Properties

Label 546.8.a.c.1.1
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0000 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -10.0000 q^{5} +216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -80.0000 q^{10} +1508.00 q^{11} +1728.00 q^{12} +2197.00 q^{13} -2744.00 q^{14} -270.000 q^{15} +4096.00 q^{16} -1042.00 q^{17} +5832.00 q^{18} -9068.00 q^{19} -640.000 q^{20} -9261.00 q^{21} +12064.0 q^{22} -98988.0 q^{23} +13824.0 q^{24} -78025.0 q^{25} +17576.0 q^{26} +19683.0 q^{27} -21952.0 q^{28} -213642. q^{29} -2160.00 q^{30} -22048.0 q^{31} +32768.0 q^{32} +40716.0 q^{33} -8336.00 q^{34} +3430.00 q^{35} +46656.0 q^{36} +418246. q^{37} -72544.0 q^{38} +59319.0 q^{39} -5120.00 q^{40} -76414.0 q^{41} -74088.0 q^{42} -177524. q^{43} +96512.0 q^{44} -7290.00 q^{45} -791904. q^{46} +631916. q^{47} +110592. q^{48} +117649. q^{49} -624200. q^{50} -28134.0 q^{51} +140608. q^{52} -982354. q^{53} +157464. q^{54} -15080.0 q^{55} -175616. q^{56} -244836. q^{57} -1.70914e6 q^{58} -596384. q^{59} -17280.0 q^{60} +1.86341e6 q^{61} -176384. q^{62} -250047. q^{63} +262144. q^{64} -21970.0 q^{65} +325728. q^{66} -1.84565e6 q^{67} -66688.0 q^{68} -2.67268e6 q^{69} +27440.0 q^{70} +1.63228e6 q^{71} +373248. q^{72} +216650. q^{73} +3.34597e6 q^{74} -2.10668e6 q^{75} -580352. q^{76} -517244. q^{77} +474552. q^{78} -6.46017e6 q^{79} -40960.0 q^{80} +531441. q^{81} -611312. q^{82} +3.59215e6 q^{83} -592704. q^{84} +10420.0 q^{85} -1.42019e6 q^{86} -5.76833e6 q^{87} +772096. q^{88} -9.48266e6 q^{89} -58320.0 q^{90} -753571. q^{91} -6.33523e6 q^{92} -595296. q^{93} +5.05533e6 q^{94} +90680.0 q^{95} +884736. q^{96} +840226. q^{97} +941192. q^{98} +1.09933e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −10.0000 −0.0357771 −0.0178885 0.999840i \(-0.505694\pi\)
−0.0178885 + 0.999840i \(0.505694\pi\)
\(6\) 216.000 0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −80.0000 −0.0252982
\(11\) 1508.00 0.341607 0.170804 0.985305i \(-0.445364\pi\)
0.170804 + 0.985305i \(0.445364\pi\)
\(12\) 1728.00 0.288675
\(13\) 2197.00 0.277350
\(14\) −2744.00 −0.267261
\(15\) −270.000 −0.0206559
\(16\) 4096.00 0.250000
\(17\) −1042.00 −0.0514395 −0.0257197 0.999669i \(-0.508188\pi\)
−0.0257197 + 0.999669i \(0.508188\pi\)
\(18\) 5832.00 0.235702
\(19\) −9068.00 −0.303301 −0.151651 0.988434i \(-0.548459\pi\)
−0.151651 + 0.988434i \(0.548459\pi\)
\(20\) −640.000 −0.0178885
\(21\) −9261.00 −0.218218
\(22\) 12064.0 0.241553
\(23\) −98988.0 −1.69643 −0.848213 0.529655i \(-0.822322\pi\)
−0.848213 + 0.529655i \(0.822322\pi\)
\(24\) 13824.0 0.204124
\(25\) −78025.0 −0.998720
\(26\) 17576.0 0.196116
\(27\) 19683.0 0.192450
\(28\) −21952.0 −0.188982
\(29\) −213642. −1.62665 −0.813324 0.581811i \(-0.802344\pi\)
−0.813324 + 0.581811i \(0.802344\pi\)
\(30\) −2160.00 −0.0146059
\(31\) −22048.0 −0.132924 −0.0664620 0.997789i \(-0.521171\pi\)
−0.0664620 + 0.997789i \(0.521171\pi\)
\(32\) 32768.0 0.176777
\(33\) 40716.0 0.197227
\(34\) −8336.00 −0.0363732
\(35\) 3430.00 0.0135225
\(36\) 46656.0 0.166667
\(37\) 418246. 1.35746 0.678728 0.734390i \(-0.262532\pi\)
0.678728 + 0.734390i \(0.262532\pi\)
\(38\) −72544.0 −0.214466
\(39\) 59319.0 0.160128
\(40\) −5120.00 −0.0126491
\(41\) −76414.0 −0.173153 −0.0865764 0.996245i \(-0.527593\pi\)
−0.0865764 + 0.996245i \(0.527593\pi\)
\(42\) −74088.0 −0.154303
\(43\) −177524. −0.340500 −0.170250 0.985401i \(-0.554458\pi\)
−0.170250 + 0.985401i \(0.554458\pi\)
\(44\) 96512.0 0.170804
\(45\) −7290.00 −0.0119257
\(46\) −791904. −1.19955
\(47\) 631916. 0.887803 0.443902 0.896075i \(-0.353594\pi\)
0.443902 + 0.896075i \(0.353594\pi\)
\(48\) 110592. 0.144338
\(49\) 117649. 0.142857
\(50\) −624200. −0.706202
\(51\) −28134.0 −0.0296986
\(52\) 140608. 0.138675
\(53\) −982354. −0.906364 −0.453182 0.891418i \(-0.649711\pi\)
−0.453182 + 0.891418i \(0.649711\pi\)
\(54\) 157464. 0.136083
\(55\) −15080.0 −0.0122217
\(56\) −175616. −0.133631
\(57\) −244836. −0.175111
\(58\) −1.70914e6 −1.15021
\(59\) −596384. −0.378045 −0.189023 0.981973i \(-0.560532\pi\)
−0.189023 + 0.981973i \(0.560532\pi\)
\(60\) −17280.0 −0.0103280
\(61\) 1.86341e6 1.05112 0.525561 0.850756i \(-0.323855\pi\)
0.525561 + 0.850756i \(0.323855\pi\)
\(62\) −176384. −0.0939914
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) −21970.0 −0.00992278
\(66\) 325728. 0.139461
\(67\) −1.84565e6 −0.749701 −0.374850 0.927085i \(-0.622306\pi\)
−0.374850 + 0.927085i \(0.622306\pi\)
\(68\) −66688.0 −0.0257197
\(69\) −2.67268e6 −0.979432
\(70\) 27440.0 0.00956183
\(71\) 1.63228e6 0.541241 0.270620 0.962686i \(-0.412771\pi\)
0.270620 + 0.962686i \(0.412771\pi\)
\(72\) 373248. 0.117851
\(73\) 216650. 0.0651822 0.0325911 0.999469i \(-0.489624\pi\)
0.0325911 + 0.999469i \(0.489624\pi\)
\(74\) 3.34597e6 0.959866
\(75\) −2.10668e6 −0.576611
\(76\) −580352. −0.151651
\(77\) −517244. −0.129115
\(78\) 474552. 0.113228
\(79\) −6.46017e6 −1.47417 −0.737087 0.675797i \(-0.763799\pi\)
−0.737087 + 0.675797i \(0.763799\pi\)
\(80\) −40960.0 −0.00894427
\(81\) 531441. 0.111111
\(82\) −611312. −0.122437
\(83\) 3.59215e6 0.689575 0.344787 0.938681i \(-0.387951\pi\)
0.344787 + 0.938681i \(0.387951\pi\)
\(84\) −592704. −0.109109
\(85\) 10420.0 0.00184035
\(86\) −1.42019e6 −0.240770
\(87\) −5.76833e6 −0.939146
\(88\) 772096. 0.120776
\(89\) −9.48266e6 −1.42582 −0.712911 0.701255i \(-0.752624\pi\)
−0.712911 + 0.701255i \(0.752624\pi\)
\(90\) −58320.0 −0.00843274
\(91\) −753571. −0.104828
\(92\) −6.33523e6 −0.848213
\(93\) −595296. −0.0767437
\(94\) 5.05533e6 0.627772
\(95\) 90680.0 0.0108512
\(96\) 884736. 0.102062
\(97\) 840226. 0.0934749 0.0467375 0.998907i \(-0.485118\pi\)
0.0467375 + 0.998907i \(0.485118\pi\)
\(98\) 941192. 0.101015
\(99\) 1.09933e6 0.113869
\(100\) −4.99360e6 −0.499360
\(101\) −8.66694e6 −0.837030 −0.418515 0.908210i \(-0.637449\pi\)
−0.418515 + 0.908210i \(0.637449\pi\)
\(102\) −225072. −0.0210001
\(103\) −9.63078e6 −0.868422 −0.434211 0.900811i \(-0.642973\pi\)
−0.434211 + 0.900811i \(0.642973\pi\)
\(104\) 1.12486e6 0.0980581
\(105\) 92610.0 0.00780720
\(106\) −7.85883e6 −0.640896
\(107\) 3.51576e6 0.277444 0.138722 0.990331i \(-0.455700\pi\)
0.138722 + 0.990331i \(0.455700\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 1.15869e6 0.0856991 0.0428495 0.999082i \(-0.486356\pi\)
0.0428495 + 0.999082i \(0.486356\pi\)
\(110\) −120640. −0.00864205
\(111\) 1.12926e7 0.783728
\(112\) −1.40493e6 −0.0944911
\(113\) −7.50697e6 −0.489429 −0.244715 0.969595i \(-0.578694\pi\)
−0.244715 + 0.969595i \(0.578694\pi\)
\(114\) −1.95869e6 −0.123822
\(115\) 989880. 0.0606932
\(116\) −1.36731e7 −0.813324
\(117\) 1.60161e6 0.0924500
\(118\) −4.77107e6 −0.267318
\(119\) 357406. 0.0194423
\(120\) −138240. −0.00730297
\(121\) −1.72131e7 −0.883305
\(122\) 1.49072e7 0.743255
\(123\) −2.06318e6 −0.0999698
\(124\) −1.41107e6 −0.0664620
\(125\) 1.56150e6 0.0715084
\(126\) −2.00038e6 −0.0890871
\(127\) −2.50594e7 −1.08557 −0.542784 0.839872i \(-0.682630\pi\)
−0.542784 + 0.839872i \(0.682630\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −4.79315e6 −0.196588
\(130\) −175760. −0.00701646
\(131\) 1.65072e7 0.641540 0.320770 0.947157i \(-0.396058\pi\)
0.320770 + 0.947157i \(0.396058\pi\)
\(132\) 2.60582e6 0.0986135
\(133\) 3.11032e6 0.114637
\(134\) −1.47652e7 −0.530119
\(135\) −196830. −0.00688530
\(136\) −533504. −0.0181866
\(137\) −2.64726e6 −0.0879578 −0.0439789 0.999032i \(-0.514003\pi\)
−0.0439789 + 0.999032i \(0.514003\pi\)
\(138\) −2.13814e7 −0.692563
\(139\) −4.79145e7 −1.51327 −0.756634 0.653839i \(-0.773157\pi\)
−0.756634 + 0.653839i \(0.773157\pi\)
\(140\) 219520. 0.00676123
\(141\) 1.70617e7 0.512574
\(142\) 1.30582e7 0.382715
\(143\) 3.31308e6 0.0947448
\(144\) 2.98598e6 0.0833333
\(145\) 2.13642e6 0.0581967
\(146\) 1.73320e6 0.0460907
\(147\) 3.17652e6 0.0824786
\(148\) 2.67677e7 0.678728
\(149\) −1.62956e7 −0.403569 −0.201784 0.979430i \(-0.564674\pi\)
−0.201784 + 0.979430i \(0.564674\pi\)
\(150\) −1.68534e7 −0.407726
\(151\) 6.98150e7 1.65017 0.825086 0.565007i \(-0.191126\pi\)
0.825086 + 0.565007i \(0.191126\pi\)
\(152\) −4.64282e6 −0.107233
\(153\) −759618. −0.0171465
\(154\) −4.13795e6 −0.0912983
\(155\) 220480. 0.00475563
\(156\) 3.79642e6 0.0800641
\(157\) −3.05107e7 −0.629221 −0.314611 0.949221i \(-0.601874\pi\)
−0.314611 + 0.949221i \(0.601874\pi\)
\(158\) −5.16813e7 −1.04240
\(159\) −2.65236e7 −0.523289
\(160\) −327680. −0.00632456
\(161\) 3.39529e7 0.641189
\(162\) 4.25153e6 0.0785674
\(163\) 1.03245e6 0.0186730 0.00933648 0.999956i \(-0.497028\pi\)
0.00933648 + 0.999956i \(0.497028\pi\)
\(164\) −4.89050e6 −0.0865764
\(165\) −407160. −0.00705621
\(166\) 2.87372e7 0.487603
\(167\) −5.59050e7 −0.928845 −0.464423 0.885614i \(-0.653738\pi\)
−0.464423 + 0.885614i \(0.653738\pi\)
\(168\) −4.74163e6 −0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 83360.0 0.00130133
\(171\) −6.61057e6 −0.101100
\(172\) −1.13615e7 −0.170250
\(173\) −4.10563e7 −0.602863 −0.301431 0.953488i \(-0.597464\pi\)
−0.301431 + 0.953488i \(0.597464\pi\)
\(174\) −4.61467e7 −0.664076
\(175\) 2.67626e7 0.377481
\(176\) 6.17677e6 0.0854018
\(177\) −1.61024e7 −0.218265
\(178\) −7.58613e7 −1.00821
\(179\) −7.65930e7 −0.998167 −0.499083 0.866554i \(-0.666330\pi\)
−0.499083 + 0.866554i \(0.666330\pi\)
\(180\) −466560. −0.00596285
\(181\) −7.49751e7 −0.939815 −0.469907 0.882716i \(-0.655713\pi\)
−0.469907 + 0.882716i \(0.655713\pi\)
\(182\) −6.02857e6 −0.0741249
\(183\) 5.03120e7 0.606865
\(184\) −5.06819e7 −0.599777
\(185\) −4.18246e6 −0.0485658
\(186\) −4.76237e6 −0.0542660
\(187\) −1.57134e6 −0.0175721
\(188\) 4.04426e7 0.443902
\(189\) −6.75127e6 −0.0727393
\(190\) 725440. 0.00767298
\(191\) 1.50516e8 1.56303 0.781514 0.623888i \(-0.214448\pi\)
0.781514 + 0.623888i \(0.214448\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 1.26286e8 1.26445 0.632227 0.774783i \(-0.282141\pi\)
0.632227 + 0.774783i \(0.282141\pi\)
\(194\) 6.72181e6 0.0660967
\(195\) −593190. −0.00572892
\(196\) 7.52954e6 0.0714286
\(197\) −8.50143e7 −0.792246 −0.396123 0.918197i \(-0.629645\pi\)
−0.396123 + 0.918197i \(0.629645\pi\)
\(198\) 8.79466e6 0.0805176
\(199\) −7.77938e7 −0.699776 −0.349888 0.936792i \(-0.613780\pi\)
−0.349888 + 0.936792i \(0.613780\pi\)
\(200\) −3.99488e7 −0.353101
\(201\) −4.98326e7 −0.432840
\(202\) −6.93355e7 −0.591870
\(203\) 7.32792e7 0.614815
\(204\) −1.80058e6 −0.0148493
\(205\) 764140. 0.00619490
\(206\) −7.70462e7 −0.614067
\(207\) −7.21623e7 −0.565476
\(208\) 8.99891e6 0.0693375
\(209\) −1.36745e7 −0.103610
\(210\) 740880. 0.00552052
\(211\) 1.51198e8 1.10805 0.554023 0.832501i \(-0.313092\pi\)
0.554023 + 0.832501i \(0.313092\pi\)
\(212\) −6.28707e7 −0.453182
\(213\) 4.40716e7 0.312485
\(214\) 2.81261e7 0.196183
\(215\) 1.77524e6 0.0121821
\(216\) 1.00777e7 0.0680414
\(217\) 7.56246e6 0.0502405
\(218\) 9.26955e6 0.0605984
\(219\) 5.84955e6 0.0376329
\(220\) −965120. −0.00611085
\(221\) −2.28927e6 −0.0142667
\(222\) 9.03411e7 0.554179
\(223\) −9.82783e7 −0.593459 −0.296729 0.954962i \(-0.595896\pi\)
−0.296729 + 0.954962i \(0.595896\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −5.68802e7 −0.332907
\(226\) −6.00557e7 −0.346079
\(227\) −1.80043e8 −1.02161 −0.510806 0.859696i \(-0.670653\pi\)
−0.510806 + 0.859696i \(0.670653\pi\)
\(228\) −1.56695e7 −0.0875555
\(229\) 3.10223e7 0.170707 0.0853533 0.996351i \(-0.472798\pi\)
0.0853533 + 0.996351i \(0.472798\pi\)
\(230\) 7.91904e6 0.0429166
\(231\) −1.39656e7 −0.0745448
\(232\) −1.09385e8 −0.575107
\(233\) −8.98629e7 −0.465409 −0.232704 0.972548i \(-0.574757\pi\)
−0.232704 + 0.972548i \(0.574757\pi\)
\(234\) 1.28129e7 0.0653720
\(235\) −6.31916e6 −0.0317630
\(236\) −3.81686e7 −0.189023
\(237\) −1.74425e8 −0.851115
\(238\) 2.85925e6 0.0137478
\(239\) 5.22535e6 0.0247584 0.0123792 0.999923i \(-0.496059\pi\)
0.0123792 + 0.999923i \(0.496059\pi\)
\(240\) −1.10592e6 −0.00516398
\(241\) 4.35619e7 0.200469 0.100234 0.994964i \(-0.468041\pi\)
0.100234 + 0.994964i \(0.468041\pi\)
\(242\) −1.37705e8 −0.624591
\(243\) 1.43489e7 0.0641500
\(244\) 1.19258e8 0.525561
\(245\) −1.17649e6 −0.00511101
\(246\) −1.65054e7 −0.0706893
\(247\) −1.99224e7 −0.0841206
\(248\) −1.12886e7 −0.0469957
\(249\) 9.69881e7 0.398126
\(250\) 1.24920e7 0.0505641
\(251\) 3.21053e8 1.28150 0.640750 0.767750i \(-0.278624\pi\)
0.640750 + 0.767750i \(0.278624\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) −1.49274e8 −0.579512
\(254\) −2.00475e8 −0.767613
\(255\) 281340. 0.00106253
\(256\) 1.67772e7 0.0625000
\(257\) 5.27063e7 0.193685 0.0968427 0.995300i \(-0.469126\pi\)
0.0968427 + 0.995300i \(0.469126\pi\)
\(258\) −3.83452e7 −0.139009
\(259\) −1.43458e8 −0.513070
\(260\) −1.40608e6 −0.00496139
\(261\) −1.55745e8 −0.542216
\(262\) 1.32058e8 0.453638
\(263\) −4.90103e8 −1.66128 −0.830638 0.556812i \(-0.812024\pi\)
−0.830638 + 0.556812i \(0.812024\pi\)
\(264\) 2.08466e7 0.0697303
\(265\) 9.82354e6 0.0324271
\(266\) 2.48826e7 0.0810606
\(267\) −2.56032e8 −0.823199
\(268\) −1.18122e8 −0.374850
\(269\) 3.43105e8 1.07472 0.537358 0.843354i \(-0.319422\pi\)
0.537358 + 0.843354i \(0.319422\pi\)
\(270\) −1.57464e6 −0.00486864
\(271\) 1.83046e8 0.558686 0.279343 0.960191i \(-0.409883\pi\)
0.279343 + 0.960191i \(0.409883\pi\)
\(272\) −4.26803e6 −0.0128599
\(273\) −2.03464e7 −0.0605228
\(274\) −2.11781e7 −0.0621955
\(275\) −1.17662e8 −0.341170
\(276\) −1.71051e8 −0.489716
\(277\) 3.26261e8 0.922328 0.461164 0.887315i \(-0.347432\pi\)
0.461164 + 0.887315i \(0.347432\pi\)
\(278\) −3.83316e8 −1.07004
\(279\) −1.60730e7 −0.0443080
\(280\) 1.75616e6 0.00478091
\(281\) −3.37042e8 −0.906173 −0.453087 0.891466i \(-0.649677\pi\)
−0.453087 + 0.891466i \(0.649677\pi\)
\(282\) 1.36494e8 0.362444
\(283\) 3.08528e8 0.809175 0.404587 0.914499i \(-0.367415\pi\)
0.404587 + 0.914499i \(0.367415\pi\)
\(284\) 1.04466e8 0.270620
\(285\) 2.44836e6 0.00626496
\(286\) 2.65046e7 0.0669947
\(287\) 2.62100e7 0.0654456
\(288\) 2.38879e7 0.0589256
\(289\) −4.09253e8 −0.997354
\(290\) 1.70914e7 0.0411513
\(291\) 2.26861e7 0.0539678
\(292\) 1.38656e7 0.0325911
\(293\) 4.83962e8 1.12402 0.562011 0.827130i \(-0.310028\pi\)
0.562011 + 0.827130i \(0.310028\pi\)
\(294\) 2.54122e7 0.0583212
\(295\) 5.96384e6 0.0135254
\(296\) 2.14142e8 0.479933
\(297\) 2.96820e7 0.0657423
\(298\) −1.30365e8 −0.285366
\(299\) −2.17477e8 −0.470504
\(300\) −1.34827e8 −0.288306
\(301\) 6.08907e7 0.128697
\(302\) 5.58520e8 1.16685
\(303\) −2.34007e8 −0.483260
\(304\) −3.71425e7 −0.0758253
\(305\) −1.86341e7 −0.0376061
\(306\) −6.07694e6 −0.0121244
\(307\) −6.16455e8 −1.21595 −0.607977 0.793955i \(-0.708019\pi\)
−0.607977 + 0.793955i \(0.708019\pi\)
\(308\) −3.31036e7 −0.0645577
\(309\) −2.60031e8 −0.501384
\(310\) 1.76384e6 0.00336274
\(311\) 4.63646e7 0.0874028 0.0437014 0.999045i \(-0.486085\pi\)
0.0437014 + 0.999045i \(0.486085\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) −8.55946e8 −1.57776 −0.788881 0.614546i \(-0.789339\pi\)
−0.788881 + 0.614546i \(0.789339\pi\)
\(314\) −2.44086e8 −0.444927
\(315\) 2.50047e6 0.00450749
\(316\) −4.13451e8 −0.737087
\(317\) 3.66851e8 0.646819 0.323409 0.946259i \(-0.395171\pi\)
0.323409 + 0.946259i \(0.395171\pi\)
\(318\) −2.12188e8 −0.370021
\(319\) −3.22172e8 −0.555675
\(320\) −2.62144e6 −0.00447214
\(321\) 9.49255e7 0.160183
\(322\) 2.71623e8 0.453389
\(323\) 9.44886e6 0.0156016
\(324\) 3.40122e7 0.0555556
\(325\) −1.71421e8 −0.276995
\(326\) 8.25962e6 0.0132038
\(327\) 3.12847e7 0.0494784
\(328\) −3.91240e7 −0.0612187
\(329\) −2.16747e8 −0.335558
\(330\) −3.25728e6 −0.00498949
\(331\) 3.26194e8 0.494399 0.247200 0.968965i \(-0.420490\pi\)
0.247200 + 0.968965i \(0.420490\pi\)
\(332\) 2.29898e8 0.344787
\(333\) 3.04901e8 0.452485
\(334\) −4.47240e8 −0.656793
\(335\) 1.84565e7 0.0268221
\(336\) −3.79331e7 −0.0545545
\(337\) 7.22325e8 1.02808 0.514042 0.857765i \(-0.328148\pi\)
0.514042 + 0.857765i \(0.328148\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) −2.02688e8 −0.282572
\(340\) 666880. 0.000920177 0
\(341\) −3.32484e7 −0.0454078
\(342\) −5.28846e7 −0.0714887
\(343\) −4.03536e7 −0.0539949
\(344\) −9.08923e7 −0.120385
\(345\) 2.67268e7 0.0350412
\(346\) −3.28450e8 −0.426288
\(347\) 4.03337e8 0.518220 0.259110 0.965848i \(-0.416571\pi\)
0.259110 + 0.965848i \(0.416571\pi\)
\(348\) −3.69173e8 −0.469573
\(349\) 6.62276e8 0.833969 0.416985 0.908914i \(-0.363087\pi\)
0.416985 + 0.908914i \(0.363087\pi\)
\(350\) 2.14101e8 0.266919
\(351\) 4.32436e7 0.0533761
\(352\) 4.94141e7 0.0603882
\(353\) −1.33432e8 −0.161453 −0.0807267 0.996736i \(-0.525724\pi\)
−0.0807267 + 0.996736i \(0.525724\pi\)
\(354\) −1.28819e8 −0.154336
\(355\) −1.63228e7 −0.0193640
\(356\) −6.06890e8 −0.712911
\(357\) 9.64996e6 0.0112250
\(358\) −6.12744e8 −0.705811
\(359\) −3.89725e8 −0.444557 −0.222279 0.974983i \(-0.571349\pi\)
−0.222279 + 0.974983i \(0.571349\pi\)
\(360\) −3.73248e6 −0.00421637
\(361\) −8.11643e8 −0.908008
\(362\) −5.99801e8 −0.664549
\(363\) −4.64754e8 −0.509976
\(364\) −4.82285e7 −0.0524142
\(365\) −2.16650e6 −0.00233203
\(366\) 4.02496e8 0.429119
\(367\) 7.13690e8 0.753666 0.376833 0.926281i \(-0.377013\pi\)
0.376833 + 0.926281i \(0.377013\pi\)
\(368\) −4.05455e8 −0.424107
\(369\) −5.57058e7 −0.0577176
\(370\) −3.34597e7 −0.0343412
\(371\) 3.36947e8 0.342573
\(372\) −3.80989e7 −0.0383718
\(373\) −8.19941e8 −0.818091 −0.409046 0.912514i \(-0.634138\pi\)
−0.409046 + 0.912514i \(0.634138\pi\)
\(374\) −1.25707e7 −0.0124253
\(375\) 4.21605e7 0.0412854
\(376\) 3.23541e8 0.313886
\(377\) −4.69371e8 −0.451151
\(378\) −5.40102e7 −0.0514344
\(379\) 3.57866e8 0.337663 0.168831 0.985645i \(-0.446001\pi\)
0.168831 + 0.985645i \(0.446001\pi\)
\(380\) 5.80352e6 0.00542561
\(381\) −6.76603e8 −0.626753
\(382\) 1.20413e9 1.10523
\(383\) 8.40713e8 0.764632 0.382316 0.924032i \(-0.375127\pi\)
0.382316 + 0.924032i \(0.375127\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 5.17244e6 0.00461937
\(386\) 1.01029e9 0.894105
\(387\) −1.29415e8 −0.113500
\(388\) 5.37745e7 0.0467375
\(389\) 1.30488e9 1.12395 0.561975 0.827154i \(-0.310042\pi\)
0.561975 + 0.827154i \(0.310042\pi\)
\(390\) −4.74552e6 −0.00405096
\(391\) 1.03145e8 0.0872633
\(392\) 6.02363e7 0.0505076
\(393\) 4.45695e8 0.370393
\(394\) −6.80115e8 −0.560203
\(395\) 6.46017e7 0.0527417
\(396\) 7.03572e7 0.0569345
\(397\) −1.07773e9 −0.864458 −0.432229 0.901764i \(-0.642273\pi\)
−0.432229 + 0.901764i \(0.642273\pi\)
\(398\) −6.22350e8 −0.494816
\(399\) 8.39787e7 0.0661857
\(400\) −3.19590e8 −0.249680
\(401\) −1.80045e9 −1.39436 −0.697181 0.716895i \(-0.745563\pi\)
−0.697181 + 0.716895i \(0.745563\pi\)
\(402\) −3.98661e8 −0.306064
\(403\) −4.84395e7 −0.0368665
\(404\) −5.54684e8 −0.418515
\(405\) −5.31441e6 −0.00397523
\(406\) 5.86234e8 0.434740
\(407\) 6.30715e8 0.463717
\(408\) −1.44046e7 −0.0105000
\(409\) −2.50445e9 −1.81001 −0.905004 0.425402i \(-0.860133\pi\)
−0.905004 + 0.425402i \(0.860133\pi\)
\(410\) 6.11312e6 0.00438046
\(411\) −7.14760e7 −0.0507825
\(412\) −6.16370e8 −0.434211
\(413\) 2.04560e8 0.142888
\(414\) −5.77298e8 −0.399852
\(415\) −3.59215e7 −0.0246710
\(416\) 7.19913e7 0.0490290
\(417\) −1.29369e9 −0.873685
\(418\) −1.09396e8 −0.0732632
\(419\) −4.49332e8 −0.298414 −0.149207 0.988806i \(-0.547672\pi\)
−0.149207 + 0.988806i \(0.547672\pi\)
\(420\) 5.92704e6 0.00390360
\(421\) 7.67923e8 0.501569 0.250784 0.968043i \(-0.419311\pi\)
0.250784 + 0.968043i \(0.419311\pi\)
\(422\) 1.20959e9 0.783507
\(423\) 4.60667e8 0.295934
\(424\) −5.02965e8 −0.320448
\(425\) 8.13020e7 0.0513736
\(426\) 3.52572e8 0.220961
\(427\) −6.39148e8 −0.397287
\(428\) 2.25009e8 0.138722
\(429\) 8.94531e7 0.0547009
\(430\) 1.42019e7 0.00861405
\(431\) −7.03828e8 −0.423444 −0.211722 0.977330i \(-0.567907\pi\)
−0.211722 + 0.977330i \(0.567907\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.67590e9 0.992063 0.496032 0.868304i \(-0.334790\pi\)
0.496032 + 0.868304i \(0.334790\pi\)
\(434\) 6.04997e7 0.0355254
\(435\) 5.76833e7 0.0335999
\(436\) 7.41564e7 0.0428495
\(437\) 8.97623e8 0.514528
\(438\) 4.67964e7 0.0266105
\(439\) 2.12336e9 1.19784 0.598918 0.800811i \(-0.295598\pi\)
0.598918 + 0.800811i \(0.295598\pi\)
\(440\) −7.72096e6 −0.00432103
\(441\) 8.57661e7 0.0476190
\(442\) −1.83142e7 −0.0100881
\(443\) 1.19425e9 0.652654 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(444\) 7.22729e8 0.391864
\(445\) 9.48266e7 0.0510118
\(446\) −7.86226e8 −0.419639
\(447\) −4.39980e8 −0.233001
\(448\) −8.99154e7 −0.0472456
\(449\) 1.62441e9 0.846903 0.423451 0.905919i \(-0.360818\pi\)
0.423451 + 0.905919i \(0.360818\pi\)
\(450\) −4.55042e8 −0.235401
\(451\) −1.15232e8 −0.0591502
\(452\) −4.80446e8 −0.244715
\(453\) 1.88500e9 0.952728
\(454\) −1.44034e9 −0.722388
\(455\) 7.53571e6 0.00375046
\(456\) −1.25356e8 −0.0619111
\(457\) 1.96761e8 0.0964343 0.0482172 0.998837i \(-0.484646\pi\)
0.0482172 + 0.998837i \(0.484646\pi\)
\(458\) 2.48179e8 0.120708
\(459\) −2.05097e7 −0.00989953
\(460\) 6.33523e7 0.0303466
\(461\) 1.32676e9 0.630722 0.315361 0.948972i \(-0.397874\pi\)
0.315361 + 0.948972i \(0.397874\pi\)
\(462\) −1.11725e8 −0.0527111
\(463\) 2.87807e9 1.34762 0.673810 0.738904i \(-0.264656\pi\)
0.673810 + 0.738904i \(0.264656\pi\)
\(464\) −8.75078e8 −0.406662
\(465\) 5.95296e6 0.00274567
\(466\) −7.18903e8 −0.329094
\(467\) 2.01274e9 0.914487 0.457244 0.889342i \(-0.348837\pi\)
0.457244 + 0.889342i \(0.348837\pi\)
\(468\) 1.02503e8 0.0462250
\(469\) 6.33059e8 0.283360
\(470\) −5.05533e7 −0.0224598
\(471\) −8.23789e8 −0.363281
\(472\) −3.05349e8 −0.133659
\(473\) −2.67706e8 −0.116317
\(474\) −1.39540e9 −0.601829
\(475\) 7.07531e8 0.302913
\(476\) 2.28740e7 0.00972115
\(477\) −7.16136e8 −0.302121
\(478\) 4.18028e7 0.0175068
\(479\) −1.72294e9 −0.716302 −0.358151 0.933664i \(-0.616593\pi\)
−0.358151 + 0.933664i \(0.616593\pi\)
\(480\) −8.84736e6 −0.00365148
\(481\) 9.18886e8 0.376491
\(482\) 3.48495e8 0.141753
\(483\) 9.16728e8 0.370191
\(484\) −1.10164e9 −0.441652
\(485\) −8.40226e6 −0.00334426
\(486\) 1.14791e8 0.0453609
\(487\) −3.30049e9 −1.29487 −0.647437 0.762119i \(-0.724159\pi\)
−0.647437 + 0.762119i \(0.724159\pi\)
\(488\) 9.54064e8 0.371628
\(489\) 2.78762e7 0.0107808
\(490\) −9.41192e6 −0.00361403
\(491\) −2.46868e8 −0.0941194 −0.0470597 0.998892i \(-0.514985\pi\)
−0.0470597 + 0.998892i \(0.514985\pi\)
\(492\) −1.32043e8 −0.0499849
\(493\) 2.22615e8 0.0836739
\(494\) −1.59379e8 −0.0594822
\(495\) −1.09933e7 −0.00407390
\(496\) −9.03086e7 −0.0332310
\(497\) −5.59872e8 −0.204570
\(498\) 7.75905e8 0.281518
\(499\) 3.62691e8 0.130673 0.0653364 0.997863i \(-0.479188\pi\)
0.0653364 + 0.997863i \(0.479188\pi\)
\(500\) 9.99360e7 0.0357542
\(501\) −1.50944e9 −0.536269
\(502\) 2.56842e9 0.906157
\(503\) −1.48371e9 −0.519829 −0.259914 0.965632i \(-0.583694\pi\)
−0.259914 + 0.965632i \(0.583694\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 8.66694e7 0.0299465
\(506\) −1.19419e9 −0.409777
\(507\) 1.30324e8 0.0444116
\(508\) −1.60380e9 −0.542784
\(509\) −4.96412e8 −0.166851 −0.0834257 0.996514i \(-0.526586\pi\)
−0.0834257 + 0.996514i \(0.526586\pi\)
\(510\) 2.25072e6 0.000751322 0
\(511\) −7.43110e7 −0.0246365
\(512\) 1.34218e8 0.0441942
\(513\) −1.78485e8 −0.0583703
\(514\) 4.21651e8 0.136956
\(515\) 9.63078e7 0.0310696
\(516\) −3.06761e8 −0.0982940
\(517\) 9.52929e8 0.303280
\(518\) −1.14767e9 −0.362795
\(519\) −1.10852e9 −0.348063
\(520\) −1.12486e7 −0.00350823
\(521\) 9.52711e8 0.295141 0.147570 0.989052i \(-0.452855\pi\)
0.147570 + 0.989052i \(0.452855\pi\)
\(522\) −1.24596e9 −0.383405
\(523\) −4.02099e9 −1.22907 −0.614535 0.788890i \(-0.710656\pi\)
−0.614535 + 0.788890i \(0.710656\pi\)
\(524\) 1.05646e9 0.320770
\(525\) 7.22590e8 0.217939
\(526\) −3.92082e9 −1.17470
\(527\) 2.29740e7 0.00683754
\(528\) 1.66773e8 0.0493067
\(529\) 6.39380e9 1.87786
\(530\) 7.85883e7 0.0229294
\(531\) −4.34764e8 −0.126015
\(532\) 1.99061e8 0.0573185
\(533\) −1.67882e8 −0.0480239
\(534\) −2.04825e9 −0.582089
\(535\) −3.51576e7 −0.00992615
\(536\) −9.44974e8 −0.265059
\(537\) −2.06801e9 −0.576292
\(538\) 2.74484e9 0.759939
\(539\) 1.77415e8 0.0488010
\(540\) −1.25971e7 −0.00344265
\(541\) 1.54687e9 0.420013 0.210006 0.977700i \(-0.432652\pi\)
0.210006 + 0.977700i \(0.432652\pi\)
\(542\) 1.46437e9 0.395051
\(543\) −2.02433e9 −0.542602
\(544\) −3.41443e7 −0.00909330
\(545\) −1.15869e7 −0.00306606
\(546\) −1.62771e8 −0.0427960
\(547\) −2.54655e9 −0.665267 −0.332634 0.943056i \(-0.607937\pi\)
−0.332634 + 0.943056i \(0.607937\pi\)
\(548\) −1.69425e8 −0.0439789
\(549\) 1.35842e9 0.350374
\(550\) −9.41294e8 −0.241244
\(551\) 1.93731e9 0.493364
\(552\) −1.36841e9 −0.346282
\(553\) 2.21584e9 0.557186
\(554\) 2.61008e9 0.652184
\(555\) −1.12926e8 −0.0280395
\(556\) −3.06653e9 −0.756634
\(557\) 6.56728e9 1.61025 0.805123 0.593107i \(-0.202099\pi\)
0.805123 + 0.593107i \(0.202099\pi\)
\(558\) −1.28584e8 −0.0313305
\(559\) −3.90020e8 −0.0944378
\(560\) 1.40493e7 0.00338062
\(561\) −4.24261e7 −0.0101453
\(562\) −2.69633e9 −0.640761
\(563\) −1.05889e9 −0.250075 −0.125038 0.992152i \(-0.539905\pi\)
−0.125038 + 0.992152i \(0.539905\pi\)
\(564\) 1.09195e9 0.256287
\(565\) 7.50697e7 0.0175104
\(566\) 2.46823e9 0.572173
\(567\) −1.82284e8 −0.0419961
\(568\) 8.35727e8 0.191357
\(569\) 3.21980e9 0.732717 0.366358 0.930474i \(-0.380604\pi\)
0.366358 + 0.930474i \(0.380604\pi\)
\(570\) 1.95869e7 0.00443000
\(571\) −1.98851e9 −0.446995 −0.223497 0.974705i \(-0.571747\pi\)
−0.223497 + 0.974705i \(0.571747\pi\)
\(572\) 2.12037e8 0.0473724
\(573\) 4.06394e9 0.902415
\(574\) 2.09680e8 0.0462770
\(575\) 7.72354e9 1.69426
\(576\) 1.91103e8 0.0416667
\(577\) 4.08039e9 0.884272 0.442136 0.896948i \(-0.354221\pi\)
0.442136 + 0.896948i \(0.354221\pi\)
\(578\) −3.27402e9 −0.705236
\(579\) 3.40971e9 0.730033
\(580\) 1.36731e8 0.0290984
\(581\) −1.23211e9 −0.260635
\(582\) 1.81489e8 0.0381610
\(583\) −1.48139e9 −0.309620
\(584\) 1.10925e8 0.0230454
\(585\) −1.60161e7 −0.00330759
\(586\) 3.87170e9 0.794803
\(587\) −2.96773e9 −0.605606 −0.302803 0.953053i \(-0.597922\pi\)
−0.302803 + 0.953053i \(0.597922\pi\)
\(588\) 2.03297e8 0.0412393
\(589\) 1.99931e8 0.0403160
\(590\) 4.77107e7 0.00956388
\(591\) −2.29539e9 −0.457404
\(592\) 1.71314e9 0.339364
\(593\) −8.04326e9 −1.58395 −0.791973 0.610556i \(-0.790946\pi\)
−0.791973 + 0.610556i \(0.790946\pi\)
\(594\) 2.37456e8 0.0464868
\(595\) −3.57406e6 −0.000695589 0
\(596\) −1.04292e9 −0.201784
\(597\) −2.10043e9 −0.404016
\(598\) −1.73981e9 −0.332697
\(599\) 4.02244e9 0.764708 0.382354 0.924016i \(-0.375113\pi\)
0.382354 + 0.924016i \(0.375113\pi\)
\(600\) −1.07862e9 −0.203863
\(601\) 8.88903e9 1.67030 0.835149 0.550025i \(-0.185382\pi\)
0.835149 + 0.550025i \(0.185382\pi\)
\(602\) 4.87126e8 0.0910025
\(603\) −1.34548e9 −0.249900
\(604\) 4.46816e9 0.825086
\(605\) 1.72131e8 0.0316021
\(606\) −1.87206e9 −0.341716
\(607\) −1.99678e9 −0.362384 −0.181192 0.983448i \(-0.557996\pi\)
−0.181192 + 0.983448i \(0.557996\pi\)
\(608\) −2.97140e8 −0.0536166
\(609\) 1.97854e9 0.354964
\(610\) −1.49072e8 −0.0265915
\(611\) 1.38832e9 0.246232
\(612\) −4.86156e7 −0.00857325
\(613\) 2.57276e9 0.451115 0.225557 0.974230i \(-0.427580\pi\)
0.225557 + 0.974230i \(0.427580\pi\)
\(614\) −4.93164e9 −0.859809
\(615\) 2.06318e7 0.00357663
\(616\) −2.64829e8 −0.0456492
\(617\) −2.39839e9 −0.411076 −0.205538 0.978649i \(-0.565894\pi\)
−0.205538 + 0.978649i \(0.565894\pi\)
\(618\) −2.08025e9 −0.354532
\(619\) 7.51379e9 1.27333 0.636666 0.771140i \(-0.280313\pi\)
0.636666 + 0.771140i \(0.280313\pi\)
\(620\) 1.41107e7 0.00237782
\(621\) −1.94838e9 −0.326477
\(622\) 3.70917e8 0.0618031
\(623\) 3.25255e9 0.538910
\(624\) 2.42971e8 0.0400320
\(625\) 6.08009e9 0.996162
\(626\) −6.84757e9 −1.11565
\(627\) −3.69213e8 −0.0598191
\(628\) −1.95269e9 −0.314611
\(629\) −4.35812e8 −0.0698268
\(630\) 2.00038e7 0.00318728
\(631\) 1.12189e10 1.77766 0.888828 0.458241i \(-0.151520\pi\)
0.888828 + 0.458241i \(0.151520\pi\)
\(632\) −3.30761e9 −0.521200
\(633\) 4.08235e9 0.639731
\(634\) 2.93481e9 0.457370
\(635\) 2.50594e8 0.0388385
\(636\) −1.69751e9 −0.261645
\(637\) 2.58475e8 0.0396214
\(638\) −2.57738e9 −0.392921
\(639\) 1.18993e9 0.180414
\(640\) −2.09715e7 −0.00316228
\(641\) −4.73617e9 −0.710271 −0.355135 0.934815i \(-0.615565\pi\)
−0.355135 + 0.934815i \(0.615565\pi\)
\(642\) 7.59404e8 0.113266
\(643\) 2.70605e9 0.401418 0.200709 0.979651i \(-0.435675\pi\)
0.200709 + 0.979651i \(0.435675\pi\)
\(644\) 2.17298e9 0.320595
\(645\) 4.79315e7 0.00703334
\(646\) 7.55908e7 0.0110320
\(647\) 1.87404e9 0.272028 0.136014 0.990707i \(-0.456571\pi\)
0.136014 + 0.990707i \(0.456571\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) −8.99347e8 −0.129143
\(650\) −1.37137e9 −0.195865
\(651\) 2.04187e8 0.0290064
\(652\) 6.60769e7 0.00933648
\(653\) 4.27498e9 0.600811 0.300406 0.953812i \(-0.402878\pi\)
0.300406 + 0.953812i \(0.402878\pi\)
\(654\) 2.50278e8 0.0349865
\(655\) −1.65072e8 −0.0229524
\(656\) −3.12992e8 −0.0432882
\(657\) 1.57938e8 0.0217274
\(658\) −1.73398e9 −0.237275
\(659\) 5.27465e9 0.717951 0.358975 0.933347i \(-0.383126\pi\)
0.358975 + 0.933347i \(0.383126\pi\)
\(660\) −2.60582e7 −0.00352810
\(661\) 8.72606e9 1.17520 0.587602 0.809150i \(-0.300072\pi\)
0.587602 + 0.809150i \(0.300072\pi\)
\(662\) 2.60955e9 0.349593
\(663\) −6.18104e7 −0.00823691
\(664\) 1.83918e9 0.243802
\(665\) −3.11032e7 −0.00410138
\(666\) 2.43921e9 0.319955
\(667\) 2.11480e10 2.75949
\(668\) −3.57792e9 −0.464423
\(669\) −2.65351e9 −0.342634
\(670\) 1.47652e8 0.0189661
\(671\) 2.81002e9 0.359071
\(672\) −3.03464e8 −0.0385758
\(673\) 1.36345e10 1.72420 0.862098 0.506742i \(-0.169150\pi\)
0.862098 + 0.506742i \(0.169150\pi\)
\(674\) 5.77860e9 0.726965
\(675\) −1.53577e9 −0.192204
\(676\) 3.08916e8 0.0384615
\(677\) 4.89007e9 0.605696 0.302848 0.953039i \(-0.402063\pi\)
0.302848 + 0.953039i \(0.402063\pi\)
\(678\) −1.62150e9 −0.199809
\(679\) −2.88198e8 −0.0353302
\(680\) 5.33504e6 0.000650664 0
\(681\) −4.86116e9 −0.589828
\(682\) −2.65987e8 −0.0321081
\(683\) −7.61689e9 −0.914756 −0.457378 0.889272i \(-0.651211\pi\)
−0.457378 + 0.889272i \(0.651211\pi\)
\(684\) −4.23077e8 −0.0505502
\(685\) 2.64726e7 0.00314687
\(686\) −3.22829e8 −0.0381802
\(687\) 8.37603e8 0.0985575
\(688\) −7.27138e8 −0.0851251
\(689\) −2.15823e9 −0.251380
\(690\) 2.13814e8 0.0247779
\(691\) 1.12162e10 1.29322 0.646611 0.762820i \(-0.276186\pi\)
0.646611 + 0.762820i \(0.276186\pi\)
\(692\) −2.62760e9 −0.301431
\(693\) −3.77071e8 −0.0430385
\(694\) 3.22669e9 0.366437
\(695\) 4.79145e8 0.0541403
\(696\) −2.95339e9 −0.332038
\(697\) 7.96234e7 0.00890689
\(698\) 5.29821e9 0.589705
\(699\) −2.42630e9 −0.268704
\(700\) 1.71280e9 0.188740
\(701\) −8.80671e8 −0.0965607 −0.0482804 0.998834i \(-0.515374\pi\)
−0.0482804 + 0.998834i \(0.515374\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) −3.79265e9 −0.411718
\(704\) 3.95313e8 0.0427009
\(705\) −1.70617e8 −0.0183384
\(706\) −1.06745e9 −0.114165
\(707\) 2.97276e9 0.316368
\(708\) −1.03055e9 −0.109132
\(709\) −6.35616e9 −0.669781 −0.334890 0.942257i \(-0.608699\pi\)
−0.334890 + 0.942257i \(0.608699\pi\)
\(710\) −1.30582e8 −0.0136924
\(711\) −4.70946e9 −0.491392
\(712\) −4.85512e9 −0.504104
\(713\) 2.18249e9 0.225496
\(714\) 7.71997e7 0.00793728
\(715\) −3.31308e7 −0.00338969
\(716\) −4.90195e9 −0.499083
\(717\) 1.41085e8 0.0142943
\(718\) −3.11780e9 −0.314349
\(719\) 3.57315e9 0.358509 0.179255 0.983803i \(-0.442631\pi\)
0.179255 + 0.983803i \(0.442631\pi\)
\(720\) −2.98598e7 −0.00298142
\(721\) 3.30336e9 0.328233
\(722\) −6.49314e9 −0.642059
\(723\) 1.17617e9 0.115741
\(724\) −4.79841e9 −0.469907
\(725\) 1.66694e10 1.62457
\(726\) −3.71803e9 −0.360608
\(727\) −6.46393e9 −0.623916 −0.311958 0.950096i \(-0.600985\pi\)
−0.311958 + 0.950096i \(0.600985\pi\)
\(728\) −3.85828e8 −0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −1.73320e7 −0.00164899
\(731\) 1.84980e8 0.0175152
\(732\) 3.21997e9 0.303433
\(733\) 5.70905e9 0.535426 0.267713 0.963499i \(-0.413732\pi\)
0.267713 + 0.963499i \(0.413732\pi\)
\(734\) 5.70952e9 0.532922
\(735\) −3.17652e7 −0.00295084
\(736\) −3.24364e9 −0.299889
\(737\) −2.78324e9 −0.256103
\(738\) −4.45646e8 −0.0408125
\(739\) −1.05430e10 −0.960966 −0.480483 0.877004i \(-0.659539\pi\)
−0.480483 + 0.877004i \(0.659539\pi\)
\(740\) −2.67677e8 −0.0242829
\(741\) −5.37905e8 −0.0485670
\(742\) 2.69558e9 0.242236
\(743\) −1.55085e9 −0.138710 −0.0693551 0.997592i \(-0.522094\pi\)
−0.0693551 + 0.997592i \(0.522094\pi\)
\(744\) −3.04792e8 −0.0271330
\(745\) 1.62956e8 0.0144385
\(746\) −6.55953e9 −0.578478
\(747\) 2.61868e9 0.229858
\(748\) −1.00566e8 −0.00878605
\(749\) −1.20591e9 −0.104864
\(750\) 3.37284e8 0.0291932
\(751\) −9.50917e8 −0.0819224 −0.0409612 0.999161i \(-0.513042\pi\)
−0.0409612 + 0.999161i \(0.513042\pi\)
\(752\) 2.58833e9 0.221951
\(753\) 8.66842e9 0.739874
\(754\) −3.75497e9 −0.319012
\(755\) −6.98150e8 −0.0590384
\(756\) −4.32081e8 −0.0363696
\(757\) 1.00840e10 0.844887 0.422443 0.906389i \(-0.361173\pi\)
0.422443 + 0.906389i \(0.361173\pi\)
\(758\) 2.86293e9 0.238763
\(759\) −4.03040e9 −0.334581
\(760\) 4.64282e7 0.00383649
\(761\) −7.21075e9 −0.593109 −0.296554 0.955016i \(-0.595838\pi\)
−0.296554 + 0.955016i \(0.595838\pi\)
\(762\) −5.41283e9 −0.443182
\(763\) −3.97432e8 −0.0323912
\(764\) 9.63304e9 0.781514
\(765\) 7.59618e6 0.000613451 0
\(766\) 6.72571e9 0.540676
\(767\) −1.31026e9 −0.104851
\(768\) 4.52985e8 0.0360844
\(769\) 1.73356e10 1.37467 0.687334 0.726341i \(-0.258781\pi\)
0.687334 + 0.726341i \(0.258781\pi\)
\(770\) 4.13795e7 0.00326639
\(771\) 1.42307e9 0.111824
\(772\) 8.08228e9 0.632227
\(773\) −1.81631e10 −1.41436 −0.707182 0.707031i \(-0.750034\pi\)
−0.707182 + 0.707031i \(0.750034\pi\)
\(774\) −1.03532e9 −0.0802567
\(775\) 1.72030e9 0.132754
\(776\) 4.30196e8 0.0330484
\(777\) −3.87338e9 −0.296221
\(778\) 1.04390e10 0.794753
\(779\) 6.92922e8 0.0525174
\(780\) −3.79642e7 −0.00286446
\(781\) 2.46148e9 0.184892
\(782\) 8.25164e8 0.0617045
\(783\) −4.20512e9 −0.313049
\(784\) 4.81890e8 0.0357143
\(785\) 3.05107e8 0.0225117
\(786\) 3.56556e9 0.261908
\(787\) 1.72323e10 1.26018 0.630090 0.776522i \(-0.283018\pi\)
0.630090 + 0.776522i \(0.283018\pi\)
\(788\) −5.44092e9 −0.396123
\(789\) −1.32328e10 −0.959139
\(790\) 5.16813e8 0.0372940
\(791\) 2.57489e9 0.184987
\(792\) 5.62858e8 0.0402588
\(793\) 4.09390e9 0.291529
\(794\) −8.62185e9 −0.611264
\(795\) 2.65236e8 0.0187218
\(796\) −4.97880e9 −0.349888
\(797\) −1.51855e9 −0.106249 −0.0531246 0.998588i \(-0.516918\pi\)
−0.0531246 + 0.998588i \(0.516918\pi\)
\(798\) 6.71830e8 0.0468004
\(799\) −6.58456e8 −0.0456681
\(800\) −2.55672e9 −0.176550
\(801\) −6.91286e9 −0.475274
\(802\) −1.44036e10 −0.985963
\(803\) 3.26708e8 0.0222667
\(804\) −3.18929e9 −0.216420
\(805\) −3.39529e8 −0.0229399
\(806\) −3.87516e8 −0.0260685
\(807\) 9.26383e9 0.620488
\(808\) −4.43747e9 −0.295935
\(809\) −2.11304e10 −1.40310 −0.701548 0.712623i \(-0.747507\pi\)
−0.701548 + 0.712623i \(0.747507\pi\)
\(810\) −4.25153e7 −0.00281091
\(811\) −1.32957e10 −0.875260 −0.437630 0.899155i \(-0.644182\pi\)
−0.437630 + 0.899155i \(0.644182\pi\)
\(812\) 4.68987e9 0.307408
\(813\) 4.94225e9 0.322558
\(814\) 5.04572e9 0.327897
\(815\) −1.03245e7 −0.000668064 0
\(816\) −1.15237e8 −0.00742465
\(817\) 1.60979e9 0.103274
\(818\) −2.00356e10 −1.27987
\(819\) −5.49353e8 −0.0349428
\(820\) 4.89050e7 0.00309745
\(821\) 1.02411e10 0.645869 0.322934 0.946421i \(-0.395331\pi\)
0.322934 + 0.946421i \(0.395331\pi\)
\(822\) −5.71808e8 −0.0359086
\(823\) 1.53728e9 0.0961285 0.0480643 0.998844i \(-0.484695\pi\)
0.0480643 + 0.998844i \(0.484695\pi\)
\(824\) −4.93096e9 −0.307034
\(825\) −3.17687e9 −0.196975
\(826\) 1.63648e9 0.101037
\(827\) 1.29037e10 0.793316 0.396658 0.917966i \(-0.370170\pi\)
0.396658 + 0.917966i \(0.370170\pi\)
\(828\) −4.61838e9 −0.282738
\(829\) 1.47174e10 0.897200 0.448600 0.893733i \(-0.351923\pi\)
0.448600 + 0.893733i \(0.351923\pi\)
\(830\) −2.87372e8 −0.0174450
\(831\) 8.80903e9 0.532506
\(832\) 5.75930e8 0.0346688
\(833\) −1.22590e8 −0.00734850
\(834\) −1.03495e10 −0.617789
\(835\) 5.59050e8 0.0332314
\(836\) −8.75171e8 −0.0518049
\(837\) −4.33971e8 −0.0255812
\(838\) −3.59466e9 −0.211010
\(839\) −1.05302e10 −0.615557 −0.307778 0.951458i \(-0.599586\pi\)
−0.307778 + 0.951458i \(0.599586\pi\)
\(840\) 4.74163e7 0.00276026
\(841\) 2.83930e10 1.64598
\(842\) 6.14339e9 0.354663
\(843\) −9.10012e9 −0.523179
\(844\) 9.67668e9 0.554023
\(845\) −4.82681e7 −0.00275208
\(846\) 3.68533e9 0.209257
\(847\) 5.90410e9 0.333858
\(848\) −4.02372e9 −0.226591
\(849\) 8.33026e9 0.467177
\(850\) 6.50416e8 0.0363266
\(851\) −4.14013e10 −2.30282
\(852\) 2.82058e9 0.156243
\(853\) −2.70736e10 −1.49357 −0.746784 0.665067i \(-0.768403\pi\)
−0.746784 + 0.665067i \(0.768403\pi\)
\(854\) −5.11319e9 −0.280924
\(855\) 6.61057e7 0.00361708
\(856\) 1.80007e9 0.0980914
\(857\) 2.17919e10 1.18267 0.591333 0.806427i \(-0.298602\pi\)
0.591333 + 0.806427i \(0.298602\pi\)
\(858\) 7.15624e8 0.0386794
\(859\) −7.41255e8 −0.0399017 −0.0199509 0.999801i \(-0.506351\pi\)
−0.0199509 + 0.999801i \(0.506351\pi\)
\(860\) 1.13615e8 0.00609105
\(861\) 7.07670e8 0.0377850
\(862\) −5.63063e9 −0.299420
\(863\) −1.06003e10 −0.561410 −0.280705 0.959794i \(-0.590568\pi\)
−0.280705 + 0.959794i \(0.590568\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 4.10563e8 0.0215687
\(866\) 1.34072e10 0.701495
\(867\) −1.10498e10 −0.575823
\(868\) 4.83998e8 0.0251203
\(869\) −9.74193e9 −0.503589
\(870\) 4.61467e8 0.0237587
\(871\) −4.05490e9 −0.207930
\(872\) 5.93251e8 0.0302992
\(873\) 6.12525e8 0.0311583
\(874\) 7.18099e9 0.363826
\(875\) −5.35594e8 −0.0270276
\(876\) 3.74371e8 0.0188165
\(877\) 1.79764e10 0.899921 0.449960 0.893049i \(-0.351438\pi\)
0.449960 + 0.893049i \(0.351438\pi\)
\(878\) 1.69868e10 0.846997
\(879\) 1.30670e10 0.648954
\(880\) −6.17677e7 −0.00305543
\(881\) −3.82110e10 −1.88266 −0.941332 0.337482i \(-0.890425\pi\)
−0.941332 + 0.337482i \(0.890425\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 1.27600e10 0.623718 0.311859 0.950128i \(-0.399048\pi\)
0.311859 + 0.950128i \(0.399048\pi\)
\(884\) −1.46514e8 −0.00713337
\(885\) 1.61024e8 0.00780887
\(886\) 9.55402e9 0.461496
\(887\) −2.94961e10 −1.41916 −0.709582 0.704623i \(-0.751116\pi\)
−0.709582 + 0.704623i \(0.751116\pi\)
\(888\) 5.78183e9 0.277090
\(889\) 8.59537e9 0.410306
\(890\) 7.58613e8 0.0360708
\(891\) 8.01413e8 0.0379563
\(892\) −6.28981e9 −0.296729
\(893\) −5.73021e9 −0.269272
\(894\) −3.51984e9 −0.164756
\(895\) 7.65930e8 0.0357115
\(896\) −7.19323e8 −0.0334077
\(897\) −5.87187e9 −0.271646
\(898\) 1.29953e10 0.598851
\(899\) 4.71038e9 0.216221
\(900\) −3.64033e9 −0.166453
\(901\) 1.02361e9 0.0466229
\(902\) −9.21858e8 −0.0418255
\(903\) 1.64405e9 0.0743033
\(904\) −3.84357e9 −0.173039
\(905\) 7.49751e8 0.0336238
\(906\) 1.50800e10 0.673680
\(907\) 8.81674e9 0.392358 0.196179 0.980568i \(-0.437147\pi\)
0.196179 + 0.980568i \(0.437147\pi\)
\(908\) −1.15228e10 −0.510806
\(909\) −6.31820e9 −0.279010
\(910\) 6.02857e7 0.00265197
\(911\) −4.40436e10 −1.93005 −0.965026 0.262156i \(-0.915567\pi\)
−0.965026 + 0.262156i \(0.915567\pi\)
\(912\) −1.00285e9 −0.0437777
\(913\) 5.41697e9 0.235564
\(914\) 1.57409e9 0.0681894
\(915\) −5.03120e8 −0.0217119
\(916\) 1.98543e9 0.0853533
\(917\) −5.66197e9 −0.242479
\(918\) −1.64077e8 −0.00700003
\(919\) 1.62626e10 0.691171 0.345586 0.938387i \(-0.387680\pi\)
0.345586 + 0.938387i \(0.387680\pi\)
\(920\) 5.06819e8 0.0214583
\(921\) −1.66443e10 −0.702031
\(922\) 1.06141e10 0.445988
\(923\) 3.58612e9 0.150113
\(924\) −8.93798e8 −0.0372724
\(925\) −3.26336e10 −1.35572
\(926\) 2.30246e10 0.952912
\(927\) −7.02084e9 −0.289474
\(928\) −7.00062e9 −0.287553
\(929\) −2.37038e10 −0.969979 −0.484989 0.874520i \(-0.661176\pi\)
−0.484989 + 0.874520i \(0.661176\pi\)
\(930\) 4.76237e7 0.00194148
\(931\) −1.06684e9 −0.0433287
\(932\) −5.75122e9 −0.232704
\(933\) 1.25184e9 0.0504620
\(934\) 1.61019e10 0.646640
\(935\) 1.57134e7 0.000628678 0
\(936\) 8.20026e8 0.0326860
\(937\) 1.56896e10 0.623052 0.311526 0.950238i \(-0.399160\pi\)
0.311526 + 0.950238i \(0.399160\pi\)
\(938\) 5.06447e9 0.200366
\(939\) −2.31106e10 −0.910921
\(940\) −4.04426e8 −0.0158815
\(941\) 1.84030e10 0.719986 0.359993 0.932955i \(-0.382779\pi\)
0.359993 + 0.932955i \(0.382779\pi\)
\(942\) −6.59031e9 −0.256879
\(943\) 7.56407e9 0.293741
\(944\) −2.44279e9 −0.0945113
\(945\) 6.75127e7 0.00260240
\(946\) −2.14165e9 −0.0822488
\(947\) −1.22185e10 −0.467513 −0.233756 0.972295i \(-0.575102\pi\)
−0.233756 + 0.972295i \(0.575102\pi\)
\(948\) −1.11632e10 −0.425558
\(949\) 4.75980e8 0.0180783
\(950\) 5.66025e9 0.214192
\(951\) 9.90498e9 0.373441
\(952\) 1.82992e8 0.00687389
\(953\) 3.37697e10 1.26387 0.631934 0.775022i \(-0.282261\pi\)
0.631934 + 0.775022i \(0.282261\pi\)
\(954\) −5.72909e9 −0.213632
\(955\) −1.50516e9 −0.0559206
\(956\) 3.34423e8 0.0123792
\(957\) −8.69865e9 −0.320819
\(958\) −1.37835e10 −0.506502
\(959\) 9.08009e8 0.0332449
\(960\) −7.07789e7 −0.00258199
\(961\) −2.70265e10 −0.982331
\(962\) 7.35109e9 0.266219
\(963\) 2.56299e9 0.0924815
\(964\) 2.78796e9 0.100234
\(965\) −1.26286e9 −0.0452385
\(966\) 7.33382e9 0.261764
\(967\) −3.09840e10 −1.10191 −0.550954 0.834536i \(-0.685736\pi\)
−0.550954 + 0.834536i \(0.685736\pi\)
\(968\) −8.81311e9 −0.312295
\(969\) 2.55119e8 0.00900761
\(970\) −6.72181e7 −0.00236475
\(971\) −4.82560e10 −1.69155 −0.845773 0.533542i \(-0.820860\pi\)
−0.845773 + 0.533542i \(0.820860\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) 1.64347e10 0.571961
\(974\) −2.64039e10 −0.915614
\(975\) −4.62836e9 −0.159923
\(976\) 7.63251e9 0.262780
\(977\) −1.49627e10 −0.513308 −0.256654 0.966503i \(-0.582620\pi\)
−0.256654 + 0.966503i \(0.582620\pi\)
\(978\) 2.23010e8 0.00762321
\(979\) −1.42999e10 −0.487071
\(980\) −7.52954e7 −0.00255551
\(981\) 8.44688e8 0.0285664
\(982\) −1.97494e9 −0.0665525
\(983\) −2.24569e10 −0.754071 −0.377036 0.926199i \(-0.623057\pi\)
−0.377036 + 0.926199i \(0.623057\pi\)
\(984\) −1.05635e9 −0.0353447
\(985\) 8.50143e8 0.0283443
\(986\) 1.78092e9 0.0591664
\(987\) −5.85217e9 −0.193735
\(988\) −1.27503e9 −0.0420603
\(989\) 1.75727e10 0.577634
\(990\) −8.79466e7 −0.00288068
\(991\) 7.33554e9 0.239427 0.119714 0.992808i \(-0.461802\pi\)
0.119714 + 0.992808i \(0.461802\pi\)
\(992\) −7.22469e8 −0.0234979
\(993\) 8.80723e9 0.285441
\(994\) −4.47898e9 −0.144653
\(995\) 7.77938e8 0.0250359
\(996\) 6.20724e9 0.199063
\(997\) −2.09002e10 −0.667909 −0.333955 0.942589i \(-0.608383\pi\)
−0.333955 + 0.942589i \(0.608383\pi\)
\(998\) 2.90153e9 0.0923996
\(999\) 8.23234e9 0.261243
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 546.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.c.1.1 1 1.1 even 1 trivial