36891e7 −0.295265
\(156\) −1.99828e8 −4.21425
\(157\) 9.38899e7 1.93629 0.968144 0.250392i \(-0.0805596\pi\)
0.968144 + 0.250392i \(0.0805596\pi\)
\(158\) −4.16285e7 −0.839636
\(159\) −9.46833e7 −1.86803
\(160\) −8.67696e6 −0.167474
\(161\) −4.23919e7 −0.800557
\(162\) 3.36150e8 6.21198
\(163\) 4.02174e7 0.727373 0.363686 0.931521i \(-0.381518\pi\)
0.363686 + 0.931521i \(0.381518\pi\)
\(164\) 4.94211e7 0.874900
\(165\) 9.21839e6 0.159757
\(166\) −1.62401e8 −2.75557
\(167\) 1.22868e7 0.204141 0.102071 0.994777i \(-0.467453\pi\)
0.102071 + 0.994777i \(0.467453\pi\)
\(168\) −1.79761e8 −2.92491
\(169\) 3.21424e7 0.512242
\(170\) 2.82142e7 0.440450
\(171\) 2.70200e8 4.13237
\(172\) −1.80544e7 −0.270541
\(173\) −5.25459e6 −0.0771574 −0.0385787 0.999256i \(-0.512283\pi\)
−0.0385787 + 0.999256i \(0.512283\pi\)
\(174\) −1.81865e8 −2.61714
\(175\) −7.80274e7 −1.10056
\(176\) −8.89923e6 −0.123043
\(177\) −2.37140e8 −3.21439
\(178\) −8.47081e7 −1.12578
\(179\) 5.70630e7 0.743651 0.371826 0.928303i \(-0.378732\pi\)
0.371826 + 0.928303i \(0.378732\pi\)
\(180\) 9.51143e7 1.21560
\(181\) −6.26864e7 −0.785775 −0.392888 0.919586i \(-0.628524\pi\)
−0.392888 + 0.919586i \(0.628524\pi\)
\(182\) 1.95641e8 2.40552
\(183\) −2.76179e7 −0.333129
\(184\) −7.42580e7 −0.878782
\(185\) −2.43823e6 −0.0283121
\(186\) 3.32344e8 3.78698
\(187\) −3.10786e7 −0.347549
\(188\) 2.07704e8 2.27978
\(189\) −3.64610e8 −3.92836
\(190\) 5.97600e7 0.632081
\(191\) −7.15929e7 −0.743453 −0.371726 0.928342i \(-0.621234\pi\)
−0.371726 + 0.928342i \(0.621234\pi\)
\(192\) 2.81366e8 2.86892
\(193\) −6.80335e7 −0.681196 −0.340598 0.940209i \(-0.610630\pi\)
−0.340598 + 0.940209i \(0.610630\pi\)
\(194\) 2.09356e8 2.05863
\(195\) −6.17014e7 −0.595901
\(196\) 7.09463e7 0.673029
\(197\) 1.26363e8 1.17758 0.588788 0.808288i \(-0.299605\pi\)
0.588788 + 0.808288i \(0.299605\pi\)
\(198\) −1.63828e8 −1.49989
\(199\) −2.86067e7 −0.257325 −0.128662 0.991688i \(-0.541068\pi\)
−0.128662 + 0.991688i \(0.541068\pi\)
\(200\) −1.36681e8 −1.20810
\(201\) 1.36459e8 1.18527
\(202\) −8.75354e7 −0.747229
\(203\) 1.13869e8 0.955362
\(204\) −4.38060e8 −3.61267
\(205\) 1.52599e7 0.123712
\(206\) −3.23401e7 −0.257754
\(207\) −2.37603e8 −1.86190
\(208\) 5.95652e7 0.458956
\(209\) −6.58271e7 −0.498761
\(210\) −1.27212e8 −0.947898
\(211\) −1.14904e8 −0.842066 −0.421033 0.907045i \(-0.638332\pi\)
−0.421033 + 0.907045i \(0.638332\pi\)
\(212\) 2.38003e8 1.71557
\(213\) 2.73906e8 1.94211
\(214\) −1.32112e8 −0.921495
\(215\) −5.57469e6 −0.0382548
\(216\) −6.38687e8 −4.31221
\(217\) −2.08086e8 −1.38240
\(218\) 1.87235e8 1.22402
\(219\) 4.41450e8 2.84006
\(220\) −2.31721e7 −0.146719
\(221\) 2.08018e8 1.29637
\(222\) 5.91955e7 0.363122
\(223\) −8.16498e7 −0.493047 −0.246523 0.969137i \(-0.579288\pi\)
−0.246523 + 0.969137i \(0.579288\pi\)
\(224\) −1.31897e8 −0.784094
\(225\) −4.37337e8 −2.55963
\(226\) −1.93941e8 −1.11761
\(227\) 3.41280e8 1.93651 0.968257 0.249958i \(-0.0804169\pi\)
0.968257 + 0.249958i \(0.0804169\pi\)
\(228\) −9.27848e8 −5.18448
\(229\) −1.15657e8 −0.636427 −0.318213 0.948019i \(-0.603083\pi\)
−0.318213 + 0.948019i \(0.603083\pi\)
\(230\) −5.25505e7 −0.284793
\(231\) 1.40128e8 0.747966
\(232\) 1.99464e8 1.04871
\(233\) −1.82084e8 −0.943029 −0.471514 0.881858i \(-0.656292\pi\)
−0.471514 + 0.881858i \(0.656292\pi\)
\(234\) 1.09655e9 5.59465
\(235\) 6.41334e7 0.322364
\(236\) 5.96094e8 2.95204
\(237\) 1.99570e8 0.973815
\(238\) 4.28880e8 2.06213
\(239\) −3.58475e6 −0.0169850 −0.00849252 0.999964i \(-0.502703\pi\)
−0.00849252 + 0.999964i \(0.502703\pi\)
\(240\) −3.87314e7 −0.180852
\(241\) 1.50219e8 0.691299 0.345650 0.938364i \(-0.387659\pi\)
0.345650 + 0.938364i \(0.387659\pi\)
\(242\) −3.27295e8 −1.48452
\(243\) −8.63369e8 −3.85989
\(244\) 6.94226e7 0.305940
\(245\) 2.19063e7 0.0951671
\(246\) −3.70480e8 −1.58669
\(247\) 4.40601e8 1.86040
\(248\) −3.64504e8 −1.51748
\(249\) 7.78564e8 3.19593
\(250\) −1.99946e8 −0.809327
\(251\) −2.63281e8 −1.05090 −0.525450 0.850824i \(-0.676103\pi\)
−0.525450 + 0.850824i \(0.676103\pi\)
\(252\) 1.44582e9 5.69131
\(253\) 5.78857e7 0.224724
\(254\) −4.86059e8 −1.86111
\(255\) −1.35261e8 −0.510836
\(256\) −4.08778e8 −1.52282
\(257\) 2.87010e8 1.05470 0.527352 0.849647i \(-0.323185\pi\)
0.527352 + 0.849647i \(0.323185\pi\)
\(258\) 1.35343e8 0.490644
\(259\) −3.70632e7 −0.132554
\(260\) 1.55098e8 0.547266
\(261\) 6.38224e8 2.22194
\(262\) −4.12307e6 −0.0141633
\(263\) 4.14761e8 1.40589 0.702947 0.711242i \(-0.251867\pi\)
0.702947 + 0.711242i \(0.251867\pi\)
\(264\) 2.45462e8 0.821051
\(265\) 7.34888e7 0.242583
\(266\) 9.08404e8 2.95933
\(267\) 4.06097e8 1.30569
\(268\) −3.43014e8 −1.08853
\(269\) −1.11250e8 −0.348470 −0.174235 0.984704i \(-0.555745\pi\)
−0.174235 + 0.984704i \(0.555745\pi\)
\(270\) −4.51983e8 −1.39749
\(271\) 1.01854e8 0.310874 0.155437 0.987846i \(-0.450321\pi\)
0.155437 + 0.987846i \(0.450321\pi\)
\(272\) 1.30578e8 0.393440
\(273\) −9.37916e8 −2.78994
\(274\) 3.19623e8 0.938667
\(275\) 1.06546e8 0.308938
\(276\) 8.15912e8 2.33594
\(277\) 3.01221e8 0.851543 0.425772 0.904831i \(-0.360003\pi\)
0.425772 + 0.904831i \(0.360003\pi\)
\(278\) −8.86767e8 −2.47544
\(279\) −1.16630e9 −3.21512
\(280\) 1.39522e8 0.379831
\(281\) 539783. 0.00145127 0.000725633 1.00000i \(-0.499769\pi\)
0.000725633 1.00000i \(0.499769\pi\)
\(282\) −1.55704e9 −4.13453
\(283\) 7.75243e7 0.203322 0.101661 0.994819i \(-0.467584\pi\)
0.101661 + 0.994819i \(0.467584\pi\)
\(284\) −6.88512e8 −1.78360
\(285\) −2.86494e8 −0.733092
\(286\) −2.67145e8 −0.675252
\(287\) 2.31963e8 0.579205
\(288\) −7.39275e8 −1.82361
\(289\) 4.56762e7 0.111313
\(290\) 1.41156e8 0.339864
\(291\) −1.00367e9 −2.38761
\(292\) −1.10966e9 −2.60826
\(293\) −3.93510e8 −0.913943 −0.456971 0.889481i \(-0.651066\pi\)
−0.456971 + 0.889481i \(0.651066\pi\)
\(294\) −5.31842e8 −1.22058
\(295\) 1.84057e8 0.417423
\(296\) −6.49236e7 −0.145506
\(297\) 4.97870e8 1.10273
\(298\) 6.38133e8 1.39687
\(299\) −3.87446e8 −0.838228
\(300\) 1.50178e9 3.21132
\(301\) −8.47402e7 −0.179105
\(302\) −1.07187e9 −2.23932
\(303\) 4.19651e8 0.866641
\(304\) 2.76575e8 0.564619
\(305\) 2.14358e7 0.0432603
\(306\) 2.40384e9 4.79601
\(307\) 5.00982e8 0.988184 0.494092 0.869409i \(-0.335501\pi\)
0.494092 + 0.869409i \(0.335501\pi\)
\(308\) −3.52236e8 −0.686919
\(309\) 1.55041e8 0.298945
\(310\) −2.57950e8 −0.491779
\(311\) −4.05811e8 −0.765002 −0.382501 0.923955i \(-0.624937\pi\)
−0.382501 + 0.923955i \(0.624937\pi\)
\(312\) −1.64295e9 −3.06255
\(313\) −8.16758e7 −0.150553 −0.0752763 0.997163i \(-0.523984\pi\)
−0.0752763 + 0.997163i \(0.523984\pi\)
\(314\) 1.76922e9 3.22499
\(315\) 4.46429e8 0.804759
\(316\) −5.01655e8 −0.894336
\(317\) −3.74278e8 −0.659913 −0.329956 0.943996i \(-0.607034\pi\)
−0.329956 + 0.943996i \(0.607034\pi\)
\(318\) −1.78417e9 −3.11129
\(319\) −1.55486e8 −0.268179
\(320\) −2.18384e8 −0.372559
\(321\) 6.33353e8 1.06876
\(322\) −7.98814e8 −1.33337
\(323\) 9.65878e8 1.59483
\(324\) 4.05086e9 6.61667
\(325\) −7.13141e8 −1.15235
\(326\) 7.57838e8 1.21148
\(327\) −8.97621e8 −1.41963
\(328\) 4.06330e8 0.635801
\(329\) 9.74883e8 1.50927
\(330\) 1.73707e8 0.266084
\(331\) 8.80018e7 0.133381 0.0666904 0.997774i \(-0.478756\pi\)
0.0666904 + 0.997774i \(0.478756\pi\)
\(332\) −1.95706e9 −2.93509
\(333\) −2.07736e8 −0.308288
\(334\) 2.31527e8 0.340007
\(335\) −1.05913e8 −0.153919
\(336\) −5.88751e8 −0.846728
\(337\) −2.56628e8 −0.365257 −0.182629 0.983182i \(-0.558461\pi\)
−0.182629 + 0.983182i \(0.558461\pi\)
\(338\) 6.05677e8 0.853165
\(339\) 9.29769e8 1.29621
\(340\) 3.40002e8 0.469144
\(341\) 2.84139e8 0.388052
\(342\) 5.09153e9 6.88267
\(343\) −5.44755e8 −0.728906
\(344\) −1.48440e8 −0.196605
\(345\) 2.51931e8 0.330305
\(346\) −9.90151e7 −0.128510
\(347\) −3.07698e8 −0.395341 −0.197670 0.980269i \(-0.563338\pi\)
−0.197670 + 0.980269i \(0.563338\pi\)
\(348\) −2.19162e9 −2.78764
\(349\) −1.28877e9 −1.62288 −0.811439 0.584437i \(-0.801316\pi\)
−0.811439 + 0.584437i \(0.801316\pi\)
\(350\) −1.47031e9 −1.83304
\(351\) −3.33239e9 −4.11322
\(352\) 1.80104e8 0.220103
\(353\) 1.44420e9 1.74749 0.873747 0.486380i \(-0.161683\pi\)
0.873747 + 0.486380i \(0.161683\pi\)
\(354\) −4.46856e9 −5.35373
\(355\) −2.12593e8 −0.252203
\(356\) −1.02080e9 −1.19912
\(357\) −2.05608e9 −2.39167
\(358\) 1.07527e9 1.23859
\(359\) 8.83030e8 1.00727 0.503634 0.863917i \(-0.331996\pi\)
0.503634 + 0.863917i \(0.331996\pi\)
\(360\) 7.82011e8 0.883394
\(361\) 1.15194e9 1.28871
\(362\) −1.18123e9 −1.30875
\(363\) 1.56908e9 1.72175
\(364\) 2.35762e9 2.56223
\(365\) −3.42633e8 −0.368811
\(366\) −5.20420e8 −0.554843
\(367\) −1.60796e8 −0.169803 −0.0849015 0.996389i \(-0.527058\pi\)
−0.0849015 + 0.996389i \(0.527058\pi\)
\(368\) −2.43209e8 −0.254397
\(369\) 1.30014e9 1.34709
\(370\) −4.59448e7 −0.0471553
\(371\) 1.11709e9 1.13575
\(372\) 4.00500e9 4.03369
\(373\) 1.15384e8 0.115124 0.0575620 0.998342i \(-0.481667\pi\)
0.0575620 + 0.998342i \(0.481667\pi\)
\(374\) −5.85631e8 −0.578860
\(375\) 9.58559e8 0.938662
\(376\) 1.70770e9 1.65674
\(377\) 1.04072e9 1.00032
\(378\) −6.87054e9 −6.54288
\(379\) 1.76476e9 1.66513 0.832565 0.553928i \(-0.186872\pi\)
0.832565 + 0.553928i \(0.186872\pi\)
\(380\) 7.20153e8 0.673259
\(381\) 2.33020e9 2.15852
\(382\) −1.34906e9 −1.23826
\(383\) 2.52128e8 0.229311 0.114656 0.993405i \(-0.463424\pi\)
0.114656 + 0.993405i \(0.463424\pi\)
\(384\) 3.87097e9 3.48867
\(385\) −1.08761e8 −0.0971313
\(386\) −1.28199e9 −1.13457
\(387\) −4.74962e8 −0.416553
\(388\) 2.52289e9 2.19274
\(389\) −1.17424e9 −1.01142 −0.505710 0.862703i \(-0.668770\pi\)
−0.505710 + 0.862703i \(0.668770\pi\)
\(390\) −1.16267e9 −0.992502
\(391\) −8.49354e8 −0.718571
\(392\) 5.83307e8 0.489098
\(393\) 1.97663e7 0.0164267
\(394\) 2.38113e9 1.96131
\(395\) −1.54897e8 −0.126460
\(396\) −1.97425e9 −1.59761
\(397\) −1.20026e9 −0.962738 −0.481369 0.876518i \(-0.659860\pi\)
−0.481369 + 0.876518i \(0.659860\pi\)
\(398\) −5.39051e8 −0.428587
\(399\) −4.35496e9 −3.43225
\(400\) −4.47655e8 −0.349730
\(401\) −1.92289e8 −0.148919 −0.0744593 0.997224i \(-0.523723\pi\)
−0.0744593 + 0.997224i \(0.523723\pi\)
\(402\) 2.57137e9 1.97412
\(403\) −1.90183e9 −1.44745
\(404\) −1.05487e9 −0.795909
\(405\) 1.25079e9 0.935606
\(406\) 2.14569e9 1.59120
\(407\) 5.06093e7 0.0372092
\(408\) −3.60165e9 −2.62537
\(409\) 4.65392e8 0.336347 0.168173 0.985757i \(-0.446213\pi\)
0.168173 + 0.985757i \(0.446213\pi\)
\(410\) 2.87550e8 0.206049
\(411\) −1.53230e9 −1.08867
\(412\) −3.89723e8 −0.274546
\(413\) 2.79783e9 1.95432
\(414\) −4.47729e9 −3.10109
\(415\) −6.04286e8 −0.415025
\(416\) −1.20549e9 −0.820990
\(417\) 4.25123e9 2.87103
\(418\) −1.24042e9 −0.830712
\(419\) −2.24682e9 −1.49217 −0.746086 0.665850i \(-0.768069\pi\)
−0.746086 + 0.665850i \(0.768069\pi\)
\(420\) −1.53301e9 −1.00965
\(421\) 6.62749e8 0.432874 0.216437 0.976297i \(-0.430556\pi\)
0.216437 + 0.976297i \(0.430556\pi\)
\(422\) −2.16520e9 −1.40250
\(423\) 5.46414e9 3.51019
\(424\) 1.95682e9 1.24672
\(425\) −1.56334e9 −0.987851
\(426\) 5.16136e9 3.23467
\(427\) 3.25843e8 0.202540
\(428\) −1.59205e9 −0.981528
\(429\) 1.28071e9 0.783162
\(430\) −1.05047e8 −0.0637153
\(431\) −4.38748e8 −0.263964 −0.131982 0.991252i \(-0.542134\pi\)
−0.131982 + 0.991252i \(0.542134\pi\)
\(432\) −2.09182e9 −1.24833
\(433\) 1.08042e8 0.0639567 0.0319784 0.999489i \(-0.489819\pi\)
0.0319784 + 0.999489i \(0.489819\pi\)
\(434\) −3.92107e9 −2.30245
\(435\) −6.76710e8 −0.394176
\(436\) 2.25633e9 1.30377
\(437\) −1.79900e9 −1.03121
\(438\) 8.31848e9 4.73025
\(439\) −2.59454e9 −1.46364 −0.731820 0.681498i \(-0.761329\pi\)
−0.731820 + 0.681498i \(0.761329\pi\)
\(440\) −1.90516e8 −0.106622
\(441\) 1.86641e9 1.03627
\(442\) 3.91980e9 2.15917
\(443\) 1.32379e9 0.723446 0.361723 0.932286i \(-0.382189\pi\)
0.361723 + 0.932286i \(0.382189\pi\)
\(444\) 7.13350e8 0.386779
\(445\) −3.15194e8 −0.169558
\(446\) −1.53857e9 −0.821194
\(447\) −3.05926e9 −1.62009
\(448\) −3.31962e9 −1.74428
\(449\) 1.73729e9 0.905753 0.452876 0.891573i \(-0.350398\pi\)
0.452876 + 0.891573i \(0.350398\pi\)
\(450\) −8.24098e9 −4.26320
\(451\) −3.16743e8 −0.162588
\(452\) −2.33714e9 −1.19042
\(453\) 5.13861e9 2.59718
\(454\) 6.43093e9 3.22536
\(455\) 7.27967e8 0.362303
\(456\) −7.62859e9 −3.76762
\(457\) −1.37475e9 −0.673776 −0.336888 0.941545i \(-0.609374\pi\)
−0.336888 + 0.941545i \(0.609374\pi\)
\(458\) −2.17939e9 −1.06000
\(459\) −7.30522e9 −3.52606
\(460\) −6.33274e8 −0.303347
\(461\) −1.34864e9 −0.641124 −0.320562 0.947228i \(-0.603872\pi\)
−0.320562 + 0.947228i \(0.603872\pi\)
\(462\) 2.64050e9 1.24577
\(463\) −1.82505e9 −0.854557 −0.427279 0.904120i \(-0.640528\pi\)
−0.427279 + 0.904120i \(0.640528\pi\)
\(464\) 6.53281e8 0.303590
\(465\) 1.23663e9 0.570369
\(466\) −3.43110e9 −1.57066
\(467\) 1.36048e9 0.618135 0.309067 0.951040i \(-0.399983\pi\)
0.309067 + 0.951040i \(0.399983\pi\)
\(468\) 1.32143e10 5.95912
\(469\) −1.60997e9 −0.720633
\(470\) 1.20850e9 0.536913
\(471\) −8.48177e9 −3.74036
\(472\) 4.90097e9 2.14529
\(473\) 1.15712e8 0.0502763
\(474\) 3.76061e9 1.62194
\(475\) −3.31128e9 −1.41765
\(476\) 5.16833e9 2.19647
\(477\) 6.26122e9 2.64147
\(478\) −6.75494e7 −0.0282894
\(479\) 3.05531e9 1.27023 0.635113 0.772419i \(-0.280954\pi\)
0.635113 + 0.772419i \(0.280954\pi\)
\(480\) 7.83854e8 0.323512
\(481\) −3.38743e8 −0.138792
\(482\) 2.83066e9 1.15139
\(483\) 3.82957e9 1.54645
\(484\) −3.94415e9 −1.58123
\(485\) 7.79000e8 0.310057
\(486\) −1.62689e10 −6.42883
\(487\) −3.84229e9 −1.50743 −0.753717 0.657199i \(-0.771741\pi\)
−0.753717 + 0.657199i \(0.771741\pi\)
\(488\) 5.70779e8 0.222330
\(489\) −3.63313e9 −1.40508
\(490\) 4.12792e8 0.158506
\(491\) −9.60708e7 −0.0366274 −0.0183137 0.999832i \(-0.505830\pi\)
−0.0183137 + 0.999832i \(0.505830\pi\)
\(492\) −4.46457e9 −1.69006
\(493\) 2.28144e9 0.857522
\(494\) 8.30247e9 3.09858
\(495\) −6.09594e8 −0.225903
\(496\) −1.19382e9 −0.439291
\(497\) −3.23160e9 −1.18079
\(498\) 1.46709e10 5.32297
\(499\) 4.64015e8 0.167178 0.0835892 0.996500i \(-0.473362\pi\)
0.0835892 + 0.996500i \(0.473362\pi\)
\(500\) −2.40951e9 −0.862052
\(501\) −1.10996e9 −0.394343
\(502\) −4.96115e9 −1.75033
\(503\) 3.05552e9 1.07052 0.535262 0.844686i \(-0.320213\pi\)
0.535262 + 0.844686i \(0.320213\pi\)
\(504\) 1.18873e10 4.13595
\(505\) −3.25714e8 −0.112543
\(506\) 1.09077e9 0.374289
\(507\) −2.90366e9 −0.989506
\(508\) −5.85738e9 −1.98235
\(509\) −2.74667e9 −0.923198 −0.461599 0.887089i \(-0.652724\pi\)
−0.461599 + 0.887089i \(0.652724\pi\)
\(510\) −2.54879e9 −0.850823
\(511\) −5.20832e9 −1.72673
\(512\) −2.21800e9 −0.730326
\(513\) −1.54731e10 −5.06018
\(514\) 5.40828e9 1.75666
\(515\) −1.20336e8 −0.0388212
\(516\) 1.63098e9 0.522608
\(517\) −1.33119e9 −0.423666
\(518\) −6.98401e8 −0.220775
\(519\) 4.74686e8 0.149046
\(520\) 1.27518e9 0.397704
\(521\) −3.04048e9 −0.941912 −0.470956 0.882157i \(-0.656091\pi\)
−0.470956 + 0.882157i \(0.656091\pi\)
\(522\) 1.20264e10 3.70075
\(523\) 3.17164e9 0.969457 0.484729 0.874665i \(-0.338918\pi\)
0.484729 + 0.874665i \(0.338918\pi\)
\(524\) −4.96861e7 −0.0150860
\(525\) 7.04879e9 2.12597
\(526\) 7.81557e9 2.34159
\(527\) −4.16915e9 −1.24082
\(528\) 8.03933e8 0.237685
\(529\) −1.82286e9 −0.535374
\(530\) 1.38479e9 0.404034
\(531\) 1.56816e10 4.54528
\(532\) 1.09470e10 3.15212
\(533\) 2.12006e9 0.606460
\(534\) 7.65230e9 2.17469
\(535\) −4.91580e8 −0.138789
\(536\) −2.82020e9 −0.791047
\(537\) −5.15492e9 −1.43652
\(538\) −2.09634e9 −0.580394
\(539\) −4.54700e8 −0.125073
\(540\) −5.44674e9 −1.48853
\(541\) 4.49965e9 1.22177 0.610884 0.791720i \(-0.290814\pi\)
0.610884 + 0.791720i \(0.290814\pi\)
\(542\) 1.91929e9 0.517777
\(543\) 5.66292e9 1.51789
\(544\) −2.64266e9 −0.703794
\(545\) 6.96692e8 0.184354
\(546\) −1.76737e10 −4.64678
\(547\) 4.37317e8 0.114246 0.0571229 0.998367i \(-0.481807\pi\)
0.0571229 + 0.998367i \(0.481807\pi\)
\(548\) 3.85170e9 0.999818
\(549\) 1.82632e9 0.471058
\(550\) 2.00770e9 0.514551
\(551\) 4.83229e9 1.23061
\(552\) 6.70827e9 1.69756
\(553\) −2.35457e9 −0.592072
\(554\) 5.67608e9 1.41829
\(555\) 2.20263e8 0.0546910
\(556\) −1.06862e10 −2.63671
\(557\) −7.67789e9 −1.88256 −0.941280 0.337627i \(-0.890376\pi\)
−0.941280 + 0.337627i \(0.890376\pi\)
\(558\) −2.19773e10 −5.35494
\(559\) −7.74493e8 −0.187532
\(560\) 4.56962e8 0.109957
\(561\) 2.80756e9 0.671366
\(562\) 1.01714e7 0.00241716
\(563\) 7.00323e8 0.165394 0.0826968 0.996575i \(-0.473647\pi\)
0.0826968 + 0.996575i \(0.473647\pi\)
\(564\) −1.87635e10 −4.40389
\(565\) −7.21644e8 −0.168327
\(566\) 1.46083e9 0.338643
\(567\) 1.90132e10 4.38040
\(568\) −5.66081e9 −1.29616
\(569\) 1.86146e9 0.423605 0.211803 0.977312i \(-0.432067\pi\)
0.211803 + 0.977312i \(0.432067\pi\)
\(570\) −5.39856e9 −1.22100
\(571\) 6.78752e9 1.52575 0.762877 0.646544i \(-0.223786\pi\)
0.762877 + 0.646544i \(0.223786\pi\)
\(572\) −3.21930e9 −0.719243
\(573\) 6.46752e9 1.43614
\(574\) 4.37101e9 0.964695
\(575\) 2.91180e9 0.638741
\(576\) −1.86062e10 −4.05676
\(577\) −4.99855e8 −0.108325 −0.0541625 0.998532i \(-0.517249\pi\)
−0.0541625 + 0.998532i \(0.517249\pi\)
\(578\) 8.60702e8 0.185398
\(579\) 6.14597e9 1.31588
\(580\) 1.70103e9 0.362005
\(581\) −9.18567e9 −1.94310
\(582\) −1.89126e10 −3.97669
\(583\) −1.52538e9 −0.318814
\(584\) −9.12343e9 −1.89545
\(585\) 4.08020e9 0.842628
\(586\) −7.41512e9 −1.52222
\(587\) 7.26067e9 1.48164 0.740821 0.671703i \(-0.234437\pi\)
0.740821 + 0.671703i \(0.234437\pi\)
\(588\) −6.40910e9 −1.30010
\(589\) −8.83061e9 −1.78069
\(590\) 3.46829e9 0.695238
\(591\) −1.14153e10 −2.27474
\(592\) −2.12637e8 −0.0421224
\(593\) 6.06525e9 1.19442 0.597210 0.802085i \(-0.296276\pi\)
0.597210 + 0.802085i \(0.296276\pi\)
\(594\) 9.38164e9 1.83665
\(595\) 1.59584e9 0.310584
\(596\) 7.68999e9 1.48787
\(597\) 2.58425e9 0.497078
\(598\) −7.30086e9 −1.39611
\(599\) −5.97141e8 −0.113523 −0.0567614 0.998388i \(-0.518077\pi\)
−0.0567614 + 0.998388i \(0.518077\pi\)
\(600\) 1.23474e10 2.33370
\(601\) 2.66025e8 0.0499876 0.0249938 0.999688i \(-0.492043\pi\)
0.0249938 + 0.999688i \(0.492043\pi\)
\(602\) −1.59681e9 −0.298308
\(603\) −9.02378e9 −1.67601
\(604\) −1.29168e10 −2.38521
\(605\) −1.21785e9 −0.223588
\(606\) 7.90771e9 1.44343
\(607\) 5.53315e9 1.00418 0.502090 0.864815i \(-0.332564\pi\)
0.502090 + 0.864815i \(0.332564\pi\)
\(608\) −5.59738e9 −1.01000
\(609\) −1.02866e10 −1.84549
\(610\) 4.03926e8 0.0720522
\(611\) 8.91007e9 1.58029
\(612\) 2.89681e10 5.10846
\(613\) 3.58911e9 0.629325 0.314662 0.949204i \(-0.398109\pi\)
0.314662 + 0.949204i \(0.398109\pi\)
\(614\) 9.44028e9 1.64587
\(615\) −1.37854e9 −0.238976
\(616\) −2.89601e9 −0.499192
\(617\) −3.27338e9 −0.561046 −0.280523 0.959847i \(-0.590508\pi\)
−0.280523 + 0.959847i \(0.590508\pi\)
\(618\) 2.92152e9 0.497908
\(619\) −9.05095e9 −1.53383 −0.766914 0.641749i \(-0.778209\pi\)
−0.766914 + 0.641749i \(0.778209\pi\)
\(620\) −3.10850e9 −0.523817
\(621\) 1.36064e10 2.27993
\(622\) −7.64691e9 −1.27415
\(623\) −4.79122e9 −0.793850
\(624\) −5.38096e9 −0.886572
\(625\) 4.97545e9 0.815177
\(626\) −1.53906e9 −0.250753
\(627\) 5.94665e9 0.963465
\(628\) 2.13204e10 3.43508
\(629\) −7.42588e8 −0.118979
\(630\) 8.41231e9 1.34037
\(631\) −4.70226e9 −0.745082 −0.372541 0.928016i \(-0.621513\pi\)
−0.372541 + 0.928016i \(0.621513\pi\)
\(632\) −4.12451e9 −0.649925
\(633\) 1.03801e10 1.62663
\(634\) −7.05271e9 −1.09912
\(635\) −1.80860e9 −0.280307
\(636\) −2.15006e10 −3.31398
\(637\) 3.04344e9 0.466528
\(638\) −2.92991e9 −0.446665
\(639\) −1.81129e10 −2.74622
\(640\) −3.00447e9 −0.453041
\(641\) 8.63472e9 1.29493 0.647463 0.762097i \(-0.275830\pi\)
0.647463 + 0.762097i \(0.275830\pi\)
\(642\) 1.19346e10 1.78007
\(643\) 1.85229e9 0.274770 0.137385 0.990518i \(-0.456130\pi\)
0.137385 + 0.990518i \(0.456130\pi\)
\(644\) −9.62632e9 −1.42023
\(645\) 5.03603e8 0.0738974
\(646\) 1.82006e10 2.65626
\(647\) 4.30999e9 0.625622 0.312811 0.949815i \(-0.398729\pi\)
0.312811 + 0.949815i \(0.398729\pi\)
\(648\) 3.33054e10 4.80841
\(649\) −3.82041e9 −0.548597
\(650\) −1.34381e10 −1.91929
\(651\) 1.87979e10 2.67040
\(652\) 9.13252e9 1.29040
\(653\) −5.51577e9 −0.775193 −0.387597 0.921829i \(-0.626695\pi\)
−0.387597 + 0.921829i \(0.626695\pi\)
\(654\) −1.69144e10 −2.36447
\(655\) −1.53417e7 −0.00213319
\(656\) 1.33081e9 0.184057
\(657\) −2.91922e10 −4.01595
\(658\) 1.83703e10 2.51376
\(659\) 1.13199e10 1.54079 0.770397 0.637564i \(-0.220058\pi\)
0.770397 + 0.637564i \(0.220058\pi\)
\(660\) 2.09330e9 0.283419
\(661\) 5.19183e9 0.699222 0.349611 0.936895i \(-0.386314\pi\)
0.349611 + 0.936895i \(0.386314\pi\)
\(662\) 1.65826e9 0.222152
\(663\) −1.87918e10 −2.50422
\(664\) −1.60906e10 −2.13296
\(665\) 3.38012e9 0.445714
\(666\) −3.91448e9 −0.513469
\(667\) −4.24932e9 −0.554471
\(668\) 2.79007e9 0.362158
\(669\) 7.37603e9 0.952426
\(670\) −1.99578e9 −0.256360
\(671\) −4.44934e8 −0.0568548
\(672\) 1.19153e10 1.51465
\(673\) −7.58270e9 −0.958895 −0.479447 0.877571i \(-0.659163\pi\)
−0.479447 + 0.877571i \(0.659163\pi\)
\(674\) −4.83577e9 −0.608354
\(675\) 2.50442e10 3.13432
\(676\) 7.29887e9 0.908746
\(677\) 4.96130e9 0.614519 0.307259 0.951626i \(-0.400588\pi\)
0.307259 + 0.951626i \(0.400588\pi\)
\(678\) 1.75201e10 2.15891
\(679\) 1.18415e10 1.45165
\(680\) 2.79543e9 0.340932
\(681\) −3.08303e10 −3.74079
\(682\) 5.35418e9 0.646320
\(683\) −1.05886e10 −1.27165 −0.635823 0.771835i \(-0.719339\pi\)
−0.635823 + 0.771835i \(0.719339\pi\)
\(684\) 6.13569e10 7.33106
\(685\) 1.18930e9 0.141376
\(686\) −1.02651e10 −1.21403
\(687\) 1.04482e10 1.22939
\(688\) −4.86167e8 −0.0569149
\(689\) 1.02098e10 1.18919
\(690\) 4.74727e9 0.550139
\(691\) −1.26917e10 −1.46335 −0.731674 0.681655i \(-0.761261\pi\)
−0.731674 + 0.681655i \(0.761261\pi\)
\(692\) −1.19321e9 −0.136882
\(693\) −9.26637e9 −1.05765
\(694\) −5.79812e9 −0.658460
\(695\) −3.29961e9 −0.372834
\(696\) −1.80190e10 −2.02581
\(697\) 4.64756e9 0.519888
\(698\) −2.42850e10 −2.70298
\(699\) 1.64489e10 1.82166
\(700\) −1.77184e10 −1.95246
\(701\) −1.15011e10 −1.26104 −0.630519 0.776174i \(-0.717158\pi\)
−0.630519 + 0.776174i \(0.717158\pi\)
\(702\) −6.27941e10 −6.85076
\(703\) −1.57286e9 −0.170745
\(704\) 4.53291e9 0.489635
\(705\) −5.79364e9 −0.622715
\(706\) 2.72138e10 2.91054
\(707\) −4.95114e9 −0.526911
\(708\) −5.38496e10 −5.70251
\(709\) −1.09458e9 −0.115342 −0.0576709 0.998336i \(-0.518367\pi\)
−0.0576709 + 0.998336i \(0.518367\pi\)
\(710\) −4.00601e9 −0.420057
\(711\) −1.31972e10 −1.37701
\(712\) −8.39280e9 −0.871418
\(713\) 7.76528e9 0.802313
\(714\) −3.87439e10 −3.98345
\(715\) −9.94031e8 −0.101702
\(716\) 1.29578e10 1.31928
\(717\) 3.23837e8 0.0328102
\(718\) 1.66394e10 1.67765
\(719\) −6.22726e9 −0.624807 −0.312403 0.949950i \(-0.601134\pi\)
−0.312403 + 0.949950i \(0.601134\pi\)
\(720\) 2.56123e9 0.255732
\(721\) −1.82921e9 −0.181756
\(722\) 2.17066e10 2.14640
\(723\) −1.35704e10 −1.33539
\(724\) −1.42348e10 −1.39401
\(725\) −7.82138e9 −0.762255
\(726\) 2.95670e10 2.86767
\(727\) 1.91377e10 1.84722 0.923611 0.383332i \(-0.125223\pi\)
0.923611 + 0.383332i \(0.125223\pi\)
\(728\) 1.93839e10 1.86201
\(729\) 3.89806e10 3.72651
\(730\) −6.45642e9 −0.614273
\(731\) −1.69783e9 −0.160762
\(732\) −6.27145e9 −0.590989
\(733\) 9.03821e9 0.847654 0.423827 0.905743i \(-0.360686\pi\)
0.423827 + 0.905743i \(0.360686\pi\)
\(734\) −3.02997e9 −0.282815
\(735\) −1.97895e9 −0.183836
\(736\) 4.92211e9 0.455071
\(737\) 2.19840e9 0.202288
\(738\) 2.44991e10 2.24364
\(739\) −1.24567e9 −0.113540 −0.0567698 0.998387i \(-0.518080\pi\)
−0.0567698 + 0.998387i \(0.518080\pi\)
\(740\) −5.53670e8 −0.0502273
\(741\) −3.98027e10 −3.59376
\(742\) 2.10500e10 1.89164
\(743\) 1.95885e10 1.75203 0.876013 0.482287i \(-0.160194\pi\)
0.876013 + 0.482287i \(0.160194\pi\)
\(744\) 3.29283e10 2.93133
\(745\) 2.37446e9 0.210386
\(746\) 2.17425e9 0.191745
\(747\) −5.14850e10 −4.51917
\(748\) −7.05730e9 −0.616571
\(749\) −7.47244e9 −0.649795
\(750\) 1.80626e10 1.56339
\(751\) 1.10559e10 0.952473 0.476236 0.879317i \(-0.342001\pi\)
0.476236 + 0.879317i \(0.342001\pi\)
\(752\) 5.59305e9 0.479608
\(753\) 2.37841e10 2.03004
\(754\) 1.96108e10 1.66608
\(755\) −3.98835e9 −0.337271
\(756\) −8.27952e10 −6.96913
\(757\) −1.11431e10 −0.933617 −0.466808 0.884358i \(-0.654596\pi\)
−0.466808 + 0.884358i \(0.654596\pi\)
\(758\) 3.32543e10 2.77335
\(759\) −5.22924e9 −0.434103
\(760\) 5.92096e9 0.489266
\(761\) −1.64914e10 −1.35647 −0.678236 0.734844i \(-0.737255\pi\)
−0.678236 + 0.734844i \(0.737255\pi\)
\(762\) 4.39093e10 3.59513
\(763\) 1.05903e10 0.863125
\(764\) −1.62573e10 −1.31893
\(765\) 8.94455e9 0.722343
\(766\) 4.75099e9 0.381929
\(767\) 2.55712e10 2.04629
\(768\) 3.69279e10 2.94165
\(769\) −2.19764e10 −1.74266 −0.871332 0.490694i \(-0.836743\pi\)
−0.871332 + 0.490694i \(0.836743\pi\)
\(770\) −2.04943e9 −0.161777
\(771\) −2.59277e10 −2.03739
\(772\) −1.54490e10 −1.20848
\(773\) 1.65022e10 1.28503 0.642517 0.766271i \(-0.277890\pi\)
0.642517 + 0.766271i \(0.277890\pi\)
\(774\) −8.94996e9 −0.693790
\(775\) 1.42929e10 1.10297
\(776\) 2.07427e10 1.59349
\(777\) 3.34819e9 0.256057
\(778\) −2.21268e10 −1.68457
\(779\) 9.84391e9 0.746082
\(780\) −1.40111e10 −1.05716
\(781\) 4.41272e9 0.331457
\(782\) −1.60048e10 −1.19682
\(783\) −3.65480e10 −2.72081
\(784\) 1.91044e9 0.141588
\(785\) 6.58316e9 0.485725
\(786\) 3.72467e8 0.0273595
\(787\) 1.33119e10 0.973483 0.486741 0.873546i \(-0.338185\pi\)
0.486741 + 0.873546i \(0.338185\pi\)
\(788\) 2.86944e10 2.08908
\(789\) −3.74684e10 −2.71579
\(790\) −2.91881e9 −0.210626
\(791\) −1.09696e10 −0.788087
\(792\) −1.62319e10 −1.16100
\(793\) 2.97808e9 0.212070
\(794\) −2.26171e10 −1.60349
\(795\) −6.63878e9 −0.468601
\(796\) −6.49597e9 −0.456508
\(797\) 6.86024e9 0.479993 0.239997 0.970774i \(-0.422854\pi\)
0.239997 + 0.970774i \(0.422854\pi\)
\(798\) −8.20628e10 −5.71658
\(799\) 1.95325e10 1.35470
\(800\) 9.05974e9 0.625606
\(801\) −2.68544e10 −1.84630
\(802\) −3.62340e9 −0.248031
\(803\) 7.11191e9 0.484710
\(804\) 3.09870e10 2.10273
\(805\) −2.97234e9 −0.200823
\(806\) −3.58371e10 −2.41080
\(807\) 1.00500e10 0.673145
\(808\) −8.67292e9 −0.578396
\(809\) 7.51249e9 0.498843 0.249422 0.968395i \(-0.419760\pi\)
0.249422 + 0.968395i \(0.419760\pi\)
\(810\) 2.35694e10 1.55830
\(811\) 1.36176e10 0.896453 0.448226 0.893920i \(-0.352056\pi\)
0.448226 + 0.893920i \(0.352056\pi\)
\(812\) 2.58572e10 1.69486
\(813\) −9.20121e9 −0.600521
\(814\) 9.53659e8 0.0619737
\(815\) 2.81987e9 0.182464
\(816\) −1.17961e10 −0.760014
\(817\) −3.59615e9 −0.230707
\(818\) 8.76964e9 0.560202
\(819\) 6.20226e10 3.94508
\(820\) 3.46519e9 0.219472
\(821\) −8.07643e8 −0.0509353 −0.0254676 0.999676i \(-0.508107\pi\)
−0.0254676 + 0.999676i \(0.508107\pi\)
\(822\) −2.88739e10 −1.81324
\(823\) −2.23299e10 −1.39633 −0.698163 0.715938i \(-0.745999\pi\)
−0.698163 + 0.715938i \(0.745999\pi\)
\(824\) −3.20423e9 −0.199516
\(825\) −9.62504e9 −0.596780
\(826\) 5.27211e10 3.25502
\(827\) 1.07538e9 0.0661136 0.0330568 0.999453i \(-0.489476\pi\)
0.0330568 + 0.999453i \(0.489476\pi\)
\(828\) −5.39547e10 −3.30311
\(829\) 1.30676e10 0.796628 0.398314 0.917249i \(-0.369595\pi\)
0.398314 + 0.917249i \(0.369595\pi\)
\(830\) −1.13869e10 −0.691244
\(831\) −2.72115e10 −1.64494
\(832\) −3.03401e10 −1.82636
\(833\) 6.67179e9 0.399931
\(834\) 8.01082e10 4.78185
\(835\) 8.61497e8 0.0512096
\(836\) −1.49480e10 −0.884830
\(837\) 6.67886e10 3.93698
\(838\) −4.23380e10 −2.48529
\(839\) −7.30739e9 −0.427165 −0.213582 0.976925i \(-0.568513\pi\)
−0.213582 + 0.976925i \(0.568513\pi\)
\(840\) −1.26041e10 −0.733725
\(841\) −5.83582e9 −0.338311
\(842\) 1.24885e10 0.720974
\(843\) −4.87626e7 −0.00280343
\(844\) −2.60923e10 −1.49387
\(845\) 2.25369e9 0.128498
\(846\) 1.02964e11 5.84640
\(847\) −1.85123e10 −1.04681
\(848\) 6.40894e9 0.360911
\(849\) −7.00334e9 −0.392761
\(850\) −2.94588e10 −1.64531
\(851\) 1.38311e9 0.0769315
\(852\) 6.21983e10 3.44540
\(853\) −2.19746e10 −1.21227 −0.606134 0.795363i \(-0.707280\pi\)
−0.606134 + 0.795363i \(0.707280\pi\)
\(854\) 6.14003e9 0.337340
\(855\) 1.89453e10 1.03662
\(856\) −1.30895e10 −0.713288
\(857\) −4.09566e9 −0.222275 −0.111138 0.993805i \(-0.535449\pi\)
−0.111138 + 0.993805i \(0.535449\pi\)
\(858\) 2.41332e10 1.30439
\(859\) −2.48200e10 −1.33606 −0.668031 0.744134i \(-0.732863\pi\)
−0.668031 + 0.744134i \(0.732863\pi\)
\(860\) −1.26590e9 −0.0678661
\(861\) −2.09549e10 −1.11886
\(862\) −8.26757e9 −0.439645
\(863\) −9.32141e9 −0.493678 −0.246839 0.969056i \(-0.579392\pi\)
−0.246839 + 0.969056i \(0.579392\pi\)
\(864\) 4.23347e10 2.23305
\(865\) −3.68430e8 −0.0193552
\(866\) 2.03590e9 0.106523
\(867\) −4.12627e9 −0.215026
\(868\) −4.72519e10 −2.45245
\(869\) 3.21515e9 0.166200
\(870\) −1.27516e10 −0.656520
\(871\) −1.47146e10 −0.754543
\(872\) 1.85511e10 0.947463
\(873\) 6.63705e10 3.37618
\(874\) −3.38996e10 −1.71753
\(875\) −1.13093e10 −0.570699
\(876\) 1.00244e11 5.03841
\(877\) −1.06318e10 −0.532239 −0.266120 0.963940i \(-0.585742\pi\)
−0.266120 + 0.963940i \(0.585742\pi\)
\(878\) −4.88903e10 −2.43777
\(879\) 3.55486e10 1.76548
\(880\) −6.23976e8 −0.0308659
\(881\) 8.99181e9 0.443028 0.221514 0.975157i \(-0.428900\pi\)
0.221514 + 0.975157i \(0.428900\pi\)
\(882\) 3.51697e10 1.72595
\(883\) 7.28785e9 0.356235 0.178118 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(884\) 4.72366e10 2.29983
\(885\) −1.66273e10 −0.806342
\(886\) 2.49449e10 1.20494
\(887\) 2.86600e10 1.37893 0.689467 0.724317i \(-0.257845\pi\)
0.689467 + 0.724317i \(0.257845\pi\)
\(888\) 5.86503e9 0.281077
\(889\) −2.74923e10 −1.31236
\(890\) −5.93937e9 −0.282407
\(891\) −2.59622e10 −1.22962
\(892\) −1.85410e10 −0.874692
\(893\) 4.13715e10 1.94411
\(894\) −5.76473e10 −2.69834
\(895\) 4.00102e9 0.186548
\(896\) −4.56706e10 −2.12109
\(897\) 3.50009e10 1.61922
\(898\) 3.27367e10 1.50858
\(899\) −2.08583e10 −0.957457
\(900\) −9.93102e10 −4.54093
\(901\) 2.23818e10 1.01943
\(902\) −5.96856e9 −0.270799
\(903\) 7.65520e9 0.345979
\(904\) −1.92155e10 −0.865092
\(905\) −4.39530e9 −0.197115
\(906\) 9.68296e10 4.32573
\(907\) 2.38033e9 0.105928 0.0529642 0.998596i \(-0.483133\pi\)
0.0529642 + 0.998596i \(0.483133\pi\)
\(908\) 7.74976e10 3.43548
\(909\) −2.77507e10 −1.22546
\(910\) 1.37175e10 0.603434
\(911\) −3.27857e10 −1.43671 −0.718356 0.695676i \(-0.755105\pi\)
−0.718356 + 0.695676i \(0.755105\pi\)
\(912\) −2.49851e10 −1.09068
\(913\) 1.25429e10 0.545446
\(914\) −2.59051e10 −1.12221
\(915\) −1.93645e9 −0.0835666
\(916\) −2.62633e10 −1.12906
\(917\) −2.33207e8 −0.00998732
\(918\) −1.37656e11 −5.87282
\(919\) 1.71738e10 0.729899 0.364950 0.931027i \(-0.381086\pi\)
0.364950 + 0.931027i \(0.381086\pi\)
\(920\) −5.20665e9 −0.220446
\(921\) −4.52574e10 −1.90889
\(922\) −2.54131e10 −1.06782
\(923\) −2.95357e10 −1.23635
\(924\) 3.18200e10 1.32693
\(925\) 2.54578e9 0.105761
\(926\) −3.43904e10 −1.42331
\(927\) −1.02526e10 −0.422721
\(928\) −1.32212e10 −0.543068
\(929\) −4.09833e9 −0.167707 −0.0838536 0.996478i \(-0.526723\pi\)
−0.0838536 + 0.996478i \(0.526723\pi\)
\(930\) 2.33025e10 0.949977
\(931\) 1.41314e10 0.573934
\(932\) −4.13474e10 −1.67299
\(933\) 3.66599e10 1.47776
\(934\) 2.56363e10 1.02953
\(935\) −2.17910e9 −0.0871839
\(936\) 1.08645e11 4.33057
\(937\) 3.03563e10 1.20548 0.602740 0.797938i \(-0.294076\pi\)
0.602740 + 0.797938i \(0.294076\pi\)
\(938\) −3.03376e10 −1.20025
\(939\) 7.37838e9 0.290825
\(940\) 1.45633e10 0.571891
\(941\) −3.15855e10 −1.23573 −0.617867 0.786283i \(-0.712003\pi\)
−0.617867 + 0.786283i \(0.712003\pi\)
\(942\) −1.59827e11 −6.22975
\(943\) −8.65634e9 −0.336158
\(944\) 1.60516e10 0.621035
\(945\) −2.55649e10 −0.985444
\(946\) 2.18042e9 0.0837377
\(947\) −1.44552e10 −0.553095 −0.276548 0.961000i \(-0.589190\pi\)
−0.276548 + 0.961000i \(0.589190\pi\)
\(948\) 4.53182e10 1.72760
\(949\) −4.76021e10 −1.80798
\(950\) −6.23962e10 −2.36116
\(951\) 3.38112e10 1.27476
\(952\) 4.24930e10 1.59620
\(953\) −8.95669e9 −0.335214 −0.167607 0.985854i \(-0.553604\pi\)
−0.167607 + 0.985854i \(0.553604\pi\)
\(954\) 1.17984e11 4.39949
\(955\) −5.01979e9 −0.186498
\(956\) −8.14022e8 −0.0301324
\(957\) 1.40462e10 0.518045
\(958\) 5.75729e10 2.11562
\(959\) 1.80784e10 0.661904
\(960\) 1.97282e10 0.719678
\(961\) 1.06042e10 0.385430
\(962\) −6.38313e9 −0.231164
\(963\) −4.18824e10 −1.51126
\(964\) 3.41117e10 1.22640
\(965\) −4.77022e9 −0.170881
\(966\) 7.21627e10 2.57569
\(967\) 3.78882e10 1.34745 0.673723 0.738984i \(-0.264694\pi\)
0.673723 + 0.738984i \(0.264694\pi\)
\(968\) −3.24281e10 −1.14910
\(969\) −8.72548e10 −3.08075
\(970\) 1.46791e10 0.516415
\(971\) 3.53468e10 1.23903 0.619516 0.784984i \(-0.287329\pi\)
0.619516 + 0.784984i \(0.287329\pi\)
\(972\) −1.96053e11 −6.84765
\(973\) −5.01570e10 −1.74557
\(974\) −7.24023e10 −2.51071
\(975\) 6.44233e10 2.22601
\(976\) 1.86941e9 0.0643620
\(977\) 4.28246e10 1.46914 0.734568 0.678535i \(-0.237385\pi\)
0.734568 + 0.678535i \(0.237385\pi\)
\(978\) −6.84610e10 −2.34022
\(979\) 6.54236e9 0.222841
\(980\) 4.97445e9 0.168832
\(981\) 5.93579e10 2.00742
\(982\) −1.81031e9 −0.0610048
\(983\) 1.30981e10 0.439815 0.219908 0.975521i \(-0.429424\pi\)
0.219908 + 0.975521i \(0.429424\pi\)
\(984\) −3.67068e10 −1.22819
\(985\) 8.86005e9 0.295399
\(986\) 4.29905e10 1.42825
\(987\) −8.80684e10 −2.91548
\(988\) 1.00051e11 3.30045
\(989\) 3.16231e9 0.103948
\(990\) −1.14869e10 −0.376253
\(991\) −2.48042e10 −0.809593 −0.404796 0.914407i \(-0.632658\pi\)
−0.404796 + 0.914407i \(0.632658\pi\)
\(992\) 2.41608e10 0.785814
\(993\) −7.94985e9 −0.257654
\(994\) −6.08949e10 −1.96666
\(995\) −2.00578e9 −0.0645508
\(996\) 1.76796e11 5.66975
\(997\) 4.36888e10 1.39617 0.698083 0.716017i \(-0.254037\pi\)
0.698083 + 0.716017i \(0.254037\pi\)
\(998\) 8.74368e9 0.278444
\(999\) 1.18960e10 0.377506
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 43.8.a.b.1.12 13
3.2 odd 2 387.8.a.d.1.2 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.12 13 1.1 even 1 trivial
387.8.a.d.1.2 13 3.2 odd 2