Properties

Label 273.12.a.c.1.12
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 1.12
Root \(41.2629\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q+37.2629 q^{2} -243.000 q^{3} -659.475 q^{4} +3035.84 q^{5} -9054.89 q^{6} +16807.0 q^{7} -100888. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+37.2629 q^{2} -243.000 q^{3} -659.475 q^{4} +3035.84 q^{5} -9054.89 q^{6} +16807.0 q^{7} -100888. q^{8} +59049.0 q^{9} +113124. q^{10} +349838. q^{11} +160252. q^{12} -371293. q^{13} +626278. q^{14} -737710. q^{15} -2.40879e6 q^{16} +4.65704e6 q^{17} +2.20034e6 q^{18} -4.50229e6 q^{19} -2.00206e6 q^{20} -4.08410e6 q^{21} +1.30360e7 q^{22} -3.89018e7 q^{23} +2.45159e7 q^{24} -3.96118e7 q^{25} -1.38355e7 q^{26} -1.43489e7 q^{27} -1.10838e7 q^{28} +7.68779e7 q^{29} -2.74892e7 q^{30} -5.73477e7 q^{31} +1.16861e8 q^{32} -8.50107e7 q^{33} +1.73535e8 q^{34} +5.10234e7 q^{35} -3.89413e7 q^{36} +2.31992e8 q^{37} -1.67768e8 q^{38} +9.02242e7 q^{39} -3.06281e8 q^{40} +4.48764e7 q^{41} -1.52186e8 q^{42} +6.56941e8 q^{43} -2.30710e8 q^{44} +1.79263e8 q^{45} -1.44960e9 q^{46} +2.23582e9 q^{47} +5.85336e8 q^{48} +2.82475e8 q^{49} -1.47605e9 q^{50} -1.13166e9 q^{51} +2.44858e8 q^{52} +4.17552e9 q^{53} -5.34682e8 q^{54} +1.06205e9 q^{55} -1.69563e9 q^{56} +1.09406e9 q^{57} +2.86469e9 q^{58} +5.16620e9 q^{59} +4.86501e8 q^{60} -9.97103e9 q^{61} -2.13694e9 q^{62} +9.92437e8 q^{63} +9.28778e9 q^{64} -1.12719e9 q^{65} -3.16775e9 q^{66} -3.36147e9 q^{67} -3.07120e9 q^{68} +9.45315e9 q^{69} +1.90128e9 q^{70} -1.55923e10 q^{71} -5.95736e9 q^{72} -1.68502e10 q^{73} +8.64471e9 q^{74} +9.62566e9 q^{75} +2.96915e9 q^{76} +5.87973e9 q^{77} +3.36202e9 q^{78} -3.64367e10 q^{79} -7.31271e9 q^{80} +3.48678e9 q^{81} +1.67222e9 q^{82} +2.09943e10 q^{83} +2.69336e9 q^{84} +1.41380e10 q^{85} +2.44795e10 q^{86} -1.86813e10 q^{87} -3.52946e10 q^{88} -3.68249e10 q^{89} +6.67988e9 q^{90} -6.24032e9 q^{91} +2.56548e10 q^{92} +1.39355e10 q^{93} +8.33131e10 q^{94} -1.36682e10 q^{95} -2.83972e10 q^{96} +1.38985e11 q^{97} +1.05259e10 q^{98} +2.06576e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 37.2629 0.823402 0.411701 0.911319i \(-0.364935\pi\)
0.411701 + 0.911319i \(0.364935\pi\)
\(3\) −243.000 −0.577350
\(4\) −659.475 −0.322009
\(5\) 3035.84 0.434454 0.217227 0.976121i \(-0.430299\pi\)
0.217227 + 0.976121i \(0.430299\pi\)
\(6\) −9054.89 −0.475391
\(7\) 16807.0 0.377964
\(8\) −100888. −1.08854
\(9\) 59049.0 0.333333
\(10\) 113124. 0.357731
\(11\) 349838. 0.654949 0.327474 0.944860i \(-0.393802\pi\)
0.327474 + 0.944860i \(0.393802\pi\)
\(12\) 160252. 0.185912
\(13\) −371293. −0.277350
\(14\) 626278. 0.311217
\(15\) −737710. −0.250832
\(16\) −2.40879e6 −0.574301
\(17\) 4.65704e6 0.795502 0.397751 0.917493i \(-0.369791\pi\)
0.397751 + 0.917493i \(0.369791\pi\)
\(18\) 2.20034e6 0.274467
\(19\) −4.50229e6 −0.417146 −0.208573 0.978007i \(-0.566882\pi\)
−0.208573 + 0.978007i \(0.566882\pi\)
\(20\) −2.00206e6 −0.139898
\(21\) −4.08410e6 −0.218218
\(22\) 1.30360e7 0.539286
\(23\) −3.89018e7 −1.26028 −0.630140 0.776481i \(-0.717003\pi\)
−0.630140 + 0.776481i \(0.717003\pi\)
\(24\) 2.45159e7 0.628472
\(25\) −3.96118e7 −0.811249
\(26\) −1.38355e7 −0.228371
\(27\) −1.43489e7 −0.192450
\(28\) −1.10838e7 −0.121708
\(29\) 7.68779e7 0.696005 0.348002 0.937494i \(-0.386860\pi\)
0.348002 + 0.937494i \(0.386860\pi\)
\(30\) −2.74892e7 −0.206536
\(31\) −5.73477e7 −0.359771 −0.179886 0.983688i \(-0.557573\pi\)
−0.179886 + 0.983688i \(0.557573\pi\)
\(32\) 1.16861e8 0.615665
\(33\) −8.50107e7 −0.378135
\(34\) 1.73535e8 0.655018
\(35\) 5.10234e7 0.164208
\(36\) −3.89413e7 −0.107336
\(37\) 2.31992e8 0.550002 0.275001 0.961444i \(-0.411322\pi\)
0.275001 + 0.961444i \(0.411322\pi\)
\(38\) −1.67768e8 −0.343479
\(39\) 9.02242e7 0.160128
\(40\) −3.06281e8 −0.472923
\(41\) 4.48764e7 0.0604932 0.0302466 0.999542i \(-0.490371\pi\)
0.0302466 + 0.999542i \(0.490371\pi\)
\(42\) −1.52186e8 −0.179681
\(43\) 6.56941e8 0.681475 0.340737 0.940158i \(-0.389323\pi\)
0.340737 + 0.940158i \(0.389323\pi\)
\(44\) −2.30710e8 −0.210900
\(45\) 1.79263e8 0.144818
\(46\) −1.44960e9 −1.03772
\(47\) 2.23582e9 1.42200 0.710998 0.703194i \(-0.248244\pi\)
0.710998 + 0.703194i \(0.248244\pi\)
\(48\) 5.85336e8 0.331573
\(49\) 2.82475e8 0.142857
\(50\) −1.47605e9 −0.667984
\(51\) −1.13166e9 −0.459283
\(52\) 2.44858e8 0.0893093
\(53\) 4.17552e9 1.37149 0.685746 0.727841i \(-0.259476\pi\)
0.685746 + 0.727841i \(0.259476\pi\)
\(54\) −5.34682e8 −0.158464
\(55\) 1.06205e9 0.284545
\(56\) −1.69563e9 −0.411431
\(57\) 1.09406e9 0.240839
\(58\) 2.86469e9 0.573092
\(59\) 5.16620e9 0.940773 0.470387 0.882460i \(-0.344114\pi\)
0.470387 + 0.882460i \(0.344114\pi\)
\(60\) 4.86501e8 0.0807704
\(61\) −9.97103e9 −1.51156 −0.755781 0.654824i \(-0.772743\pi\)
−0.755781 + 0.654824i \(0.772743\pi\)
\(62\) −2.13694e9 −0.296236
\(63\) 9.92437e8 0.125988
\(64\) 9.28778e9 1.08124
\(65\) −1.12719e9 −0.120496
\(66\) −3.16775e9 −0.311357
\(67\) −3.36147e9 −0.304171 −0.152086 0.988367i \(-0.548599\pi\)
−0.152086 + 0.988367i \(0.548599\pi\)
\(68\) −3.07120e9 −0.256159
\(69\) 9.45315e9 0.727623
\(70\) 1.90128e9 0.135209
\(71\) −1.55923e10 −1.02563 −0.512813 0.858500i \(-0.671397\pi\)
−0.512813 + 0.858500i \(0.671397\pi\)
\(72\) −5.95736e9 −0.362848
\(73\) −1.68502e10 −0.951327 −0.475663 0.879627i \(-0.657792\pi\)
−0.475663 + 0.879627i \(0.657792\pi\)
\(74\) 8.64471e9 0.452873
\(75\) 9.62566e9 0.468375
\(76\) 2.96915e9 0.134325
\(77\) 5.87973e9 0.247547
\(78\) 3.36202e9 0.131850
\(79\) −3.64367e10 −1.33226 −0.666132 0.745834i \(-0.732051\pi\)
−0.666132 + 0.745834i \(0.732051\pi\)
\(80\) −7.31271e9 −0.249507
\(81\) 3.48678e9 0.111111
\(82\) 1.67222e9 0.0498102
\(83\) 2.09943e10 0.585020 0.292510 0.956262i \(-0.405509\pi\)
0.292510 + 0.956262i \(0.405509\pi\)
\(84\) 2.69336e9 0.0702682
\(85\) 1.41380e10 0.345609
\(86\) 2.44795e10 0.561128
\(87\) −1.86813e10 −0.401839
\(88\) −3.52946e10 −0.712941
\(89\) −3.68249e10 −0.699032 −0.349516 0.936930i \(-0.613654\pi\)
−0.349516 + 0.936930i \(0.613654\pi\)
\(90\) 6.67988e9 0.119244
\(91\) −6.24032e9 −0.104828
\(92\) 2.56548e10 0.405822
\(93\) 1.39355e10 0.207714
\(94\) 8.33131e10 1.17087
\(95\) −1.36682e10 −0.181231
\(96\) −2.83972e10 −0.355454
\(97\) 1.38985e11 1.64332 0.821660 0.569978i \(-0.193048\pi\)
0.821660 + 0.569978i \(0.193048\pi\)
\(98\) 1.05259e10 0.117629
\(99\) 2.06576e10 0.218316
\(100\) 2.61230e10 0.261230
\(101\) −1.83966e11 −1.74168 −0.870842 0.491564i \(-0.836426\pi\)
−0.870842 + 0.491564i \(0.836426\pi\)
\(102\) −4.21690e10 −0.378175
\(103\) −1.80056e10 −0.153040 −0.0765198 0.997068i \(-0.524381\pi\)
−0.0765198 + 0.997068i \(0.524381\pi\)
\(104\) 3.74592e10 0.301908
\(105\) −1.23987e10 −0.0948057
\(106\) 1.55592e11 1.12929
\(107\) 1.92018e11 1.32352 0.661759 0.749716i \(-0.269810\pi\)
0.661759 + 0.749716i \(0.269810\pi\)
\(108\) 9.46275e9 0.0619707
\(109\) 1.61795e11 1.00721 0.503605 0.863934i \(-0.332007\pi\)
0.503605 + 0.863934i \(0.332007\pi\)
\(110\) 3.95752e10 0.234295
\(111\) −5.63742e10 −0.317544
\(112\) −4.04846e10 −0.217065
\(113\) −3.27978e11 −1.67461 −0.837303 0.546739i \(-0.815869\pi\)
−0.837303 + 0.546739i \(0.815869\pi\)
\(114\) 4.07677e10 0.198308
\(115\) −1.18100e11 −0.547534
\(116\) −5.06990e10 −0.224120
\(117\) −2.19245e10 −0.0924500
\(118\) 1.92508e11 0.774635
\(119\) 7.82709e10 0.300671
\(120\) 7.44264e10 0.273042
\(121\) −1.62925e11 −0.571042
\(122\) −3.71550e11 −1.24462
\(123\) −1.09050e10 −0.0349258
\(124\) 3.78194e10 0.115850
\(125\) −2.68490e11 −0.786905
\(126\) 3.69811e10 0.103739
\(127\) 4.72055e11 1.26786 0.633932 0.773389i \(-0.281440\pi\)
0.633932 + 0.773389i \(0.281440\pi\)
\(128\) 1.06759e11 0.274630
\(129\) −1.59637e11 −0.393450
\(130\) −4.20023e10 −0.0992166
\(131\) −3.40224e11 −0.770499 −0.385250 0.922812i \(-0.625885\pi\)
−0.385250 + 0.922812i \(0.625885\pi\)
\(132\) 5.60624e10 0.121763
\(133\) −7.56699e10 −0.157666
\(134\) −1.25258e11 −0.250455
\(135\) −4.35610e10 −0.0836108
\(136\) −4.69842e11 −0.865939
\(137\) 3.24673e11 0.574755 0.287378 0.957817i \(-0.407217\pi\)
0.287378 + 0.957817i \(0.407217\pi\)
\(138\) 3.52252e11 0.599126
\(139\) −9.62198e11 −1.57284 −0.786418 0.617695i \(-0.788067\pi\)
−0.786418 + 0.617695i \(0.788067\pi\)
\(140\) −3.36487e10 −0.0528766
\(141\) −5.43304e11 −0.820989
\(142\) −5.81014e11 −0.844503
\(143\) −1.29892e11 −0.181650
\(144\) −1.42237e11 −0.191434
\(145\) 2.33389e11 0.302382
\(146\) −6.27888e11 −0.783324
\(147\) −6.86415e10 −0.0824786
\(148\) −1.52993e11 −0.177106
\(149\) 1.15790e12 1.29165 0.645825 0.763485i \(-0.276513\pi\)
0.645825 + 0.763485i \(0.276513\pi\)
\(150\) 3.58680e11 0.385661
\(151\) −1.82109e12 −1.88781 −0.943903 0.330223i \(-0.892876\pi\)
−0.943903 + 0.330223i \(0.892876\pi\)
\(152\) 4.54228e11 0.454082
\(153\) 2.74994e11 0.265167
\(154\) 2.19096e11 0.203831
\(155\) −1.74098e11 −0.156304
\(156\) −5.95006e10 −0.0515628
\(157\) 8.12422e11 0.679725 0.339863 0.940475i \(-0.389619\pi\)
0.339863 + 0.940475i \(0.389619\pi\)
\(158\) −1.35774e12 −1.09699
\(159\) −1.01465e12 −0.791832
\(160\) 3.54771e11 0.267478
\(161\) −6.53823e11 −0.476341
\(162\) 1.29928e11 0.0914891
\(163\) −2.19880e12 −1.49677 −0.748384 0.663266i \(-0.769170\pi\)
−0.748384 + 0.663266i \(0.769170\pi\)
\(164\) −2.95949e10 −0.0194794
\(165\) −2.58079e11 −0.164282
\(166\) 7.82307e11 0.481707
\(167\) 1.76362e12 1.05067 0.525334 0.850896i \(-0.323940\pi\)
0.525334 + 0.850896i \(0.323940\pi\)
\(168\) 4.12038e11 0.237540
\(169\) 1.37858e11 0.0769231
\(170\) 5.26825e11 0.284575
\(171\) −2.65855e11 −0.139049
\(172\) −4.33236e11 −0.219441
\(173\) −4.54586e11 −0.223029 −0.111515 0.993763i \(-0.535570\pi\)
−0.111515 + 0.993763i \(0.535570\pi\)
\(174\) −6.96120e11 −0.330875
\(175\) −6.65755e11 −0.306623
\(176\) −8.42687e11 −0.376138
\(177\) −1.25539e12 −0.543156
\(178\) −1.37220e12 −0.575584
\(179\) −4.18301e12 −1.70136 −0.850681 0.525682i \(-0.823810\pi\)
−0.850681 + 0.525682i \(0.823810\pi\)
\(180\) −1.18220e11 −0.0466328
\(181\) −4.41868e12 −1.69068 −0.845338 0.534232i \(-0.820601\pi\)
−0.845338 + 0.534232i \(0.820601\pi\)
\(182\) −2.32533e11 −0.0863160
\(183\) 2.42296e12 0.872701
\(184\) 3.92475e12 1.37187
\(185\) 7.04292e11 0.238951
\(186\) 5.19277e11 0.171032
\(187\) 1.62921e12 0.521013
\(188\) −1.47447e12 −0.457896
\(189\) −2.41162e11 −0.0727393
\(190\) −5.09318e11 −0.149226
\(191\) −2.38573e12 −0.679107 −0.339554 0.940587i \(-0.610276\pi\)
−0.339554 + 0.940587i \(0.610276\pi\)
\(192\) −2.25693e12 −0.624254
\(193\) −1.64628e12 −0.442527 −0.221263 0.975214i \(-0.571018\pi\)
−0.221263 + 0.975214i \(0.571018\pi\)
\(194\) 5.17897e12 1.35311
\(195\) 2.73906e11 0.0695684
\(196\) −1.86285e11 −0.0460013
\(197\) −4.20144e12 −1.00887 −0.504434 0.863450i \(-0.668299\pi\)
−0.504434 + 0.863450i \(0.668299\pi\)
\(198\) 7.69762e11 0.179762
\(199\) −6.43492e12 −1.46168 −0.730839 0.682550i \(-0.760871\pi\)
−0.730839 + 0.682550i \(0.760871\pi\)
\(200\) 3.99637e12 0.883081
\(201\) 8.16838e11 0.175613
\(202\) −6.85510e12 −1.43411
\(203\) 1.29209e12 0.263065
\(204\) 7.46302e11 0.147893
\(205\) 1.36238e11 0.0262815
\(206\) −6.70943e11 −0.126013
\(207\) −2.29712e12 −0.420093
\(208\) 8.94367e11 0.159282
\(209\) −1.57507e12 −0.273209
\(210\) −4.62011e11 −0.0780632
\(211\) −4.83623e12 −0.796074 −0.398037 0.917369i \(-0.630308\pi\)
−0.398037 + 0.917369i \(0.630308\pi\)
\(212\) −2.75365e12 −0.441633
\(213\) 3.78893e12 0.592146
\(214\) 7.15513e12 1.08979
\(215\) 1.99437e12 0.296070
\(216\) 1.44764e12 0.209491
\(217\) −9.63842e11 −0.135981
\(218\) 6.02896e12 0.829339
\(219\) 4.09460e12 0.549249
\(220\) −7.00398e11 −0.0916263
\(221\) −1.72913e12 −0.220632
\(222\) −2.10067e12 −0.261466
\(223\) −6.49076e12 −0.788167 −0.394084 0.919075i \(-0.628938\pi\)
−0.394084 + 0.919075i \(0.628938\pi\)
\(224\) 1.96408e12 0.232699
\(225\) −2.33904e12 −0.270416
\(226\) −1.22214e13 −1.37887
\(227\) −1.02175e13 −1.12513 −0.562563 0.826754i \(-0.690185\pi\)
−0.562563 + 0.826754i \(0.690185\pi\)
\(228\) −7.21502e11 −0.0775525
\(229\) −1.74772e13 −1.83390 −0.916952 0.398997i \(-0.869358\pi\)
−0.916952 + 0.398997i \(0.869358\pi\)
\(230\) −4.40075e12 −0.450841
\(231\) −1.42877e12 −0.142922
\(232\) −7.75609e12 −0.757633
\(233\) 1.96809e13 1.87753 0.938766 0.344556i \(-0.111970\pi\)
0.938766 + 0.344556i \(0.111970\pi\)
\(234\) −8.16970e11 −0.0761235
\(235\) 6.78759e12 0.617792
\(236\) −3.40698e12 −0.302938
\(237\) 8.85412e12 0.769183
\(238\) 2.91660e12 0.247573
\(239\) −2.64938e12 −0.219764 −0.109882 0.993945i \(-0.535047\pi\)
−0.109882 + 0.993945i \(0.535047\pi\)
\(240\) 1.77699e12 0.144053
\(241\) 1.73879e13 1.37769 0.688846 0.724907i \(-0.258117\pi\)
0.688846 + 0.724907i \(0.258117\pi\)
\(242\) −6.07106e12 −0.470197
\(243\) −8.47289e11 −0.0641500
\(244\) 6.57565e12 0.486737
\(245\) 8.57550e11 0.0620649
\(246\) −4.06351e11 −0.0287579
\(247\) 1.67167e12 0.115696
\(248\) 5.78571e12 0.391627
\(249\) −5.10160e12 −0.337762
\(250\) −1.00047e13 −0.647939
\(251\) −4.59100e11 −0.0290872 −0.0145436 0.999894i \(-0.504630\pi\)
−0.0145436 + 0.999894i \(0.504630\pi\)
\(252\) −6.54487e11 −0.0405694
\(253\) −1.36093e13 −0.825419
\(254\) 1.75902e13 1.04396
\(255\) −3.43555e12 −0.199538
\(256\) −1.50432e13 −0.855109
\(257\) 2.55683e13 1.42255 0.711277 0.702911i \(-0.248117\pi\)
0.711277 + 0.702911i \(0.248117\pi\)
\(258\) −5.94853e12 −0.323967
\(259\) 3.89910e12 0.207881
\(260\) 7.43352e11 0.0388008
\(261\) 4.53956e12 0.232002
\(262\) −1.26777e13 −0.634431
\(263\) 9.19464e11 0.0450586 0.0225293 0.999746i \(-0.492828\pi\)
0.0225293 + 0.999746i \(0.492828\pi\)
\(264\) 8.57659e12 0.411617
\(265\) 1.26762e13 0.595851
\(266\) −2.81968e12 −0.129823
\(267\) 8.94846e12 0.403586
\(268\) 2.21681e12 0.0979460
\(269\) 4.54900e12 0.196915 0.0984575 0.995141i \(-0.468609\pi\)
0.0984575 + 0.995141i \(0.468609\pi\)
\(270\) −1.62321e12 −0.0688453
\(271\) 4.01167e13 1.66723 0.833613 0.552349i \(-0.186268\pi\)
0.833613 + 0.552349i \(0.186268\pi\)
\(272\) −1.12178e13 −0.456857
\(273\) 1.51640e12 0.0605228
\(274\) 1.20983e13 0.473255
\(275\) −1.38577e13 −0.531327
\(276\) −6.23412e12 −0.234301
\(277\) −2.94514e13 −1.08509 −0.542547 0.840025i \(-0.682540\pi\)
−0.542547 + 0.840025i \(0.682540\pi\)
\(278\) −3.58543e13 −1.29508
\(279\) −3.38632e12 −0.119924
\(280\) −5.14767e12 −0.178748
\(281\) 2.64894e13 0.901959 0.450980 0.892534i \(-0.351075\pi\)
0.450980 + 0.892534i \(0.351075\pi\)
\(282\) −2.02451e13 −0.676004
\(283\) −4.45950e12 −0.146036 −0.0730182 0.997331i \(-0.523263\pi\)
−0.0730182 + 0.997331i \(0.523263\pi\)
\(284\) 1.02827e13 0.330261
\(285\) 3.32138e12 0.104634
\(286\) −4.84017e12 −0.149571
\(287\) 7.54237e11 0.0228643
\(288\) 6.90052e12 0.205222
\(289\) −1.25839e13 −0.367177
\(290\) 8.69676e12 0.248982
\(291\) −3.37733e13 −0.948771
\(292\) 1.11123e13 0.306336
\(293\) −5.48510e13 −1.48393 −0.741963 0.670440i \(-0.766105\pi\)
−0.741963 + 0.670440i \(0.766105\pi\)
\(294\) −2.55778e12 −0.0679130
\(295\) 1.56838e13 0.408723
\(296\) −2.34053e13 −0.598702
\(297\) −5.01979e12 −0.126045
\(298\) 4.31466e13 1.06355
\(299\) 1.44440e13 0.349539
\(300\) −6.34789e12 −0.150821
\(301\) 1.10412e13 0.257573
\(302\) −6.78590e13 −1.55442
\(303\) 4.47037e13 1.00556
\(304\) 1.08451e13 0.239567
\(305\) −3.02705e13 −0.656705
\(306\) 1.02471e13 0.218339
\(307\) −8.62310e13 −1.80469 −0.902344 0.431017i \(-0.858155\pi\)
−0.902344 + 0.431017i \(0.858155\pi\)
\(308\) −3.87754e12 −0.0797126
\(309\) 4.37537e12 0.0883575
\(310\) −6.48742e12 −0.128701
\(311\) −6.54436e13 −1.27551 −0.637756 0.770238i \(-0.720137\pi\)
−0.637756 + 0.770238i \(0.720137\pi\)
\(312\) −9.10258e12 −0.174307
\(313\) −6.19679e13 −1.16593 −0.582965 0.812497i \(-0.698108\pi\)
−0.582965 + 0.812497i \(0.698108\pi\)
\(314\) 3.02732e13 0.559687
\(315\) 3.01288e12 0.0547361
\(316\) 2.40291e13 0.429001
\(317\) 7.69685e13 1.35048 0.675238 0.737600i \(-0.264041\pi\)
0.675238 + 0.737600i \(0.264041\pi\)
\(318\) −3.78089e13 −0.651996
\(319\) 2.68948e13 0.455848
\(320\) 2.81962e13 0.469750
\(321\) −4.66603e13 −0.764134
\(322\) −2.43634e13 −0.392220
\(323\) −2.09673e13 −0.331840
\(324\) −2.29945e12 −0.0357788
\(325\) 1.47076e13 0.225000
\(326\) −8.19338e13 −1.23244
\(327\) −3.93162e13 −0.581513
\(328\) −4.52751e12 −0.0658496
\(329\) 3.75774e13 0.537464
\(330\) −9.61678e12 −0.135270
\(331\) −8.49103e13 −1.17464 −0.587322 0.809353i \(-0.699818\pi\)
−0.587322 + 0.809353i \(0.699818\pi\)
\(332\) −1.38452e13 −0.188382
\(333\) 1.36989e13 0.183334
\(334\) 6.57178e13 0.865122
\(335\) −1.02049e13 −0.132149
\(336\) 9.83775e12 0.125323
\(337\) −3.43643e13 −0.430668 −0.215334 0.976540i \(-0.569084\pi\)
−0.215334 + 0.976540i \(0.569084\pi\)
\(338\) 5.13701e12 0.0633386
\(339\) 7.96986e13 0.966834
\(340\) −9.32369e12 −0.111289
\(341\) −2.00624e13 −0.235632
\(342\) −9.90655e12 −0.114493
\(343\) 4.74756e12 0.0539949
\(344\) −6.62777e13 −0.741816
\(345\) 2.86983e13 0.316119
\(346\) −1.69392e13 −0.183643
\(347\) 1.50594e14 1.60693 0.803463 0.595354i \(-0.202988\pi\)
0.803463 + 0.595354i \(0.202988\pi\)
\(348\) 1.23199e13 0.129396
\(349\) 7.44243e13 0.769440 0.384720 0.923033i \(-0.374298\pi\)
0.384720 + 0.923033i \(0.374298\pi\)
\(350\) −2.48080e13 −0.252474
\(351\) 5.32765e12 0.0533761
\(352\) 4.08824e13 0.403229
\(353\) −6.82587e13 −0.662822 −0.331411 0.943486i \(-0.607525\pi\)
−0.331411 + 0.943486i \(0.607525\pi\)
\(354\) −4.67794e13 −0.447236
\(355\) −4.73358e13 −0.445588
\(356\) 2.42851e13 0.225095
\(357\) −1.90198e13 −0.173593
\(358\) −1.55871e14 −1.40091
\(359\) −1.31323e14 −1.16231 −0.581153 0.813794i \(-0.697398\pi\)
−0.581153 + 0.813794i \(0.697398\pi\)
\(360\) −1.80856e13 −0.157641
\(361\) −9.62197e13 −0.825989
\(362\) −1.64653e14 −1.39211
\(363\) 3.95908e13 0.329691
\(364\) 4.11534e12 0.0337558
\(365\) −5.11546e13 −0.413308
\(366\) 9.02866e13 0.718584
\(367\) 5.27754e13 0.413779 0.206890 0.978364i \(-0.433666\pi\)
0.206890 + 0.978364i \(0.433666\pi\)
\(368\) 9.37064e13 0.723780
\(369\) 2.64991e12 0.0201644
\(370\) 2.62440e13 0.196753
\(371\) 7.01780e13 0.518376
\(372\) −9.19010e12 −0.0668858
\(373\) 2.63210e13 0.188757 0.0943787 0.995536i \(-0.469914\pi\)
0.0943787 + 0.995536i \(0.469914\pi\)
\(374\) 6.07091e13 0.429003
\(375\) 6.52430e13 0.454320
\(376\) −2.25568e14 −1.54791
\(377\) −2.85442e13 −0.193037
\(378\) −8.98640e12 −0.0598937
\(379\) −2.17845e13 −0.143097 −0.0715486 0.997437i \(-0.522794\pi\)
−0.0715486 + 0.997437i \(0.522794\pi\)
\(380\) 9.01386e12 0.0583581
\(381\) −1.14709e14 −0.732001
\(382\) −8.88994e13 −0.559178
\(383\) 2.25785e14 1.39991 0.699956 0.714185i \(-0.253203\pi\)
0.699956 + 0.714185i \(0.253203\pi\)
\(384\) −2.59424e13 −0.158558
\(385\) 1.78499e13 0.107548
\(386\) −6.13454e13 −0.364378
\(387\) 3.87917e13 0.227158
\(388\) −9.16569e13 −0.529164
\(389\) −4.75630e13 −0.270737 −0.135368 0.990795i \(-0.543222\pi\)
−0.135368 + 0.990795i \(0.543222\pi\)
\(390\) 1.02066e13 0.0572827
\(391\) −1.81168e14 −1.00255
\(392\) −2.84985e13 −0.155506
\(393\) 8.26743e13 0.444848
\(394\) −1.56558e14 −0.830704
\(395\) −1.10616e14 −0.578808
\(396\) −1.36232e13 −0.0702999
\(397\) 1.45288e14 0.739404 0.369702 0.929150i \(-0.379460\pi\)
0.369702 + 0.929150i \(0.379460\pi\)
\(398\) −2.39784e14 −1.20355
\(399\) 1.83878e13 0.0910288
\(400\) 9.54165e13 0.465901
\(401\) 6.42242e12 0.0309318 0.0154659 0.999880i \(-0.495077\pi\)
0.0154659 + 0.999880i \(0.495077\pi\)
\(402\) 3.04378e13 0.144600
\(403\) 2.12928e13 0.0997826
\(404\) 1.21321e14 0.560838
\(405\) 1.05853e13 0.0482727
\(406\) 4.81469e13 0.216608
\(407\) 8.11598e13 0.360223
\(408\) 1.14172e14 0.499950
\(409\) −2.25782e14 −0.975463 −0.487731 0.872994i \(-0.662175\pi\)
−0.487731 + 0.872994i \(0.662175\pi\)
\(410\) 5.07661e12 0.0216403
\(411\) −7.88955e13 −0.331835
\(412\) 1.18743e13 0.0492802
\(413\) 8.68283e13 0.355579
\(414\) −8.55972e13 −0.345906
\(415\) 6.37353e13 0.254165
\(416\) −4.33896e13 −0.170755
\(417\) 2.33814e14 0.908077
\(418\) −5.86917e13 −0.224961
\(419\) −2.18264e14 −0.825666 −0.412833 0.910807i \(-0.635461\pi\)
−0.412833 + 0.910807i \(0.635461\pi\)
\(420\) 8.17663e12 0.0305283
\(421\) −3.00322e14 −1.10671 −0.553357 0.832944i \(-0.686653\pi\)
−0.553357 + 0.832944i \(0.686653\pi\)
\(422\) −1.80212e14 −0.655489
\(423\) 1.32023e14 0.473998
\(424\) −4.21262e14 −1.49293
\(425\) −1.84474e14 −0.645350
\(426\) 1.41187e14 0.487574
\(427\) −1.67583e14 −0.571317
\(428\) −1.26631e14 −0.426185
\(429\) 3.15639e13 0.104876
\(430\) 7.43160e13 0.243784
\(431\) 5.60779e14 1.81621 0.908106 0.418740i \(-0.137528\pi\)
0.908106 + 0.418740i \(0.137528\pi\)
\(432\) 3.45635e13 0.110524
\(433\) 3.66243e14 1.15634 0.578170 0.815917i \(-0.303767\pi\)
0.578170 + 0.815917i \(0.303767\pi\)
\(434\) −3.59156e13 −0.111967
\(435\) −5.67135e13 −0.174581
\(436\) −1.06700e14 −0.324331
\(437\) 1.75147e14 0.525721
\(438\) 1.52577e14 0.452253
\(439\) −3.40906e14 −0.997882 −0.498941 0.866636i \(-0.666278\pi\)
−0.498941 + 0.866636i \(0.666278\pi\)
\(440\) −1.07149e14 −0.309740
\(441\) 1.66799e13 0.0476190
\(442\) −6.44323e13 −0.181669
\(443\) −3.75147e13 −0.104468 −0.0522338 0.998635i \(-0.516634\pi\)
−0.0522338 + 0.998635i \(0.516634\pi\)
\(444\) 3.71774e13 0.102252
\(445\) −1.11795e14 −0.303698
\(446\) −2.41864e14 −0.648979
\(447\) −2.81369e14 −0.745735
\(448\) 1.56100e14 0.408670
\(449\) −6.49981e14 −1.68092 −0.840458 0.541877i \(-0.817714\pi\)
−0.840458 + 0.541877i \(0.817714\pi\)
\(450\) −8.71593e13 −0.222661
\(451\) 1.56995e13 0.0396200
\(452\) 2.16293e14 0.539239
\(453\) 4.42524e14 1.08993
\(454\) −3.80733e14 −0.926431
\(455\) −1.89446e13 −0.0455432
\(456\) −1.10378e14 −0.262165
\(457\) −6.61580e14 −1.55254 −0.776270 0.630400i \(-0.782891\pi\)
−0.776270 + 0.630400i \(0.782891\pi\)
\(458\) −6.51251e14 −1.51004
\(459\) −6.68235e13 −0.153094
\(460\) 7.78839e13 0.176311
\(461\) 1.19329e14 0.266926 0.133463 0.991054i \(-0.457390\pi\)
0.133463 + 0.991054i \(0.457390\pi\)
\(462\) −5.32403e13 −0.117682
\(463\) −1.81496e14 −0.396435 −0.198217 0.980158i \(-0.563515\pi\)
−0.198217 + 0.980158i \(0.563515\pi\)
\(464\) −1.85183e14 −0.399716
\(465\) 4.23059e13 0.0902423
\(466\) 7.33368e14 1.54596
\(467\) −6.47256e14 −1.34845 −0.674223 0.738528i \(-0.735521\pi\)
−0.674223 + 0.738528i \(0.735521\pi\)
\(468\) 1.44586e13 0.0297698
\(469\) −5.64963e13 −0.114966
\(470\) 2.52925e14 0.508691
\(471\) −1.97418e14 −0.392440
\(472\) −5.21210e14 −1.02407
\(473\) 2.29823e14 0.446331
\(474\) 3.29930e14 0.633346
\(475\) 1.78344e14 0.338410
\(476\) −5.16177e13 −0.0968190
\(477\) 2.46560e14 0.457164
\(478\) −9.87237e13 −0.180954
\(479\) 8.54467e14 1.54828 0.774141 0.633014i \(-0.218182\pi\)
0.774141 + 0.633014i \(0.218182\pi\)
\(480\) −8.62094e13 −0.154429
\(481\) −8.61372e13 −0.152543
\(482\) 6.47922e14 1.13439
\(483\) 1.58879e14 0.275016
\(484\) 1.07445e14 0.183881
\(485\) 4.21935e14 0.713948
\(486\) −3.15724e13 −0.0528213
\(487\) 6.82021e14 1.12821 0.564103 0.825705i \(-0.309222\pi\)
0.564103 + 0.825705i \(0.309222\pi\)
\(488\) 1.00596e15 1.64540
\(489\) 5.34309e14 0.864159
\(490\) 3.19548e13 0.0511044
\(491\) 1.03914e15 1.64334 0.821668 0.569967i \(-0.193044\pi\)
0.821668 + 0.569967i \(0.193044\pi\)
\(492\) 7.19155e12 0.0112464
\(493\) 3.58023e14 0.553673
\(494\) 6.22912e13 0.0952639
\(495\) 6.27132e13 0.0948485
\(496\) 1.38139e14 0.206617
\(497\) −2.62060e14 −0.387650
\(498\) −1.90101e14 −0.278114
\(499\) 1.34100e15 1.94033 0.970164 0.242448i \(-0.0779505\pi\)
0.970164 + 0.242448i \(0.0779505\pi\)
\(500\) 1.77062e14 0.253391
\(501\) −4.28561e14 −0.606603
\(502\) −1.71074e13 −0.0239504
\(503\) 5.26904e14 0.729638 0.364819 0.931078i \(-0.381131\pi\)
0.364819 + 0.931078i \(0.381131\pi\)
\(504\) −1.00125e14 −0.137144
\(505\) −5.58491e14 −0.756682
\(506\) −5.07124e14 −0.679652
\(507\) −3.34996e13 −0.0444116
\(508\) −3.11309e14 −0.408264
\(509\) 7.55877e14 0.980625 0.490313 0.871547i \(-0.336883\pi\)
0.490313 + 0.871547i \(0.336883\pi\)
\(510\) −1.28018e14 −0.164300
\(511\) −2.83202e14 −0.359568
\(512\) −7.79197e14 −0.978729
\(513\) 6.46029e13 0.0802798
\(514\) 9.52748e14 1.17133
\(515\) −5.46623e13 −0.0664888
\(516\) 1.05276e14 0.126694
\(517\) 7.82174e14 0.931334
\(518\) 1.45292e14 0.171170
\(519\) 1.10464e14 0.128766
\(520\) 1.13720e14 0.131165
\(521\) 1.55318e15 1.77261 0.886306 0.463101i \(-0.153263\pi\)
0.886306 + 0.463101i \(0.153263\pi\)
\(522\) 1.69157e14 0.191031
\(523\) 6.01398e14 0.672051 0.336026 0.941853i \(-0.390917\pi\)
0.336026 + 0.941853i \(0.390917\pi\)
\(524\) 2.24369e14 0.248108
\(525\) 1.61779e14 0.177029
\(526\) 3.42619e13 0.0371014
\(527\) −2.67070e14 −0.286199
\(528\) 2.04773e14 0.217163
\(529\) 5.60544e14 0.588306
\(530\) 4.72353e14 0.490625
\(531\) 3.05059e14 0.313591
\(532\) 4.99024e13 0.0507701
\(533\) −1.66623e13 −0.0167778
\(534\) 3.33446e14 0.332314
\(535\) 5.82935e14 0.575009
\(536\) 3.39134e14 0.331104
\(537\) 1.01647e15 0.982282
\(538\) 1.69509e14 0.162140
\(539\) 9.88206e13 0.0935641
\(540\) 2.87274e13 0.0269235
\(541\) −1.89035e15 −1.75371 −0.876854 0.480757i \(-0.840362\pi\)
−0.876854 + 0.480757i \(0.840362\pi\)
\(542\) 1.49487e15 1.37280
\(543\) 1.07374e15 0.976112
\(544\) 5.44226e14 0.489762
\(545\) 4.91185e14 0.437587
\(546\) 5.65054e13 0.0498345
\(547\) 1.41581e15 1.23616 0.618079 0.786116i \(-0.287911\pi\)
0.618079 + 0.786116i \(0.287911\pi\)
\(548\) −2.14114e14 −0.185077
\(549\) −5.88780e14 −0.503854
\(550\) −5.16379e14 −0.437495
\(551\) −3.46126e14 −0.290336
\(552\) −9.53713e14 −0.792050
\(553\) −6.12392e14 −0.503548
\(554\) −1.09745e15 −0.893468
\(555\) −1.71143e14 −0.137958
\(556\) 6.34546e14 0.506468
\(557\) −6.60429e14 −0.521943 −0.260971 0.965347i \(-0.584043\pi\)
−0.260971 + 0.965347i \(0.584043\pi\)
\(558\) −1.26184e14 −0.0987454
\(559\) −2.43918e14 −0.189007
\(560\) −1.22905e14 −0.0943050
\(561\) −3.95898e14 −0.300807
\(562\) 9.87071e14 0.742675
\(563\) −2.54912e15 −1.89931 −0.949653 0.313305i \(-0.898564\pi\)
−0.949653 + 0.313305i \(0.898564\pi\)
\(564\) 3.58295e14 0.264366
\(565\) −9.95688e14 −0.727540
\(566\) −1.66174e14 −0.120247
\(567\) 5.86024e13 0.0419961
\(568\) 1.57308e15 1.11644
\(569\) −5.20210e14 −0.365646 −0.182823 0.983146i \(-0.558524\pi\)
−0.182823 + 0.983146i \(0.558524\pi\)
\(570\) 1.23764e14 0.0861556
\(571\) 5.35292e14 0.369056 0.184528 0.982827i \(-0.440924\pi\)
0.184528 + 0.982827i \(0.440924\pi\)
\(572\) 8.56608e13 0.0584930
\(573\) 5.79733e14 0.392083
\(574\) 2.81051e13 0.0188265
\(575\) 1.54097e15 1.02240
\(576\) 5.48434e14 0.360413
\(577\) −2.67229e14 −0.173947 −0.0869735 0.996211i \(-0.527720\pi\)
−0.0869735 + 0.996211i \(0.527720\pi\)
\(578\) −4.68911e14 −0.302334
\(579\) 4.00047e14 0.255493
\(580\) −1.53914e14 −0.0973700
\(581\) 3.52850e14 0.221117
\(582\) −1.25849e15 −0.781220
\(583\) 1.46076e15 0.898258
\(584\) 1.69999e15 1.03556
\(585\) −6.65593e13 −0.0401653
\(586\) −2.04391e15 −1.22187
\(587\) −1.19351e15 −0.706831 −0.353415 0.935466i \(-0.614980\pi\)
−0.353415 + 0.935466i \(0.614980\pi\)
\(588\) 4.52674e13 0.0265589
\(589\) 2.58196e14 0.150077
\(590\) 5.84423e14 0.336543
\(591\) 1.02095e15 0.582470
\(592\) −5.58821e14 −0.315867
\(593\) 2.08117e15 1.16549 0.582743 0.812657i \(-0.301980\pi\)
0.582743 + 0.812657i \(0.301980\pi\)
\(594\) −1.87052e14 −0.103786
\(595\) 2.37618e14 0.130628
\(596\) −7.63604e14 −0.415924
\(597\) 1.56369e15 0.843900
\(598\) 5.38225e14 0.287811
\(599\) −5.53856e13 −0.0293460 −0.0146730 0.999892i \(-0.504671\pi\)
−0.0146730 + 0.999892i \(0.504671\pi\)
\(600\) −9.71118e14 −0.509847
\(601\) −1.82715e14 −0.0950527 −0.0475263 0.998870i \(-0.515134\pi\)
−0.0475263 + 0.998870i \(0.515134\pi\)
\(602\) 4.11428e14 0.212086
\(603\) −1.98492e14 −0.101390
\(604\) 1.20096e15 0.607891
\(605\) −4.94615e14 −0.248092
\(606\) 1.66579e15 0.827981
\(607\) −7.64019e14 −0.376328 −0.188164 0.982138i \(-0.560254\pi\)
−0.188164 + 0.982138i \(0.560254\pi\)
\(608\) −5.26141e14 −0.256822
\(609\) −3.13977e14 −0.151881
\(610\) −1.12797e15 −0.540732
\(611\) −8.30143e14 −0.394391
\(612\) −1.81351e14 −0.0853863
\(613\) −3.22110e15 −1.50304 −0.751522 0.659708i \(-0.770680\pi\)
−0.751522 + 0.659708i \(0.770680\pi\)
\(614\) −3.21322e15 −1.48598
\(615\) −3.31057e13 −0.0151737
\(616\) −5.93197e14 −0.269466
\(617\) −1.29898e14 −0.0584835 −0.0292417 0.999572i \(-0.509309\pi\)
−0.0292417 + 0.999572i \(0.509309\pi\)
\(618\) 1.63039e14 0.0727537
\(619\) 1.59258e15 0.704373 0.352186 0.935930i \(-0.385438\pi\)
0.352186 + 0.935930i \(0.385438\pi\)
\(620\) 1.14814e14 0.0503314
\(621\) 5.58199e14 0.242541
\(622\) −2.43862e15 −1.05026
\(623\) −6.18917e14 −0.264209
\(624\) −2.17331e14 −0.0919617
\(625\) 1.11908e15 0.469375
\(626\) −2.30910e15 −0.960029
\(627\) 3.82742e14 0.157738
\(628\) −5.35772e14 −0.218878
\(629\) 1.08040e15 0.437528
\(630\) 1.12269e14 0.0450698
\(631\) 2.13905e15 0.851255 0.425627 0.904898i \(-0.360053\pi\)
0.425627 + 0.904898i \(0.360053\pi\)
\(632\) 3.67604e15 1.45023
\(633\) 1.17520e15 0.459614
\(634\) 2.86807e15 1.11198
\(635\) 1.43309e15 0.550829
\(636\) 6.69138e14 0.254977
\(637\) −1.04881e14 −0.0396214
\(638\) 1.00218e15 0.375346
\(639\) −9.20710e14 −0.341875
\(640\) 3.24103e14 0.119314
\(641\) 4.33157e15 1.58098 0.790490 0.612475i \(-0.209826\pi\)
0.790490 + 0.612475i \(0.209826\pi\)
\(642\) −1.73870e15 −0.629189
\(643\) −1.76449e15 −0.633082 −0.316541 0.948579i \(-0.602521\pi\)
−0.316541 + 0.948579i \(0.602521\pi\)
\(644\) 4.31180e14 0.153386
\(645\) −4.84632e14 −0.170936
\(646\) −7.81304e14 −0.273238
\(647\) 1.41745e15 0.491511 0.245756 0.969332i \(-0.420964\pi\)
0.245756 + 0.969332i \(0.420964\pi\)
\(648\) −3.51776e14 −0.120949
\(649\) 1.80733e15 0.616158
\(650\) 5.48047e14 0.185265
\(651\) 2.34214e14 0.0785085
\(652\) 1.45006e15 0.481973
\(653\) −3.09019e15 −1.01850 −0.509252 0.860618i \(-0.670078\pi\)
−0.509252 + 0.860618i \(0.670078\pi\)
\(654\) −1.46504e15 −0.478819
\(655\) −1.03287e15 −0.334747
\(656\) −1.08098e14 −0.0347413
\(657\) −9.94988e14 −0.317109
\(658\) 1.40024e15 0.442549
\(659\) −1.36283e15 −0.427141 −0.213570 0.976928i \(-0.568509\pi\)
−0.213570 + 0.976928i \(0.568509\pi\)
\(660\) 1.70197e14 0.0529005
\(661\) 6.24406e15 1.92468 0.962341 0.271845i \(-0.0876340\pi\)
0.962341 + 0.271845i \(0.0876340\pi\)
\(662\) −3.16400e15 −0.967204
\(663\) 4.20178e14 0.127382
\(664\) −2.11808e15 −0.636821
\(665\) −2.29722e14 −0.0684989
\(666\) 5.10462e14 0.150958
\(667\) −2.99069e15 −0.877161
\(668\) −1.16307e15 −0.338325
\(669\) 1.57725e15 0.455049
\(670\) −3.80265e14 −0.108811
\(671\) −3.48825e15 −0.989996
\(672\) −4.77272e14 −0.134349
\(673\) 2.71811e15 0.758900 0.379450 0.925212i \(-0.376113\pi\)
0.379450 + 0.925212i \(0.376113\pi\)
\(674\) −1.28051e15 −0.354613
\(675\) 5.68386e14 0.156125
\(676\) −9.09142e13 −0.0247699
\(677\) −2.84608e15 −0.769148 −0.384574 0.923094i \(-0.625652\pi\)
−0.384574 + 0.923094i \(0.625652\pi\)
\(678\) 2.96980e15 0.796093
\(679\) 2.33591e15 0.621117
\(680\) −1.42637e15 −0.376211
\(681\) 2.48285e15 0.649592
\(682\) −7.47583e14 −0.194020
\(683\) −1.43943e15 −0.370577 −0.185288 0.982684i \(-0.559322\pi\)
−0.185288 + 0.982684i \(0.559322\pi\)
\(684\) 1.75325e14 0.0447750
\(685\) 9.85656e14 0.249705
\(686\) 1.76908e14 0.0444595
\(687\) 4.24696e15 1.05881
\(688\) −1.58243e15 −0.391372
\(689\) −1.55034e15 −0.380384
\(690\) 1.06938e15 0.260293
\(691\) −6.64085e14 −0.160359 −0.0801797 0.996780i \(-0.525549\pi\)
−0.0801797 + 0.996780i \(0.525549\pi\)
\(692\) 2.99788e14 0.0718176
\(693\) 3.47192e14 0.0825158
\(694\) 5.61158e15 1.32315
\(695\) −2.92108e15 −0.683325
\(696\) 1.88473e15 0.437419
\(697\) 2.08991e14 0.0481224
\(698\) 2.77327e15 0.633558
\(699\) −4.78246e15 −1.08399
\(700\) 4.39049e14 0.0987356
\(701\) −5.62068e15 −1.25412 −0.627061 0.778970i \(-0.715742\pi\)
−0.627061 + 0.778970i \(0.715742\pi\)
\(702\) 1.98524e14 0.0439499
\(703\) −1.04450e15 −0.229431
\(704\) 3.24922e15 0.708157
\(705\) −1.64938e15 −0.356682
\(706\) −2.54352e15 −0.545769
\(707\) −3.09191e15 −0.658294
\(708\) 8.27896e14 0.174901
\(709\) −2.48632e15 −0.521199 −0.260599 0.965447i \(-0.583920\pi\)
−0.260599 + 0.965447i \(0.583920\pi\)
\(710\) −1.76387e15 −0.366898
\(711\) −2.15155e15 −0.444088
\(712\) 3.71521e15 0.760928
\(713\) 2.23093e15 0.453412
\(714\) −7.08734e14 −0.142937
\(715\) −3.94333e14 −0.0789187
\(716\) 2.75859e15 0.547855
\(717\) 6.43800e14 0.126881
\(718\) −4.89347e15 −0.957045
\(719\) −7.41628e14 −0.143939 −0.0719693 0.997407i \(-0.522928\pi\)
−0.0719693 + 0.997407i \(0.522928\pi\)
\(720\) −4.31808e14 −0.0831692
\(721\) −3.02621e14 −0.0578436
\(722\) −3.58543e15 −0.680121
\(723\) −4.22525e15 −0.795411
\(724\) 2.91401e15 0.544413
\(725\) −3.04527e15 −0.564634
\(726\) 1.47527e15 0.271468
\(727\) 3.94395e15 0.720264 0.360132 0.932901i \(-0.382732\pi\)
0.360132 + 0.932901i \(0.382732\pi\)
\(728\) 6.29576e14 0.114111
\(729\) 2.05891e14 0.0370370
\(730\) −1.90617e15 −0.340319
\(731\) 3.05940e15 0.542114
\(732\) −1.59788e15 −0.281018
\(733\) 5.50036e15 0.960107 0.480053 0.877239i \(-0.340617\pi\)
0.480053 + 0.877239i \(0.340617\pi\)
\(734\) 1.96657e15 0.340706
\(735\) −2.08385e14 −0.0358332
\(736\) −4.54610e15 −0.775910
\(737\) −1.17597e15 −0.199217
\(738\) 9.87432e13 0.0166034
\(739\) −4.77070e15 −0.796229 −0.398114 0.917336i \(-0.630335\pi\)
−0.398114 + 0.917336i \(0.630335\pi\)
\(740\) −4.64463e14 −0.0769444
\(741\) −4.06215e14 −0.0667968
\(742\) 2.61504e15 0.426831
\(743\) −1.04075e16 −1.68619 −0.843096 0.537763i \(-0.819270\pi\)
−0.843096 + 0.537763i \(0.819270\pi\)
\(744\) −1.40593e15 −0.226106
\(745\) 3.51519e15 0.561163
\(746\) 9.80798e14 0.155423
\(747\) 1.23969e15 0.195007
\(748\) −1.07442e15 −0.167771
\(749\) 3.22724e15 0.500243
\(750\) 2.43114e15 0.374088
\(751\) −4.26891e15 −0.652074 −0.326037 0.945357i \(-0.605713\pi\)
−0.326037 + 0.945357i \(0.605713\pi\)
\(752\) −5.38562e15 −0.816653
\(753\) 1.11561e14 0.0167935
\(754\) −1.06364e15 −0.158947
\(755\) −5.52853e15 −0.820166
\(756\) 1.59040e14 0.0234227
\(757\) −1.13376e16 −1.65765 −0.828825 0.559508i \(-0.810990\pi\)
−0.828825 + 0.559508i \(0.810990\pi\)
\(758\) −8.11752e14 −0.117826
\(759\) 3.30707e15 0.476556
\(760\) 1.37897e15 0.197278
\(761\) −2.83667e15 −0.402896 −0.201448 0.979499i \(-0.564565\pi\)
−0.201448 + 0.979499i \(0.564565\pi\)
\(762\) −4.27441e15 −0.602731
\(763\) 2.71929e15 0.380690
\(764\) 1.57333e15 0.218679
\(765\) 8.34837e14 0.115203
\(766\) 8.41339e15 1.15269
\(767\) −1.91817e15 −0.260924
\(768\) 3.65551e15 0.493697
\(769\) −6.87447e15 −0.921816 −0.460908 0.887448i \(-0.652476\pi\)
−0.460908 + 0.887448i \(0.652476\pi\)
\(770\) 6.65141e14 0.0885553
\(771\) −6.21309e15 −0.821312
\(772\) 1.08568e15 0.142498
\(773\) 3.05904e15 0.398656 0.199328 0.979933i \(-0.436124\pi\)
0.199328 + 0.979933i \(0.436124\pi\)
\(774\) 1.44549e15 0.187043
\(775\) 2.27164e15 0.291864
\(776\) −1.40219e16 −1.78883
\(777\) −9.47480e14 −0.120020
\(778\) −1.77234e15 −0.222925
\(779\) −2.02046e14 −0.0252345
\(780\) −1.80635e14 −0.0224017
\(781\) −5.45478e15 −0.671733
\(782\) −6.75083e15 −0.825506
\(783\) −1.10311e15 −0.133946
\(784\) −6.80424e14 −0.0820430
\(785\) 2.46638e15 0.295310
\(786\) 3.08069e15 0.366289
\(787\) 1.15821e16 1.36749 0.683745 0.729721i \(-0.260350\pi\)
0.683745 + 0.729721i \(0.260350\pi\)
\(788\) 2.77075e15 0.324865
\(789\) −2.23430e14 −0.0260146
\(790\) −4.12188e15 −0.476591
\(791\) −5.51232e15 −0.632942
\(792\) −2.08411e15 −0.237647
\(793\) 3.70218e15 0.419232
\(794\) 5.41385e15 0.608827
\(795\) −3.08032e15 −0.344015
\(796\) 4.24367e15 0.470674
\(797\) −1.48374e16 −1.63431 −0.817157 0.576415i \(-0.804451\pi\)
−0.817157 + 0.576415i \(0.804451\pi\)
\(798\) 6.85183e14 0.0749532
\(799\) 1.04123e16 1.13120
\(800\) −4.62907e15 −0.499458
\(801\) −2.17448e15 −0.233011
\(802\) 2.39318e14 0.0254693
\(803\) −5.89485e15 −0.623070
\(804\) −5.38684e14 −0.0565492
\(805\) −1.98490e15 −0.206949
\(806\) 7.93431e14 0.0821612
\(807\) −1.10541e15 −0.113689
\(808\) 1.85600e16 1.89590
\(809\) −6.62670e15 −0.672327 −0.336163 0.941804i \(-0.609129\pi\)
−0.336163 + 0.941804i \(0.609129\pi\)
\(810\) 3.94440e14 0.0397478
\(811\) 6.81374e15 0.681979 0.340990 0.940067i \(-0.389238\pi\)
0.340990 + 0.940067i \(0.389238\pi\)
\(812\) −8.52099e14 −0.0847094
\(813\) −9.74837e15 −0.962574
\(814\) 3.02425e15 0.296608
\(815\) −6.67522e15 −0.650277
\(816\) 2.72594e15 0.263767
\(817\) −2.95774e15 −0.284275
\(818\) −8.41329e15 −0.803198
\(819\) −3.68485e14 −0.0349428
\(820\) −8.98453e13 −0.00846290
\(821\) 1.09253e16 1.02222 0.511110 0.859515i \(-0.329234\pi\)
0.511110 + 0.859515i \(0.329234\pi\)
\(822\) −2.93988e15 −0.273234
\(823\) −1.18483e16 −1.09384 −0.546922 0.837183i \(-0.684201\pi\)
−0.546922 + 0.837183i \(0.684201\pi\)
\(824\) 1.81656e15 0.166591
\(825\) 3.36742e15 0.306762
\(826\) 3.23548e15 0.292784
\(827\) −8.89785e15 −0.799843 −0.399922 0.916549i \(-0.630963\pi\)
−0.399922 + 0.916549i \(0.630963\pi\)
\(828\) 1.51489e15 0.135274
\(829\) −5.95278e15 −0.528043 −0.264022 0.964517i \(-0.585049\pi\)
−0.264022 + 0.964517i \(0.585049\pi\)
\(830\) 2.37496e15 0.209280
\(831\) 7.15669e15 0.626479
\(832\) −3.44849e15 −0.299882
\(833\) 1.31550e15 0.113643
\(834\) 8.71260e15 0.747712
\(835\) 5.35409e15 0.456467
\(836\) 1.03872e15 0.0879760
\(837\) 8.22876e14 0.0692380
\(838\) −8.13314e15 −0.679855
\(839\) 2.17822e16 1.80888 0.904442 0.426596i \(-0.140287\pi\)
0.904442 + 0.426596i \(0.140287\pi\)
\(840\) 1.25088e15 0.103200
\(841\) −6.29030e15 −0.515577
\(842\) −1.11909e16 −0.911270
\(843\) −6.43692e15 −0.520747
\(844\) 3.18937e15 0.256343
\(845\) 4.18517e14 0.0334196
\(846\) 4.91955e15 0.390291
\(847\) −2.73828e15 −0.215834
\(848\) −1.00580e16 −0.787649
\(849\) 1.08366e15 0.0843141
\(850\) −6.87403e15 −0.531383
\(851\) −9.02493e15 −0.693157
\(852\) −2.49870e15 −0.190676
\(853\) 2.15102e16 1.63089 0.815446 0.578834i \(-0.196492\pi\)
0.815446 + 0.578834i \(0.196492\pi\)
\(854\) −6.24464e15 −0.470423
\(855\) −8.07095e14 −0.0604103
\(856\) −1.93723e16 −1.44071
\(857\) 1.11459e16 0.823607 0.411803 0.911273i \(-0.364899\pi\)
0.411803 + 0.911273i \(0.364899\pi\)
\(858\) 1.17616e15 0.0863549
\(859\) 9.59422e15 0.699918 0.349959 0.936765i \(-0.386196\pi\)
0.349959 + 0.936765i \(0.386196\pi\)
\(860\) −1.31524e15 −0.0953372
\(861\) −1.83280e14 −0.0132007
\(862\) 2.08963e16 1.49547
\(863\) −1.35396e16 −0.962826 −0.481413 0.876494i \(-0.659876\pi\)
−0.481413 + 0.876494i \(0.659876\pi\)
\(864\) −1.67683e15 −0.118485
\(865\) −1.38005e15 −0.0968961
\(866\) 1.36473e16 0.952132
\(867\) 3.05788e15 0.211990
\(868\) 6.35630e14 0.0437871
\(869\) −1.27469e16 −0.872564
\(870\) −2.11331e15 −0.143750
\(871\) 1.24809e15 0.0843619
\(872\) −1.63233e16 −1.09639
\(873\) 8.20690e15 0.547773
\(874\) 6.52650e15 0.432880
\(875\) −4.51251e15 −0.297422
\(876\) −2.70029e15 −0.176863
\(877\) 1.04172e16 0.678034 0.339017 0.940780i \(-0.389906\pi\)
0.339017 + 0.940780i \(0.389906\pi\)
\(878\) −1.27031e16 −0.821658
\(879\) 1.33288e16 0.856746
\(880\) −2.55827e15 −0.163415
\(881\) −1.25439e16 −0.796278 −0.398139 0.917325i \(-0.630344\pi\)
−0.398139 + 0.917325i \(0.630344\pi\)
\(882\) 6.21541e14 0.0392096
\(883\) −3.98189e15 −0.249635 −0.124817 0.992180i \(-0.539834\pi\)
−0.124817 + 0.992180i \(0.539834\pi\)
\(884\) 1.14032e15 0.0710457
\(885\) −3.81116e15 −0.235976
\(886\) −1.39791e15 −0.0860188
\(887\) −3.03520e16 −1.85612 −0.928061 0.372427i \(-0.878526\pi\)
−0.928061 + 0.372427i \(0.878526\pi\)
\(888\) 5.68750e15 0.345661
\(889\) 7.93383e15 0.479207
\(890\) −4.16580e15 −0.250065
\(891\) 1.21981e15 0.0727721
\(892\) 4.28049e15 0.253797
\(893\) −1.00663e16 −0.593180
\(894\) −1.04846e16 −0.614039
\(895\) −1.26990e16 −0.739165
\(896\) 1.79429e15 0.103801
\(897\) −3.50989e15 −0.201806
\(898\) −2.42202e16 −1.38407
\(899\) −4.40876e15 −0.250402
\(900\) 1.54254e15 0.0870766
\(901\) 1.94456e16 1.09102
\(902\) 5.85008e14 0.0326231
\(903\) −2.68301e15 −0.148710
\(904\) 3.30891e16 1.82288
\(905\) −1.34144e16 −0.734522
\(906\) 1.64897e16 0.897447
\(907\) −2.77273e16 −1.49992 −0.749959 0.661484i \(-0.769927\pi\)
−0.749959 + 0.661484i \(0.769927\pi\)
\(908\) 6.73817e15 0.362301
\(909\) −1.08630e16 −0.580561
\(910\) −7.05932e14 −0.0375004
\(911\) −1.34276e16 −0.709002 −0.354501 0.935056i \(-0.615349\pi\)
−0.354501 + 0.935056i \(0.615349\pi\)
\(912\) −2.63535e15 −0.138314
\(913\) 7.34459e15 0.383158
\(914\) −2.46524e16 −1.27836
\(915\) 7.35573e15 0.379149
\(916\) 1.15258e16 0.590534
\(917\) −5.71814e15 −0.291221
\(918\) −2.49004e15 −0.126058
\(919\) 1.05159e16 0.529187 0.264594 0.964360i \(-0.414762\pi\)
0.264594 + 0.964360i \(0.414762\pi\)
\(920\) 1.19149e16 0.596016
\(921\) 2.09541e16 1.04194
\(922\) 4.44655e15 0.219788
\(923\) 5.78931e15 0.284458
\(924\) 9.42241e14 0.0460221
\(925\) −9.18963e15 −0.446189
\(926\) −6.76307e15 −0.326425
\(927\) −1.06322e15 −0.0510132
\(928\) 8.98401e15 0.428506
\(929\) 1.11643e15 0.0529354 0.0264677 0.999650i \(-0.491574\pi\)
0.0264677 + 0.999650i \(0.491574\pi\)
\(930\) 1.57644e15 0.0743056
\(931\) −1.27178e15 −0.0595923
\(932\) −1.29791e16 −0.604583
\(933\) 1.59028e16 0.736418
\(934\) −2.41186e16 −1.11031
\(935\) 4.94603e15 0.226356
\(936\) 2.21193e15 0.100636
\(937\) 3.32493e16 1.50389 0.751943 0.659229i \(-0.229117\pi\)
0.751943 + 0.659229i \(0.229117\pi\)
\(938\) −2.10522e15 −0.0946632
\(939\) 1.50582e16 0.673150
\(940\) −4.47625e15 −0.198935
\(941\) −5.82322e15 −0.257289 −0.128644 0.991691i \(-0.541063\pi\)
−0.128644 + 0.991691i \(0.541063\pi\)
\(942\) −7.35639e15 −0.323136
\(943\) −1.74577e15 −0.0762384
\(944\) −1.24443e16 −0.540287
\(945\) −7.32130e14 −0.0316019
\(946\) 8.56388e15 0.367510
\(947\) 2.20671e16 0.941499 0.470750 0.882267i \(-0.343984\pi\)
0.470750 + 0.882267i \(0.343984\pi\)
\(948\) −5.83907e15 −0.247684
\(949\) 6.25637e15 0.263851
\(950\) 6.64560e15 0.278647
\(951\) −1.87033e16 −0.779698
\(952\) −7.89663e15 −0.327294
\(953\) 1.18910e16 0.490011 0.245005 0.969522i \(-0.421210\pi\)
0.245005 + 0.969522i \(0.421210\pi\)
\(954\) 9.18756e15 0.376430
\(955\) −7.24271e15 −0.295041
\(956\) 1.74720e15 0.0707660
\(957\) −6.53544e15 −0.263184
\(958\) 3.18399e16 1.27486
\(959\) 5.45678e15 0.217237
\(960\) −6.85169e15 −0.271210
\(961\) −2.21197e16 −0.870565
\(962\) −3.20972e15 −0.125604
\(963\) 1.13384e16 0.441173
\(964\) −1.14669e16 −0.443630
\(965\) −4.99786e15 −0.192258
\(966\) 5.92030e15 0.226448
\(967\) −4.59256e16 −1.74666 −0.873332 0.487126i \(-0.838045\pi\)
−0.873332 + 0.487126i \(0.838045\pi\)
\(968\) 1.64372e16 0.621605
\(969\) 5.09506e15 0.191588
\(970\) 1.57225e16 0.587866
\(971\) 3.49556e16 1.29960 0.649802 0.760103i \(-0.274852\pi\)
0.649802 + 0.760103i \(0.274852\pi\)
\(972\) 5.58766e14 0.0206569
\(973\) −1.61717e16 −0.594476
\(974\) 2.54141e16 0.928966
\(975\) −3.57394e15 −0.129904
\(976\) 2.40181e16 0.868091
\(977\) 2.00159e16 0.719376 0.359688 0.933073i \(-0.382883\pi\)
0.359688 + 0.933073i \(0.382883\pi\)
\(978\) 1.99099e16 0.711550
\(979\) −1.28828e16 −0.457830
\(980\) −5.65533e14 −0.0199855
\(981\) 9.55385e15 0.335737
\(982\) 3.87214e16 1.35313
\(983\) 2.88215e14 0.0100155 0.00500774 0.999987i \(-0.498406\pi\)
0.00500774 + 0.999987i \(0.498406\pi\)
\(984\) 1.10018e15 0.0380183
\(985\) −1.27549e16 −0.438307
\(986\) 1.33410e16 0.455895
\(987\) −9.13131e15 −0.310305
\(988\) −1.10242e15 −0.0372550
\(989\) −2.55562e16 −0.858849
\(990\) 2.33688e15 0.0780984
\(991\) −2.39038e16 −0.794440 −0.397220 0.917723i \(-0.630025\pi\)
−0.397220 + 0.917723i \(0.630025\pi\)
\(992\) −6.70170e15 −0.221498
\(993\) 2.06332e16 0.678181
\(994\) −9.76511e15 −0.319192
\(995\) −1.95354e16 −0.635032
\(996\) 3.36438e15 0.108762
\(997\) 7.30428e15 0.234830 0.117415 0.993083i \(-0.462539\pi\)
0.117415 + 0.993083i \(0.462539\pi\)
\(998\) 4.99695e16 1.59767
\(999\) −3.32884e15 −0.105848
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.1.12 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.12 16 1.1 even 1 trivial