Properties

Label 273.12.a.c.1.6
Level $273$
Weight $12$
Character 273.1
Self dual yes
Analytic conductor $209.758$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 273.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(209.757688293\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - x^{15} - 26640 x^{14} - 38138 x^{13} + 279954048 x^{12} + 940095168 x^{11} - 1473489774048 x^{10} - 6951601887328 x^{9} + 4072481535854976 x^{8} + 21355761647276288 x^{7} - 5703452025696599040 x^{6} - 25041700255043457024 x^{5} + 3737994671267475603456 x^{4} + 6711995200063092490240 x^{3} - 963518272786595267739648 x^{2} + 1287697466669064032092160 x + 45555354651594891689197568\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{12}\cdot 5\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-37.3707\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q-41.3707 q^{2} -243.000 q^{3} -336.467 q^{4} -335.043 q^{5} +10053.1 q^{6} +16807.0 q^{7} +98647.0 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-41.3707 q^{2} -243.000 q^{3} -336.467 q^{4} -335.043 q^{5} +10053.1 q^{6} +16807.0 q^{7} +98647.0 q^{8} +59049.0 q^{9} +13861.0 q^{10} -167805. q^{11} +81761.4 q^{12} -371293. q^{13} -695317. q^{14} +81415.4 q^{15} -3.39201e6 q^{16} -48979.8 q^{17} -2.44290e6 q^{18} +1.41962e7 q^{19} +112731. q^{20} -4.08410e6 q^{21} +6.94222e6 q^{22} +3.84648e7 q^{23} -2.39712e7 q^{24} -4.87159e7 q^{25} +1.53606e7 q^{26} -1.43489e7 q^{27} -5.65500e6 q^{28} -8.99484e7 q^{29} -3.36821e6 q^{30} +1.22353e8 q^{31} -6.16993e7 q^{32} +4.07767e7 q^{33} +2.02633e6 q^{34} -5.63107e6 q^{35} -1.98680e7 q^{36} -6.35049e8 q^{37} -5.87307e8 q^{38} +9.02242e7 q^{39} -3.30510e7 q^{40} -4.35832e8 q^{41} +1.68962e8 q^{42} -7.21546e8 q^{43} +5.64609e7 q^{44} -1.97840e7 q^{45} -1.59131e9 q^{46} -1.84403e9 q^{47} +8.24259e8 q^{48} +2.82475e8 q^{49} +2.01541e9 q^{50} +1.19021e7 q^{51} +1.24928e8 q^{52} +2.91076e9 q^{53} +5.93624e8 q^{54} +5.62220e7 q^{55} +1.65796e9 q^{56} -3.44968e9 q^{57} +3.72122e9 q^{58} +3.37446e9 q^{59} -2.73936e7 q^{60} -2.50761e9 q^{61} -5.06181e9 q^{62} +9.92437e8 q^{63} +9.49938e9 q^{64} +1.24399e8 q^{65} -1.68696e9 q^{66} +1.35442e10 q^{67} +1.64801e7 q^{68} -9.34694e9 q^{69} +2.32961e8 q^{70} -1.32563e10 q^{71} +5.82501e9 q^{72} -8.63655e9 q^{73} +2.62724e10 q^{74} +1.18380e10 q^{75} -4.77656e9 q^{76} -2.82030e9 q^{77} -3.73264e9 q^{78} +1.81183e10 q^{79} +1.13647e9 q^{80} +3.48678e9 q^{81} +1.80307e10 q^{82} +2.48333e10 q^{83} +1.37416e9 q^{84} +1.64103e7 q^{85} +2.98508e10 q^{86} +2.18575e10 q^{87} -1.65535e10 q^{88} +5.46685e10 q^{89} +8.18476e8 q^{90} -6.24032e9 q^{91} -1.29421e10 q^{92} -2.97317e10 q^{93} +7.62888e10 q^{94} -4.75634e9 q^{95} +1.49929e10 q^{96} -3.67985e10 q^{97} -1.16862e10 q^{98} -9.90874e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −41.3707 −0.914172 −0.457086 0.889423i \(-0.651107\pi\)
−0.457086 + 0.889423i \(0.651107\pi\)
\(3\) −243.000 −0.577350
\(4\) −336.467 −0.164290
\(5\) −335.043 −0.0479474 −0.0239737 0.999713i \(-0.507632\pi\)
−0.0239737 + 0.999713i \(0.507632\pi\)
\(6\) 10053.1 0.527797
\(7\) 16807.0 0.377964
\(8\) 98647.0 1.06436
\(9\) 59049.0 0.333333
\(10\) 13861.0 0.0438322
\(11\) −167805. −0.314157 −0.157078 0.987586i \(-0.550207\pi\)
−0.157078 + 0.987586i \(0.550207\pi\)
\(12\) 81761.4 0.0948531
\(13\) −371293. −0.277350
\(14\) −695317. −0.345524
\(15\) 81415.4 0.0276825
\(16\) −3.39201e6 −0.808718
\(17\) −48979.8 −0.00836658 −0.00418329 0.999991i \(-0.501332\pi\)
−0.00418329 + 0.999991i \(0.501332\pi\)
\(18\) −2.44290e6 −0.304724
\(19\) 1.41962e7 1.31531 0.657655 0.753319i \(-0.271549\pi\)
0.657655 + 0.753319i \(0.271549\pi\)
\(20\) 112731. 0.00787730
\(21\) −4.08410e6 −0.218218
\(22\) 6.94222e6 0.287193
\(23\) 3.84648e7 1.24612 0.623060 0.782174i \(-0.285889\pi\)
0.623060 + 0.782174i \(0.285889\pi\)
\(24\) −2.39712e7 −0.614509
\(25\) −4.87159e7 −0.997701
\(26\) 1.53606e7 0.253546
\(27\) −1.43489e7 −0.192450
\(28\) −5.65500e6 −0.0620959
\(29\) −8.99484e7 −0.814337 −0.407169 0.913353i \(-0.633484\pi\)
−0.407169 + 0.913353i \(0.633484\pi\)
\(30\) −3.36821e6 −0.0253065
\(31\) 1.22353e8 0.767580 0.383790 0.923420i \(-0.374619\pi\)
0.383790 + 0.923420i \(0.374619\pi\)
\(32\) −6.16993e7 −0.325054
\(33\) 4.07767e7 0.181378
\(34\) 2.02633e6 0.00764849
\(35\) −5.63107e6 −0.0181224
\(36\) −1.98680e7 −0.0547635
\(37\) −6.35049e8 −1.50556 −0.752779 0.658273i \(-0.771287\pi\)
−0.752779 + 0.658273i \(0.771287\pi\)
\(38\) −5.87307e8 −1.20242
\(39\) 9.02242e7 0.160128
\(40\) −3.30510e7 −0.0510334
\(41\) −4.35832e8 −0.587500 −0.293750 0.955882i \(-0.594903\pi\)
−0.293750 + 0.955882i \(0.594903\pi\)
\(42\) 1.68962e8 0.199489
\(43\) −7.21546e8 −0.748492 −0.374246 0.927329i \(-0.622099\pi\)
−0.374246 + 0.927329i \(0.622099\pi\)
\(44\) 5.64609e7 0.0516129
\(45\) −1.97840e7 −0.0159825
\(46\) −1.59131e9 −1.13917
\(47\) −1.84403e9 −1.17282 −0.586408 0.810016i \(-0.699459\pi\)
−0.586408 + 0.810016i \(0.699459\pi\)
\(48\) 8.24259e8 0.466914
\(49\) 2.82475e8 0.142857
\(50\) 2.01541e9 0.912070
\(51\) 1.19021e7 0.00483045
\(52\) 1.24928e8 0.0455660
\(53\) 2.91076e9 0.956068 0.478034 0.878341i \(-0.341349\pi\)
0.478034 + 0.878341i \(0.341349\pi\)
\(54\) 5.93624e8 0.175932
\(55\) 5.62220e7 0.0150630
\(56\) 1.65796e9 0.402291
\(57\) −3.44968e9 −0.759394
\(58\) 3.72122e9 0.744444
\(59\) 3.37446e9 0.614494 0.307247 0.951630i \(-0.400592\pi\)
0.307247 + 0.951630i \(0.400592\pi\)
\(60\) −2.73936e7 −0.00454796
\(61\) −2.50761e9 −0.380142 −0.190071 0.981770i \(-0.560872\pi\)
−0.190071 + 0.981770i \(0.560872\pi\)
\(62\) −5.06181e9 −0.701700
\(63\) 9.92437e8 0.125988
\(64\) 9.49938e9 1.10587
\(65\) 1.24399e8 0.0132982
\(66\) −1.68696e9 −0.165811
\(67\) 1.35442e10 1.22558 0.612791 0.790245i \(-0.290047\pi\)
0.612791 + 0.790245i \(0.290047\pi\)
\(68\) 1.64801e7 0.00137455
\(69\) −9.34694e9 −0.719448
\(70\) 2.32961e8 0.0165670
\(71\) −1.32563e10 −0.871967 −0.435984 0.899955i \(-0.643599\pi\)
−0.435984 + 0.899955i \(0.643599\pi\)
\(72\) 5.82501e9 0.354787
\(73\) −8.63655e9 −0.487601 −0.243801 0.969825i \(-0.578394\pi\)
−0.243801 + 0.969825i \(0.578394\pi\)
\(74\) 2.62724e10 1.37634
\(75\) 1.18380e10 0.576023
\(76\) −4.77656e9 −0.216093
\(77\) −2.82030e9 −0.118740
\(78\) −3.73264e9 −0.146385
\(79\) 1.81183e10 0.662472 0.331236 0.943548i \(-0.392534\pi\)
0.331236 + 0.943548i \(0.392534\pi\)
\(80\) 1.13647e9 0.0387760
\(81\) 3.48678e9 0.111111
\(82\) 1.80307e10 0.537076
\(83\) 2.48333e10 0.691999 0.346000 0.938235i \(-0.387540\pi\)
0.346000 + 0.938235i \(0.387540\pi\)
\(84\) 1.37416e9 0.0358511
\(85\) 1.64103e7 0.000401156 0
\(86\) 2.98508e10 0.684250
\(87\) 2.18575e10 0.470158
\(88\) −1.65535e10 −0.334376
\(89\) 5.46685e10 1.03775 0.518874 0.854851i \(-0.326351\pi\)
0.518874 + 0.854851i \(0.326351\pi\)
\(90\) 8.18476e8 0.0146107
\(91\) −6.24032e9 −0.104828
\(92\) −1.29421e10 −0.204726
\(93\) −2.97317e10 −0.443163
\(94\) 7.62888e10 1.07216
\(95\) −4.75634e9 −0.0630657
\(96\) 1.49929e10 0.187670
\(97\) −3.67985e10 −0.435097 −0.217548 0.976050i \(-0.569806\pi\)
−0.217548 + 0.976050i \(0.569806\pi\)
\(98\) −1.16862e10 −0.130596
\(99\) −9.90874e9 −0.104719
\(100\) 1.63913e10 0.163913
\(101\) −1.28433e10 −0.121593 −0.0607967 0.998150i \(-0.519364\pi\)
−0.0607967 + 0.998150i \(0.519364\pi\)
\(102\) −4.92398e8 −0.00441586
\(103\) −1.49949e11 −1.27450 −0.637248 0.770659i \(-0.719927\pi\)
−0.637248 + 0.770659i \(0.719927\pi\)
\(104\) −3.66269e10 −0.295201
\(105\) 1.36835e9 0.0104630
\(106\) −1.20420e11 −0.874011
\(107\) 1.96667e11 1.35557 0.677785 0.735261i \(-0.262940\pi\)
0.677785 + 0.735261i \(0.262940\pi\)
\(108\) 4.82793e9 0.0316177
\(109\) 3.02081e11 1.88052 0.940261 0.340456i \(-0.110581\pi\)
0.940261 + 0.340456i \(0.110581\pi\)
\(110\) −2.32594e9 −0.0137702
\(111\) 1.54317e11 0.869234
\(112\) −5.70095e10 −0.305667
\(113\) −4.14817e10 −0.211799 −0.105900 0.994377i \(-0.533772\pi\)
−0.105900 + 0.994377i \(0.533772\pi\)
\(114\) 1.42716e11 0.694217
\(115\) −1.28874e10 −0.0597483
\(116\) 3.02646e10 0.133788
\(117\) −2.19245e10 −0.0924500
\(118\) −1.39604e11 −0.561753
\(119\) −8.23204e8 −0.00316227
\(120\) 8.03139e9 0.0294641
\(121\) −2.57153e11 −0.901306
\(122\) 1.03742e11 0.347515
\(123\) 1.05907e11 0.339193
\(124\) −4.11676e10 −0.126106
\(125\) 3.26814e10 0.0957847
\(126\) −4.10578e10 −0.115175
\(127\) 4.18372e11 1.12368 0.561839 0.827247i \(-0.310094\pi\)
0.561839 + 0.827247i \(0.310094\pi\)
\(128\) −2.66636e11 −0.685904
\(129\) 1.75336e11 0.432142
\(130\) −5.14648e9 −0.0121569
\(131\) 1.54067e11 0.348913 0.174456 0.984665i \(-0.444183\pi\)
0.174456 + 0.984665i \(0.444183\pi\)
\(132\) −1.37200e10 −0.0297987
\(133\) 2.38596e11 0.497140
\(134\) −5.60333e11 −1.12039
\(135\) 4.80750e9 0.00922749
\(136\) −4.83171e9 −0.00890507
\(137\) 5.72343e11 1.01320 0.506598 0.862182i \(-0.330903\pi\)
0.506598 + 0.862182i \(0.330903\pi\)
\(138\) 3.86689e11 0.657699
\(139\) 3.12240e11 0.510395 0.255198 0.966889i \(-0.417859\pi\)
0.255198 + 0.966889i \(0.417859\pi\)
\(140\) 1.89467e9 0.00297734
\(141\) 4.48100e11 0.677126
\(142\) 5.48421e11 0.797128
\(143\) 6.23050e10 0.0871313
\(144\) −2.00295e11 −0.269573
\(145\) 3.01366e10 0.0390454
\(146\) 3.57300e11 0.445751
\(147\) −6.86415e10 −0.0824786
\(148\) 2.13673e11 0.247349
\(149\) −1.45681e12 −1.62509 −0.812546 0.582897i \(-0.801919\pi\)
−0.812546 + 0.582897i \(0.801919\pi\)
\(150\) −4.89744e11 −0.526584
\(151\) 8.43380e11 0.874279 0.437140 0.899394i \(-0.355992\pi\)
0.437140 + 0.899394i \(0.355992\pi\)
\(152\) 1.40041e12 1.39996
\(153\) −2.89221e9 −0.00278886
\(154\) 1.16678e11 0.108549
\(155\) −4.09934e10 −0.0368035
\(156\) −3.03574e10 −0.0263075
\(157\) −2.28089e12 −1.90834 −0.954172 0.299260i \(-0.903260\pi\)
−0.954172 + 0.299260i \(0.903260\pi\)
\(158\) −7.49565e11 −0.605613
\(159\) −7.07314e11 −0.551986
\(160\) 2.06719e10 0.0155855
\(161\) 6.46477e11 0.470989
\(162\) −1.44251e11 −0.101575
\(163\) 9.22369e11 0.627874 0.313937 0.949444i \(-0.398352\pi\)
0.313937 + 0.949444i \(0.398352\pi\)
\(164\) 1.46643e11 0.0965206
\(165\) −1.36619e10 −0.00869663
\(166\) −1.02737e12 −0.632606
\(167\) 3.11967e12 1.85852 0.929262 0.369422i \(-0.120444\pi\)
0.929262 + 0.369422i \(0.120444\pi\)
\(168\) −4.02884e11 −0.232263
\(169\) 1.37858e11 0.0769231
\(170\) −6.78907e8 −0.000366726 0
\(171\) 8.38273e11 0.438436
\(172\) 2.42776e11 0.122970
\(173\) −2.40095e12 −1.17796 −0.588978 0.808149i \(-0.700469\pi\)
−0.588978 + 0.808149i \(0.700469\pi\)
\(174\) −9.04258e11 −0.429805
\(175\) −8.18768e11 −0.377096
\(176\) 5.69198e11 0.254064
\(177\) −8.19993e11 −0.354778
\(178\) −2.26167e12 −0.948680
\(179\) 2.43237e12 0.989323 0.494662 0.869086i \(-0.335292\pi\)
0.494662 + 0.869086i \(0.335292\pi\)
\(180\) 6.65664e9 0.00262577
\(181\) −1.36684e12 −0.522980 −0.261490 0.965206i \(-0.584214\pi\)
−0.261490 + 0.965206i \(0.584214\pi\)
\(182\) 2.58166e11 0.0958312
\(183\) 6.09350e11 0.219475
\(184\) 3.79443e12 1.32632
\(185\) 2.12769e11 0.0721877
\(186\) 1.23002e12 0.405127
\(187\) 8.21908e9 0.00262842
\(188\) 6.20455e11 0.192683
\(189\) −2.41162e11 −0.0727393
\(190\) 1.96773e11 0.0576529
\(191\) −3.69529e12 −1.05188 −0.525938 0.850523i \(-0.676286\pi\)
−0.525938 + 0.850523i \(0.676286\pi\)
\(192\) −2.30835e12 −0.638476
\(193\) −4.85389e12 −1.30474 −0.652371 0.757900i \(-0.726226\pi\)
−0.652371 + 0.757900i \(0.726226\pi\)
\(194\) 1.52238e12 0.397753
\(195\) −3.02290e10 −0.00767774
\(196\) −9.50435e10 −0.0234701
\(197\) −2.33085e12 −0.559694 −0.279847 0.960045i \(-0.590284\pi\)
−0.279847 + 0.960045i \(0.590284\pi\)
\(198\) 4.09931e11 0.0957310
\(199\) 6.85105e12 1.55620 0.778099 0.628141i \(-0.216184\pi\)
0.778099 + 0.628141i \(0.216184\pi\)
\(200\) −4.80568e12 −1.06191
\(201\) −3.29124e12 −0.707590
\(202\) 5.31337e11 0.111157
\(203\) −1.51176e12 −0.307791
\(204\) −4.00466e9 −0.000793596 0
\(205\) 1.46022e11 0.0281691
\(206\) 6.20348e12 1.16511
\(207\) 2.27131e12 0.415374
\(208\) 1.25943e12 0.224298
\(209\) −2.38220e12 −0.413213
\(210\) −5.66095e10 −0.00956497
\(211\) 7.09731e12 1.16826 0.584131 0.811660i \(-0.301435\pi\)
0.584131 + 0.811660i \(0.301435\pi\)
\(212\) −9.79374e11 −0.157073
\(213\) 3.22127e12 0.503430
\(214\) −8.13627e12 −1.23922
\(215\) 2.41749e11 0.0358883
\(216\) −1.41548e12 −0.204836
\(217\) 2.05638e12 0.290118
\(218\) −1.24973e13 −1.71912
\(219\) 2.09868e12 0.281517
\(220\) −1.89168e10 −0.00247471
\(221\) 1.81859e10 0.00232047
\(222\) −6.38419e12 −0.794629
\(223\) 1.02373e13 1.24310 0.621551 0.783374i \(-0.286503\pi\)
0.621551 + 0.783374i \(0.286503\pi\)
\(224\) −1.03698e12 −0.122859
\(225\) −2.87662e12 −0.332567
\(226\) 1.71612e12 0.193621
\(227\) −1.06971e13 −1.17794 −0.588968 0.808156i \(-0.700466\pi\)
−0.588968 + 0.808156i \(0.700466\pi\)
\(228\) 1.16070e12 0.124761
\(229\) −1.41550e13 −1.48530 −0.742648 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(230\) 5.33159e11 0.0546202
\(231\) 6.85334e11 0.0685546
\(232\) −8.87314e12 −0.866749
\(233\) −7.50958e12 −0.716404 −0.358202 0.933644i \(-0.616610\pi\)
−0.358202 + 0.933644i \(0.616610\pi\)
\(234\) 9.07031e11 0.0845152
\(235\) 6.17830e11 0.0562336
\(236\) −1.13539e12 −0.100955
\(237\) −4.40274e12 −0.382478
\(238\) 3.40565e10 0.00289086
\(239\) −9.42879e12 −0.782109 −0.391055 0.920367i \(-0.627890\pi\)
−0.391055 + 0.920367i \(0.627890\pi\)
\(240\) −2.76162e11 −0.0223873
\(241\) 1.91573e13 1.51789 0.758945 0.651155i \(-0.225715\pi\)
0.758945 + 0.651155i \(0.225715\pi\)
\(242\) 1.06386e13 0.823948
\(243\) −8.47289e11 −0.0641500
\(244\) 8.43728e11 0.0624538
\(245\) −9.46413e10 −0.00684964
\(246\) −4.38145e12 −0.310081
\(247\) −5.27096e12 −0.364801
\(248\) 1.20697e13 0.816983
\(249\) −6.03450e12 −0.399526
\(250\) −1.35205e12 −0.0875636
\(251\) −1.43679e13 −0.910307 −0.455154 0.890413i \(-0.650416\pi\)
−0.455154 + 0.890413i \(0.650416\pi\)
\(252\) −3.33922e11 −0.0206986
\(253\) −6.45459e12 −0.391477
\(254\) −1.73083e13 −1.02723
\(255\) −3.98771e9 −0.000231608 0
\(256\) −8.42383e12 −0.478839
\(257\) 1.62702e13 0.905236 0.452618 0.891705i \(-0.350490\pi\)
0.452618 + 0.891705i \(0.350490\pi\)
\(258\) −7.25375e12 −0.395052
\(259\) −1.06733e13 −0.569047
\(260\) −4.18562e10 −0.00218477
\(261\) −5.31136e12 −0.271446
\(262\) −6.37385e12 −0.318966
\(263\) 8.77423e12 0.429984 0.214992 0.976616i \(-0.431027\pi\)
0.214992 + 0.976616i \(0.431027\pi\)
\(264\) 4.02250e12 0.193052
\(265\) −9.75229e11 −0.0458410
\(266\) −9.87087e12 −0.454471
\(267\) −1.32844e13 −0.599144
\(268\) −4.55718e12 −0.201351
\(269\) 7.78663e12 0.337064 0.168532 0.985696i \(-0.446097\pi\)
0.168532 + 0.985696i \(0.446097\pi\)
\(270\) −1.98890e11 −0.00843551
\(271\) −2.54766e13 −1.05879 −0.529396 0.848375i \(-0.677581\pi\)
−0.529396 + 0.848375i \(0.677581\pi\)
\(272\) 1.66140e11 0.00676621
\(273\) 1.51640e12 0.0605228
\(274\) −2.36782e13 −0.926235
\(275\) 8.17478e12 0.313434
\(276\) 3.14493e12 0.118198
\(277\) −3.70111e13 −1.36362 −0.681810 0.731529i \(-0.738807\pi\)
−0.681810 + 0.731529i \(0.738807\pi\)
\(278\) −1.29176e13 −0.466589
\(279\) 7.22480e12 0.255860
\(280\) −5.55488e11 −0.0192888
\(281\) 2.87923e13 0.980375 0.490187 0.871617i \(-0.336928\pi\)
0.490187 + 0.871617i \(0.336928\pi\)
\(282\) −1.85382e13 −0.619009
\(283\) −1.79126e13 −0.586589 −0.293295 0.956022i \(-0.594752\pi\)
−0.293295 + 0.956022i \(0.594752\pi\)
\(284\) 4.46029e12 0.143256
\(285\) 1.15579e12 0.0364110
\(286\) −2.57760e12 −0.0796530
\(287\) −7.32502e12 −0.222054
\(288\) −3.64328e12 −0.108351
\(289\) −3.42695e13 −0.999930
\(290\) −1.24677e12 −0.0356942
\(291\) 8.94204e12 0.251203
\(292\) 2.90591e12 0.0801082
\(293\) −5.92929e13 −1.60410 −0.802049 0.597259i \(-0.796257\pi\)
−0.802049 + 0.597259i \(0.796257\pi\)
\(294\) 2.83975e12 0.0753996
\(295\) −1.13059e12 −0.0294634
\(296\) −6.26456e13 −1.60246
\(297\) 2.40782e12 0.0604594
\(298\) 6.02692e13 1.48561
\(299\) −1.42817e13 −0.345612
\(300\) −3.98308e12 −0.0946350
\(301\) −1.21270e13 −0.282904
\(302\) −3.48912e13 −0.799241
\(303\) 3.12093e12 0.0702020
\(304\) −4.81537e13 −1.06371
\(305\) 8.40158e11 0.0182269
\(306\) 1.19653e11 0.00254950
\(307\) 7.45676e13 1.56059 0.780295 0.625412i \(-0.215069\pi\)
0.780295 + 0.625412i \(0.215069\pi\)
\(308\) 9.48939e11 0.0195078
\(309\) 3.64375e13 0.735830
\(310\) 1.69592e12 0.0336447
\(311\) −7.14810e13 −1.39318 −0.696592 0.717467i \(-0.745301\pi\)
−0.696592 + 0.717467i \(0.745301\pi\)
\(312\) 8.90035e12 0.170434
\(313\) −3.33154e12 −0.0626832 −0.0313416 0.999509i \(-0.509978\pi\)
−0.0313416 + 0.999509i \(0.509978\pi\)
\(314\) 9.43620e13 1.74455
\(315\) −3.32509e11 −0.00604081
\(316\) −6.09619e12 −0.108838
\(317\) 3.24500e13 0.569363 0.284681 0.958622i \(-0.408112\pi\)
0.284681 + 0.958622i \(0.408112\pi\)
\(318\) 2.92621e13 0.504610
\(319\) 1.50938e13 0.255829
\(320\) −3.18270e12 −0.0530238
\(321\) −4.77902e13 −0.782638
\(322\) −2.67452e13 −0.430565
\(323\) −6.95328e11 −0.0110046
\(324\) −1.17319e12 −0.0182545
\(325\) 1.80879e13 0.276712
\(326\) −3.81590e13 −0.573985
\(327\) −7.34058e13 −1.08572
\(328\) −4.29935e13 −0.625312
\(329\) −3.09926e13 −0.443283
\(330\) 5.65204e11 0.00795021
\(331\) 9.52552e12 0.131776 0.0658878 0.997827i \(-0.479012\pi\)
0.0658878 + 0.997827i \(0.479012\pi\)
\(332\) −8.35559e12 −0.113689
\(333\) −3.74990e13 −0.501853
\(334\) −1.29063e14 −1.69901
\(335\) −4.53789e12 −0.0587635
\(336\) 1.38533e13 0.176477
\(337\) 1.20827e14 1.51426 0.757129 0.653266i \(-0.226602\pi\)
0.757129 + 0.653266i \(0.226602\pi\)
\(338\) −5.70330e12 −0.0703209
\(339\) 1.00800e13 0.122282
\(340\) −5.52154e9 −6.59061e−5 0
\(341\) −2.05314e13 −0.241140
\(342\) −3.46799e13 −0.400806
\(343\) 4.74756e12 0.0539949
\(344\) −7.11783e13 −0.796666
\(345\) 3.13163e12 0.0344957
\(346\) 9.93288e13 1.07685
\(347\) −9.23674e13 −0.985613 −0.492807 0.870139i \(-0.664029\pi\)
−0.492807 + 0.870139i \(0.664029\pi\)
\(348\) −7.35430e12 −0.0772424
\(349\) −1.22711e14 −1.26866 −0.634330 0.773062i \(-0.718724\pi\)
−0.634330 + 0.773062i \(0.718724\pi\)
\(350\) 3.38730e13 0.344730
\(351\) 5.32765e12 0.0533761
\(352\) 1.03535e13 0.102118
\(353\) −1.52789e14 −1.48365 −0.741826 0.670592i \(-0.766040\pi\)
−0.741826 + 0.670592i \(0.766040\pi\)
\(354\) 3.39237e13 0.324328
\(355\) 4.44142e12 0.0418086
\(356\) −1.83941e13 −0.170492
\(357\) 2.00039e11 0.00182574
\(358\) −1.00629e14 −0.904411
\(359\) −1.51944e14 −1.34482 −0.672408 0.740181i \(-0.734740\pi\)
−0.672408 + 0.740181i \(0.734740\pi\)
\(360\) −1.95163e12 −0.0170111
\(361\) 8.50424e13 0.730039
\(362\) 5.65471e13 0.478094
\(363\) 6.24882e13 0.520369
\(364\) 2.09966e12 0.0172223
\(365\) 2.89362e12 0.0233792
\(366\) −2.52092e13 −0.200638
\(367\) −9.94466e13 −0.779698 −0.389849 0.920879i \(-0.627473\pi\)
−0.389849 + 0.920879i \(0.627473\pi\)
\(368\) −1.30473e14 −1.00776
\(369\) −2.57354e13 −0.195833
\(370\) −8.80238e12 −0.0659919
\(371\) 4.89211e13 0.361360
\(372\) 1.00037e13 0.0728074
\(373\) 1.04403e14 0.748710 0.374355 0.927285i \(-0.377864\pi\)
0.374355 + 0.927285i \(0.377864\pi\)
\(374\) −3.40029e11 −0.00240282
\(375\) −7.94159e12 −0.0553013
\(376\) −1.81908e14 −1.24830
\(377\) 3.33972e13 0.225856
\(378\) 9.97704e12 0.0664962
\(379\) 1.25664e13 0.0825456 0.0412728 0.999148i \(-0.486859\pi\)
0.0412728 + 0.999148i \(0.486859\pi\)
\(380\) 1.60035e12 0.0103611
\(381\) −1.01664e14 −0.648756
\(382\) 1.52876e14 0.961595
\(383\) 2.19835e14 1.36302 0.681511 0.731808i \(-0.261323\pi\)
0.681511 + 0.731808i \(0.261323\pi\)
\(384\) 6.47925e13 0.396007
\(385\) 9.44923e11 0.00569328
\(386\) 2.00809e14 1.19276
\(387\) −4.26066e13 −0.249497
\(388\) 1.23815e13 0.0714822
\(389\) −3.80170e13 −0.216399 −0.108200 0.994129i \(-0.534509\pi\)
−0.108200 + 0.994129i \(0.534509\pi\)
\(390\) 1.25059e12 0.00701877
\(391\) −1.88400e12 −0.0104258
\(392\) 2.78653e13 0.152052
\(393\) −3.74382e13 −0.201445
\(394\) 9.64290e13 0.511657
\(395\) −6.07039e12 −0.0317638
\(396\) 3.33396e12 0.0172043
\(397\) 2.29382e14 1.16738 0.583688 0.811978i \(-0.301609\pi\)
0.583688 + 0.811978i \(0.301609\pi\)
\(398\) −2.83433e14 −1.42263
\(399\) −5.79788e13 −0.287024
\(400\) 1.65245e14 0.806859
\(401\) −3.84869e14 −1.85361 −0.926806 0.375541i \(-0.877457\pi\)
−0.926806 + 0.375541i \(0.877457\pi\)
\(402\) 1.36161e14 0.646859
\(403\) −4.54286e13 −0.212888
\(404\) 4.32135e12 0.0199766
\(405\) −1.16822e12 −0.00532749
\(406\) 6.25426e13 0.281373
\(407\) 1.06565e14 0.472981
\(408\) 1.17411e12 0.00514134
\(409\) 1.30983e14 0.565898 0.282949 0.959135i \(-0.408687\pi\)
0.282949 + 0.959135i \(0.408687\pi\)
\(410\) −6.04104e12 −0.0257514
\(411\) −1.39079e14 −0.584969
\(412\) 5.04528e13 0.209387
\(413\) 5.67145e13 0.232257
\(414\) −9.39655e13 −0.379723
\(415\) −8.32024e12 −0.0331796
\(416\) 2.29085e13 0.0901537
\(417\) −7.58742e13 −0.294677
\(418\) 9.85533e13 0.377748
\(419\) −5.28075e13 −0.199764 −0.0998822 0.994999i \(-0.531847\pi\)
−0.0998822 + 0.994999i \(0.531847\pi\)
\(420\) −4.60404e11 −0.00171897
\(421\) −1.14300e14 −0.421207 −0.210603 0.977572i \(-0.567543\pi\)
−0.210603 + 0.977572i \(0.567543\pi\)
\(422\) −2.93620e14 −1.06799
\(423\) −1.08888e14 −0.390939
\(424\) 2.87138e14 1.01760
\(425\) 2.38609e12 0.00834735
\(426\) −1.33266e14 −0.460222
\(427\) −4.21454e13 −0.143680
\(428\) −6.61721e13 −0.222707
\(429\) −1.51401e13 −0.0503053
\(430\) −1.00013e13 −0.0328081
\(431\) 6.52231e13 0.211240 0.105620 0.994407i \(-0.466317\pi\)
0.105620 + 0.994407i \(0.466317\pi\)
\(432\) 4.86716e13 0.155638
\(433\) 2.39836e14 0.757235 0.378617 0.925553i \(-0.376400\pi\)
0.378617 + 0.925553i \(0.376400\pi\)
\(434\) −8.50738e13 −0.265218
\(435\) −7.32319e12 −0.0225429
\(436\) −1.01640e14 −0.308952
\(437\) 5.46054e14 1.63903
\(438\) −8.68239e13 −0.257355
\(439\) 2.26264e14 0.662309 0.331154 0.943577i \(-0.392562\pi\)
0.331154 + 0.943577i \(0.392562\pi\)
\(440\) 5.54613e12 0.0160325
\(441\) 1.66799e13 0.0476190
\(442\) −7.52362e11 −0.00212131
\(443\) −1.76041e14 −0.490222 −0.245111 0.969495i \(-0.578824\pi\)
−0.245111 + 0.969495i \(0.578824\pi\)
\(444\) −5.19225e13 −0.142807
\(445\) −1.83163e13 −0.0497574
\(446\) −4.23522e14 −1.13641
\(447\) 3.54005e14 0.938248
\(448\) 1.59656e14 0.417981
\(449\) 4.90170e14 1.26763 0.633814 0.773486i \(-0.281489\pi\)
0.633814 + 0.773486i \(0.281489\pi\)
\(450\) 1.19008e14 0.304023
\(451\) 7.31349e13 0.184567
\(452\) 1.39572e13 0.0347966
\(453\) −2.04941e14 −0.504765
\(454\) 4.42544e14 1.07684
\(455\) 2.09078e12 0.00502626
\(456\) −3.40301e14 −0.808270
\(457\) 3.01761e14 0.708148 0.354074 0.935217i \(-0.384796\pi\)
0.354074 + 0.935217i \(0.384796\pi\)
\(458\) 5.85600e14 1.35782
\(459\) 7.02807e11 0.00161015
\(460\) 4.33616e12 0.00981607
\(461\) 3.91642e14 0.876060 0.438030 0.898960i \(-0.355676\pi\)
0.438030 + 0.898960i \(0.355676\pi\)
\(462\) −2.83527e13 −0.0626706
\(463\) 4.34843e14 0.949811 0.474905 0.880037i \(-0.342482\pi\)
0.474905 + 0.880037i \(0.342482\pi\)
\(464\) 3.05106e14 0.658569
\(465\) 9.96139e12 0.0212485
\(466\) 3.10676e14 0.654916
\(467\) −4.21299e14 −0.877703 −0.438852 0.898560i \(-0.644615\pi\)
−0.438852 + 0.898560i \(0.644615\pi\)
\(468\) 7.37686e12 0.0151887
\(469\) 2.27638e14 0.463226
\(470\) −2.55600e13 −0.0514071
\(471\) 5.54257e14 1.10178
\(472\) 3.32880e14 0.654044
\(473\) 1.21079e14 0.235144
\(474\) 1.82144e14 0.349651
\(475\) −6.91581e14 −1.31229
\(476\) 2.76981e11 0.000519531 0
\(477\) 1.71877e14 0.318689
\(478\) 3.90075e14 0.714982
\(479\) −3.72285e14 −0.674575 −0.337287 0.941402i \(-0.609509\pi\)
−0.337287 + 0.941402i \(0.609509\pi\)
\(480\) −5.02328e12 −0.00899830
\(481\) 2.35789e14 0.417567
\(482\) −7.92550e14 −1.38761
\(483\) −1.57094e14 −0.271926
\(484\) 8.65234e13 0.148076
\(485\) 1.23291e13 0.0208618
\(486\) 3.50529e13 0.0586441
\(487\) −5.84872e14 −0.967502 −0.483751 0.875206i \(-0.660726\pi\)
−0.483751 + 0.875206i \(0.660726\pi\)
\(488\) −2.47369e14 −0.404609
\(489\) −2.24136e14 −0.362503
\(490\) 3.91538e12 0.00626174
\(491\) −1.07279e15 −1.69655 −0.848274 0.529558i \(-0.822358\pi\)
−0.848274 + 0.529558i \(0.822358\pi\)
\(492\) −3.56342e13 −0.0557262
\(493\) 4.40565e12 0.00681322
\(494\) 2.18063e14 0.333491
\(495\) 3.31985e12 0.00502100
\(496\) −4.15021e14 −0.620756
\(497\) −2.22798e14 −0.329573
\(498\) 2.49652e14 0.365235
\(499\) 1.05041e15 1.51987 0.759937 0.649997i \(-0.225230\pi\)
0.759937 + 0.649997i \(0.225230\pi\)
\(500\) −1.09962e13 −0.0157365
\(501\) −7.58080e14 −1.07302
\(502\) 5.94410e14 0.832177
\(503\) −9.32550e14 −1.29136 −0.645681 0.763607i \(-0.723427\pi\)
−0.645681 + 0.763607i \(0.723427\pi\)
\(504\) 9.79009e13 0.134097
\(505\) 4.30307e12 0.00583009
\(506\) 2.67031e14 0.357877
\(507\) −3.34996e13 −0.0444116
\(508\) −1.40768e14 −0.184609
\(509\) 1.21853e15 1.58084 0.790418 0.612568i \(-0.209863\pi\)
0.790418 + 0.612568i \(0.209863\pi\)
\(510\) 1.64974e11 0.000211729 0
\(511\) −1.45155e14 −0.184296
\(512\) 8.94569e14 1.12365
\(513\) −2.03700e14 −0.253131
\(514\) −6.73111e14 −0.827541
\(515\) 5.02393e13 0.0611088
\(516\) −5.89946e13 −0.0709968
\(517\) 3.09438e14 0.368448
\(518\) 4.41560e14 0.520207
\(519\) 5.83430e14 0.680093
\(520\) 1.22716e13 0.0141541
\(521\) −1.21714e15 −1.38910 −0.694551 0.719443i \(-0.744397\pi\)
−0.694551 + 0.719443i \(0.744397\pi\)
\(522\) 2.19735e14 0.248148
\(523\) 1.58024e15 1.76589 0.882945 0.469477i \(-0.155558\pi\)
0.882945 + 0.469477i \(0.155558\pi\)
\(524\) −5.18384e13 −0.0573230
\(525\) 1.98961e14 0.217716
\(526\) −3.62996e14 −0.393079
\(527\) −5.99281e12 −0.00642202
\(528\) −1.38315e14 −0.146684
\(529\) 5.26729e14 0.552816
\(530\) 4.03459e13 0.0419066
\(531\) 1.99258e14 0.204831
\(532\) −8.02796e13 −0.0816754
\(533\) 1.61821e14 0.162943
\(534\) 5.49586e14 0.547721
\(535\) −6.58920e13 −0.0649961
\(536\) 1.33610e15 1.30446
\(537\) −5.91066e14 −0.571186
\(538\) −3.22138e14 −0.308134
\(539\) −4.74009e13 −0.0448795
\(540\) −1.61756e12 −0.00151599
\(541\) −1.10910e15 −1.02893 −0.514465 0.857511i \(-0.672010\pi\)
−0.514465 + 0.857511i \(0.672010\pi\)
\(542\) 1.05398e15 0.967917
\(543\) 3.32142e14 0.301943
\(544\) 3.02202e12 0.00271959
\(545\) −1.01210e14 −0.0901662
\(546\) −6.27344e13 −0.0553282
\(547\) −5.12831e14 −0.447759 −0.223879 0.974617i \(-0.571872\pi\)
−0.223879 + 0.974617i \(0.571872\pi\)
\(548\) −1.92574e14 −0.166458
\(549\) −1.48072e14 −0.126714
\(550\) −3.38196e14 −0.286533
\(551\) −1.27693e15 −1.07111
\(552\) −9.22048e14 −0.765753
\(553\) 3.04514e14 0.250391
\(554\) 1.53117e15 1.24658
\(555\) −5.17028e13 −0.0416776
\(556\) −1.05058e14 −0.0838531
\(557\) −1.05328e15 −0.832414 −0.416207 0.909270i \(-0.636641\pi\)
−0.416207 + 0.909270i \(0.636641\pi\)
\(558\) −2.98895e14 −0.233900
\(559\) 2.67905e14 0.207594
\(560\) 1.91006e13 0.0146559
\(561\) −1.99724e12 −0.00151752
\(562\) −1.19116e15 −0.896231
\(563\) −6.85740e14 −0.510932 −0.255466 0.966818i \(-0.582229\pi\)
−0.255466 + 0.966818i \(0.582229\pi\)
\(564\) −1.50771e14 −0.111245
\(565\) 1.38981e13 0.0101552
\(566\) 7.41058e14 0.536243
\(567\) 5.86024e13 0.0419961
\(568\) −1.30769e15 −0.928088
\(569\) −1.33364e15 −0.937392 −0.468696 0.883360i \(-0.655276\pi\)
−0.468696 + 0.883360i \(0.655276\pi\)
\(570\) −4.78159e13 −0.0332859
\(571\) 2.27533e15 1.56872 0.784361 0.620305i \(-0.212991\pi\)
0.784361 + 0.620305i \(0.212991\pi\)
\(572\) −2.09635e13 −0.0143148
\(573\) 8.97954e14 0.607301
\(574\) 3.03041e14 0.202995
\(575\) −1.87384e15 −1.24326
\(576\) 5.60929e14 0.368624
\(577\) 1.97105e13 0.0128301 0.00641505 0.999979i \(-0.497958\pi\)
0.00641505 + 0.999979i \(0.497958\pi\)
\(578\) 1.41775e15 0.914108
\(579\) 1.17950e15 0.753293
\(580\) −1.01400e13 −0.00641478
\(581\) 4.17374e14 0.261551
\(582\) −3.69938e14 −0.229643
\(583\) −4.88441e14 −0.300355
\(584\) −8.51970e14 −0.518984
\(585\) 7.34564e12 0.00443274
\(586\) 2.45299e15 1.46642
\(587\) 2.33373e15 1.38211 0.691053 0.722804i \(-0.257147\pi\)
0.691053 + 0.722804i \(0.257147\pi\)
\(588\) 2.30956e13 0.0135504
\(589\) 1.73694e15 1.00961
\(590\) 4.67732e13 0.0269346
\(591\) 5.66398e14 0.323140
\(592\) 2.15409e15 1.21757
\(593\) −1.46795e15 −0.822074 −0.411037 0.911619i \(-0.634833\pi\)
−0.411037 + 0.911619i \(0.634833\pi\)
\(594\) −9.96133e13 −0.0552703
\(595\) 2.75809e11 0.000151623 0
\(596\) 4.90168e14 0.266987
\(597\) −1.66480e15 −0.898472
\(598\) 5.90844e14 0.315948
\(599\) −1.25723e15 −0.666140 −0.333070 0.942902i \(-0.608085\pi\)
−0.333070 + 0.942902i \(0.608085\pi\)
\(600\) 1.16778e15 0.613096
\(601\) −1.97279e15 −1.02629 −0.513147 0.858301i \(-0.671520\pi\)
−0.513147 + 0.858301i \(0.671520\pi\)
\(602\) 5.01703e14 0.258622
\(603\) 7.99772e14 0.408527
\(604\) −2.83769e14 −0.143636
\(605\) 8.61573e13 0.0432153
\(606\) −1.29115e14 −0.0641766
\(607\) −3.46544e15 −1.70695 −0.853476 0.521133i \(-0.825510\pi\)
−0.853476 + 0.521133i \(0.825510\pi\)
\(608\) −8.75897e14 −0.427546
\(609\) 3.67358e14 0.177703
\(610\) −3.47579e13 −0.0166625
\(611\) 6.84676e14 0.325281
\(612\) 9.73132e11 0.000458183 0
\(613\) 2.07469e15 0.968101 0.484051 0.875040i \(-0.339165\pi\)
0.484051 + 0.875040i \(0.339165\pi\)
\(614\) −3.08491e15 −1.42665
\(615\) −3.54834e13 −0.0162634
\(616\) −2.78215e14 −0.126382
\(617\) −7.15598e14 −0.322182 −0.161091 0.986940i \(-0.551501\pi\)
−0.161091 + 0.986940i \(0.551501\pi\)
\(618\) −1.50745e15 −0.672675
\(619\) 3.32049e15 1.46860 0.734300 0.678825i \(-0.237511\pi\)
0.734300 + 0.678825i \(0.237511\pi\)
\(620\) 1.37929e13 0.00604646
\(621\) −5.51927e14 −0.239816
\(622\) 2.95722e15 1.27361
\(623\) 9.18813e14 0.392232
\(624\) −3.06041e14 −0.129499
\(625\) 2.36775e15 0.993108
\(626\) 1.37828e14 0.0573032
\(627\) 5.78875e14 0.238569
\(628\) 7.67444e14 0.313522
\(629\) 3.11046e13 0.0125964
\(630\) 1.37561e13 0.00552234
\(631\) 1.41030e15 0.561242 0.280621 0.959819i \(-0.409460\pi\)
0.280621 + 0.959819i \(0.409460\pi\)
\(632\) 1.78731e15 0.705109
\(633\) −1.72465e15 −0.674496
\(634\) −1.34248e15 −0.520495
\(635\) −1.40172e14 −0.0538775
\(636\) 2.37988e14 0.0906861
\(637\) −1.04881e14 −0.0396214
\(638\) −6.24441e14 −0.233872
\(639\) −7.82769e14 −0.290656
\(640\) 8.93344e13 0.0328873
\(641\) −3.17728e15 −1.15967 −0.579837 0.814733i \(-0.696884\pi\)
−0.579837 + 0.814733i \(0.696884\pi\)
\(642\) 1.97711e15 0.715466
\(643\) −2.15907e15 −0.774650 −0.387325 0.921943i \(-0.626601\pi\)
−0.387325 + 0.921943i \(0.626601\pi\)
\(644\) −2.17518e14 −0.0773790
\(645\) −5.87450e13 −0.0207201
\(646\) 2.87662e13 0.0100601
\(647\) 5.05672e14 0.175346 0.0876729 0.996149i \(-0.472057\pi\)
0.0876729 + 0.996149i \(0.472057\pi\)
\(648\) 3.43961e14 0.118262
\(649\) −5.66252e14 −0.193047
\(650\) −7.48307e14 −0.252963
\(651\) −4.99700e14 −0.167500
\(652\) −3.10346e14 −0.103154
\(653\) 9.87951e14 0.325621 0.162811 0.986657i \(-0.447944\pi\)
0.162811 + 0.986657i \(0.447944\pi\)
\(654\) 3.03685e15 0.992534
\(655\) −5.16190e13 −0.0167295
\(656\) 1.47835e15 0.475122
\(657\) −5.09980e14 −0.162534
\(658\) 1.28219e15 0.405237
\(659\) −5.39788e15 −1.69182 −0.845908 0.533329i \(-0.820941\pi\)
−0.845908 + 0.533329i \(0.820941\pi\)
\(660\) 4.59679e12 0.00142877
\(661\) 1.62009e15 0.499381 0.249691 0.968326i \(-0.419671\pi\)
0.249691 + 0.968326i \(0.419671\pi\)
\(662\) −3.94077e14 −0.120465
\(663\) −4.41917e12 −0.00133973
\(664\) 2.44974e15 0.736537
\(665\) −7.99399e13 −0.0238366
\(666\) 1.55136e15 0.458779
\(667\) −3.45984e15 −1.01476
\(668\) −1.04967e15 −0.305338
\(669\) −2.48765e15 −0.717705
\(670\) 1.87736e14 0.0537199
\(671\) 4.20791e14 0.119424
\(672\) 2.51986e14 0.0709326
\(673\) −4.65049e15 −1.29842 −0.649211 0.760608i \(-0.724901\pi\)
−0.649211 + 0.760608i \(0.724901\pi\)
\(674\) −4.99870e15 −1.38429
\(675\) 6.99020e14 0.192008
\(676\) −4.63848e13 −0.0126377
\(677\) −1.86212e15 −0.503233 −0.251616 0.967827i \(-0.580962\pi\)
−0.251616 + 0.967827i \(0.580962\pi\)
\(678\) −4.17018e14 −0.111787
\(679\) −6.18472e14 −0.164451
\(680\) 1.61883e12 0.000426975 0
\(681\) 2.59938e15 0.680082
\(682\) 8.49399e14 0.220444
\(683\) 2.33245e15 0.600480 0.300240 0.953864i \(-0.402933\pi\)
0.300240 + 0.953864i \(0.402933\pi\)
\(684\) −2.82051e14 −0.0720309
\(685\) −1.91760e14 −0.0485802
\(686\) −1.96410e14 −0.0493606
\(687\) 3.43965e15 0.857537
\(688\) 2.44749e15 0.605320
\(689\) −1.08074e15 −0.265166
\(690\) −1.29558e14 −0.0315350
\(691\) 4.93407e15 1.19145 0.595725 0.803188i \(-0.296865\pi\)
0.595725 + 0.803188i \(0.296865\pi\)
\(692\) 8.07838e14 0.193527
\(693\) −1.66536e14 −0.0395800
\(694\) 3.82130e15 0.901020
\(695\) −1.04614e14 −0.0244722
\(696\) 2.15617e15 0.500418
\(697\) 2.13470e13 0.00491537
\(698\) 5.07666e15 1.15977
\(699\) 1.82483e15 0.413616
\(700\) 2.75488e14 0.0619532
\(701\) 5.07900e14 0.113326 0.0566629 0.998393i \(-0.481954\pi\)
0.0566629 + 0.998393i \(0.481954\pi\)
\(702\) −2.20408e14 −0.0487949
\(703\) −9.01529e15 −1.98027
\(704\) −1.59405e15 −0.347417
\(705\) −1.50133e14 −0.0324665
\(706\) 6.32100e15 1.35631
\(707\) −2.15858e14 −0.0459580
\(708\) 2.75900e14 0.0582867
\(709\) −3.82699e15 −0.802238 −0.401119 0.916026i \(-0.631379\pi\)
−0.401119 + 0.916026i \(0.631379\pi\)
\(710\) −1.83744e14 −0.0382202
\(711\) 1.06986e15 0.220824
\(712\) 5.39288e15 1.10454
\(713\) 4.70626e15 0.956497
\(714\) −8.27573e12 −0.00166904
\(715\) −2.08748e13 −0.00417773
\(716\) −8.18412e14 −0.162536
\(717\) 2.29120e15 0.451551
\(718\) 6.28601e15 1.22939
\(719\) −4.52496e15 −0.878225 −0.439112 0.898432i \(-0.644707\pi\)
−0.439112 + 0.898432i \(0.644707\pi\)
\(720\) 6.71074e13 0.0129253
\(721\) −2.52019e15 −0.481714
\(722\) −3.51826e15 −0.667381
\(723\) −4.65522e15 −0.876354
\(724\) 4.59896e14 0.0859206
\(725\) 4.38191e15 0.812465
\(726\) −2.58518e15 −0.475707
\(727\) 4.60621e15 0.841210 0.420605 0.907244i \(-0.361818\pi\)
0.420605 + 0.907244i \(0.361818\pi\)
\(728\) −6.15589e14 −0.111575
\(729\) 2.05891e14 0.0370370
\(730\) −1.19711e14 −0.0213726
\(731\) 3.53412e13 0.00626232
\(732\) −2.05026e14 −0.0360577
\(733\) 4.27153e15 0.745610 0.372805 0.927910i \(-0.378396\pi\)
0.372805 + 0.927910i \(0.378396\pi\)
\(734\) 4.11417e15 0.712778
\(735\) 2.29978e13 0.00395464
\(736\) −2.37325e15 −0.405056
\(737\) −2.27279e15 −0.385024
\(738\) 1.06469e15 0.179025
\(739\) −1.24314e15 −0.207479 −0.103740 0.994604i \(-0.533081\pi\)
−0.103740 + 0.994604i \(0.533081\pi\)
\(740\) −7.15895e13 −0.0118597
\(741\) 1.28084e15 0.210618
\(742\) −2.02390e15 −0.330345
\(743\) 4.37560e15 0.708923 0.354461 0.935071i \(-0.384664\pi\)
0.354461 + 0.935071i \(0.384664\pi\)
\(744\) −2.93294e15 −0.471685
\(745\) 4.88094e14 0.0779190
\(746\) −4.31922e15 −0.684450
\(747\) 1.46638e15 0.230666
\(748\) −2.76545e12 −0.000431824 0
\(749\) 3.30539e15 0.512357
\(750\) 3.28549e14 0.0505549
\(751\) −5.32766e15 −0.813798 −0.406899 0.913473i \(-0.633390\pi\)
−0.406899 + 0.913473i \(0.633390\pi\)
\(752\) 6.25497e15 0.948478
\(753\) 3.49140e15 0.525566
\(754\) −1.38166e15 −0.206472
\(755\) −2.82569e14 −0.0419195
\(756\) 8.11430e13 0.0119504
\(757\) −2.89374e15 −0.423090 −0.211545 0.977368i \(-0.567850\pi\)
−0.211545 + 0.977368i \(0.567850\pi\)
\(758\) −5.19879e14 −0.0754609
\(759\) 1.56847e15 0.226019
\(760\) −4.69199e14 −0.0671247
\(761\) 3.90539e15 0.554687 0.277344 0.960771i \(-0.410546\pi\)
0.277344 + 0.960771i \(0.410546\pi\)
\(762\) 4.20592e15 0.593074
\(763\) 5.07708e15 0.710770
\(764\) 1.24334e15 0.172813
\(765\) 9.69015e11 0.000133719 0
\(766\) −9.09471e15 −1.24604
\(767\) −1.25291e15 −0.170430
\(768\) 2.04699e15 0.276458
\(769\) −1.21703e16 −1.63195 −0.815977 0.578085i \(-0.803800\pi\)
−0.815977 + 0.578085i \(0.803800\pi\)
\(770\) −3.90921e13 −0.00520463
\(771\) −3.95367e15 −0.522638
\(772\) 1.63317e15 0.214357
\(773\) −2.92041e15 −0.380590 −0.190295 0.981727i \(-0.560944\pi\)
−0.190295 + 0.981727i \(0.560944\pi\)
\(774\) 1.76266e15 0.228083
\(775\) −5.96051e15 −0.765816
\(776\) −3.63006e15 −0.463100
\(777\) 2.59360e15 0.328540
\(778\) 1.57279e15 0.197826
\(779\) −6.18716e15 −0.772744
\(780\) 1.01710e13 0.00126138
\(781\) 2.22447e15 0.273934
\(782\) 7.79423e13 0.00953094
\(783\) 1.29066e15 0.156719
\(784\) −9.58159e14 −0.115531
\(785\) 7.64197e14 0.0915002
\(786\) 1.54885e15 0.184155
\(787\) −9.47089e15 −1.11823 −0.559113 0.829091i \(-0.688858\pi\)
−0.559113 + 0.829091i \(0.688858\pi\)
\(788\) 7.84255e14 0.0919524
\(789\) −2.13214e15 −0.248251
\(790\) 2.51136e14 0.0290376
\(791\) −6.97182e14 −0.0800526
\(792\) −9.77467e14 −0.111459
\(793\) 9.31059e14 0.105433
\(794\) −9.48967e15 −1.06718
\(795\) 2.36981e14 0.0264663
\(796\) −2.30515e15 −0.255668
\(797\) −7.51916e15 −0.828225 −0.414113 0.910226i \(-0.635908\pi\)
−0.414113 + 0.910226i \(0.635908\pi\)
\(798\) 2.39862e15 0.262389
\(799\) 9.03203e13 0.00981247
\(800\) 3.00574e15 0.324307
\(801\) 3.22812e15 0.345916
\(802\) 1.59223e16 1.69452
\(803\) 1.44926e15 0.153183
\(804\) 1.10739e15 0.116250
\(805\) −2.16598e14 −0.0225827
\(806\) 1.87941e15 0.194617
\(807\) −1.89215e15 −0.194604
\(808\) −1.26696e15 −0.129419
\(809\) 3.62657e15 0.367942 0.183971 0.982932i \(-0.441105\pi\)
0.183971 + 0.982932i \(0.441105\pi\)
\(810\) 4.83302e13 0.00487024
\(811\) −2.00263e15 −0.200440 −0.100220 0.994965i \(-0.531955\pi\)
−0.100220 + 0.994965i \(0.531955\pi\)
\(812\) 5.08658e14 0.0505670
\(813\) 6.19082e15 0.611294
\(814\) −4.40865e15 −0.432386
\(815\) −3.09033e14 −0.0301050
\(816\) −4.03720e13 −0.00390647
\(817\) −1.02432e16 −0.984499
\(818\) −5.41888e15 −0.517328
\(819\) −3.68485e14 −0.0349428
\(820\) −4.91317e13 −0.00462791
\(821\) −1.90716e16 −1.78443 −0.892217 0.451608i \(-0.850851\pi\)
−0.892217 + 0.451608i \(0.850851\pi\)
\(822\) 5.75381e15 0.534762
\(823\) 5.30225e15 0.489509 0.244755 0.969585i \(-0.421293\pi\)
0.244755 + 0.969585i \(0.421293\pi\)
\(824\) −1.47920e16 −1.35652
\(825\) −1.98647e15 −0.180961
\(826\) −2.34632e15 −0.212323
\(827\) −5.28776e15 −0.475326 −0.237663 0.971348i \(-0.576381\pi\)
−0.237663 + 0.971348i \(0.576381\pi\)
\(828\) −7.64219e14 −0.0682419
\(829\) −1.08973e16 −0.966653 −0.483327 0.875440i \(-0.660572\pi\)
−0.483327 + 0.875440i \(0.660572\pi\)
\(830\) 3.44214e14 0.0303319
\(831\) 8.99370e15 0.787286
\(832\) −3.52705e15 −0.306714
\(833\) −1.38356e13 −0.00119523
\(834\) 3.13897e15 0.269385
\(835\) −1.04522e15 −0.0891115
\(836\) 8.01532e14 0.0678869
\(837\) −1.75563e15 −0.147721
\(838\) 2.18468e15 0.182619
\(839\) 2.82658e15 0.234731 0.117365 0.993089i \(-0.462555\pi\)
0.117365 + 0.993089i \(0.462555\pi\)
\(840\) 1.34984e14 0.0111364
\(841\) −4.10980e15 −0.336855
\(842\) 4.72868e15 0.385055
\(843\) −6.99654e15 −0.566020
\(844\) −2.38801e15 −0.191934
\(845\) −4.61885e13 −0.00368827
\(846\) 4.50478e15 0.357385
\(847\) −4.32197e15 −0.340662
\(848\) −9.87332e15 −0.773190
\(849\) 4.35277e15 0.338667
\(850\) −9.87144e13 −0.00763091
\(851\) −2.44270e16 −1.87611
\(852\) −1.08385e15 −0.0827088
\(853\) −1.07402e16 −0.814314 −0.407157 0.913358i \(-0.633480\pi\)
−0.407157 + 0.913358i \(0.633480\pi\)
\(854\) 1.74359e15 0.131348
\(855\) −2.80857e14 −0.0210219
\(856\) 1.94007e16 1.44282
\(857\) −1.52126e16 −1.12411 −0.562055 0.827100i \(-0.689989\pi\)
−0.562055 + 0.827100i \(0.689989\pi\)
\(858\) 6.26356e14 0.0459877
\(859\) 7.54484e15 0.550412 0.275206 0.961385i \(-0.411254\pi\)
0.275206 + 0.961385i \(0.411254\pi\)
\(860\) −8.13404e13 −0.00589610
\(861\) 1.77998e15 0.128203
\(862\) −2.69833e15 −0.193110
\(863\) −2.70835e16 −1.92595 −0.962976 0.269586i \(-0.913113\pi\)
−0.962976 + 0.269586i \(0.913113\pi\)
\(864\) 8.85318e14 0.0625567
\(865\) 8.04420e14 0.0564799
\(866\) −9.92217e15 −0.692243
\(867\) 8.32749e15 0.577310
\(868\) −6.91903e14 −0.0476636
\(869\) −3.04034e15 −0.208120
\(870\) 3.02965e14 0.0206080
\(871\) −5.02887e15 −0.339915
\(872\) 2.97994e16 2.00155
\(873\) −2.17291e15 −0.145032
\(874\) −2.25906e16 −1.49836
\(875\) 5.49277e14 0.0362032
\(876\) −7.06137e14 −0.0462505
\(877\) −2.46809e16 −1.60643 −0.803217 0.595686i \(-0.796880\pi\)
−0.803217 + 0.595686i \(0.796880\pi\)
\(878\) −9.36070e15 −0.605464
\(879\) 1.44082e16 0.926126
\(880\) −1.90706e14 −0.0121817
\(881\) −8.87922e15 −0.563647 −0.281824 0.959466i \(-0.590939\pi\)
−0.281824 + 0.959466i \(0.590939\pi\)
\(882\) −6.90058e14 −0.0435320
\(883\) −1.36250e16 −0.854188 −0.427094 0.904207i \(-0.640463\pi\)
−0.427094 + 0.904207i \(0.640463\pi\)
\(884\) −6.11894e12 −0.000381231 0
\(885\) 2.74733e14 0.0170107
\(886\) 7.28293e15 0.448147
\(887\) 1.64891e15 0.100837 0.0504183 0.998728i \(-0.483945\pi\)
0.0504183 + 0.998728i \(0.483945\pi\)
\(888\) 1.52229e16 0.925179
\(889\) 7.03157e15 0.424710
\(890\) 7.57757e14 0.0454868
\(891\) −5.85101e14 −0.0349063
\(892\) −3.44449e15 −0.204230
\(893\) −2.61783e16 −1.54262
\(894\) −1.46454e16 −0.857719
\(895\) −8.14949e14 −0.0474355
\(896\) −4.48134e15 −0.259247
\(897\) 3.47045e15 0.199539
\(898\) −2.02787e16 −1.15883
\(899\) −1.10054e16 −0.625069
\(900\) 9.67888e14 0.0546376
\(901\) −1.42568e14 −0.00799903
\(902\) −3.02564e15 −0.168726
\(903\) 2.94687e15 0.163334
\(904\) −4.09204e15 −0.225431
\(905\) 4.57950e14 0.0250756
\(906\) 8.47857e15 0.461442
\(907\) 1.02159e16 0.552634 0.276317 0.961066i \(-0.410886\pi\)
0.276317 + 0.961066i \(0.410886\pi\)
\(908\) 3.59920e15 0.193524
\(909\) −7.58386e14 −0.0405311
\(910\) −8.64968e13 −0.00459486
\(911\) −8.31181e14 −0.0438879 −0.0219439 0.999759i \(-0.506986\pi\)
−0.0219439 + 0.999759i \(0.506986\pi\)
\(912\) 1.17014e16 0.614136
\(913\) −4.16717e15 −0.217396
\(914\) −1.24841e16 −0.647369
\(915\) −2.04158e14 −0.0105233
\(916\) 4.76267e15 0.244020
\(917\) 2.58940e15 0.131877
\(918\) −2.90756e13 −0.00147195
\(919\) 1.89639e15 0.0954318 0.0477159 0.998861i \(-0.484806\pi\)
0.0477159 + 0.998861i \(0.484806\pi\)
\(920\) −1.27130e15 −0.0635938
\(921\) −1.81199e16 −0.901007
\(922\) −1.62025e16 −0.800869
\(923\) 4.92196e15 0.241840
\(924\) −2.30592e14 −0.0112629
\(925\) 3.09369e16 1.50210
\(926\) −1.79898e16 −0.868290
\(927\) −8.85432e15 −0.424832
\(928\) 5.54975e15 0.264703
\(929\) −2.21251e16 −1.04906 −0.524528 0.851393i \(-0.675758\pi\)
−0.524528 + 0.851393i \(0.675758\pi\)
\(930\) −4.12109e14 −0.0194248
\(931\) 4.01008e15 0.187901
\(932\) 2.52672e15 0.117698
\(933\) 1.73699e16 0.804355
\(934\) 1.74294e16 0.802371
\(935\) −2.75374e12 −0.000126026 0
\(936\) −2.16278e15 −0.0984002
\(937\) −3.28940e15 −0.148782 −0.0743908 0.997229i \(-0.523701\pi\)
−0.0743908 + 0.997229i \(0.523701\pi\)
\(938\) −9.41752e15 −0.423468
\(939\) 8.09564e14 0.0361902
\(940\) −2.07879e14 −0.00923864
\(941\) 1.06884e16 0.472248 0.236124 0.971723i \(-0.424123\pi\)
0.236124 + 0.971723i \(0.424123\pi\)
\(942\) −2.29300e16 −1.00722
\(943\) −1.67642e16 −0.732095
\(944\) −1.14462e16 −0.496953
\(945\) 8.07997e13 0.00348766
\(946\) −5.00913e15 −0.214962
\(947\) −8.74933e15 −0.373293 −0.186646 0.982427i \(-0.559762\pi\)
−0.186646 + 0.982427i \(0.559762\pi\)
\(948\) 1.48137e15 0.0628375
\(949\) 3.20669e15 0.135236
\(950\) 2.86112e16 1.19965
\(951\) −7.88536e15 −0.328722
\(952\) −8.12066e13 −0.00336580
\(953\) 3.98003e16 1.64012 0.820060 0.572277i \(-0.193940\pi\)
0.820060 + 0.572277i \(0.193940\pi\)
\(954\) −7.11069e15 −0.291337
\(955\) 1.23808e15 0.0504348
\(956\) 3.17247e15 0.128493
\(957\) −3.66780e15 −0.147703
\(958\) 1.54017e16 0.616677
\(959\) 9.61937e15 0.382952
\(960\) 7.73396e14 0.0306133
\(961\) −1.04383e16 −0.410821
\(962\) −9.75475e15 −0.381727
\(963\) 1.16130e16 0.451856
\(964\) −6.44579e15 −0.249375
\(965\) 1.62626e15 0.0625591
\(966\) 6.49909e15 0.248587
\(967\) 7.95672e15 0.302614 0.151307 0.988487i \(-0.451652\pi\)
0.151307 + 0.988487i \(0.451652\pi\)
\(968\) −2.53674e16 −0.959315
\(969\) 1.68965e14 0.00635354
\(970\) −5.10062e14 −0.0190712
\(971\) −1.39864e16 −0.519998 −0.259999 0.965609i \(-0.583722\pi\)
−0.259999 + 0.965609i \(0.583722\pi\)
\(972\) 2.85084e14 0.0105392
\(973\) 5.24781e15 0.192911
\(974\) 2.41966e16 0.884462
\(975\) −4.39535e15 −0.159760
\(976\) 8.50585e15 0.307428
\(977\) 7.04353e13 0.00253145 0.00126573 0.999999i \(-0.499597\pi\)
0.00126573 + 0.999999i \(0.499597\pi\)
\(978\) 9.27264e15 0.331390
\(979\) −9.17366e15 −0.326015
\(980\) 3.18437e13 0.00112533
\(981\) 1.78376e16 0.626840
\(982\) 4.43820e16 1.55094
\(983\) −2.51147e16 −0.872737 −0.436369 0.899768i \(-0.643736\pi\)
−0.436369 + 0.899768i \(0.643736\pi\)
\(984\) 1.04474e16 0.361024
\(985\) 7.80936e14 0.0268359
\(986\) −1.82265e14 −0.00622845
\(987\) 7.53121e15 0.255930
\(988\) 1.77350e15 0.0599333
\(989\) −2.77541e16 −0.932712
\(990\) −1.37345e14 −0.00459006
\(991\) 1.59790e16 0.531060 0.265530 0.964103i \(-0.414453\pi\)
0.265530 + 0.964103i \(0.414453\pi\)
\(992\) −7.54907e15 −0.249505
\(993\) −2.31470e15 −0.0760806
\(994\) 9.21730e15 0.301286
\(995\) −2.29540e15 −0.0746158
\(996\) 2.03041e15 0.0656383
\(997\) −2.69128e16 −0.865239 −0.432619 0.901577i \(-0.642411\pi\)
−0.432619 + 0.901577i \(0.642411\pi\)
\(998\) −4.34563e16 −1.38943
\(999\) 9.11225e15 0.289745
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 273.12.a.c.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.6 16 1.1 even 1 trivial