Properties

Label 273.12.a.c.1.3
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 1.3
Root \(-68.0202\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q-72.0202 q^{2} -243.000 q^{3} +3138.91 q^{4} +10386.5 q^{5} +17500.9 q^{6} +16807.0 q^{7} -78567.7 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-72.0202 q^{2} -243.000 q^{3} +3138.91 q^{4} +10386.5 q^{5} +17500.9 q^{6} +16807.0 q^{7} -78567.7 q^{8} +59049.0 q^{9} -748037. q^{10} -54869.8 q^{11} -762756. q^{12} -371293. q^{13} -1.21044e6 q^{14} -2.52391e6 q^{15} -770028. q^{16} +8.38294e6 q^{17} -4.25272e6 q^{18} +2.28143e6 q^{19} +3.26022e7 q^{20} -4.08410e6 q^{21} +3.95173e6 q^{22} +3.46075e7 q^{23} +1.90920e7 q^{24} +5.90508e7 q^{25} +2.67406e7 q^{26} -1.43489e7 q^{27} +5.27557e7 q^{28} +4.87020e7 q^{29} +1.81773e8 q^{30} -1.98801e8 q^{31} +2.16364e8 q^{32} +1.33333e7 q^{33} -6.03741e8 q^{34} +1.74566e8 q^{35} +1.85350e8 q^{36} -2.17672e8 q^{37} -1.64309e8 q^{38} +9.02242e7 q^{39} -8.16042e8 q^{40} -5.94586e8 q^{41} +2.94138e8 q^{42} -9.67503e7 q^{43} -1.72231e8 q^{44} +6.13311e8 q^{45} -2.49244e9 q^{46} -1.28528e9 q^{47} +1.87117e8 q^{48} +2.82475e8 q^{49} -4.25285e9 q^{50} -2.03706e9 q^{51} -1.16546e9 q^{52} -2.30912e9 q^{53} +1.03341e9 q^{54} -5.69904e8 q^{55} -1.32049e9 q^{56} -5.54386e8 q^{57} -3.50753e9 q^{58} -3.33247e9 q^{59} -7.92235e9 q^{60} -5.51641e8 q^{61} +1.43177e10 q^{62} +9.92437e8 q^{63} -1.40056e10 q^{64} -3.85643e9 q^{65} -9.60271e8 q^{66} -1.91638e10 q^{67} +2.63133e10 q^{68} -8.40963e9 q^{69} -1.25722e10 q^{70} -2.26355e10 q^{71} -4.63934e9 q^{72} +1.35333e10 q^{73} +1.56768e10 q^{74} -1.43494e10 q^{75} +7.16119e9 q^{76} -9.22196e8 q^{77} -6.49797e9 q^{78} -1.95395e10 q^{79} -7.99788e9 q^{80} +3.48678e9 q^{81} +4.28222e10 q^{82} -2.23053e10 q^{83} -1.28196e10 q^{84} +8.70693e10 q^{85} +6.96798e9 q^{86} -1.18346e10 q^{87} +4.31099e9 q^{88} -8.54763e9 q^{89} -4.41708e10 q^{90} -6.24032e9 q^{91} +1.08630e11 q^{92} +4.83087e10 q^{93} +9.25662e10 q^{94} +2.36960e10 q^{95} -5.25765e10 q^{96} -1.38639e11 q^{97} -2.03439e10 q^{98} -3.24000e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −72.0202 −1.59144 −0.795719 0.605667i \(-0.792907\pi\)
−0.795719 + 0.605667i \(0.792907\pi\)
\(3\) −243.000 −0.577350
\(4\) 3138.91 1.53267
\(5\) 10386.5 1.48639 0.743196 0.669074i \(-0.233309\pi\)
0.743196 + 0.669074i \(0.233309\pi\)
\(6\) 17500.9 0.918817
\(7\) 16807.0 0.377964
\(8\) −78567.7 −0.847714
\(9\) 59049.0 0.333333
\(10\) −748037. −2.36550
\(11\) −54869.8 −0.102724 −0.0513622 0.998680i \(-0.516356\pi\)
−0.0513622 + 0.998680i \(0.516356\pi\)
\(12\) −762756. −0.884888
\(13\) −371293. −0.277350
\(14\) −1.21044e6 −0.601507
\(15\) −2.52391e6 −0.858169
\(16\) −770028. −0.183589
\(17\) 8.38294e6 1.43195 0.715974 0.698127i \(-0.245983\pi\)
0.715974 + 0.698127i \(0.245983\pi\)
\(18\) −4.25272e6 −0.530479
\(19\) 2.28143e6 0.211379 0.105689 0.994399i \(-0.466295\pi\)
0.105689 + 0.994399i \(0.466295\pi\)
\(20\) 3.26022e7 2.27815
\(21\) −4.08410e6 −0.218218
\(22\) 3.95173e6 0.163479
\(23\) 3.46075e7 1.12116 0.560580 0.828100i \(-0.310578\pi\)
0.560580 + 0.828100i \(0.310578\pi\)
\(24\) 1.90920e7 0.489428
\(25\) 5.90508e7 1.20936
\(26\) 2.67406e7 0.441385
\(27\) −1.43489e7 −0.192450
\(28\) 5.27557e7 0.579296
\(29\) 4.87020e7 0.440918 0.220459 0.975396i \(-0.429244\pi\)
0.220459 + 0.975396i \(0.429244\pi\)
\(30\) 1.81773e8 1.36572
\(31\) −1.98801e8 −1.24718 −0.623591 0.781751i \(-0.714327\pi\)
−0.623591 + 0.781751i \(0.714327\pi\)
\(32\) 2.16364e8 1.13988
\(33\) 1.33333e7 0.0593079
\(34\) −6.03741e8 −2.27886
\(35\) 1.74566e8 0.561803
\(36\) 1.85350e8 0.510891
\(37\) −2.17672e8 −0.516052 −0.258026 0.966138i \(-0.583072\pi\)
−0.258026 + 0.966138i \(0.583072\pi\)
\(38\) −1.64309e8 −0.336396
\(39\) 9.02242e7 0.160128
\(40\) −8.16042e8 −1.26003
\(41\) −5.94586e8 −0.801500 −0.400750 0.916187i \(-0.631250\pi\)
−0.400750 + 0.916187i \(0.631250\pi\)
\(42\) 2.94138e8 0.347280
\(43\) −9.67503e7 −0.100363 −0.0501817 0.998740i \(-0.515980\pi\)
−0.0501817 + 0.998740i \(0.515980\pi\)
\(44\) −1.72231e8 −0.157443
\(45\) 6.13311e8 0.495464
\(46\) −2.49244e9 −1.78426
\(47\) −1.28528e9 −0.817447 −0.408724 0.912658i \(-0.634026\pi\)
−0.408724 + 0.912658i \(0.634026\pi\)
\(48\) 1.87117e8 0.105995
\(49\) 2.82475e8 0.142857
\(50\) −4.25285e9 −1.92462
\(51\) −2.03706e9 −0.826736
\(52\) −1.16546e9 −0.425087
\(53\) −2.30912e9 −0.758454 −0.379227 0.925304i \(-0.623810\pi\)
−0.379227 + 0.925304i \(0.623810\pi\)
\(54\) 1.03341e9 0.306272
\(55\) −5.69904e8 −0.152689
\(56\) −1.32049e9 −0.320406
\(57\) −5.54386e8 −0.122040
\(58\) −3.50753e9 −0.701693
\(59\) −3.33247e9 −0.606848 −0.303424 0.952856i \(-0.598130\pi\)
−0.303424 + 0.952856i \(0.598130\pi\)
\(60\) −7.92235e9 −1.31529
\(61\) −5.51641e8 −0.0836262 −0.0418131 0.999125i \(-0.513313\pi\)
−0.0418131 + 0.999125i \(0.513313\pi\)
\(62\) 1.43177e10 1.98481
\(63\) 9.92437e8 0.125988
\(64\) −1.40056e10 −1.63046
\(65\) −3.85643e9 −0.412251
\(66\) −9.60271e8 −0.0943848
\(67\) −1.91638e10 −1.73409 −0.867043 0.498234i \(-0.833982\pi\)
−0.867043 + 0.498234i \(0.833982\pi\)
\(68\) 2.63133e10 2.19471
\(69\) −8.40963e9 −0.647302
\(70\) −1.25722e10 −0.894075
\(71\) −2.26355e10 −1.48891 −0.744455 0.667673i \(-0.767291\pi\)
−0.744455 + 0.667673i \(0.767291\pi\)
\(72\) −4.63934e9 −0.282571
\(73\) 1.35333e10 0.764060 0.382030 0.924150i \(-0.375225\pi\)
0.382030 + 0.924150i \(0.375225\pi\)
\(74\) 1.56768e10 0.821264
\(75\) −1.43494e10 −0.698225
\(76\) 7.16119e9 0.323974
\(77\) −9.22196e8 −0.0388261
\(78\) −6.49797e9 −0.254834
\(79\) −1.95395e10 −0.714436 −0.357218 0.934021i \(-0.616275\pi\)
−0.357218 + 0.934021i \(0.616275\pi\)
\(80\) −7.99788e9 −0.272885
\(81\) 3.48678e9 0.111111
\(82\) 4.28222e10 1.27554
\(83\) −2.23053e10 −0.621554 −0.310777 0.950483i \(-0.600589\pi\)
−0.310777 + 0.950483i \(0.600589\pi\)
\(84\) −1.28196e10 −0.334456
\(85\) 8.70693e10 2.12844
\(86\) 6.96798e9 0.159722
\(87\) −1.18346e10 −0.254564
\(88\) 4.31099e9 0.0870808
\(89\) −8.54763e9 −0.162256 −0.0811280 0.996704i \(-0.525852\pi\)
−0.0811280 + 0.996704i \(0.525852\pi\)
\(90\) −4.41708e10 −0.788500
\(91\) −6.24032e9 −0.104828
\(92\) 1.08630e11 1.71837
\(93\) 4.83087e10 0.720061
\(94\) 9.25662e10 1.30092
\(95\) 2.36960e10 0.314192
\(96\) −5.25765e10 −0.658112
\(97\) −1.38639e11 −1.63923 −0.819617 0.572911i \(-0.805814\pi\)
−0.819617 + 0.572911i \(0.805814\pi\)
\(98\) −2.03439e10 −0.227348
\(99\) −3.24000e9 −0.0342414
\(100\) 1.85355e11 1.85355
\(101\) 1.02717e11 0.972466 0.486233 0.873829i \(-0.338371\pi\)
0.486233 + 0.873829i \(0.338371\pi\)
\(102\) 1.46709e11 1.31570
\(103\) −1.89741e10 −0.161271 −0.0806357 0.996744i \(-0.525695\pi\)
−0.0806357 + 0.996744i \(0.525695\pi\)
\(104\) 2.91716e10 0.235113
\(105\) −4.24194e10 −0.324357
\(106\) 1.66303e11 1.20703
\(107\) 6.90189e10 0.475727 0.237863 0.971299i \(-0.423553\pi\)
0.237863 + 0.971299i \(0.423553\pi\)
\(108\) −4.50400e10 −0.294963
\(109\) −1.66287e11 −1.03517 −0.517585 0.855632i \(-0.673169\pi\)
−0.517585 + 0.855632i \(0.673169\pi\)
\(110\) 4.10446e10 0.242994
\(111\) 5.28943e10 0.297943
\(112\) −1.29419e10 −0.0693901
\(113\) −8.46513e10 −0.432217 −0.216109 0.976369i \(-0.569337\pi\)
−0.216109 + 0.976369i \(0.569337\pi\)
\(114\) 3.99270e10 0.194218
\(115\) 3.59451e11 1.66648
\(116\) 1.52871e11 0.675783
\(117\) −2.19245e10 −0.0924500
\(118\) 2.40005e11 0.965760
\(119\) 1.40892e11 0.541226
\(120\) 1.98298e11 0.727481
\(121\) −2.82301e11 −0.989448
\(122\) 3.97293e10 0.133086
\(123\) 1.44484e11 0.462746
\(124\) −6.24020e11 −1.91152
\(125\) 1.06178e11 0.311192
\(126\) −7.14755e10 −0.200502
\(127\) 6.05760e11 1.62697 0.813486 0.581584i \(-0.197567\pi\)
0.813486 + 0.581584i \(0.197567\pi\)
\(128\) 5.65571e11 1.45490
\(129\) 2.35103e10 0.0579449
\(130\) 2.77741e11 0.656071
\(131\) −3.67870e11 −0.833109 −0.416555 0.909111i \(-0.636763\pi\)
−0.416555 + 0.909111i \(0.636763\pi\)
\(132\) 4.18522e10 0.0908996
\(133\) 3.83439e10 0.0798937
\(134\) 1.38018e12 2.75969
\(135\) −1.49035e11 −0.286056
\(136\) −6.58629e11 −1.21388
\(137\) −2.88591e11 −0.510881 −0.255440 0.966825i \(-0.582220\pi\)
−0.255440 + 0.966825i \(0.582220\pi\)
\(138\) 6.05664e11 1.03014
\(139\) −4.45630e11 −0.728439 −0.364219 0.931313i \(-0.618664\pi\)
−0.364219 + 0.931313i \(0.618664\pi\)
\(140\) 5.47946e11 0.861060
\(141\) 3.12323e11 0.471953
\(142\) 1.63021e12 2.36951
\(143\) 2.03728e10 0.0284906
\(144\) −4.54694e10 −0.0611963
\(145\) 5.05842e11 0.655377
\(146\) −9.74670e11 −1.21595
\(147\) −6.86415e10 −0.0824786
\(148\) −6.83253e11 −0.790938
\(149\) 4.94332e11 0.551435 0.275718 0.961239i \(-0.411085\pi\)
0.275718 + 0.961239i \(0.411085\pi\)
\(150\) 1.03344e12 1.11118
\(151\) −6.45835e11 −0.669496 −0.334748 0.942308i \(-0.608651\pi\)
−0.334748 + 0.942308i \(0.608651\pi\)
\(152\) −1.79246e11 −0.179189
\(153\) 4.95004e11 0.477316
\(154\) 6.64167e10 0.0617894
\(155\) −2.06485e12 −1.85380
\(156\) 2.83206e11 0.245424
\(157\) 2.25304e12 1.88504 0.942522 0.334144i \(-0.108447\pi\)
0.942522 + 0.334144i \(0.108447\pi\)
\(158\) 1.40724e12 1.13698
\(159\) 5.61116e11 0.437894
\(160\) 2.24726e12 1.69431
\(161\) 5.81649e11 0.423759
\(162\) −2.51119e11 −0.176826
\(163\) −1.69121e12 −1.15124 −0.575619 0.817718i \(-0.695239\pi\)
−0.575619 + 0.817718i \(0.695239\pi\)
\(164\) −1.86635e12 −1.22844
\(165\) 1.38487e11 0.0881548
\(166\) 1.60643e12 0.989163
\(167\) −2.14256e10 −0.0127642 −0.00638209 0.999980i \(-0.502031\pi\)
−0.00638209 + 0.999980i \(0.502031\pi\)
\(168\) 3.20878e11 0.184986
\(169\) 1.37858e11 0.0769231
\(170\) −6.27075e12 −3.38727
\(171\) 1.34716e11 0.0704596
\(172\) −3.03691e11 −0.153824
\(173\) −8.20851e11 −0.402727 −0.201363 0.979517i \(-0.564537\pi\)
−0.201363 + 0.979517i \(0.564537\pi\)
\(174\) 8.52330e11 0.405123
\(175\) 9.92467e11 0.457095
\(176\) 4.22512e10 0.0188590
\(177\) 8.09790e11 0.350364
\(178\) 6.15602e11 0.258220
\(179\) −2.51132e11 −0.102143 −0.0510716 0.998695i \(-0.516264\pi\)
−0.0510716 + 0.998695i \(0.516264\pi\)
\(180\) 1.92513e12 0.759384
\(181\) −2.70122e12 −1.03354 −0.516770 0.856124i \(-0.672866\pi\)
−0.516770 + 0.856124i \(0.672866\pi\)
\(182\) 4.49429e11 0.166828
\(183\) 1.34049e11 0.0482816
\(184\) −2.71904e12 −0.950423
\(185\) −2.26085e12 −0.767055
\(186\) −3.47921e12 −1.14593
\(187\) −4.59970e11 −0.147096
\(188\) −4.03438e12 −1.25288
\(189\) −2.41162e11 −0.0727393
\(190\) −1.70659e12 −0.500016
\(191\) 3.09580e12 0.881230 0.440615 0.897696i \(-0.354760\pi\)
0.440615 + 0.897696i \(0.354760\pi\)
\(192\) 3.40336e12 0.941349
\(193\) 4.71878e12 1.26842 0.634212 0.773159i \(-0.281325\pi\)
0.634212 + 0.773159i \(0.281325\pi\)
\(194\) 9.98482e12 2.60874
\(195\) 9.37112e11 0.238013
\(196\) 8.86665e11 0.218953
\(197\) −2.27060e12 −0.545227 −0.272613 0.962124i \(-0.587888\pi\)
−0.272613 + 0.962124i \(0.587888\pi\)
\(198\) 2.33346e11 0.0544931
\(199\) 1.25545e12 0.285171 0.142586 0.989782i \(-0.454458\pi\)
0.142586 + 0.989782i \(0.454458\pi\)
\(200\) −4.63949e12 −1.02519
\(201\) 4.65681e12 1.00117
\(202\) −7.39769e12 −1.54762
\(203\) 8.18535e11 0.166651
\(204\) −6.39414e12 −1.26712
\(205\) −6.17566e12 −1.19134
\(206\) 1.36652e12 0.256653
\(207\) 2.04354e12 0.373720
\(208\) 2.85906e11 0.0509184
\(209\) −1.25181e11 −0.0217137
\(210\) 3.05506e12 0.516194
\(211\) −7.68169e12 −1.26445 −0.632227 0.774783i \(-0.717859\pi\)
−0.632227 + 0.774783i \(0.717859\pi\)
\(212\) −7.24813e12 −1.16246
\(213\) 5.50042e12 0.859622
\(214\) −4.97076e12 −0.757089
\(215\) −1.00489e12 −0.149179
\(216\) 1.12736e12 0.163143
\(217\) −3.34125e12 −0.471391
\(218\) 1.19760e13 1.64741
\(219\) −3.28859e12 −0.441130
\(220\) −1.78888e12 −0.234021
\(221\) −3.11253e12 −0.397151
\(222\) −3.80946e12 −0.474157
\(223\) −9.73111e11 −0.118164 −0.0590821 0.998253i \(-0.518817\pi\)
−0.0590821 + 0.998253i \(0.518817\pi\)
\(224\) 3.63643e12 0.430836
\(225\) 3.48689e12 0.403120
\(226\) 6.09661e12 0.687847
\(227\) 2.99000e12 0.329253 0.164626 0.986356i \(-0.447358\pi\)
0.164626 + 0.986356i \(0.447358\pi\)
\(228\) −1.74017e12 −0.187047
\(229\) 1.32586e13 1.39124 0.695622 0.718408i \(-0.255129\pi\)
0.695622 + 0.718408i \(0.255129\pi\)
\(230\) −2.58877e13 −2.65210
\(231\) 2.24094e11 0.0224163
\(232\) −3.82641e12 −0.373772
\(233\) −1.04644e13 −0.998293 −0.499147 0.866518i \(-0.666353\pi\)
−0.499147 + 0.866518i \(0.666353\pi\)
\(234\) 1.57901e12 0.147128
\(235\) −1.33495e13 −1.21505
\(236\) −1.04603e13 −0.930099
\(237\) 4.74809e12 0.412480
\(238\) −1.01471e13 −0.861327
\(239\) 3.09290e12 0.256554 0.128277 0.991738i \(-0.459055\pi\)
0.128277 + 0.991738i \(0.459055\pi\)
\(240\) 1.94348e12 0.157550
\(241\) 3.96827e12 0.314418 0.157209 0.987565i \(-0.449750\pi\)
0.157209 + 0.987565i \(0.449750\pi\)
\(242\) 2.03314e13 1.57464
\(243\) −8.47289e11 −0.0641500
\(244\) −1.73155e12 −0.128172
\(245\) 2.93392e12 0.212342
\(246\) −1.04058e13 −0.736432
\(247\) −8.47077e11 −0.0586259
\(248\) 1.56194e13 1.05725
\(249\) 5.42019e12 0.358854
\(250\) −7.64695e12 −0.495243
\(251\) −1.83874e13 −1.16497 −0.582486 0.812841i \(-0.697920\pi\)
−0.582486 + 0.812841i \(0.697920\pi\)
\(252\) 3.11517e12 0.193099
\(253\) −1.89891e12 −0.115170
\(254\) −4.36270e13 −2.58922
\(255\) −2.11578e13 −1.22885
\(256\) −1.20491e13 −0.684914
\(257\) −1.81562e13 −1.01017 −0.505083 0.863071i \(-0.668538\pi\)
−0.505083 + 0.863071i \(0.668538\pi\)
\(258\) −1.69322e12 −0.0922156
\(259\) −3.65841e12 −0.195049
\(260\) −1.21050e13 −0.631845
\(261\) 2.87580e12 0.146973
\(262\) 2.64941e13 1.32584
\(263\) 1.31793e13 0.645854 0.322927 0.946424i \(-0.395333\pi\)
0.322927 + 0.946424i \(0.395333\pi\)
\(264\) −1.04757e12 −0.0502761
\(265\) −2.39836e13 −1.12736
\(266\) −2.76154e12 −0.127146
\(267\) 2.07707e12 0.0936785
\(268\) −6.01535e13 −2.65778
\(269\) −1.95765e13 −0.847416 −0.423708 0.905799i \(-0.639272\pi\)
−0.423708 + 0.905799i \(0.639272\pi\)
\(270\) 1.07335e13 0.455241
\(271\) 6.04703e12 0.251311 0.125655 0.992074i \(-0.459897\pi\)
0.125655 + 0.992074i \(0.459897\pi\)
\(272\) −6.45510e12 −0.262890
\(273\) 1.51640e12 0.0605228
\(274\) 2.07844e13 0.813035
\(275\) −3.24010e12 −0.124231
\(276\) −2.63971e13 −0.992102
\(277\) 4.19521e13 1.54566 0.772832 0.634610i \(-0.218839\pi\)
0.772832 + 0.634610i \(0.218839\pi\)
\(278\) 3.20944e13 1.15926
\(279\) −1.17390e13 −0.415728
\(280\) −1.37152e13 −0.476248
\(281\) 2.28404e13 0.777712 0.388856 0.921299i \(-0.372870\pi\)
0.388856 + 0.921299i \(0.372870\pi\)
\(282\) −2.24936e13 −0.751084
\(283\) 4.50394e13 1.47492 0.737458 0.675393i \(-0.236026\pi\)
0.737458 + 0.675393i \(0.236026\pi\)
\(284\) −7.10507e13 −2.28201
\(285\) −5.75812e12 −0.181399
\(286\) −1.46725e12 −0.0453410
\(287\) −9.99321e12 −0.302939
\(288\) 1.27761e13 0.379961
\(289\) 3.60019e13 1.05048
\(290\) −3.64309e13 −1.04299
\(291\) 3.36893e13 0.946413
\(292\) 4.24798e13 1.17105
\(293\) 1.09077e12 0.0295094 0.0147547 0.999891i \(-0.495303\pi\)
0.0147547 + 0.999891i \(0.495303\pi\)
\(294\) 4.94357e12 0.131260
\(295\) −3.46126e13 −0.902014
\(296\) 1.71020e13 0.437464
\(297\) 7.87321e11 0.0197693
\(298\) −3.56019e13 −0.877575
\(299\) −1.28495e13 −0.310954
\(300\) −4.50413e13 −1.07015
\(301\) −1.62608e12 −0.0379338
\(302\) 4.65132e13 1.06546
\(303\) −2.49602e13 −0.561453
\(304\) −1.75676e12 −0.0388068
\(305\) −5.72961e12 −0.124301
\(306\) −3.56503e13 −0.759619
\(307\) 5.11602e11 0.0107071 0.00535354 0.999986i \(-0.498296\pi\)
0.00535354 + 0.999986i \(0.498296\pi\)
\(308\) −2.89469e12 −0.0595077
\(309\) 4.61072e12 0.0931101
\(310\) 1.48711e14 2.95021
\(311\) −7.70245e13 −1.50123 −0.750615 0.660740i \(-0.770242\pi\)
−0.750615 + 0.660740i \(0.770242\pi\)
\(312\) −7.08871e12 −0.135743
\(313\) 2.41275e13 0.453960 0.226980 0.973899i \(-0.427115\pi\)
0.226980 + 0.973899i \(0.427115\pi\)
\(314\) −1.62265e14 −2.99993
\(315\) 1.03079e13 0.187268
\(316\) −6.13326e13 −1.09500
\(317\) −9.86801e13 −1.73143 −0.865713 0.500541i \(-0.833134\pi\)
−0.865713 + 0.500541i \(0.833134\pi\)
\(318\) −4.04117e13 −0.696880
\(319\) −2.67227e12 −0.0452930
\(320\) −1.45469e14 −2.42351
\(321\) −1.67716e13 −0.274661
\(322\) −4.18905e13 −0.674385
\(323\) 1.91251e13 0.302684
\(324\) 1.09447e13 0.170297
\(325\) −2.19252e13 −0.335416
\(326\) 1.21801e14 1.83212
\(327\) 4.04076e13 0.597655
\(328\) 4.67153e13 0.679443
\(329\) −2.16017e13 −0.308966
\(330\) −9.97383e12 −0.140293
\(331\) 1.74708e13 0.241691 0.120845 0.992671i \(-0.461439\pi\)
0.120845 + 0.992671i \(0.461439\pi\)
\(332\) −7.00144e13 −0.952638
\(333\) −1.28533e13 −0.172017
\(334\) 1.54308e12 0.0203134
\(335\) −1.99045e14 −2.57753
\(336\) 3.14487e12 0.0400624
\(337\) −3.26051e13 −0.408622 −0.204311 0.978906i \(-0.565495\pi\)
−0.204311 + 0.978906i \(0.565495\pi\)
\(338\) −9.92860e12 −0.122418
\(339\) 2.05703e13 0.249541
\(340\) 2.73303e14 3.26220
\(341\) 1.09082e13 0.128116
\(342\) −9.70227e12 −0.112132
\(343\) 4.74756e12 0.0539949
\(344\) 7.60145e12 0.0850795
\(345\) −8.73465e13 −0.962145
\(346\) 5.91179e13 0.640915
\(347\) −1.75533e14 −1.87304 −0.936519 0.350617i \(-0.885972\pi\)
−0.936519 + 0.350617i \(0.885972\pi\)
\(348\) −3.71477e13 −0.390163
\(349\) −1.62935e13 −0.168452 −0.0842258 0.996447i \(-0.526842\pi\)
−0.0842258 + 0.996447i \(0.526842\pi\)
\(350\) −7.14777e13 −0.727439
\(351\) 5.32765e12 0.0533761
\(352\) −1.18719e13 −0.117094
\(353\) −1.30829e14 −1.27041 −0.635203 0.772345i \(-0.719084\pi\)
−0.635203 + 0.772345i \(0.719084\pi\)
\(354\) −5.83212e13 −0.557582
\(355\) −2.35103e14 −2.21310
\(356\) −2.68302e13 −0.248685
\(357\) −3.42368e13 −0.312477
\(358\) 1.80865e13 0.162555
\(359\) 1.06773e14 0.945019 0.472509 0.881326i \(-0.343348\pi\)
0.472509 + 0.881326i \(0.343348\pi\)
\(360\) −4.81865e13 −0.420012
\(361\) −1.11285e14 −0.955319
\(362\) 1.94542e14 1.64481
\(363\) 6.85991e13 0.571258
\(364\) −1.95878e13 −0.160668
\(365\) 1.40563e14 1.13569
\(366\) −9.65422e12 −0.0768371
\(367\) −2.16483e13 −0.169730 −0.0848652 0.996392i \(-0.527046\pi\)
−0.0848652 + 0.996392i \(0.527046\pi\)
\(368\) −2.66488e13 −0.205833
\(369\) −3.51097e13 −0.267167
\(370\) 1.62827e14 1.22072
\(371\) −3.88094e13 −0.286669
\(372\) 1.51637e14 1.10362
\(373\) 1.24228e14 0.890881 0.445441 0.895311i \(-0.353047\pi\)
0.445441 + 0.895311i \(0.353047\pi\)
\(374\) 3.31271e13 0.234094
\(375\) −2.58012e13 −0.179667
\(376\) 1.00982e14 0.692961
\(377\) −1.80827e13 −0.122289
\(378\) 1.73685e13 0.115760
\(379\) 2.17734e14 1.43025 0.715124 0.698998i \(-0.246370\pi\)
0.715124 + 0.698998i \(0.246370\pi\)
\(380\) 7.43796e13 0.481553
\(381\) −1.47200e14 −0.939333
\(382\) −2.22960e14 −1.40242
\(383\) 9.73543e13 0.603617 0.301809 0.953368i \(-0.402410\pi\)
0.301809 + 0.953368i \(0.402410\pi\)
\(384\) −1.37434e14 −0.839986
\(385\) −9.57837e12 −0.0577109
\(386\) −3.39847e14 −2.01862
\(387\) −5.71301e12 −0.0334545
\(388\) −4.35176e14 −2.51241
\(389\) 6.00957e13 0.342074 0.171037 0.985265i \(-0.445288\pi\)
0.171037 + 0.985265i \(0.445288\pi\)
\(390\) −6.74910e13 −0.378783
\(391\) 2.90113e14 1.60544
\(392\) −2.21934e13 −0.121102
\(393\) 8.93923e13 0.480996
\(394\) 1.63529e14 0.867694
\(395\) −2.02946e14 −1.06193
\(396\) −1.01701e13 −0.0524809
\(397\) 2.20423e14 1.12178 0.560891 0.827889i \(-0.310459\pi\)
0.560891 + 0.827889i \(0.310459\pi\)
\(398\) −9.04175e13 −0.453832
\(399\) −9.31757e12 −0.0461266
\(400\) −4.54708e13 −0.222025
\(401\) −1.19980e14 −0.577850 −0.288925 0.957352i \(-0.593298\pi\)
−0.288925 + 0.957352i \(0.593298\pi\)
\(402\) −3.35384e14 −1.59331
\(403\) 7.38136e13 0.345906
\(404\) 3.22419e14 1.49047
\(405\) 3.62154e13 0.165155
\(406\) −5.89510e13 −0.265215
\(407\) 1.19436e13 0.0530110
\(408\) 1.60047e14 0.700835
\(409\) 2.83112e14 1.22315 0.611575 0.791187i \(-0.290536\pi\)
0.611575 + 0.791187i \(0.290536\pi\)
\(410\) 4.44772e14 1.89595
\(411\) 7.01276e13 0.294957
\(412\) −5.95582e13 −0.247176
\(413\) −5.60088e13 −0.229367
\(414\) −1.47176e14 −0.594752
\(415\) −2.31674e14 −0.923872
\(416\) −8.03345e13 −0.316147
\(417\) 1.08288e14 0.420564
\(418\) 9.01558e12 0.0345561
\(419\) 5.05660e14 1.91285 0.956427 0.291972i \(-0.0943115\pi\)
0.956427 + 0.291972i \(0.0943115\pi\)
\(420\) −1.33151e14 −0.497133
\(421\) 4.07129e14 1.50031 0.750154 0.661263i \(-0.229979\pi\)
0.750154 + 0.661263i \(0.229979\pi\)
\(422\) 5.53237e14 2.01230
\(423\) −7.58945e13 −0.272482
\(424\) 1.81422e14 0.642952
\(425\) 4.95020e14 1.73174
\(426\) −3.96141e14 −1.36803
\(427\) −9.27143e12 −0.0316077
\(428\) 2.16644e14 0.729133
\(429\) −4.95058e12 −0.0164491
\(430\) 7.23727e13 0.237410
\(431\) −1.02067e14 −0.330567 −0.165283 0.986246i \(-0.552854\pi\)
−0.165283 + 0.986246i \(0.552854\pi\)
\(432\) 1.10491e13 0.0353317
\(433\) 3.71741e14 1.17370 0.586850 0.809696i \(-0.300368\pi\)
0.586850 + 0.809696i \(0.300368\pi\)
\(434\) 2.40638e14 0.750189
\(435\) −1.22920e14 −0.378382
\(436\) −5.21959e14 −1.58658
\(437\) 7.89545e13 0.236990
\(438\) 2.36845e14 0.702031
\(439\) 2.76343e13 0.0808899 0.0404449 0.999182i \(-0.487122\pi\)
0.0404449 + 0.999182i \(0.487122\pi\)
\(440\) 4.47760e13 0.129436
\(441\) 1.66799e13 0.0476190
\(442\) 2.24165e14 0.632041
\(443\) 5.21741e14 1.45290 0.726448 0.687222i \(-0.241170\pi\)
0.726448 + 0.687222i \(0.241170\pi\)
\(444\) 1.66031e14 0.456648
\(445\) −8.87798e13 −0.241176
\(446\) 7.00837e13 0.188051
\(447\) −1.20123e14 −0.318371
\(448\) −2.35392e14 −0.616258
\(449\) −3.32038e12 −0.00858682 −0.00429341 0.999991i \(-0.501367\pi\)
−0.00429341 + 0.999991i \(0.501367\pi\)
\(450\) −2.51127e14 −0.641541
\(451\) 3.26248e13 0.0823335
\(452\) −2.65713e14 −0.662448
\(453\) 1.56938e14 0.386534
\(454\) −2.15341e14 −0.523985
\(455\) −6.48150e13 −0.155816
\(456\) 4.35569e13 0.103455
\(457\) −9.43177e13 −0.221337 −0.110669 0.993857i \(-0.535299\pi\)
−0.110669 + 0.993857i \(0.535299\pi\)
\(458\) −9.54889e14 −2.21408
\(459\) −1.20286e14 −0.275579
\(460\) 1.12828e15 2.55417
\(461\) −5.47693e13 −0.122513 −0.0612565 0.998122i \(-0.519511\pi\)
−0.0612565 + 0.998122i \(0.519511\pi\)
\(462\) −1.61393e13 −0.0356741
\(463\) −4.18485e14 −0.914081 −0.457040 0.889446i \(-0.651090\pi\)
−0.457040 + 0.889446i \(0.651090\pi\)
\(464\) −3.75019e13 −0.0809477
\(465\) 5.01758e14 1.07029
\(466\) 7.53651e14 1.58872
\(467\) −6.64068e14 −1.38347 −0.691735 0.722151i \(-0.743153\pi\)
−0.691735 + 0.722151i \(0.743153\pi\)
\(468\) −6.88190e13 −0.141696
\(469\) −3.22086e14 −0.655423
\(470\) 9.61437e14 1.93367
\(471\) −5.47490e14 −1.08833
\(472\) 2.61824e14 0.514433
\(473\) 5.30866e12 0.0103098
\(474\) −3.41958e14 −0.656436
\(475\) 1.34720e14 0.255633
\(476\) 4.42248e14 0.829522
\(477\) −1.36351e14 −0.252818
\(478\) −2.22752e14 −0.408289
\(479\) 3.72281e14 0.674568 0.337284 0.941403i \(-0.390492\pi\)
0.337284 + 0.941403i \(0.390492\pi\)
\(480\) −5.46085e14 −0.978213
\(481\) 8.08201e13 0.143127
\(482\) −2.85795e14 −0.500376
\(483\) −1.41341e14 −0.244657
\(484\) −8.86118e14 −1.51650
\(485\) −1.43997e15 −2.43655
\(486\) 6.10219e13 0.102091
\(487\) 6.95896e13 0.115116 0.0575579 0.998342i \(-0.481669\pi\)
0.0575579 + 0.998342i \(0.481669\pi\)
\(488\) 4.33412e13 0.0708911
\(489\) 4.10964e14 0.664668
\(490\) −2.11302e14 −0.337928
\(491\) 1.14475e15 1.81034 0.905171 0.425047i \(-0.139742\pi\)
0.905171 + 0.425047i \(0.139742\pi\)
\(492\) 4.53524e14 0.709238
\(493\) 4.08266e14 0.631372
\(494\) 6.10067e13 0.0932995
\(495\) −3.36522e13 −0.0508962
\(496\) 1.53083e14 0.228969
\(497\) −3.80434e14 −0.562755
\(498\) −3.90363e14 −0.571094
\(499\) −9.79136e14 −1.41674 −0.708370 0.705842i \(-0.750569\pi\)
−0.708370 + 0.705842i \(0.750569\pi\)
\(500\) 3.33283e14 0.476956
\(501\) 5.20643e12 0.00736940
\(502\) 1.32427e15 1.85398
\(503\) 8.22443e13 0.113889 0.0569445 0.998377i \(-0.481864\pi\)
0.0569445 + 0.998377i \(0.481864\pi\)
\(504\) −7.79735e13 −0.106802
\(505\) 1.06687e15 1.44547
\(506\) 1.36760e14 0.183286
\(507\) −3.34996e13 −0.0444116
\(508\) 1.90143e15 2.49361
\(509\) 4.86891e14 0.631660 0.315830 0.948816i \(-0.397717\pi\)
0.315830 + 0.948816i \(0.397717\pi\)
\(510\) 1.52379e15 1.95564
\(511\) 2.27454e14 0.288788
\(512\) −2.90509e14 −0.364901
\(513\) −3.27360e13 −0.0406799
\(514\) 1.30761e15 1.60762
\(515\) −1.97075e14 −0.239713
\(516\) 7.37968e13 0.0888105
\(517\) 7.05230e13 0.0839717
\(518\) 2.63480e14 0.310409
\(519\) 1.99467e14 0.232515
\(520\) 3.02991e14 0.349471
\(521\) −1.86024e14 −0.212306 −0.106153 0.994350i \(-0.533853\pi\)
−0.106153 + 0.994350i \(0.533853\pi\)
\(522\) −2.07116e14 −0.233898
\(523\) −1.18067e15 −1.31938 −0.659688 0.751540i \(-0.729311\pi\)
−0.659688 + 0.751540i \(0.729311\pi\)
\(524\) −1.15471e15 −1.27688
\(525\) −2.41170e14 −0.263904
\(526\) −9.49173e14 −1.02784
\(527\) −1.66654e15 −1.78590
\(528\) −1.02670e13 −0.0108883
\(529\) 2.44873e14 0.257001
\(530\) 1.72731e15 1.79412
\(531\) −1.96779e14 −0.202283
\(532\) 1.20358e14 0.122451
\(533\) 2.20766e14 0.222296
\(534\) −1.49591e14 −0.149083
\(535\) 7.16864e14 0.707116
\(536\) 1.50566e15 1.47001
\(537\) 6.10250e13 0.0589724
\(538\) 1.40990e15 1.34861
\(539\) −1.54993e13 −0.0146749
\(540\) −4.67807e14 −0.438430
\(541\) −1.21630e15 −1.12838 −0.564191 0.825645i \(-0.690812\pi\)
−0.564191 + 0.825645i \(0.690812\pi\)
\(542\) −4.35508e14 −0.399945
\(543\) 6.56396e14 0.596715
\(544\) 1.81377e15 1.63226
\(545\) −1.72713e15 −1.53867
\(546\) −1.09211e14 −0.0963182
\(547\) 1.61350e15 1.40877 0.704384 0.709819i \(-0.251223\pi\)
0.704384 + 0.709819i \(0.251223\pi\)
\(548\) −9.05861e14 −0.783013
\(549\) −3.25738e13 −0.0278754
\(550\) 2.33353e14 0.197705
\(551\) 1.11110e14 0.0932007
\(552\) 6.60726e14 0.548727
\(553\) −3.28400e14 −0.270032
\(554\) −3.02140e15 −2.45983
\(555\) 5.49386e14 0.442859
\(556\) −1.39879e15 −1.11646
\(557\) −4.11057e14 −0.324862 −0.162431 0.986720i \(-0.551933\pi\)
−0.162431 + 0.986720i \(0.551933\pi\)
\(558\) 8.45447e14 0.661604
\(559\) 3.59227e13 0.0278358
\(560\) −1.34420e14 −0.103141
\(561\) 1.11773e14 0.0849259
\(562\) −1.64497e15 −1.23768
\(563\) −2.10559e15 −1.56883 −0.784417 0.620234i \(-0.787038\pi\)
−0.784417 + 0.620234i \(0.787038\pi\)
\(564\) 9.80355e14 0.723350
\(565\) −8.79229e14 −0.642445
\(566\) −3.24375e15 −2.34724
\(567\) 5.86024e13 0.0419961
\(568\) 1.77842e15 1.26217
\(569\) −2.68415e15 −1.88664 −0.943319 0.331886i \(-0.892315\pi\)
−0.943319 + 0.331886i \(0.892315\pi\)
\(570\) 4.14701e14 0.288685
\(571\) −9.50779e14 −0.655512 −0.327756 0.944762i \(-0.606292\pi\)
−0.327756 + 0.944762i \(0.606292\pi\)
\(572\) 6.39483e13 0.0436667
\(573\) −7.52279e14 −0.508778
\(574\) 7.19713e14 0.482108
\(575\) 2.04360e15 1.35589
\(576\) −8.27016e14 −0.543488
\(577\) −2.98377e15 −1.94222 −0.971111 0.238630i \(-0.923302\pi\)
−0.971111 + 0.238630i \(0.923302\pi\)
\(578\) −2.59286e15 −1.67177
\(579\) −1.14666e15 −0.732325
\(580\) 1.58779e15 1.00448
\(581\) −3.74885e14 −0.234925
\(582\) −2.42631e15 −1.50616
\(583\) 1.26701e14 0.0779117
\(584\) −1.06328e15 −0.647704
\(585\) −2.27718e14 −0.137417
\(586\) −7.85572e13 −0.0469623
\(587\) −2.34472e15 −1.38861 −0.694307 0.719679i \(-0.744289\pi\)
−0.694307 + 0.719679i \(0.744289\pi\)
\(588\) −2.15460e14 −0.126413
\(589\) −4.53550e14 −0.263628
\(590\) 2.49281e15 1.43550
\(591\) 5.51757e14 0.314787
\(592\) 1.67613e14 0.0947414
\(593\) 1.69092e15 0.946942 0.473471 0.880809i \(-0.343001\pi\)
0.473471 + 0.880809i \(0.343001\pi\)
\(594\) −5.67030e13 −0.0314616
\(595\) 1.46337e15 0.804474
\(596\) 1.55167e15 0.845169
\(597\) −3.05073e14 −0.164644
\(598\) 9.25427e14 0.494864
\(599\) −1.61129e13 −0.00853739 −0.00426869 0.999991i \(-0.501359\pi\)
−0.00426869 + 0.999991i \(0.501359\pi\)
\(600\) 1.12740e15 0.591895
\(601\) 1.63144e15 0.848713 0.424357 0.905495i \(-0.360500\pi\)
0.424357 + 0.905495i \(0.360500\pi\)
\(602\) 1.17111e14 0.0603693
\(603\) −1.13160e15 −0.578028
\(604\) −2.02722e15 −1.02612
\(605\) −2.93211e15 −1.47071
\(606\) 1.79764e15 0.893518
\(607\) −2.08154e15 −1.02529 −0.512646 0.858600i \(-0.671335\pi\)
−0.512646 + 0.858600i \(0.671335\pi\)
\(608\) 4.93619e14 0.240947
\(609\) −1.98904e14 −0.0962162
\(610\) 4.12648e14 0.197818
\(611\) 4.77216e14 0.226719
\(612\) 1.55378e15 0.731569
\(613\) −1.27712e15 −0.595933 −0.297967 0.954576i \(-0.596308\pi\)
−0.297967 + 0.954576i \(0.596308\pi\)
\(614\) −3.68457e13 −0.0170396
\(615\) 1.50068e15 0.687822
\(616\) 7.24548e13 0.0329134
\(617\) −2.00426e15 −0.902372 −0.451186 0.892430i \(-0.648999\pi\)
−0.451186 + 0.892430i \(0.648999\pi\)
\(618\) −3.32065e14 −0.148179
\(619\) −2.69075e15 −1.19008 −0.595038 0.803698i \(-0.702863\pi\)
−0.595038 + 0.803698i \(0.702863\pi\)
\(620\) −6.48137e15 −2.84127
\(621\) −4.96580e14 −0.215767
\(622\) 5.54732e15 2.38911
\(623\) −1.43660e14 −0.0613270
\(624\) −6.94751e13 −0.0293977
\(625\) −1.78053e15 −0.746807
\(626\) −1.73767e15 −0.722449
\(627\) 3.04190e13 0.0125364
\(628\) 7.07210e15 2.88915
\(629\) −1.82473e15 −0.738960
\(630\) −7.42379e14 −0.298025
\(631\) 2.25077e15 0.895716 0.447858 0.894105i \(-0.352187\pi\)
0.447858 + 0.894105i \(0.352187\pi\)
\(632\) 1.53517e15 0.605637
\(633\) 1.86665e15 0.730033
\(634\) 7.10696e15 2.75545
\(635\) 6.29172e15 2.41832
\(636\) 1.76129e15 0.671147
\(637\) −1.04881e14 −0.0396214
\(638\) 1.92457e14 0.0720810
\(639\) −1.33660e15 −0.496303
\(640\) 5.87429e15 2.16255
\(641\) 2.47916e15 0.904867 0.452434 0.891798i \(-0.350556\pi\)
0.452434 + 0.891798i \(0.350556\pi\)
\(642\) 1.20789e15 0.437106
\(643\) −2.98504e14 −0.107100 −0.0535501 0.998565i \(-0.517054\pi\)
−0.0535501 + 0.998565i \(0.517054\pi\)
\(644\) 1.82575e15 0.649483
\(645\) 2.44189e14 0.0861288
\(646\) −1.37739e15 −0.481702
\(647\) −1.41307e15 −0.489993 −0.244996 0.969524i \(-0.578787\pi\)
−0.244996 + 0.969524i \(0.578787\pi\)
\(648\) −2.73949e14 −0.0941904
\(649\) 1.82852e14 0.0623380
\(650\) 1.57905e15 0.533794
\(651\) 8.11925e14 0.272158
\(652\) −5.30855e15 −1.76447
\(653\) 4.39715e15 1.44927 0.724635 0.689133i \(-0.242008\pi\)
0.724635 + 0.689133i \(0.242008\pi\)
\(654\) −2.91017e15 −0.951131
\(655\) −3.82087e15 −1.23833
\(656\) 4.57848e14 0.147147
\(657\) 7.99127e14 0.254687
\(658\) 1.55576e15 0.491700
\(659\) −1.80382e14 −0.0565358 −0.0282679 0.999600i \(-0.508999\pi\)
−0.0282679 + 0.999600i \(0.508999\pi\)
\(660\) 4.34697e14 0.135112
\(661\) −3.59188e15 −1.10717 −0.553585 0.832793i \(-0.686740\pi\)
−0.553585 + 0.832793i \(0.686740\pi\)
\(662\) −1.25825e15 −0.384636
\(663\) 7.56344e14 0.229295
\(664\) 1.75248e15 0.526899
\(665\) 3.98258e14 0.118753
\(666\) 9.25699e14 0.273755
\(667\) 1.68546e15 0.494340
\(668\) −6.72531e13 −0.0195633
\(669\) 2.36466e14 0.0682221
\(670\) 1.43352e16 4.10198
\(671\) 3.02684e13 0.00859044
\(672\) −8.83653e14 −0.248743
\(673\) 5.31465e14 0.148386 0.0741928 0.997244i \(-0.476362\pi\)
0.0741928 + 0.997244i \(0.476362\pi\)
\(674\) 2.34823e15 0.650295
\(675\) −8.47315e14 −0.232742
\(676\) 4.32726e14 0.117898
\(677\) −1.52468e15 −0.412041 −0.206020 0.978548i \(-0.566051\pi\)
−0.206020 + 0.978548i \(0.566051\pi\)
\(678\) −1.48148e15 −0.397129
\(679\) −2.33011e15 −0.619573
\(680\) −6.84083e15 −1.80431
\(681\) −7.26571e14 −0.190094
\(682\) −7.85610e14 −0.203888
\(683\) −1.28588e15 −0.331043 −0.165522 0.986206i \(-0.552931\pi\)
−0.165522 + 0.986206i \(0.552931\pi\)
\(684\) 4.22861e14 0.107991
\(685\) −2.99744e15 −0.759369
\(686\) −3.41920e14 −0.0859295
\(687\) −3.22185e15 −0.803235
\(688\) 7.45004e13 0.0184256
\(689\) 8.57360e14 0.210357
\(690\) 6.29071e15 1.53119
\(691\) −2.93689e14 −0.0709183 −0.0354592 0.999371i \(-0.511289\pi\)
−0.0354592 + 0.999371i \(0.511289\pi\)
\(692\) −2.57658e15 −0.617248
\(693\) −5.44547e13 −0.0129420
\(694\) 1.26419e16 2.98082
\(695\) −4.62853e15 −1.08275
\(696\) 9.29817e14 0.215798
\(697\) −4.98438e15 −1.14771
\(698\) 1.17346e15 0.268080
\(699\) 2.54286e15 0.576365
\(700\) 3.11527e15 0.700577
\(701\) 6.49037e15 1.44817 0.724087 0.689709i \(-0.242262\pi\)
0.724087 + 0.689709i \(0.242262\pi\)
\(702\) −3.83698e14 −0.0849446
\(703\) −4.96603e14 −0.109082
\(704\) 7.68483e14 0.167488
\(705\) 3.24394e15 0.701508
\(706\) 9.42232e15 2.02177
\(707\) 1.72636e15 0.367558
\(708\) 2.54186e15 0.536993
\(709\) 6.48647e15 1.35973 0.679867 0.733335i \(-0.262037\pi\)
0.679867 + 0.733335i \(0.262037\pi\)
\(710\) 1.69321e16 3.52201
\(711\) −1.15379e15 −0.238145
\(712\) 6.71568e14 0.137547
\(713\) −6.88003e15 −1.39829
\(714\) 2.46574e15 0.497287
\(715\) 2.11601e14 0.0423482
\(716\) −7.88280e14 −0.156552
\(717\) −7.51576e14 −0.148121
\(718\) −7.68979e15 −1.50394
\(719\) 1.10548e15 0.214556 0.107278 0.994229i \(-0.465786\pi\)
0.107278 + 0.994229i \(0.465786\pi\)
\(720\) −4.72267e14 −0.0909617
\(721\) −3.18898e14 −0.0609549
\(722\) 8.01480e15 1.52033
\(723\) −9.64289e14 −0.181529
\(724\) −8.47889e15 −1.58408
\(725\) 2.87589e15 0.533229
\(726\) −4.94052e15 −0.909121
\(727\) −4.93429e15 −0.901126 −0.450563 0.892745i \(-0.648777\pi\)
−0.450563 + 0.892745i \(0.648777\pi\)
\(728\) 4.90288e14 0.0888645
\(729\) 2.05891e14 0.0370370
\(730\) −1.01234e16 −1.80738
\(731\) −8.11052e14 −0.143715
\(732\) 4.20767e14 0.0739999
\(733\) −1.02451e16 −1.78832 −0.894159 0.447749i \(-0.852226\pi\)
−0.894159 + 0.447749i \(0.852226\pi\)
\(734\) 1.55911e15 0.270115
\(735\) −7.12943e14 −0.122596
\(736\) 7.48784e15 1.27799
\(737\) 1.05151e15 0.178133
\(738\) 2.52861e15 0.425179
\(739\) −5.63366e15 −0.940256 −0.470128 0.882598i \(-0.655792\pi\)
−0.470128 + 0.882598i \(0.655792\pi\)
\(740\) −7.09660e15 −1.17564
\(741\) 2.05840e14 0.0338477
\(742\) 2.79506e15 0.456215
\(743\) −7.62785e15 −1.23584 −0.617922 0.786240i \(-0.712025\pi\)
−0.617922 + 0.786240i \(0.712025\pi\)
\(744\) −3.79551e15 −0.610406
\(745\) 5.13437e15 0.819649
\(746\) −8.94691e15 −1.41778
\(747\) −1.31711e15 −0.207185
\(748\) −1.44381e15 −0.225450
\(749\) 1.16000e15 0.179808
\(750\) 1.85821e15 0.285929
\(751\) −1.18592e16 −1.81149 −0.905743 0.423827i \(-0.860686\pi\)
−0.905743 + 0.423827i \(0.860686\pi\)
\(752\) 9.89701e14 0.150074
\(753\) 4.46814e15 0.672597
\(754\) 1.30232e15 0.194615
\(755\) −6.70795e15 −0.995134
\(756\) −7.56987e14 −0.111485
\(757\) 3.75419e15 0.548894 0.274447 0.961602i \(-0.411505\pi\)
0.274447 + 0.961602i \(0.411505\pi\)
\(758\) −1.56813e16 −2.27615
\(759\) 4.61435e14 0.0664937
\(760\) −1.86174e15 −0.266345
\(761\) −4.79297e15 −0.680752 −0.340376 0.940289i \(-0.610554\pi\)
−0.340376 + 0.940289i \(0.610554\pi\)
\(762\) 1.06014e16 1.49489
\(763\) −2.79478e15 −0.391257
\(764\) 9.71744e15 1.35064
\(765\) 5.14135e15 0.709479
\(766\) −7.01148e15 −0.960619
\(767\) 1.23732e15 0.168309
\(768\) 2.92794e15 0.395435
\(769\) −1.07639e16 −1.44336 −0.721680 0.692227i \(-0.756630\pi\)
−0.721680 + 0.692227i \(0.756630\pi\)
\(770\) 6.89836e14 0.0918432
\(771\) 4.41196e15 0.583220
\(772\) 1.48118e16 1.94408
\(773\) −3.60788e15 −0.470181 −0.235091 0.971973i \(-0.575539\pi\)
−0.235091 + 0.971973i \(0.575539\pi\)
\(774\) 4.11452e14 0.0532407
\(775\) −1.17394e16 −1.50829
\(776\) 1.08926e16 1.38960
\(777\) 8.88995e14 0.112612
\(778\) −4.32810e15 −0.544390
\(779\) −1.35650e15 −0.169420
\(780\) 2.94151e15 0.364796
\(781\) 1.24200e15 0.152947
\(782\) −2.08940e16 −2.55496
\(783\) −6.98821e14 −0.0848547
\(784\) −2.17514e14 −0.0262270
\(785\) 2.34012e16 2.80191
\(786\) −6.43806e15 −0.765474
\(787\) 1.99937e15 0.236065 0.118033 0.993010i \(-0.462341\pi\)
0.118033 + 0.993010i \(0.462341\pi\)
\(788\) −7.12723e15 −0.835654
\(789\) −3.20256e15 −0.372884
\(790\) 1.46162e16 1.69000
\(791\) −1.42273e15 −0.163363
\(792\) 2.54560e14 0.0290269
\(793\) 2.04820e14 0.0231937
\(794\) −1.58749e16 −1.78525
\(795\) 5.82802e15 0.650882
\(796\) 3.94073e15 0.437074
\(797\) 2.19280e14 0.0241534 0.0120767 0.999927i \(-0.496156\pi\)
0.0120767 + 0.999927i \(0.496156\pi\)
\(798\) 6.71054e14 0.0734076
\(799\) −1.07744e16 −1.17054
\(800\) 1.27765e16 1.37853
\(801\) −5.04729e14 −0.0540853
\(802\) 8.64099e15 0.919611
\(803\) −7.42568e14 −0.0784876
\(804\) 1.46173e16 1.53447
\(805\) 6.04129e15 0.629872
\(806\) −5.31607e15 −0.550488
\(807\) 4.75708e15 0.489256
\(808\) −8.07023e15 −0.824373
\(809\) 1.84828e16 1.87521 0.937605 0.347701i \(-0.113038\pi\)
0.937605 + 0.347701i \(0.113038\pi\)
\(810\) −2.60824e15 −0.262833
\(811\) 1.30720e16 1.30837 0.654183 0.756337i \(-0.273013\pi\)
0.654183 + 0.756337i \(0.273013\pi\)
\(812\) 2.56931e15 0.255422
\(813\) −1.46943e15 −0.145094
\(814\) −8.60181e14 −0.0843637
\(815\) −1.75657e16 −1.71119
\(816\) 1.56859e15 0.151780
\(817\) −2.20729e14 −0.0212147
\(818\) −2.03898e16 −1.94657
\(819\) −3.68485e14 −0.0349428
\(820\) −1.93848e16 −1.82594
\(821\) 5.54032e15 0.518379 0.259190 0.965826i \(-0.416545\pi\)
0.259190 + 0.965826i \(0.416545\pi\)
\(822\) −5.05060e15 −0.469406
\(823\) −1.33182e16 −1.22955 −0.614775 0.788702i \(-0.710753\pi\)
−0.614775 + 0.788702i \(0.710753\pi\)
\(824\) 1.49076e15 0.136712
\(825\) 7.87345e14 0.0717247
\(826\) 4.03376e15 0.365023
\(827\) −3.48404e14 −0.0313186 −0.0156593 0.999877i \(-0.504985\pi\)
−0.0156593 + 0.999877i \(0.504985\pi\)
\(828\) 6.41450e15 0.572790
\(829\) 8.61235e15 0.763962 0.381981 0.924170i \(-0.375242\pi\)
0.381981 + 0.924170i \(0.375242\pi\)
\(830\) 1.66852e16 1.47028
\(831\) −1.01944e16 −0.892390
\(832\) 5.20017e15 0.452209
\(833\) 2.36797e15 0.204564
\(834\) −7.79893e15 −0.669302
\(835\) −2.22537e14 −0.0189726
\(836\) −3.92933e14 −0.0332800
\(837\) 2.85258e15 0.240020
\(838\) −3.64178e16 −3.04419
\(839\) 3.52097e15 0.292396 0.146198 0.989255i \(-0.453296\pi\)
0.146198 + 0.989255i \(0.453296\pi\)
\(840\) 3.33280e15 0.274962
\(841\) −9.82862e15 −0.805591
\(842\) −2.93215e16 −2.38764
\(843\) −5.55021e15 −0.449012
\(844\) −2.41122e16 −1.93799
\(845\) 1.43186e15 0.114338
\(846\) 5.46594e15 0.433639
\(847\) −4.74463e15 −0.373976
\(848\) 1.77809e15 0.139244
\(849\) −1.09446e16 −0.851543
\(850\) −3.56514e16 −2.75596
\(851\) −7.53310e15 −0.578577
\(852\) 1.72653e16 1.31752
\(853\) 5.94306e15 0.450599 0.225300 0.974290i \(-0.427664\pi\)
0.225300 + 0.974290i \(0.427664\pi\)
\(854\) 6.67730e14 0.0503017
\(855\) 1.39922e15 0.104731
\(856\) −5.42266e15 −0.403280
\(857\) −1.70470e16 −1.25966 −0.629829 0.776734i \(-0.716875\pi\)
−0.629829 + 0.776734i \(0.716875\pi\)
\(858\) 3.56542e14 0.0261776
\(859\) −4.41501e15 −0.322084 −0.161042 0.986948i \(-0.551486\pi\)
−0.161042 + 0.986948i \(0.551486\pi\)
\(860\) −3.15428e15 −0.228643
\(861\) 2.42835e15 0.174902
\(862\) 7.35087e15 0.526076
\(863\) 5.48468e15 0.390024 0.195012 0.980801i \(-0.437525\pi\)
0.195012 + 0.980801i \(0.437525\pi\)
\(864\) −3.10459e15 −0.219371
\(865\) −8.52575e15 −0.598610
\(866\) −2.67729e16 −1.86787
\(867\) −8.74845e15 −0.606493
\(868\) −1.04879e16 −0.722487
\(869\) 1.07213e15 0.0733900
\(870\) 8.85270e15 0.602171
\(871\) 7.11539e15 0.480949
\(872\) 1.30648e16 0.877527
\(873\) −8.18650e15 −0.546412
\(874\) −5.68632e15 −0.377154
\(875\) 1.78453e15 0.117620
\(876\) −1.03226e16 −0.676108
\(877\) −2.63676e16 −1.71622 −0.858109 0.513467i \(-0.828361\pi\)
−0.858109 + 0.513467i \(0.828361\pi\)
\(878\) −1.99023e15 −0.128731
\(879\) −2.65056e14 −0.0170372
\(880\) 4.38841e14 0.0280319
\(881\) 1.60495e16 1.01881 0.509406 0.860526i \(-0.329865\pi\)
0.509406 + 0.860526i \(0.329865\pi\)
\(882\) −1.20129e15 −0.0757827
\(883\) 2.51258e16 1.57520 0.787602 0.616184i \(-0.211322\pi\)
0.787602 + 0.616184i \(0.211322\pi\)
\(884\) −9.76995e15 −0.608702
\(885\) 8.41086e15 0.520778
\(886\) −3.75759e16 −2.31219
\(887\) −2.20145e16 −1.34626 −0.673130 0.739524i \(-0.735051\pi\)
−0.673130 + 0.739524i \(0.735051\pi\)
\(888\) −4.15578e15 −0.252570
\(889\) 1.01810e16 0.614938
\(890\) 6.39394e15 0.383816
\(891\) −1.91319e14 −0.0114138
\(892\) −3.05451e15 −0.181107
\(893\) −2.93227e15 −0.172791
\(894\) 8.65127e15 0.506668
\(895\) −2.60837e15 −0.151825
\(896\) 9.50556e15 0.549900
\(897\) 3.12244e15 0.179529
\(898\) 2.39134e14 0.0136654
\(899\) −9.68203e15 −0.549905
\(900\) 1.09450e16 0.617851
\(901\) −1.93572e16 −1.08607
\(902\) −2.34964e15 −0.131029
\(903\) 3.95138e14 0.0219011
\(904\) 6.65086e15 0.366397
\(905\) −2.80561e16 −1.53625
\(906\) −1.13027e16 −0.615144
\(907\) −2.39382e16 −1.29495 −0.647473 0.762089i \(-0.724174\pi\)
−0.647473 + 0.762089i \(0.724174\pi\)
\(908\) 9.38536e15 0.504637
\(909\) 6.06533e15 0.324155
\(910\) 4.66799e15 0.247972
\(911\) 2.18881e16 1.15573 0.577866 0.816132i \(-0.303886\pi\)
0.577866 + 0.816132i \(0.303886\pi\)
\(912\) 4.26893e14 0.0224051
\(913\) 1.22389e15 0.0638487
\(914\) 6.79278e15 0.352244
\(915\) 1.39229e15 0.0717654
\(916\) 4.16177e16 2.13232
\(917\) −6.18279e15 −0.314886
\(918\) 8.66303e15 0.438566
\(919\) −2.30665e16 −1.16077 −0.580385 0.814342i \(-0.697098\pi\)
−0.580385 + 0.814342i \(0.697098\pi\)
\(920\) −2.82412e16 −1.41270
\(921\) −1.24319e14 −0.00618173
\(922\) 3.94450e15 0.194972
\(923\) 8.40439e15 0.412949
\(924\) 7.03410e14 0.0343568
\(925\) −1.28537e16 −0.624093
\(926\) 3.01394e16 1.45470
\(927\) −1.12040e15 −0.0537571
\(928\) 1.05374e16 0.502595
\(929\) 2.27055e16 1.07657 0.538287 0.842761i \(-0.319072\pi\)
0.538287 + 0.842761i \(0.319072\pi\)
\(930\) −3.61367e16 −1.70330
\(931\) 6.44446e14 0.0301970
\(932\) −3.28469e16 −1.53006
\(933\) 1.87170e16 0.866735
\(934\) 4.78263e16 2.20171
\(935\) −4.77747e15 −0.218642
\(936\) 1.72256e15 0.0783712
\(937\) −2.97054e16 −1.34359 −0.671796 0.740736i \(-0.734477\pi\)
−0.671796 + 0.740736i \(0.734477\pi\)
\(938\) 2.31967e16 1.04306
\(939\) −5.86297e15 −0.262094
\(940\) −4.19030e16 −1.86227
\(941\) 3.25186e16 1.43678 0.718388 0.695642i \(-0.244880\pi\)
0.718388 + 0.695642i \(0.244880\pi\)
\(942\) 3.94303e16 1.73201
\(943\) −2.05772e16 −0.898610
\(944\) 2.56609e15 0.111411
\(945\) −2.50482e15 −0.108119
\(946\) −3.82331e14 −0.0164073
\(947\) 6.73968e15 0.287551 0.143775 0.989610i \(-0.454076\pi\)
0.143775 + 0.989610i \(0.454076\pi\)
\(948\) 1.49038e16 0.632197
\(949\) −5.02481e15 −0.211912
\(950\) −9.70257e15 −0.406824
\(951\) 2.39793e16 0.999639
\(952\) −1.10696e16 −0.458805
\(953\) 9.97765e15 0.411166 0.205583 0.978640i \(-0.434091\pi\)
0.205583 + 0.978640i \(0.434091\pi\)
\(954\) 9.82005e15 0.402344
\(955\) 3.21544e16 1.30985
\(956\) 9.70836e15 0.393212
\(957\) 6.49361e14 0.0261499
\(958\) −2.68118e16 −1.07353
\(959\) −4.85035e15 −0.193095
\(960\) 3.53489e16 1.39921
\(961\) 1.41135e16 0.555464
\(962\) −5.82068e15 −0.227778
\(963\) 4.07550e15 0.158576
\(964\) 1.24560e16 0.481899
\(965\) 4.90115e16 1.88537
\(966\) 1.01794e16 0.389357
\(967\) −2.22025e16 −0.844416 −0.422208 0.906499i \(-0.638745\pi\)
−0.422208 + 0.906499i \(0.638745\pi\)
\(968\) 2.21797e16 0.838768
\(969\) −4.64739e15 −0.174754
\(970\) 1.03707e17 3.87761
\(971\) −3.61492e16 −1.34398 −0.671990 0.740560i \(-0.734560\pi\)
−0.671990 + 0.740560i \(0.734560\pi\)
\(972\) −2.65956e15 −0.0983209
\(973\) −7.48970e15 −0.275324
\(974\) −5.01186e15 −0.183200
\(975\) 5.32781e15 0.193653
\(976\) 4.24779e14 0.0153528
\(977\) 2.03981e16 0.733112 0.366556 0.930396i \(-0.380537\pi\)
0.366556 + 0.930396i \(0.380537\pi\)
\(978\) −2.95977e16 −1.05778
\(979\) 4.69006e14 0.0166676
\(980\) 9.20933e15 0.325450
\(981\) −9.81906e15 −0.345056
\(982\) −8.24448e16 −2.88105
\(983\) −3.81698e15 −0.132640 −0.0663202 0.997798i \(-0.521126\pi\)
−0.0663202 + 0.997798i \(0.521126\pi\)
\(984\) −1.13518e16 −0.392276
\(985\) −2.35836e16 −0.810421
\(986\) −2.94034e16 −1.00479
\(987\) 5.24922e15 0.178382
\(988\) −2.65890e15 −0.0898543
\(989\) −3.34829e15 −0.112524
\(990\) 2.42364e15 0.0809981
\(991\) 4.37998e16 1.45568 0.727841 0.685746i \(-0.240524\pi\)
0.727841 + 0.685746i \(0.240524\pi\)
\(992\) −4.30135e16 −1.42164
\(993\) −4.24542e15 −0.139540
\(994\) 2.73989e16 0.895589
\(995\) 1.30397e16 0.423877
\(996\) 1.70135e16 0.550006
\(997\) −1.61014e15 −0.0517656 −0.0258828 0.999665i \(-0.508240\pi\)
−0.0258828 + 0.999665i \(0.508240\pi\)
\(998\) 7.05176e16 2.25465
\(999\) 3.12336e15 0.0993142
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.1.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.3 16 1.1 even 1 trivial