Properties

Label 273.12.a.c.1.14
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 1.14
Root \(72.4378\) of defining polynomial
Character \(\chi\) \(=\) 273.1

$q$-expansion

\(f(q)\) \(=\) \(q+68.4378 q^{2} -243.000 q^{3} +2635.73 q^{4} +3481.49 q^{5} -16630.4 q^{6} +16807.0 q^{7} +40223.3 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+68.4378 q^{2} -243.000 q^{3} +2635.73 q^{4} +3481.49 q^{5} -16630.4 q^{6} +16807.0 q^{7} +40223.3 q^{8} +59049.0 q^{9} +238265. q^{10} +564893. q^{11} -640484. q^{12} -371293. q^{13} +1.15023e6 q^{14} -846001. q^{15} -2.64519e6 q^{16} -7.43323e6 q^{17} +4.04118e6 q^{18} -5.25778e6 q^{19} +9.17628e6 q^{20} -4.08410e6 q^{21} +3.86601e7 q^{22} +6.43385e6 q^{23} -9.77426e6 q^{24} -3.67074e7 q^{25} -2.54105e7 q^{26} -1.43489e7 q^{27} +4.42988e7 q^{28} -6.05266e7 q^{29} -5.78985e7 q^{30} +1.86892e8 q^{31} -2.63408e8 q^{32} -1.37269e8 q^{33} -5.08714e8 q^{34} +5.85134e7 q^{35} +1.55638e8 q^{36} -2.81894e7 q^{37} -3.59831e8 q^{38} +9.02242e7 q^{39} +1.40037e8 q^{40} -9.05699e8 q^{41} -2.79507e8 q^{42} -9.23656e8 q^{43} +1.48891e9 q^{44} +2.05578e8 q^{45} +4.40319e8 q^{46} +1.20837e9 q^{47} +6.42781e8 q^{48} +2.82475e8 q^{49} -2.51217e9 q^{50} +1.80628e9 q^{51} -9.78630e8 q^{52} +7.69035e8 q^{53} -9.82008e8 q^{54} +1.96667e9 q^{55} +6.76033e8 q^{56} +1.27764e9 q^{57} -4.14231e9 q^{58} +3.00954e9 q^{59} -2.22984e9 q^{60} +6.04594e9 q^{61} +1.27905e10 q^{62} +9.92437e8 q^{63} -1.26097e10 q^{64} -1.29265e9 q^{65} -9.39440e9 q^{66} -1.34769e10 q^{67} -1.95920e10 q^{68} -1.56343e9 q^{69} +4.00453e9 q^{70} -1.16893e10 q^{71} +2.37515e9 q^{72} +1.04851e9 q^{73} -1.92922e9 q^{74} +8.91989e9 q^{75} -1.38581e10 q^{76} +9.49416e9 q^{77} +6.17475e9 q^{78} +1.51686e10 q^{79} -9.20920e9 q^{80} +3.48678e9 q^{81} -6.19840e10 q^{82} +3.27992e10 q^{83} -1.07646e10 q^{84} -2.58787e10 q^{85} -6.32130e10 q^{86} +1.47080e10 q^{87} +2.27219e10 q^{88} -6.18269e10 q^{89} +1.40693e10 q^{90} -6.24032e9 q^{91} +1.69579e10 q^{92} -4.54147e10 q^{93} +8.26980e10 q^{94} -1.83049e10 q^{95} +6.40082e10 q^{96} -5.58660e10 q^{97} +1.93320e10 q^{98} +3.33564e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + O(q^{10}) \) \( 16q - 63q^{2} - 3888q^{3} + 20761q^{4} - 3168q^{5} + 15309q^{6} + 268912q^{7} - 187965q^{8} + 944784q^{9} + 1056711q^{10} - 466884q^{11} - 5044923q^{12} - 5940688q^{13} - 1058841q^{14} + 769824q^{15} + 40058265q^{16} + 1753452q^{17} - 3720087q^{18} + 4237800q^{19} - 20710503q^{20} - 65345616q^{21} - 37479661q^{22} - 150481440q^{23} + 45675495q^{24} + 150983176q^{25} + 23391459q^{26} - 229582512q^{27} + 348930127q^{28} - 111411432q^{29} - 256780773q^{30} - 90536236q^{31} - 609941313q^{32} + 113452812q^{33} + 443745577q^{34} - 53244576q^{35} + 1225916289q^{36} - 1756337900q^{37} - 4378868973q^{38} + 1443587184q^{39} + 57535383q^{40} + 649575720q^{41} + 257298363q^{42} - 1889554520q^{43} - 4979587689q^{44} - 187067232q^{45} - 3204000867q^{46} - 4036082940q^{47} - 9734158395q^{48} + 4519603984q^{49} - 15928886862q^{50} - 426088836q^{51} - 7708413973q^{52} + 511144020q^{53} + 903981141q^{54} - 8519365572q^{55} - 3159127755q^{56} - 1029785400q^{57} - 7851149577q^{58} + 3728287296q^{59} + 5032652229q^{60} + 26227205052q^{61} + 2996618490q^{62} + 15878984688q^{63} + 51104408465q^{64} + 1176256224q^{65} + 9107557623q^{66} - 5295680024q^{67} + 7919540325q^{68} + 36566989920q^{69} + 17760141777q^{70} + 17082800928q^{71} - 11099145285q^{72} + 21076301488q^{73} + 52794333675q^{74} - 36688911768q^{75} + 81459179177q^{76} - 7846919388q^{77} - 5684124537q^{78} - 83677977852q^{79} - 163988465427q^{80} + 55788550416q^{81} - 61543728760q^{82} - 72857072340q^{83} - 84790020861q^{84} - 12608565060q^{85} - 20177818011q^{86} + 27072977976q^{87} - 202213679643q^{88} + 32238476676q^{89} + 62397727839q^{90} - 99845143216q^{91} - 487057197033q^{92} + 22000305348q^{93} + 183904776602q^{94} - 143022915504q^{95} + 148215739059q^{96} + 6973535140q^{97} - 17795940687q^{98} - 27569033316q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 68.4378 1.51228 0.756138 0.654412i \(-0.227084\pi\)
0.756138 + 0.654412i \(0.227084\pi\)
\(3\) −243.000 −0.577350
\(4\) 2635.73 1.28698
\(5\) 3481.49 0.498230 0.249115 0.968474i \(-0.419860\pi\)
0.249115 + 0.968474i \(0.419860\pi\)
\(6\) −16630.4 −0.873113
\(7\) 16807.0 0.377964
\(8\) 40223.3 0.433993
\(9\) 59049.0 0.333333
\(10\) 238265. 0.753461
\(11\) 564893. 1.05756 0.528782 0.848758i \(-0.322649\pi\)
0.528782 + 0.848758i \(0.322649\pi\)
\(12\) −640484. −0.743038
\(13\) −371293. −0.277350
\(14\) 1.15023e6 0.571587
\(15\) −846001. −0.287653
\(16\) −2.64519e6 −0.630663
\(17\) −7.43323e6 −1.26972 −0.634861 0.772626i \(-0.718943\pi\)
−0.634861 + 0.772626i \(0.718943\pi\)
\(18\) 4.04118e6 0.504092
\(19\) −5.25778e6 −0.487144 −0.243572 0.969883i \(-0.578319\pi\)
−0.243572 + 0.969883i \(0.578319\pi\)
\(20\) 9.17628e6 0.641212
\(21\) −4.08410e6 −0.218218
\(22\) 3.86601e7 1.59933
\(23\) 6.43385e6 0.208434 0.104217 0.994555i \(-0.466766\pi\)
0.104217 + 0.994555i \(0.466766\pi\)
\(24\) −9.77426e6 −0.250566
\(25\) −3.67074e7 −0.751767
\(26\) −2.54105e7 −0.419430
\(27\) −1.43489e7 −0.192450
\(28\) 4.42988e7 0.486433
\(29\) −6.05266e7 −0.547971 −0.273985 0.961734i \(-0.588342\pi\)
−0.273985 + 0.961734i \(0.588342\pi\)
\(30\) −5.78985e7 −0.435011
\(31\) 1.86892e8 1.17247 0.586234 0.810142i \(-0.300610\pi\)
0.586234 + 0.810142i \(0.300610\pi\)
\(32\) −2.63408e8 −1.38773
\(33\) −1.37269e8 −0.610585
\(34\) −5.08714e8 −1.92017
\(35\) 5.85134e7 0.188313
\(36\) 1.55638e8 0.428993
\(37\) −2.81894e7 −0.0668307 −0.0334154 0.999442i \(-0.510638\pi\)
−0.0334154 + 0.999442i \(0.510638\pi\)
\(38\) −3.59831e8 −0.736697
\(39\) 9.02242e7 0.160128
\(40\) 1.40037e8 0.216228
\(41\) −9.05699e8 −1.22088 −0.610439 0.792063i \(-0.709007\pi\)
−0.610439 + 0.792063i \(0.709007\pi\)
\(42\) −2.79507e8 −0.330006
\(43\) −9.23656e8 −0.958150 −0.479075 0.877774i \(-0.659028\pi\)
−0.479075 + 0.877774i \(0.659028\pi\)
\(44\) 1.48891e9 1.36106
\(45\) 2.05578e8 0.166077
\(46\) 4.40319e8 0.315209
\(47\) 1.20837e9 0.768529 0.384265 0.923223i \(-0.374455\pi\)
0.384265 + 0.923223i \(0.374455\pi\)
\(48\) 6.42781e8 0.364113
\(49\) 2.82475e8 0.142857
\(50\) −2.51217e9 −1.13688
\(51\) 1.80628e9 0.733074
\(52\) −9.78630e8 −0.356944
\(53\) 7.69035e8 0.252597 0.126299 0.991992i \(-0.459690\pi\)
0.126299 + 0.991992i \(0.459690\pi\)
\(54\) −9.82008e8 −0.291038
\(55\) 1.96667e9 0.526910
\(56\) 6.76033e8 0.164034
\(57\) 1.27764e9 0.281253
\(58\) −4.14231e9 −0.828683
\(59\) 3.00954e9 0.548042 0.274021 0.961724i \(-0.411646\pi\)
0.274021 + 0.961724i \(0.411646\pi\)
\(60\) −2.22984e9 −0.370204
\(61\) 6.04594e9 0.916537 0.458268 0.888814i \(-0.348470\pi\)
0.458268 + 0.888814i \(0.348470\pi\)
\(62\) 1.27905e10 1.77310
\(63\) 9.92437e8 0.125988
\(64\) −1.26097e10 −1.46797
\(65\) −1.29265e9 −0.138184
\(66\) −9.39440e9 −0.923373
\(67\) −1.34769e10 −1.21949 −0.609746 0.792597i \(-0.708729\pi\)
−0.609746 + 0.792597i \(0.708729\pi\)
\(68\) −1.95920e10 −1.63411
\(69\) −1.56343e9 −0.120339
\(70\) 4.00453e9 0.284782
\(71\) −1.16893e10 −0.768895 −0.384447 0.923147i \(-0.625608\pi\)
−0.384447 + 0.923147i \(0.625608\pi\)
\(72\) 2.37515e9 0.144664
\(73\) 1.04851e9 0.0591965 0.0295982 0.999562i \(-0.490577\pi\)
0.0295982 + 0.999562i \(0.490577\pi\)
\(74\) −1.92922e9 −0.101067
\(75\) 8.91989e9 0.434033
\(76\) −1.38581e10 −0.626945
\(77\) 9.49416e9 0.399722
\(78\) 6.17475e9 0.242158
\(79\) 1.51686e10 0.554621 0.277311 0.960780i \(-0.410557\pi\)
0.277311 + 0.960780i \(0.410557\pi\)
\(80\) −9.20920e9 −0.314215
\(81\) 3.48678e9 0.111111
\(82\) −6.19840e10 −1.84631
\(83\) 3.27992e10 0.913974 0.456987 0.889473i \(-0.348929\pi\)
0.456987 + 0.889473i \(0.348929\pi\)
\(84\) −1.07646e10 −0.280842
\(85\) −2.58787e10 −0.632613
\(86\) −6.32130e10 −1.44899
\(87\) 1.47080e10 0.316371
\(88\) 2.27219e10 0.458975
\(89\) −6.18269e10 −1.17363 −0.586816 0.809720i \(-0.699619\pi\)
−0.586816 + 0.809720i \(0.699619\pi\)
\(90\) 1.40693e10 0.251154
\(91\) −6.24032e9 −0.104828
\(92\) 1.69579e10 0.268250
\(93\) −4.54147e10 −0.676924
\(94\) 8.26980e10 1.16223
\(95\) −1.83049e10 −0.242710
\(96\) 6.40082e10 0.801206
\(97\) −5.58660e10 −0.660546 −0.330273 0.943885i \(-0.607141\pi\)
−0.330273 + 0.943885i \(0.607141\pi\)
\(98\) 1.93320e10 0.216039
\(99\) 3.33564e10 0.352521
\(100\) −9.67509e10 −0.967509
\(101\) −1.20562e11 −1.14142 −0.570709 0.821153i \(-0.693331\pi\)
−0.570709 + 0.821153i \(0.693331\pi\)
\(102\) 1.23618e11 1.10861
\(103\) −1.91328e11 −1.62620 −0.813101 0.582122i \(-0.802222\pi\)
−0.813101 + 0.582122i \(0.802222\pi\)
\(104\) −1.49346e10 −0.120368
\(105\) −1.42187e10 −0.108723
\(106\) 5.26310e10 0.381997
\(107\) −2.48403e11 −1.71216 −0.856082 0.516840i \(-0.827108\pi\)
−0.856082 + 0.516840i \(0.827108\pi\)
\(108\) −3.78199e10 −0.247679
\(109\) −1.88296e11 −1.17218 −0.586090 0.810246i \(-0.699333\pi\)
−0.586090 + 0.810246i \(0.699333\pi\)
\(110\) 1.34595e11 0.796834
\(111\) 6.85002e9 0.0385847
\(112\) −4.44577e10 −0.238368
\(113\) 3.09322e11 1.57935 0.789677 0.613523i \(-0.210248\pi\)
0.789677 + 0.613523i \(0.210248\pi\)
\(114\) 8.74389e10 0.425332
\(115\) 2.23994e10 0.103848
\(116\) −1.59532e11 −0.705227
\(117\) −2.19245e10 −0.0924500
\(118\) 2.05966e11 0.828791
\(119\) −1.24930e11 −0.479910
\(120\) −3.40290e10 −0.124839
\(121\) 3.37930e10 0.118442
\(122\) 4.13771e11 1.38606
\(123\) 2.20085e11 0.704875
\(124\) 4.92597e11 1.50894
\(125\) −2.97791e11 −0.872783
\(126\) 6.79202e10 0.190529
\(127\) 2.25536e11 0.605752 0.302876 0.953030i \(-0.402053\pi\)
0.302876 + 0.953030i \(0.402053\pi\)
\(128\) −3.23523e11 −0.832243
\(129\) 2.24448e11 0.553188
\(130\) −8.84663e10 −0.208973
\(131\) 7.60909e11 1.72322 0.861610 0.507571i \(-0.169457\pi\)
0.861610 + 0.507571i \(0.169457\pi\)
\(132\) −3.61805e11 −0.785811
\(133\) −8.83675e10 −0.184123
\(134\) −9.22331e11 −1.84421
\(135\) −4.99555e10 −0.0958844
\(136\) −2.98989e11 −0.551050
\(137\) −5.74354e11 −1.01676 −0.508378 0.861134i \(-0.669755\pi\)
−0.508378 + 0.861134i \(0.669755\pi\)
\(138\) −1.06997e11 −0.181986
\(139\) 8.10846e11 1.32543 0.662715 0.748871i \(-0.269404\pi\)
0.662715 + 0.748871i \(0.269404\pi\)
\(140\) 1.54226e11 0.242355
\(141\) −2.93633e11 −0.443711
\(142\) −7.99989e11 −1.16278
\(143\) −2.09741e11 −0.293316
\(144\) −1.56196e11 −0.210221
\(145\) −2.10723e11 −0.273015
\(146\) 7.17576e10 0.0895214
\(147\) −6.86415e10 −0.0824786
\(148\) −7.42997e10 −0.0860098
\(149\) −1.03895e12 −1.15896 −0.579481 0.814986i \(-0.696745\pi\)
−0.579481 + 0.814986i \(0.696745\pi\)
\(150\) 6.10458e11 0.656378
\(151\) −1.27401e12 −1.32069 −0.660343 0.750964i \(-0.729589\pi\)
−0.660343 + 0.750964i \(0.729589\pi\)
\(152\) −2.11485e11 −0.211417
\(153\) −4.38925e11 −0.423241
\(154\) 6.49760e11 0.604490
\(155\) 6.50661e11 0.584158
\(156\) 2.37807e11 0.206082
\(157\) −1.75079e12 −1.46482 −0.732412 0.680862i \(-0.761605\pi\)
−0.732412 + 0.680862i \(0.761605\pi\)
\(158\) 1.03811e12 0.838741
\(159\) −1.86875e11 −0.145837
\(160\) −9.17053e11 −0.691408
\(161\) 1.08134e11 0.0787805
\(162\) 2.38628e11 0.168031
\(163\) 1.96458e12 1.33733 0.668664 0.743565i \(-0.266867\pi\)
0.668664 + 0.743565i \(0.266867\pi\)
\(164\) −2.38718e12 −1.57125
\(165\) −4.77901e11 −0.304212
\(166\) 2.24470e12 1.38218
\(167\) 4.65300e10 0.0277200 0.0138600 0.999904i \(-0.495588\pi\)
0.0138600 + 0.999904i \(0.495588\pi\)
\(168\) −1.64276e11 −0.0947050
\(169\) 1.37858e11 0.0769231
\(170\) −1.77108e12 −0.956686
\(171\) −3.10467e11 −0.162381
\(172\) −2.43451e12 −1.23312
\(173\) −3.15039e12 −1.54565 −0.772825 0.634619i \(-0.781157\pi\)
−0.772825 + 0.634619i \(0.781157\pi\)
\(174\) 1.00658e12 0.478441
\(175\) −6.16941e11 −0.284141
\(176\) −1.49425e12 −0.666966
\(177\) −7.31318e11 −0.316412
\(178\) −4.23130e12 −1.77486
\(179\) −1.12504e12 −0.457588 −0.228794 0.973475i \(-0.573478\pi\)
−0.228794 + 0.973475i \(0.573478\pi\)
\(180\) 5.41850e11 0.213737
\(181\) 1.35256e11 0.0517518 0.0258759 0.999665i \(-0.491763\pi\)
0.0258759 + 0.999665i \(0.491763\pi\)
\(182\) −4.27074e11 −0.158530
\(183\) −1.46916e12 −0.529163
\(184\) 2.58791e11 0.0904588
\(185\) −9.81410e10 −0.0332971
\(186\) −3.10808e12 −1.02370
\(187\) −4.19898e12 −1.34281
\(188\) 3.18493e12 0.989082
\(189\) −2.41162e11 −0.0727393
\(190\) −1.25275e12 −0.367044
\(191\) 8.49260e11 0.241745 0.120872 0.992668i \(-0.461431\pi\)
0.120872 + 0.992668i \(0.461431\pi\)
\(192\) 3.06417e12 0.847531
\(193\) −3.57384e12 −0.960661 −0.480330 0.877088i \(-0.659483\pi\)
−0.480330 + 0.877088i \(0.659483\pi\)
\(194\) −3.82335e12 −0.998929
\(195\) 3.14114e11 0.0797806
\(196\) 7.44530e11 0.183854
\(197\) 7.02036e8 0.000168576 0 8.42879e−5 1.00000i \(-0.499973\pi\)
8.42879e−5 1.00000i \(0.499973\pi\)
\(198\) 2.28284e12 0.533110
\(199\) −1.20032e12 −0.272650 −0.136325 0.990664i \(-0.543529\pi\)
−0.136325 + 0.990664i \(0.543529\pi\)
\(200\) −1.47649e12 −0.326262
\(201\) 3.27489e12 0.704074
\(202\) −8.25103e12 −1.72614
\(203\) −1.01727e12 −0.207114
\(204\) 4.76086e12 0.943452
\(205\) −3.15318e12 −0.608278
\(206\) −1.30941e13 −2.45927
\(207\) 3.79913e11 0.0694779
\(208\) 9.82141e11 0.174914
\(209\) −2.97009e12 −0.515186
\(210\) −9.73100e11 −0.164419
\(211\) −5.23462e12 −0.861652 −0.430826 0.902435i \(-0.641778\pi\)
−0.430826 + 0.902435i \(0.641778\pi\)
\(212\) 2.02697e12 0.325088
\(213\) 2.84050e12 0.443922
\(214\) −1.70001e13 −2.58926
\(215\) −3.21570e12 −0.477379
\(216\) −5.77160e11 −0.0835220
\(217\) 3.14109e12 0.443151
\(218\) −1.28865e13 −1.77266
\(219\) −2.54787e11 −0.0341771
\(220\) 5.18362e12 0.678123
\(221\) 2.75991e12 0.352157
\(222\) 4.68800e11 0.0583508
\(223\) 3.57089e12 0.433610 0.216805 0.976215i \(-0.430436\pi\)
0.216805 + 0.976215i \(0.430436\pi\)
\(224\) −4.42710e12 −0.524512
\(225\) −2.16753e12 −0.250589
\(226\) 2.11693e13 2.38842
\(227\) −1.63188e12 −0.179699 −0.0898494 0.995955i \(-0.528639\pi\)
−0.0898494 + 0.995955i \(0.528639\pi\)
\(228\) 3.36752e12 0.361967
\(229\) 9.85983e12 1.03460 0.517302 0.855803i \(-0.326936\pi\)
0.517302 + 0.855803i \(0.326936\pi\)
\(230\) 1.53296e12 0.157047
\(231\) −2.30708e12 −0.230779
\(232\) −2.43458e12 −0.237815
\(233\) −1.30585e13 −1.24577 −0.622883 0.782315i \(-0.714039\pi\)
−0.622883 + 0.782315i \(0.714039\pi\)
\(234\) −1.50046e12 −0.139810
\(235\) 4.20691e12 0.382904
\(236\) 7.93234e12 0.705319
\(237\) −3.68597e12 −0.320211
\(238\) −8.54996e12 −0.725756
\(239\) 7.45534e12 0.618414 0.309207 0.950995i \(-0.399936\pi\)
0.309207 + 0.950995i \(0.399936\pi\)
\(240\) 2.23784e12 0.181412
\(241\) −1.10955e12 −0.0879129 −0.0439565 0.999033i \(-0.513996\pi\)
−0.0439565 + 0.999033i \(0.513996\pi\)
\(242\) 2.31272e12 0.179118
\(243\) −8.47289e11 −0.0641500
\(244\) 1.59355e13 1.17956
\(245\) 9.83434e11 0.0711757
\(246\) 1.50621e13 1.06597
\(247\) 1.95218e12 0.135110
\(248\) 7.51740e12 0.508843
\(249\) −7.97020e12 −0.527683
\(250\) −2.03801e13 −1.31989
\(251\) 1.24080e13 0.786133 0.393066 0.919510i \(-0.371414\pi\)
0.393066 + 0.919510i \(0.371414\pi\)
\(252\) 2.61580e12 0.162144
\(253\) 3.63444e12 0.220432
\(254\) 1.54352e13 0.916064
\(255\) 6.28852e12 0.365240
\(256\) 3.68355e12 0.209386
\(257\) 2.13373e12 0.118716 0.0593578 0.998237i \(-0.481095\pi\)
0.0593578 + 0.998237i \(0.481095\pi\)
\(258\) 1.53608e13 0.836574
\(259\) −4.73779e11 −0.0252596
\(260\) −3.40709e12 −0.177840
\(261\) −3.57404e12 −0.182657
\(262\) 5.20750e13 2.60598
\(263\) 2.97653e13 1.45866 0.729328 0.684164i \(-0.239833\pi\)
0.729328 + 0.684164i \(0.239833\pi\)
\(264\) −5.52142e12 −0.264990
\(265\) 2.67738e12 0.125851
\(266\) −6.04768e12 −0.278445
\(267\) 1.50239e13 0.677597
\(268\) −3.55216e13 −1.56946
\(269\) −3.33130e13 −1.44203 −0.721017 0.692917i \(-0.756325\pi\)
−0.721017 + 0.692917i \(0.756325\pi\)
\(270\) −3.41885e12 −0.145004
\(271\) 1.57794e13 0.655781 0.327890 0.944716i \(-0.393662\pi\)
0.327890 + 0.944716i \(0.393662\pi\)
\(272\) 1.96623e13 0.800766
\(273\) 1.51640e12 0.0605228
\(274\) −3.93075e13 −1.53761
\(275\) −2.07358e13 −0.795042
\(276\) −4.12078e12 −0.154874
\(277\) 3.54997e13 1.30794 0.653968 0.756523i \(-0.273103\pi\)
0.653968 + 0.756523i \(0.273103\pi\)
\(278\) 5.54925e13 2.00442
\(279\) 1.10358e13 0.390823
\(280\) 2.35360e12 0.0817266
\(281\) −4.92480e13 −1.67689 −0.838443 0.544989i \(-0.816534\pi\)
−0.838443 + 0.544989i \(0.816534\pi\)
\(282\) −2.00956e13 −0.671013
\(283\) −1.79600e13 −0.588141 −0.294071 0.955784i \(-0.595010\pi\)
−0.294071 + 0.955784i \(0.595010\pi\)
\(284\) −3.08099e13 −0.989552
\(285\) 4.44809e12 0.140129
\(286\) −1.43542e13 −0.443574
\(287\) −1.52221e13 −0.461449
\(288\) −1.55540e13 −0.462576
\(289\) 2.09810e13 0.612194
\(290\) −1.44214e13 −0.412875
\(291\) 1.35754e13 0.381367
\(292\) 2.76359e12 0.0761847
\(293\) 5.39376e13 1.45922 0.729608 0.683865i \(-0.239702\pi\)
0.729608 + 0.683865i \(0.239702\pi\)
\(294\) −4.69767e12 −0.124730
\(295\) 1.04777e13 0.273051
\(296\) −1.13387e12 −0.0290041
\(297\) −8.10560e12 −0.203528
\(298\) −7.11033e13 −1.75267
\(299\) −2.38884e12 −0.0578091
\(300\) 2.35105e13 0.558592
\(301\) −1.55239e13 −0.362147
\(302\) −8.71905e13 −1.99724
\(303\) 2.92967e13 0.658998
\(304\) 1.39078e13 0.307224
\(305\) 2.10489e13 0.456646
\(306\) −3.00391e13 −0.640057
\(307\) −2.56574e13 −0.536972 −0.268486 0.963284i \(-0.586523\pi\)
−0.268486 + 0.963284i \(0.586523\pi\)
\(308\) 2.50241e13 0.514434
\(309\) 4.64928e13 0.938888
\(310\) 4.45298e13 0.883409
\(311\) −1.84395e13 −0.359391 −0.179695 0.983722i \(-0.557511\pi\)
−0.179695 + 0.983722i \(0.557511\pi\)
\(312\) 3.62911e12 0.0694945
\(313\) −1.83257e13 −0.344800 −0.172400 0.985027i \(-0.555152\pi\)
−0.172400 + 0.985027i \(0.555152\pi\)
\(314\) −1.19820e14 −2.21522
\(315\) 3.45516e12 0.0627711
\(316\) 3.99804e13 0.713786
\(317\) −5.12731e13 −0.899629 −0.449814 0.893122i \(-0.648510\pi\)
−0.449814 + 0.893122i \(0.648510\pi\)
\(318\) −1.27893e13 −0.220546
\(319\) −3.41911e13 −0.579514
\(320\) −4.39007e13 −0.731385
\(321\) 6.03618e13 0.988518
\(322\) 7.40044e12 0.119138
\(323\) 3.90823e13 0.618538
\(324\) 9.19024e12 0.142998
\(325\) 1.36292e13 0.208503
\(326\) 1.34452e14 2.02241
\(327\) 4.57558e13 0.676758
\(328\) −3.64302e13 −0.529853
\(329\) 2.03090e13 0.290477
\(330\) −3.27065e13 −0.460052
\(331\) −1.41462e13 −0.195697 −0.0978486 0.995201i \(-0.531196\pi\)
−0.0978486 + 0.995201i \(0.531196\pi\)
\(332\) 8.64500e13 1.17627
\(333\) −1.66456e12 −0.0222769
\(334\) 3.18441e12 0.0419202
\(335\) −4.69197e13 −0.607587
\(336\) 1.08032e13 0.137622
\(337\) 2.17233e13 0.272246 0.136123 0.990692i \(-0.456536\pi\)
0.136123 + 0.990692i \(0.456536\pi\)
\(338\) 9.43473e12 0.116329
\(339\) −7.51653e13 −0.911840
\(340\) −6.82094e13 −0.814161
\(341\) 1.05574e14 1.23996
\(342\) −2.12477e13 −0.245566
\(343\) 4.74756e12 0.0539949
\(344\) −3.71525e13 −0.415830
\(345\) −5.44305e12 −0.0599566
\(346\) −2.15606e14 −2.33745
\(347\) 1.86177e13 0.198662 0.0993308 0.995054i \(-0.468330\pi\)
0.0993308 + 0.995054i \(0.468330\pi\)
\(348\) 3.87663e13 0.407163
\(349\) −1.13332e14 −1.17169 −0.585847 0.810421i \(-0.699238\pi\)
−0.585847 + 0.810421i \(0.699238\pi\)
\(350\) −4.22221e13 −0.429700
\(351\) 5.32765e12 0.0533761
\(352\) −1.48798e14 −1.46761
\(353\) 6.63649e13 0.644433 0.322216 0.946666i \(-0.395572\pi\)
0.322216 + 0.946666i \(0.395572\pi\)
\(354\) −5.00498e13 −0.478503
\(355\) −4.06961e13 −0.383086
\(356\) −1.62959e14 −1.51044
\(357\) 3.03581e13 0.277076
\(358\) −7.69950e13 −0.692000
\(359\) 1.92535e14 1.70408 0.852039 0.523479i \(-0.175366\pi\)
0.852039 + 0.523479i \(0.175366\pi\)
\(360\) 8.26904e12 0.0720761
\(361\) −8.88460e13 −0.762690
\(362\) 9.25665e12 0.0782630
\(363\) −8.21169e12 −0.0683827
\(364\) −1.64478e13 −0.134912
\(365\) 3.65037e12 0.0294935
\(366\) −1.00546e14 −0.800240
\(367\) −5.64786e13 −0.442813 −0.221407 0.975182i \(-0.571065\pi\)
−0.221407 + 0.975182i \(0.571065\pi\)
\(368\) −1.70188e13 −0.131451
\(369\) −5.34806e13 −0.406960
\(370\) −6.71656e12 −0.0503544
\(371\) 1.29252e13 0.0954728
\(372\) −1.19701e14 −0.871188
\(373\) −2.00343e14 −1.43673 −0.718365 0.695666i \(-0.755109\pi\)
−0.718365 + 0.695666i \(0.755109\pi\)
\(374\) −2.87369e14 −2.03070
\(375\) 7.23632e13 0.503901
\(376\) 4.86045e13 0.333536
\(377\) 2.24731e13 0.151980
\(378\) −1.65046e13 −0.110002
\(379\) −6.63639e13 −0.435929 −0.217965 0.975957i \(-0.569942\pi\)
−0.217965 + 0.975957i \(0.569942\pi\)
\(380\) −4.82468e13 −0.312363
\(381\) −5.48052e13 −0.349731
\(382\) 5.81215e13 0.365585
\(383\) 2.46460e14 1.52811 0.764053 0.645154i \(-0.223207\pi\)
0.764053 + 0.645154i \(0.223207\pi\)
\(384\) 7.86161e13 0.480496
\(385\) 3.30538e13 0.199153
\(386\) −2.44586e14 −1.45278
\(387\) −5.45409e13 −0.319383
\(388\) −1.47248e14 −0.850110
\(389\) 3.76084e13 0.214073 0.107036 0.994255i \(-0.465864\pi\)
0.107036 + 0.994255i \(0.465864\pi\)
\(390\) 2.14973e13 0.120650
\(391\) −4.78243e13 −0.264653
\(392\) 1.13621e13 0.0619990
\(393\) −1.84901e14 −0.994901
\(394\) 4.80458e10 0.000254933 0
\(395\) 5.28093e13 0.276329
\(396\) 8.79186e13 0.453688
\(397\) 2.05786e14 1.04729 0.523645 0.851936i \(-0.324572\pi\)
0.523645 + 0.851936i \(0.324572\pi\)
\(398\) −8.21473e13 −0.412322
\(399\) 2.14733e13 0.106304
\(400\) 9.70980e13 0.474111
\(401\) −2.11851e14 −1.02032 −0.510160 0.860079i \(-0.670414\pi\)
−0.510160 + 0.860079i \(0.670414\pi\)
\(402\) 2.24126e14 1.06475
\(403\) −6.93916e13 −0.325184
\(404\) −3.17771e14 −1.46898
\(405\) 1.21392e13 0.0553589
\(406\) −6.96198e13 −0.313213
\(407\) −1.59240e13 −0.0706778
\(408\) 7.26543e13 0.318149
\(409\) −9.92496e13 −0.428796 −0.214398 0.976746i \(-0.568779\pi\)
−0.214398 + 0.976746i \(0.568779\pi\)
\(410\) −2.15797e14 −0.919885
\(411\) 1.39568e14 0.587024
\(412\) −5.04291e14 −2.09289
\(413\) 5.05813e13 0.207140
\(414\) 2.60004e13 0.105070
\(415\) 1.14190e14 0.455369
\(416\) 9.78017e13 0.384887
\(417\) −1.97036e14 −0.765238
\(418\) −2.03266e14 −0.779104
\(419\) 4.76296e14 1.80177 0.900885 0.434057i \(-0.142918\pi\)
0.900885 + 0.434057i \(0.142918\pi\)
\(420\) −3.74768e13 −0.139924
\(421\) −6.89369e13 −0.254039 −0.127019 0.991900i \(-0.540541\pi\)
−0.127019 + 0.991900i \(0.540541\pi\)
\(422\) −3.58246e14 −1.30306
\(423\) 7.13528e13 0.256176
\(424\) 3.09331e13 0.109625
\(425\) 2.72854e14 0.954535
\(426\) 1.94397e14 0.671332
\(427\) 1.01614e14 0.346418
\(428\) −6.54723e14 −2.20352
\(429\) 5.09671e13 0.169346
\(430\) −2.20075e14 −0.721929
\(431\) −8.62118e13 −0.279217 −0.139608 0.990207i \(-0.544584\pi\)
−0.139608 + 0.990207i \(0.544584\pi\)
\(432\) 3.79556e13 0.121371
\(433\) −3.47451e14 −1.09701 −0.548504 0.836148i \(-0.684802\pi\)
−0.548504 + 0.836148i \(0.684802\pi\)
\(434\) 2.14969e14 0.670167
\(435\) 5.12056e13 0.157626
\(436\) −4.96297e14 −1.50857
\(437\) −3.38278e13 −0.101537
\(438\) −1.74371e13 −0.0516852
\(439\) 1.22043e14 0.357238 0.178619 0.983918i \(-0.442837\pi\)
0.178619 + 0.983918i \(0.442837\pi\)
\(440\) 7.91059e13 0.228675
\(441\) 1.66799e13 0.0476190
\(442\) 1.88882e14 0.532559
\(443\) −6.92479e13 −0.192835 −0.0964175 0.995341i \(-0.530738\pi\)
−0.0964175 + 0.995341i \(0.530738\pi\)
\(444\) 1.80548e13 0.0496578
\(445\) −2.15250e14 −0.584739
\(446\) 2.44384e14 0.655739
\(447\) 2.52464e14 0.669127
\(448\) −2.11932e14 −0.554840
\(449\) 6.35502e14 1.64347 0.821736 0.569868i \(-0.193006\pi\)
0.821736 + 0.569868i \(0.193006\pi\)
\(450\) −1.48341e14 −0.378960
\(451\) −5.11623e14 −1.29116
\(452\) 8.15291e14 2.03260
\(453\) 3.09585e14 0.762499
\(454\) −1.11682e14 −0.271754
\(455\) −2.17256e13 −0.0522287
\(456\) 5.13909e13 0.122062
\(457\) 4.55857e14 1.06977 0.534884 0.844925i \(-0.320355\pi\)
0.534884 + 0.844925i \(0.320355\pi\)
\(458\) 6.74785e14 1.56461
\(459\) 1.06659e14 0.244358
\(460\) 5.90388e13 0.133650
\(461\) 7.15733e14 1.60102 0.800508 0.599322i \(-0.204563\pi\)
0.800508 + 0.599322i \(0.204563\pi\)
\(462\) −1.57892e14 −0.349002
\(463\) −6.87620e14 −1.50194 −0.750971 0.660335i \(-0.770414\pi\)
−0.750971 + 0.660335i \(0.770414\pi\)
\(464\) 1.60104e14 0.345585
\(465\) −1.58111e14 −0.337264
\(466\) −8.93697e14 −1.88394
\(467\) 1.37772e14 0.287025 0.143512 0.989649i \(-0.454160\pi\)
0.143512 + 0.989649i \(0.454160\pi\)
\(468\) −5.77871e13 −0.118981
\(469\) −2.26506e14 −0.460925
\(470\) 2.87912e14 0.579057
\(471\) 4.25441e14 0.845716
\(472\) 1.21053e14 0.237846
\(473\) −5.21767e14 −1.01331
\(474\) −2.52260e14 −0.484247
\(475\) 1.92999e14 0.366219
\(476\) −3.29283e14 −0.617634
\(477\) 4.54107e13 0.0841991
\(478\) 5.10227e14 0.935212
\(479\) 4.62840e13 0.0838659 0.0419329 0.999120i \(-0.486648\pi\)
0.0419329 + 0.999120i \(0.486648\pi\)
\(480\) 2.22844e14 0.399185
\(481\) 1.04665e13 0.0185355
\(482\) −7.59351e13 −0.132949
\(483\) −2.62765e13 −0.0454840
\(484\) 8.90693e13 0.152433
\(485\) −1.94497e14 −0.329104
\(486\) −5.79866e13 −0.0970126
\(487\) 4.35951e14 0.721154 0.360577 0.932729i \(-0.382580\pi\)
0.360577 + 0.932729i \(0.382580\pi\)
\(488\) 2.43188e14 0.397771
\(489\) −4.77393e14 −0.772106
\(490\) 6.73041e13 0.107637
\(491\) 6.19791e14 0.980161 0.490080 0.871677i \(-0.336967\pi\)
0.490080 + 0.871677i \(0.336967\pi\)
\(492\) 5.80085e14 0.907159
\(493\) 4.49908e14 0.695771
\(494\) 1.33603e14 0.204323
\(495\) 1.16130e14 0.175637
\(496\) −4.94364e14 −0.739431
\(497\) −1.96462e14 −0.290615
\(498\) −5.45463e14 −0.798002
\(499\) −1.52673e14 −0.220906 −0.110453 0.993881i \(-0.535230\pi\)
−0.110453 + 0.993881i \(0.535230\pi\)
\(500\) −7.84897e14 −1.12325
\(501\) −1.13068e13 −0.0160041
\(502\) 8.49176e14 1.18885
\(503\) 1.18696e15 1.64366 0.821831 0.569731i \(-0.192952\pi\)
0.821831 + 0.569731i \(0.192952\pi\)
\(504\) 3.99191e13 0.0546780
\(505\) −4.19737e14 −0.568688
\(506\) 2.48733e14 0.333354
\(507\) −3.34996e13 −0.0444116
\(508\) 5.94452e14 0.779590
\(509\) 1.39272e15 1.80682 0.903412 0.428773i \(-0.141054\pi\)
0.903412 + 0.428773i \(0.141054\pi\)
\(510\) 4.30373e14 0.552343
\(511\) 1.76223e13 0.0223742
\(512\) 9.14669e14 1.14889
\(513\) 7.54434e13 0.0937510
\(514\) 1.46028e14 0.179531
\(515\) −6.66107e14 −0.810223
\(516\) 5.91586e14 0.711942
\(517\) 6.82598e14 0.812769
\(518\) −3.24244e13 −0.0381996
\(519\) 7.65546e14 0.892382
\(520\) −5.19947e13 −0.0599709
\(521\) −1.03833e14 −0.118502 −0.0592511 0.998243i \(-0.518871\pi\)
−0.0592511 + 0.998243i \(0.518871\pi\)
\(522\) −2.44599e14 −0.276228
\(523\) −2.26986e14 −0.253653 −0.126827 0.991925i \(-0.540479\pi\)
−0.126827 + 0.991925i \(0.540479\pi\)
\(524\) 2.00555e15 2.21775
\(525\) 1.49917e14 0.164049
\(526\) 2.03707e15 2.20589
\(527\) −1.38921e15 −1.48871
\(528\) 3.63103e14 0.385073
\(529\) −9.11415e14 −0.956555
\(530\) 1.83234e14 0.190322
\(531\) 1.77710e14 0.182681
\(532\) −2.32913e14 −0.236963
\(533\) 3.36280e14 0.338611
\(534\) 1.02821e15 1.02471
\(535\) −8.64810e14 −0.853051
\(536\) −5.42086e14 −0.529251
\(537\) 2.73384e14 0.264189
\(538\) −2.27987e15 −2.18076
\(539\) 1.59568e14 0.151081
\(540\) −1.31670e14 −0.123401
\(541\) 4.53144e14 0.420389 0.210194 0.977660i \(-0.432590\pi\)
0.210194 + 0.977660i \(0.432590\pi\)
\(542\) 1.07991e15 0.991722
\(543\) −3.28673e13 −0.0298789
\(544\) 1.95798e15 1.76203
\(545\) −6.55549e14 −0.584015
\(546\) 1.03779e14 0.0915271
\(547\) 7.41296e14 0.647234 0.323617 0.946188i \(-0.395101\pi\)
0.323617 + 0.946188i \(0.395101\pi\)
\(548\) −1.51384e15 −1.30854
\(549\) 3.57007e14 0.305512
\(550\) −1.41911e15 −1.20232
\(551\) 3.18236e14 0.266941
\(552\) −6.28861e13 −0.0522264
\(553\) 2.54939e14 0.209627
\(554\) 2.42952e15 1.97796
\(555\) 2.38483e13 0.0192241
\(556\) 2.13718e15 1.70580
\(557\) −1.05234e15 −0.831676 −0.415838 0.909439i \(-0.636512\pi\)
−0.415838 + 0.909439i \(0.636512\pi\)
\(558\) 7.55264e14 0.591032
\(559\) 3.42947e14 0.265743
\(560\) −1.54779e14 −0.118762
\(561\) 1.02035e15 0.775273
\(562\) −3.37042e15 −2.53592
\(563\) 3.00877e14 0.224178 0.112089 0.993698i \(-0.464246\pi\)
0.112089 + 0.993698i \(0.464246\pi\)
\(564\) −7.73939e14 −0.571047
\(565\) 1.07690e15 0.786881
\(566\) −1.22914e15 −0.889432
\(567\) 5.86024e13 0.0419961
\(568\) −4.70181e14 −0.333695
\(569\) 9.20184e14 0.646781 0.323390 0.946266i \(-0.395177\pi\)
0.323390 + 0.946266i \(0.395177\pi\)
\(570\) 3.04418e14 0.211913
\(571\) 1.94242e15 1.33920 0.669598 0.742724i \(-0.266466\pi\)
0.669598 + 0.742724i \(0.266466\pi\)
\(572\) −5.52822e14 −0.377491
\(573\) −2.06370e14 −0.139571
\(574\) −1.04177e15 −0.697838
\(575\) −2.36170e14 −0.156694
\(576\) −7.44593e14 −0.489322
\(577\) 8.72958e14 0.568233 0.284117 0.958790i \(-0.408300\pi\)
0.284117 + 0.958790i \(0.408300\pi\)
\(578\) 1.43590e15 0.925806
\(579\) 8.68443e14 0.554638
\(580\) −5.55409e14 −0.351365
\(581\) 5.51256e14 0.345450
\(582\) 9.29074e14 0.576732
\(583\) 4.34423e14 0.267138
\(584\) 4.21744e13 0.0256909
\(585\) −7.63298e13 −0.0460614
\(586\) 3.69137e15 2.20674
\(587\) −2.91423e15 −1.72589 −0.862946 0.505296i \(-0.831383\pi\)
−0.862946 + 0.505296i \(0.831383\pi\)
\(588\) −1.80921e14 −0.106148
\(589\) −9.82636e14 −0.571161
\(590\) 7.17069e14 0.412928
\(591\) −1.70595e11 −9.73273e−5 0
\(592\) 7.45663e13 0.0421476
\(593\) 1.40975e15 0.789479 0.394740 0.918793i \(-0.370835\pi\)
0.394740 + 0.918793i \(0.370835\pi\)
\(594\) −5.54730e14 −0.307791
\(595\) −4.34943e14 −0.239105
\(596\) −2.73839e15 −1.49156
\(597\) 2.91678e14 0.157414
\(598\) −1.63487e14 −0.0874234
\(599\) −1.24793e14 −0.0661217 −0.0330608 0.999453i \(-0.510526\pi\)
−0.0330608 + 0.999453i \(0.510526\pi\)
\(600\) 3.58787e14 0.188367
\(601\) 1.03295e15 0.537367 0.268683 0.963229i \(-0.413411\pi\)
0.268683 + 0.963229i \(0.413411\pi\)
\(602\) −1.06242e15 −0.547666
\(603\) −7.95798e14 −0.406497
\(604\) −3.35795e15 −1.69970
\(605\) 1.17650e14 0.0590115
\(606\) 2.00500e15 0.996587
\(607\) 2.63492e14 0.129787 0.0648934 0.997892i \(-0.479329\pi\)
0.0648934 + 0.997892i \(0.479329\pi\)
\(608\) 1.38494e15 0.676024
\(609\) 2.47197e14 0.119577
\(610\) 1.44054e15 0.690575
\(611\) −4.48658e14 −0.213152
\(612\) −1.15689e15 −0.544702
\(613\) 3.61668e14 0.168763 0.0843815 0.996434i \(-0.473109\pi\)
0.0843815 + 0.996434i \(0.473109\pi\)
\(614\) −1.75594e15 −0.812051
\(615\) 7.66222e14 0.351190
\(616\) 3.81887e14 0.173476
\(617\) −1.47671e15 −0.664854 −0.332427 0.943129i \(-0.607868\pi\)
−0.332427 + 0.943129i \(0.607868\pi\)
\(618\) 3.18187e15 1.41986
\(619\) −1.16495e15 −0.515240 −0.257620 0.966246i \(-0.582938\pi\)
−0.257620 + 0.966246i \(0.582938\pi\)
\(620\) 1.71497e15 0.751800
\(621\) −9.23187e13 −0.0401131
\(622\) −1.26196e15 −0.543499
\(623\) −1.03912e15 −0.443592
\(624\) −2.38660e14 −0.100987
\(625\) 7.55597e14 0.316920
\(626\) −1.25417e15 −0.521433
\(627\) 7.21731e14 0.297443
\(628\) −4.61461e15 −1.88520
\(629\) 2.09538e14 0.0848564
\(630\) 2.36463e14 0.0949272
\(631\) 4.09811e15 1.63088 0.815440 0.578842i \(-0.196495\pi\)
0.815440 + 0.578842i \(0.196495\pi\)
\(632\) 6.10131e14 0.240702
\(633\) 1.27201e15 0.497475
\(634\) −3.50902e15 −1.36049
\(635\) 7.85199e14 0.301804
\(636\) −4.92554e14 −0.187689
\(637\) −1.04881e14 −0.0396214
\(638\) −2.33996e15 −0.876386
\(639\) −6.90241e14 −0.256298
\(640\) −1.12634e15 −0.414649
\(641\) −6.73941e14 −0.245982 −0.122991 0.992408i \(-0.539249\pi\)
−0.122991 + 0.992408i \(0.539249\pi\)
\(642\) 4.13103e15 1.49491
\(643\) 3.72880e15 1.33785 0.668927 0.743328i \(-0.266754\pi\)
0.668927 + 0.743328i \(0.266754\pi\)
\(644\) 2.85012e14 0.101389
\(645\) 7.81414e14 0.275615
\(646\) 2.67471e15 0.935400
\(647\) 3.53635e15 1.22626 0.613129 0.789983i \(-0.289911\pi\)
0.613129 + 0.789983i \(0.289911\pi\)
\(648\) 1.40250e14 0.0482214
\(649\) 1.70007e15 0.579589
\(650\) 9.32752e14 0.315314
\(651\) −7.63285e14 −0.255853
\(652\) 5.17811e15 1.72111
\(653\) −5.87842e15 −1.93748 −0.968742 0.248072i \(-0.920203\pi\)
−0.968742 + 0.248072i \(0.920203\pi\)
\(654\) 3.13143e15 1.02345
\(655\) 2.64910e15 0.858560
\(656\) 2.39575e15 0.769963
\(657\) 6.19133e13 0.0197322
\(658\) 1.38990e15 0.439281
\(659\) −4.25033e15 −1.33215 −0.666074 0.745886i \(-0.732026\pi\)
−0.666074 + 0.745886i \(0.732026\pi\)
\(660\) −1.25962e15 −0.391514
\(661\) 5.20976e15 1.60587 0.802933 0.596069i \(-0.203272\pi\)
0.802933 + 0.596069i \(0.203272\pi\)
\(662\) −9.68133e14 −0.295948
\(663\) −6.70657e14 −0.203318
\(664\) 1.31929e15 0.396658
\(665\) −3.07650e14 −0.0917357
\(666\) −1.13919e14 −0.0336888
\(667\) −3.89419e14 −0.114216
\(668\) 1.22641e14 0.0356750
\(669\) −8.67726e14 −0.250345
\(670\) −3.21108e15 −0.918840
\(671\) 3.41531e15 0.969297
\(672\) 1.07579e15 0.302827
\(673\) −5.75002e15 −1.60541 −0.802706 0.596375i \(-0.796607\pi\)
−0.802706 + 0.596375i \(0.796607\pi\)
\(674\) 1.48670e15 0.411712
\(675\) 5.26711e14 0.144678
\(676\) 3.63358e14 0.0989985
\(677\) 3.42405e15 0.925342 0.462671 0.886530i \(-0.346891\pi\)
0.462671 + 0.886530i \(0.346891\pi\)
\(678\) −5.14415e15 −1.37895
\(679\) −9.38941e14 −0.249663
\(680\) −1.04093e15 −0.274550
\(681\) 3.96546e14 0.103749
\(682\) 7.22525e15 1.87516
\(683\) −6.36803e15 −1.63942 −0.819712 0.572775i \(-0.805867\pi\)
−0.819712 + 0.572775i \(0.805867\pi\)
\(684\) −8.18308e14 −0.208982
\(685\) −1.99961e15 −0.506578
\(686\) 3.24913e14 0.0816553
\(687\) −2.39594e15 −0.597329
\(688\) 2.44325e15 0.604270
\(689\) −2.85537e14 −0.0700579
\(690\) −3.72510e14 −0.0906710
\(691\) −3.43644e15 −0.829810 −0.414905 0.909865i \(-0.636185\pi\)
−0.414905 + 0.909865i \(0.636185\pi\)
\(692\) −8.30360e15 −1.98922
\(693\) 5.60621e14 0.133241
\(694\) 1.27415e15 0.300431
\(695\) 2.82295e15 0.660369
\(696\) 5.91603e14 0.137303
\(697\) 6.73227e15 1.55018
\(698\) −7.75623e15 −1.77193
\(699\) 3.17322e15 0.719243
\(700\) −1.62609e15 −0.365684
\(701\) 3.08865e15 0.689159 0.344580 0.938757i \(-0.388021\pi\)
0.344580 + 0.938757i \(0.388021\pi\)
\(702\) 3.64613e14 0.0807193
\(703\) 1.48214e14 0.0325562
\(704\) −7.12316e15 −1.55247
\(705\) −1.02228e15 −0.221070
\(706\) 4.54187e15 0.974561
\(707\) −2.02629e15 −0.431415
\(708\) −1.92756e15 −0.407216
\(709\) 1.67007e15 0.350090 0.175045 0.984560i \(-0.443993\pi\)
0.175045 + 0.984560i \(0.443993\pi\)
\(710\) −2.78515e15 −0.579333
\(711\) 8.95691e14 0.184874
\(712\) −2.48688e15 −0.509348
\(713\) 1.20243e15 0.244382
\(714\) 2.07764e15 0.419016
\(715\) −7.30211e14 −0.146139
\(716\) −2.96530e15 −0.588907
\(717\) −1.81165e15 −0.357041
\(718\) 1.31766e16 2.57704
\(719\) −1.28059e15 −0.248542 −0.124271 0.992248i \(-0.539659\pi\)
−0.124271 + 0.992248i \(0.539659\pi\)
\(720\) −5.43794e14 −0.104738
\(721\) −3.21566e15 −0.614647
\(722\) −6.08043e15 −1.15340
\(723\) 2.69620e14 0.0507566
\(724\) 3.56500e14 0.0666035
\(725\) 2.22177e15 0.411946
\(726\) −5.61990e14 −0.103414
\(727\) −5.21108e15 −0.951674 −0.475837 0.879533i \(-0.657855\pi\)
−0.475837 + 0.879533i \(0.657855\pi\)
\(728\) −2.51006e14 −0.0454948
\(729\) 2.05891e14 0.0370370
\(730\) 2.49823e14 0.0446023
\(731\) 6.86575e15 1.21658
\(732\) −3.87233e15 −0.681022
\(733\) 6.95004e15 1.21315 0.606576 0.795025i \(-0.292543\pi\)
0.606576 + 0.795025i \(0.292543\pi\)
\(734\) −3.86527e15 −0.669656
\(735\) −2.38974e14 −0.0410933
\(736\) −1.69473e15 −0.289250
\(737\) −7.61302e15 −1.28969
\(738\) −3.66010e15 −0.615435
\(739\) 2.58294e15 0.431091 0.215546 0.976494i \(-0.430847\pi\)
0.215546 + 0.976494i \(0.430847\pi\)
\(740\) −2.58674e14 −0.0428527
\(741\) −4.74379e14 −0.0780055
\(742\) 8.84570e14 0.144381
\(743\) −4.84381e15 −0.784782 −0.392391 0.919799i \(-0.628352\pi\)
−0.392391 + 0.919799i \(0.628352\pi\)
\(744\) −1.82673e15 −0.293780
\(745\) −3.61708e15 −0.577429
\(746\) −1.37110e16 −2.17273
\(747\) 1.93676e15 0.304658
\(748\) −1.10674e16 −1.72817
\(749\) −4.17490e15 −0.647137
\(750\) 4.95238e15 0.762038
\(751\) −4.72235e15 −0.721338 −0.360669 0.932694i \(-0.617452\pi\)
−0.360669 + 0.932694i \(0.617452\pi\)
\(752\) −3.19636e15 −0.484683
\(753\) −3.01514e15 −0.453874
\(754\) 1.53801e15 0.229835
\(755\) −4.43545e15 −0.658006
\(756\) −6.35639e14 −0.0936140
\(757\) 4.22447e15 0.617653 0.308826 0.951118i \(-0.400064\pi\)
0.308826 + 0.951118i \(0.400064\pi\)
\(758\) −4.54180e15 −0.659246
\(759\) −8.83169e14 −0.127267
\(760\) −7.36283e14 −0.105334
\(761\) −1.46113e15 −0.207526 −0.103763 0.994602i \(-0.533088\pi\)
−0.103763 + 0.994602i \(0.533088\pi\)
\(762\) −3.75075e15 −0.528890
\(763\) −3.16468e15 −0.443042
\(764\) 2.23842e15 0.311121
\(765\) −1.52811e15 −0.210871
\(766\) 1.68672e16 2.31092
\(767\) −1.11742e15 −0.151999
\(768\) −8.95102e14 −0.120889
\(769\) 4.54971e15 0.610083 0.305041 0.952339i \(-0.401330\pi\)
0.305041 + 0.952339i \(0.401330\pi\)
\(770\) 2.26213e15 0.301175
\(771\) −5.18497e14 −0.0685405
\(772\) −9.41970e15 −1.23635
\(773\) −1.50519e15 −0.196157 −0.0980783 0.995179i \(-0.531270\pi\)
−0.0980783 + 0.995179i \(0.531270\pi\)
\(774\) −3.73266e15 −0.482996
\(775\) −6.86031e15 −0.881422
\(776\) −2.24712e15 −0.286672
\(777\) 1.15128e14 0.0145837
\(778\) 2.57384e15 0.323738
\(779\) 4.76196e15 0.594744
\(780\) 8.27922e14 0.102676
\(781\) −6.60320e15 −0.813156
\(782\) −3.27299e15 −0.400228
\(783\) 8.68491e14 0.105457
\(784\) −7.47201e14 −0.0900947
\(785\) −6.09534e15 −0.729819
\(786\) −1.26542e16 −1.50457
\(787\) −3.40004e15 −0.401442 −0.200721 0.979648i \(-0.564329\pi\)
−0.200721 + 0.979648i \(0.564329\pi\)
\(788\) 1.85038e12 0.000216954 0
\(789\) −7.23296e15 −0.842156
\(790\) 3.61415e15 0.417886
\(791\) 5.19878e15 0.596940
\(792\) 1.34170e15 0.152992
\(793\) −2.24482e15 −0.254202
\(794\) 1.40835e16 1.58379
\(795\) −6.50604e14 −0.0726604
\(796\) −3.16373e15 −0.350895
\(797\) 4.93714e14 0.0543820 0.0271910 0.999630i \(-0.491344\pi\)
0.0271910 + 0.999630i \(0.491344\pi\)
\(798\) 1.46959e15 0.160760
\(799\) −8.98207e15 −0.975819
\(800\) 9.66903e15 1.04325
\(801\) −3.65082e15 −0.391211
\(802\) −1.44986e16 −1.54301
\(803\) 5.92295e14 0.0626041
\(804\) 8.63174e15 0.906129
\(805\) 3.76466e14 0.0392508
\(806\) −4.74901e15 −0.491768
\(807\) 8.09505e15 0.832559
\(808\) −4.84942e15 −0.495367
\(809\) 7.00301e15 0.710506 0.355253 0.934770i \(-0.384395\pi\)
0.355253 + 0.934770i \(0.384395\pi\)
\(810\) 8.30780e14 0.0837179
\(811\) −1.19498e16 −1.19604 −0.598022 0.801480i \(-0.704046\pi\)
−0.598022 + 0.801480i \(0.704046\pi\)
\(812\) −2.68126e15 −0.266551
\(813\) −3.83439e15 −0.378615
\(814\) −1.08980e15 −0.106884
\(815\) 6.83966e15 0.666296
\(816\) −4.77794e15 −0.462323
\(817\) 4.85638e15 0.466757
\(818\) −6.79242e15 −0.648458
\(819\) −3.68485e14 −0.0349428
\(820\) −8.31094e15 −0.782842
\(821\) −1.00099e16 −0.936576 −0.468288 0.883576i \(-0.655129\pi\)
−0.468288 + 0.883576i \(0.655129\pi\)
\(822\) 9.55173e15 0.887742
\(823\) −1.16216e16 −1.07292 −0.536459 0.843927i \(-0.680238\pi\)
−0.536459 + 0.843927i \(0.680238\pi\)
\(824\) −7.69586e15 −0.705760
\(825\) 5.03879e15 0.459018
\(826\) 3.46167e15 0.313253
\(827\) −1.16988e16 −1.05163 −0.525814 0.850600i \(-0.676239\pi\)
−0.525814 + 0.850600i \(0.676239\pi\)
\(828\) 1.00135e15 0.0894167
\(829\) 1.61662e16 1.43403 0.717014 0.697059i \(-0.245508\pi\)
0.717014 + 0.697059i \(0.245508\pi\)
\(830\) 7.81491e15 0.688644
\(831\) −8.62643e15 −0.755137
\(832\) 4.68191e15 0.407141
\(833\) −2.09970e15 −0.181389
\(834\) −1.34847e16 −1.15725
\(835\) 1.61994e14 0.0138109
\(836\) −7.82836e15 −0.663035
\(837\) −2.68169e15 −0.225641
\(838\) 3.25966e16 2.72478
\(839\) −2.92069e15 −0.242546 −0.121273 0.992619i \(-0.538698\pi\)
−0.121273 + 0.992619i \(0.538698\pi\)
\(840\) −5.71925e14 −0.0471849
\(841\) −8.53704e15 −0.699728
\(842\) −4.71789e15 −0.384177
\(843\) 1.19673e16 0.968151
\(844\) −1.37971e16 −1.10893
\(845\) 4.79953e14 0.0383254
\(846\) 4.88323e15 0.387410
\(847\) 5.67959e14 0.0447670
\(848\) −2.03424e15 −0.159304
\(849\) 4.36429e15 0.339563
\(850\) 1.86736e16 1.44352
\(851\) −1.81366e14 −0.0139298
\(852\) 7.48679e15 0.571318
\(853\) 5.41837e15 0.410817 0.205409 0.978676i \(-0.434148\pi\)
0.205409 + 0.978676i \(0.434148\pi\)
\(854\) 6.95425e15 0.523880
\(855\) −1.08089e15 −0.0809033
\(856\) −9.99157e15 −0.743067
\(857\) 7.90458e15 0.584096 0.292048 0.956404i \(-0.405663\pi\)
0.292048 + 0.956404i \(0.405663\pi\)
\(858\) 3.48807e15 0.256098
\(859\) 7.29614e15 0.532269 0.266134 0.963936i \(-0.414254\pi\)
0.266134 + 0.963936i \(0.414254\pi\)
\(860\) −8.47572e15 −0.614377
\(861\) 3.69896e15 0.266418
\(862\) −5.90015e15 −0.422253
\(863\) 7.92090e15 0.563268 0.281634 0.959522i \(-0.409124\pi\)
0.281634 + 0.959522i \(0.409124\pi\)
\(864\) 3.77962e15 0.267069
\(865\) −1.09681e16 −0.770089
\(866\) −2.37788e16 −1.65898
\(867\) −5.09839e15 −0.353450
\(868\) 8.27908e15 0.570327
\(869\) 8.56864e15 0.586548
\(870\) 3.50440e15 0.238373
\(871\) 5.00388e15 0.338226
\(872\) −7.57387e15 −0.508718
\(873\) −3.29883e15 −0.220182
\(874\) −2.31510e15 −0.153552
\(875\) −5.00497e15 −0.329881
\(876\) −6.71552e14 −0.0439852
\(877\) 1.31560e16 0.856303 0.428151 0.903707i \(-0.359165\pi\)
0.428151 + 0.903707i \(0.359165\pi\)
\(878\) 8.35236e15 0.540243
\(879\) −1.31068e16 −0.842479
\(880\) −5.20222e15 −0.332303
\(881\) 1.35898e16 0.862672 0.431336 0.902191i \(-0.358042\pi\)
0.431336 + 0.902191i \(0.358042\pi\)
\(882\) 1.14153e15 0.0720132
\(883\) 1.94005e15 0.121627 0.0608133 0.998149i \(-0.480631\pi\)
0.0608133 + 0.998149i \(0.480631\pi\)
\(884\) 7.27438e15 0.453220
\(885\) −2.54607e15 −0.157646
\(886\) −4.73917e15 −0.291620
\(887\) 2.27701e16 1.39247 0.696234 0.717815i \(-0.254858\pi\)
0.696234 + 0.717815i \(0.254858\pi\)
\(888\) 2.75530e14 0.0167455
\(889\) 3.79058e15 0.228953
\(890\) −1.47312e16 −0.884287
\(891\) 1.96966e15 0.117507
\(892\) 9.41192e15 0.558048
\(893\) −6.35333e15 −0.374385
\(894\) 1.72781e16 1.01190
\(895\) −3.91680e15 −0.227984
\(896\) −5.43745e15 −0.314558
\(897\) 5.80489e14 0.0333761
\(898\) 4.34924e16 2.48538
\(899\) −1.13119e16 −0.642478
\(900\) −5.71304e15 −0.322503
\(901\) −5.71641e15 −0.320728
\(902\) −3.50144e16 −1.95259
\(903\) 3.77230e15 0.209086
\(904\) 1.24420e16 0.685428
\(905\) 4.70893e14 0.0257843
\(906\) 2.11873e16 1.15311
\(907\) 3.17945e16 1.71993 0.859966 0.510351i \(-0.170484\pi\)
0.859966 + 0.510351i \(0.170484\pi\)
\(908\) −4.30120e15 −0.231269
\(909\) −7.11909e15 −0.380472
\(910\) −1.48685e15 −0.0789842
\(911\) 1.55297e15 0.0819994 0.0409997 0.999159i \(-0.486946\pi\)
0.0409997 + 0.999159i \(0.486946\pi\)
\(912\) −3.37960e15 −0.177376
\(913\) 1.85280e16 0.966586
\(914\) 3.11979e16 1.61779
\(915\) −5.11488e15 −0.263645
\(916\) 2.59879e16 1.33151
\(917\) 1.27886e16 0.651316
\(918\) 7.29949e15 0.369537
\(919\) 1.18817e16 0.597923 0.298961 0.954265i \(-0.403360\pi\)
0.298961 + 0.954265i \(0.403360\pi\)
\(920\) 9.00977e14 0.0450693
\(921\) 6.23475e15 0.310021
\(922\) 4.89832e16 2.42118
\(923\) 4.34015e15 0.213253
\(924\) −6.08086e15 −0.297009
\(925\) 1.03476e15 0.0502411
\(926\) −4.70592e16 −2.27135
\(927\) −1.12977e16 −0.542067
\(928\) 1.59432e16 0.760435
\(929\) −2.21695e16 −1.05116 −0.525580 0.850744i \(-0.676152\pi\)
−0.525580 + 0.850744i \(0.676152\pi\)
\(930\) −1.08208e16 −0.510036
\(931\) −1.48519e15 −0.0695920
\(932\) −3.44188e16 −1.60328
\(933\) 4.48080e15 0.207494
\(934\) 9.42884e15 0.434061
\(935\) −1.46187e16 −0.669029
\(936\) −8.81875e14 −0.0401227
\(937\) 2.52413e16 1.14168 0.570839 0.821062i \(-0.306618\pi\)
0.570839 + 0.821062i \(0.306618\pi\)
\(938\) −1.55016e16 −0.697046
\(939\) 4.45316e15 0.199071
\(940\) 1.10883e16 0.492790
\(941\) −1.97665e15 −0.0873347 −0.0436673 0.999046i \(-0.513904\pi\)
−0.0436673 + 0.999046i \(0.513904\pi\)
\(942\) 2.91163e16 1.27896
\(943\) −5.82713e15 −0.254472
\(944\) −7.96080e15 −0.345629
\(945\) −8.39603e14 −0.0362409
\(946\) −3.57086e16 −1.53240
\(947\) 1.56686e16 0.668504 0.334252 0.942484i \(-0.391516\pi\)
0.334252 + 0.942484i \(0.391516\pi\)
\(948\) −9.71524e15 −0.412105
\(949\) −3.89304e14 −0.0164182
\(950\) 1.32084e16 0.553824
\(951\) 1.24594e16 0.519401
\(952\) −5.02511e15 −0.208277
\(953\) 3.87197e15 0.159559 0.0797795 0.996813i \(-0.474578\pi\)
0.0797795 + 0.996813i \(0.474578\pi\)
\(954\) 3.10781e15 0.127332
\(955\) 2.95669e15 0.120444
\(956\) 1.96503e16 0.795886
\(957\) 8.30844e15 0.334583
\(958\) 3.16757e15 0.126828
\(959\) −9.65316e15 −0.384297
\(960\) 1.06679e16 0.422265
\(961\) 9.52005e15 0.374680
\(962\) 7.16306e14 0.0280308
\(963\) −1.46679e16 −0.570721
\(964\) −2.92448e15 −0.113142
\(965\) −1.24423e16 −0.478630
\(966\) −1.79831e15 −0.0687843
\(967\) −4.48172e16 −1.70451 −0.852254 0.523129i \(-0.824765\pi\)
−0.852254 + 0.523129i \(0.824765\pi\)
\(968\) 1.35926e15 0.0514031
\(969\) −9.49700e15 −0.357113
\(970\) −1.33109e16 −0.497696
\(971\) −4.31794e16 −1.60535 −0.802676 0.596415i \(-0.796591\pi\)
−0.802676 + 0.596415i \(0.796591\pi\)
\(972\) −2.23323e15 −0.0825598
\(973\) 1.36279e16 0.500966
\(974\) 2.98355e16 1.09058
\(975\) −3.31189e15 −0.120379
\(976\) −1.59927e16 −0.578026
\(977\) −3.39705e16 −1.22091 −0.610453 0.792053i \(-0.709012\pi\)
−0.610453 + 0.792053i \(0.709012\pi\)
\(978\) −3.26717e16 −1.16764
\(979\) −3.49256e16 −1.24119
\(980\) 2.59207e15 0.0916017
\(981\) −1.11187e16 −0.390727
\(982\) 4.24172e16 1.48227
\(983\) 4.91544e16 1.70812 0.854059 0.520175i \(-0.174133\pi\)
0.854059 + 0.520175i \(0.174133\pi\)
\(984\) 8.85253e15 0.305911
\(985\) 2.44413e12 8.39895e−5 0
\(986\) 3.07907e16 1.05220
\(987\) −4.93509e15 −0.167707
\(988\) 5.14542e15 0.173883
\(989\) −5.94266e15 −0.199711
\(990\) 7.94767e15 0.265611
\(991\) 1.74861e16 0.581149 0.290575 0.956852i \(-0.406154\pi\)
0.290575 + 0.956852i \(0.406154\pi\)
\(992\) −4.92289e16 −1.62707
\(993\) 3.43752e15 0.112986
\(994\) −1.34454e16 −0.439490
\(995\) −4.17890e15 −0.135842
\(996\) −2.10073e16 −0.679117
\(997\) −3.59421e16 −1.15553 −0.577763 0.816205i \(-0.696074\pi\)
−0.577763 + 0.816205i \(0.696074\pi\)
\(998\) −1.04486e16 −0.334072
\(999\) 4.04487e14 0.0128616
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.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.12.a.c.1.14 16 1.1 even 1 trivial