Properties

Label 10.16.a.c.1.1
Level $10$
Weight $16$
Character 10.1
Self dual yes
Analytic conductor $14.269$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10,16,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.2693505100\)
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 \(\chi\) \(=\) 10.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+128.000 q^{2} -1302.00 q^{3} +16384.0 q^{4} -78125.0 q^{5} -166656. q^{6} -90706.0 q^{7} +2.09715e6 q^{8} -1.26537e7 q^{9} +O(q^{10})\) \(q+128.000 q^{2} -1302.00 q^{3} +16384.0 q^{4} -78125.0 q^{5} -166656. q^{6} -90706.0 q^{7} +2.09715e6 q^{8} -1.26537e7 q^{9} -1.00000e7 q^{10} -6.94116e7 q^{11} -2.13320e7 q^{12} -3.70575e7 q^{13} -1.16104e7 q^{14} +1.01719e8 q^{15} +2.68435e8 q^{16} -1.37203e9 q^{17} -1.61967e9 q^{18} -5.06876e9 q^{19} -1.28000e9 q^{20} +1.18099e8 q^{21} -8.88469e9 q^{22} -1.23426e10 q^{23} -2.73049e9 q^{24} +6.10352e9 q^{25} -4.74336e9 q^{26} +3.51574e10 q^{27} -1.48613e9 q^{28} +5.78257e10 q^{29} +1.30200e10 q^{30} +2.33728e11 q^{31} +3.43597e10 q^{32} +9.03740e10 q^{33} -1.75620e11 q^{34} +7.08641e9 q^{35} -2.07318e11 q^{36} +7.42248e11 q^{37} -6.48801e11 q^{38} +4.82488e10 q^{39} -1.63840e11 q^{40} -7.72700e11 q^{41} +1.51167e10 q^{42} -4.05994e11 q^{43} -1.13724e12 q^{44} +9.88571e11 q^{45} -1.57986e12 q^{46} +1.62301e12 q^{47} -3.49503e11 q^{48} -4.73933e12 q^{49} +7.81250e11 q^{50} +1.78638e12 q^{51} -6.07149e11 q^{52} -5.36560e12 q^{53} +4.50015e12 q^{54} +5.42278e12 q^{55} -1.90224e11 q^{56} +6.59953e12 q^{57} +7.40169e12 q^{58} +9.24016e12 q^{59} +1.66656e12 q^{60} +9.44308e11 q^{61} +2.99172e13 q^{62} +1.14777e12 q^{63} +4.39805e12 q^{64} +2.89511e12 q^{65} +1.15679e13 q^{66} -7.05676e13 q^{67} -2.24793e13 q^{68} +1.60701e13 q^{69} +9.07060e11 q^{70} -8.25347e13 q^{71} -2.65367e13 q^{72} -1.78432e14 q^{73} +9.50077e13 q^{74} -7.94678e12 q^{75} -8.30466e13 q^{76} +6.29605e12 q^{77} +6.17585e12 q^{78} +3.07261e14 q^{79} -2.09715e13 q^{80} +1.35792e14 q^{81} -9.89056e13 q^{82} +4.31597e13 q^{83} +1.93494e12 q^{84} +1.07190e14 q^{85} -5.19673e13 q^{86} -7.52891e13 q^{87} -1.45567e14 q^{88} +6.81678e14 q^{89} +1.26537e14 q^{90} +3.36133e12 q^{91} -2.02222e14 q^{92} -3.04314e14 q^{93} +2.07745e14 q^{94} +3.95997e14 q^{95} -4.47364e13 q^{96} -2.36520e14 q^{97} -6.06635e14 q^{98} +8.78314e14 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 128.000 0.707107
\(3\) −1302.00 −0.343717 −0.171859 0.985122i \(-0.554977\pi\)
−0.171859 + 0.985122i \(0.554977\pi\)
\(4\) 16384.0 0.500000
\(5\) −78125.0 −0.447214
\(6\) −166656. −0.243045
\(7\) −90706.0 −0.0416295 −0.0208147 0.999783i \(-0.506626\pi\)
−0.0208147 + 0.999783i \(0.506626\pi\)
\(8\) 2.09715e6 0.353553
\(9\) −1.26537e7 −0.881858
\(10\) −1.00000e7 −0.316228
\(11\) −6.94116e7 −1.07396 −0.536979 0.843596i \(-0.680435\pi\)
−0.536979 + 0.843596i \(0.680435\pi\)
\(12\) −2.13320e7 −0.171859
\(13\) −3.70575e7 −0.163795 −0.0818975 0.996641i \(-0.526098\pi\)
−0.0818975 + 0.996641i \(0.526098\pi\)
\(14\) −1.16104e7 −0.0294365
\(15\) 1.01719e8 0.153715
\(16\) 2.68435e8 0.250000
\(17\) −1.37203e9 −0.810954 −0.405477 0.914105i \(-0.632894\pi\)
−0.405477 + 0.914105i \(0.632894\pi\)
\(18\) −1.61967e9 −0.623568
\(19\) −5.06876e9 −1.30092 −0.650459 0.759542i \(-0.725423\pi\)
−0.650459 + 0.759542i \(0.725423\pi\)
\(20\) −1.28000e9 −0.223607
\(21\) 1.18099e8 0.0143088
\(22\) −8.88469e9 −0.759403
\(23\) −1.23426e10 −0.755873 −0.377937 0.925831i \(-0.623366\pi\)
−0.377937 + 0.925831i \(0.623366\pi\)
\(24\) −2.73049e9 −0.121522
\(25\) 6.10352e9 0.200000
\(26\) −4.74336e9 −0.115821
\(27\) 3.51574e10 0.646828
\(28\) −1.48613e9 −0.0208147
\(29\) 5.78257e10 0.622495 0.311248 0.950329i \(-0.399253\pi\)
0.311248 + 0.950329i \(0.399253\pi\)
\(30\) 1.30200e10 0.108693
\(31\) 2.33728e11 1.52580 0.762901 0.646516i \(-0.223775\pi\)
0.762901 + 0.646516i \(0.223775\pi\)
\(32\) 3.43597e10 0.176777
\(33\) 9.03740e10 0.369138
\(34\) −1.75620e11 −0.573431
\(35\) 7.08641e9 0.0186173
\(36\) −2.07318e11 −0.440929
\(37\) 7.42248e11 1.28539 0.642696 0.766121i \(-0.277816\pi\)
0.642696 + 0.766121i \(0.277816\pi\)
\(38\) −6.48801e11 −0.919888
\(39\) 4.82488e10 0.0562992
\(40\) −1.63840e11 −0.158114
\(41\) −7.72700e11 −0.619629 −0.309815 0.950797i \(-0.600267\pi\)
−0.309815 + 0.950797i \(0.600267\pi\)
\(42\) 1.51167e10 0.0101178
\(43\) −4.05994e11 −0.227775 −0.113888 0.993494i \(-0.536330\pi\)
−0.113888 + 0.993494i \(0.536330\pi\)
\(44\) −1.13724e12 −0.536979
\(45\) 9.88571e11 0.394379
\(46\) −1.57986e12 −0.534483
\(47\) 1.62301e12 0.467291 0.233646 0.972322i \(-0.424934\pi\)
0.233646 + 0.972322i \(0.424934\pi\)
\(48\) −3.49503e11 −0.0859294
\(49\) −4.73933e12 −0.998267
\(50\) 7.81250e11 0.141421
\(51\) 1.78638e12 0.278739
\(52\) −6.07149e11 −0.0818975
\(53\) −5.36560e12 −0.627407 −0.313703 0.949521i \(-0.601570\pi\)
−0.313703 + 0.949521i \(0.601570\pi\)
\(54\) 4.50015e12 0.457376
\(55\) 5.42278e12 0.480289
\(56\) −1.90224e11 −0.0147182
\(57\) 6.59953e12 0.447148
\(58\) 7.40169e12 0.440171
\(59\) 9.24016e12 0.483381 0.241690 0.970353i \(-0.422298\pi\)
0.241690 + 0.970353i \(0.422298\pi\)
\(60\) 1.66656e12 0.0768576
\(61\) 9.44308e11 0.0384716 0.0192358 0.999815i \(-0.493877\pi\)
0.0192358 + 0.999815i \(0.493877\pi\)
\(62\) 2.99172e13 1.07890
\(63\) 1.14777e12 0.0367113
\(64\) 4.39805e12 0.125000
\(65\) 2.89511e12 0.0732513
\(66\) 1.15679e13 0.261020
\(67\) −7.05676e13 −1.42247 −0.711237 0.702953i \(-0.751865\pi\)
−0.711237 + 0.702953i \(0.751865\pi\)
\(68\) −2.24793e13 −0.405477
\(69\) 1.60701e13 0.259807
\(70\) 9.07060e11 0.0131644
\(71\) −8.25347e13 −1.07696 −0.538479 0.842639i \(-0.681001\pi\)
−0.538479 + 0.842639i \(0.681001\pi\)
\(72\) −2.65367e13 −0.311784
\(73\) −1.78432e14 −1.89039 −0.945197 0.326502i \(-0.894130\pi\)
−0.945197 + 0.326502i \(0.894130\pi\)
\(74\) 9.50077e13 0.908910
\(75\) −7.94678e12 −0.0687435
\(76\) −8.30466e13 −0.650459
\(77\) 6.29605e12 0.0447083
\(78\) 6.17585e12 0.0398095
\(79\) 3.07261e14 1.80013 0.900067 0.435752i \(-0.143518\pi\)
0.900067 + 0.435752i \(0.143518\pi\)
\(80\) −2.09715e13 −0.111803
\(81\) 1.35792e14 0.659532
\(82\) −9.89056e13 −0.438144
\(83\) 4.31597e13 0.174579 0.0872897 0.996183i \(-0.472179\pi\)
0.0872897 + 0.996183i \(0.472179\pi\)
\(84\) 1.93494e12 0.00715438
\(85\) 1.07190e14 0.362669
\(86\) −5.19673e13 −0.161061
\(87\) −7.52891e13 −0.213962
\(88\) −1.45567e14 −0.379701
\(89\) 6.81678e14 1.63363 0.816816 0.576899i \(-0.195737\pi\)
0.816816 + 0.576899i \(0.195737\pi\)
\(90\) 1.26537e14 0.278868
\(91\) 3.36133e12 0.00681869
\(92\) −2.02222e14 −0.377937
\(93\) −3.04314e14 −0.524445
\(94\) 2.07745e14 0.330425
\(95\) 3.95997e14 0.581788
\(96\) −4.47364e13 −0.0607612
\(97\) −2.36520e14 −0.297222 −0.148611 0.988896i \(-0.547480\pi\)
−0.148611 + 0.988896i \(0.547480\pi\)
\(98\) −6.06635e14 −0.705881
\(99\) 8.78314e14 0.947079
\(100\) 1.00000e14 0.100000
\(101\) −1.39643e15 −1.29601 −0.648005 0.761636i \(-0.724397\pi\)
−0.648005 + 0.761636i \(0.724397\pi\)
\(102\) 2.28657e14 0.197098
\(103\) −1.46886e15 −1.17680 −0.588398 0.808572i \(-0.700241\pi\)
−0.588398 + 0.808572i \(0.700241\pi\)
\(104\) −7.77151e13 −0.0579103
\(105\) −9.22650e12 −0.00639908
\(106\) −6.86797e14 −0.443644
\(107\) 1.19960e15 0.722203 0.361102 0.932526i \(-0.382401\pi\)
0.361102 + 0.932526i \(0.382401\pi\)
\(108\) 5.76019e14 0.323414
\(109\) 2.63914e14 0.138281 0.0691407 0.997607i \(-0.477974\pi\)
0.0691407 + 0.997607i \(0.477974\pi\)
\(110\) 6.94116e14 0.339615
\(111\) −9.66406e14 −0.441812
\(112\) −2.43487e13 −0.0104074
\(113\) −4.67332e15 −1.86869 −0.934346 0.356368i \(-0.884015\pi\)
−0.934346 + 0.356368i \(0.884015\pi\)
\(114\) 8.44739e14 0.316181
\(115\) 9.64268e14 0.338037
\(116\) 9.47417e14 0.311248
\(117\) 4.68914e14 0.144444
\(118\) 1.18274e15 0.341802
\(119\) 1.24451e14 0.0337596
\(120\) 2.13320e14 0.0543465
\(121\) 6.40729e14 0.153385
\(122\) 1.20871e14 0.0272035
\(123\) 1.00606e15 0.212977
\(124\) 3.82940e15 0.762901
\(125\) −4.76837e14 −0.0894427
\(126\) 1.46914e14 0.0259588
\(127\) 5.53712e15 0.922053 0.461027 0.887386i \(-0.347481\pi\)
0.461027 + 0.887386i \(0.347481\pi\)
\(128\) 5.62950e14 0.0883883
\(129\) 5.28605e14 0.0782903
\(130\) 3.70575e14 0.0517965
\(131\) 7.83095e15 1.03343 0.516714 0.856158i \(-0.327155\pi\)
0.516714 + 0.856158i \(0.327155\pi\)
\(132\) 1.48069e15 0.184569
\(133\) 4.59767e14 0.0541565
\(134\) −9.03265e15 −1.00584
\(135\) −2.74667e15 −0.289270
\(136\) −2.87735e15 −0.286715
\(137\) −1.88872e16 −1.78140 −0.890702 0.454587i \(-0.849787\pi\)
−0.890702 + 0.454587i \(0.849787\pi\)
\(138\) 2.05697e15 0.183711
\(139\) 6.08888e15 0.515142 0.257571 0.966259i \(-0.417078\pi\)
0.257571 + 0.966259i \(0.417078\pi\)
\(140\) 1.16104e14 0.00930863
\(141\) −2.11316e15 −0.160616
\(142\) −1.05644e16 −0.761525
\(143\) 2.57222e15 0.175909
\(144\) −3.39670e15 −0.220465
\(145\) −4.51763e15 −0.278388
\(146\) −2.28393e16 −1.33671
\(147\) 6.17061e15 0.343122
\(148\) 1.21610e16 0.642696
\(149\) −2.97847e16 −1.49657 −0.748285 0.663378i \(-0.769122\pi\)
−0.748285 + 0.663378i \(0.769122\pi\)
\(150\) −1.01719e15 −0.0486090
\(151\) 3.43750e16 1.56284 0.781421 0.624004i \(-0.214495\pi\)
0.781421 + 0.624004i \(0.214495\pi\)
\(152\) −1.06300e16 −0.459944
\(153\) 1.73612e16 0.715146
\(154\) 8.05895e14 0.0316135
\(155\) −1.82600e16 −0.682359
\(156\) 7.90509e14 0.0281496
\(157\) 3.24674e16 1.10205 0.551024 0.834490i \(-0.314237\pi\)
0.551024 + 0.834490i \(0.314237\pi\)
\(158\) 3.93294e16 1.27289
\(159\) 6.98601e15 0.215651
\(160\) −2.68435e15 −0.0790569
\(161\) 1.11955e15 0.0314666
\(162\) 1.73814e16 0.466360
\(163\) 2.42386e16 0.621013 0.310506 0.950571i \(-0.399501\pi\)
0.310506 + 0.950571i \(0.399501\pi\)
\(164\) −1.26599e16 −0.309815
\(165\) −7.06047e15 −0.165084
\(166\) 5.52445e15 0.123446
\(167\) 3.26384e15 0.0697196 0.0348598 0.999392i \(-0.488902\pi\)
0.0348598 + 0.999392i \(0.488902\pi\)
\(168\) 2.47672e14 0.00505891
\(169\) −4.98126e16 −0.973171
\(170\) 1.37203e16 0.256446
\(171\) 6.41386e16 1.14722
\(172\) −6.65181e15 −0.113888
\(173\) −9.54205e16 −1.56421 −0.782106 0.623145i \(-0.785855\pi\)
−0.782106 + 0.623145i \(0.785855\pi\)
\(174\) −9.63700e15 −0.151294
\(175\) −5.53625e14 −0.00832589
\(176\) −1.86325e16 −0.268489
\(177\) −1.20307e16 −0.166146
\(178\) 8.72548e16 1.15515
\(179\) −1.08211e17 −1.37365 −0.686823 0.726824i \(-0.740995\pi\)
−0.686823 + 0.726824i \(0.740995\pi\)
\(180\) 1.61967e16 0.197190
\(181\) 2.21984e16 0.259258 0.129629 0.991563i \(-0.458621\pi\)
0.129629 + 0.991563i \(0.458621\pi\)
\(182\) 4.30251e14 0.00482155
\(183\) −1.22949e15 −0.0132234
\(184\) −2.58844e16 −0.267242
\(185\) −5.79881e16 −0.574845
\(186\) −3.89522e16 −0.370838
\(187\) 9.52347e16 0.870930
\(188\) 2.65914e16 0.233646
\(189\) −3.18899e15 −0.0269271
\(190\) 5.06876e16 0.411386
\(191\) −3.42389e16 −0.267159 −0.133580 0.991038i \(-0.542647\pi\)
−0.133580 + 0.991038i \(0.542647\pi\)
\(192\) −5.72626e15 −0.0429647
\(193\) −2.27131e17 −1.63907 −0.819533 0.573031i \(-0.805767\pi\)
−0.819533 + 0.573031i \(0.805767\pi\)
\(194\) −3.02746e16 −0.210167
\(195\) −3.76944e15 −0.0251778
\(196\) −7.76492e16 −0.499133
\(197\) 2.63870e16 0.163265 0.0816325 0.996662i \(-0.473987\pi\)
0.0816325 + 0.996662i \(0.473987\pi\)
\(198\) 1.12424e17 0.669686
\(199\) 1.99493e17 1.14427 0.572137 0.820158i \(-0.306115\pi\)
0.572137 + 0.820158i \(0.306115\pi\)
\(200\) 1.28000e16 0.0707107
\(201\) 9.18790e16 0.488929
\(202\) −1.78743e17 −0.916418
\(203\) −5.24514e15 −0.0259141
\(204\) 2.92681e16 0.139369
\(205\) 6.03672e16 0.277107
\(206\) −1.88014e17 −0.832120
\(207\) 1.56180e17 0.666573
\(208\) −9.94754e15 −0.0409487
\(209\) 3.51831e17 1.39713
\(210\) −1.18099e15 −0.00452483
\(211\) 6.45604e16 0.238697 0.119349 0.992852i \(-0.461919\pi\)
0.119349 + 0.992852i \(0.461919\pi\)
\(212\) −8.79100e16 −0.313703
\(213\) 1.07460e17 0.370170
\(214\) 1.53549e17 0.510675
\(215\) 3.17183e16 0.101864
\(216\) 7.37304e16 0.228688
\(217\) −2.12005e16 −0.0635183
\(218\) 3.37810e16 0.0977797
\(219\) 2.32319e17 0.649761
\(220\) 8.88469e16 0.240144
\(221\) 5.08439e16 0.132830
\(222\) −1.23700e17 −0.312408
\(223\) −6.71970e17 −1.64083 −0.820415 0.571769i \(-0.806257\pi\)
−0.820415 + 0.571769i \(0.806257\pi\)
\(224\) −3.11663e15 −0.00735912
\(225\) −7.72321e16 −0.176372
\(226\) −5.98185e17 −1.32136
\(227\) −5.40955e17 −1.15602 −0.578012 0.816028i \(-0.696171\pi\)
−0.578012 + 0.816028i \(0.696171\pi\)
\(228\) 1.08127e17 0.223574
\(229\) 7.52317e17 1.50534 0.752671 0.658397i \(-0.228765\pi\)
0.752671 + 0.658397i \(0.228765\pi\)
\(230\) 1.23426e17 0.239028
\(231\) −8.19746e15 −0.0153670
\(232\) 1.21269e17 0.220085
\(233\) −7.64780e17 −1.34390 −0.671951 0.740596i \(-0.734543\pi\)
−0.671951 + 0.740596i \(0.734543\pi\)
\(234\) 6.00210e16 0.102137
\(235\) −1.26798e17 −0.208979
\(236\) 1.51391e17 0.241690
\(237\) −4.00054e17 −0.618737
\(238\) 1.59297e16 0.0238716
\(239\) −6.74103e17 −0.978908 −0.489454 0.872029i \(-0.662804\pi\)
−0.489454 + 0.872029i \(0.662804\pi\)
\(240\) 2.73049e16 0.0384288
\(241\) 3.16938e17 0.432361 0.216180 0.976353i \(-0.430640\pi\)
0.216180 + 0.976353i \(0.430640\pi\)
\(242\) 8.20133e16 0.108460
\(243\) −6.81271e17 −0.873520
\(244\) 1.54715e16 0.0192358
\(245\) 3.70260e17 0.446439
\(246\) 1.28775e17 0.150598
\(247\) 1.87835e17 0.213084
\(248\) 4.90163e17 0.539452
\(249\) −5.61940e16 −0.0600060
\(250\) −6.10352e16 −0.0632456
\(251\) −1.54964e17 −0.155840 −0.0779200 0.996960i \(-0.524828\pi\)
−0.0779200 + 0.996960i \(0.524828\pi\)
\(252\) 1.88050e16 0.0183556
\(253\) 8.56722e17 0.811776
\(254\) 7.08751e17 0.651990
\(255\) −1.39561e17 −0.124656
\(256\) 7.20576e16 0.0625000
\(257\) 2.25253e18 1.89746 0.948731 0.316086i \(-0.102369\pi\)
0.948731 + 0.316086i \(0.102369\pi\)
\(258\) 6.76614e16 0.0553596
\(259\) −6.73263e16 −0.0535102
\(260\) 4.74336e16 0.0366257
\(261\) −7.31710e17 −0.548953
\(262\) 1.00236e18 0.730744
\(263\) 1.00925e18 0.715044 0.357522 0.933905i \(-0.383622\pi\)
0.357522 + 0.933905i \(0.383622\pi\)
\(264\) 1.89528e17 0.130510
\(265\) 4.19187e17 0.280585
\(266\) 5.88502e16 0.0382944
\(267\) −8.87544e17 −0.561508
\(268\) −1.15618e18 −0.711237
\(269\) −9.29818e17 −0.556232 −0.278116 0.960548i \(-0.589710\pi\)
−0.278116 + 0.960548i \(0.589710\pi\)
\(270\) −3.51574e17 −0.204545
\(271\) −6.51376e16 −0.0368606 −0.0184303 0.999830i \(-0.505867\pi\)
−0.0184303 + 0.999830i \(0.505867\pi\)
\(272\) −3.68301e17 −0.202738
\(273\) −4.37646e15 −0.00234370
\(274\) −2.41756e18 −1.25964
\(275\) −4.23655e17 −0.214792
\(276\) 2.63293e17 0.129903
\(277\) −8.74228e16 −0.0419784 −0.0209892 0.999780i \(-0.506682\pi\)
−0.0209892 + 0.999780i \(0.506682\pi\)
\(278\) 7.79377e17 0.364260
\(279\) −2.95752e18 −1.34554
\(280\) 1.48613e16 0.00658219
\(281\) 1.10585e18 0.476867 0.238433 0.971159i \(-0.423366\pi\)
0.238433 + 0.971159i \(0.423366\pi\)
\(282\) −2.70484e17 −0.113573
\(283\) 2.15159e18 0.879756 0.439878 0.898058i \(-0.355022\pi\)
0.439878 + 0.898058i \(0.355022\pi\)
\(284\) −1.35225e18 −0.538479
\(285\) −5.15588e17 −0.199971
\(286\) 3.29244e17 0.124386
\(287\) 7.00885e16 0.0257948
\(288\) −4.34778e17 −0.155892
\(289\) −9.79963e17 −0.342354
\(290\) −5.78257e17 −0.196850
\(291\) 3.07949e17 0.102160
\(292\) −2.92344e18 −0.945197
\(293\) 5.52332e18 1.74058 0.870288 0.492543i \(-0.163933\pi\)
0.870288 + 0.492543i \(0.163933\pi\)
\(294\) 7.89838e17 0.242624
\(295\) −7.21887e17 −0.216174
\(296\) 1.55661e18 0.454455
\(297\) −2.44033e18 −0.694665
\(298\) −3.81245e18 −1.05823
\(299\) 4.57387e17 0.123808
\(300\) −1.30200e17 −0.0343717
\(301\) 3.68261e16 0.00948216
\(302\) 4.40000e18 1.10510
\(303\) 1.81815e18 0.445461
\(304\) −1.36064e18 −0.325229
\(305\) −7.37741e16 −0.0172050
\(306\) 2.22224e18 0.505685
\(307\) 9.87586e17 0.219299 0.109650 0.993970i \(-0.465027\pi\)
0.109650 + 0.993970i \(0.465027\pi\)
\(308\) 1.03155e17 0.0223541
\(309\) 1.91245e18 0.404485
\(310\) −2.33728e18 −0.482501
\(311\) −3.09644e18 −0.623965 −0.311982 0.950088i \(-0.600993\pi\)
−0.311982 + 0.950088i \(0.600993\pi\)
\(312\) 1.01185e17 0.0199048
\(313\) 1.19355e18 0.229224 0.114612 0.993410i \(-0.463438\pi\)
0.114612 + 0.993410i \(0.463438\pi\)
\(314\) 4.15583e18 0.779265
\(315\) −8.96693e16 −0.0164178
\(316\) 5.03417e18 0.900067
\(317\) −2.70108e18 −0.471621 −0.235811 0.971799i \(-0.575774\pi\)
−0.235811 + 0.971799i \(0.575774\pi\)
\(318\) 8.94209e17 0.152488
\(319\) −4.01378e18 −0.668534
\(320\) −3.43597e17 −0.0559017
\(321\) −1.56188e18 −0.248234
\(322\) 1.43302e17 0.0222502
\(323\) 6.95448e18 1.05498
\(324\) 2.22481e18 0.329766
\(325\) −2.26181e17 −0.0327590
\(326\) 3.10254e18 0.439122
\(327\) −3.43616e17 −0.0475297
\(328\) −1.62047e18 −0.219072
\(329\) −1.47217e17 −0.0194531
\(330\) −9.03740e17 −0.116732
\(331\) −9.04958e17 −0.114266 −0.0571332 0.998367i \(-0.518196\pi\)
−0.0571332 + 0.998367i \(0.518196\pi\)
\(332\) 7.07129e17 0.0872897
\(333\) −9.39218e18 −1.13353
\(334\) 4.17771e17 0.0492992
\(335\) 5.51309e18 0.636150
\(336\) 3.17020e16 0.00357719
\(337\) −6.84140e17 −0.0754955 −0.0377477 0.999287i \(-0.512018\pi\)
−0.0377477 + 0.999287i \(0.512018\pi\)
\(338\) −6.37602e18 −0.688136
\(339\) 6.08467e18 0.642302
\(340\) 1.75620e18 0.181335
\(341\) −1.62234e19 −1.63865
\(342\) 8.20974e18 0.811211
\(343\) 8.60518e17 0.0831868
\(344\) −8.51432e17 −0.0805307
\(345\) −1.25548e18 −0.116189
\(346\) −1.22138e19 −1.10607
\(347\) −1.18426e19 −1.04948 −0.524742 0.851261i \(-0.675838\pi\)
−0.524742 + 0.851261i \(0.675838\pi\)
\(348\) −1.23354e18 −0.106981
\(349\) −4.67865e18 −0.397127 −0.198564 0.980088i \(-0.563628\pi\)
−0.198564 + 0.980088i \(0.563628\pi\)
\(350\) −7.08641e16 −0.00588729
\(351\) −1.30284e18 −0.105947
\(352\) −2.38497e18 −0.189851
\(353\) −6.21571e18 −0.484374 −0.242187 0.970230i \(-0.577865\pi\)
−0.242187 + 0.970230i \(0.577865\pi\)
\(354\) −1.53993e18 −0.117483
\(355\) 6.44803e18 0.481631
\(356\) 1.11686e19 0.816816
\(357\) −1.62035e17 −0.0116037
\(358\) −1.38511e19 −0.971315
\(359\) 8.67422e18 0.595693 0.297846 0.954614i \(-0.403732\pi\)
0.297846 + 0.954614i \(0.403732\pi\)
\(360\) 2.07318e18 0.139434
\(361\) 1.05112e19 0.692386
\(362\) 2.84139e18 0.183323
\(363\) −8.34229e17 −0.0527212
\(364\) 5.50721e16 0.00340935
\(365\) 1.39400e19 0.845410
\(366\) −1.57375e17 −0.00935032
\(367\) 2.60076e19 1.51393 0.756964 0.653457i \(-0.226682\pi\)
0.756964 + 0.653457i \(0.226682\pi\)
\(368\) −3.31320e18 −0.188968
\(369\) 9.77751e18 0.546425
\(370\) −7.42248e18 −0.406477
\(371\) 4.86692e17 0.0261186
\(372\) −4.98588e18 −0.262222
\(373\) 3.24522e18 0.167274 0.0836369 0.996496i \(-0.473346\pi\)
0.0836369 + 0.996496i \(0.473346\pi\)
\(374\) 1.21900e19 0.615840
\(375\) 6.20842e17 0.0307430
\(376\) 3.40370e18 0.165212
\(377\) −2.14287e18 −0.101962
\(378\) −4.08190e17 −0.0190403
\(379\) 3.97275e19 1.81676 0.908379 0.418148i \(-0.137321\pi\)
0.908379 + 0.418148i \(0.137321\pi\)
\(380\) 6.48801e18 0.290894
\(381\) −7.20933e18 −0.316926
\(382\) −4.38258e18 −0.188910
\(383\) −1.39053e19 −0.587747 −0.293874 0.955844i \(-0.594944\pi\)
−0.293874 + 0.955844i \(0.594944\pi\)
\(384\) −7.32961e17 −0.0303806
\(385\) −4.91879e17 −0.0199941
\(386\) −2.90728e19 −1.15900
\(387\) 5.13733e18 0.200866
\(388\) −3.87515e18 −0.148611
\(389\) −2.40519e19 −0.904749 −0.452374 0.891828i \(-0.649423\pi\)
−0.452374 + 0.891828i \(0.649423\pi\)
\(390\) −4.82488e17 −0.0178034
\(391\) 1.69344e19 0.612978
\(392\) −9.93910e18 −0.352941
\(393\) −1.01959e19 −0.355207
\(394\) 3.37754e18 0.115446
\(395\) −2.40048e19 −0.805044
\(396\) 1.43903e19 0.473539
\(397\) 4.68747e17 0.0151359 0.00756797 0.999971i \(-0.497591\pi\)
0.00756797 + 0.999971i \(0.497591\pi\)
\(398\) 2.55351e19 0.809124
\(399\) −5.98617e17 −0.0186145
\(400\) 1.63840e18 0.0500000
\(401\) −4.59281e19 −1.37561 −0.687806 0.725894i \(-0.741426\pi\)
−0.687806 + 0.725894i \(0.741426\pi\)
\(402\) 1.17605e19 0.345725
\(403\) −8.66137e18 −0.249919
\(404\) −2.28791e19 −0.648005
\(405\) −1.06087e19 −0.294952
\(406\) −6.71378e17 −0.0183241
\(407\) −5.15206e19 −1.38046
\(408\) 3.74631e18 0.0985491
\(409\) −2.87071e18 −0.0741421 −0.0370710 0.999313i \(-0.511803\pi\)
−0.0370710 + 0.999313i \(0.511803\pi\)
\(410\) 7.72700e18 0.195944
\(411\) 2.45911e19 0.612300
\(412\) −2.40658e19 −0.588398
\(413\) −8.38138e17 −0.0201229
\(414\) 1.99910e19 0.471338
\(415\) −3.37185e18 −0.0780743
\(416\) −1.27328e18 −0.0289551
\(417\) −7.92773e18 −0.177063
\(418\) 4.50344e19 0.987920
\(419\) −1.95181e19 −0.420563 −0.210282 0.977641i \(-0.567438\pi\)
−0.210282 + 0.977641i \(0.567438\pi\)
\(420\) −1.51167e17 −0.00319954
\(421\) 7.84258e19 1.63058 0.815292 0.579049i \(-0.196576\pi\)
0.815292 + 0.579049i \(0.196576\pi\)
\(422\) 8.26373e18 0.168785
\(423\) −2.05371e19 −0.412084
\(424\) −1.12525e19 −0.221822
\(425\) −8.37419e18 −0.162191
\(426\) 1.37549e19 0.261749
\(427\) −8.56544e16 −0.00160155
\(428\) 1.96543e19 0.361102
\(429\) −3.34903e18 −0.0604629
\(430\) 4.05994e18 0.0720289
\(431\) −5.24106e19 −0.913776 −0.456888 0.889524i \(-0.651036\pi\)
−0.456888 + 0.889524i \(0.651036\pi\)
\(432\) 9.43749e18 0.161707
\(433\) 6.72089e19 1.13179 0.565897 0.824476i \(-0.308530\pi\)
0.565897 + 0.824476i \(0.308530\pi\)
\(434\) −2.71367e18 −0.0449142
\(435\) 5.88196e18 0.0956869
\(436\) 4.32397e18 0.0691407
\(437\) 6.25618e19 0.983329
\(438\) 2.97368e19 0.459450
\(439\) −6.41938e19 −0.975010 −0.487505 0.873120i \(-0.662093\pi\)
−0.487505 + 0.873120i \(0.662093\pi\)
\(440\) 1.13724e19 0.169808
\(441\) 5.99701e19 0.880330
\(442\) 6.50801e18 0.0939251
\(443\) 8.09420e19 1.14854 0.574270 0.818666i \(-0.305286\pi\)
0.574270 + 0.818666i \(0.305286\pi\)
\(444\) −1.58336e19 −0.220906
\(445\) −5.32561e19 −0.730582
\(446\) −8.60122e19 −1.16024
\(447\) 3.87797e19 0.514397
\(448\) −3.98929e17 −0.00520368
\(449\) −5.63441e19 −0.722771 −0.361386 0.932416i \(-0.617696\pi\)
−0.361386 + 0.932416i \(0.617696\pi\)
\(450\) −9.88571e18 −0.124714
\(451\) 5.36344e19 0.665456
\(452\) −7.65677e19 −0.934346
\(453\) −4.47562e19 −0.537176
\(454\) −6.92422e19 −0.817433
\(455\) −2.62604e17 −0.00304941
\(456\) 1.38402e19 0.158091
\(457\) 8.12666e19 0.913147 0.456573 0.889686i \(-0.349077\pi\)
0.456573 + 0.889686i \(0.349077\pi\)
\(458\) 9.62966e19 1.06444
\(459\) −4.82369e19 −0.524547
\(460\) 1.57986e19 0.169018
\(461\) 1.12727e20 1.18651 0.593255 0.805015i \(-0.297843\pi\)
0.593255 + 0.805015i \(0.297843\pi\)
\(462\) −1.04927e18 −0.0108661
\(463\) −1.79852e20 −1.83256 −0.916278 0.400542i \(-0.868822\pi\)
−0.916278 + 0.400542i \(0.868822\pi\)
\(464\) 1.55225e19 0.155624
\(465\) 2.37745e19 0.234539
\(466\) −9.78919e19 −0.950282
\(467\) 3.50796e19 0.335103 0.167552 0.985863i \(-0.446414\pi\)
0.167552 + 0.985863i \(0.446414\pi\)
\(468\) 7.68269e18 0.0722220
\(469\) 6.40090e18 0.0592168
\(470\) −1.62301e19 −0.147770
\(471\) −4.22726e19 −0.378793
\(472\) 1.93780e19 0.170901
\(473\) 2.81807e19 0.244621
\(474\) −5.12069e19 −0.437513
\(475\) −3.09373e19 −0.260184
\(476\) 2.03901e18 0.0168798
\(477\) 6.78947e19 0.553284
\(478\) −8.62852e19 −0.692192
\(479\) 1.32716e19 0.104811 0.0524053 0.998626i \(-0.483311\pi\)
0.0524053 + 0.998626i \(0.483311\pi\)
\(480\) 3.49503e18 0.0271732
\(481\) −2.75058e19 −0.210541
\(482\) 4.05681e19 0.305725
\(483\) −1.45766e18 −0.0108156
\(484\) 1.04977e19 0.0766927
\(485\) 1.84781e19 0.132922
\(486\) −8.72027e19 −0.617672
\(487\) −2.27726e20 −1.58835 −0.794173 0.607692i \(-0.792096\pi\)
−0.794173 + 0.607692i \(0.792096\pi\)
\(488\) 1.98036e18 0.0136018
\(489\) −3.15587e19 −0.213453
\(490\) 4.73933e19 0.315680
\(491\) 2.06518e19 0.135471 0.0677356 0.997703i \(-0.478423\pi\)
0.0677356 + 0.997703i \(0.478423\pi\)
\(492\) 1.64832e19 0.106489
\(493\) −7.93385e19 −0.504815
\(494\) 2.40429e19 0.150673
\(495\) −6.86183e19 −0.423546
\(496\) 6.27409e19 0.381450
\(497\) 7.48639e18 0.0448332
\(498\) −7.19283e18 −0.0424306
\(499\) 2.08538e20 1.21180 0.605901 0.795540i \(-0.292813\pi\)
0.605901 + 0.795540i \(0.292813\pi\)
\(500\) −7.81250e18 −0.0447214
\(501\) −4.24952e18 −0.0239638
\(502\) −1.98354e19 −0.110196
\(503\) 1.61975e20 0.886516 0.443258 0.896394i \(-0.353822\pi\)
0.443258 + 0.896394i \(0.353822\pi\)
\(504\) 2.40704e18 0.0129794
\(505\) 1.09096e20 0.579594
\(506\) 1.09660e20 0.574012
\(507\) 6.48561e19 0.334496
\(508\) 9.07202e19 0.461027
\(509\) 1.84945e20 0.926102 0.463051 0.886332i \(-0.346755\pi\)
0.463051 + 0.886332i \(0.346755\pi\)
\(510\) −1.78638e19 −0.0881450
\(511\) 1.61849e19 0.0786960
\(512\) 9.22337e18 0.0441942
\(513\) −1.78204e20 −0.841469
\(514\) 2.88324e20 1.34171
\(515\) 1.14755e20 0.526279
\(516\) 8.66066e18 0.0391452
\(517\) −1.12656e20 −0.501851
\(518\) −8.61777e18 −0.0378374
\(519\) 1.24237e20 0.537647
\(520\) 6.07149e18 0.0258983
\(521\) −1.19106e20 −0.500785 −0.250392 0.968144i \(-0.580560\pi\)
−0.250392 + 0.968144i \(0.580560\pi\)
\(522\) −9.36588e19 −0.388168
\(523\) −2.07240e20 −0.846663 −0.423331 0.905975i \(-0.639139\pi\)
−0.423331 + 0.905975i \(0.639139\pi\)
\(524\) 1.28302e20 0.516714
\(525\) 7.20820e17 0.00286175
\(526\) 1.29185e20 0.505612
\(527\) −3.20681e20 −1.23735
\(528\) 2.42596e19 0.0922845
\(529\) −1.14295e20 −0.428656
\(530\) 5.36560e19 0.198403
\(531\) −1.16922e20 −0.426273
\(532\) 7.53282e18 0.0270782
\(533\) 2.86343e19 0.101492
\(534\) −1.13606e20 −0.397046
\(535\) −9.37190e19 −0.322979
\(536\) −1.47991e20 −0.502920
\(537\) 1.40891e20 0.472146
\(538\) −1.19017e20 −0.393315
\(539\) 3.28965e20 1.07210
\(540\) −4.50015e19 −0.144635
\(541\) −4.26438e20 −1.35169 −0.675844 0.737045i \(-0.736221\pi\)
−0.675844 + 0.737045i \(0.736221\pi\)
\(542\) −8.33761e18 −0.0260644
\(543\) −2.89023e19 −0.0891114
\(544\) −4.71425e19 −0.143358
\(545\) −2.06183e19 −0.0618413
\(546\) −5.60187e17 −0.00165725
\(547\) −1.46134e20 −0.426427 −0.213214 0.977006i \(-0.568393\pi\)
−0.213214 + 0.977006i \(0.568393\pi\)
\(548\) −3.09447e20 −0.890702
\(549\) −1.19490e19 −0.0339265
\(550\) −5.42278e19 −0.151881
\(551\) −2.93105e20 −0.809815
\(552\) 3.37015e19 0.0918556
\(553\) −2.78704e19 −0.0749386
\(554\) −1.11901e19 −0.0296832
\(555\) 7.55005e19 0.197584
\(556\) 9.97603e19 0.257571
\(557\) 3.01056e20 0.766891 0.383445 0.923564i \(-0.374737\pi\)
0.383445 + 0.923564i \(0.374737\pi\)
\(558\) −3.78563e20 −0.951441
\(559\) 1.50451e19 0.0373084
\(560\) 1.90224e18 0.00465431
\(561\) −1.23996e20 −0.299354
\(562\) 1.41548e20 0.337196
\(563\) 5.43653e20 1.27793 0.638967 0.769234i \(-0.279362\pi\)
0.638967 + 0.769234i \(0.279362\pi\)
\(564\) −3.46220e19 −0.0803080
\(565\) 3.65103e20 0.835704
\(566\) 2.75404e20 0.622081
\(567\) −1.23171e19 −0.0274560
\(568\) −1.73088e20 −0.380762
\(569\) 2.77606e19 0.0602681 0.0301340 0.999546i \(-0.490407\pi\)
0.0301340 + 0.999546i \(0.490407\pi\)
\(570\) −6.59953e19 −0.141401
\(571\) 7.86073e20 1.66223 0.831116 0.556098i \(-0.187702\pi\)
0.831116 + 0.556098i \(0.187702\pi\)
\(572\) 4.21432e19 0.0879544
\(573\) 4.45791e19 0.0918272
\(574\) 8.97133e18 0.0182397
\(575\) −7.53334e19 −0.151175
\(576\) −5.56516e19 −0.110232
\(577\) −9.30552e19 −0.181937 −0.0909687 0.995854i \(-0.528996\pi\)
−0.0909687 + 0.995854i \(0.528996\pi\)
\(578\) −1.25435e20 −0.242081
\(579\) 2.95725e20 0.563376
\(580\) −7.40169e19 −0.139194
\(581\) −3.91485e18 −0.00726765
\(582\) 3.94175e19 0.0722382
\(583\) 3.72435e20 0.673808
\(584\) −3.74200e20 −0.668355
\(585\) −3.66339e19 −0.0645973
\(586\) 7.06985e20 1.23077
\(587\) 9.75604e20 1.67682 0.838412 0.545036i \(-0.183484\pi\)
0.838412 + 0.545036i \(0.183484\pi\)
\(588\) 1.01099e20 0.171561
\(589\) −1.18471e21 −1.98494
\(590\) −9.24016e19 −0.152858
\(591\) −3.43559e19 −0.0561171
\(592\) 1.99246e20 0.321348
\(593\) 1.11954e20 0.178292 0.0891459 0.996019i \(-0.471586\pi\)
0.0891459 + 0.996019i \(0.471586\pi\)
\(594\) −3.12363e20 −0.491203
\(595\) −9.72275e18 −0.0150977
\(596\) −4.87993e20 −0.748285
\(597\) −2.59740e20 −0.393307
\(598\) 5.85455e19 0.0875456
\(599\) −7.49099e20 −1.10621 −0.553106 0.833111i \(-0.686558\pi\)
−0.553106 + 0.833111i \(0.686558\pi\)
\(600\) −1.66656e19 −0.0243045
\(601\) −1.59205e20 −0.229297 −0.114649 0.993406i \(-0.536574\pi\)
−0.114649 + 0.993406i \(0.536574\pi\)
\(602\) 4.71374e18 0.00670490
\(603\) 8.92941e20 1.25442
\(604\) 5.63199e20 0.781421
\(605\) −5.00569e19 −0.0685960
\(606\) 2.32723e20 0.314989
\(607\) −8.11583e20 −1.08497 −0.542485 0.840065i \(-0.682517\pi\)
−0.542485 + 0.840065i \(0.682517\pi\)
\(608\) −1.74161e20 −0.229972
\(609\) 6.82917e18 0.00890714
\(610\) −9.44308e18 −0.0121658
\(611\) −6.01447e19 −0.0765399
\(612\) 2.84446e20 0.357573
\(613\) 5.55939e19 0.0690357 0.0345179 0.999404i \(-0.489010\pi\)
0.0345179 + 0.999404i \(0.489010\pi\)
\(614\) 1.26411e20 0.155068
\(615\) −7.85981e19 −0.0952464
\(616\) 1.32038e19 0.0158068
\(617\) −6.97947e19 −0.0825437 −0.0412718 0.999148i \(-0.513141\pi\)
−0.0412718 + 0.999148i \(0.513141\pi\)
\(618\) 2.44794e20 0.286014
\(619\) −6.03520e20 −0.696646 −0.348323 0.937375i \(-0.613249\pi\)
−0.348323 + 0.937375i \(0.613249\pi\)
\(620\) −2.99172e20 −0.341180
\(621\) −4.33935e20 −0.488920
\(622\) −3.96344e20 −0.441210
\(623\) −6.18323e19 −0.0680072
\(624\) 1.29517e19 0.0140748
\(625\) 3.72529e19 0.0400000
\(626\) 1.52775e20 0.162086
\(627\) −4.58084e20 −0.480218
\(628\) 5.31946e20 0.551024
\(629\) −1.01838e21 −1.04239
\(630\) −1.14777e19 −0.0116091
\(631\) −3.64878e20 −0.364693 −0.182346 0.983234i \(-0.558369\pi\)
−0.182346 + 0.983234i \(0.558369\pi\)
\(632\) 6.44374e20 0.636443
\(633\) −8.40576e19 −0.0820445
\(634\) −3.45739e20 −0.333487
\(635\) −4.32587e20 −0.412355
\(636\) 1.14459e20 0.107825
\(637\) 1.75628e20 0.163511
\(638\) −5.13764e20 −0.472725
\(639\) 1.04437e21 0.949725
\(640\) −4.39805e19 −0.0395285
\(641\) 2.06269e20 0.183231 0.0916156 0.995794i \(-0.470797\pi\)
0.0916156 + 0.995794i \(0.470797\pi\)
\(642\) −1.99921e20 −0.175528
\(643\) 2.11836e21 1.83830 0.919152 0.393903i \(-0.128875\pi\)
0.919152 + 0.393903i \(0.128875\pi\)
\(644\) 1.83427e19 0.0157333
\(645\) −4.12972e19 −0.0350125
\(646\) 8.90173e20 0.745986
\(647\) −2.29485e21 −1.90096 −0.950480 0.310787i \(-0.899407\pi\)
−0.950480 + 0.310787i \(0.899407\pi\)
\(648\) 2.84776e20 0.233180
\(649\) −6.41375e20 −0.519130
\(650\) −2.89511e19 −0.0231641
\(651\) 2.76031e19 0.0218323
\(652\) 3.97125e20 0.310506
\(653\) −5.84612e20 −0.451876 −0.225938 0.974142i \(-0.572545\pi\)
−0.225938 + 0.974142i \(0.572545\pi\)
\(654\) −4.39829e19 −0.0336086
\(655\) −6.11793e20 −0.462163
\(656\) −2.07420e20 −0.154907
\(657\) 2.25783e21 1.66706
\(658\) −1.88438e19 −0.0137554
\(659\) 1.79922e21 1.29850 0.649252 0.760573i \(-0.275082\pi\)
0.649252 + 0.760573i \(0.275082\pi\)
\(660\) −1.15679e20 −0.0825418
\(661\) 1.48620e21 1.04849 0.524247 0.851566i \(-0.324347\pi\)
0.524247 + 0.851566i \(0.324347\pi\)
\(662\) −1.15835e20 −0.0807985
\(663\) −6.61987e19 −0.0456560
\(664\) 9.05125e19 0.0617231
\(665\) −3.59193e19 −0.0242195
\(666\) −1.20220e21 −0.801530
\(667\) −7.13722e20 −0.470528
\(668\) 5.34747e19 0.0348598
\(669\) 8.74905e20 0.563982
\(670\) 7.05676e20 0.449826
\(671\) −6.55460e19 −0.0413168
\(672\) 4.05786e18 0.00252946
\(673\) −1.01586e21 −0.626214 −0.313107 0.949718i \(-0.601370\pi\)
−0.313107 + 0.949718i \(0.601370\pi\)
\(674\) −8.75700e19 −0.0533834
\(675\) 2.14584e20 0.129366
\(676\) −8.16130e20 −0.486586
\(677\) 2.64661e21 1.56054 0.780269 0.625445i \(-0.215082\pi\)
0.780269 + 0.625445i \(0.215082\pi\)
\(678\) 7.78837e20 0.454176
\(679\) 2.14538e19 0.0123732
\(680\) 2.24793e20 0.128223
\(681\) 7.04323e20 0.397346
\(682\) −2.07660e21 −1.15870
\(683\) −1.97807e21 −1.09166 −0.545830 0.837896i \(-0.683786\pi\)
−0.545830 + 0.837896i \(0.683786\pi\)
\(684\) 1.05085e21 0.573612
\(685\) 1.47556e21 0.796668
\(686\) 1.10146e20 0.0588219
\(687\) −9.79517e20 −0.517412
\(688\) −1.08983e20 −0.0569438
\(689\) 1.98835e20 0.102766
\(690\) −1.60701e20 −0.0821581
\(691\) 2.52634e21 1.27763 0.638817 0.769359i \(-0.279424\pi\)
0.638817 + 0.769359i \(0.279424\pi\)
\(692\) −1.56337e21 −0.782106
\(693\) −7.96684e19 −0.0394264
\(694\) −1.51585e21 −0.742097
\(695\) −4.75694e20 −0.230378
\(696\) −1.57893e20 −0.0756472
\(697\) 1.06017e21 0.502491
\(698\) −5.98867e20 −0.280811
\(699\) 9.95744e20 0.461922
\(700\) −9.07060e18 −0.00416295
\(701\) −2.18514e21 −0.992190 −0.496095 0.868268i \(-0.665233\pi\)
−0.496095 + 0.868268i \(0.665233\pi\)
\(702\) −1.66764e20 −0.0749159
\(703\) −3.76228e21 −1.67219
\(704\) −3.05276e20 −0.134245
\(705\) 1.65091e20 0.0718297
\(706\) −7.95611e20 −0.342504
\(707\) 1.26664e20 0.0539522
\(708\) −1.97111e20 −0.0830732
\(709\) −2.66228e19 −0.0111021 −0.00555107 0.999985i \(-0.501767\pi\)
−0.00555107 + 0.999985i \(0.501767\pi\)
\(710\) 8.25347e20 0.340564
\(711\) −3.88799e21 −1.58746
\(712\) 1.42958e21 0.577576
\(713\) −2.88482e21 −1.15331
\(714\) −2.07405e19 −0.00820509
\(715\) −2.00955e20 −0.0786688
\(716\) −1.77293e21 −0.686823
\(717\) 8.77682e20 0.336468
\(718\) 1.11030e21 0.421218
\(719\) 9.86146e20 0.370233 0.185116 0.982717i \(-0.440734\pi\)
0.185116 + 0.982717i \(0.440734\pi\)
\(720\) 2.65367e20 0.0985948
\(721\) 1.33234e20 0.0489894
\(722\) 1.34543e21 0.489591
\(723\) −4.12653e20 −0.148610
\(724\) 3.63698e20 0.129629
\(725\) 3.52940e20 0.124499
\(726\) −1.06781e20 −0.0372795
\(727\) −2.40230e21 −0.830078 −0.415039 0.909804i \(-0.636232\pi\)
−0.415039 + 0.909804i \(0.636232\pi\)
\(728\) 7.04923e18 0.00241077
\(729\) −1.06145e21 −0.359288
\(730\) 1.78432e21 0.597795
\(731\) 5.57036e20 0.184715
\(732\) −2.01440e19 −0.00661168
\(733\) 1.87874e21 0.610361 0.305181 0.952295i \(-0.401283\pi\)
0.305181 + 0.952295i \(0.401283\pi\)
\(734\) 3.32898e21 1.07051
\(735\) −4.82079e20 −0.153449
\(736\) −4.24090e20 −0.133621
\(737\) 4.89821e21 1.52768
\(738\) 1.25152e21 0.386381
\(739\) 5.37988e21 1.64414 0.822072 0.569384i \(-0.192818\pi\)
0.822072 + 0.569384i \(0.192818\pi\)
\(740\) −9.50077e20 −0.287423
\(741\) −2.44562e20 −0.0732406
\(742\) 6.22966e19 0.0184686
\(743\) −2.67434e21 −0.784876 −0.392438 0.919778i \(-0.628368\pi\)
−0.392438 + 0.919778i \(0.628368\pi\)
\(744\) −6.38192e20 −0.185419
\(745\) 2.32693e21 0.669286
\(746\) 4.15388e20 0.118280
\(747\) −5.46130e20 −0.153954
\(748\) 1.56033e21 0.435465
\(749\) −1.08811e20 −0.0300649
\(750\) 7.94678e19 0.0217386
\(751\) 4.11797e21 1.11528 0.557640 0.830083i \(-0.311707\pi\)
0.557640 + 0.830083i \(0.311707\pi\)
\(752\) 4.35674e20 0.116823
\(753\) 2.01764e20 0.0535649
\(754\) −2.74288e20 −0.0720977
\(755\) −2.68554e21 −0.698924
\(756\) −5.22484e19 −0.0134635
\(757\) −3.83231e21 −0.977781 −0.488890 0.872345i \(-0.662598\pi\)
−0.488890 + 0.872345i \(0.662598\pi\)
\(758\) 5.08512e21 1.28464
\(759\) −1.11545e21 −0.279022
\(760\) 8.30466e20 0.205693
\(761\) 3.19772e21 0.784253 0.392126 0.919911i \(-0.371740\pi\)
0.392126 + 0.919911i \(0.371740\pi\)
\(762\) −9.22794e20 −0.224100
\(763\) −2.39386e19 −0.00575658
\(764\) −5.60971e20 −0.133580
\(765\) −1.35635e21 −0.319823
\(766\) −1.77988e21 −0.415600
\(767\) −3.42417e20 −0.0791753
\(768\) −9.38190e19 −0.0214823
\(769\) −5.42196e21 −1.22944 −0.614722 0.788744i \(-0.710732\pi\)
−0.614722 + 0.788744i \(0.710732\pi\)
\(770\) −6.29605e19 −0.0141380
\(771\) −2.93280e21 −0.652191
\(772\) −3.72132e21 −0.819533
\(773\) −4.69088e21 −1.02308 −0.511538 0.859261i \(-0.670924\pi\)
−0.511538 + 0.859261i \(0.670924\pi\)
\(774\) 6.57579e20 0.142033
\(775\) 1.42656e21 0.305160
\(776\) −4.96019e20 −0.105084
\(777\) 8.76589e19 0.0183924
\(778\) −3.07865e21 −0.639754
\(779\) 3.91663e21 0.806087
\(780\) −6.17585e19 −0.0125889
\(781\) 5.72887e21 1.15661
\(782\) 2.16761e21 0.433441
\(783\) 2.03300e21 0.402647
\(784\) −1.27221e21 −0.249567
\(785\) −2.53652e21 −0.492851
\(786\) −1.30508e21 −0.251169
\(787\) −9.27131e21 −1.76738 −0.883691 0.468070i \(-0.844949\pi\)
−0.883691 + 0.468070i \(0.844949\pi\)
\(788\) 4.32325e20 0.0816325
\(789\) −1.31405e21 −0.245773
\(790\) −3.07261e21 −0.569252
\(791\) 4.23899e20 0.0777926
\(792\) 1.84196e21 0.334843
\(793\) −3.49937e19 −0.00630145
\(794\) 5.99996e19 0.0107027
\(795\) −5.45782e20 −0.0964419
\(796\) 3.26850e21 0.572137
\(797\) 8.35314e21 1.44848 0.724240 0.689548i \(-0.242191\pi\)
0.724240 + 0.689548i \(0.242191\pi\)
\(798\) −7.66229e19 −0.0131625
\(799\) −2.22682e21 −0.378951
\(800\) 2.09715e20 0.0353553
\(801\) −8.62575e21 −1.44063
\(802\) −5.87880e21 −0.972705
\(803\) 1.23853e22 2.03020
\(804\) 1.50535e21 0.244464
\(805\) −8.74649e19 −0.0140723
\(806\) −1.10865e21 −0.176719
\(807\) 1.21062e21 0.191187
\(808\) −2.92852e21 −0.458209
\(809\) −5.86882e21 −0.909781 −0.454890 0.890547i \(-0.650322\pi\)
−0.454890 + 0.890547i \(0.650322\pi\)
\(810\) −1.35792e21 −0.208562
\(811\) 9.20718e21 1.40110 0.700552 0.713602i \(-0.252937\pi\)
0.700552 + 0.713602i \(0.252937\pi\)
\(812\) −8.59364e19 −0.0129571
\(813\) 8.48091e19 0.0126696
\(814\) −6.59464e21 −0.976131
\(815\) −1.89364e21 −0.277725
\(816\) 4.79528e20 0.0696847
\(817\) 2.05789e21 0.296317
\(818\) −3.67451e20 −0.0524264
\(819\) −4.25333e19 −0.00601312
\(820\) 9.89056e20 0.138553
\(821\) 2.78078e21 0.386005 0.193003 0.981198i \(-0.438177\pi\)
0.193003 + 0.981198i \(0.438177\pi\)
\(822\) 3.14766e21 0.432961
\(823\) 1.08620e22 1.48051 0.740257 0.672324i \(-0.234704\pi\)
0.740257 + 0.672324i \(0.234704\pi\)
\(824\) −3.08042e21 −0.416060
\(825\) 5.51599e20 0.0738276
\(826\) −1.07282e20 −0.0142290
\(827\) −4.48033e21 −0.588869 −0.294435 0.955672i \(-0.595131\pi\)
−0.294435 + 0.955672i \(0.595131\pi\)
\(828\) 2.55885e21 0.333287
\(829\) −1.15562e22 −1.49161 −0.745807 0.666163i \(-0.767936\pi\)
−0.745807 + 0.666163i \(0.767936\pi\)
\(830\) −4.31597e20 −0.0552069
\(831\) 1.13824e20 0.0144287
\(832\) −1.62980e20 −0.0204744
\(833\) 6.50250e21 0.809548
\(834\) −1.01475e21 −0.125203
\(835\) −2.54987e20 −0.0311796
\(836\) 5.76440e21 0.698565
\(837\) 8.21727e21 0.986930
\(838\) −2.49831e21 −0.297383
\(839\) 1.16410e22 1.37333 0.686666 0.726973i \(-0.259074\pi\)
0.686666 + 0.726973i \(0.259074\pi\)
\(840\) −1.93494e19 −0.00226242
\(841\) −5.28537e21 −0.612500
\(842\) 1.00385e22 1.15300
\(843\) −1.43981e21 −0.163907
\(844\) 1.05776e21 0.119349
\(845\) 3.89161e21 0.435215
\(846\) −2.62875e21 −0.291388
\(847\) −5.81179e19 −0.00638535
\(848\) −1.44032e21 −0.156852
\(849\) −2.80138e21 −0.302387
\(850\) −1.07190e21 −0.114686
\(851\) −9.16129e21 −0.971594
\(852\) 1.76063e21 0.185085
\(853\) −8.05337e21 −0.839189 −0.419595 0.907712i \(-0.637828\pi\)
−0.419595 + 0.907712i \(0.637828\pi\)
\(854\) −1.09638e19 −0.00113247
\(855\) −5.01083e21 −0.513055
\(856\) 2.51575e21 0.255337
\(857\) 3.51278e21 0.353423 0.176712 0.984263i \(-0.443454\pi\)
0.176712 + 0.984263i \(0.443454\pi\)
\(858\) −4.28676e20 −0.0427538
\(859\) 4.79620e21 0.474186 0.237093 0.971487i \(-0.423805\pi\)
0.237093 + 0.971487i \(0.423805\pi\)
\(860\) 5.19673e20 0.0509321
\(861\) −9.12552e19 −0.00886613
\(862\) −6.70856e21 −0.646137
\(863\) 6.27772e21 0.599406 0.299703 0.954033i \(-0.403112\pi\)
0.299703 + 0.954033i \(0.403112\pi\)
\(864\) 1.20800e21 0.114344
\(865\) 7.45472e21 0.699537
\(866\) 8.60274e21 0.800300
\(867\) 1.27591e21 0.117673
\(868\) −3.47349e20 −0.0317591
\(869\) −2.13275e22 −1.93327
\(870\) 7.52891e20 0.0676609
\(871\) 2.61506e21 0.232994
\(872\) 5.53468e20 0.0488899
\(873\) 2.99286e21 0.262107
\(874\) 8.00792e21 0.695319
\(875\) 4.32520e19 0.00372345
\(876\) 3.80631e21 0.324881
\(877\) −4.69963e21 −0.397711 −0.198855 0.980029i \(-0.563722\pi\)
−0.198855 + 0.980029i \(0.563722\pi\)
\(878\) −8.21680e21 −0.689436
\(879\) −7.19136e21 −0.598266
\(880\) 1.45567e21 0.120072
\(881\) −1.89876e22 −1.55293 −0.776465 0.630161i \(-0.782989\pi\)
−0.776465 + 0.630161i \(0.782989\pi\)
\(882\) 7.67618e21 0.622487
\(883\) −5.43744e21 −0.437209 −0.218605 0.975814i \(-0.570151\pi\)
−0.218605 + 0.975814i \(0.570151\pi\)
\(884\) 8.33026e20 0.0664150
\(885\) 9.39897e20 0.0743029
\(886\) 1.03606e22 0.812140
\(887\) −1.43529e22 −1.11561 −0.557807 0.829971i \(-0.688357\pi\)
−0.557807 + 0.829971i \(0.688357\pi\)
\(888\) −2.02670e21 −0.156204
\(889\) −5.02250e20 −0.0383846
\(890\) −6.81678e21 −0.516600
\(891\) −9.42554e21 −0.708310
\(892\) −1.10096e22 −0.820415
\(893\) −8.22665e21 −0.607907
\(894\) 4.96381e21 0.363734
\(895\) 8.45401e21 0.614314
\(896\) −5.10629e19 −0.00367956
\(897\) −5.95517e20 −0.0425551
\(898\) −7.21204e21 −0.511076
\(899\) 1.35155e22 0.949804
\(900\) −1.26537e21 −0.0881858
\(901\) 7.36175e21 0.508798
\(902\) 6.86520e21 0.470548
\(903\) −4.79476e19 −0.00325918
\(904\) −9.80067e21 −0.660682
\(905\) −1.73425e21 −0.115944
\(906\) −5.72879e21 −0.379841
\(907\) −1.44404e22 −0.949564 −0.474782 0.880103i \(-0.657473\pi\)
−0.474782 + 0.880103i \(0.657473\pi\)
\(908\) −8.86301e21 −0.578012
\(909\) 1.76700e22 1.14290
\(910\) −3.36133e19 −0.00215626
\(911\) −6.00661e21 −0.382157 −0.191078 0.981575i \(-0.561198\pi\)
−0.191078 + 0.981575i \(0.561198\pi\)
\(912\) 1.77155e21 0.111787
\(913\) −2.99579e21 −0.187491
\(914\) 1.04021e22 0.645692
\(915\) 9.60538e19 0.00591366
\(916\) 1.23260e22 0.752671
\(917\) −7.10315e20 −0.0430210
\(918\) −6.17433e21 −0.370911
\(919\) 2.25875e22 1.34586 0.672932 0.739704i \(-0.265034\pi\)
0.672932 + 0.739704i \(0.265034\pi\)
\(920\) 2.02222e21 0.119514
\(921\) −1.28584e21 −0.0753770
\(922\) 1.44291e22 0.838989
\(923\) 3.05853e21 0.176400
\(924\) −1.34307e20 −0.00768351
\(925\) 4.53032e21 0.257079
\(926\) −2.30210e22 −1.29581
\(927\) 1.85865e22 1.03777
\(928\) 1.98688e21 0.110043
\(929\) 3.26962e22 1.79630 0.898151 0.439686i \(-0.144910\pi\)
0.898151 + 0.439686i \(0.144910\pi\)
\(930\) 3.04314e21 0.165844
\(931\) 2.40225e22 1.29866
\(932\) −1.25302e22 −0.671951
\(933\) 4.03157e21 0.214468
\(934\) 4.49019e21 0.236954
\(935\) −7.44021e21 −0.389492
\(936\) 9.83384e20 0.0510686
\(937\) 2.67083e21 0.137594 0.0687970 0.997631i \(-0.478084\pi\)
0.0687970 + 0.997631i \(0.478084\pi\)
\(938\) 8.19316e20 0.0418726
\(939\) −1.55401e21 −0.0787881
\(940\) −2.07745e21 −0.104489
\(941\) −2.61162e22 −1.30313 −0.651565 0.758593i \(-0.725887\pi\)
−0.651565 + 0.758593i \(0.725887\pi\)
\(942\) −5.41089e21 −0.267847
\(943\) 9.53715e21 0.468361
\(944\) 2.48039e21 0.120845
\(945\) 2.49140e20 0.0120422
\(946\) 3.60713e21 0.172973
\(947\) −1.03859e22 −0.494105 −0.247053 0.969002i \(-0.579462\pi\)
−0.247053 + 0.969002i \(0.579462\pi\)
\(948\) −6.55449e21 −0.309369
\(949\) 6.61225e21 0.309637
\(950\) −3.95997e21 −0.183978
\(951\) 3.51681e21 0.162104
\(952\) 2.60993e20 0.0119358
\(953\) 1.16131e20 0.00526929 0.00263464 0.999997i \(-0.499161\pi\)
0.00263464 + 0.999997i \(0.499161\pi\)
\(954\) 8.69052e21 0.391231
\(955\) 2.67492e21 0.119477
\(956\) −1.10445e22 −0.489454
\(957\) 5.22594e21 0.229787
\(958\) 1.69876e21 0.0741123
\(959\) 1.71318e21 0.0741589
\(960\) 4.47364e20 0.0192144
\(961\) 3.11635e22 1.32807
\(962\) −3.52074e21 −0.148875
\(963\) −1.51794e22 −0.636881
\(964\) 5.19271e21 0.216180
\(965\) 1.77446e22 0.733013
\(966\) −1.86580e20 −0.00764780
\(967\) −3.55619e22 −1.44639 −0.723197 0.690642i \(-0.757328\pi\)
−0.723197 + 0.690642i \(0.757328\pi\)
\(968\) 1.34371e21 0.0542299
\(969\) −9.05473e21 −0.362616
\(970\) 2.36520e21 0.0939897
\(971\) −4.08883e22 −1.61233 −0.806166 0.591690i \(-0.798461\pi\)
−0.806166 + 0.591690i \(0.798461\pi\)
\(972\) −1.11619e22 −0.436760
\(973\) −5.52298e20 −0.0214451
\(974\) −2.91489e22 −1.12313
\(975\) 2.94487e20 0.0112598
\(976\) 2.53486e20 0.00961789
\(977\) 1.47056e22 0.553700 0.276850 0.960913i \(-0.410709\pi\)
0.276850 + 0.960913i \(0.410709\pi\)
\(978\) −4.03951e21 −0.150934
\(979\) −4.73164e22 −1.75445
\(980\) 6.06635e21 0.223219
\(981\) −3.33949e21 −0.121945
\(982\) 2.64343e21 0.0957926
\(983\) −2.02443e22 −0.728034 −0.364017 0.931392i \(-0.618595\pi\)
−0.364017 + 0.931392i \(0.618595\pi\)
\(984\) 2.10985e21 0.0752989
\(985\) −2.06148e21 −0.0730144
\(986\) −1.01553e22 −0.356958
\(987\) 1.91676e20 0.00668636
\(988\) 3.07750e21 0.106542
\(989\) 5.01104e21 0.172169
\(990\) −8.78314e21 −0.299493
\(991\) 1.99356e22 0.674649 0.337324 0.941389i \(-0.390478\pi\)
0.337324 + 0.941389i \(0.390478\pi\)
\(992\) 8.03083e21 0.269726
\(993\) 1.17825e21 0.0392753
\(994\) 9.58259e20 0.0317019
\(995\) −1.55854e22 −0.511735
\(996\) −9.20682e20 −0.0300030
\(997\) −4.00115e21 −0.129411 −0.0647056 0.997904i \(-0.520611\pi\)
−0.0647056 + 0.997904i \(0.520611\pi\)
\(998\) 2.66929e22 0.856874
\(999\) 2.60955e22 0.831427
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 10.16.a.c.1.1 1
3.2 odd 2 90.16.a.d.1.1 1
4.3 odd 2 80.16.a.b.1.1 1
5.2 odd 4 50.16.b.b.49.2 2
5.3 odd 4 50.16.b.b.49.1 2
5.4 even 2 50.16.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.16.a.c.1.1 1 1.1 even 1 trivial
50.16.a.a.1.1 1 5.4 even 2
50.16.b.b.49.1 2 5.3 odd 4
50.16.b.b.49.2 2 5.2 odd 4
80.16.a.b.1.1 1 4.3 odd 2
90.16.a.d.1.1 1 3.2 odd 2