Properties

Label 6.26.a.a.1.1
Level $6$
Weight $26$
Character 6.1
Self dual yes
Analytic conductor $23.760$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.7598067971\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6.1

$q$-expansion

\(f(q)\) \(=\) \(q-4096.00 q^{2} -531441. q^{3} +1.67772e7 q^{4} -2.92755e8 q^{5} +2.17678e9 q^{6} +3.58064e9 q^{7} -6.87195e10 q^{8} +2.82430e11 q^{9} +O(q^{10})\) \(q-4096.00 q^{2} -531441. q^{3} +1.67772e7 q^{4} -2.92755e8 q^{5} +2.17678e9 q^{6} +3.58064e9 q^{7} -6.87195e10 q^{8} +2.82430e11 q^{9} +1.19912e12 q^{10} +1.51116e13 q^{11} -8.91610e12 q^{12} +1.22107e12 q^{13} -1.46663e13 q^{14} +1.55582e14 q^{15} +2.81475e14 q^{16} +2.51825e15 q^{17} -1.15683e15 q^{18} -7.99269e15 q^{19} -4.91161e15 q^{20} -1.90290e15 q^{21} -6.18970e16 q^{22} -9.96456e16 q^{23} +3.65203e16 q^{24} -2.12318e17 q^{25} -5.00151e15 q^{26} -1.50095e17 q^{27} +6.00732e16 q^{28} -2.08067e18 q^{29} -6.37264e17 q^{30} -4.93767e18 q^{31} -1.15292e18 q^{32} -8.03091e18 q^{33} -1.03148e19 q^{34} -1.04825e18 q^{35} +4.73838e18 q^{36} +1.98292e19 q^{37} +3.27381e19 q^{38} -6.48928e17 q^{39} +2.01180e19 q^{40} +2.24696e20 q^{41} +7.79428e18 q^{42} -7.22210e19 q^{43} +2.53530e20 q^{44} -8.26826e19 q^{45} +4.08149e20 q^{46} +1.89872e20 q^{47} -1.49587e20 q^{48} -1.32825e21 q^{49} +8.69654e20 q^{50} -1.33830e21 q^{51} +2.04862e19 q^{52} -2.64568e21 q^{53} +6.14788e20 q^{54} -4.42399e21 q^{55} -2.46060e20 q^{56} +4.24764e21 q^{57} +8.52244e21 q^{58} -1.64546e22 q^{59} +2.61023e21 q^{60} -3.55470e22 q^{61} +2.02247e22 q^{62} +1.01128e21 q^{63} +4.72237e21 q^{64} -3.57475e20 q^{65} +3.28946e22 q^{66} +1.06704e23 q^{67} +4.22492e22 q^{68} +5.29558e22 q^{69} +4.29364e21 q^{70} +7.36720e22 q^{71} -1.94084e22 q^{72} -2.62403e23 q^{73} -8.12202e22 q^{74} +1.12834e23 q^{75} -1.34095e23 q^{76} +5.41092e22 q^{77} +2.65801e21 q^{78} -1.00264e24 q^{79} -8.24032e22 q^{80} +7.97664e22 q^{81} -9.20355e23 q^{82} +1.55859e24 q^{83} -3.19254e22 q^{84} -7.37230e23 q^{85} +2.95817e23 q^{86} +1.10575e24 q^{87} -1.03846e24 q^{88} +2.18167e24 q^{89} +3.38668e23 q^{90} +4.37222e21 q^{91} -1.67178e24 q^{92} +2.62408e24 q^{93} -7.77717e23 q^{94} +2.33990e24 q^{95} +6.12710e23 q^{96} -4.40165e23 q^{97} +5.44050e24 q^{98} +4.26795e24 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4096.00 −0.707107
\(3\) −531441. −0.577350
\(4\) 1.67772e7 0.500000
\(5\) −2.92755e8 −0.536264 −0.268132 0.963382i \(-0.586406\pi\)
−0.268132 + 0.963382i \(0.586406\pi\)
\(6\) 2.17678e9 0.408248
\(7\) 3.58064e9 0.0977768 0.0488884 0.998804i \(-0.484432\pi\)
0.0488884 + 0.998804i \(0.484432\pi\)
\(8\) −6.87195e10 −0.353553
\(9\) 2.82430e11 0.333333
\(10\) 1.19912e12 0.379196
\(11\) 1.51116e13 1.45178 0.725891 0.687810i \(-0.241428\pi\)
0.725891 + 0.687810i \(0.241428\pi\)
\(12\) −8.91610e12 −0.288675
\(13\) 1.22107e12 0.0145361 0.00726807 0.999974i \(-0.497686\pi\)
0.00726807 + 0.999974i \(0.497686\pi\)
\(14\) −1.46663e13 −0.0691386
\(15\) 1.55582e14 0.309612
\(16\) 2.81475e14 0.250000
\(17\) 2.51825e15 1.04830 0.524152 0.851625i \(-0.324382\pi\)
0.524152 + 0.851625i \(0.324382\pi\)
\(18\) −1.15683e15 −0.235702
\(19\) −7.99269e15 −0.828463 −0.414232 0.910172i \(-0.635950\pi\)
−0.414232 + 0.910172i \(0.635950\pi\)
\(20\) −4.91161e15 −0.268132
\(21\) −1.90290e15 −0.0564514
\(22\) −6.18970e16 −1.02656
\(23\) −9.96456e16 −0.948114 −0.474057 0.880494i \(-0.657211\pi\)
−0.474057 + 0.880494i \(0.657211\pi\)
\(24\) 3.65203e16 0.204124
\(25\) −2.12318e17 −0.712420
\(26\) −5.00151e15 −0.0102786
\(27\) −1.50095e17 −0.192450
\(28\) 6.00732e16 0.0488884
\(29\) −2.08067e18 −1.09202 −0.546008 0.837780i \(-0.683853\pi\)
−0.546008 + 0.837780i \(0.683853\pi\)
\(30\) −6.37264e17 −0.218929
\(31\) −4.93767e18 −1.12590 −0.562952 0.826490i \(-0.690334\pi\)
−0.562952 + 0.826490i \(0.690334\pi\)
\(32\) −1.15292e18 −0.176777
\(33\) −8.03091e18 −0.838186
\(34\) −1.03148e19 −0.741263
\(35\) −1.04825e18 −0.0524342
\(36\) 4.73838e18 0.166667
\(37\) 1.98292e19 0.495202 0.247601 0.968862i \(-0.420358\pi\)
0.247601 + 0.968862i \(0.420358\pi\)
\(38\) 3.27381e19 0.585812
\(39\) −6.48928e17 −0.00839245
\(40\) 2.01180e19 0.189598
\(41\) 2.24696e20 1.55524 0.777620 0.628735i \(-0.216427\pi\)
0.777620 + 0.628735i \(0.216427\pi\)
\(42\) 7.79428e18 0.0399172
\(43\) −7.22210e19 −0.275618 −0.137809 0.990459i \(-0.544006\pi\)
−0.137809 + 0.990459i \(0.544006\pi\)
\(44\) 2.53530e20 0.725891
\(45\) −8.26826e19 −0.178755
\(46\) 4.08149e20 0.670418
\(47\) 1.89872e20 0.238363 0.119181 0.992872i \(-0.461973\pi\)
0.119181 + 0.992872i \(0.461973\pi\)
\(48\) −1.49587e20 −0.144338
\(49\) −1.32825e21 −0.990440
\(50\) 8.69654e20 0.503757
\(51\) −1.33830e21 −0.605239
\(52\) 2.04862e19 0.00726807
\(53\) −2.64568e21 −0.739756 −0.369878 0.929080i \(-0.620600\pi\)
−0.369878 + 0.929080i \(0.620600\pi\)
\(54\) 6.14788e20 0.136083
\(55\) −4.42399e21 −0.778539
\(56\) −2.46060e20 −0.0345693
\(57\) 4.24764e21 0.478313
\(58\) 8.52244e21 0.772172
\(59\) −1.64546e22 −1.20403 −0.602015 0.798485i \(-0.705635\pi\)
−0.602015 + 0.798485i \(0.705635\pi\)
\(60\) 2.61023e21 0.154806
\(61\) −3.55470e22 −1.71467 −0.857333 0.514762i \(-0.827880\pi\)
−0.857333 + 0.514762i \(0.827880\pi\)
\(62\) 2.02247e22 0.796134
\(63\) 1.01128e21 0.0325923
\(64\) 4.72237e21 0.125000
\(65\) −3.57475e20 −0.00779522
\(66\) 3.28946e22 0.592687
\(67\) 1.06704e23 1.59311 0.796553 0.604569i \(-0.206655\pi\)
0.796553 + 0.604569i \(0.206655\pi\)
\(68\) 4.22492e22 0.524152
\(69\) 5.29558e22 0.547394
\(70\) 4.29364e21 0.0370766
\(71\) 7.36720e22 0.532811 0.266405 0.963861i \(-0.414164\pi\)
0.266405 + 0.963861i \(0.414164\pi\)
\(72\) −1.94084e22 −0.117851
\(73\) −2.62403e23 −1.34101 −0.670506 0.741904i \(-0.733923\pi\)
−0.670506 + 0.741904i \(0.733923\pi\)
\(74\) −8.12202e22 −0.350161
\(75\) 1.12834e23 0.411316
\(76\) −1.34095e23 −0.414232
\(77\) 5.41092e22 0.141950
\(78\) 2.65801e21 0.00593436
\(79\) −1.00264e24 −1.90901 −0.954503 0.298201i \(-0.903613\pi\)
−0.954503 + 0.298201i \(0.903613\pi\)
\(80\) −8.24032e22 −0.134066
\(81\) 7.97664e22 0.111111
\(82\) −9.20355e23 −1.09972
\(83\) 1.55859e24 1.60050 0.800249 0.599667i \(-0.204700\pi\)
0.800249 + 0.599667i \(0.204700\pi\)
\(84\) −3.19254e22 −0.0282257
\(85\) −7.37230e23 −0.562169
\(86\) 2.95817e23 0.194891
\(87\) 1.10575e24 0.630475
\(88\) −1.03846e24 −0.513282
\(89\) 2.18167e24 0.936298 0.468149 0.883650i \(-0.344921\pi\)
0.468149 + 0.883650i \(0.344921\pi\)
\(90\) 3.38668e23 0.126399
\(91\) 4.37222e21 0.00142130
\(92\) −1.67178e24 −0.474057
\(93\) 2.62408e24 0.650041
\(94\) −7.77717e23 −0.168548
\(95\) 2.33990e24 0.444275
\(96\) 6.12710e23 0.102062
\(97\) −4.40165e23 −0.0644123 −0.0322062 0.999481i \(-0.510253\pi\)
−0.0322062 + 0.999481i \(0.510253\pi\)
\(98\) 5.44050e24 0.700347
\(99\) 4.26795e24 0.483927
\(100\) −3.56210e24 −0.356210
\(101\) −5.02582e24 −0.443802 −0.221901 0.975069i \(-0.571226\pi\)
−0.221901 + 0.975069i \(0.571226\pi\)
\(102\) 5.48168e24 0.427969
\(103\) −1.52311e25 −1.05261 −0.526304 0.850296i \(-0.676423\pi\)
−0.526304 + 0.850296i \(0.676423\pi\)
\(104\) −8.39114e22 −0.00513930
\(105\) 5.57084e23 0.0302729
\(106\) 1.08367e25 0.523086
\(107\) −2.47232e25 −1.06123 −0.530613 0.847614i \(-0.678038\pi\)
−0.530613 + 0.847614i \(0.678038\pi\)
\(108\) −2.51817e24 −0.0962250
\(109\) −2.85771e25 −0.973166 −0.486583 0.873634i \(-0.661757\pi\)
−0.486583 + 0.873634i \(0.661757\pi\)
\(110\) 1.81206e25 0.550510
\(111\) −1.05380e25 −0.285905
\(112\) 1.00786e24 0.0244442
\(113\) −1.73240e25 −0.375982 −0.187991 0.982171i \(-0.560198\pi\)
−0.187991 + 0.982171i \(0.560198\pi\)
\(114\) −1.73984e25 −0.338219
\(115\) 2.91717e25 0.508440
\(116\) −3.49079e25 −0.546008
\(117\) 3.44867e23 0.00484538
\(118\) 6.73981e25 0.851378
\(119\) 9.01696e24 0.102500
\(120\) −1.06915e25 −0.109465
\(121\) 1.20013e26 1.10767
\(122\) 1.45600e26 1.21245
\(123\) −1.19413e26 −0.897918
\(124\) −8.28404e25 −0.562952
\(125\) 1.49405e26 0.918310
\(126\) −4.14220e24 −0.0230462
\(127\) 3.23349e26 1.62976 0.814882 0.579627i \(-0.196802\pi\)
0.814882 + 0.579627i \(0.196802\pi\)
\(128\) −1.93428e25 −0.0883883
\(129\) 3.83812e25 0.159128
\(130\) 1.46422e24 0.00551205
\(131\) −3.63273e26 −1.24263 −0.621315 0.783561i \(-0.713401\pi\)
−0.621315 + 0.783561i \(0.713401\pi\)
\(132\) −1.34736e26 −0.419093
\(133\) −2.86190e25 −0.0810044
\(134\) −4.37059e26 −1.12650
\(135\) 4.39409e25 0.103204
\(136\) −1.73053e26 −0.370632
\(137\) −6.56742e26 −1.28348 −0.641738 0.766924i \(-0.721786\pi\)
−0.641738 + 0.766924i \(0.721786\pi\)
\(138\) −2.16907e26 −0.387066
\(139\) 6.72509e26 1.09651 0.548256 0.836311i \(-0.315292\pi\)
0.548256 + 0.836311i \(0.315292\pi\)
\(140\) −1.75867e25 −0.0262171
\(141\) −1.00906e26 −0.137619
\(142\) −3.01761e26 −0.376754
\(143\) 1.84523e25 0.0211033
\(144\) 7.94968e25 0.0833333
\(145\) 6.09127e26 0.585609
\(146\) 1.07480e27 0.948239
\(147\) 7.05885e26 0.571831
\(148\) 3.32678e26 0.247601
\(149\) −7.98627e26 −0.546406 −0.273203 0.961956i \(-0.588083\pi\)
−0.273203 + 0.961956i \(0.588083\pi\)
\(150\) −4.62170e26 −0.290844
\(151\) −1.03633e27 −0.600187 −0.300093 0.953910i \(-0.597018\pi\)
−0.300093 + 0.953910i \(0.597018\pi\)
\(152\) 5.49254e26 0.292906
\(153\) 7.11228e26 0.349435
\(154\) −2.21631e26 −0.100374
\(155\) 1.44553e27 0.603782
\(156\) −1.08872e25 −0.00419622
\(157\) −3.36346e27 −1.19685 −0.598427 0.801178i \(-0.704207\pi\)
−0.598427 + 0.801178i \(0.704207\pi\)
\(158\) 4.10682e27 1.34987
\(159\) 1.40602e27 0.427098
\(160\) 3.37523e26 0.0947991
\(161\) −3.56796e26 −0.0927035
\(162\) −3.26723e26 −0.0785674
\(163\) 8.00443e26 0.178232 0.0891159 0.996021i \(-0.471596\pi\)
0.0891159 + 0.996021i \(0.471596\pi\)
\(164\) 3.76977e27 0.777620
\(165\) 2.35109e27 0.449489
\(166\) −6.38398e27 −1.13172
\(167\) −5.17982e27 −0.851842 −0.425921 0.904760i \(-0.640050\pi\)
−0.425921 + 0.904760i \(0.640050\pi\)
\(168\) 1.30766e26 0.0199586
\(169\) −7.05492e27 −0.999789
\(170\) 3.01969e27 0.397513
\(171\) −2.25737e27 −0.276154
\(172\) −1.21167e27 −0.137809
\(173\) −3.40880e27 −0.360599 −0.180300 0.983612i \(-0.557707\pi\)
−0.180300 + 0.983612i \(0.557707\pi\)
\(174\) −4.52917e27 −0.445813
\(175\) −7.60235e26 −0.0696582
\(176\) 4.25353e27 0.362945
\(177\) 8.74465e27 0.695147
\(178\) −8.93612e27 −0.662063
\(179\) −1.23610e28 −0.853870 −0.426935 0.904282i \(-0.640407\pi\)
−0.426935 + 0.904282i \(0.640407\pi\)
\(180\) −1.38718e27 −0.0893774
\(181\) 3.38516e27 0.203516 0.101758 0.994809i \(-0.467553\pi\)
0.101758 + 0.994809i \(0.467553\pi\)
\(182\) −1.79086e25 −0.00100501
\(183\) 1.88911e28 0.989963
\(184\) 6.84760e27 0.335209
\(185\) −5.80508e27 −0.265559
\(186\) −1.07482e28 −0.459648
\(187\) 3.80547e28 1.52191
\(188\) 3.18553e27 0.119181
\(189\) −5.37435e26 −0.0188171
\(190\) −9.58423e27 −0.314150
\(191\) 3.81140e28 1.16995 0.584975 0.811052i \(-0.301104\pi\)
0.584975 + 0.811052i \(0.301104\pi\)
\(192\) −2.50966e27 −0.0721688
\(193\) 6.38842e28 1.72158 0.860790 0.508961i \(-0.169970\pi\)
0.860790 + 0.508961i \(0.169970\pi\)
\(194\) 1.80292e27 0.0455464
\(195\) 1.89977e26 0.00450057
\(196\) −2.22843e28 −0.495220
\(197\) −8.73680e28 −1.82190 −0.910950 0.412517i \(-0.864650\pi\)
−0.910950 + 0.412517i \(0.864650\pi\)
\(198\) −1.74815e28 −0.342188
\(199\) 2.76766e28 0.508686 0.254343 0.967114i \(-0.418141\pi\)
0.254343 + 0.967114i \(0.418141\pi\)
\(200\) 1.45904e28 0.251879
\(201\) −5.67067e28 −0.919780
\(202\) 2.05857e28 0.313816
\(203\) −7.45015e27 −0.106774
\(204\) −2.24530e28 −0.302619
\(205\) −6.57809e28 −0.834020
\(206\) 6.23867e28 0.744307
\(207\) −2.81429e28 −0.316038
\(208\) 3.43701e26 0.00363404
\(209\) −1.20782e29 −1.20275
\(210\) −2.28181e27 −0.0214062
\(211\) −2.08479e28 −0.184303 −0.0921513 0.995745i \(-0.529374\pi\)
−0.0921513 + 0.995745i \(0.529374\pi\)
\(212\) −4.43871e28 −0.369878
\(213\) −3.91523e28 −0.307619
\(214\) 1.01266e29 0.750400
\(215\) 2.11431e28 0.147804
\(216\) 1.03144e28 0.0680414
\(217\) −1.76800e28 −0.110087
\(218\) 1.17052e29 0.688132
\(219\) 1.39452e29 0.774234
\(220\) −7.42222e28 −0.389269
\(221\) 3.07496e27 0.0152383
\(222\) 4.31638e28 0.202165
\(223\) 2.90506e29 1.28630 0.643152 0.765739i \(-0.277626\pi\)
0.643152 + 0.765739i \(0.277626\pi\)
\(224\) −4.12820e27 −0.0172847
\(225\) −5.99648e28 −0.237473
\(226\) 7.09591e28 0.265860
\(227\) 3.23765e29 1.14791 0.573954 0.818887i \(-0.305409\pi\)
0.573954 + 0.818887i \(0.305409\pi\)
\(228\) 7.12637e28 0.239157
\(229\) −5.11099e29 −1.62391 −0.811954 0.583721i \(-0.801596\pi\)
−0.811954 + 0.583721i \(0.801596\pi\)
\(230\) −1.19487e29 −0.359521
\(231\) −2.87558e28 −0.0819551
\(232\) 1.42983e29 0.386086
\(233\) −1.20130e29 −0.307399 −0.153700 0.988118i \(-0.549119\pi\)
−0.153700 + 0.988118i \(0.549119\pi\)
\(234\) −1.41257e27 −0.00342620
\(235\) −5.55861e28 −0.127826
\(236\) −2.76063e29 −0.602015
\(237\) 5.32845e29 1.10217
\(238\) −3.69335e28 −0.0724783
\(239\) 3.34916e29 0.623681 0.311840 0.950135i \(-0.399055\pi\)
0.311840 + 0.950135i \(0.399055\pi\)
\(240\) 4.37924e28 0.0774031
\(241\) −3.53238e29 −0.592727 −0.296363 0.955075i \(-0.595774\pi\)
−0.296363 + 0.955075i \(0.595774\pi\)
\(242\) −4.91572e29 −0.783240
\(243\) −4.23912e28 −0.0641500
\(244\) −5.96379e29 −0.857333
\(245\) 3.88851e29 0.531138
\(246\) 4.89114e29 0.634924
\(247\) −9.75965e27 −0.0120427
\(248\) 3.39314e29 0.398067
\(249\) −8.28298e29 −0.924048
\(250\) −6.11962e29 −0.649343
\(251\) 1.07603e30 1.08619 0.543093 0.839673i \(-0.317253\pi\)
0.543093 + 0.839673i \(0.317253\pi\)
\(252\) 1.69665e28 0.0162961
\(253\) −1.50580e30 −1.37645
\(254\) −1.32444e30 −1.15242
\(255\) 3.91794e29 0.324568
\(256\) 7.92282e28 0.0625000
\(257\) 6.56258e27 0.00493073 0.00246536 0.999997i \(-0.499215\pi\)
0.00246536 + 0.999997i \(0.499215\pi\)
\(258\) −1.57209e29 −0.112521
\(259\) 7.10011e28 0.0484193
\(260\) −5.99743e27 −0.00389761
\(261\) −5.87643e29 −0.364005
\(262\) 1.48797e30 0.878673
\(263\) 2.03818e30 1.14761 0.573807 0.818990i \(-0.305466\pi\)
0.573807 + 0.818990i \(0.305466\pi\)
\(264\) 5.51880e29 0.296344
\(265\) 7.74534e29 0.396705
\(266\) 1.17223e29 0.0572788
\(267\) −1.15943e30 −0.540572
\(268\) 1.79019e30 0.796553
\(269\) 1.30216e30 0.553046 0.276523 0.961007i \(-0.410818\pi\)
0.276523 + 0.961007i \(0.410818\pi\)
\(270\) −1.79982e29 −0.0729764
\(271\) 2.73384e30 1.05842 0.529209 0.848492i \(-0.322489\pi\)
0.529209 + 0.848492i \(0.322489\pi\)
\(272\) 7.08825e29 0.262076
\(273\) −2.32358e27 −0.000820586 0
\(274\) 2.69002e30 0.907554
\(275\) −3.20846e30 −1.03428
\(276\) 8.88451e29 0.273697
\(277\) −4.83846e29 −0.142466 −0.0712329 0.997460i \(-0.522693\pi\)
−0.0712329 + 0.997460i \(0.522693\pi\)
\(278\) −2.75460e30 −0.775351
\(279\) −1.39454e30 −0.375301
\(280\) 7.20353e28 0.0185383
\(281\) −7.57818e30 −1.86525 −0.932623 0.360853i \(-0.882486\pi\)
−0.932623 + 0.360853i \(0.882486\pi\)
\(282\) 4.13311e29 0.0973113
\(283\) −1.30443e30 −0.293827 −0.146914 0.989149i \(-0.546934\pi\)
−0.146914 + 0.989149i \(0.546934\pi\)
\(284\) 1.23601e30 0.266405
\(285\) −1.24352e30 −0.256502
\(286\) −7.55807e28 −0.0149223
\(287\) 8.04557e29 0.152066
\(288\) −3.25619e29 −0.0589256
\(289\) 5.70960e29 0.0989424
\(290\) −2.49498e30 −0.414088
\(291\) 2.33922e29 0.0371885
\(292\) −4.40239e30 −0.670506
\(293\) −4.04021e29 −0.0589601 −0.0294800 0.999565i \(-0.509385\pi\)
−0.0294800 + 0.999565i \(0.509385\pi\)
\(294\) −2.89131e30 −0.404345
\(295\) 4.81717e30 0.645679
\(296\) −1.36265e30 −0.175080
\(297\) −2.26817e30 −0.279395
\(298\) 3.27118e30 0.386367
\(299\) −1.21674e29 −0.0137819
\(300\) 1.89305e30 0.205658
\(301\) −2.58598e29 −0.0269491
\(302\) 4.24481e30 0.424396
\(303\) 2.67092e30 0.256229
\(304\) −2.24974e30 −0.207116
\(305\) 1.04065e31 0.919514
\(306\) −2.91319e30 −0.247088
\(307\) 6.95636e30 0.566438 0.283219 0.959055i \(-0.408598\pi\)
0.283219 + 0.959055i \(0.408598\pi\)
\(308\) 9.07801e29 0.0709752
\(309\) 8.09445e30 0.607724
\(310\) −5.92088e30 −0.426938
\(311\) 2.74913e30 0.190410 0.0952051 0.995458i \(-0.469649\pi\)
0.0952051 + 0.995458i \(0.469649\pi\)
\(312\) 4.45940e28 0.00296718
\(313\) 9.39383e30 0.600535 0.300267 0.953855i \(-0.402924\pi\)
0.300267 + 0.953855i \(0.402924\pi\)
\(314\) 1.37767e31 0.846303
\(315\) −2.96057e29 −0.0174781
\(316\) −1.68215e31 −0.954503
\(317\) 1.86543e31 1.01751 0.508755 0.860912i \(-0.330106\pi\)
0.508755 + 0.860912i \(0.330106\pi\)
\(318\) −5.75906e30 −0.302004
\(319\) −3.14422e31 −1.58537
\(320\) −1.38250e30 −0.0670331
\(321\) 1.31389e31 0.612699
\(322\) 1.46143e30 0.0655513
\(323\) −2.01276e31 −0.868482
\(324\) 1.33826e30 0.0555556
\(325\) −2.59255e29 −0.0103558
\(326\) −3.27861e30 −0.126029
\(327\) 1.51870e31 0.561857
\(328\) −1.54410e31 −0.549860
\(329\) 6.79866e29 0.0233064
\(330\) −9.63006e30 −0.317837
\(331\) −4.71638e31 −1.49885 −0.749426 0.662088i \(-0.769670\pi\)
−0.749426 + 0.662088i \(0.769670\pi\)
\(332\) 2.61488e31 0.800249
\(333\) 5.60034e30 0.165067
\(334\) 2.12166e31 0.602343
\(335\) −3.12380e31 −0.854326
\(336\) −5.35619e29 −0.0141129
\(337\) 1.85792e31 0.471686 0.235843 0.971791i \(-0.424215\pi\)
0.235843 + 0.971791i \(0.424215\pi\)
\(338\) 2.88969e31 0.706957
\(339\) 9.20668e30 0.217074
\(340\) −1.23687e31 −0.281084
\(341\) −7.46160e31 −1.63456
\(342\) 9.24620e30 0.195271
\(343\) −9.55787e30 −0.194619
\(344\) 4.96299e30 0.0974457
\(345\) −1.55031e31 −0.293548
\(346\) 1.39624e31 0.254982
\(347\) −7.27635e31 −1.28173 −0.640863 0.767655i \(-0.721423\pi\)
−0.640863 + 0.767655i \(0.721423\pi\)
\(348\) 1.85515e31 0.315238
\(349\) 1.19713e32 1.96256 0.981282 0.192574i \(-0.0616835\pi\)
0.981282 + 0.192574i \(0.0616835\pi\)
\(350\) 3.11392e30 0.0492558
\(351\) −1.83276e29 −0.00279748
\(352\) −1.74225e31 −0.256641
\(353\) 6.99722e31 0.994813 0.497407 0.867517i \(-0.334286\pi\)
0.497407 + 0.867517i \(0.334286\pi\)
\(354\) −3.58181e31 −0.491543
\(355\) −2.15678e31 −0.285728
\(356\) 3.66024e31 0.468149
\(357\) −4.79198e30 −0.0591783
\(358\) 5.06308e31 0.603777
\(359\) 6.06364e31 0.698317 0.349159 0.937064i \(-0.386467\pi\)
0.349159 + 0.937064i \(0.386467\pi\)
\(360\) 5.68191e30 0.0631994
\(361\) −2.91933e31 −0.313649
\(362\) −1.38656e31 −0.143907
\(363\) −6.37796e31 −0.639513
\(364\) 7.33537e28 0.000710649 0
\(365\) 7.68197e31 0.719137
\(366\) −7.73780e31 −0.700009
\(367\) 2.22539e32 1.94572 0.972860 0.231394i \(-0.0743287\pi\)
0.972860 + 0.231394i \(0.0743287\pi\)
\(368\) −2.80478e31 −0.237028
\(369\) 6.34608e31 0.518413
\(370\) 2.37776e31 0.187779
\(371\) −9.47322e30 −0.0723309
\(372\) 4.40248e31 0.325020
\(373\) −1.15379e32 −0.823695 −0.411848 0.911253i \(-0.635116\pi\)
−0.411848 + 0.911253i \(0.635116\pi\)
\(374\) −1.55872e32 −1.07615
\(375\) −7.93998e31 −0.530187
\(376\) −1.30479e31 −0.0842740
\(377\) −2.54065e30 −0.0158737
\(378\) 2.20134e30 0.0133057
\(379\) 5.47213e31 0.320012 0.160006 0.987116i \(-0.448849\pi\)
0.160006 + 0.987116i \(0.448849\pi\)
\(380\) 3.92570e31 0.222138
\(381\) −1.71841e32 −0.940945
\(382\) −1.56115e32 −0.827279
\(383\) −1.98130e32 −1.01617 −0.508085 0.861307i \(-0.669646\pi\)
−0.508085 + 0.861307i \(0.669646\pi\)
\(384\) 1.02796e31 0.0510310
\(385\) −1.58407e31 −0.0761230
\(386\) −2.61670e32 −1.21734
\(387\) −2.03973e31 −0.0918727
\(388\) −7.38475e30 −0.0322062
\(389\) 4.19098e31 0.176989 0.0884945 0.996077i \(-0.471794\pi\)
0.0884945 + 0.996077i \(0.471794\pi\)
\(390\) −7.78145e29 −0.00318238
\(391\) −2.50933e32 −0.993912
\(392\) 9.12765e31 0.350173
\(393\) 1.93058e32 0.717433
\(394\) 3.57859e32 1.28828
\(395\) 2.93528e32 1.02373
\(396\) 7.16044e31 0.241964
\(397\) 2.64500e32 0.866053 0.433027 0.901381i \(-0.357446\pi\)
0.433027 + 0.901381i \(0.357446\pi\)
\(398\) −1.13363e32 −0.359695
\(399\) 1.52093e31 0.0467679
\(400\) −5.97622e31 −0.178105
\(401\) 2.33767e32 0.675273 0.337636 0.941277i \(-0.390373\pi\)
0.337636 + 0.941277i \(0.390373\pi\)
\(402\) 2.32271e32 0.650383
\(403\) −6.02925e30 −0.0163663
\(404\) −8.43192e31 −0.221901
\(405\) −2.33520e31 −0.0595849
\(406\) 3.05158e31 0.0755004
\(407\) 2.99650e32 0.718925
\(408\) 9.19674e31 0.213984
\(409\) −5.48497e32 −1.23775 −0.618875 0.785490i \(-0.712411\pi\)
−0.618875 + 0.785490i \(0.712411\pi\)
\(410\) 2.69438e32 0.589741
\(411\) 3.49020e32 0.741015
\(412\) −2.55536e32 −0.526304
\(413\) −5.89181e31 −0.117726
\(414\) 1.15273e32 0.223473
\(415\) −4.56284e32 −0.858291
\(416\) −1.40780e30 −0.00256965
\(417\) −3.57399e32 −0.633071
\(418\) 4.94724e32 0.850471
\(419\) 3.41488e31 0.0569771 0.0284885 0.999594i \(-0.490931\pi\)
0.0284885 + 0.999594i \(0.490931\pi\)
\(420\) 9.34631e30 0.0151365
\(421\) −5.71127e32 −0.897856 −0.448928 0.893568i \(-0.648194\pi\)
−0.448928 + 0.893568i \(0.648194\pi\)
\(422\) 8.53930e31 0.130322
\(423\) 5.36256e31 0.0794543
\(424\) 1.81809e32 0.261543
\(425\) −5.34670e32 −0.746834
\(426\) 1.60368e32 0.217519
\(427\) −1.27281e32 −0.167654
\(428\) −4.14787e32 −0.530613
\(429\) −9.80632e30 −0.0121840
\(430\) −8.66019e31 −0.104513
\(431\) 5.26390e32 0.617081 0.308540 0.951211i \(-0.400160\pi\)
0.308540 + 0.951211i \(0.400160\pi\)
\(432\) −4.22479e31 −0.0481125
\(433\) −7.00633e32 −0.775160 −0.387580 0.921836i \(-0.626689\pi\)
−0.387580 + 0.921836i \(0.626689\pi\)
\(434\) 7.24175e31 0.0778434
\(435\) −3.23715e32 −0.338102
\(436\) −4.79444e32 −0.486583
\(437\) 7.96437e32 0.785477
\(438\) −5.71194e32 −0.547466
\(439\) 5.29542e32 0.493280 0.246640 0.969107i \(-0.420673\pi\)
0.246640 + 0.969107i \(0.420673\pi\)
\(440\) 3.04014e32 0.275255
\(441\) −3.75136e32 −0.330147
\(442\) −1.25951e31 −0.0107751
\(443\) 4.49792e32 0.374080 0.187040 0.982352i \(-0.440111\pi\)
0.187040 + 0.982352i \(0.440111\pi\)
\(444\) −1.76799e32 −0.142953
\(445\) −6.38695e32 −0.502103
\(446\) −1.18991e33 −0.909554
\(447\) 4.24423e32 0.315468
\(448\) 1.69091e31 0.0122221
\(449\) −9.88741e32 −0.695030 −0.347515 0.937674i \(-0.612974\pi\)
−0.347515 + 0.937674i \(0.612974\pi\)
\(450\) 2.45616e32 0.167919
\(451\) 3.39551e33 2.25787
\(452\) −2.90648e32 −0.187991
\(453\) 5.50748e32 0.346518
\(454\) −1.32614e33 −0.811694
\(455\) −1.27999e30 −0.000762191 0
\(456\) −2.91896e32 −0.169109
\(457\) 2.93791e33 1.65610 0.828050 0.560654i \(-0.189450\pi\)
0.828050 + 0.560654i \(0.189450\pi\)
\(458\) 2.09346e33 1.14828
\(459\) −3.77976e32 −0.201746
\(460\) 4.89421e32 0.254220
\(461\) −2.14053e33 −1.08208 −0.541039 0.840997i \(-0.681969\pi\)
−0.541039 + 0.840997i \(0.681969\pi\)
\(462\) 1.17784e32 0.0579510
\(463\) 8.71982e32 0.417585 0.208793 0.977960i \(-0.433047\pi\)
0.208793 + 0.977960i \(0.433047\pi\)
\(464\) −5.85657e32 −0.273004
\(465\) −7.68213e32 −0.348594
\(466\) 4.92053e32 0.217364
\(467\) −6.01153e32 −0.258538 −0.129269 0.991610i \(-0.541263\pi\)
−0.129269 + 0.991610i \(0.541263\pi\)
\(468\) 5.78590e30 0.00242269
\(469\) 3.82068e32 0.155769
\(470\) 2.27681e32 0.0903863
\(471\) 1.78748e33 0.691003
\(472\) 1.13075e33 0.425689
\(473\) −1.09137e33 −0.400137
\(474\) −2.18253e33 −0.779348
\(475\) 1.69699e33 0.590214
\(476\) 1.51279e32 0.0512499
\(477\) −7.47217e32 −0.246585
\(478\) −1.37182e33 −0.441009
\(479\) −3.01870e33 −0.945423 −0.472711 0.881217i \(-0.656725\pi\)
−0.472711 + 0.881217i \(0.656725\pi\)
\(480\) −1.79374e32 −0.0547323
\(481\) 2.42128e31 0.00719833
\(482\) 1.44686e33 0.419121
\(483\) 1.89616e32 0.0535224
\(484\) 2.01348e33 0.553834
\(485\) 1.28861e32 0.0345421
\(486\) 1.73634e32 0.0453609
\(487\) −5.24137e33 −1.33454 −0.667272 0.744814i \(-0.732538\pi\)
−0.667272 + 0.744814i \(0.732538\pi\)
\(488\) 2.44277e33 0.606226
\(489\) −4.25388e32 −0.102902
\(490\) −1.59273e33 −0.375571
\(491\) −3.08172e33 −0.708393 −0.354197 0.935171i \(-0.615246\pi\)
−0.354197 + 0.935171i \(0.615246\pi\)
\(492\) −2.00341e33 −0.448959
\(493\) −5.23966e33 −1.14476
\(494\) 3.99755e31 0.00851545
\(495\) −1.24946e33 −0.259513
\(496\) −1.38983e33 −0.281476
\(497\) 2.63793e32 0.0520965
\(498\) 3.39271e33 0.653401
\(499\) −1.64910e33 −0.309734 −0.154867 0.987935i \(-0.549495\pi\)
−0.154867 + 0.987935i \(0.549495\pi\)
\(500\) 2.50660e33 0.459155
\(501\) 2.75277e33 0.491811
\(502\) −4.40743e33 −0.768049
\(503\) −2.47162e32 −0.0420128 −0.0210064 0.999779i \(-0.506687\pi\)
−0.0210064 + 0.999779i \(0.506687\pi\)
\(504\) −6.94946e31 −0.0115231
\(505\) 1.47133e33 0.237995
\(506\) 6.16777e33 0.973299
\(507\) 3.74927e33 0.577228
\(508\) 5.42489e33 0.814882
\(509\) 6.26740e33 0.918577 0.459288 0.888287i \(-0.348105\pi\)
0.459288 + 0.888287i \(0.348105\pi\)
\(510\) −1.60479e33 −0.229504
\(511\) −9.39571e32 −0.131120
\(512\) −3.24519e32 −0.0441942
\(513\) 1.19966e33 0.159438
\(514\) −2.68803e31 −0.00348655
\(515\) 4.45899e33 0.564477
\(516\) 6.43930e32 0.0795641
\(517\) 2.86927e33 0.346051
\(518\) −2.90821e32 −0.0342376
\(519\) 1.81158e33 0.208192
\(520\) 2.45655e31 0.00275603
\(521\) 1.64678e34 1.80370 0.901850 0.432049i \(-0.142209\pi\)
0.901850 + 0.432049i \(0.142209\pi\)
\(522\) 2.40699e33 0.257391
\(523\) −3.45969e33 −0.361216 −0.180608 0.983555i \(-0.557806\pi\)
−0.180608 + 0.983555i \(0.557806\pi\)
\(524\) −6.09471e33 −0.621315
\(525\) 4.04020e32 0.0402172
\(526\) −8.34839e33 −0.811486
\(527\) −1.24343e34 −1.18029
\(528\) −2.26050e33 −0.209547
\(529\) −1.11651e33 −0.101081
\(530\) −3.17249e33 −0.280513
\(531\) −4.64727e33 −0.401343
\(532\) −4.80147e32 −0.0405022
\(533\) 2.74370e32 0.0226072
\(534\) 4.74902e33 0.382242
\(535\) 7.23785e33 0.569098
\(536\) −7.33263e33 −0.563248
\(537\) 6.56916e33 0.492982
\(538\) −5.33365e33 −0.391062
\(539\) −2.00719e34 −1.43790
\(540\) 7.37207e32 0.0516021
\(541\) −2.31144e33 −0.158094 −0.0790472 0.996871i \(-0.525188\pi\)
−0.0790472 + 0.996871i \(0.525188\pi\)
\(542\) −1.11978e34 −0.748414
\(543\) −1.79902e33 −0.117500
\(544\) −2.90335e33 −0.185316
\(545\) 8.36608e33 0.521874
\(546\) 9.51738e30 0.000580242 0
\(547\) −4.04733e33 −0.241172 −0.120586 0.992703i \(-0.538477\pi\)
−0.120586 + 0.992703i \(0.538477\pi\)
\(548\) −1.10183e34 −0.641738
\(549\) −1.00395e34 −0.571555
\(550\) 1.31418e34 0.731345
\(551\) 1.66302e34 0.904694
\(552\) −3.63909e33 −0.193533
\(553\) −3.59010e33 −0.186656
\(554\) 1.98183e33 0.100738
\(555\) 3.08506e33 0.153321
\(556\) 1.12828e34 0.548256
\(557\) 2.32181e34 1.10315 0.551577 0.834124i \(-0.314026\pi\)
0.551577 + 0.834124i \(0.314026\pi\)
\(558\) 5.71205e33 0.265378
\(559\) −8.81870e31 −0.00400643
\(560\) −2.95056e32 −0.0131086
\(561\) −2.02238e34 −0.878674
\(562\) 3.10402e34 1.31893
\(563\) −2.70103e34 −1.12247 −0.561235 0.827656i \(-0.689674\pi\)
−0.561235 + 0.827656i \(0.689674\pi\)
\(564\) −1.69292e33 −0.0688094
\(565\) 5.07168e33 0.201626
\(566\) 5.34296e33 0.207767
\(567\) 2.85615e32 0.0108641
\(568\) −5.06270e33 −0.188377
\(569\) −2.32147e33 −0.0845007 −0.0422504 0.999107i \(-0.513453\pi\)
−0.0422504 + 0.999107i \(0.513453\pi\)
\(570\) 5.09345e33 0.181375
\(571\) 2.55837e34 0.891276 0.445638 0.895213i \(-0.352977\pi\)
0.445638 + 0.895213i \(0.352977\pi\)
\(572\) 3.09578e32 0.0105516
\(573\) −2.02553e34 −0.675470
\(574\) −3.29546e33 −0.107527
\(575\) 2.11565e34 0.675455
\(576\) 1.33374e33 0.0416667
\(577\) 1.95952e34 0.599035 0.299517 0.954091i \(-0.403174\pi\)
0.299517 + 0.954091i \(0.403174\pi\)
\(578\) −2.33865e33 −0.0699629
\(579\) −3.39507e34 −0.993954
\(580\) 1.02195e34 0.292805
\(581\) 5.58075e33 0.156492
\(582\) −9.58144e32 −0.0262962
\(583\) −3.99803e34 −1.07396
\(584\) 1.80322e34 0.474119
\(585\) −1.00961e32 −0.00259841
\(586\) 1.65487e33 0.0416911
\(587\) 5.98783e33 0.147670 0.0738352 0.997270i \(-0.476476\pi\)
0.0738352 + 0.997270i \(0.476476\pi\)
\(588\) 1.18428e34 0.285915
\(589\) 3.94653e34 0.932769
\(590\) −1.97311e34 −0.456564
\(591\) 4.64309e34 1.05187
\(592\) 5.58141e33 0.123801
\(593\) 1.57169e34 0.341338 0.170669 0.985328i \(-0.445407\pi\)
0.170669 + 0.985328i \(0.445407\pi\)
\(594\) 9.29041e33 0.197562
\(595\) −2.63976e33 −0.0549670
\(596\) −1.33987e34 −0.273203
\(597\) −1.47085e34 −0.293690
\(598\) 4.98379e32 0.00974529
\(599\) −7.90989e34 −1.51473 −0.757364 0.652992i \(-0.773513\pi\)
−0.757364 + 0.652992i \(0.773513\pi\)
\(600\) −7.75392e33 −0.145422
\(601\) −9.50067e34 −1.74511 −0.872556 0.488514i \(-0.837539\pi\)
−0.872556 + 0.488514i \(0.837539\pi\)
\(602\) 1.05922e33 0.0190559
\(603\) 3.01363e34 0.531035
\(604\) −1.73867e34 −0.300093
\(605\) −3.51343e34 −0.594003
\(606\) −1.09401e34 −0.181181
\(607\) −3.98588e32 −0.00646644 −0.00323322 0.999995i \(-0.501029\pi\)
−0.00323322 + 0.999995i \(0.501029\pi\)
\(608\) 9.21495e33 0.146453
\(609\) 3.95931e33 0.0616458
\(610\) −4.26252e34 −0.650195
\(611\) 2.31848e32 0.00346488
\(612\) 1.19324e34 0.174717
\(613\) −6.82592e34 −0.979276 −0.489638 0.871926i \(-0.662871\pi\)
−0.489638 + 0.871926i \(0.662871\pi\)
\(614\) −2.84932e34 −0.400532
\(615\) 3.49586e34 0.481521
\(616\) −3.71835e33 −0.0501871
\(617\) 1.15084e35 1.52212 0.761062 0.648679i \(-0.224678\pi\)
0.761062 + 0.648679i \(0.224678\pi\)
\(618\) −3.31549e34 −0.429726
\(619\) 4.34788e34 0.552261 0.276131 0.961120i \(-0.410948\pi\)
0.276131 + 0.961120i \(0.410948\pi\)
\(620\) 2.42519e34 0.301891
\(621\) 1.49563e34 0.182465
\(622\) −1.12604e34 −0.134640
\(623\) 7.81178e33 0.0915482
\(624\) −1.82657e32 −0.00209811
\(625\) 1.95367e34 0.219963
\(626\) −3.84771e34 −0.424642
\(627\) 6.41886e34 0.694406
\(628\) −5.64295e34 −0.598427
\(629\) 4.99348e34 0.519123
\(630\) 1.21265e33 0.0123589
\(631\) 1.27302e35 1.27195 0.635974 0.771711i \(-0.280599\pi\)
0.635974 + 0.771711i \(0.280599\pi\)
\(632\) 6.89010e34 0.674936
\(633\) 1.10794e34 0.106407
\(634\) −7.64080e34 −0.719488
\(635\) −9.46619e34 −0.873985
\(636\) 2.35891e34 0.213549
\(637\) −1.62189e33 −0.0143972
\(638\) 1.28787e35 1.12102
\(639\) 2.08072e34 0.177604
\(640\) 5.66270e33 0.0473995
\(641\) −9.45454e33 −0.0776096 −0.0388048 0.999247i \(-0.512355\pi\)
−0.0388048 + 0.999247i \(0.512355\pi\)
\(642\) −5.38171e34 −0.433244
\(643\) 8.31941e34 0.656834 0.328417 0.944533i \(-0.393485\pi\)
0.328417 + 0.944533i \(0.393485\pi\)
\(644\) −5.98604e33 −0.0463517
\(645\) −1.12363e34 −0.0853348
\(646\) 8.24427e34 0.614109
\(647\) 2.34992e35 1.71692 0.858460 0.512880i \(-0.171421\pi\)
0.858460 + 0.512880i \(0.171421\pi\)
\(648\) −5.48151e33 −0.0392837
\(649\) −2.48655e35 −1.74799
\(650\) 1.06191e33 0.00732269
\(651\) 9.39590e33 0.0635589
\(652\) 1.34292e34 0.0891159
\(653\) −1.75276e35 −1.14106 −0.570530 0.821277i \(-0.693262\pi\)
−0.570530 + 0.821277i \(0.693262\pi\)
\(654\) −6.22061e34 −0.397293
\(655\) 1.06350e35 0.666379
\(656\) 6.32463e34 0.388810
\(657\) −7.41103e34 −0.447004
\(658\) −2.78473e33 −0.0164801
\(659\) −2.48211e35 −1.44130 −0.720648 0.693301i \(-0.756156\pi\)
−0.720648 + 0.693301i \(0.756156\pi\)
\(660\) 3.94447e34 0.224745
\(661\) −3.37345e35 −1.88606 −0.943030 0.332707i \(-0.892038\pi\)
−0.943030 + 0.332707i \(0.892038\pi\)
\(662\) 1.93183e35 1.05985
\(663\) −1.63416e33 −0.00879784
\(664\) −1.07105e35 −0.565862
\(665\) 8.37835e33 0.0434398
\(666\) −2.29390e34 −0.116720
\(667\) 2.07330e35 1.03535
\(668\) −8.69030e34 −0.425921
\(669\) −1.54387e35 −0.742648
\(670\) 1.27951e35 0.604100
\(671\) −5.37170e35 −2.48932
\(672\) 2.19390e33 0.00997930
\(673\) 2.99045e35 1.33521 0.667604 0.744517i \(-0.267320\pi\)
0.667604 + 0.744517i \(0.267320\pi\)
\(674\) −7.61004e34 −0.333533
\(675\) 3.18678e34 0.137105
\(676\) −1.18362e35 −0.499894
\(677\) 5.69058e34 0.235938 0.117969 0.993017i \(-0.462362\pi\)
0.117969 + 0.993017i \(0.462362\pi\)
\(678\) −3.77106e34 −0.153494
\(679\) −1.57608e33 −0.00629803
\(680\) 5.06621e34 0.198757
\(681\) −1.72062e35 −0.662745
\(682\) 3.05627e35 1.15581
\(683\) 1.38059e35 0.512633 0.256317 0.966593i \(-0.417491\pi\)
0.256317 + 0.966593i \(0.417491\pi\)
\(684\) −3.78724e34 −0.138077
\(685\) 1.92264e35 0.688283
\(686\) 3.91490e34 0.137616
\(687\) 2.71619e35 0.937564
\(688\) −2.03284e34 −0.0689045
\(689\) −3.23056e33 −0.0107532
\(690\) 6.35005e34 0.207570
\(691\) 1.16343e35 0.373479 0.186739 0.982409i \(-0.440208\pi\)
0.186739 + 0.982409i \(0.440208\pi\)
\(692\) −5.71901e34 −0.180300
\(693\) 1.52820e34 0.0473168
\(694\) 2.98039e35 0.906317
\(695\) −1.96880e35 −0.588020
\(696\) −7.59869e34 −0.222907
\(697\) 5.65841e35 1.63036
\(698\) −4.90345e35 −1.38774
\(699\) 6.38421e34 0.177477
\(700\) −1.27546e34 −0.0348291
\(701\) 6.30496e35 1.69125 0.845624 0.533779i \(-0.179229\pi\)
0.845624 + 0.533779i \(0.179229\pi\)
\(702\) 7.50700e32 0.00197812
\(703\) −1.58488e35 −0.410257
\(704\) 7.13624e34 0.181473
\(705\) 2.95407e34 0.0738001
\(706\) −2.86606e35 −0.703439
\(707\) −1.79957e34 −0.0433935
\(708\) 1.46711e35 0.347574
\(709\) −7.52844e35 −1.75238 −0.876188 0.481970i \(-0.839921\pi\)
−0.876188 + 0.481970i \(0.839921\pi\)
\(710\) 8.83419e34 0.202040
\(711\) −2.83176e35 −0.636335
\(712\) −1.49923e35 −0.331031
\(713\) 4.92018e35 1.06748
\(714\) 1.96280e34 0.0418454
\(715\) −5.40200e33 −0.0113170
\(716\) −2.07384e35 −0.426935
\(717\) −1.77988e35 −0.360082
\(718\) −2.48367e35 −0.493785
\(719\) −1.28708e35 −0.251475 −0.125738 0.992064i \(-0.540130\pi\)
−0.125738 + 0.992064i \(0.540130\pi\)
\(720\) −2.32731e34 −0.0446887
\(721\) −5.45373e34 −0.102921
\(722\) 1.19576e35 0.221783
\(723\) 1.87725e35 0.342211
\(724\) 5.67936e34 0.101758
\(725\) 4.41764e35 0.777974
\(726\) 2.61241e35 0.452204
\(727\) 4.39640e35 0.748027 0.374014 0.927423i \(-0.377981\pi\)
0.374014 + 0.927423i \(0.377981\pi\)
\(728\) −3.00457e32 −0.000502504 0
\(729\) 2.25284e34 0.0370370
\(730\) −3.14654e35 −0.508507
\(731\) −1.81871e35 −0.288932
\(732\) 3.16940e35 0.494981
\(733\) −1.18297e36 −1.81625 −0.908124 0.418702i \(-0.862485\pi\)
−0.908124 + 0.418702i \(0.862485\pi\)
\(734\) −9.11520e35 −1.37583
\(735\) −2.06651e35 −0.306652
\(736\) 1.14884e35 0.167604
\(737\) 1.61246e36 2.31284
\(738\) −2.59935e35 −0.366573
\(739\) −8.13836e35 −1.12845 −0.564224 0.825622i \(-0.690825\pi\)
−0.564224 + 0.825622i \(0.690825\pi\)
\(740\) −9.73931e34 −0.132780
\(741\) 5.18668e33 0.00695283
\(742\) 3.88023e34 0.0511457
\(743\) −1.02640e36 −1.33033 −0.665164 0.746698i \(-0.731638\pi\)
−0.665164 + 0.746698i \(0.731638\pi\)
\(744\) −1.80325e35 −0.229824
\(745\) 2.33802e35 0.293018
\(746\) 4.72593e35 0.582440
\(747\) 4.40191e35 0.533500
\(748\) 6.38452e35 0.760954
\(749\) −8.85251e34 −0.103763
\(750\) 3.25222e35 0.374899
\(751\) −1.43129e36 −1.62266 −0.811332 0.584586i \(-0.801257\pi\)
−0.811332 + 0.584586i \(0.801257\pi\)
\(752\) 5.34443e34 0.0595907
\(753\) −5.71848e35 −0.627110
\(754\) 1.04065e34 0.0112244
\(755\) 3.03391e35 0.321859
\(756\) −9.01667e33 −0.00940857
\(757\) 8.68697e35 0.891600 0.445800 0.895133i \(-0.352919\pi\)
0.445800 + 0.895133i \(0.352919\pi\)
\(758\) −2.24138e35 −0.226283
\(759\) 8.00245e35 0.794696
\(760\) −1.60797e35 −0.157075
\(761\) 5.51250e34 0.0529713 0.0264857 0.999649i \(-0.491568\pi\)
0.0264857 + 0.999649i \(0.491568\pi\)
\(762\) 7.03860e35 0.665348
\(763\) −1.02324e35 −0.0951530
\(764\) 6.39446e35 0.584975
\(765\) −2.08216e35 −0.187390
\(766\) 8.11541e35 0.718540
\(767\) −2.00923e34 −0.0175020
\(768\) −4.21051e34 −0.0360844
\(769\) −4.98681e35 −0.420479 −0.210239 0.977650i \(-0.567424\pi\)
−0.210239 + 0.977650i \(0.567424\pi\)
\(770\) 6.48836e34 0.0538271
\(771\) −3.48763e33 −0.00284676
\(772\) 1.07180e36 0.860790
\(773\) 4.79467e35 0.378892 0.189446 0.981891i \(-0.439331\pi\)
0.189446 + 0.981891i \(0.439331\pi\)
\(774\) 8.35475e34 0.0649638
\(775\) 1.04836e36 0.802116
\(776\) 3.02479e34 0.0227732
\(777\) −3.77329e34 −0.0279549
\(778\) −1.71663e35 −0.125150
\(779\) −1.79593e36 −1.28846
\(780\) 3.18728e33 0.00225029
\(781\) 1.11330e36 0.773525
\(782\) 1.02782e36 0.702802
\(783\) 3.12298e35 0.210158
\(784\) −3.73868e35 −0.247610
\(785\) 9.84670e35 0.641830
\(786\) −7.90767e35 −0.507302
\(787\) 1.22599e36 0.774111 0.387055 0.922056i \(-0.373492\pi\)
0.387055 + 0.922056i \(0.373492\pi\)
\(788\) −1.46579e36 −0.910950
\(789\) −1.08317e36 −0.662576
\(790\) −1.20229e36 −0.723888
\(791\) −6.20311e34 −0.0367623
\(792\) −2.93292e35 −0.171094
\(793\) −4.34054e34 −0.0249246
\(794\) −1.08339e36 −0.612392
\(795\) −4.11619e35 −0.229038
\(796\) 4.64337e35 0.254343
\(797\) −3.12403e36 −1.68456 −0.842278 0.539043i \(-0.818786\pi\)
−0.842278 + 0.539043i \(0.818786\pi\)
\(798\) −6.22973e34 −0.0330699
\(799\) 4.78146e35 0.249877
\(800\) 2.44786e35 0.125939
\(801\) 6.16168e35 0.312099
\(802\) −9.57511e35 −0.477490
\(803\) −3.96532e36 −1.94686
\(804\) −9.51381e35 −0.459890
\(805\) 1.04454e35 0.0497136
\(806\) 2.46958e34 0.0115727
\(807\) −6.92021e35 −0.319301
\(808\) 3.45371e35 0.156908
\(809\) 4.16982e36 1.86535 0.932675 0.360717i \(-0.117468\pi\)
0.932675 + 0.360717i \(0.117468\pi\)
\(810\) 9.56498e34 0.0421329
\(811\) 7.93102e35 0.344008 0.172004 0.985096i \(-0.444976\pi\)
0.172004 + 0.985096i \(0.444976\pi\)
\(812\) −1.24993e35 −0.0533869
\(813\) −1.45287e36 −0.611078
\(814\) −1.22737e36 −0.508357
\(815\) −2.34334e35 −0.0955794
\(816\) −3.76698e35 −0.151310
\(817\) 5.77240e35 0.228339
\(818\) 2.24664e36 0.875221
\(819\) 1.23484e33 0.000473766 0
\(820\) −1.10362e36 −0.417010
\(821\) 2.22198e35 0.0826896 0.0413448 0.999145i \(-0.486836\pi\)
0.0413448 + 0.999145i \(0.486836\pi\)
\(822\) −1.42958e36 −0.523977
\(823\) 4.63197e36 1.67212 0.836061 0.548637i \(-0.184853\pi\)
0.836061 + 0.548637i \(0.184853\pi\)
\(824\) 1.04668e36 0.372153
\(825\) 1.70511e36 0.597141
\(826\) 2.41329e35 0.0832450
\(827\) 5.60825e36 1.90550 0.952748 0.303760i \(-0.0982422\pi\)
0.952748 + 0.303760i \(0.0982422\pi\)
\(828\) −4.72159e35 −0.158019
\(829\) −3.61387e36 −1.19135 −0.595677 0.803224i \(-0.703116\pi\)
−0.595677 + 0.803224i \(0.703116\pi\)
\(830\) 1.86894e36 0.606903
\(831\) 2.57136e35 0.0822526
\(832\) 5.76635e33 0.00181702
\(833\) −3.34486e36 −1.03828
\(834\) 1.46391e36 0.447649
\(835\) 1.51642e36 0.456813
\(836\) −2.02639e36 −0.601373
\(837\) 7.41118e35 0.216680
\(838\) −1.39873e35 −0.0402889
\(839\) −3.53738e36 −1.00382 −0.501911 0.864919i \(-0.667370\pi\)
−0.501911 + 0.864919i \(0.667370\pi\)
\(840\) −3.82825e34 −0.0107031
\(841\) 6.98837e35 0.192498
\(842\) 2.33934e36 0.634880
\(843\) 4.02736e36 1.07690
\(844\) −3.49770e35 −0.0921513
\(845\) 2.06536e36 0.536151
\(846\) −2.19650e35 −0.0561827
\(847\) 4.29722e35 0.108304
\(848\) −7.44692e35 −0.184939
\(849\) 6.93230e35 0.169641
\(850\) 2.19001e36 0.528091
\(851\) −1.97589e36 −0.469508
\(852\) −6.56867e35 −0.153809
\(853\) −5.36485e36 −1.23792 −0.618962 0.785421i \(-0.712447\pi\)
−0.618962 + 0.785421i \(0.712447\pi\)
\(854\) 5.21343e35 0.118550
\(855\) 6.60857e35 0.148092
\(856\) 1.69897e36 0.375200
\(857\) 1.12850e36 0.245607 0.122803 0.992431i \(-0.460812\pi\)
0.122803 + 0.992431i \(0.460812\pi\)
\(858\) 4.01667e34 0.00861539
\(859\) −6.05620e36 −1.28022 −0.640111 0.768282i \(-0.721112\pi\)
−0.640111 + 0.768282i \(0.721112\pi\)
\(860\) 3.54722e35 0.0739021
\(861\) −4.27574e35 −0.0877955
\(862\) −2.15609e36 −0.436342
\(863\) 5.30956e35 0.105907 0.0529533 0.998597i \(-0.483137\pi\)
0.0529533 + 0.998597i \(0.483137\pi\)
\(864\) 1.73047e35 0.0340207
\(865\) 9.97942e35 0.193377
\(866\) 2.86979e36 0.548121
\(867\) −3.03432e35 −0.0571244
\(868\) −2.96622e35 −0.0550436
\(869\) −1.51515e37 −2.77146
\(870\) 1.32594e36 0.239074
\(871\) 1.30293e35 0.0231576
\(872\) 1.96380e36 0.344066
\(873\) −1.24316e35 −0.0214708
\(874\) −3.26221e36 −0.555416
\(875\) 5.34965e35 0.0897894
\(876\) 2.33961e36 0.387117
\(877\) −5.83031e36 −0.951035 −0.475518 0.879706i \(-0.657739\pi\)
−0.475518 + 0.879706i \(0.657739\pi\)
\(878\) −2.16900e36 −0.348802
\(879\) 2.14713e35 0.0340406
\(880\) −1.24524e36 −0.194635
\(881\) 7.57293e36 1.16698 0.583492 0.812119i \(-0.301686\pi\)
0.583492 + 0.812119i \(0.301686\pi\)
\(882\) 1.53656e36 0.233449
\(883\) −2.59184e36 −0.388239 −0.194119 0.980978i \(-0.562185\pi\)
−0.194119 + 0.980978i \(0.562185\pi\)
\(884\) 5.15893e34 0.00761915
\(885\) −2.56004e36 −0.372783
\(886\) −1.84235e36 −0.264515
\(887\) 7.96686e36 1.12782 0.563912 0.825835i \(-0.309296\pi\)
0.563912 + 0.825835i \(0.309296\pi\)
\(888\) 7.24168e35 0.101083
\(889\) 1.15780e36 0.159353
\(890\) 2.61609e36 0.355041
\(891\) 1.20540e36 0.161309
\(892\) 4.87387e36 0.643152
\(893\) −1.51759e36 −0.197475
\(894\) −1.73844e36 −0.223069
\(895\) 3.61875e36 0.457900
\(896\) −6.92597e34 −0.00864233
\(897\) 6.46628e34 0.00795699
\(898\) 4.04988e36 0.491460
\(899\) 1.02737e37 1.22950
\(900\) −1.00604e36 −0.118737
\(901\) −6.66248e36 −0.775489
\(902\) −1.39080e37 −1.59655
\(903\) 1.37429e35 0.0155590
\(904\) 1.19050e36 0.132930
\(905\) −9.91023e35 −0.109138
\(906\) −2.25586e36 −0.245025
\(907\) −4.86506e36 −0.521191 −0.260596 0.965448i \(-0.583919\pi\)
−0.260596 + 0.965448i \(0.583919\pi\)
\(908\) 5.43188e36 0.573954
\(909\) −1.41944e36 −0.147934
\(910\) 5.24284e33 0.000538951 0
\(911\) 1.10036e37 1.11572 0.557859 0.829936i \(-0.311623\pi\)
0.557859 + 0.829936i \(0.311623\pi\)
\(912\) 1.19561e36 0.119578
\(913\) 2.35527e37 2.32357
\(914\) −1.20337e37 −1.17104
\(915\) −5.53046e36 −0.530882
\(916\) −8.57482e36 −0.811954
\(917\) −1.30075e36 −0.121500
\(918\) 1.54819e36 0.142656
\(919\) 1.10577e37 1.00513 0.502564 0.864540i \(-0.332390\pi\)
0.502564 + 0.864540i \(0.332390\pi\)
\(920\) −2.00467e36 −0.179761
\(921\) −3.69689e36 −0.327033
\(922\) 8.76759e36 0.765145
\(923\) 8.99588e34 0.00774502
\(924\) −4.82443e35 −0.0409776
\(925\) −4.21008e36 −0.352792
\(926\) −3.57164e36 −0.295277
\(927\) −4.30172e36 −0.350870
\(928\) 2.39885e36 0.193043
\(929\) 1.82435e37 1.44848 0.724239 0.689549i \(-0.242191\pi\)
0.724239 + 0.689549i \(0.242191\pi\)
\(930\) 3.14660e36 0.246493
\(931\) 1.06163e37 0.820543
\(932\) −2.01545e36 −0.153700
\(933\) −1.46100e36 −0.109933
\(934\) 2.46232e36 0.182814
\(935\) −1.11407e37 −0.816146
\(936\) −2.36991e34 −0.00171310
\(937\) 9.54074e36 0.680514 0.340257 0.940332i \(-0.389486\pi\)
0.340257 + 0.940332i \(0.389486\pi\)
\(938\) −1.56495e36 −0.110145
\(939\) −4.99226e36 −0.346719
\(940\) −9.32580e35 −0.0639128
\(941\) −7.72127e36 −0.522178 −0.261089 0.965315i \(-0.584082\pi\)
−0.261089 + 0.965315i \(0.584082\pi\)
\(942\) −7.32152e36 −0.488613
\(943\) −2.23900e37 −1.47454
\(944\) −4.63156e36 −0.301008
\(945\) 1.57337e35 0.0100910
\(946\) 4.47026e36 0.282940
\(947\) −1.89183e37 −1.18170 −0.590849 0.806782i \(-0.701207\pi\)
−0.590849 + 0.806782i \(0.701207\pi\)
\(948\) 8.93966e36 0.551083
\(949\) −3.20413e35 −0.0194932
\(950\) −6.95088e36 −0.417344
\(951\) −9.91366e36 −0.587459
\(952\) −6.19641e35 −0.0362392
\(953\) −1.61494e37 −0.932172 −0.466086 0.884739i \(-0.654336\pi\)
−0.466086 + 0.884739i \(0.654336\pi\)
\(954\) 3.06060e36 0.174362
\(955\) −1.11580e37 −0.627402
\(956\) 5.61896e36 0.311840
\(957\) 1.67097e37 0.915312
\(958\) 1.23646e37 0.668515
\(959\) −2.35156e36 −0.125494
\(960\) 7.34715e35 0.0387016
\(961\) 5.14781e36 0.267658
\(962\) −9.91757e34 −0.00508999
\(963\) −6.98257e36 −0.353742
\(964\) −5.92635e36 −0.296363
\(965\) −1.87024e37 −0.923222
\(966\) −7.76666e35 −0.0378460
\(967\) −2.64645e37 −1.27301 −0.636507 0.771271i \(-0.719621\pi\)
−0.636507 + 0.771271i \(0.719621\pi\)
\(968\) −8.24720e36 −0.391620
\(969\) 1.06966e37 0.501418
\(970\) −5.27813e35 −0.0244249
\(971\) 2.73601e37 1.24990 0.624952 0.780663i \(-0.285118\pi\)
0.624952 + 0.780663i \(0.285118\pi\)
\(972\) −7.11206e35 −0.0320750
\(973\) 2.40802e36 0.107213
\(974\) 2.14687e37 0.943665
\(975\) 1.37779e35 0.00597895
\(976\) −1.00056e37 −0.428666
\(977\) 3.04743e37 1.28900 0.644498 0.764606i \(-0.277066\pi\)
0.644498 + 0.764606i \(0.277066\pi\)
\(978\) 1.74239e36 0.0727629
\(979\) 3.29685e37 1.35930
\(980\) 6.52384e36 0.265569
\(981\) −8.07101e36 −0.324389
\(982\) 1.26227e37 0.500910
\(983\) 3.84516e36 0.150659 0.0753296 0.997159i \(-0.475999\pi\)
0.0753296 + 0.997159i \(0.475999\pi\)
\(984\) 8.20598e36 0.317462
\(985\) 2.55774e37 0.977020
\(986\) 2.14616e37 0.809471
\(987\) −3.61308e35 −0.0134559
\(988\) −1.63740e35 −0.00602133
\(989\) 7.19651e36 0.261317
\(990\) 5.11781e36 0.183503
\(991\) −4.30043e37 −1.52262 −0.761309 0.648389i \(-0.775443\pi\)
−0.761309 + 0.648389i \(0.775443\pi\)
\(992\) 5.69275e36 0.199033
\(993\) 2.50648e37 0.865363
\(994\) −1.08050e36 −0.0368378
\(995\) −8.10246e36 −0.272790
\(996\) −1.38965e37 −0.462024
\(997\) −1.52317e37 −0.500103 −0.250051 0.968233i \(-0.580447\pi\)
−0.250051 + 0.968233i \(0.580447\pi\)
\(998\) 6.75470e36 0.219015
\(999\) −2.97625e36 −0.0953017
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 6.26.a.a.1.1 1
3.2 odd 2 18.26.a.d.1.1 1
4.3 odd 2 48.26.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.26.a.a.1.1 1 1.1 even 1 trivial
18.26.a.d.1.1 1 3.2 odd 2
48.26.a.c.1.1 1 4.3 odd 2