Properties

Label 43.8.a.b.1.3
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 1.3
Root \(-16.1540\) 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

\(f(q)\) \(=\) \(q-15.1540 q^{2} +48.8090 q^{3} +101.645 q^{4} +210.950 q^{5} -739.653 q^{6} +1100.87 q^{7} +399.390 q^{8} +195.321 q^{9} +O(q^{10})\) \(q-15.1540 q^{2} +48.8090 q^{3} +101.645 q^{4} +210.950 q^{5} -739.653 q^{6} +1100.87 q^{7} +399.390 q^{8} +195.321 q^{9} -3196.74 q^{10} -3136.09 q^{11} +4961.17 q^{12} -5743.36 q^{13} -16682.6 q^{14} +10296.3 q^{15} -19062.9 q^{16} +27231.5 q^{17} -2959.90 q^{18} +54685.6 q^{19} +21441.9 q^{20} +53732.2 q^{21} +47524.4 q^{22} +12024.9 q^{23} +19493.9 q^{24} -33625.2 q^{25} +87035.1 q^{26} -97211.9 q^{27} +111897. q^{28} +132127. q^{29} -156030. q^{30} +281517. q^{31} +237758. q^{32} -153070. q^{33} -412666. q^{34} +232228. q^{35} +19853.3 q^{36} -214511. q^{37} -828707. q^{38} -280328. q^{39} +84251.4 q^{40} +302212. q^{41} -814260. q^{42} -79507.0 q^{43} -318767. q^{44} +41202.9 q^{45} -182226. q^{46} +235041. q^{47} -930441. q^{48} +388365. q^{49} +509557. q^{50} +1.32914e6 q^{51} -583782. q^{52} +55032.5 q^{53} +1.47315e6 q^{54} -661558. q^{55} +439676. q^{56} +2.66915e6 q^{57} -2.00226e6 q^{58} +150144. q^{59} +1.04656e6 q^{60} -1.65598e6 q^{61} -4.26612e6 q^{62} +215022. q^{63} -1.16294e6 q^{64} -1.21156e6 q^{65} +2.31962e6 q^{66} +1.76796e6 q^{67} +2.76793e6 q^{68} +586923. q^{69} -3.51919e6 q^{70} +3.57159e6 q^{71} +78009.3 q^{72} -430044. q^{73} +3.25070e6 q^{74} -1.64121e6 q^{75} +5.55849e6 q^{76} -3.45242e6 q^{77} +4.24810e6 q^{78} -7.44014e6 q^{79} -4.02131e6 q^{80} -5.17199e6 q^{81} -4.57974e6 q^{82} +5.50388e6 q^{83} +5.46159e6 q^{84} +5.74447e6 q^{85} +1.20485e6 q^{86} +6.44900e6 q^{87} -1.25252e6 q^{88} -3.53700e6 q^{89} -624390. q^{90} -6.32268e6 q^{91} +1.22227e6 q^{92} +1.37406e7 q^{93} -3.56182e6 q^{94} +1.15359e7 q^{95} +1.16047e7 q^{96} -2.08749e6 q^{97} -5.88530e6 q^{98} -612544. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.1540 −1.33944 −0.669720 0.742614i \(-0.733586\pi\)
−0.669720 + 0.742614i \(0.733586\pi\)
\(3\) 48.8090 1.04370 0.521850 0.853037i \(-0.325242\pi\)
0.521850 + 0.853037i \(0.325242\pi\)
\(4\) 101.645 0.794098
\(5\) 210.950 0.754717 0.377359 0.926067i \(-0.376832\pi\)
0.377359 + 0.926067i \(0.376832\pi\)
\(6\) −739.653 −1.39797
\(7\) 1100.87 1.21309 0.606543 0.795051i \(-0.292556\pi\)
0.606543 + 0.795051i \(0.292556\pi\)
\(8\) 399.390 0.275793
\(9\) 195.321 0.0893100
\(10\) −3196.74 −1.01090
\(11\) −3136.09 −0.710419 −0.355209 0.934787i \(-0.615590\pi\)
−0.355209 + 0.934787i \(0.615590\pi\)
\(12\) 4961.17 0.828801
\(13\) −5743.36 −0.725044 −0.362522 0.931975i \(-0.618084\pi\)
−0.362522 + 0.931975i \(0.618084\pi\)
\(14\) −16682.6 −1.62486
\(15\) 10296.3 0.787698
\(16\) −19062.9 −1.16351
\(17\) 27231.5 1.34431 0.672155 0.740410i \(-0.265369\pi\)
0.672155 + 0.740410i \(0.265369\pi\)
\(18\) −2959.90 −0.119625
\(19\) 54685.6 1.82909 0.914545 0.404484i \(-0.132549\pi\)
0.914545 + 0.404484i \(0.132549\pi\)
\(20\) 21441.9 0.599320
\(21\) 53732.2 1.26610
\(22\) 47524.4 0.951563
\(23\) 12024.9 0.206079 0.103040 0.994677i \(-0.467143\pi\)
0.103040 + 0.994677i \(0.467143\pi\)
\(24\) 19493.9 0.287845
\(25\) −33625.2 −0.430402
\(26\) 87035.1 0.971153
\(27\) −97211.9 −0.950487
\(28\) 111897. 0.963310
\(29\) 132127. 1.00600 0.503001 0.864286i \(-0.332229\pi\)
0.503001 + 0.864286i \(0.332229\pi\)
\(30\) −156030. −1.05507
\(31\) 281517. 1.69722 0.848612 0.529016i \(-0.177439\pi\)
0.848612 + 0.529016i \(0.177439\pi\)
\(32\) 237758. 1.28265
\(33\) −153070. −0.741464
\(34\) −412666. −1.80062
\(35\) 232228. 0.915537
\(36\) 19853.3 0.0709209
\(37\) −214511. −0.696215 −0.348107 0.937455i \(-0.613176\pi\)
−0.348107 + 0.937455i \(0.613176\pi\)
\(38\) −828707. −2.44996
\(39\) −280328. −0.756729
\(40\) 84251.4 0.208145
\(41\) 302212. 0.684808 0.342404 0.939553i \(-0.388759\pi\)
0.342404 + 0.939553i \(0.388759\pi\)
\(42\) −814260. −1.69586
\(43\) −79507.0 −0.152499
\(44\) −318767. −0.564142
\(45\) 41202.9 0.0674038
\(46\) −182226. −0.276030
\(47\) 235041. 0.330218 0.165109 0.986275i \(-0.447202\pi\)
0.165109 + 0.986275i \(0.447202\pi\)
\(48\) −930441. −1.21435
\(49\) 388365. 0.471578
\(50\) 509557. 0.576498
\(51\) 1.32914e6 1.40306
\(52\) −583782. −0.575757
\(53\) 55032.5 0.0507754 0.0253877 0.999678i \(-0.491918\pi\)
0.0253877 + 0.999678i \(0.491918\pi\)
\(54\) 1.47315e6 1.27312
\(55\) −661558. −0.536165
\(56\) 439676. 0.334560
\(57\) 2.66915e6 1.90902
\(58\) −2.00226e6 −1.34748
\(59\) 150144. 0.0951754 0.0475877 0.998867i \(-0.484847\pi\)
0.0475877 + 0.998867i \(0.484847\pi\)
\(60\) 1.04656e6 0.625510
\(61\) −1.65598e6 −0.934113 −0.467057 0.884227i \(-0.654686\pi\)
−0.467057 + 0.884227i \(0.654686\pi\)
\(62\) −4.26612e6 −2.27333
\(63\) 215022. 0.108341
\(64\) −1.16294e6 −0.554531
\(65\) −1.21156e6 −0.547203
\(66\) 2.31962e6 0.993146
\(67\) 1.76796e6 0.718144 0.359072 0.933310i \(-0.383093\pi\)
0.359072 + 0.933310i \(0.383093\pi\)
\(68\) 2.76793e6 1.06752
\(69\) 586923. 0.215085
\(70\) −3.51919e6 −1.22631
\(71\) 3.57159e6 1.18429 0.592144 0.805832i \(-0.298282\pi\)
0.592144 + 0.805832i \(0.298282\pi\)
\(72\) 78009.3 0.0246310
\(73\) −430044. −0.129385 −0.0646923 0.997905i \(-0.520607\pi\)
−0.0646923 + 0.997905i \(0.520607\pi\)
\(74\) 3.25070e6 0.932538
\(75\) −1.64121e6 −0.449211
\(76\) 5.55849e6 1.45248
\(77\) −3.45242e6 −0.861799
\(78\) 4.24810e6 1.01359
\(79\) −7.44014e6 −1.69780 −0.848900 0.528554i \(-0.822734\pi\)
−0.848900 + 0.528554i \(0.822734\pi\)
\(80\) −4.02131e6 −0.878118
\(81\) −5.17199e6 −1.08133
\(82\) −4.57974e6 −0.917259
\(83\) 5.50388e6 1.05656 0.528282 0.849069i \(-0.322837\pi\)
0.528282 + 0.849069i \(0.322837\pi\)
\(84\) 5.46159e6 1.00541
\(85\) 5.74447e6 1.01457
\(86\) 1.20485e6 0.204263
\(87\) 6.44900e6 1.04996
\(88\) −1.25252e6 −0.195928
\(89\) −3.53700e6 −0.531826 −0.265913 0.963997i \(-0.585673\pi\)
−0.265913 + 0.963997i \(0.585673\pi\)
\(90\) −624390. −0.0902833
\(91\) −6.32268e6 −0.879541
\(92\) 1.22227e6 0.163647
\(93\) 1.37406e7 1.77139
\(94\) −3.56182e6 −0.442308
\(95\) 1.15359e7 1.38045
\(96\) 1.16047e7 1.33871
\(97\) −2.08749e6 −0.232233 −0.116117 0.993236i \(-0.537045\pi\)
−0.116117 + 0.993236i \(0.537045\pi\)
\(98\) −5.88530e6 −0.631651
\(99\) −612544. −0.0634475
\(100\) −3.41782e6 −0.341782
\(101\) 1.59947e7 1.54473 0.772364 0.635180i \(-0.219074\pi\)
0.772364 + 0.635180i \(0.219074\pi\)
\(102\) −2.01418e7 −1.87931
\(103\) −1.14391e7 −1.03148 −0.515739 0.856746i \(-0.672483\pi\)
−0.515739 + 0.856746i \(0.672483\pi\)
\(104\) −2.29384e6 −0.199962
\(105\) 1.13348e7 0.955546
\(106\) −833964. −0.0680106
\(107\) 5.13105e6 0.404914 0.202457 0.979291i \(-0.435107\pi\)
0.202457 + 0.979291i \(0.435107\pi\)
\(108\) −9.88107e6 −0.754781
\(109\) −2.10496e7 −1.55687 −0.778433 0.627727i \(-0.783985\pi\)
−0.778433 + 0.627727i \(0.783985\pi\)
\(110\) 1.00253e7 0.718161
\(111\) −1.04701e7 −0.726639
\(112\) −2.09857e7 −1.41143
\(113\) −1.44050e7 −0.939156 −0.469578 0.882891i \(-0.655594\pi\)
−0.469578 + 0.882891i \(0.655594\pi\)
\(114\) −4.04484e7 −2.55702
\(115\) 2.53665e6 0.155531
\(116\) 1.34300e7 0.798865
\(117\) −1.12180e6 −0.0647537
\(118\) −2.27528e6 −0.127482
\(119\) 2.99782e7 1.63077
\(120\) 4.11223e6 0.217241
\(121\) −9.65210e6 −0.495306
\(122\) 2.50947e7 1.25119
\(123\) 1.47507e7 0.714734
\(124\) 2.86147e7 1.34776
\(125\) −2.35737e7 −1.07955
\(126\) −3.25845e6 −0.145116
\(127\) −2.66397e7 −1.15403 −0.577014 0.816734i \(-0.695782\pi\)
−0.577014 + 0.816734i \(0.695782\pi\)
\(128\) −1.28098e7 −0.539893
\(129\) −3.88066e6 −0.159163
\(130\) 1.83600e7 0.732946
\(131\) −1.53052e7 −0.594825 −0.297413 0.954749i \(-0.596124\pi\)
−0.297413 + 0.954749i \(0.596124\pi\)
\(132\) −1.55587e7 −0.588795
\(133\) 6.02015e7 2.21884
\(134\) −2.67918e7 −0.961911
\(135\) −2.05068e7 −0.717349
\(136\) 1.08760e7 0.370751
\(137\) −2.88018e7 −0.956968 −0.478484 0.878096i \(-0.658814\pi\)
−0.478484 + 0.878096i \(0.658814\pi\)
\(138\) −8.89425e6 −0.288093
\(139\) 2.41973e7 0.764213 0.382106 0.924118i \(-0.375199\pi\)
0.382106 + 0.924118i \(0.375199\pi\)
\(140\) 2.36047e7 0.727026
\(141\) 1.14721e7 0.344649
\(142\) −5.41240e7 −1.58628
\(143\) 1.80117e7 0.515085
\(144\) −3.72338e6 −0.103913
\(145\) 2.78722e7 0.759247
\(146\) 6.51690e6 0.173303
\(147\) 1.89557e7 0.492186
\(148\) −2.18039e7 −0.552863
\(149\) 1.88108e7 0.465859 0.232929 0.972494i \(-0.425169\pi\)
0.232929 + 0.972494i \(0.425169\pi\)
\(150\) 2.48710e7 0.601691
\(151\) −5.28673e7 −1.24959 −0.624796 0.780788i \(-0.714818\pi\)
−0.624796 + 0.780788i \(0.714818\pi\)
\(152\) 2.18409e7 0.504450
\(153\) 5.31887e6 0.120060
\(154\) 5.23181e7 1.15433
\(155\) 5.93860e7 1.28092
\(156\) −2.84938e7 −0.600917
\(157\) −5.80087e7 −1.19631 −0.598156 0.801380i \(-0.704100\pi\)
−0.598156 + 0.801380i \(0.704100\pi\)
\(158\) 1.12748e8 2.27410
\(159\) 2.68608e6 0.0529943
\(160\) 5.01549e7 0.968040
\(161\) 1.32378e7 0.249992
\(162\) 7.83764e7 1.44838
\(163\) 9.85826e7 1.78297 0.891484 0.453051i \(-0.149665\pi\)
0.891484 + 0.453051i \(0.149665\pi\)
\(164\) 3.07183e7 0.543805
\(165\) −3.22900e7 −0.559595
\(166\) −8.34059e7 −1.41520
\(167\) 3.30309e7 0.548798 0.274399 0.961616i \(-0.411521\pi\)
0.274399 + 0.961616i \(0.411521\pi\)
\(168\) 2.14601e7 0.349181
\(169\) −2.97623e7 −0.474311
\(170\) −8.70519e7 −1.35896
\(171\) 1.06812e7 0.163356
\(172\) −8.08146e6 −0.121099
\(173\) −9.21797e7 −1.35355 −0.676774 0.736191i \(-0.736623\pi\)
−0.676774 + 0.736191i \(0.736623\pi\)
\(174\) −9.77283e7 −1.40636
\(175\) −3.70168e7 −0.522115
\(176\) 5.97829e7 0.826576
\(177\) 7.32836e6 0.0993346
\(178\) 5.35998e7 0.712349
\(179\) 8.22086e7 1.07135 0.535675 0.844424i \(-0.320057\pi\)
0.535675 + 0.844424i \(0.320057\pi\)
\(180\) 4.18805e6 0.0535252
\(181\) −4.53246e7 −0.568144 −0.284072 0.958803i \(-0.591686\pi\)
−0.284072 + 0.958803i \(0.591686\pi\)
\(182\) 9.58141e7 1.17809
\(183\) −8.08266e7 −0.974934
\(184\) 4.80263e6 0.0568351
\(185\) −4.52510e7 −0.525445
\(186\) −2.08225e8 −2.37267
\(187\) −8.54003e7 −0.955023
\(188\) 2.38907e7 0.262226
\(189\) −1.07017e8 −1.15302
\(190\) −1.74816e8 −1.84902
\(191\) −1.21596e8 −1.26270 −0.631352 0.775496i \(-0.717500\pi\)
−0.631352 + 0.775496i \(0.717500\pi\)
\(192\) −5.67617e7 −0.578764
\(193\) 1.40361e8 1.40538 0.702691 0.711495i \(-0.251981\pi\)
0.702691 + 0.711495i \(0.251981\pi\)
\(194\) 3.16339e7 0.311062
\(195\) −5.91351e7 −0.571116
\(196\) 3.94752e7 0.374480
\(197\) −5.68563e7 −0.529842 −0.264921 0.964270i \(-0.585346\pi\)
−0.264921 + 0.964270i \(0.585346\pi\)
\(198\) 9.28251e6 0.0849840
\(199\) 3.29464e7 0.296361 0.148181 0.988960i \(-0.452658\pi\)
0.148181 + 0.988960i \(0.452658\pi\)
\(200\) −1.34296e7 −0.118702
\(201\) 8.62926e7 0.749527
\(202\) −2.42385e8 −2.06907
\(203\) 1.45454e8 1.22037
\(204\) 1.35100e8 1.11417
\(205\) 6.37517e7 0.516836
\(206\) 1.73348e8 1.38160
\(207\) 2.34871e6 0.0184049
\(208\) 1.09485e8 0.843593
\(209\) −1.71499e8 −1.29942
\(210\) −1.71768e8 −1.27990
\(211\) −2.35087e8 −1.72282 −0.861411 0.507908i \(-0.830419\pi\)
−0.861411 + 0.507908i \(0.830419\pi\)
\(212\) 5.59375e6 0.0403207
\(213\) 1.74326e8 1.23604
\(214\) −7.77561e7 −0.542358
\(215\) −1.67720e7 −0.115093
\(216\) −3.88255e7 −0.262137
\(217\) 3.09913e8 2.05888
\(218\) 3.18986e8 2.08533
\(219\) −2.09900e7 −0.135039
\(220\) −6.72438e7 −0.425768
\(221\) −1.56400e8 −0.974685
\(222\) 1.58664e8 0.973290
\(223\) 6.86931e7 0.414807 0.207404 0.978255i \(-0.433499\pi\)
0.207404 + 0.978255i \(0.433499\pi\)
\(224\) 2.61739e8 1.55597
\(225\) −6.56770e6 −0.0384392
\(226\) 2.18293e8 1.25794
\(227\) −3.33942e8 −1.89487 −0.947436 0.319945i \(-0.896336\pi\)
−0.947436 + 0.319945i \(0.896336\pi\)
\(228\) 2.71305e8 1.51595
\(229\) 3.08822e8 1.69935 0.849677 0.527303i \(-0.176797\pi\)
0.849677 + 0.527303i \(0.176797\pi\)
\(230\) −3.84405e7 −0.208325
\(231\) −1.68509e8 −0.899460
\(232\) 5.27703e7 0.277448
\(233\) 1.57739e8 0.816947 0.408473 0.912770i \(-0.366061\pi\)
0.408473 + 0.912770i \(0.366061\pi\)
\(234\) 1.69998e7 0.0867337
\(235\) 4.95819e7 0.249221
\(236\) 1.52613e7 0.0755787
\(237\) −3.63146e8 −1.77199
\(238\) −4.54291e8 −2.18431
\(239\) −1.84694e8 −0.875105 −0.437553 0.899193i \(-0.644155\pi\)
−0.437553 + 0.899193i \(0.644155\pi\)
\(240\) −1.96276e8 −0.916492
\(241\) 4.79565e7 0.220692 0.110346 0.993893i \(-0.464804\pi\)
0.110346 + 0.993893i \(0.464804\pi\)
\(242\) 1.46268e8 0.663432
\(243\) −3.98371e7 −0.178101
\(244\) −1.68321e8 −0.741778
\(245\) 8.19256e7 0.355908
\(246\) −2.23532e8 −0.957343
\(247\) −3.14079e8 −1.32617
\(248\) 1.12435e8 0.468082
\(249\) 2.68639e8 1.10274
\(250\) 3.57236e8 1.44599
\(251\) −4.64988e8 −1.85602 −0.928012 0.372550i \(-0.878483\pi\)
−0.928012 + 0.372550i \(0.878483\pi\)
\(252\) 2.18559e7 0.0860332
\(253\) −3.77112e7 −0.146402
\(254\) 4.03699e8 1.54575
\(255\) 2.80382e8 1.05891
\(256\) 3.42976e8 1.27768
\(257\) 2.59186e8 0.952456 0.476228 0.879322i \(-0.342004\pi\)
0.476228 + 0.879322i \(0.342004\pi\)
\(258\) 5.88076e7 0.213189
\(259\) −2.36148e8 −0.844569
\(260\) −1.23149e8 −0.434533
\(261\) 2.58072e7 0.0898460
\(262\) 2.31935e8 0.796732
\(263\) −2.19827e8 −0.745137 −0.372569 0.928005i \(-0.621523\pi\)
−0.372569 + 0.928005i \(0.621523\pi\)
\(264\) −6.11345e7 −0.204490
\(265\) 1.16091e7 0.0383211
\(266\) −9.12296e8 −2.97201
\(267\) −1.72637e8 −0.555067
\(268\) 1.79704e8 0.570277
\(269\) 5.19950e8 1.62865 0.814327 0.580407i \(-0.197107\pi\)
0.814327 + 0.580407i \(0.197107\pi\)
\(270\) 3.10761e8 0.960846
\(271\) 2.80890e8 0.857323 0.428661 0.903465i \(-0.358985\pi\)
0.428661 + 0.903465i \(0.358985\pi\)
\(272\) −5.19110e8 −1.56411
\(273\) −3.08604e8 −0.917977
\(274\) 4.36463e8 1.28180
\(275\) 1.05452e8 0.305766
\(276\) 5.96576e7 0.170798
\(277\) 2.46843e8 0.697818 0.348909 0.937157i \(-0.386552\pi\)
0.348909 + 0.937157i \(0.386552\pi\)
\(278\) −3.66686e8 −1.02362
\(279\) 5.49862e7 0.151579
\(280\) 9.27495e7 0.252498
\(281\) 1.74389e8 0.468863 0.234432 0.972133i \(-0.424677\pi\)
0.234432 + 0.972133i \(0.424677\pi\)
\(282\) −1.73849e8 −0.461636
\(283\) −5.30249e8 −1.39068 −0.695341 0.718680i \(-0.744746\pi\)
−0.695341 + 0.718680i \(0.744746\pi\)
\(284\) 3.63033e8 0.940441
\(285\) 5.63057e8 1.44077
\(286\) −2.72950e8 −0.689925
\(287\) 3.32696e8 0.830731
\(288\) 4.64390e7 0.114554
\(289\) 3.31214e8 0.807172
\(290\) −4.22376e8 −1.01697
\(291\) −1.01889e8 −0.242382
\(292\) −4.37116e7 −0.102744
\(293\) 2.11533e8 0.491294 0.245647 0.969359i \(-0.421000\pi\)
0.245647 + 0.969359i \(0.421000\pi\)
\(294\) −2.87256e8 −0.659254
\(295\) 3.16728e7 0.0718305
\(296\) −8.56736e7 −0.192011
\(297\) 3.04865e8 0.675244
\(298\) −2.85059e8 −0.623990
\(299\) −6.90633e7 −0.149416
\(300\) −1.66820e8 −0.356718
\(301\) −8.75266e7 −0.184994
\(302\) 8.01153e8 1.67375
\(303\) 7.80687e8 1.61223
\(304\) −1.04246e9 −2.12816
\(305\) −3.49328e8 −0.704991
\(306\) −8.06024e7 −0.160814
\(307\) −3.83496e8 −0.756444 −0.378222 0.925715i \(-0.623464\pi\)
−0.378222 + 0.925715i \(0.623464\pi\)
\(308\) −3.50920e8 −0.684353
\(309\) −5.58330e8 −1.07655
\(310\) −8.99937e8 −1.71572
\(311\) 2.71805e8 0.512386 0.256193 0.966626i \(-0.417532\pi\)
0.256193 + 0.966626i \(0.417532\pi\)
\(312\) −1.11960e8 −0.208700
\(313\) 2.82814e8 0.521309 0.260655 0.965432i \(-0.416062\pi\)
0.260655 + 0.965432i \(0.416062\pi\)
\(314\) 8.79065e8 1.60239
\(315\) 4.53589e7 0.0817666
\(316\) −7.56250e8 −1.34822
\(317\) 1.50905e8 0.266070 0.133035 0.991111i \(-0.457528\pi\)
0.133035 + 0.991111i \(0.457528\pi\)
\(318\) −4.07050e7 −0.0709827
\(319\) −4.14363e8 −0.714683
\(320\) −2.45321e8 −0.418514
\(321\) 2.50442e8 0.422609
\(322\) −2.00606e8 −0.334849
\(323\) 1.48917e9 2.45887
\(324\) −5.25704e8 −0.858685
\(325\) 1.93122e8 0.312061
\(326\) −1.49392e9 −2.38818
\(327\) −1.02741e9 −1.62490
\(328\) 1.20701e8 0.188865
\(329\) 2.58749e8 0.400583
\(330\) 4.89324e8 0.749544
\(331\) 4.82795e8 0.731754 0.365877 0.930663i \(-0.380769\pi\)
0.365877 + 0.930663i \(0.380769\pi\)
\(332\) 5.59439e8 0.839015
\(333\) −4.18985e7 −0.0621789
\(334\) −5.00551e8 −0.735082
\(335\) 3.72952e8 0.541996
\(336\) −1.02429e9 −1.47311
\(337\) −5.19140e8 −0.738891 −0.369445 0.929252i \(-0.620452\pi\)
−0.369445 + 0.929252i \(0.620452\pi\)
\(338\) 4.51019e8 0.635311
\(339\) −7.03092e8 −0.980197
\(340\) 5.83895e8 0.805672
\(341\) −8.82863e8 −1.20574
\(342\) −1.61864e8 −0.218805
\(343\) −4.79073e8 −0.641021
\(344\) −3.17543e7 −0.0420580
\(345\) 1.23811e8 0.162328
\(346\) 1.39689e9 1.81300
\(347\) 2.17439e8 0.279373 0.139686 0.990196i \(-0.455391\pi\)
0.139686 + 0.990196i \(0.455391\pi\)
\(348\) 6.55506e8 0.833775
\(349\) −1.23345e8 −0.155322 −0.0776609 0.996980i \(-0.524745\pi\)
−0.0776609 + 0.996980i \(0.524745\pi\)
\(350\) 5.60954e8 0.699341
\(351\) 5.58323e8 0.689145
\(352\) −7.45629e8 −0.911221
\(353\) 3.57356e8 0.432404 0.216202 0.976349i \(-0.430633\pi\)
0.216202 + 0.976349i \(0.430633\pi\)
\(354\) −1.11054e8 −0.133053
\(355\) 7.53426e8 0.893802
\(356\) −3.59517e8 −0.422322
\(357\) 1.46321e9 1.70203
\(358\) −1.24579e9 −1.43501
\(359\) −2.89591e8 −0.330335 −0.165168 0.986266i \(-0.552816\pi\)
−0.165168 + 0.986266i \(0.552816\pi\)
\(360\) 1.64561e7 0.0185895
\(361\) 2.09664e9 2.34557
\(362\) 6.86850e8 0.760995
\(363\) −4.71110e8 −0.516950
\(364\) −6.42666e8 −0.698442
\(365\) −9.07177e7 −0.0976488
\(366\) 1.22485e9 1.30587
\(367\) −8.92000e8 −0.941963 −0.470981 0.882143i \(-0.656100\pi\)
−0.470981 + 0.882143i \(0.656100\pi\)
\(368\) −2.29229e8 −0.239774
\(369\) 5.90284e7 0.0611602
\(370\) 6.85735e8 0.703802
\(371\) 6.05834e7 0.0615950
\(372\) 1.39666e9 1.40666
\(373\) 1.71138e9 1.70752 0.853761 0.520665i \(-0.174316\pi\)
0.853761 + 0.520665i \(0.174316\pi\)
\(374\) 1.29416e9 1.27920
\(375\) −1.15061e9 −1.12673
\(376\) 9.38732e7 0.0910718
\(377\) −7.58854e8 −0.729396
\(378\) 1.62174e9 1.54440
\(379\) 4.74180e7 0.0447410 0.0223705 0.999750i \(-0.492879\pi\)
0.0223705 + 0.999750i \(0.492879\pi\)
\(380\) 1.17256e9 1.09621
\(381\) −1.30026e9 −1.20446
\(382\) 1.84267e9 1.69132
\(383\) −3.02938e8 −0.275523 −0.137761 0.990465i \(-0.543991\pi\)
−0.137761 + 0.990465i \(0.543991\pi\)
\(384\) −6.25234e8 −0.563486
\(385\) −7.28287e8 −0.650414
\(386\) −2.12703e9 −1.88243
\(387\) −1.55294e7 −0.0136196
\(388\) −2.12182e8 −0.184416
\(389\) 2.48110e8 0.213708 0.106854 0.994275i \(-0.465922\pi\)
0.106854 + 0.994275i \(0.465922\pi\)
\(390\) 8.96136e8 0.764976
\(391\) 3.27455e8 0.277034
\(392\) 1.55109e8 0.130058
\(393\) −7.47032e8 −0.620819
\(394\) 8.61601e8 0.709691
\(395\) −1.56950e9 −1.28136
\(396\) −6.22618e7 −0.0503835
\(397\) −1.17389e9 −0.941589 −0.470794 0.882243i \(-0.656033\pi\)
−0.470794 + 0.882243i \(0.656033\pi\)
\(398\) −4.99270e8 −0.396958
\(399\) 2.93838e9 2.31581
\(400\) 6.40993e8 0.500776
\(401\) −4.69179e8 −0.363357 −0.181679 0.983358i \(-0.558153\pi\)
−0.181679 + 0.983358i \(0.558153\pi\)
\(402\) −1.30768e9 −1.00395
\(403\) −1.61686e9 −1.23056
\(404\) 1.62578e9 1.22667
\(405\) −1.09103e9 −0.816101
\(406\) −2.20422e9 −1.63461
\(407\) 6.72726e8 0.494604
\(408\) 5.30846e8 0.386953
\(409\) −2.36052e9 −1.70599 −0.852995 0.521918i \(-0.825217\pi\)
−0.852995 + 0.521918i \(0.825217\pi\)
\(410\) −9.66094e8 −0.692271
\(411\) −1.40579e9 −0.998788
\(412\) −1.16272e9 −0.819096
\(413\) 1.65288e8 0.115456
\(414\) −3.55925e7 −0.0246523
\(415\) 1.16104e9 0.797406
\(416\) −1.36553e9 −0.929981
\(417\) 1.18104e9 0.797609
\(418\) 2.59890e9 1.74049
\(419\) −2.55941e9 −1.69977 −0.849886 0.526967i \(-0.823329\pi\)
−0.849886 + 0.526967i \(0.823329\pi\)
\(420\) 1.15212e9 0.758798
\(421\) −3.64111e7 −0.0237819 −0.0118909 0.999929i \(-0.503785\pi\)
−0.0118909 + 0.999929i \(0.503785\pi\)
\(422\) 3.56252e9 2.30762
\(423\) 4.59084e7 0.0294918
\(424\) 2.19794e7 0.0140035
\(425\) −9.15662e8 −0.578594
\(426\) −2.64174e9 −1.65560
\(427\) −1.82301e9 −1.13316
\(428\) 5.21543e8 0.321542
\(429\) 8.79134e8 0.537594
\(430\) 2.54163e8 0.154160
\(431\) −1.88575e9 −1.13452 −0.567262 0.823538i \(-0.691997\pi\)
−0.567262 + 0.823538i \(0.691997\pi\)
\(432\) 1.85314e9 1.10590
\(433\) −4.74195e8 −0.280705 −0.140352 0.990102i \(-0.544824\pi\)
−0.140352 + 0.990102i \(0.544824\pi\)
\(434\) −4.69643e9 −2.75774
\(435\) 1.36041e9 0.792426
\(436\) −2.13958e9 −1.23631
\(437\) 6.57588e8 0.376937
\(438\) 3.18083e8 0.180876
\(439\) 1.83494e9 1.03513 0.517565 0.855644i \(-0.326838\pi\)
0.517565 + 0.855644i \(0.326838\pi\)
\(440\) −2.64220e8 −0.147870
\(441\) 7.58558e7 0.0421167
\(442\) 2.37009e9 1.30553
\(443\) 2.78161e9 1.52014 0.760070 0.649841i \(-0.225164\pi\)
0.760070 + 0.649841i \(0.225164\pi\)
\(444\) −1.06423e9 −0.577023
\(445\) −7.46129e8 −0.401378
\(446\) −1.04098e9 −0.555609
\(447\) 9.18135e8 0.486217
\(448\) −1.28024e9 −0.672694
\(449\) −2.43689e9 −1.27050 −0.635250 0.772307i \(-0.719103\pi\)
−0.635250 + 0.772307i \(0.719103\pi\)
\(450\) 9.95271e7 0.0514870
\(451\) −9.47766e8 −0.486500
\(452\) −1.46419e9 −0.745782
\(453\) −2.58040e9 −1.30420
\(454\) 5.06056e9 2.53807
\(455\) −1.33377e9 −0.663805
\(456\) 1.06603e9 0.526494
\(457\) 3.65375e9 1.79074 0.895369 0.445324i \(-0.146912\pi\)
0.895369 + 0.445324i \(0.146912\pi\)
\(458\) −4.67990e9 −2.27618
\(459\) −2.64722e9 −1.27775
\(460\) 2.57837e8 0.123507
\(461\) −1.07049e9 −0.508897 −0.254449 0.967086i \(-0.581894\pi\)
−0.254449 + 0.967086i \(0.581894\pi\)
\(462\) 2.55359e9 1.20477
\(463\) 3.98983e9 1.86819 0.934095 0.357025i \(-0.116209\pi\)
0.934095 + 0.357025i \(0.116209\pi\)
\(464\) −2.51872e9 −1.17049
\(465\) 2.89857e9 1.33690
\(466\) −2.39038e9 −1.09425
\(467\) −3.19032e9 −1.44952 −0.724762 0.688999i \(-0.758050\pi\)
−0.724762 + 0.688999i \(0.758050\pi\)
\(468\) −1.14025e8 −0.0514208
\(469\) 1.94629e9 0.871171
\(470\) −7.51365e8 −0.333817
\(471\) −2.83135e9 −1.24859
\(472\) 5.99659e7 0.0262487
\(473\) 2.49341e8 0.108338
\(474\) 5.50313e9 2.37348
\(475\) −1.83881e9 −0.787244
\(476\) 3.04712e9 1.29499
\(477\) 1.07490e7 0.00453475
\(478\) 2.79886e9 1.17215
\(479\) −2.57106e9 −1.06890 −0.534450 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(480\) 2.44801e9 1.01034
\(481\) 1.23201e9 0.504787
\(482\) −7.26734e8 −0.295604
\(483\) 6.46125e8 0.260916
\(484\) −9.81084e8 −0.393321
\(485\) −4.40356e8 −0.175270
\(486\) 6.03693e8 0.238555
\(487\) 2.60295e9 1.02121 0.510604 0.859816i \(-0.329422\pi\)
0.510604 + 0.859816i \(0.329422\pi\)
\(488\) −6.61381e8 −0.257622
\(489\) 4.81172e9 1.86088
\(490\) −1.24150e9 −0.476718
\(491\) 1.12697e9 0.429662 0.214831 0.976651i \(-0.431080\pi\)
0.214831 + 0.976651i \(0.431080\pi\)
\(492\) 1.49933e9 0.567569
\(493\) 3.59801e9 1.35238
\(494\) 4.75956e9 1.77633
\(495\) −1.29216e8 −0.0478849
\(496\) −5.36653e9 −1.97473
\(497\) 3.93184e9 1.43664
\(498\) −4.07096e9 −1.47705
\(499\) −4.16790e9 −1.50164 −0.750820 0.660507i \(-0.770341\pi\)
−0.750820 + 0.660507i \(0.770341\pi\)
\(500\) −2.39614e9 −0.857268
\(501\) 1.61221e9 0.572781
\(502\) 7.04644e9 2.48603
\(503\) 4.24704e9 1.48799 0.743993 0.668187i \(-0.232930\pi\)
0.743993 + 0.668187i \(0.232930\pi\)
\(504\) 8.58779e7 0.0298796
\(505\) 3.37409e9 1.16583
\(506\) 5.71476e8 0.196097
\(507\) −1.45267e9 −0.495038
\(508\) −2.70778e9 −0.916412
\(509\) 4.51429e9 1.51732 0.758661 0.651486i \(-0.225854\pi\)
0.758661 + 0.651486i \(0.225854\pi\)
\(510\) −4.24892e9 −1.41835
\(511\) −4.73421e8 −0.156955
\(512\) −3.55781e9 −1.17149
\(513\) −5.31609e9 −1.73853
\(514\) −3.92771e9 −1.27576
\(515\) −2.41307e9 −0.778475
\(516\) −3.94448e8 −0.126391
\(517\) −7.37110e8 −0.234593
\(518\) 3.57859e9 1.13125
\(519\) −4.49920e9 −1.41270
\(520\) −4.83886e8 −0.150915
\(521\) −1.82834e9 −0.566402 −0.283201 0.959061i \(-0.591396\pi\)
−0.283201 + 0.959061i \(0.591396\pi\)
\(522\) −3.91083e8 −0.120343
\(523\) −8.74437e8 −0.267284 −0.133642 0.991030i \(-0.542667\pi\)
−0.133642 + 0.991030i \(0.542667\pi\)
\(524\) −1.55569e9 −0.472350
\(525\) −1.80676e9 −0.544931
\(526\) 3.33127e9 0.998066
\(527\) 7.66612e9 2.28160
\(528\) 2.91795e9 0.862698
\(529\) −3.26023e9 −0.957531
\(530\) −1.75924e8 −0.0513288
\(531\) 2.93262e7 0.00850012
\(532\) 6.11916e9 1.76198
\(533\) −1.73572e9 −0.496516
\(534\) 2.61615e9 0.743479
\(535\) 1.08239e9 0.305596
\(536\) 7.06108e8 0.198059
\(537\) 4.01252e9 1.11817
\(538\) −7.87934e9 −2.18148
\(539\) −1.21795e9 −0.335018
\(540\) −2.08441e9 −0.569646
\(541\) 4.52769e9 1.22938 0.614690 0.788769i \(-0.289281\pi\)
0.614690 + 0.788769i \(0.289281\pi\)
\(542\) −4.25662e9 −1.14833
\(543\) −2.21225e9 −0.592972
\(544\) 6.47448e9 1.72429
\(545\) −4.44041e9 −1.17499
\(546\) 4.67659e9 1.22958
\(547\) 2.38403e9 0.622811 0.311406 0.950277i \(-0.399200\pi\)
0.311406 + 0.950277i \(0.399200\pi\)
\(548\) −2.92755e9 −0.759927
\(549\) −3.23447e8 −0.0834257
\(550\) −1.59802e9 −0.409555
\(551\) 7.22545e9 1.84007
\(552\) 2.34412e8 0.0593188
\(553\) −8.19061e9 −2.05958
\(554\) −3.74067e9 −0.934685
\(555\) −2.20866e9 −0.548407
\(556\) 2.45952e9 0.606860
\(557\) −2.04626e9 −0.501727 −0.250864 0.968022i \(-0.580715\pi\)
−0.250864 + 0.968022i \(0.580715\pi\)
\(558\) −8.33262e8 −0.203031
\(559\) 4.56638e8 0.110568
\(560\) −4.42693e9 −1.06523
\(561\) −4.16831e9 −0.996758
\(562\) −2.64269e9 −0.628014
\(563\) 3.85093e9 0.909465 0.454733 0.890628i \(-0.349735\pi\)
0.454733 + 0.890628i \(0.349735\pi\)
\(564\) 1.16608e9 0.273685
\(565\) −3.03873e9 −0.708797
\(566\) 8.03541e9 1.86273
\(567\) −5.69367e9 −1.31175
\(568\) 1.42646e9 0.326618
\(569\) −1.30991e9 −0.298091 −0.149045 0.988830i \(-0.547620\pi\)
−0.149045 + 0.988830i \(0.547620\pi\)
\(570\) −8.53258e9 −1.92983
\(571\) 3.44368e9 0.774098 0.387049 0.922059i \(-0.373494\pi\)
0.387049 + 0.922059i \(0.373494\pi\)
\(572\) 1.83079e9 0.409028
\(573\) −5.93497e9 −1.31788
\(574\) −5.04168e9 −1.11271
\(575\) −4.04339e8 −0.0886969
\(576\) −2.27146e8 −0.0495251
\(577\) 2.53520e9 0.549411 0.274706 0.961528i \(-0.411420\pi\)
0.274706 + 0.961528i \(0.411420\pi\)
\(578\) −5.01922e9 −1.08116
\(579\) 6.85086e9 1.46680
\(580\) 2.83306e9 0.602917
\(581\) 6.05904e9 1.28170
\(582\) 1.54402e9 0.324656
\(583\) −1.72587e8 −0.0360718
\(584\) −1.71755e8 −0.0356833
\(585\) −2.36643e8 −0.0488707
\(586\) −3.20558e9 −0.658059
\(587\) −7.98017e9 −1.62847 −0.814233 0.580539i \(-0.802842\pi\)
−0.814233 + 0.580539i \(0.802842\pi\)
\(588\) 1.92675e9 0.390845
\(589\) 1.53949e10 3.10437
\(590\) −4.79970e8 −0.0962127
\(591\) −2.77510e9 −0.552996
\(592\) 4.08920e9 0.810050
\(593\) 4.65675e8 0.0917047 0.0458523 0.998948i \(-0.485400\pi\)
0.0458523 + 0.998948i \(0.485400\pi\)
\(594\) −4.61994e9 −0.904448
\(595\) 6.32390e9 1.23077
\(596\) 1.91201e9 0.369938
\(597\) 1.60808e9 0.309312
\(598\) 1.04659e9 0.200134
\(599\) 6.41918e9 1.22035 0.610177 0.792265i \(-0.291098\pi\)
0.610177 + 0.792265i \(0.291098\pi\)
\(600\) −6.55484e8 −0.123889
\(601\) −5.93318e9 −1.11488 −0.557438 0.830219i \(-0.688216\pi\)
−0.557438 + 0.830219i \(0.688216\pi\)
\(602\) 1.32638e9 0.247788
\(603\) 3.45320e8 0.0641374
\(604\) −5.37368e9 −0.992299
\(605\) −2.03611e9 −0.373816
\(606\) −1.18306e10 −2.15949
\(607\) −8.51996e9 −1.54624 −0.773121 0.634259i \(-0.781305\pi\)
−0.773121 + 0.634259i \(0.781305\pi\)
\(608\) 1.30019e10 2.34609
\(609\) 7.09949e9 1.27370
\(610\) 5.29373e9 0.944293
\(611\) −1.34993e9 −0.239423
\(612\) 5.40635e8 0.0953398
\(613\) 5.16503e9 0.905652 0.452826 0.891599i \(-0.350416\pi\)
0.452826 + 0.891599i \(0.350416\pi\)
\(614\) 5.81152e9 1.01321
\(615\) 3.11166e9 0.539422
\(616\) −1.37886e9 −0.237678
\(617\) −2.47374e9 −0.423991 −0.211995 0.977271i \(-0.567996\pi\)
−0.211995 + 0.977271i \(0.567996\pi\)
\(618\) 8.46094e9 1.44198
\(619\) −5.19636e9 −0.880607 −0.440304 0.897849i \(-0.645129\pi\)
−0.440304 + 0.897849i \(0.645129\pi\)
\(620\) 6.03627e9 1.01718
\(621\) −1.16896e9 −0.195876
\(622\) −4.11895e9 −0.686309
\(623\) −3.89376e9 −0.645151
\(624\) 5.34386e9 0.880459
\(625\) −2.34590e9 −0.384352
\(626\) −4.28577e9 −0.698263
\(627\) −8.37070e9 −1.35620
\(628\) −5.89627e9 −0.949989
\(629\) −5.84144e9 −0.935929
\(630\) −6.87371e8 −0.109521
\(631\) 4.50866e9 0.714405 0.357203 0.934027i \(-0.383731\pi\)
0.357203 + 0.934027i \(0.383731\pi\)
\(632\) −2.97152e9 −0.468241
\(633\) −1.14744e10 −1.79811
\(634\) −2.28681e9 −0.356384
\(635\) −5.61964e9 −0.870965
\(636\) 2.73026e8 0.0420827
\(637\) −2.23052e9 −0.341915
\(638\) 6.27926e9 0.957274
\(639\) 6.97606e8 0.105769
\(640\) −2.70223e9 −0.407466
\(641\) −7.93187e8 −0.118952 −0.0594761 0.998230i \(-0.518943\pi\)
−0.0594761 + 0.998230i \(0.518943\pi\)
\(642\) −3.79520e9 −0.566059
\(643\) −1.11969e10 −1.66095 −0.830477 0.557053i \(-0.811932\pi\)
−0.830477 + 0.557053i \(0.811932\pi\)
\(644\) 1.34555e9 0.198518
\(645\) −8.18624e8 −0.120123
\(646\) −2.25669e10 −3.29350
\(647\) 1.19259e10 1.73111 0.865554 0.500815i \(-0.166966\pi\)
0.865554 + 0.500815i \(0.166966\pi\)
\(648\) −2.06564e9 −0.298224
\(649\) −4.70864e8 −0.0676144
\(650\) −2.92657e9 −0.417986
\(651\) 1.51265e10 2.14885
\(652\) 1.00204e10 1.41585
\(653\) −2.22608e9 −0.312857 −0.156428 0.987689i \(-0.549998\pi\)
−0.156428 + 0.987689i \(0.549998\pi\)
\(654\) 1.55694e10 2.17646
\(655\) −3.22863e9 −0.448925
\(656\) −5.76104e9 −0.796778
\(657\) −8.39965e7 −0.0115553
\(658\) −3.92109e9 −0.536557
\(659\) −8.32960e7 −0.0113377 −0.00566885 0.999984i \(-0.501804\pi\)
−0.00566885 + 0.999984i \(0.501804\pi\)
\(660\) −3.28210e9 −0.444374
\(661\) 6.64097e9 0.894389 0.447194 0.894437i \(-0.352423\pi\)
0.447194 + 0.894437i \(0.352423\pi\)
\(662\) −7.31629e9 −0.980140
\(663\) −7.63374e9 −1.01728
\(664\) 2.19820e9 0.291392
\(665\) 1.26995e10 1.67460
\(666\) 6.34930e8 0.0832849
\(667\) 1.58881e9 0.207316
\(668\) 3.35741e9 0.435800
\(669\) 3.35284e9 0.432934
\(670\) −5.65172e9 −0.725970
\(671\) 5.19329e9 0.663612
\(672\) 1.27752e10 1.62397
\(673\) 3.72281e9 0.470780 0.235390 0.971901i \(-0.424363\pi\)
0.235390 + 0.971901i \(0.424363\pi\)
\(674\) 7.86707e9 0.989699
\(675\) 3.26877e9 0.409092
\(676\) −3.02518e9 −0.376649
\(677\) 1.49598e10 1.85296 0.926480 0.376343i \(-0.122819\pi\)
0.926480 + 0.376343i \(0.122819\pi\)
\(678\) 1.06547e10 1.31291
\(679\) −2.29805e9 −0.281719
\(680\) 2.29429e9 0.279812
\(681\) −1.62994e10 −1.97768
\(682\) 1.33789e10 1.61501
\(683\) 6.36282e9 0.764148 0.382074 0.924132i \(-0.375210\pi\)
0.382074 + 0.924132i \(0.375210\pi\)
\(684\) 1.08569e9 0.129721
\(685\) −6.07573e9 −0.722240
\(686\) 7.25989e9 0.858609
\(687\) 1.50733e10 1.77362
\(688\) 1.51563e9 0.177433
\(689\) −3.16071e8 −0.0368144
\(690\) −1.87624e9 −0.217429
\(691\) 1.18625e10 1.36774 0.683870 0.729604i \(-0.260296\pi\)
0.683870 + 0.729604i \(0.260296\pi\)
\(692\) −9.36957e9 −1.07485
\(693\) −6.74330e8 −0.0769672
\(694\) −3.29507e9 −0.374203
\(695\) 5.10441e9 0.576764
\(696\) 2.57567e9 0.289573
\(697\) 8.22969e9 0.920595
\(698\) 1.86917e9 0.208044
\(699\) 7.69909e9 0.852647
\(700\) −3.76256e9 −0.414611
\(701\) 8.69584e9 0.953451 0.476725 0.879052i \(-0.341824\pi\)
0.476725 + 0.879052i \(0.341824\pi\)
\(702\) −8.46085e9 −0.923069
\(703\) −1.17306e10 −1.27344
\(704\) 3.64707e9 0.393949
\(705\) 2.42004e9 0.260112
\(706\) −5.41539e9 −0.579179
\(707\) 1.76081e10 1.87389
\(708\) 7.44889e8 0.0788815
\(709\) −6.50844e9 −0.685828 −0.342914 0.939367i \(-0.611414\pi\)
−0.342914 + 0.939367i \(0.611414\pi\)
\(710\) −1.14174e10 −1.19719
\(711\) −1.45322e9 −0.151630
\(712\) −1.41264e9 −0.146674
\(713\) 3.38521e9 0.349762
\(714\) −2.21735e10 −2.27977
\(715\) 3.79957e9 0.388743
\(716\) 8.35606e9 0.850758
\(717\) −9.01474e9 −0.913348
\(718\) 4.38847e9 0.442464
\(719\) −3.90453e9 −0.391757 −0.195879 0.980628i \(-0.562756\pi\)
−0.195879 + 0.980628i \(0.562756\pi\)
\(720\) −7.85446e8 −0.0784247
\(721\) −1.25929e10 −1.25127
\(722\) −3.17725e10 −3.14175
\(723\) 2.34071e9 0.230337
\(724\) −4.60700e9 −0.451163
\(725\) −4.44280e9 −0.432986
\(726\) 7.13921e9 0.692424
\(727\) −1.72837e10 −1.66827 −0.834133 0.551563i \(-0.814032\pi\)
−0.834133 + 0.551563i \(0.814032\pi\)
\(728\) −2.52522e9 −0.242571
\(729\) 9.36672e9 0.895450
\(730\) 1.37474e9 0.130795
\(731\) −2.16509e9 −0.205005
\(732\) −8.21559e9 −0.774194
\(733\) 8.26113e9 0.774775 0.387387 0.921917i \(-0.373378\pi\)
0.387387 + 0.921917i \(0.373378\pi\)
\(734\) 1.35174e10 1.26170
\(735\) 3.99871e9 0.371462
\(736\) 2.85901e9 0.264328
\(737\) −5.54450e9 −0.510183
\(738\) −8.94518e8 −0.0819204
\(739\) 6.43474e9 0.586510 0.293255 0.956034i \(-0.405261\pi\)
0.293255 + 0.956034i \(0.405261\pi\)
\(740\) −4.59952e9 −0.417255
\(741\) −1.53299e10 −1.38413
\(742\) −9.18083e8 −0.0825027
\(743\) 9.72354e9 0.869689 0.434844 0.900506i \(-0.356803\pi\)
0.434844 + 0.900506i \(0.356803\pi\)
\(744\) 5.48786e9 0.488537
\(745\) 3.96813e9 0.351592
\(746\) −2.59344e10 −2.28712
\(747\) 1.07502e9 0.0943616
\(748\) −8.68048e9 −0.758383
\(749\) 5.64860e9 0.491196
\(750\) 1.74364e10 1.50918
\(751\) −1.14363e10 −0.985252 −0.492626 0.870241i \(-0.663963\pi\)
−0.492626 + 0.870241i \(0.663963\pi\)
\(752\) −4.48056e9 −0.384211
\(753\) −2.26956e10 −1.93713
\(754\) 1.14997e10 0.976982
\(755\) −1.11524e10 −0.943088
\(756\) −1.08777e10 −0.915614
\(757\) −7.15016e9 −0.599073 −0.299537 0.954085i \(-0.596832\pi\)
−0.299537 + 0.954085i \(0.596832\pi\)
\(758\) −7.18573e8 −0.0599278
\(759\) −1.84065e9 −0.152800
\(760\) 4.60733e9 0.380717
\(761\) 3.88824e9 0.319821 0.159910 0.987132i \(-0.448879\pi\)
0.159910 + 0.987132i \(0.448879\pi\)
\(762\) 1.97042e10 1.61330
\(763\) −2.31728e10 −1.88861
\(764\) −1.23596e10 −1.00271
\(765\) 1.12202e9 0.0906116
\(766\) 4.59073e9 0.369046
\(767\) −8.62330e8 −0.0690064
\(768\) 1.67403e10 1.33352
\(769\) −1.58580e10 −1.25750 −0.628748 0.777609i \(-0.716432\pi\)
−0.628748 + 0.777609i \(0.716432\pi\)
\(770\) 1.10365e10 0.871191
\(771\) 1.26506e10 0.994079
\(772\) 1.42669e10 1.11601
\(773\) −6.91144e9 −0.538196 −0.269098 0.963113i \(-0.586726\pi\)
−0.269098 + 0.963113i \(0.586726\pi\)
\(774\) 2.35333e8 0.0182427
\(775\) −9.46606e9 −0.730488
\(776\) −8.33725e8 −0.0640482
\(777\) −1.15261e10 −0.881476
\(778\) −3.75987e9 −0.286249
\(779\) 1.65267e10 1.25258
\(780\) −6.01077e9 −0.453522
\(781\) −1.12008e10 −0.841340
\(782\) −4.96227e9 −0.371071
\(783\) −1.28443e10 −0.956192
\(784\) −7.40336e9 −0.548684
\(785\) −1.22369e10 −0.902877
\(786\) 1.13205e10 0.831550
\(787\) −8.31218e9 −0.607860 −0.303930 0.952694i \(-0.598299\pi\)
−0.303930 + 0.952694i \(0.598299\pi\)
\(788\) −5.77913e9 −0.420747
\(789\) −1.07296e10 −0.777700
\(790\) 2.37842e10 1.71630
\(791\) −1.58580e10 −1.13928
\(792\) −2.44644e8 −0.0174983
\(793\) 9.51087e9 0.677274
\(794\) 1.77892e10 1.26120
\(795\) 5.66628e8 0.0399957
\(796\) 3.34882e9 0.235340
\(797\) 2.22972e10 1.56008 0.780038 0.625733i \(-0.215200\pi\)
0.780038 + 0.625733i \(0.215200\pi\)
\(798\) −4.45283e10 −3.10189
\(799\) 6.40051e9 0.443916
\(800\) −7.99464e9 −0.552057
\(801\) −6.90850e8 −0.0474974
\(802\) 7.10996e9 0.486695
\(803\) 1.34866e9 0.0919172
\(804\) 8.77118e9 0.595198
\(805\) 2.79251e9 0.188673
\(806\) 2.45019e10 1.64826
\(807\) 2.53783e10 1.69983
\(808\) 6.38814e9 0.426025
\(809\) 1.32718e10 0.881275 0.440638 0.897685i \(-0.354752\pi\)
0.440638 + 0.897685i \(0.354752\pi\)
\(810\) 1.65335e10 1.09312
\(811\) 4.23818e9 0.279002 0.139501 0.990222i \(-0.455450\pi\)
0.139501 + 0.990222i \(0.455450\pi\)
\(812\) 1.47847e10 0.969092
\(813\) 1.37100e10 0.894788
\(814\) −1.01945e10 −0.662492
\(815\) 2.07960e10 1.34564
\(816\) −2.53373e10 −1.63247
\(817\) −4.34789e9 −0.278934
\(818\) 3.57714e10 2.28507
\(819\) −1.23495e9 −0.0785518
\(820\) 6.48001e9 0.410419
\(821\) −1.61244e10 −1.01691 −0.508455 0.861089i \(-0.669783\pi\)
−0.508455 + 0.861089i \(0.669783\pi\)
\(822\) 2.13033e10 1.33782
\(823\) −1.67540e9 −0.104766 −0.0523829 0.998627i \(-0.516682\pi\)
−0.0523829 + 0.998627i \(0.516682\pi\)
\(824\) −4.56865e9 −0.284474
\(825\) 5.14699e9 0.319128
\(826\) −2.50478e9 −0.154646
\(827\) −2.26903e10 −1.39499 −0.697496 0.716588i \(-0.745703\pi\)
−0.697496 + 0.716588i \(0.745703\pi\)
\(828\) 2.38734e8 0.0146153
\(829\) −2.14017e10 −1.30469 −0.652345 0.757922i \(-0.726215\pi\)
−0.652345 + 0.757922i \(0.726215\pi\)
\(830\) −1.75945e10 −1.06808
\(831\) 1.20482e10 0.728313
\(832\) 6.67916e9 0.402059
\(833\) 1.05758e10 0.633948
\(834\) −1.78976e10 −1.06835
\(835\) 6.96786e9 0.414187
\(836\) −1.74319e10 −1.03187
\(837\) −2.73668e10 −1.61319
\(838\) 3.87854e10 2.27674
\(839\) 3.21757e10 1.88088 0.940439 0.339962i \(-0.110414\pi\)
0.940439 + 0.339962i \(0.110414\pi\)
\(840\) 4.52701e9 0.263533
\(841\) 2.07698e8 0.0120406
\(842\) 5.51774e8 0.0318544
\(843\) 8.51174e9 0.489353
\(844\) −2.38953e10 −1.36809
\(845\) −6.27835e9 −0.357970
\(846\) −6.95698e8 −0.0395025
\(847\) −1.06257e10 −0.600848
\(848\) −1.04908e9 −0.0590775
\(849\) −2.58810e10 −1.45145
\(850\) 1.38760e10 0.774992
\(851\) −2.57947e9 −0.143475
\(852\) 1.77193e10 0.981539
\(853\) −8.66660e9 −0.478109 −0.239055 0.971006i \(-0.576838\pi\)
−0.239055 + 0.971006i \(0.576838\pi\)
\(854\) 2.76259e10 1.51780
\(855\) 2.25320e9 0.123288
\(856\) 2.04929e9 0.111672
\(857\) 4.64158e9 0.251903 0.125951 0.992036i \(-0.459802\pi\)
0.125951 + 0.992036i \(0.459802\pi\)
\(858\) −1.33224e10 −0.720075
\(859\) 1.96063e10 1.05541 0.527703 0.849429i \(-0.323053\pi\)
0.527703 + 0.849429i \(0.323053\pi\)
\(860\) −1.70478e9 −0.0913954
\(861\) 1.62386e10 0.867034
\(862\) 2.85767e10 1.51963
\(863\) 2.94733e10 1.56096 0.780478 0.625183i \(-0.214976\pi\)
0.780478 + 0.625183i \(0.214976\pi\)
\(864\) −2.31129e10 −1.21915
\(865\) −1.94453e10 −1.02155
\(866\) 7.18597e9 0.375987
\(867\) 1.61662e10 0.842445
\(868\) 3.15010e10 1.63495
\(869\) 2.33330e10 1.20615
\(870\) −2.06158e10 −1.06141
\(871\) −1.01541e10 −0.520686
\(872\) −8.40702e9 −0.429372
\(873\) −4.07731e8 −0.0207407
\(874\) −9.96511e9 −0.504885
\(875\) −2.59515e10 −1.30959
\(876\) −2.13352e9 −0.107234
\(877\) −2.55789e9 −0.128051 −0.0640256 0.997948i \(-0.520394\pi\)
−0.0640256 + 0.997948i \(0.520394\pi\)
\(878\) −2.78067e10 −1.38650
\(879\) 1.03247e10 0.512764
\(880\) 1.26112e10 0.623831
\(881\) −2.66718e10 −1.31412 −0.657062 0.753837i \(-0.728201\pi\)
−0.657062 + 0.753837i \(0.728201\pi\)
\(882\) −1.14952e9 −0.0564127
\(883\) −4.99020e9 −0.243924 −0.121962 0.992535i \(-0.538919\pi\)
−0.121962 + 0.992535i \(0.538919\pi\)
\(884\) −1.58972e10 −0.773996
\(885\) 1.54592e9 0.0749695
\(886\) −4.21527e10 −2.03614
\(887\) −2.25283e10 −1.08392 −0.541959 0.840405i \(-0.682317\pi\)
−0.541959 + 0.840405i \(0.682317\pi\)
\(888\) −4.18164e9 −0.200402
\(889\) −2.93268e10 −1.39994
\(890\) 1.13069e10 0.537622
\(891\) 1.62198e10 0.768199
\(892\) 6.98228e9 0.329398
\(893\) 1.28534e10 0.603999
\(894\) −1.39134e10 −0.651258
\(895\) 1.73419e10 0.808567
\(896\) −1.41019e10 −0.654937
\(897\) −3.37091e9 −0.155946
\(898\) 3.69288e10 1.70176
\(899\) 3.71960e10 1.70741
\(900\) −6.67571e8 −0.0305245
\(901\) 1.49861e9 0.0682579
\(902\) 1.43625e10 0.651638
\(903\) −4.27209e9 −0.193078
\(904\) −5.75321e9 −0.259012
\(905\) −9.56121e9 −0.428788
\(906\) 3.91035e10 1.74690
\(907\) 2.20475e10 0.981145 0.490573 0.871400i \(-0.336788\pi\)
0.490573 + 0.871400i \(0.336788\pi\)
\(908\) −3.39434e10 −1.50472
\(909\) 3.12411e9 0.137960
\(910\) 2.02120e10 0.889127
\(911\) 4.20990e9 0.184484 0.0922418 0.995737i \(-0.470597\pi\)
0.0922418 + 0.995737i \(0.470597\pi\)
\(912\) −5.08817e10 −2.22116
\(913\) −1.72607e10 −0.750602
\(914\) −5.53690e10 −2.39859
\(915\) −1.70504e10 −0.735800
\(916\) 3.13901e10 1.34945
\(917\) −1.68490e10 −0.721574
\(918\) 4.01161e10 1.71147
\(919\) −1.55285e10 −0.659972 −0.329986 0.943986i \(-0.607044\pi\)
−0.329986 + 0.943986i \(0.607044\pi\)
\(920\) 1.01311e9 0.0428944
\(921\) −1.87181e10 −0.789501
\(922\) 1.62223e10 0.681637
\(923\) −2.05129e10 −0.858661
\(924\) −1.71281e10 −0.714260
\(925\) 7.21296e9 0.299652
\(926\) −6.04620e10 −2.50233
\(927\) −2.23429e9 −0.0921213
\(928\) 3.14142e10 1.29035
\(929\) −2.69496e10 −1.10280 −0.551401 0.834240i \(-0.685907\pi\)
−0.551401 + 0.834240i \(0.685907\pi\)
\(930\) −4.39251e10 −1.79070
\(931\) 2.12380e10 0.862560
\(932\) 1.60333e10 0.648736
\(933\) 1.32666e10 0.534777
\(934\) 4.83462e10 1.94155
\(935\) −1.80152e10 −0.720772
\(936\) −4.48036e8 −0.0178586
\(937\) −1.04584e10 −0.415315 −0.207657 0.978202i \(-0.566584\pi\)
−0.207657 + 0.978202i \(0.566584\pi\)
\(938\) −2.94942e10 −1.16688
\(939\) 1.38039e10 0.544091
\(940\) 5.03973e9 0.197906
\(941\) 2.29124e10 0.896412 0.448206 0.893930i \(-0.352063\pi\)
0.448206 + 0.893930i \(0.352063\pi\)
\(942\) 4.29063e10 1.67241
\(943\) 3.63407e9 0.141125
\(944\) −2.86217e9 −0.110737
\(945\) −2.25753e10 −0.870206
\(946\) −3.77852e9 −0.145112
\(947\) −2.22008e10 −0.849460 −0.424730 0.905320i \(-0.639631\pi\)
−0.424730 + 0.905320i \(0.639631\pi\)
\(948\) −3.69118e10 −1.40714
\(949\) 2.46990e9 0.0938096
\(950\) 2.78654e10 1.05447
\(951\) 7.36551e9 0.277697
\(952\) 1.19730e10 0.449753
\(953\) 1.45448e10 0.544357 0.272178 0.962247i \(-0.412256\pi\)
0.272178 + 0.962247i \(0.412256\pi\)
\(954\) −1.62891e8 −0.00607402
\(955\) −2.56506e10 −0.952985
\(956\) −1.87732e10 −0.694920
\(957\) −2.02246e10 −0.745914
\(958\) 3.89619e10 1.43173
\(959\) −3.17069e10 −1.16088
\(960\) −1.19739e10 −0.436803
\(961\) 5.17393e10 1.88057
\(962\) −1.86700e10 −0.676131
\(963\) 1.00220e9 0.0361629
\(964\) 4.87452e9 0.175252
\(965\) 2.96090e10 1.06067
\(966\) −9.79139e9 −0.349482
\(967\) −4.60488e10 −1.63767 −0.818833 0.574032i \(-0.805379\pi\)
−0.818833 + 0.574032i \(0.805379\pi\)
\(968\) −3.85496e9 −0.136602
\(969\) 7.26848e10 2.56632
\(970\) 6.67317e9 0.234764
\(971\) 2.30696e10 0.808671 0.404335 0.914611i \(-0.367503\pi\)
0.404335 + 0.914611i \(0.367503\pi\)
\(972\) −4.04923e9 −0.141430
\(973\) 2.66380e10 0.927056
\(974\) −3.94451e10 −1.36785
\(975\) 9.42607e9 0.325698
\(976\) 3.15677e10 1.08685
\(977\) 5.54001e10 1.90055 0.950276 0.311408i \(-0.100801\pi\)
0.950276 + 0.311408i \(0.100801\pi\)
\(978\) −7.29170e10 −2.49254
\(979\) 1.10923e10 0.377819
\(980\) 8.32729e9 0.282626
\(981\) −4.11143e9 −0.139044
\(982\) −1.70781e10 −0.575506
\(983\) −3.63783e10 −1.22153 −0.610766 0.791811i \(-0.709139\pi\)
−0.610766 + 0.791811i \(0.709139\pi\)
\(984\) 5.89129e9 0.197118
\(985\) −1.19938e10 −0.399881
\(986\) −5.45244e10 −1.81143
\(987\) 1.26293e10 0.418089
\(988\) −3.19244e10 −1.05311
\(989\) −9.56063e8 −0.0314268
\(990\) 1.95814e9 0.0641389
\(991\) 1.33718e9 0.0436446 0.0218223 0.999762i \(-0.493053\pi\)
0.0218223 + 0.999762i \(0.493053\pi\)
\(992\) 6.69328e10 2.17695
\(993\) 2.35648e10 0.763731
\(994\) −5.95833e10 −1.92430
\(995\) 6.95003e9 0.223669
\(996\) 2.73057e10 0.875680
\(997\) 2.40234e10 0.767718 0.383859 0.923392i \(-0.374595\pi\)
0.383859 + 0.923392i \(0.374595\pi\)
\(998\) 6.31605e10 2.01135
\(999\) 2.08530e10 0.661743
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.1.3 13
3.2 odd 2 387.8.a.d.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.3 13 1.1 even 1 trivial
387.8.a.d.1.11 13 3.2 odd 2