Properties

Label 43.8.a.b.1.10
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

Related objects

Downloads

Learn more about

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 1.10
Root \(15.0703\) 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

\(f(q)\) \(=\) \(q+16.0703 q^{2} +41.5098 q^{3} +130.256 q^{4} +431.884 q^{5} +667.077 q^{6} +218.970 q^{7} +36.2566 q^{8} -463.938 q^{9} +O(q^{10})\) \(q+16.0703 q^{2} +41.5098 q^{3} +130.256 q^{4} +431.884 q^{5} +667.077 q^{6} +218.970 q^{7} +36.2566 q^{8} -463.938 q^{9} +6940.53 q^{10} -7017.15 q^{11} +5406.90 q^{12} +7017.50 q^{13} +3518.93 q^{14} +17927.4 q^{15} -16090.1 q^{16} +37064.5 q^{17} -7455.64 q^{18} -39841.8 q^{19} +56255.5 q^{20} +9089.40 q^{21} -112768. q^{22} +29837.9 q^{23} +1505.01 q^{24} +108399. q^{25} +112774. q^{26} -110040. q^{27} +28522.2 q^{28} -72817.2 q^{29} +288100. q^{30} -103069. q^{31} -263215. q^{32} -291280. q^{33} +595639. q^{34} +94569.7 q^{35} -60430.7 q^{36} +330208. q^{37} -640272. q^{38} +291295. q^{39} +15658.7 q^{40} +387979. q^{41} +146070. q^{42} -79507.0 q^{43} -914026. q^{44} -200367. q^{45} +479505. q^{46} -1.39939e6 q^{47} -667898. q^{48} -775595. q^{49} +1.74201e6 q^{50} +1.53854e6 q^{51} +914072. q^{52} -300329. q^{53} -1.76838e6 q^{54} -3.03059e6 q^{55} +7939.12 q^{56} -1.65382e6 q^{57} -1.17020e6 q^{58} -440915. q^{59} +2.33515e6 q^{60} +1.72200e6 q^{61} -1.65635e6 q^{62} -101588. q^{63} -2.17042e6 q^{64} +3.03075e6 q^{65} -4.68098e6 q^{66} +3.45225e6 q^{67} +4.82787e6 q^{68} +1.23856e6 q^{69} +1.51977e6 q^{70} -1.85700e6 q^{71} -16820.8 q^{72} -673491. q^{73} +5.30655e6 q^{74} +4.49961e6 q^{75} -5.18964e6 q^{76} -1.53655e6 q^{77} +4.68121e6 q^{78} +5.44201e6 q^{79} -6.94907e6 q^{80} -3.55310e6 q^{81} +6.23496e6 q^{82} +4.84142e6 q^{83} +1.18395e6 q^{84} +1.60075e7 q^{85} -1.27771e6 q^{86} -3.02263e6 q^{87} -254418. q^{88} +8.98649e6 q^{89} -3.21997e6 q^{90} +1.53662e6 q^{91} +3.88656e6 q^{92} -4.27836e6 q^{93} -2.24887e7 q^{94} -1.72070e7 q^{95} -1.09260e7 q^{96} -7.64277e6 q^{97} -1.24641e7 q^{98} +3.25552e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0703 1.42043 0.710216 0.703984i \(-0.248597\pi\)
0.710216 + 0.703984i \(0.248597\pi\)
\(3\) 41.5098 0.887618 0.443809 0.896121i \(-0.353627\pi\)
0.443809 + 0.896121i \(0.353627\pi\)
\(4\) 130.256 1.01763
\(5\) 431.884 1.54515 0.772577 0.634921i \(-0.218967\pi\)
0.772577 + 0.634921i \(0.218967\pi\)
\(6\) 667.077 1.26080
\(7\) 218.970 0.241291 0.120646 0.992696i \(-0.461504\pi\)
0.120646 + 0.992696i \(0.461504\pi\)
\(8\) 36.2566 0.0250364
\(9\) −463.938 −0.212134
\(10\) 6940.53 2.19479
\(11\) −7017.15 −1.58959 −0.794797 0.606875i \(-0.792423\pi\)
−0.794797 + 0.606875i \(0.792423\pi\)
\(12\) 5406.90 0.903263
\(13\) 7017.50 0.885892 0.442946 0.896548i \(-0.353933\pi\)
0.442946 + 0.896548i \(0.353933\pi\)
\(14\) 3518.93 0.342738
\(15\) 17927.4 1.37151
\(16\) −16090.1 −0.982063
\(17\) 37064.5 1.82973 0.914864 0.403763i \(-0.132298\pi\)
0.914864 + 0.403763i \(0.132298\pi\)
\(18\) −7455.64 −0.301322
\(19\) −39841.8 −1.33260 −0.666302 0.745682i \(-0.732124\pi\)
−0.666302 + 0.745682i \(0.732124\pi\)
\(20\) 56255.5 1.57239
\(21\) 9089.40 0.214175
\(22\) −112768. −2.25791
\(23\) 29837.9 0.511352 0.255676 0.966762i \(-0.417702\pi\)
0.255676 + 0.966762i \(0.417702\pi\)
\(24\) 1505.01 0.0222228
\(25\) 108399. 1.38750
\(26\) 112774. 1.25835
\(27\) −110040. −1.07591
\(28\) 28522.2 0.245544
\(29\) −72817.2 −0.554423 −0.277211 0.960809i \(-0.589410\pi\)
−0.277211 + 0.960809i \(0.589410\pi\)
\(30\) 288100. 1.94813
\(31\) −103069. −0.621386 −0.310693 0.950510i \(-0.600561\pi\)
−0.310693 + 0.950510i \(0.600561\pi\)
\(32\) −263215. −1.41999
\(33\) −291280. −1.41095
\(34\) 595639. 2.59900
\(35\) 94569.7 0.372832
\(36\) −60430.7 −0.215873
\(37\) 330208. 1.07172 0.535860 0.844307i \(-0.319988\pi\)
0.535860 + 0.844307i \(0.319988\pi\)
\(38\) −640272. −1.89287
\(39\) 291295. 0.786334
\(40\) 15658.7 0.0386852
\(41\) 387979. 0.879154 0.439577 0.898205i \(-0.355128\pi\)
0.439577 + 0.898205i \(0.355128\pi\)
\(42\) 146070. 0.304220
\(43\) −79507.0 −0.152499
\(44\) −914026. −1.61761
\(45\) −200367. −0.327780
\(46\) 479505. 0.726341
\(47\) −1.39939e6 −1.96606 −0.983031 0.183440i \(-0.941277\pi\)
−0.983031 + 0.183440i \(0.941277\pi\)
\(48\) −667898. −0.871697
\(49\) −775595. −0.941778
\(50\) 1.74201e6 1.97085
\(51\) 1.53854e6 1.62410
\(52\) 914072. 0.901507
\(53\) −300329. −0.277097 −0.138548 0.990356i \(-0.544244\pi\)
−0.138548 + 0.990356i \(0.544244\pi\)
\(54\) −1.76838e6 −1.52826
\(55\) −3.03059e6 −2.45617
\(56\) 7939.12 0.00604108
\(57\) −1.65382e6 −1.18284
\(58\) −1.17020e6 −0.787520
\(59\) −440915. −0.279494 −0.139747 0.990187i \(-0.544629\pi\)
−0.139747 + 0.990187i \(0.544629\pi\)
\(60\) 2.33515e6 1.39568
\(61\) 1.72200e6 0.971356 0.485678 0.874138i \(-0.338573\pi\)
0.485678 + 0.874138i \(0.338573\pi\)
\(62\) −1.65635e6 −0.882636
\(63\) −101588. −0.0511862
\(64\) −2.17042e6 −1.03494
\(65\) 3.03075e6 1.36884
\(66\) −4.68098e6 −2.00416
\(67\) 3.45225e6 1.40230 0.701149 0.713015i \(-0.252671\pi\)
0.701149 + 0.713015i \(0.252671\pi\)
\(68\) 4.82787e6 1.86198
\(69\) 1.23856e6 0.453885
\(70\) 1.51977e6 0.529583
\(71\) −1.85700e6 −0.615756 −0.307878 0.951426i \(-0.599619\pi\)
−0.307878 + 0.951426i \(0.599619\pi\)
\(72\) −16820.8 −0.00531109
\(73\) −673491. −0.202629 −0.101315 0.994854i \(-0.532305\pi\)
−0.101315 + 0.994854i \(0.532305\pi\)
\(74\) 5.30655e6 1.52230
\(75\) 4.49961e6 1.23157
\(76\) −5.18964e6 −1.35609
\(77\) −1.53655e6 −0.383555
\(78\) 4.68121e6 1.11693
\(79\) 5.44201e6 1.24184 0.620918 0.783875i \(-0.286760\pi\)
0.620918 + 0.783875i \(0.286760\pi\)
\(80\) −6.94907e6 −1.51744
\(81\) −3.55310e6 −0.742865
\(82\) 6.23496e6 1.24878
\(83\) 4.84142e6 0.929394 0.464697 0.885470i \(-0.346163\pi\)
0.464697 + 0.885470i \(0.346163\pi\)
\(84\) 1.18395e6 0.217950
\(85\) 1.60075e7 2.82721
\(86\) −1.27771e6 −0.216614
\(87\) −3.02263e6 −0.492116
\(88\) −254418. −0.0397978
\(89\) 8.98649e6 1.35122 0.675609 0.737260i \(-0.263881\pi\)
0.675609 + 0.737260i \(0.263881\pi\)
\(90\) −3.21997e6 −0.465589
\(91\) 1.53662e6 0.213758
\(92\) 3.88656e6 0.520365
\(93\) −4.27836e6 −0.551553
\(94\) −2.24887e7 −2.79266
\(95\) −1.72070e7 −2.05908
\(96\) −1.09260e7 −1.26041
\(97\) −7.64277e6 −0.850256 −0.425128 0.905133i \(-0.639771\pi\)
−0.425128 + 0.905133i \(0.639771\pi\)
\(98\) −1.24641e7 −1.33773
\(99\) 3.25552e6 0.337207
\(100\) 1.41196e7 1.41196
\(101\) −4.72174e6 −0.456013 −0.228006 0.973660i \(-0.573221\pi\)
−0.228006 + 0.973660i \(0.573221\pi\)
\(102\) 2.47248e7 2.30692
\(103\) 5.07090e6 0.457251 0.228625 0.973514i \(-0.426577\pi\)
0.228625 + 0.973514i \(0.426577\pi\)
\(104\) 254431. 0.0221796
\(105\) 3.92557e6 0.330933
\(106\) −4.82639e6 −0.393597
\(107\) −3.42527e6 −0.270304 −0.135152 0.990825i \(-0.543152\pi\)
−0.135152 + 0.990825i \(0.543152\pi\)
\(108\) −1.43334e7 −1.09488
\(109\) −4.64829e6 −0.343795 −0.171898 0.985115i \(-0.554990\pi\)
−0.171898 + 0.985115i \(0.554990\pi\)
\(110\) −4.87027e7 −3.48882
\(111\) 1.37069e7 0.951278
\(112\) −3.52326e6 −0.236963
\(113\) 1.20722e7 0.787065 0.393533 0.919311i \(-0.371253\pi\)
0.393533 + 0.919311i \(0.371253\pi\)
\(114\) −2.65775e7 −1.68015
\(115\) 1.28865e7 0.790118
\(116\) −9.48489e6 −0.564195
\(117\) −3.25568e6 −0.187928
\(118\) −7.08566e6 −0.397003
\(119\) 8.11601e6 0.441497
\(120\) 649988. 0.0343377
\(121\) 2.97532e7 1.52681
\(122\) 2.76731e7 1.37974
\(123\) 1.61049e7 0.780353
\(124\) −1.34253e7 −0.632338
\(125\) 1.30747e7 0.598753
\(126\) −1.63256e6 −0.0727064
\(127\) 3.31970e6 0.143809 0.0719045 0.997412i \(-0.477092\pi\)
0.0719045 + 0.997412i \(0.477092\pi\)
\(128\) −1.18787e6 −0.0500651
\(129\) −3.30032e6 −0.135360
\(130\) 4.87051e7 1.94434
\(131\) 2.70388e7 1.05084 0.525421 0.850842i \(-0.323908\pi\)
0.525421 + 0.850842i \(0.323908\pi\)
\(132\) −3.79410e7 −1.43582
\(133\) −8.72417e6 −0.321546
\(134\) 5.54789e7 1.99187
\(135\) −4.75244e7 −1.66245
\(136\) 1.34383e6 0.0458099
\(137\) 7.47994e6 0.248529 0.124264 0.992249i \(-0.460343\pi\)
0.124264 + 0.992249i \(0.460343\pi\)
\(138\) 1.99041e7 0.644713
\(139\) 3.21430e7 1.01516 0.507580 0.861605i \(-0.330540\pi\)
0.507580 + 0.861605i \(0.330540\pi\)
\(140\) 1.23183e7 0.379404
\(141\) −5.80885e7 −1.74511
\(142\) −2.98427e7 −0.874639
\(143\) −4.92428e7 −1.40821
\(144\) 7.46481e6 0.208329
\(145\) −3.14486e7 −0.856669
\(146\) −1.08232e7 −0.287821
\(147\) −3.21948e7 −0.835940
\(148\) 4.30116e7 1.09061
\(149\) −5.81325e7 −1.43968 −0.719842 0.694138i \(-0.755786\pi\)
−0.719842 + 0.694138i \(0.755786\pi\)
\(150\) 7.23103e7 1.74937
\(151\) 3.73079e7 0.881824 0.440912 0.897550i \(-0.354655\pi\)
0.440912 + 0.897550i \(0.354655\pi\)
\(152\) −1.44453e6 −0.0333637
\(153\) −1.71956e7 −0.388148
\(154\) −2.46928e7 −0.544814
\(155\) −4.45137e7 −0.960137
\(156\) 3.79429e7 0.800193
\(157\) 1.77014e7 0.365055 0.182528 0.983201i \(-0.441572\pi\)
0.182528 + 0.983201i \(0.441572\pi\)
\(158\) 8.74550e7 1.76394
\(159\) −1.24666e7 −0.245956
\(160\) −1.13678e8 −2.19410
\(161\) 6.53360e6 0.123385
\(162\) −5.70995e7 −1.05519
\(163\) 5.62209e7 1.01681 0.508407 0.861117i \(-0.330235\pi\)
0.508407 + 0.861117i \(0.330235\pi\)
\(164\) 5.05367e7 0.894650
\(165\) −1.25799e8 −2.18014
\(166\) 7.78033e7 1.32014
\(167\) 9.11586e7 1.51457 0.757286 0.653083i \(-0.226525\pi\)
0.757286 + 0.653083i \(0.226525\pi\)
\(168\) 329551. 0.00536217
\(169\) −1.35032e7 −0.215196
\(170\) 2.57247e8 4.01586
\(171\) 1.84841e7 0.282691
\(172\) −1.03563e7 −0.155187
\(173\) −1.16610e8 −1.71228 −0.856140 0.516744i \(-0.827144\pi\)
−0.856140 + 0.516744i \(0.827144\pi\)
\(174\) −4.85747e7 −0.699017
\(175\) 2.37361e7 0.334793
\(176\) 1.12907e8 1.56108
\(177\) −1.83023e7 −0.248084
\(178\) 1.44416e8 1.91931
\(179\) 8.45696e6 0.110212 0.0551060 0.998481i \(-0.482450\pi\)
0.0551060 + 0.998481i \(0.482450\pi\)
\(180\) −2.60991e7 −0.333558
\(181\) 2.00697e7 0.251574 0.125787 0.992057i \(-0.459854\pi\)
0.125787 + 0.992057i \(0.459854\pi\)
\(182\) 2.46941e7 0.303629
\(183\) 7.14798e7 0.862193
\(184\) 1.08182e6 0.0128024
\(185\) 1.42611e8 1.65597
\(186\) −6.87548e7 −0.783443
\(187\) −2.60087e8 −2.90852
\(188\) −1.82279e8 −2.00072
\(189\) −2.40954e7 −0.259608
\(190\) −2.76523e8 −2.92478
\(191\) 1.71659e7 0.178259 0.0891293 0.996020i \(-0.471592\pi\)
0.0891293 + 0.996020i \(0.471592\pi\)
\(192\) −9.00936e7 −0.918628
\(193\) −1.58315e8 −1.58515 −0.792575 0.609775i \(-0.791260\pi\)
−0.792575 + 0.609775i \(0.791260\pi\)
\(194\) −1.22822e8 −1.20773
\(195\) 1.25806e8 1.21501
\(196\) −1.01026e8 −0.958378
\(197\) 5.27998e7 0.492040 0.246020 0.969265i \(-0.420877\pi\)
0.246020 + 0.969265i \(0.420877\pi\)
\(198\) 5.23173e7 0.478980
\(199\) −1.26904e8 −1.14154 −0.570768 0.821112i \(-0.693354\pi\)
−0.570768 + 0.821112i \(0.693354\pi\)
\(200\) 3.93017e6 0.0347382
\(201\) 1.43302e8 1.24471
\(202\) −7.58800e7 −0.647735
\(203\) −1.59448e7 −0.133777
\(204\) 2.00404e8 1.65273
\(205\) 1.67562e8 1.35843
\(206\) 8.14911e7 0.649494
\(207\) −1.38429e7 −0.108475
\(208\) −1.12912e8 −0.870002
\(209\) 2.79576e8 2.11830
\(210\) 6.30852e7 0.470067
\(211\) −1.23497e8 −0.905040 −0.452520 0.891754i \(-0.649475\pi\)
−0.452520 + 0.891754i \(0.649475\pi\)
\(212\) −3.91197e7 −0.281981
\(213\) −7.70838e7 −0.546556
\(214\) −5.50453e7 −0.383948
\(215\) −3.43378e7 −0.235634
\(216\) −3.98968e6 −0.0269370
\(217\) −2.25690e7 −0.149935
\(218\) −7.46996e7 −0.488338
\(219\) −2.79565e7 −0.179857
\(220\) −3.94753e8 −2.49946
\(221\) 2.60100e8 1.62094
\(222\) 2.20274e8 1.35122
\(223\) −2.86561e8 −1.73041 −0.865207 0.501415i \(-0.832813\pi\)
−0.865207 + 0.501415i \(0.832813\pi\)
\(224\) −5.76362e7 −0.342631
\(225\) −5.02902e7 −0.294337
\(226\) 1.94004e8 1.11797
\(227\) 1.80796e7 0.102588 0.0512941 0.998684i \(-0.483665\pi\)
0.0512941 + 0.998684i \(0.483665\pi\)
\(228\) −2.15421e8 −1.20369
\(229\) 3.18532e8 1.75279 0.876393 0.481597i \(-0.159943\pi\)
0.876393 + 0.481597i \(0.159943\pi\)
\(230\) 2.07090e8 1.12231
\(231\) −6.37817e7 −0.340451
\(232\) −2.64011e6 −0.0138808
\(233\) 38350.8 0.000198622 0 9.93112e−5 1.00000i \(-0.499968\pi\)
9.93112e−5 1.00000i \(0.499968\pi\)
\(234\) −5.23199e7 −0.266939
\(235\) −6.04375e8 −3.03787
\(236\) −5.74319e7 −0.284421
\(237\) 2.25897e8 1.10228
\(238\) 1.30427e8 0.627117
\(239\) −2.35171e7 −0.111427 −0.0557136 0.998447i \(-0.517743\pi\)
−0.0557136 + 0.998447i \(0.517743\pi\)
\(240\) −2.88454e8 −1.34691
\(241\) 2.36236e8 1.08714 0.543570 0.839364i \(-0.317072\pi\)
0.543570 + 0.839364i \(0.317072\pi\)
\(242\) 4.78144e8 2.16873
\(243\) 9.31688e7 0.416532
\(244\) 2.24301e8 0.988477
\(245\) −3.34967e8 −1.45519
\(246\) 2.58812e8 1.10844
\(247\) −2.79590e8 −1.18054
\(248\) −3.73693e6 −0.0155573
\(249\) 2.00966e8 0.824946
\(250\) 2.10115e8 0.850488
\(251\) 5.78454e7 0.230893 0.115446 0.993314i \(-0.463170\pi\)
0.115446 + 0.993314i \(0.463170\pi\)
\(252\) −1.32325e7 −0.0520884
\(253\) −2.09377e8 −0.812843
\(254\) 5.33488e7 0.204271
\(255\) 6.64470e8 2.50948
\(256\) 2.58724e8 0.963822
\(257\) 1.20884e8 0.444226 0.222113 0.975021i \(-0.428705\pi\)
0.222113 + 0.975021i \(0.428705\pi\)
\(258\) −5.30373e7 −0.192270
\(259\) 7.23056e7 0.258597
\(260\) 3.94773e8 1.39297
\(261\) 3.37826e7 0.117612
\(262\) 4.34523e8 1.49265
\(263\) −3.92747e7 −0.133127 −0.0665637 0.997782i \(-0.521204\pi\)
−0.0665637 + 0.997782i \(0.521204\pi\)
\(264\) −1.05608e7 −0.0353252
\(265\) −1.29707e8 −0.428158
\(266\) −1.40200e8 −0.456734
\(267\) 3.73027e8 1.19937
\(268\) 4.49677e8 1.42702
\(269\) −2.04982e8 −0.642070 −0.321035 0.947067i \(-0.604031\pi\)
−0.321035 + 0.947067i \(0.604031\pi\)
\(270\) −7.63734e8 −2.36140
\(271\) −3.83850e8 −1.17157 −0.585787 0.810465i \(-0.699214\pi\)
−0.585787 + 0.810465i \(0.699214\pi\)
\(272\) −5.96372e8 −1.79691
\(273\) 6.37849e7 0.189735
\(274\) 1.20205e8 0.353018
\(275\) −7.60650e8 −2.20557
\(276\) 1.61330e8 0.461886
\(277\) 5.84462e8 1.65226 0.826128 0.563483i \(-0.190539\pi\)
0.826128 + 0.563483i \(0.190539\pi\)
\(278\) 5.16549e8 1.44196
\(279\) 4.78175e7 0.131817
\(280\) 3.42878e6 0.00933440
\(281\) −1.50550e8 −0.404770 −0.202385 0.979306i \(-0.564869\pi\)
−0.202385 + 0.979306i \(0.564869\pi\)
\(282\) −9.33503e8 −2.47881
\(283\) 3.43343e8 0.900483 0.450242 0.892907i \(-0.351338\pi\)
0.450242 + 0.892907i \(0.351338\pi\)
\(284\) −2.41886e8 −0.626609
\(285\) −7.14260e8 −1.82768
\(286\) −7.91350e8 −2.00026
\(287\) 8.49558e7 0.212132
\(288\) 1.22115e8 0.301229
\(289\) 9.63435e8 2.34790
\(290\) −5.05390e8 −1.21684
\(291\) −3.17250e8 −0.754703
\(292\) −8.77263e7 −0.206201
\(293\) −4.31569e8 −1.00234 −0.501168 0.865350i \(-0.667096\pi\)
−0.501168 + 0.865350i \(0.667096\pi\)
\(294\) −5.17381e8 −1.18739
\(295\) −1.90424e8 −0.431862
\(296\) 1.19722e7 0.0268320
\(297\) 7.72166e8 1.71026
\(298\) −9.34210e8 −2.04497
\(299\) 2.09387e8 0.453003
\(300\) 5.86101e8 1.25328
\(301\) −1.74097e7 −0.0367966
\(302\) 5.99552e8 1.25257
\(303\) −1.95998e8 −0.404765
\(304\) 6.41060e8 1.30870
\(305\) 7.43703e8 1.50089
\(306\) −2.76339e8 −0.551337
\(307\) −9.63326e8 −1.90016 −0.950078 0.312013i \(-0.898997\pi\)
−0.950078 + 0.312013i \(0.898997\pi\)
\(308\) −2.00144e8 −0.390316
\(309\) 2.10492e8 0.405864
\(310\) −7.15351e8 −1.36381
\(311\) 3.93814e8 0.742386 0.371193 0.928556i \(-0.378949\pi\)
0.371193 + 0.928556i \(0.378949\pi\)
\(312\) 1.05614e7 0.0196870
\(313\) −1.05509e9 −1.94484 −0.972422 0.233227i \(-0.925071\pi\)
−0.972422 + 0.233227i \(0.925071\pi\)
\(314\) 2.84468e8 0.518536
\(315\) −4.38744e7 −0.0790905
\(316\) 7.08855e8 1.26373
\(317\) 5.36072e8 0.945182 0.472591 0.881282i \(-0.343319\pi\)
0.472591 + 0.881282i \(0.343319\pi\)
\(318\) −2.00342e8 −0.349364
\(319\) 5.10969e8 0.881307
\(320\) −9.37368e8 −1.59914
\(321\) −1.42182e8 −0.239926
\(322\) 1.04997e8 0.175260
\(323\) −1.47671e9 −2.43830
\(324\) −4.62813e8 −0.755959
\(325\) 7.60688e8 1.22918
\(326\) 9.03490e8 1.44431
\(327\) −1.92949e8 −0.305159
\(328\) 1.40668e7 0.0220109
\(329\) −3.06425e8 −0.474394
\(330\) −2.02164e9 −3.09674
\(331\) −6.83321e8 −1.03568 −0.517842 0.855476i \(-0.673264\pi\)
−0.517842 + 0.855476i \(0.673264\pi\)
\(332\) 6.30625e8 0.945775
\(333\) −1.53196e8 −0.227348
\(334\) 1.46495e9 2.15135
\(335\) 1.49097e9 2.16677
\(336\) −1.46250e8 −0.210333
\(337\) 9.38435e8 1.33567 0.667835 0.744309i \(-0.267221\pi\)
0.667835 + 0.744309i \(0.267221\pi\)
\(338\) −2.17001e8 −0.305671
\(339\) 5.01113e8 0.698613
\(340\) 2.08508e9 2.87704
\(341\) 7.23249e8 0.987751
\(342\) 2.97046e8 0.401543
\(343\) −3.50163e8 −0.468534
\(344\) −2.88266e6 −0.00381802
\(345\) 5.34915e8 0.701323
\(346\) −1.87396e9 −2.43218
\(347\) 8.64270e8 1.11044 0.555221 0.831703i \(-0.312634\pi\)
0.555221 + 0.831703i \(0.312634\pi\)
\(348\) −3.93716e8 −0.500790
\(349\) −1.00648e9 −1.26741 −0.633706 0.773574i \(-0.718467\pi\)
−0.633706 + 0.773574i \(0.718467\pi\)
\(350\) 3.81447e8 0.475550
\(351\) −7.72205e8 −0.953142
\(352\) 1.84702e9 2.25721
\(353\) −8.45245e8 −1.02275 −0.511377 0.859356i \(-0.670864\pi\)
−0.511377 + 0.859356i \(0.670864\pi\)
\(354\) −2.94124e8 −0.352387
\(355\) −8.02010e8 −0.951438
\(356\) 1.17055e9 1.37503
\(357\) 3.36894e8 0.391881
\(358\) 1.35906e8 0.156548
\(359\) −1.65237e8 −0.188485 −0.0942423 0.995549i \(-0.530043\pi\)
−0.0942423 + 0.995549i \(0.530043\pi\)
\(360\) −7.26464e6 −0.00820645
\(361\) 6.93498e8 0.775836
\(362\) 3.22527e8 0.357343
\(363\) 1.23505e9 1.35522
\(364\) 2.00155e8 0.217526
\(365\) −2.90870e8 −0.313093
\(366\) 1.14870e9 1.22469
\(367\) −8.00752e8 −0.845603 −0.422802 0.906222i \(-0.638953\pi\)
−0.422802 + 0.906222i \(0.638953\pi\)
\(368\) −4.80095e8 −0.502180
\(369\) −1.79998e8 −0.186499
\(370\) 2.29182e9 2.35220
\(371\) −6.57631e7 −0.0668611
\(372\) −5.57283e8 −0.561275
\(373\) −4.62722e8 −0.461678 −0.230839 0.972992i \(-0.574147\pi\)
−0.230839 + 0.972992i \(0.574147\pi\)
\(374\) −4.17969e9 −4.13136
\(375\) 5.42729e8 0.531464
\(376\) −5.07373e7 −0.0492232
\(377\) −5.10995e8 −0.491159
\(378\) −3.87222e8 −0.368756
\(379\) 5.59662e8 0.528066 0.264033 0.964514i \(-0.414947\pi\)
0.264033 + 0.964514i \(0.414947\pi\)
\(380\) −2.24132e9 −2.09537
\(381\) 1.37800e8 0.127647
\(382\) 2.75862e8 0.253204
\(383\) 2.55484e8 0.232364 0.116182 0.993228i \(-0.462934\pi\)
0.116182 + 0.993228i \(0.462934\pi\)
\(384\) −4.93084e7 −0.0444387
\(385\) −6.63609e8 −0.592652
\(386\) −2.54417e9 −2.25160
\(387\) 3.68863e7 0.0323502
\(388\) −9.95518e8 −0.865243
\(389\) 2.14983e9 1.85175 0.925873 0.377834i \(-0.123331\pi\)
0.925873 + 0.377834i \(0.123331\pi\)
\(390\) 2.02174e9 1.72583
\(391\) 1.10592e9 0.935635
\(392\) −2.81205e7 −0.0235788
\(393\) 1.12237e9 0.932747
\(394\) 8.48512e8 0.698910
\(395\) 2.35032e9 1.91883
\(396\) 4.24051e8 0.343151
\(397\) −7.29245e8 −0.584933 −0.292467 0.956276i \(-0.594476\pi\)
−0.292467 + 0.956276i \(0.594476\pi\)
\(398\) −2.03939e9 −1.62147
\(399\) −3.62138e8 −0.285410
\(400\) −1.74415e9 −1.36262
\(401\) −3.33153e8 −0.258011 −0.129006 0.991644i \(-0.541179\pi\)
−0.129006 + 0.991644i \(0.541179\pi\)
\(402\) 2.30292e9 1.76802
\(403\) −7.23285e8 −0.550480
\(404\) −6.15035e8 −0.464050
\(405\) −1.53453e9 −1.14784
\(406\) −2.56238e8 −0.190022
\(407\) −2.31712e9 −1.70360
\(408\) 5.57822e7 0.0406617
\(409\) −1.35727e9 −0.980923 −0.490461 0.871463i \(-0.663172\pi\)
−0.490461 + 0.871463i \(0.663172\pi\)
\(410\) 2.69278e9 1.92956
\(411\) 3.10491e8 0.220598
\(412\) 6.60516e8 0.465310
\(413\) −9.65473e7 −0.0674396
\(414\) −2.22460e8 −0.154082
\(415\) 2.09093e9 1.43606
\(416\) −1.84711e9 −1.25796
\(417\) 1.33425e9 0.901074
\(418\) 4.49288e9 3.00890
\(419\) −6.15909e8 −0.409041 −0.204521 0.978862i \(-0.565564\pi\)
−0.204521 + 0.978862i \(0.565564\pi\)
\(420\) 5.11329e8 0.336766
\(421\) −2.78917e9 −1.82174 −0.910871 0.412690i \(-0.864589\pi\)
−0.910871 + 0.412690i \(0.864589\pi\)
\(422\) −1.98464e9 −1.28555
\(423\) 6.49231e8 0.417069
\(424\) −1.08889e7 −0.00693752
\(425\) 4.01774e9 2.53875
\(426\) −1.23876e9 −0.776345
\(427\) 3.77066e8 0.234380
\(428\) −4.46163e8 −0.275068
\(429\) −2.04406e9 −1.24995
\(430\) −5.51820e8 −0.334702
\(431\) 3.90368e8 0.234857 0.117429 0.993081i \(-0.462535\pi\)
0.117429 + 0.993081i \(0.462535\pi\)
\(432\) 1.77056e9 1.05661
\(433\) 1.71968e9 1.01798 0.508991 0.860772i \(-0.330019\pi\)
0.508991 + 0.860772i \(0.330019\pi\)
\(434\) −3.62691e8 −0.212972
\(435\) −1.30542e9 −0.760395
\(436\) −6.05468e8 −0.349855
\(437\) −1.18879e9 −0.681431
\(438\) −4.49270e8 −0.255475
\(439\) −1.48222e9 −0.836154 −0.418077 0.908412i \(-0.637296\pi\)
−0.418077 + 0.908412i \(0.637296\pi\)
\(440\) −1.09879e8 −0.0614937
\(441\) 3.59828e8 0.199783
\(442\) 4.17990e9 2.30244
\(443\) −3.10398e9 −1.69631 −0.848156 0.529747i \(-0.822287\pi\)
−0.848156 + 0.529747i \(0.822287\pi\)
\(444\) 1.78540e9 0.968045
\(445\) 3.88112e9 2.08784
\(446\) −4.60513e9 −2.45793
\(447\) −2.41307e9 −1.27789
\(448\) −4.75257e8 −0.249721
\(449\) −6.30366e8 −0.328648 −0.164324 0.986406i \(-0.552544\pi\)
−0.164324 + 0.986406i \(0.552544\pi\)
\(450\) −8.08182e8 −0.418086
\(451\) −2.72251e9 −1.39750
\(452\) 1.57247e9 0.800938
\(453\) 1.54864e9 0.782723
\(454\) 2.90545e8 0.145720
\(455\) 6.63643e8 0.330289
\(456\) −5.99621e7 −0.0296142
\(457\) −3.54718e9 −1.73851 −0.869255 0.494365i \(-0.835401\pi\)
−0.869255 + 0.494365i \(0.835401\pi\)
\(458\) 5.11892e9 2.48971
\(459\) −4.07857e9 −1.96863
\(460\) 1.67854e9 0.804045
\(461\) −9.13977e8 −0.434492 −0.217246 0.976117i \(-0.569707\pi\)
−0.217246 + 0.976117i \(0.569707\pi\)
\(462\) −1.02499e9 −0.483587
\(463\) −8.31986e8 −0.389567 −0.194784 0.980846i \(-0.562400\pi\)
−0.194784 + 0.980846i \(0.562400\pi\)
\(464\) 1.17164e9 0.544478
\(465\) −1.84776e9 −0.852235
\(466\) 616310. 0.000282130 0
\(467\) 2.96829e9 1.34864 0.674322 0.738437i \(-0.264436\pi\)
0.674322 + 0.738437i \(0.264436\pi\)
\(468\) −4.24073e8 −0.191240
\(469\) 7.55940e8 0.338362
\(470\) −9.71252e9 −4.31509
\(471\) 7.34781e8 0.324030
\(472\) −1.59861e7 −0.00699755
\(473\) 5.57912e8 0.242411
\(474\) 3.63024e9 1.56571
\(475\) −4.31880e9 −1.84899
\(476\) 1.05716e9 0.449279
\(477\) 1.39334e8 0.0587817
\(478\) −3.77928e8 −0.158275
\(479\) 3.69035e9 1.53424 0.767120 0.641504i \(-0.221689\pi\)
0.767120 + 0.641504i \(0.221689\pi\)
\(480\) −4.71876e9 −1.94753
\(481\) 2.31723e9 0.949428
\(482\) 3.79639e9 1.54421
\(483\) 2.71208e8 0.109519
\(484\) 3.87554e9 1.55372
\(485\) −3.30079e9 −1.31378
\(486\) 1.49725e9 0.591655
\(487\) −2.04194e8 −0.0801107 −0.0400554 0.999197i \(-0.512753\pi\)
−0.0400554 + 0.999197i \(0.512753\pi\)
\(488\) 6.24339e7 0.0243193
\(489\) 2.33372e9 0.902542
\(490\) −5.38304e9 −2.06700
\(491\) −1.33712e9 −0.509782 −0.254891 0.966970i \(-0.582040\pi\)
−0.254891 + 0.966970i \(0.582040\pi\)
\(492\) 2.09777e9 0.794107
\(493\) −2.69893e9 −1.01444
\(494\) −4.49311e9 −1.67688
\(495\) 1.40601e9 0.521038
\(496\) 1.65839e9 0.610240
\(497\) −4.06628e8 −0.148577
\(498\) 3.22960e9 1.17178
\(499\) −1.58060e9 −0.569469 −0.284734 0.958606i \(-0.591905\pi\)
−0.284734 + 0.958606i \(0.591905\pi\)
\(500\) 1.70306e9 0.609307
\(501\) 3.78397e9 1.34436
\(502\) 9.29595e8 0.327968
\(503\) 8.52354e8 0.298629 0.149315 0.988790i \(-0.452293\pi\)
0.149315 + 0.988790i \(0.452293\pi\)
\(504\) −3.68326e6 −0.00128152
\(505\) −2.03924e9 −0.704610
\(506\) −3.36476e9 −1.15459
\(507\) −5.60515e8 −0.191011
\(508\) 4.32412e8 0.146344
\(509\) −2.02364e9 −0.680177 −0.340089 0.940393i \(-0.610457\pi\)
−0.340089 + 0.940393i \(0.610457\pi\)
\(510\) 1.06783e10 3.56455
\(511\) −1.47474e8 −0.0488926
\(512\) 4.30983e9 1.41911
\(513\) 4.38419e9 1.43377
\(514\) 1.94265e9 0.630993
\(515\) 2.19004e9 0.706524
\(516\) −4.29887e8 −0.137746
\(517\) 9.81975e9 3.12524
\(518\) 1.16198e9 0.367319
\(519\) −4.84046e9 −1.51985
\(520\) 1.09885e8 0.0342709
\(521\) 1.77856e8 0.0550980 0.0275490 0.999620i \(-0.491230\pi\)
0.0275490 + 0.999620i \(0.491230\pi\)
\(522\) 5.42899e8 0.167060
\(523\) −1.80745e9 −0.552471 −0.276236 0.961090i \(-0.589087\pi\)
−0.276236 + 0.961090i \(0.589087\pi\)
\(524\) 3.52197e9 1.06936
\(525\) 9.85280e8 0.297168
\(526\) −6.31158e8 −0.189098
\(527\) −3.82019e9 −1.13697
\(528\) 4.68674e9 1.38564
\(529\) −2.51453e9 −0.738519
\(530\) −2.08444e9 −0.608169
\(531\) 2.04557e8 0.0592903
\(532\) −1.13638e9 −0.327214
\(533\) 2.72264e9 0.778835
\(534\) 5.99468e9 1.70362
\(535\) −1.47932e9 −0.417661
\(536\) 1.25167e8 0.0351086
\(537\) 3.51046e8 0.0978261
\(538\) −3.29413e9 −0.912016
\(539\) 5.44247e9 1.49705
\(540\) −6.19035e9 −1.69175
\(541\) −1.63216e8 −0.0443172 −0.0221586 0.999754i \(-0.507054\pi\)
−0.0221586 + 0.999754i \(0.507054\pi\)
\(542\) −6.16861e9 −1.66414
\(543\) 8.33088e8 0.223301
\(544\) −9.75591e9 −2.59820
\(545\) −2.00752e9 −0.531217
\(546\) 1.02505e9 0.269506
\(547\) 1.75641e9 0.458849 0.229425 0.973326i \(-0.426316\pi\)
0.229425 + 0.973326i \(0.426316\pi\)
\(548\) 9.74308e8 0.252909
\(549\) −7.98900e8 −0.206058
\(550\) −1.22239e10 −3.13286
\(551\) 2.90117e9 0.738826
\(552\) 4.49061e7 0.0113637
\(553\) 1.19164e9 0.299644
\(554\) 9.39251e9 2.34692
\(555\) 5.91977e9 1.46987
\(556\) 4.18682e9 1.03305
\(557\) 3.96938e9 0.973262 0.486631 0.873608i \(-0.338226\pi\)
0.486631 + 0.873608i \(0.338226\pi\)
\(558\) 7.68443e8 0.187237
\(559\) −5.57940e8 −0.135097
\(560\) −1.52164e9 −0.366145
\(561\) −1.07961e10 −2.58166
\(562\) −2.41939e9 −0.574948
\(563\) −7.01695e9 −1.65718 −0.828588 0.559858i \(-0.810855\pi\)
−0.828588 + 0.559858i \(0.810855\pi\)
\(564\) −7.56638e9 −1.77587
\(565\) 5.21378e9 1.21614
\(566\) 5.51764e9 1.27907
\(567\) −7.78023e8 −0.179247
\(568\) −6.73287e7 −0.0154163
\(569\) −3.72696e9 −0.848129 −0.424064 0.905632i \(-0.639397\pi\)
−0.424064 + 0.905632i \(0.639397\pi\)
\(570\) −1.14784e10 −2.59609
\(571\) −6.36851e9 −1.43157 −0.715783 0.698323i \(-0.753930\pi\)
−0.715783 + 0.698323i \(0.753930\pi\)
\(572\) −6.41418e9 −1.43303
\(573\) 7.12554e8 0.158226
\(574\) 1.36527e9 0.301319
\(575\) 3.23439e9 0.709503
\(576\) 1.00694e9 0.219545
\(577\) −4.84407e9 −1.04977 −0.524887 0.851172i \(-0.675892\pi\)
−0.524887 + 0.851172i \(0.675892\pi\)
\(578\) 1.54827e10 3.33503
\(579\) −6.57160e9 −1.40701
\(580\) −4.09637e9 −0.871769
\(581\) 1.06013e9 0.224255
\(582\) −5.09832e9 −1.07200
\(583\) 2.10745e9 0.440472
\(584\) −2.44185e7 −0.00507311
\(585\) −1.40608e9 −0.290378
\(586\) −6.93547e9 −1.42375
\(587\) 7.07809e9 1.44438 0.722192 0.691693i \(-0.243135\pi\)
0.722192 + 0.691693i \(0.243135\pi\)
\(588\) −4.19357e9 −0.850674
\(589\) 4.10645e9 0.828061
\(590\) −3.06018e9 −0.613431
\(591\) 2.19171e9 0.436744
\(592\) −5.31308e9 −1.05250
\(593\) −8.42138e9 −1.65841 −0.829205 0.558945i \(-0.811206\pi\)
−0.829205 + 0.558945i \(0.811206\pi\)
\(594\) 1.24090e10 2.42931
\(595\) 3.50517e9 0.682182
\(596\) −7.57211e9 −1.46506
\(597\) −5.26775e9 −1.01325
\(598\) 3.36492e9 0.643460
\(599\) −3.43703e8 −0.0653416 −0.0326708 0.999466i \(-0.510401\pi\)
−0.0326708 + 0.999466i \(0.510401\pi\)
\(600\) 1.63141e8 0.0308342
\(601\) −3.45425e9 −0.649073 −0.324536 0.945873i \(-0.605208\pi\)
−0.324536 + 0.945873i \(0.605208\pi\)
\(602\) −2.79779e8 −0.0522670
\(603\) −1.60163e9 −0.297475
\(604\) 4.85959e9 0.897367
\(605\) 1.28499e10 2.35916
\(606\) −3.14976e9 −0.574941
\(607\) 1.86343e9 0.338183 0.169092 0.985600i \(-0.445917\pi\)
0.169092 + 0.985600i \(0.445917\pi\)
\(608\) 1.04870e10 1.89229
\(609\) −6.61865e8 −0.118743
\(610\) 1.19516e10 2.13192
\(611\) −9.82024e9 −1.74172
\(612\) −2.23983e9 −0.394989
\(613\) 4.59607e9 0.805888 0.402944 0.915225i \(-0.367987\pi\)
0.402944 + 0.915225i \(0.367987\pi\)
\(614\) −1.54810e10 −2.69904
\(615\) 6.95546e9 1.20577
\(616\) −5.57100e7 −0.00960286
\(617\) 9.17212e9 1.57207 0.786034 0.618183i \(-0.212131\pi\)
0.786034 + 0.618183i \(0.212131\pi\)
\(618\) 3.38268e9 0.576502
\(619\) 2.36759e9 0.401226 0.200613 0.979671i \(-0.435707\pi\)
0.200613 + 0.979671i \(0.435707\pi\)
\(620\) −5.79819e9 −0.977060
\(621\) −3.28335e9 −0.550170
\(622\) 6.32873e9 1.05451
\(623\) 1.96777e9 0.326037
\(624\) −4.68697e9 −0.772229
\(625\) −2.82188e9 −0.462337
\(626\) −1.69557e10 −2.76252
\(627\) 1.16051e10 1.88024
\(628\) 2.30571e9 0.371490
\(629\) 1.22390e10 1.96095
\(630\) −7.05077e8 −0.112343
\(631\) −4.85631e9 −0.769491 −0.384746 0.923023i \(-0.625711\pi\)
−0.384746 + 0.923023i \(0.625711\pi\)
\(632\) 1.97309e8 0.0310912
\(633\) −5.12633e9 −0.803330
\(634\) 8.61486e9 1.34257
\(635\) 1.43373e9 0.222207
\(636\) −1.62385e9 −0.250291
\(637\) −5.44274e9 −0.834314
\(638\) 8.21145e9 1.25184
\(639\) 8.61534e8 0.130623
\(640\) −5.13023e8 −0.0773583
\(641\) 4.17538e9 0.626171 0.313086 0.949725i \(-0.398637\pi\)
0.313086 + 0.949725i \(0.398637\pi\)
\(642\) −2.28492e9 −0.340799
\(643\) 6.67976e9 0.990884 0.495442 0.868641i \(-0.335006\pi\)
0.495442 + 0.868641i \(0.335006\pi\)
\(644\) 8.51041e8 0.125560
\(645\) −1.42535e9 −0.209153
\(646\) −2.37313e10 −3.46344
\(647\) 3.94085e9 0.572038 0.286019 0.958224i \(-0.407668\pi\)
0.286019 + 0.958224i \(0.407668\pi\)
\(648\) −1.28823e8 −0.0185987
\(649\) 3.09397e9 0.444283
\(650\) 1.22245e10 1.74596
\(651\) −9.36834e8 −0.133085
\(652\) 7.32312e9 1.03474
\(653\) −3.11818e9 −0.438233 −0.219116 0.975699i \(-0.570317\pi\)
−0.219116 + 0.975699i \(0.570317\pi\)
\(654\) −3.10076e9 −0.433457
\(655\) 1.16776e10 1.62371
\(656\) −6.24263e9 −0.863385
\(657\) 3.12458e8 0.0429846
\(658\) −4.92436e9 −0.673844
\(659\) 8.02376e9 1.09214 0.546071 0.837739i \(-0.316123\pi\)
0.546071 + 0.837739i \(0.316123\pi\)
\(660\) −1.63861e10 −2.21857
\(661\) 9.19114e9 1.23784 0.618920 0.785454i \(-0.287571\pi\)
0.618920 + 0.785454i \(0.287571\pi\)
\(662\) −1.09812e10 −1.47112
\(663\) 1.07967e10 1.43878
\(664\) 1.75534e8 0.0232687
\(665\) −3.76783e9 −0.496838
\(666\) −2.46191e9 −0.322933
\(667\) −2.17271e9 −0.283505
\(668\) 1.18740e10 1.54127
\(669\) −1.18951e10 −1.53595
\(670\) 2.39604e10 3.07775
\(671\) −1.20835e10 −1.54406
\(672\) −2.39247e9 −0.304126
\(673\) −4.44798e8 −0.0562484 −0.0281242 0.999604i \(-0.508953\pi\)
−0.0281242 + 0.999604i \(0.508953\pi\)
\(674\) 1.50810e10 1.89723
\(675\) −1.19282e10 −1.49283
\(676\) −1.75888e9 −0.218989
\(677\) −7.64502e8 −0.0946931 −0.0473465 0.998879i \(-0.515077\pi\)
−0.0473465 + 0.998879i \(0.515077\pi\)
\(678\) 8.05306e9 0.992333
\(679\) −1.67354e9 −0.205160
\(680\) 5.80380e8 0.0707833
\(681\) 7.50479e8 0.0910592
\(682\) 1.16229e10 1.40303
\(683\) −9.15944e9 −1.10001 −0.550005 0.835162i \(-0.685374\pi\)
−0.550005 + 0.835162i \(0.685374\pi\)
\(684\) 2.40767e9 0.287674
\(685\) 3.23047e9 0.384015
\(686\) −5.62725e9 −0.665521
\(687\) 1.32222e10 1.55580
\(688\) 1.27928e9 0.149763
\(689\) −2.10756e9 −0.245478
\(690\) 8.59628e9 0.996182
\(691\) 1.38604e10 1.59809 0.799047 0.601269i \(-0.205338\pi\)
0.799047 + 0.601269i \(0.205338\pi\)
\(692\) −1.51892e10 −1.74246
\(693\) 7.12861e8 0.0813652
\(694\) 1.38891e10 1.57731
\(695\) 1.38820e10 1.56858
\(696\) −1.09590e8 −0.0123208
\(697\) 1.43802e10 1.60861
\(698\) −1.61745e10 −1.80027
\(699\) 1.59193e6 0.000176301 0
\(700\) 3.09177e9 0.340694
\(701\) −1.37849e10 −1.51144 −0.755718 0.654897i \(-0.772712\pi\)
−0.755718 + 0.654897i \(0.772712\pi\)
\(702\) −1.24096e10 −1.35387
\(703\) −1.31561e10 −1.42818
\(704\) 1.52301e10 1.64513
\(705\) −2.50875e10 −2.69647
\(706\) −1.35834e10 −1.45275
\(707\) −1.03392e9 −0.110032
\(708\) −2.38399e9 −0.252457
\(709\) 5.66562e9 0.597015 0.298508 0.954407i \(-0.403511\pi\)
0.298508 + 0.954407i \(0.403511\pi\)
\(710\) −1.28886e10 −1.35145
\(711\) −2.52475e9 −0.263436
\(712\) 3.25820e8 0.0338297
\(713\) −3.07535e9 −0.317747
\(714\) 5.41400e9 0.556640
\(715\) −2.12672e10 −2.17590
\(716\) 1.10157e9 0.112154
\(717\) −9.76190e8 −0.0989047
\(718\) −2.65541e9 −0.267729
\(719\) 1.02176e10 1.02518 0.512588 0.858635i \(-0.328687\pi\)
0.512588 + 0.858635i \(0.328687\pi\)
\(720\) 3.22393e9 0.321901
\(721\) 1.11038e9 0.110331
\(722\) 1.11448e10 1.10202
\(723\) 9.80609e9 0.964965
\(724\) 2.61420e9 0.256008
\(725\) −7.89329e9 −0.769263
\(726\) 1.98477e10 1.92500
\(727\) −1.12746e10 −1.08826 −0.544128 0.839002i \(-0.683139\pi\)
−0.544128 + 0.839002i \(0.683139\pi\)
\(728\) 5.57128e7 0.00535174
\(729\) 1.16380e10 1.11259
\(730\) −4.67438e9 −0.444727
\(731\) −2.94688e9 −0.279031
\(732\) 9.31068e9 0.877390
\(733\) −1.48284e10 −1.39069 −0.695345 0.718676i \(-0.744749\pi\)
−0.695345 + 0.718676i \(0.744749\pi\)
\(734\) −1.28684e10 −1.20112
\(735\) −1.39044e10 −1.29166
\(736\) −7.85377e9 −0.726115
\(737\) −2.42249e10 −2.22909
\(738\) −2.89263e9 −0.264909
\(739\) 4.66823e9 0.425497 0.212749 0.977107i \(-0.431758\pi\)
0.212749 + 0.977107i \(0.431758\pi\)
\(740\) 1.85760e10 1.68516
\(741\) −1.16057e10 −1.04787
\(742\) −1.05684e9 −0.0949716
\(743\) 1.40086e10 1.25295 0.626477 0.779440i \(-0.284496\pi\)
0.626477 + 0.779440i \(0.284496\pi\)
\(744\) −1.55119e8 −0.0138089
\(745\) −2.51065e10 −2.22453
\(746\) −7.43611e9 −0.655782
\(747\) −2.24612e9 −0.197156
\(748\) −3.38779e10 −2.95979
\(749\) −7.50032e8 −0.0652219
\(750\) 8.72185e9 0.754908
\(751\) 1.52897e10 1.31722 0.658611 0.752484i \(-0.271144\pi\)
0.658611 + 0.752484i \(0.271144\pi\)
\(752\) 2.25164e10 1.93080
\(753\) 2.40115e9 0.204945
\(754\) −8.21186e9 −0.697657
\(755\) 1.61127e10 1.36255
\(756\) −3.13858e9 −0.264184
\(757\) −3.72305e9 −0.311934 −0.155967 0.987762i \(-0.549849\pi\)
−0.155967 + 0.987762i \(0.549849\pi\)
\(758\) 8.99396e9 0.750082
\(759\) −8.69118e9 −0.721494
\(760\) −6.23869e8 −0.0515521
\(761\) 1.87206e10 1.53984 0.769918 0.638143i \(-0.220297\pi\)
0.769918 + 0.638143i \(0.220297\pi\)
\(762\) 2.21450e9 0.181315
\(763\) −1.01784e9 −0.0829549
\(764\) 2.23597e9 0.181401
\(765\) −7.42650e9 −0.599749
\(766\) 4.10572e9 0.330057
\(767\) −3.09412e9 −0.247602
\(768\) 1.07396e10 0.855505
\(769\) 1.14386e10 0.907052 0.453526 0.891243i \(-0.350166\pi\)
0.453526 + 0.891243i \(0.350166\pi\)
\(770\) −1.06644e10 −0.841822
\(771\) 5.01789e9 0.394303
\(772\) −2.06214e10 −1.61309
\(773\) 2.03033e9 0.158103 0.0790513 0.996871i \(-0.474811\pi\)
0.0790513 + 0.996871i \(0.474811\pi\)
\(774\) 5.92775e8 0.0459512
\(775\) −1.11725e10 −0.862175
\(776\) −2.77101e8 −0.0212874
\(777\) 3.00139e9 0.229535
\(778\) 3.45486e10 2.63028
\(779\) −1.54578e10 −1.17156
\(780\) 1.63869e10 1.23642
\(781\) 1.30309e10 0.978802
\(782\) 1.77726e10 1.32901
\(783\) 8.01279e9 0.596510
\(784\) 1.24794e10 0.924886
\(785\) 7.64495e9 0.564067
\(786\) 1.80369e10 1.32490
\(787\) −2.52279e10 −1.84488 −0.922442 0.386135i \(-0.873810\pi\)
−0.922442 + 0.386135i \(0.873810\pi\)
\(788\) 6.87750e9 0.500713
\(789\) −1.63028e9 −0.118166
\(790\) 3.77704e10 2.72557
\(791\) 2.64344e9 0.189912
\(792\) 1.18034e8 0.00844247
\(793\) 1.20841e10 0.860516
\(794\) −1.17192e10 −0.830858
\(795\) −5.38412e9 −0.380040
\(796\) −1.65300e10 −1.16166
\(797\) 2.72322e10 1.90536 0.952682 0.303968i \(-0.0983115\pi\)
0.952682 + 0.303968i \(0.0983115\pi\)
\(798\) −5.81969e9 −0.405405
\(799\) −5.18677e10 −3.59736
\(800\) −2.85321e10 −1.97024
\(801\) −4.16917e9 −0.286640
\(802\) −5.35389e9 −0.366487
\(803\) 4.72598e9 0.322098
\(804\) 1.86660e10 1.26664
\(805\) 2.82176e9 0.190649
\(806\) −1.16234e10 −0.781920
\(807\) −8.50875e9 −0.569913
\(808\) −1.71194e8 −0.0114169
\(809\) −6.22108e9 −0.413091 −0.206546 0.978437i \(-0.566222\pi\)
−0.206546 + 0.978437i \(0.566222\pi\)
\(810\) −2.46604e10 −1.63043
\(811\) 7.83342e8 0.0515678 0.0257839 0.999668i \(-0.491792\pi\)
0.0257839 + 0.999668i \(0.491792\pi\)
\(812\) −2.07691e9 −0.136135
\(813\) −1.59335e10 −1.03991
\(814\) −3.72369e10 −2.41985
\(815\) 2.42809e10 1.57113
\(816\) −2.47553e10 −1.59497
\(817\) 3.16770e9 0.203220
\(818\) −2.18118e10 −1.39333
\(819\) −7.12897e8 −0.0453454
\(820\) 2.18260e10 1.38237
\(821\) −7.41002e9 −0.467324 −0.233662 0.972318i \(-0.575071\pi\)
−0.233662 + 0.972318i \(0.575071\pi\)
\(822\) 4.98970e9 0.313345
\(823\) −1.21111e10 −0.757328 −0.378664 0.925534i \(-0.623616\pi\)
−0.378664 + 0.925534i \(0.623616\pi\)
\(824\) 1.83854e8 0.0114479
\(825\) −3.15744e10 −1.95770
\(826\) −1.55155e9 −0.0957933
\(827\) 1.37043e10 0.842533 0.421266 0.906937i \(-0.361586\pi\)
0.421266 + 0.906937i \(0.361586\pi\)
\(828\) −1.80312e9 −0.110387
\(829\) −1.81240e8 −0.0110487 −0.00552437 0.999985i \(-0.501758\pi\)
−0.00552437 + 0.999985i \(0.501758\pi\)
\(830\) 3.36020e10 2.03982
\(831\) 2.42609e10 1.46657
\(832\) −1.52309e10 −0.916841
\(833\) −2.87470e10 −1.72320
\(834\) 2.14418e10 1.27991
\(835\) 3.93699e10 2.34025
\(836\) 3.64165e10 2.15564
\(837\) 1.13417e10 0.668556
\(838\) −9.89787e9 −0.581015
\(839\) −1.86015e9 −0.108738 −0.0543689 0.998521i \(-0.517315\pi\)
−0.0543689 + 0.998521i \(0.517315\pi\)
\(840\) 1.42328e8 0.00828538
\(841\) −1.19475e10 −0.692615
\(842\) −4.48229e10 −2.58766
\(843\) −6.24929e9 −0.359281
\(844\) −1.60862e10 −0.920992
\(845\) −5.83182e9 −0.332511
\(846\) 1.04334e10 0.592418
\(847\) 6.51506e9 0.368406
\(848\) 4.83233e9 0.272127
\(849\) 1.42521e10 0.799285
\(850\) 6.45665e10 3.60613
\(851\) 9.85269e9 0.548026
\(852\) −1.00406e10 −0.556189
\(853\) 8.00784e9 0.441768 0.220884 0.975300i \(-0.429106\pi\)
0.220884 + 0.975300i \(0.429106\pi\)
\(854\) 6.05958e9 0.332920
\(855\) 7.98299e9 0.436802
\(856\) −1.24189e8 −0.00676744
\(857\) 3.09890e10 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(858\) −3.28488e10 −1.77547
\(859\) 2.61801e10 1.40927 0.704636 0.709569i \(-0.251110\pi\)
0.704636 + 0.709569i \(0.251110\pi\)
\(860\) −4.47271e9 −0.239787
\(861\) 3.52650e9 0.188292
\(862\) 6.27335e9 0.333598
\(863\) −5.73619e9 −0.303799 −0.151899 0.988396i \(-0.548539\pi\)
−0.151899 + 0.988396i \(0.548539\pi\)
\(864\) 2.89641e10 1.52778
\(865\) −5.03620e10 −2.64574
\(866\) 2.76359e10 1.44597
\(867\) 3.99920e10 2.08404
\(868\) −2.93975e9 −0.152578
\(869\) −3.81874e10 −1.97402
\(870\) −2.09786e10 −1.08009
\(871\) 2.42262e10 1.24228
\(872\) −1.68531e8 −0.00860741
\(873\) 3.54577e9 0.180369
\(874\) −1.91043e10 −0.967926
\(875\) 2.86298e9 0.144474
\(876\) −3.64150e9 −0.183027
\(877\) −1.13135e10 −0.566369 −0.283184 0.959065i \(-0.591391\pi\)
−0.283184 + 0.959065i \(0.591391\pi\)
\(878\) −2.38198e10 −1.18770
\(879\) −1.79143e10 −0.889692
\(880\) 4.87626e10 2.41211
\(881\) 7.59277e9 0.374097 0.187049 0.982351i \(-0.440108\pi\)
0.187049 + 0.982351i \(0.440108\pi\)
\(882\) 5.78256e9 0.283779
\(883\) −1.12205e10 −0.548466 −0.274233 0.961663i \(-0.588424\pi\)
−0.274233 + 0.961663i \(0.588424\pi\)
\(884\) 3.38796e10 1.64951
\(885\) −7.90447e9 −0.383329
\(886\) −4.98820e10 −2.40949
\(887\) 6.84701e8 0.0329434 0.0164717 0.999864i \(-0.494757\pi\)
0.0164717 + 0.999864i \(0.494757\pi\)
\(888\) 4.96964e8 0.0238166
\(889\) 7.26916e8 0.0346999
\(890\) 6.23710e10 2.96563
\(891\) 2.49326e10 1.18085
\(892\) −3.73263e10 −1.76091
\(893\) 5.57543e10 2.61998
\(894\) −3.87788e10 −1.81515
\(895\) 3.65242e9 0.170294
\(896\) −2.60109e8 −0.0120803
\(897\) 8.69162e9 0.402093
\(898\) −1.01302e10 −0.466822
\(899\) 7.50518e9 0.344510
\(900\) −6.55061e9 −0.299525
\(901\) −1.11315e10 −0.507012
\(902\) −4.37516e10 −1.98505
\(903\) −7.22671e8 −0.0326613
\(904\) 4.37696e8 0.0197053
\(905\) 8.66777e9 0.388720
\(906\) 2.48873e10 1.11180
\(907\) 2.96670e8 0.0132022 0.00660112 0.999978i \(-0.497899\pi\)
0.00660112 + 0.999978i \(0.497899\pi\)
\(908\) 2.35497e9 0.104396
\(909\) 2.19059e9 0.0967359
\(910\) 1.06650e10 0.469153
\(911\) 2.64975e10 1.16115 0.580577 0.814205i \(-0.302827\pi\)
0.580577 + 0.814205i \(0.302827\pi\)
\(912\) 2.66103e10 1.16163
\(913\) −3.39730e10 −1.47736
\(914\) −5.70045e10 −2.46943
\(915\) 3.08710e10 1.33222
\(916\) 4.14907e10 1.78368
\(917\) 5.92069e9 0.253559
\(918\) −6.55440e10 −2.79630
\(919\) 9.31788e9 0.396016 0.198008 0.980200i \(-0.436553\pi\)
0.198008 + 0.980200i \(0.436553\pi\)
\(920\) 4.67221e8 0.0197818
\(921\) −3.99875e10 −1.68661
\(922\) −1.46879e10 −0.617166
\(923\) −1.30315e10 −0.545493
\(924\) −8.30796e9 −0.346451
\(925\) 3.57941e10 1.48701
\(926\) −1.33703e10 −0.553354
\(927\) −2.35258e9 −0.0969986
\(928\) 1.91666e10 0.787275
\(929\) 1.76298e10 0.721425 0.360713 0.932677i \(-0.382533\pi\)
0.360713 + 0.932677i \(0.382533\pi\)
\(930\) −2.96941e10 −1.21054
\(931\) 3.09011e10 1.25502
\(932\) 4.99542e6 0.000202123 0
\(933\) 1.63471e10 0.658956
\(934\) 4.77015e10 1.91566
\(935\) −1.12327e11 −4.49412
\(936\) −1.18040e8 −0.00470505
\(937\) 1.86944e10 0.742373 0.371187 0.928558i \(-0.378951\pi\)
0.371187 + 0.928558i \(0.378951\pi\)
\(938\) 1.21482e10 0.480621
\(939\) −4.37966e10 −1.72628
\(940\) −7.87236e10 −3.09142
\(941\) −3.37256e10 −1.31946 −0.659729 0.751504i \(-0.729329\pi\)
−0.659729 + 0.751504i \(0.729329\pi\)
\(942\) 1.18082e10 0.460262
\(943\) 1.15765e10 0.449557
\(944\) 7.09438e9 0.274481
\(945\) −1.04064e10 −0.401135
\(946\) 8.96585e9 0.344328
\(947\) 2.90728e10 1.11240 0.556202 0.831047i \(-0.312258\pi\)
0.556202 + 0.831047i \(0.312258\pi\)
\(948\) 2.94244e10 1.12171
\(949\) −4.72622e9 −0.179507
\(950\) −6.94046e10 −2.62637
\(951\) 2.22522e10 0.838960
\(952\) 2.94259e8 0.0110535
\(953\) 3.91532e10 1.46535 0.732677 0.680577i \(-0.238271\pi\)
0.732677 + 0.680577i \(0.238271\pi\)
\(954\) 2.23914e9 0.0834954
\(955\) 7.41369e9 0.275437
\(956\) −3.06325e9 −0.113391
\(957\) 2.12102e10 0.782264
\(958\) 5.93052e10 2.17928
\(959\) 1.63788e9 0.0599678
\(960\) −3.89100e10 −1.41942
\(961\) −1.68894e10 −0.613880
\(962\) 3.72387e10 1.34860
\(963\) 1.58911e9 0.0573406
\(964\) 3.07711e10 1.10630
\(965\) −6.83735e10 −2.44930
\(966\) 4.35841e9 0.155564
\(967\) 2.39640e10 0.852248 0.426124 0.904665i \(-0.359879\pi\)
0.426124 + 0.904665i \(0.359879\pi\)
\(968\) 1.07875e9 0.0382259
\(969\) −6.12981e10 −2.16428
\(970\) −5.30449e10 −1.86613
\(971\) −1.51230e10 −0.530114 −0.265057 0.964233i \(-0.585391\pi\)
−0.265057 + 0.964233i \(0.585391\pi\)
\(972\) 1.21358e10 0.423874
\(973\) 7.03835e9 0.244949
\(974\) −3.28146e9 −0.113792
\(975\) 3.15760e10 1.09104
\(976\) −2.77072e10 −0.953933
\(977\) 6.93989e9 0.238079 0.119040 0.992890i \(-0.462018\pi\)
0.119040 + 0.992890i \(0.462018\pi\)
\(978\) 3.75037e10 1.28200
\(979\) −6.30596e10 −2.14789
\(980\) −4.36315e10 −1.48084
\(981\) 2.15651e9 0.0729308
\(982\) −2.14880e10 −0.724111
\(983\) −3.16877e10 −1.06403 −0.532014 0.846736i \(-0.678565\pi\)
−0.532014 + 0.846736i \(0.678565\pi\)
\(984\) 5.83911e8 0.0195373
\(985\) 2.28034e10 0.760279
\(986\) −4.33727e10 −1.44095
\(987\) −1.27196e10 −0.421080
\(988\) −3.64183e10 −1.20135
\(989\) −2.37232e9 −0.0779805
\(990\) 2.25950e10 0.740098
\(991\) 8.97486e9 0.292934 0.146467 0.989216i \(-0.453210\pi\)
0.146467 + 0.989216i \(0.453210\pi\)
\(992\) 2.71292e10 0.882361
\(993\) −2.83645e10 −0.919291
\(994\) −6.53466e9 −0.211043
\(995\) −5.48078e10 −1.76385
\(996\) 2.61771e10 0.839487
\(997\) −3.06127e10 −0.978291 −0.489145 0.872202i \(-0.662691\pi\)
−0.489145 + 0.872202i \(0.662691\pi\)
\(998\) −2.54008e10 −0.808891
\(999\) −3.63360e10 −1.15308
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.1.10 13
3.2 odd 2 387.8.a.d.1.4 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.10 13 1.1 even 1 trivial
387.8.a.d.1.4 13 3.2 odd 2