Properties

Label 10.24.a.c.1.1
Level $10$
Weight $24$
Character 10.1
Self dual yes
Analytic conductor $33.520$
Analytic rank $1$
Dimension $2$
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] = [10,24,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, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 24 \)
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: \(33.5204037345\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1492261}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 373065 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(611.291\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2048.00 q^{2} -339122. q^{3} +4.19430e6 q^{4} -4.88281e7 q^{5} -6.94521e8 q^{6} +5.67214e9 q^{7} +8.58993e9 q^{8} +2.08602e10 q^{9} -1.00000e11 q^{10} -7.29056e10 q^{11} -1.42238e12 q^{12} +7.50969e12 q^{13} +1.16165e13 q^{14} +1.65587e13 q^{15} +1.75922e13 q^{16} -1.22795e14 q^{17} +4.27218e13 q^{18} -6.14006e14 q^{19} -2.04800e14 q^{20} -1.92354e15 q^{21} -1.49311e14 q^{22} -3.69414e15 q^{23} -2.91303e15 q^{24} +2.38419e15 q^{25} +1.53798e16 q^{26} +2.48518e16 q^{27} +2.37907e16 q^{28} -1.00053e17 q^{29} +3.39122e16 q^{30} -1.10951e17 q^{31} +3.60288e16 q^{32} +2.47238e16 q^{33} -2.51484e17 q^{34} -2.76960e17 q^{35} +8.74941e16 q^{36} -5.06294e17 q^{37} -1.25749e18 q^{38} -2.54670e18 q^{39} -4.19430e17 q^{40} +5.37240e17 q^{41} -3.93942e18 q^{42} -3.30420e18 q^{43} -3.05788e17 q^{44} -1.01857e18 q^{45} -7.56560e18 q^{46} -2.83871e19 q^{47} -5.96589e18 q^{48} +4.80439e18 q^{49} +4.88281e18 q^{50} +4.16423e19 q^{51} +3.14979e19 q^{52} -4.33879e19 q^{53} +5.08965e19 q^{54} +3.55984e18 q^{55} +4.87233e19 q^{56} +2.08223e20 q^{57} -2.04909e20 q^{58} -1.82553e20 q^{59} +6.94521e19 q^{60} +5.31488e20 q^{61} -2.27227e20 q^{62} +1.18322e20 q^{63} +7.37870e19 q^{64} -3.66684e20 q^{65} +5.06344e19 q^{66} +1.46349e21 q^{67} -5.15038e20 q^{68} +1.25276e21 q^{69} -5.67214e20 q^{70} +3.40628e21 q^{71} +1.79188e20 q^{72} -1.21932e21 q^{73} -1.03689e21 q^{74} -8.08529e20 q^{75} -2.57533e21 q^{76} -4.13530e20 q^{77} -5.21564e21 q^{78} -3.86872e21 q^{79} -8.58993e20 q^{80} -1.03916e22 q^{81} +1.10027e21 q^{82} -1.91453e22 q^{83} -8.06793e21 q^{84} +5.99584e21 q^{85} -6.76701e21 q^{86} +3.39302e22 q^{87} -6.26254e20 q^{88} +1.38166e22 q^{89} -2.08602e21 q^{90} +4.25960e22 q^{91} -1.54943e22 q^{92} +3.76258e22 q^{93} -5.81367e22 q^{94} +2.99808e22 q^{95} -1.22181e22 q^{96} +3.39080e21 q^{97} +9.83938e21 q^{98} -1.52083e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4096 q^{2} - 91884 q^{3} + 8388608 q^{4} - 97656250 q^{5} - 188178432 q^{6} + 2146058908 q^{7} + 17179869184 q^{8} - 12156555726 q^{9} - 200000000000 q^{10} - 692289571776 q^{11} - 385389428736 q^{12}+ \cdots + 18\!\cdots\!88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2048.00 0.707107
\(3\) −339122. −1.10525 −0.552626 0.833430i \(-0.686374\pi\)
−0.552626 + 0.833430i \(0.686374\pi\)
\(4\) 4.19430e6 0.500000
\(5\) −4.88281e7 −0.447214
\(6\) −6.94521e8 −0.781530
\(7\) 5.67214e9 1.08422 0.542112 0.840306i \(-0.317625\pi\)
0.542112 + 0.840306i \(0.317625\pi\)
\(8\) 8.58993e9 0.353553
\(9\) 2.08602e10 0.221580
\(10\) −1.00000e11 −0.316228
\(11\) −7.29056e10 −0.0770451 −0.0385225 0.999258i \(-0.512265\pi\)
−0.0385225 + 0.999258i \(0.512265\pi\)
\(12\) −1.42238e12 −0.552626
\(13\) 7.50969e12 1.16218 0.581090 0.813839i \(-0.302626\pi\)
0.581090 + 0.813839i \(0.302626\pi\)
\(14\) 1.16165e13 0.766663
\(15\) 1.65587e13 0.494283
\(16\) 1.75922e13 0.250000
\(17\) −1.22795e14 −0.868995 −0.434497 0.900673i \(-0.643074\pi\)
−0.434497 + 0.900673i \(0.643074\pi\)
\(18\) 4.27218e13 0.156681
\(19\) −6.14006e14 −1.20922 −0.604612 0.796520i \(-0.706672\pi\)
−0.604612 + 0.796520i \(0.706672\pi\)
\(20\) −2.04800e14 −0.223607
\(21\) −1.92354e15 −1.19834
\(22\) −1.49311e14 −0.0544791
\(23\) −3.69414e15 −0.808431 −0.404216 0.914664i \(-0.632455\pi\)
−0.404216 + 0.914664i \(0.632455\pi\)
\(24\) −2.91303e15 −0.390765
\(25\) 2.38419e15 0.200000
\(26\) 1.53798e16 0.821785
\(27\) 2.48518e16 0.860350
\(28\) 2.37907e16 0.542112
\(29\) −1.00053e17 −1.52284 −0.761419 0.648260i \(-0.775497\pi\)
−0.761419 + 0.648260i \(0.775497\pi\)
\(30\) 3.39122e16 0.349511
\(31\) −1.10951e17 −0.784280 −0.392140 0.919905i \(-0.628265\pi\)
−0.392140 + 0.919905i \(0.628265\pi\)
\(32\) 3.60288e16 0.176777
\(33\) 2.47238e16 0.0851541
\(34\) −2.51484e17 −0.614472
\(35\) −2.76960e17 −0.484880
\(36\) 8.74941e16 0.110790
\(37\) −5.06294e17 −0.467824 −0.233912 0.972258i \(-0.575153\pi\)
−0.233912 + 0.972258i \(0.575153\pi\)
\(38\) −1.25749e18 −0.855050
\(39\) −2.54670e18 −1.28450
\(40\) −4.19430e17 −0.158114
\(41\) 5.37240e17 0.152459 0.0762297 0.997090i \(-0.475712\pi\)
0.0762297 + 0.997090i \(0.475712\pi\)
\(42\) −3.93942e18 −0.847355
\(43\) −3.30420e18 −0.542225 −0.271112 0.962548i \(-0.587392\pi\)
−0.271112 + 0.962548i \(0.587392\pi\)
\(44\) −3.05788e17 −0.0385225
\(45\) −1.01857e18 −0.0990935
\(46\) −7.56560e18 −0.571647
\(47\) −2.83871e19 −1.67492 −0.837462 0.546495i \(-0.815962\pi\)
−0.837462 + 0.546495i \(0.815962\pi\)
\(48\) −5.96589e18 −0.276313
\(49\) 4.80439e18 0.175543
\(50\) 4.88281e18 0.141421
\(51\) 4.16423e19 0.960457
\(52\) 3.14979e19 0.581090
\(53\) −4.33879e19 −0.642978 −0.321489 0.946913i \(-0.604183\pi\)
−0.321489 + 0.946913i \(0.604183\pi\)
\(54\) 5.08965e19 0.608359
\(55\) 3.55984e18 0.0344556
\(56\) 4.87233e19 0.383331
\(57\) 2.08223e20 1.33650
\(58\) −2.04909e20 −1.07681
\(59\) −1.82553e20 −0.788118 −0.394059 0.919085i \(-0.628930\pi\)
−0.394059 + 0.919085i \(0.628930\pi\)
\(60\) 6.94521e19 0.247142
\(61\) 5.31488e20 1.56387 0.781934 0.623361i \(-0.214233\pi\)
0.781934 + 0.623361i \(0.214233\pi\)
\(62\) −2.27227e20 −0.554570
\(63\) 1.18322e20 0.240242
\(64\) 7.37870e19 0.125000
\(65\) −3.66684e20 −0.519743
\(66\) 5.06344e19 0.0602131
\(67\) 1.46349e21 1.46396 0.731982 0.681323i \(-0.238595\pi\)
0.731982 + 0.681323i \(0.238595\pi\)
\(68\) −5.15038e20 −0.434497
\(69\) 1.25276e21 0.893520
\(70\) −5.67214e20 −0.342862
\(71\) 3.40628e21 1.74908 0.874538 0.484957i \(-0.161165\pi\)
0.874538 + 0.484957i \(0.161165\pi\)
\(72\) 1.79188e20 0.0783403
\(73\) −1.21932e21 −0.454889 −0.227444 0.973791i \(-0.573037\pi\)
−0.227444 + 0.973791i \(0.573037\pi\)
\(74\) −1.03689e21 −0.330802
\(75\) −8.08529e20 −0.221050
\(76\) −2.57533e21 −0.604612
\(77\) −4.13530e20 −0.0835342
\(78\) −5.21564e21 −0.908279
\(79\) −3.86872e21 −0.581909 −0.290955 0.956737i \(-0.593973\pi\)
−0.290955 + 0.956737i \(0.593973\pi\)
\(80\) −8.58993e20 −0.111803
\(81\) −1.03916e22 −1.17248
\(82\) 1.10027e21 0.107805
\(83\) −1.91453e22 −1.63179 −0.815893 0.578203i \(-0.803754\pi\)
−0.815893 + 0.578203i \(0.803754\pi\)
\(84\) −8.06793e21 −0.599170
\(85\) 5.99584e21 0.388626
\(86\) −6.76701e21 −0.383411
\(87\) 3.39302e22 1.68312
\(88\) −6.26254e20 −0.0272395
\(89\) 1.38166e22 0.527737 0.263868 0.964559i \(-0.415002\pi\)
0.263868 + 0.964559i \(0.415002\pi\)
\(90\) −2.08602e21 −0.0700697
\(91\) 4.25960e22 1.26006
\(92\) −1.54943e22 −0.404216
\(93\) 3.76258e22 0.866827
\(94\) −5.81367e22 −1.18435
\(95\) 2.99808e22 0.540781
\(96\) −1.22181e22 −0.195383
\(97\) 3.39080e21 0.0481312 0.0240656 0.999710i \(-0.492339\pi\)
0.0240656 + 0.999710i \(0.492339\pi\)
\(98\) 9.83938e21 0.124128
\(99\) −1.52083e21 −0.0170716
\(100\) 1.00000e22 0.100000
\(101\) −6.91624e22 −0.616842 −0.308421 0.951250i \(-0.599801\pi\)
−0.308421 + 0.951250i \(0.599801\pi\)
\(102\) 8.52835e22 0.679146
\(103\) 2.42869e23 1.72880 0.864398 0.502809i \(-0.167700\pi\)
0.864398 + 0.502809i \(0.167700\pi\)
\(104\) 6.45077e22 0.410893
\(105\) 9.39230e22 0.535914
\(106\) −8.88584e22 −0.454654
\(107\) −3.44606e22 −0.158274 −0.0791371 0.996864i \(-0.525216\pi\)
−0.0791371 + 0.996864i \(0.525216\pi\)
\(108\) 1.04236e23 0.430175
\(109\) −4.80102e23 −1.78209 −0.891043 0.453918i \(-0.850026\pi\)
−0.891043 + 0.453918i \(0.850026\pi\)
\(110\) 7.29056e21 0.0243638
\(111\) 1.71695e23 0.517063
\(112\) 9.97853e22 0.271056
\(113\) −6.76991e23 −1.66028 −0.830139 0.557556i \(-0.811739\pi\)
−0.830139 + 0.557556i \(0.811739\pi\)
\(114\) 4.26440e23 0.945045
\(115\) 1.80378e23 0.361541
\(116\) −4.19653e23 −0.761419
\(117\) 1.56654e23 0.257516
\(118\) −3.73869e23 −0.557284
\(119\) −6.96509e23 −0.942185
\(120\) 1.42238e23 0.174756
\(121\) −8.90115e23 −0.994064
\(122\) 1.08849e24 1.10582
\(123\) −1.82190e23 −0.168506
\(124\) −4.65362e23 −0.392140
\(125\) −1.16415e23 −0.0894427
\(126\) 2.42324e23 0.169877
\(127\) −6.00234e23 −0.384218 −0.192109 0.981374i \(-0.561533\pi\)
−0.192109 + 0.981374i \(0.561533\pi\)
\(128\) 1.51116e23 0.0883883
\(129\) 1.12053e24 0.599294
\(130\) −7.50969e23 −0.367514
\(131\) 1.78064e24 0.797914 0.398957 0.916970i \(-0.369372\pi\)
0.398957 + 0.916970i \(0.369372\pi\)
\(132\) 1.03699e23 0.0425771
\(133\) −3.48273e24 −1.31107
\(134\) 2.99724e24 1.03518
\(135\) −1.21347e24 −0.384760
\(136\) −1.05480e24 −0.307236
\(137\) −3.22877e24 −0.864471 −0.432236 0.901761i \(-0.642275\pi\)
−0.432236 + 0.901761i \(0.642275\pi\)
\(138\) 2.56566e24 0.631814
\(139\) −3.01015e23 −0.0682210 −0.0341105 0.999418i \(-0.510860\pi\)
−0.0341105 + 0.999418i \(0.510860\pi\)
\(140\) −1.16165e24 −0.242440
\(141\) 9.62667e24 1.85121
\(142\) 6.97605e24 1.23678
\(143\) −5.47498e23 −0.0895402
\(144\) 3.66977e23 0.0553950
\(145\) 4.88541e24 0.681034
\(146\) −2.49717e24 −0.321655
\(147\) −1.62927e24 −0.194019
\(148\) −2.12355e24 −0.233912
\(149\) 1.49284e25 1.52185 0.760925 0.648840i \(-0.224746\pi\)
0.760925 + 0.648840i \(0.224746\pi\)
\(150\) −1.65587e24 −0.156306
\(151\) 2.25537e24 0.197235 0.0986173 0.995125i \(-0.468558\pi\)
0.0986173 + 0.995125i \(0.468558\pi\)
\(152\) −5.27427e24 −0.427525
\(153\) −2.56153e24 −0.192552
\(154\) −8.46910e23 −0.0590676
\(155\) 5.41752e24 0.350741
\(156\) −1.06816e25 −0.642250
\(157\) −2.00139e25 −1.11812 −0.559058 0.829129i \(-0.688837\pi\)
−0.559058 + 0.829129i \(0.688837\pi\)
\(158\) −7.92313e24 −0.411472
\(159\) 1.47138e25 0.710652
\(160\) −1.75922e24 −0.0790569
\(161\) −2.09537e25 −0.876521
\(162\) −2.12821e25 −0.829070
\(163\) −5.04020e24 −0.182932 −0.0914661 0.995808i \(-0.529155\pi\)
−0.0914661 + 0.995808i \(0.529155\pi\)
\(164\) 2.25335e24 0.0762297
\(165\) −1.20722e24 −0.0380821
\(166\) −3.92095e25 −1.15385
\(167\) −4.08368e25 −1.12153 −0.560767 0.827974i \(-0.689494\pi\)
−0.560767 + 0.827974i \(0.689494\pi\)
\(168\) −1.65231e25 −0.423677
\(169\) 1.46415e25 0.350663
\(170\) 1.22795e25 0.274800
\(171\) −1.28083e25 −0.267939
\(172\) −1.38588e25 −0.271112
\(173\) 8.79177e25 1.60896 0.804482 0.593978i \(-0.202443\pi\)
0.804482 + 0.593978i \(0.202443\pi\)
\(174\) 6.94890e25 1.19014
\(175\) 1.35234e25 0.216845
\(176\) −1.28257e24 −0.0192613
\(177\) 6.19077e25 0.871069
\(178\) 2.82965e25 0.373166
\(179\) 6.44672e25 0.797129 0.398565 0.917140i \(-0.369508\pi\)
0.398565 + 0.917140i \(0.369508\pi\)
\(180\) −4.27218e24 −0.0495468
\(181\) −4.53664e25 −0.493663 −0.246832 0.969058i \(-0.579389\pi\)
−0.246832 + 0.969058i \(0.579389\pi\)
\(182\) 8.72366e25 0.891000
\(183\) −1.80239e26 −1.72847
\(184\) −3.17324e25 −0.285824
\(185\) 2.47214e25 0.209217
\(186\) 7.70577e25 0.612939
\(187\) 8.95242e24 0.0669518
\(188\) −1.19064e26 −0.837462
\(189\) 1.40963e26 0.932812
\(190\) 6.14006e25 0.382390
\(191\) 4.01824e25 0.235587 0.117794 0.993038i \(-0.462418\pi\)
0.117794 + 0.993038i \(0.462418\pi\)
\(192\) −2.50228e25 −0.138156
\(193\) −3.37563e26 −1.75568 −0.877841 0.478952i \(-0.841017\pi\)
−0.877841 + 0.478952i \(0.841017\pi\)
\(194\) 6.94435e24 0.0340339
\(195\) 1.24350e26 0.574446
\(196\) 2.01511e25 0.0877714
\(197\) 3.28937e26 1.35130 0.675649 0.737223i \(-0.263863\pi\)
0.675649 + 0.737223i \(0.263863\pi\)
\(198\) −3.11465e24 −0.0120715
\(199\) 3.82254e26 1.39811 0.699055 0.715068i \(-0.253604\pi\)
0.699055 + 0.715068i \(0.253604\pi\)
\(200\) 2.04800e25 0.0707107
\(201\) −4.96302e26 −1.61805
\(202\) −1.41645e26 −0.436173
\(203\) −5.67515e26 −1.65110
\(204\) 1.74661e26 0.480229
\(205\) −2.62324e25 −0.0681819
\(206\) 4.97395e26 1.22244
\(207\) −7.70606e25 −0.179132
\(208\) 1.32112e26 0.290545
\(209\) 4.47645e25 0.0931647
\(210\) 1.92354e26 0.378948
\(211\) −1.52630e26 −0.284703 −0.142352 0.989816i \(-0.545466\pi\)
−0.142352 + 0.989816i \(0.545466\pi\)
\(212\) −1.81982e26 −0.321489
\(213\) −1.15514e27 −1.93317
\(214\) −7.05754e25 −0.111917
\(215\) 1.61338e26 0.242490
\(216\) 2.13476e26 0.304180
\(217\) −6.29329e26 −0.850336
\(218\) −9.83249e26 −1.26013
\(219\) 4.13498e26 0.502766
\(220\) 1.49311e25 0.0172278
\(221\) −9.22150e26 −1.00993
\(222\) 3.51632e26 0.365619
\(223\) 7.09326e26 0.700390 0.350195 0.936677i \(-0.386115\pi\)
0.350195 + 0.936677i \(0.386115\pi\)
\(224\) 2.04360e26 0.191666
\(225\) 4.97347e25 0.0443160
\(226\) −1.38648e27 −1.17399
\(227\) 1.34868e27 1.08545 0.542727 0.839909i \(-0.317392\pi\)
0.542727 + 0.839909i \(0.317392\pi\)
\(228\) 8.73350e26 0.668248
\(229\) 1.71030e27 1.24441 0.622207 0.782853i \(-0.286236\pi\)
0.622207 + 0.782853i \(0.286236\pi\)
\(230\) 3.69414e26 0.255648
\(231\) 1.40237e26 0.0923262
\(232\) −8.59450e26 −0.538404
\(233\) 7.74363e26 0.461691 0.230846 0.972990i \(-0.425851\pi\)
0.230846 + 0.972990i \(0.425851\pi\)
\(234\) 3.20827e26 0.182091
\(235\) 1.38609e27 0.749049
\(236\) −7.65684e26 −0.394059
\(237\) 1.31196e27 0.643156
\(238\) −1.42645e27 −0.666226
\(239\) 2.12271e27 0.944744 0.472372 0.881399i \(-0.343398\pi\)
0.472372 + 0.881399i \(0.343398\pi\)
\(240\) 2.91303e26 0.123571
\(241\) −1.93749e27 −0.783508 −0.391754 0.920070i \(-0.628132\pi\)
−0.391754 + 0.920070i \(0.628132\pi\)
\(242\) −1.82296e27 −0.702909
\(243\) 1.18440e27 0.435537
\(244\) 2.22922e27 0.781934
\(245\) −2.34589e26 −0.0785051
\(246\) −3.73125e26 −0.119152
\(247\) −4.61100e27 −1.40534
\(248\) −9.53061e26 −0.277285
\(249\) 6.49257e27 1.80353
\(250\) −2.38419e26 −0.0632456
\(251\) 4.58522e27 1.16175 0.580875 0.813993i \(-0.302711\pi\)
0.580875 + 0.813993i \(0.302711\pi\)
\(252\) 4.96279e26 0.120121
\(253\) 2.69323e26 0.0622856
\(254\) −1.22928e27 −0.271683
\(255\) −2.03332e27 −0.429530
\(256\) 3.09485e26 0.0625000
\(257\) −5.64317e27 −1.08966 −0.544832 0.838545i \(-0.683407\pi\)
−0.544832 + 0.838545i \(0.683407\pi\)
\(258\) 2.29484e27 0.423765
\(259\) −2.87177e27 −0.507227
\(260\) −1.53798e27 −0.259871
\(261\) −2.08713e27 −0.337430
\(262\) 3.64675e27 0.564210
\(263\) −1.17070e28 −1.73363 −0.866813 0.498633i \(-0.833836\pi\)
−0.866813 + 0.498633i \(0.833836\pi\)
\(264\) 2.12376e26 0.0301065
\(265\) 2.11855e27 0.287548
\(266\) −7.13263e27 −0.927066
\(267\) −4.68552e27 −0.583282
\(268\) 6.13834e27 0.731982
\(269\) −1.41632e28 −1.61812 −0.809060 0.587726i \(-0.800023\pi\)
−0.809060 + 0.587726i \(0.800023\pi\)
\(270\) −2.48518e27 −0.272066
\(271\) 1.48604e28 1.55913 0.779566 0.626320i \(-0.215439\pi\)
0.779566 + 0.626320i \(0.215439\pi\)
\(272\) −2.16023e27 −0.217249
\(273\) −1.44452e28 −1.39269
\(274\) −6.61252e27 −0.611273
\(275\) −1.73820e26 −0.0154090
\(276\) 5.25447e27 0.446760
\(277\) 1.32716e28 1.08245 0.541223 0.840879i \(-0.317962\pi\)
0.541223 + 0.840879i \(0.317962\pi\)
\(278\) −6.16479e26 −0.0482395
\(279\) −2.31446e27 −0.173781
\(280\) −2.37907e27 −0.171431
\(281\) 1.06254e28 0.734892 0.367446 0.930045i \(-0.380232\pi\)
0.367446 + 0.930045i \(0.380232\pi\)
\(282\) 1.97154e28 1.30900
\(283\) 2.71424e28 1.73023 0.865117 0.501570i \(-0.167244\pi\)
0.865117 + 0.501570i \(0.167244\pi\)
\(284\) 1.42870e28 0.874538
\(285\) −1.01671e28 −0.597699
\(286\) −1.12128e27 −0.0633145
\(287\) 3.04730e27 0.165300
\(288\) 7.51569e26 0.0391702
\(289\) −4.88902e27 −0.244848
\(290\) 1.00053e28 0.481564
\(291\) −1.14989e27 −0.0531971
\(292\) −5.11420e27 −0.227444
\(293\) −1.41085e27 −0.0603257 −0.0301629 0.999545i \(-0.509603\pi\)
−0.0301629 + 0.999545i \(0.509603\pi\)
\(294\) −3.33675e27 −0.137192
\(295\) 8.91373e27 0.352457
\(296\) −4.34903e27 −0.165401
\(297\) −1.81184e27 −0.0662857
\(298\) 3.05734e28 1.07611
\(299\) −2.77418e28 −0.939543
\(300\) −3.39122e27 −0.110525
\(301\) −1.87419e28 −0.587893
\(302\) 4.61900e27 0.139466
\(303\) 2.34544e28 0.681765
\(304\) −1.08017e28 −0.302306
\(305\) −2.59516e28 −0.699383
\(306\) −5.24601e27 −0.136155
\(307\) 4.70277e28 1.17561 0.587804 0.809003i \(-0.299993\pi\)
0.587804 + 0.809003i \(0.299993\pi\)
\(308\) −1.73447e27 −0.0417671
\(309\) −8.23620e28 −1.91075
\(310\) 1.10951e28 0.248011
\(311\) 2.33572e28 0.503126 0.251563 0.967841i \(-0.419055\pi\)
0.251563 + 0.967841i \(0.419055\pi\)
\(312\) −2.18760e28 −0.454140
\(313\) 3.37396e28 0.675119 0.337560 0.941304i \(-0.390399\pi\)
0.337560 + 0.941304i \(0.390399\pi\)
\(314\) −4.09886e28 −0.790627
\(315\) −5.77744e27 −0.107440
\(316\) −1.62266e28 −0.290955
\(317\) −2.97844e28 −0.515000 −0.257500 0.966278i \(-0.582899\pi\)
−0.257500 + 0.966278i \(0.582899\pi\)
\(318\) 3.01338e28 0.502507
\(319\) 7.29443e27 0.117327
\(320\) −3.60288e27 −0.0559017
\(321\) 1.16863e28 0.174933
\(322\) −4.29131e28 −0.619794
\(323\) 7.53968e28 1.05081
\(324\) −4.35857e28 −0.586241
\(325\) 1.79045e28 0.232436
\(326\) −1.03223e28 −0.129353
\(327\) 1.62813e29 1.96965
\(328\) 4.61486e27 0.0539025
\(329\) −1.61015e29 −1.81599
\(330\) −2.47238e27 −0.0269281
\(331\) 5.24477e28 0.551702 0.275851 0.961200i \(-0.411040\pi\)
0.275851 + 0.961200i \(0.411040\pi\)
\(332\) −8.03010e28 −0.815893
\(333\) −1.05614e28 −0.103660
\(334\) −8.36338e28 −0.793044
\(335\) −7.14597e28 −0.654705
\(336\) −3.38393e28 −0.299585
\(337\) −1.64340e29 −1.40605 −0.703024 0.711166i \(-0.748167\pi\)
−0.703024 + 0.711166i \(0.748167\pi\)
\(338\) 2.99859e28 0.247956
\(339\) 2.29582e29 1.83502
\(340\) 2.51484e28 0.194313
\(341\) 8.08894e27 0.0604249
\(342\) −2.62314e28 −0.189462
\(343\) −1.27988e29 −0.893897
\(344\) −2.83829e28 −0.191705
\(345\) −6.11700e28 −0.399594
\(346\) 1.80055e29 1.13771
\(347\) 2.04136e28 0.124776 0.0623881 0.998052i \(-0.480128\pi\)
0.0623881 + 0.998052i \(0.480128\pi\)
\(348\) 1.42313e29 0.841559
\(349\) 2.60772e29 1.49200 0.745999 0.665947i \(-0.231972\pi\)
0.745999 + 0.665947i \(0.231972\pi\)
\(350\) 2.76960e28 0.153333
\(351\) 1.86629e29 0.999881
\(352\) −2.62670e27 −0.0136198
\(353\) −3.17093e29 −1.59139 −0.795696 0.605696i \(-0.792895\pi\)
−0.795696 + 0.605696i \(0.792895\pi\)
\(354\) 1.26787e29 0.615939
\(355\) −1.66322e29 −0.782211
\(356\) 5.79512e28 0.263868
\(357\) 2.36201e29 1.04135
\(358\) 1.32029e29 0.563656
\(359\) 3.06761e27 0.0126828 0.00634138 0.999980i \(-0.497981\pi\)
0.00634138 + 0.999980i \(0.497981\pi\)
\(360\) −8.74941e27 −0.0350348
\(361\) 1.19174e29 0.462221
\(362\) −9.29104e28 −0.349073
\(363\) 3.01857e29 1.09869
\(364\) 1.78661e29 0.630032
\(365\) 5.95372e28 0.203432
\(366\) −3.69129e29 −1.22221
\(367\) 1.46745e29 0.470874 0.235437 0.971890i \(-0.424348\pi\)
0.235437 + 0.971890i \(0.424348\pi\)
\(368\) −6.49880e28 −0.202108
\(369\) 1.12070e28 0.0337819
\(370\) 5.06294e28 0.147939
\(371\) −2.46102e29 −0.697132
\(372\) 1.57814e29 0.433413
\(373\) −5.22732e29 −1.39196 −0.695981 0.718060i \(-0.745030\pi\)
−0.695981 + 0.718060i \(0.745030\pi\)
\(374\) 1.83346e28 0.0473420
\(375\) 3.94789e28 0.0988567
\(376\) −2.43843e29 −0.592175
\(377\) −7.51368e29 −1.76981
\(378\) 2.88692e29 0.659598
\(379\) −5.96093e29 −1.32118 −0.660592 0.750745i \(-0.729694\pi\)
−0.660592 + 0.750745i \(0.729694\pi\)
\(380\) 1.25749e29 0.270391
\(381\) 2.03552e29 0.424657
\(382\) 8.22935e28 0.166585
\(383\) −4.94929e29 −0.972203 −0.486101 0.873902i \(-0.661581\pi\)
−0.486101 + 0.873902i \(0.661581\pi\)
\(384\) −5.12466e28 −0.0976913
\(385\) 2.01919e28 0.0373576
\(386\) −6.91329e29 −1.24145
\(387\) −6.89264e28 −0.120146
\(388\) 1.42220e28 0.0240656
\(389\) 5.66942e28 0.0931361 0.0465681 0.998915i \(-0.485172\pi\)
0.0465681 + 0.998915i \(0.485172\pi\)
\(390\) 2.54670e29 0.406195
\(391\) 4.53621e29 0.702523
\(392\) 4.12694e28 0.0620638
\(393\) −6.03853e29 −0.881895
\(394\) 6.73663e29 0.955512
\(395\) 1.88902e29 0.260238
\(396\) −6.37881e27 −0.00853582
\(397\) 3.68342e29 0.478807 0.239403 0.970920i \(-0.423048\pi\)
0.239403 + 0.970920i \(0.423048\pi\)
\(398\) 7.82856e29 0.988613
\(399\) 1.18107e30 1.44906
\(400\) 4.19430e28 0.0500000
\(401\) −1.21847e30 −1.41142 −0.705710 0.708501i \(-0.749372\pi\)
−0.705710 + 0.708501i \(0.749372\pi\)
\(402\) −1.01643e30 −1.14413
\(403\) −8.33207e29 −0.911475
\(404\) −2.90088e29 −0.308421
\(405\) 5.07404e29 0.524350
\(406\) −1.16227e30 −1.16750
\(407\) 3.69116e28 0.0360436
\(408\) 3.57705e29 0.339573
\(409\) 1.47992e30 1.36590 0.682952 0.730464i \(-0.260696\pi\)
0.682952 + 0.730464i \(0.260696\pi\)
\(410\) −5.37240e28 −0.0482119
\(411\) 1.09495e30 0.955458
\(412\) 1.01867e30 0.864398
\(413\) −1.03547e30 −0.854497
\(414\) −1.57820e29 −0.126666
\(415\) 9.34827e29 0.729757
\(416\) 2.70565e29 0.205446
\(417\) 1.02081e29 0.0754014
\(418\) 9.16777e28 0.0658774
\(419\) 4.36760e29 0.305339 0.152669 0.988277i \(-0.451213\pi\)
0.152669 + 0.988277i \(0.451213\pi\)
\(420\) 3.93942e29 0.267957
\(421\) 2.90507e30 1.92271 0.961353 0.275319i \(-0.0887836\pi\)
0.961353 + 0.275319i \(0.0887836\pi\)
\(422\) −3.12587e29 −0.201316
\(423\) −5.92161e29 −0.371129
\(424\) −3.72699e29 −0.227327
\(425\) −2.92765e29 −0.173799
\(426\) −2.36573e30 −1.36696
\(427\) 3.01467e30 1.69558
\(428\) −1.44538e29 −0.0791371
\(429\) 1.85668e29 0.0989644
\(430\) 3.30420e29 0.171467
\(431\) 5.61010e29 0.283454 0.141727 0.989906i \(-0.454735\pi\)
0.141727 + 0.989906i \(0.454735\pi\)
\(432\) 4.37198e29 0.215087
\(433\) −2.91680e30 −1.39732 −0.698659 0.715455i \(-0.746220\pi\)
−0.698659 + 0.715455i \(0.746220\pi\)
\(434\) −1.28887e30 −0.601278
\(435\) −1.65675e30 −0.752713
\(436\) −2.01369e30 −0.891043
\(437\) 2.26822e30 0.977574
\(438\) 8.46844e29 0.355509
\(439\) 1.03028e30 0.421322 0.210661 0.977559i \(-0.432438\pi\)
0.210661 + 0.977559i \(0.432438\pi\)
\(440\) 3.05788e28 0.0121819
\(441\) 1.00221e29 0.0388967
\(442\) −1.88856e30 −0.714127
\(443\) −2.53393e30 −0.933582 −0.466791 0.884368i \(-0.654590\pi\)
−0.466791 + 0.884368i \(0.654590\pi\)
\(444\) 7.20142e29 0.258532
\(445\) −6.74640e29 −0.236011
\(446\) 1.45270e30 0.495250
\(447\) −5.06255e30 −1.68203
\(448\) 4.18530e29 0.135528
\(449\) −4.77953e29 −0.150852 −0.0754262 0.997151i \(-0.524032\pi\)
−0.0754262 + 0.997151i \(0.524032\pi\)
\(450\) 1.01857e29 0.0313361
\(451\) −3.91678e28 −0.0117462
\(452\) −2.83951e30 −0.830139
\(453\) −7.64845e29 −0.217994
\(454\) 2.76209e30 0.767531
\(455\) −2.07988e30 −0.563518
\(456\) 1.78862e30 0.472522
\(457\) 5.53212e29 0.142513 0.0712566 0.997458i \(-0.477299\pi\)
0.0712566 + 0.997458i \(0.477299\pi\)
\(458\) 3.50270e30 0.879934
\(459\) −3.05167e30 −0.747639
\(460\) 7.56560e29 0.180771
\(461\) −3.77831e30 −0.880516 −0.440258 0.897871i \(-0.645113\pi\)
−0.440258 + 0.897871i \(0.645113\pi\)
\(462\) 2.87205e29 0.0652845
\(463\) 9.57536e29 0.212312 0.106156 0.994350i \(-0.466146\pi\)
0.106156 + 0.994350i \(0.466146\pi\)
\(464\) −1.76015e30 −0.380709
\(465\) −1.83720e30 −0.387657
\(466\) 1.58590e30 0.326465
\(467\) 1.81078e30 0.363681 0.181841 0.983328i \(-0.441795\pi\)
0.181841 + 0.983328i \(0.441795\pi\)
\(468\) 6.57054e29 0.128758
\(469\) 8.30114e30 1.58727
\(470\) 2.83871e30 0.529658
\(471\) 6.78716e30 1.23580
\(472\) −1.56812e30 −0.278642
\(473\) 2.40895e29 0.0417757
\(474\) 2.68690e30 0.454780
\(475\) −1.46391e30 −0.241845
\(476\) −2.92137e30 −0.471093
\(477\) −9.05081e29 −0.142471
\(478\) 4.34730e30 0.668035
\(479\) −1.31018e31 −1.96550 −0.982751 0.184936i \(-0.940792\pi\)
−0.982751 + 0.184936i \(0.940792\pi\)
\(480\) 5.96589e29 0.0873778
\(481\) −3.80211e30 −0.543696
\(482\) −3.96798e30 −0.554024
\(483\) 7.10584e30 0.968776
\(484\) −3.73341e30 −0.497032
\(485\) −1.65566e29 −0.0215249
\(486\) 2.42565e30 0.307972
\(487\) −7.78763e30 −0.965656 −0.482828 0.875715i \(-0.660390\pi\)
−0.482828 + 0.875715i \(0.660390\pi\)
\(488\) 4.56545e30 0.552911
\(489\) 1.70924e30 0.202186
\(490\) −4.80439e29 −0.0555115
\(491\) −2.96392e28 −0.00334526 −0.00167263 0.999999i \(-0.500532\pi\)
−0.00167263 + 0.999999i \(0.500532\pi\)
\(492\) −7.64159e29 −0.0842530
\(493\) 1.22860e31 1.32334
\(494\) −9.44332e30 −0.993722
\(495\) 7.42591e28 0.00763467
\(496\) −1.95187e30 −0.196070
\(497\) 1.93209e31 1.89639
\(498\) 1.32968e31 1.27529
\(499\) −1.73299e31 −1.62420 −0.812101 0.583517i \(-0.801676\pi\)
−0.812101 + 0.583517i \(0.801676\pi\)
\(500\) −4.88281e29 −0.0447214
\(501\) 1.38486e31 1.23958
\(502\) 9.39054e30 0.821481
\(503\) 6.74488e30 0.576690 0.288345 0.957527i \(-0.406895\pi\)
0.288345 + 0.957527i \(0.406895\pi\)
\(504\) 1.01638e30 0.0849385
\(505\) 3.37707e30 0.275860
\(506\) 5.51574e29 0.0440426
\(507\) −4.96526e30 −0.387570
\(508\) −2.51756e30 −0.192109
\(509\) 1.67540e31 1.24987 0.624935 0.780677i \(-0.285126\pi\)
0.624935 + 0.780677i \(0.285126\pi\)
\(510\) −4.16423e30 −0.303723
\(511\) −6.91616e30 −0.493202
\(512\) 6.33825e29 0.0441942
\(513\) −1.52592e31 −1.04035
\(514\) −1.15572e31 −0.770509
\(515\) −1.18588e31 −0.773141
\(516\) 4.69983e30 0.299647
\(517\) 2.06958e30 0.129045
\(518\) −5.88138e30 −0.358663
\(519\) −2.98148e31 −1.77831
\(520\) −3.14979e30 −0.183757
\(521\) 9.00283e30 0.513742 0.256871 0.966446i \(-0.417308\pi\)
0.256871 + 0.966446i \(0.417308\pi\)
\(522\) −4.27444e30 −0.238599
\(523\) −1.06079e31 −0.579239 −0.289620 0.957142i \(-0.593529\pi\)
−0.289620 + 0.957142i \(0.593529\pi\)
\(524\) 7.46854e30 0.398957
\(525\) −4.58609e30 −0.239668
\(526\) −2.39760e31 −1.22586
\(527\) 1.36242e31 0.681536
\(528\) 4.34947e29 0.0212885
\(529\) −7.23380e30 −0.346439
\(530\) 4.33879e30 0.203327
\(531\) −3.80810e30 −0.174631
\(532\) −1.46076e31 −0.655535
\(533\) 4.03451e30 0.177185
\(534\) −9.59594e30 −0.412442
\(535\) 1.68265e30 0.0707823
\(536\) 1.25713e31 0.517590
\(537\) −2.18622e31 −0.881028
\(538\) −2.90063e31 −1.14418
\(539\) −3.50267e29 −0.0135247
\(540\) −5.08965e30 −0.192380
\(541\) 1.12742e31 0.417175 0.208588 0.978004i \(-0.433113\pi\)
0.208588 + 0.978004i \(0.433113\pi\)
\(542\) 3.04341e31 1.10247
\(543\) 1.53847e31 0.545622
\(544\) −4.42415e30 −0.153618
\(545\) 2.34425e31 0.796973
\(546\) −2.95838e31 −0.984779
\(547\) 4.29561e31 1.40014 0.700069 0.714075i \(-0.253153\pi\)
0.700069 + 0.714075i \(0.253153\pi\)
\(548\) −1.35424e31 −0.432236
\(549\) 1.10870e31 0.346522
\(550\) −3.55984e29 −0.0108958
\(551\) 6.14333e31 1.84145
\(552\) 1.07611e31 0.315907
\(553\) −2.19439e31 −0.630920
\(554\) 2.71803e31 0.765405
\(555\) −8.38355e30 −0.231238
\(556\) −1.26255e30 −0.0341105
\(557\) 1.28388e31 0.339772 0.169886 0.985464i \(-0.445660\pi\)
0.169886 + 0.985464i \(0.445660\pi\)
\(558\) −4.74002e30 −0.122882
\(559\) −2.48135e31 −0.630163
\(560\) −4.87233e30 −0.121220
\(561\) −3.03596e30 −0.0739985
\(562\) 2.17609e31 0.519647
\(563\) 4.67401e31 1.09356 0.546780 0.837276i \(-0.315853\pi\)
0.546780 + 0.837276i \(0.315853\pi\)
\(564\) 4.03772e31 0.925606
\(565\) 3.30562e31 0.742499
\(566\) 5.55877e31 1.22346
\(567\) −5.89428e31 −1.27123
\(568\) 2.92597e31 0.618392
\(569\) −1.56288e31 −0.323694 −0.161847 0.986816i \(-0.551745\pi\)
−0.161847 + 0.986816i \(0.551745\pi\)
\(570\) −2.08223e31 −0.422637
\(571\) −3.73702e31 −0.743378 −0.371689 0.928357i \(-0.621221\pi\)
−0.371689 + 0.928357i \(0.621221\pi\)
\(572\) −2.29637e30 −0.0447701
\(573\) −1.36267e31 −0.260383
\(574\) 6.24087e30 0.116885
\(575\) −8.80751e30 −0.161686
\(576\) 1.53921e30 0.0276975
\(577\) −9.29399e31 −1.63938 −0.819692 0.572805i \(-0.805855\pi\)
−0.819692 + 0.572805i \(0.805855\pi\)
\(578\) −1.00127e31 −0.173134
\(579\) 1.14475e32 1.94047
\(580\) 2.04909e31 0.340517
\(581\) −1.08595e32 −1.76922
\(582\) −2.35498e30 −0.0376160
\(583\) 3.16322e30 0.0495383
\(584\) −1.04739e31 −0.160827
\(585\) −7.64911e30 −0.115165
\(586\) −2.88942e30 −0.0426567
\(587\) −4.03960e31 −0.584791 −0.292395 0.956298i \(-0.594452\pi\)
−0.292395 + 0.956298i \(0.594452\pi\)
\(588\) −6.83366e30 −0.0970094
\(589\) 6.81246e31 0.948370
\(590\) 1.82553e31 0.249225
\(591\) −1.11550e32 −1.49352
\(592\) −8.90681e30 −0.116956
\(593\) 1.98679e31 0.255872 0.127936 0.991782i \(-0.459165\pi\)
0.127936 + 0.991782i \(0.459165\pi\)
\(594\) −3.71064e30 −0.0468711
\(595\) 3.40092e31 0.421358
\(596\) 6.26143e31 0.760925
\(597\) −1.29630e32 −1.54526
\(598\) −5.68153e31 −0.664357
\(599\) −4.06122e30 −0.0465852 −0.0232926 0.999729i \(-0.507415\pi\)
−0.0232926 + 0.999729i \(0.507415\pi\)
\(600\) −6.94521e30 −0.0781530
\(601\) −2.13201e31 −0.235360 −0.117680 0.993052i \(-0.537546\pi\)
−0.117680 + 0.993052i \(0.537546\pi\)
\(602\) −3.83834e31 −0.415703
\(603\) 3.05288e31 0.324385
\(604\) 9.45971e30 0.0986173
\(605\) 4.34626e31 0.444559
\(606\) 4.80347e31 0.482081
\(607\) 1.05290e32 1.03685 0.518425 0.855123i \(-0.326519\pi\)
0.518425 + 0.855123i \(0.326519\pi\)
\(608\) −2.21219e31 −0.213762
\(609\) 1.92457e32 1.82488
\(610\) −5.31488e31 −0.494539
\(611\) −2.13178e32 −1.94656
\(612\) −1.07438e31 −0.0962759
\(613\) −1.91908e32 −1.68771 −0.843855 0.536572i \(-0.819719\pi\)
−0.843855 + 0.536572i \(0.819719\pi\)
\(614\) 9.63128e31 0.831281
\(615\) 8.89598e30 0.0753581
\(616\) −3.55220e30 −0.0295338
\(617\) 1.64022e32 1.33851 0.669256 0.743032i \(-0.266613\pi\)
0.669256 + 0.743032i \(0.266613\pi\)
\(618\) −1.68677e32 −1.35111
\(619\) −9.18518e31 −0.722180 −0.361090 0.932531i \(-0.617595\pi\)
−0.361090 + 0.932531i \(0.617595\pi\)
\(620\) 2.27227e31 0.175370
\(621\) −9.18061e31 −0.695534
\(622\) 4.78356e31 0.355764
\(623\) 7.83699e31 0.572185
\(624\) −4.48020e31 −0.321125
\(625\) 5.68434e30 0.0400000
\(626\) 6.90987e31 0.477381
\(627\) −1.51806e31 −0.102970
\(628\) −8.39446e31 −0.559058
\(629\) 6.21702e31 0.406537
\(630\) −1.18322e31 −0.0759713
\(631\) 2.95348e32 1.86207 0.931035 0.364930i \(-0.118907\pi\)
0.931035 + 0.364930i \(0.118907\pi\)
\(632\) −3.32320e31 −0.205736
\(633\) 5.17602e31 0.314669
\(634\) −6.09984e31 −0.364160
\(635\) 2.93083e31 0.171827
\(636\) 6.17140e31 0.355326
\(637\) 3.60795e31 0.204012
\(638\) 1.49390e31 0.0829628
\(639\) 7.10557e31 0.387560
\(640\) −7.37870e30 −0.0395285
\(641\) −2.62963e31 −0.138365 −0.0691826 0.997604i \(-0.522039\pi\)
−0.0691826 + 0.997604i \(0.522039\pi\)
\(642\) 2.39336e31 0.123696
\(643\) 2.59835e32 1.31908 0.659541 0.751668i \(-0.270751\pi\)
0.659541 + 0.751668i \(0.270751\pi\)
\(644\) −8.78860e31 −0.438261
\(645\) −5.47132e31 −0.268013
\(646\) 1.54413e32 0.743034
\(647\) 1.64878e31 0.0779403 0.0389702 0.999240i \(-0.487592\pi\)
0.0389702 + 0.999240i \(0.487592\pi\)
\(648\) −8.92635e31 −0.414535
\(649\) 1.33091e31 0.0607206
\(650\) 3.66684e31 0.164357
\(651\) 2.13419e32 0.939835
\(652\) −2.11401e31 −0.0914661
\(653\) −2.31883e32 −0.985751 −0.492876 0.870100i \(-0.664054\pi\)
−0.492876 + 0.870100i \(0.664054\pi\)
\(654\) 3.33441e32 1.39276
\(655\) −8.69453e31 −0.356838
\(656\) 9.45123e30 0.0381149
\(657\) −2.54353e31 −0.100794
\(658\) −3.29759e32 −1.28410
\(659\) −3.55456e32 −1.36020 −0.680102 0.733118i \(-0.738064\pi\)
−0.680102 + 0.733118i \(0.738064\pi\)
\(660\) −5.06344e30 −0.0190410
\(661\) −2.00631e32 −0.741447 −0.370724 0.928743i \(-0.620890\pi\)
−0.370724 + 0.928743i \(0.620890\pi\)
\(662\) 1.07413e32 0.390112
\(663\) 3.12721e32 1.11622
\(664\) −1.64457e32 −0.576923
\(665\) 1.70055e32 0.586328
\(666\) −2.16298e31 −0.0732990
\(667\) 3.69610e32 1.23111
\(668\) −1.71282e32 −0.560767
\(669\) −2.40548e32 −0.774106
\(670\) −1.46349e32 −0.462946
\(671\) −3.87484e31 −0.120488
\(672\) −6.93030e31 −0.211839
\(673\) 3.34870e32 1.00624 0.503122 0.864215i \(-0.332185\pi\)
0.503122 + 0.864215i \(0.332185\pi\)
\(674\) −3.36569e32 −0.994226
\(675\) 5.92514e31 0.172070
\(676\) 6.14110e31 0.175331
\(677\) −9.65120e31 −0.270902 −0.135451 0.990784i \(-0.543248\pi\)
−0.135451 + 0.990784i \(0.543248\pi\)
\(678\) 4.70184e32 1.29756
\(679\) 1.92331e31 0.0521850
\(680\) 5.15038e31 0.137400
\(681\) −4.57366e32 −1.19970
\(682\) 1.65661e31 0.0427269
\(683\) −1.74605e32 −0.442811 −0.221406 0.975182i \(-0.571065\pi\)
−0.221406 + 0.975182i \(0.571065\pi\)
\(684\) −5.37220e31 −0.133970
\(685\) 1.57655e32 0.386603
\(686\) −2.62120e32 −0.632080
\(687\) −5.80001e32 −1.37539
\(688\) −5.81281e31 −0.135556
\(689\) −3.25830e32 −0.747256
\(690\) −1.25276e32 −0.282556
\(691\) 4.17873e32 0.926931 0.463465 0.886115i \(-0.346606\pi\)
0.463465 + 0.886115i \(0.346606\pi\)
\(692\) 3.68754e32 0.804482
\(693\) −8.62634e30 −0.0185095
\(694\) 4.18071e31 0.0882300
\(695\) 1.46980e31 0.0305094
\(696\) 2.91458e32 0.595072
\(697\) −6.59703e31 −0.132486
\(698\) 5.34061e32 1.05500
\(699\) −2.62603e32 −0.510285
\(700\) 5.67214e31 0.108422
\(701\) −5.54616e32 −1.04288 −0.521441 0.853288i \(-0.674605\pi\)
−0.521441 + 0.853288i \(0.674605\pi\)
\(702\) 3.82217e32 0.707023
\(703\) 3.10868e32 0.565704
\(704\) −5.37948e30 −0.00963063
\(705\) −4.70052e32 −0.827887
\(706\) −6.49406e32 −1.12528
\(707\) −3.92298e32 −0.668795
\(708\) 2.59660e32 0.435534
\(709\) −9.37126e32 −1.54656 −0.773279 0.634066i \(-0.781385\pi\)
−0.773279 + 0.634066i \(0.781385\pi\)
\(710\) −3.40628e32 −0.553107
\(711\) −8.07023e31 −0.128939
\(712\) 1.18684e32 0.186583
\(713\) 4.09868e32 0.634037
\(714\) 4.83740e32 0.736347
\(715\) 2.67333e31 0.0400436
\(716\) 2.70395e32 0.398565
\(717\) −7.19855e32 −1.04418
\(718\) 6.28246e30 0.00896806
\(719\) −7.22362e31 −0.101478 −0.0507391 0.998712i \(-0.516158\pi\)
−0.0507391 + 0.998712i \(0.516158\pi\)
\(720\) −1.79188e31 −0.0247734
\(721\) 1.37759e33 1.87440
\(722\) 2.44069e32 0.326839
\(723\) 6.57045e32 0.865973
\(724\) −1.90281e32 −0.246832
\(725\) −2.38545e32 −0.304568
\(726\) 6.18203e32 0.776891
\(727\) 9.11706e32 1.12774 0.563870 0.825863i \(-0.309312\pi\)
0.563870 + 0.825863i \(0.309312\pi\)
\(728\) 3.65897e32 0.445500
\(729\) 5.76647e32 0.691104
\(730\) 1.21932e32 0.143848
\(731\) 4.05739e32 0.471191
\(732\) −7.55977e32 −0.864234
\(733\) 1.25151e33 1.40844 0.704221 0.709981i \(-0.251296\pi\)
0.704221 + 0.709981i \(0.251296\pi\)
\(734\) 3.00535e32 0.332958
\(735\) 7.95542e31 0.0867679
\(736\) −1.33095e32 −0.142912
\(737\) −1.06697e32 −0.112791
\(738\) 2.29518e31 0.0238874
\(739\) 7.62517e32 0.781337 0.390669 0.920531i \(-0.372244\pi\)
0.390669 + 0.920531i \(0.372244\pi\)
\(740\) 1.03689e32 0.104609
\(741\) 1.56369e33 1.55325
\(742\) −5.04017e32 −0.492947
\(743\) −6.76133e32 −0.651119 −0.325560 0.945521i \(-0.605553\pi\)
−0.325560 + 0.945521i \(0.605553\pi\)
\(744\) 3.23203e32 0.306470
\(745\) −7.28927e32 −0.680592
\(746\) −1.07055e33 −0.984265
\(747\) −3.99375e32 −0.361571
\(748\) 3.75492e31 0.0334759
\(749\) −1.95465e32 −0.171605
\(750\) 8.08529e31 0.0699022
\(751\) −9.33180e32 −0.794522 −0.397261 0.917706i \(-0.630039\pi\)
−0.397261 + 0.917706i \(0.630039\pi\)
\(752\) −4.99391e32 −0.418731
\(753\) −1.55495e33 −1.28403
\(754\) −1.53880e33 −1.25145
\(755\) −1.10126e32 −0.0882060
\(756\) 5.91241e32 0.466406
\(757\) 2.41509e33 1.87642 0.938211 0.346065i \(-0.112482\pi\)
0.938211 + 0.346065i \(0.112482\pi\)
\(758\) −1.22080e33 −0.934218
\(759\) −9.13333e31 −0.0688413
\(760\) 2.57533e32 0.191195
\(761\) −1.31749e32 −0.0963435 −0.0481718 0.998839i \(-0.515339\pi\)
−0.0481718 + 0.998839i \(0.515339\pi\)
\(762\) 4.16875e32 0.300278
\(763\) −2.72320e33 −1.93218
\(764\) 1.68537e32 0.117794
\(765\) 1.25075e32 0.0861117
\(766\) −1.01361e33 −0.687451
\(767\) −1.37092e33 −0.915935
\(768\) −1.04953e32 −0.0690782
\(769\) −7.19550e32 −0.466561 −0.233280 0.972410i \(-0.574946\pi\)
−0.233280 + 0.972410i \(0.574946\pi\)
\(770\) 4.13530e31 0.0264158
\(771\) 1.91372e33 1.20435
\(772\) −1.41584e33 −0.877841
\(773\) −3.05497e32 −0.186613 −0.0933066 0.995637i \(-0.529744\pi\)
−0.0933066 + 0.995637i \(0.529744\pi\)
\(774\) −1.41161e32 −0.0849561
\(775\) −2.64528e32 −0.156856
\(776\) 2.91267e31 0.0170170
\(777\) 9.73878e32 0.560613
\(778\) 1.16110e32 0.0658572
\(779\) −3.29869e32 −0.184357
\(780\) 5.21564e32 0.287223
\(781\) −2.48336e32 −0.134758
\(782\) 9.29016e32 0.496759
\(783\) −2.48650e33 −1.31017
\(784\) 8.45197e31 0.0438857
\(785\) 9.77243e32 0.500036
\(786\) −1.23669e33 −0.623594
\(787\) −3.70046e32 −0.183885 −0.0919426 0.995764i \(-0.529308\pi\)
−0.0919426 + 0.995764i \(0.529308\pi\)
\(788\) 1.37966e33 0.675649
\(789\) 3.97011e33 1.91609
\(790\) 3.86872e32 0.184016
\(791\) −3.83999e33 −1.80011
\(792\) −1.30638e31 −0.00603573
\(793\) 3.99131e33 1.81750
\(794\) 7.54364e32 0.338567
\(795\) −7.18446e32 −0.317813
\(796\) 1.60329e33 0.699055
\(797\) 1.56557e32 0.0672823 0.0336412 0.999434i \(-0.489290\pi\)
0.0336412 + 0.999434i \(0.489290\pi\)
\(798\) 2.41883e33 1.02464
\(799\) 3.48578e33 1.45550
\(800\) 8.58993e31 0.0353553
\(801\) 2.88218e32 0.116936
\(802\) −2.49544e33 −0.998024
\(803\) 8.88953e31 0.0350469
\(804\) −2.08164e33 −0.809024
\(805\) 1.02313e33 0.391992
\(806\) −1.70641e33 −0.644510
\(807\) 4.80305e33 1.78843
\(808\) −5.94100e32 −0.218087
\(809\) −3.53778e33 −1.28033 −0.640167 0.768236i \(-0.721135\pi\)
−0.640167 + 0.768236i \(0.721135\pi\)
\(810\) 1.03916e33 0.370771
\(811\) −2.93265e33 −1.03162 −0.515811 0.856702i \(-0.672509\pi\)
−0.515811 + 0.856702i \(0.672509\pi\)
\(812\) −2.38033e33 −0.825549
\(813\) −5.03948e33 −1.72323
\(814\) 7.55950e31 0.0254866
\(815\) 2.46103e32 0.0818098
\(816\) 7.32580e32 0.240114
\(817\) 2.02880e33 0.655671
\(818\) 3.03087e33 0.965840
\(819\) 8.88562e32 0.279205
\(820\) −1.10027e32 −0.0340910
\(821\) 2.49297e33 0.761677 0.380838 0.924642i \(-0.375635\pi\)
0.380838 + 0.924642i \(0.375635\pi\)
\(822\) 2.24245e33 0.675611
\(823\) −3.98570e33 −1.18415 −0.592075 0.805883i \(-0.701691\pi\)
−0.592075 + 0.805883i \(0.701691\pi\)
\(824\) 2.08623e33 0.611222
\(825\) 5.89463e31 0.0170308
\(826\) −2.12064e33 −0.604221
\(827\) 1.70726e32 0.0479718 0.0239859 0.999712i \(-0.492364\pi\)
0.0239859 + 0.999712i \(0.492364\pi\)
\(828\) −3.23216e32 −0.0895660
\(829\) 5.89567e33 1.61123 0.805613 0.592443i \(-0.201836\pi\)
0.805613 + 0.592443i \(0.201836\pi\)
\(830\) 1.91453e33 0.516016
\(831\) −4.50069e33 −1.19637
\(832\) 5.54117e32 0.145273
\(833\) −5.89953e32 −0.152546
\(834\) 2.09061e32 0.0533168
\(835\) 1.99399e33 0.501565
\(836\) 1.87756e32 0.0465823
\(837\) −2.75733e33 −0.674755
\(838\) 8.94485e32 0.215907
\(839\) 2.81133e33 0.669345 0.334672 0.942335i \(-0.391374\pi\)
0.334672 + 0.942335i \(0.391374\pi\)
\(840\) 8.06793e32 0.189474
\(841\) 5.69391e33 1.31903
\(842\) 5.94959e33 1.35956
\(843\) −3.60331e33 −0.812241
\(844\) −6.40178e32 −0.142352
\(845\) −7.14919e32 −0.156821
\(846\) −1.21275e33 −0.262428
\(847\) −5.04885e33 −1.07779
\(848\) −7.63288e32 −0.160744
\(849\) −9.20459e33 −1.91234
\(850\) −5.99584e32 −0.122894
\(851\) 1.87032e33 0.378204
\(852\) −4.84501e33 −0.966584
\(853\) −3.10315e33 −0.610787 −0.305393 0.952226i \(-0.598788\pi\)
−0.305393 + 0.952226i \(0.598788\pi\)
\(854\) 6.17405e33 1.19896
\(855\) 6.25406e32 0.119826
\(856\) −2.96015e32 −0.0559584
\(857\) −5.09943e33 −0.951134 −0.475567 0.879679i \(-0.657757\pi\)
−0.475567 + 0.879679i \(0.657757\pi\)
\(858\) 3.80249e32 0.0699784
\(859\) 1.31743e33 0.239224 0.119612 0.992821i \(-0.461835\pi\)
0.119612 + 0.992821i \(0.461835\pi\)
\(860\) 6.76701e32 0.121245
\(861\) −1.03341e33 −0.182698
\(862\) 1.14895e33 0.200432
\(863\) −1.29184e33 −0.222375 −0.111188 0.993799i \(-0.535465\pi\)
−0.111188 + 0.993799i \(0.535465\pi\)
\(864\) 8.95381e32 0.152090
\(865\) −4.29286e33 −0.719550
\(866\) −5.97360e33 −0.988053
\(867\) 1.65797e33 0.270619
\(868\) −2.63960e33 −0.425168
\(869\) 2.82051e32 0.0448332
\(870\) −3.39302e33 −0.532249
\(871\) 1.09904e34 1.70139
\(872\) −4.12405e33 −0.630063
\(873\) 7.07328e31 0.0106649
\(874\) 4.64532e33 0.691249
\(875\) −6.60324e32 −0.0969760
\(876\) 1.73434e33 0.251383
\(877\) −1.10353e33 −0.157866 −0.0789329 0.996880i \(-0.525151\pi\)
−0.0789329 + 0.996880i \(0.525151\pi\)
\(878\) 2.11002e33 0.297920
\(879\) 4.78449e32 0.0666750
\(880\) 6.26254e31 0.00861390
\(881\) 3.42468e33 0.464940 0.232470 0.972604i \(-0.425319\pi\)
0.232470 + 0.972604i \(0.425319\pi\)
\(882\) 2.05252e32 0.0275042
\(883\) −7.33051e33 −0.969585 −0.484792 0.874629i \(-0.661105\pi\)
−0.484792 + 0.874629i \(0.661105\pi\)
\(884\) −3.86778e33 −0.504964
\(885\) −3.02284e33 −0.389554
\(886\) −5.18950e33 −0.660142
\(887\) −7.64645e33 −0.960148 −0.480074 0.877228i \(-0.659390\pi\)
−0.480074 + 0.877228i \(0.659390\pi\)
\(888\) 1.47485e33 0.182809
\(889\) −3.40461e33 −0.416578
\(890\) −1.38166e33 −0.166885
\(891\) 7.57608e32 0.0903340
\(892\) 2.97513e33 0.350195
\(893\) 1.74298e34 2.02536
\(894\) −1.03681e34 −1.18937
\(895\) −3.14781e33 −0.356487
\(896\) 8.57149e32 0.0958328
\(897\) 9.40785e33 1.03843
\(898\) −9.78848e32 −0.106669
\(899\) 1.11010e34 1.19433
\(900\) 2.08602e32 0.0221580
\(901\) 5.32780e33 0.558744
\(902\) −8.02157e31 −0.00830585
\(903\) 6.35578e33 0.649770
\(904\) −5.81531e33 −0.586997
\(905\) 2.21516e33 0.220773
\(906\) −1.56640e33 −0.154145
\(907\) 2.55586e33 0.248344 0.124172 0.992261i \(-0.460373\pi\)
0.124172 + 0.992261i \(0.460373\pi\)
\(908\) 5.65677e33 0.542727
\(909\) −1.44274e33 −0.136680
\(910\) −4.25960e33 −0.398467
\(911\) 6.46941e33 0.597590 0.298795 0.954317i \(-0.403415\pi\)
0.298795 + 0.954317i \(0.403415\pi\)
\(912\) 3.66309e33 0.334124
\(913\) 1.39580e33 0.125721
\(914\) 1.13298e33 0.100772
\(915\) 8.80073e33 0.772994
\(916\) 7.17353e33 0.622207
\(917\) 1.01000e34 0.865118
\(918\) −6.24983e33 −0.528661
\(919\) −5.58007e33 −0.466134 −0.233067 0.972461i \(-0.574876\pi\)
−0.233067 + 0.972461i \(0.574876\pi\)
\(920\) 1.54943e33 0.127824
\(921\) −1.59481e34 −1.29934
\(922\) −7.73798e33 −0.622619
\(923\) 2.55801e34 2.03274
\(924\) 5.88197e32 0.0461631
\(925\) −1.20710e33 −0.0935649
\(926\) 1.96103e33 0.150127
\(927\) 5.06630e33 0.383066
\(928\) −3.60479e33 −0.269202
\(929\) −1.37312e34 −1.01281 −0.506406 0.862295i \(-0.669026\pi\)
−0.506406 + 0.862295i \(0.669026\pi\)
\(930\) −3.76258e33 −0.274115
\(931\) −2.94992e33 −0.212270
\(932\) 3.24791e33 0.230846
\(933\) −7.92093e33 −0.556080
\(934\) 3.70847e33 0.257161
\(935\) −4.37130e32 −0.0299417
\(936\) 1.34565e33 0.0910455
\(937\) −6.15639e33 −0.411454 −0.205727 0.978609i \(-0.565956\pi\)
−0.205727 + 0.978609i \(0.565956\pi\)
\(938\) 1.70007e34 1.12237
\(939\) −1.14418e34 −0.746176
\(940\) 5.81367e33 0.374524
\(941\) −1.54804e34 −0.985147 −0.492574 0.870271i \(-0.663944\pi\)
−0.492574 + 0.870271i \(0.663944\pi\)
\(942\) 1.39001e34 0.873841
\(943\) −1.98464e33 −0.123253
\(944\) −3.21151e33 −0.197030
\(945\) −6.88296e33 −0.417166
\(946\) 4.93352e32 0.0295399
\(947\) −1.09161e34 −0.645720 −0.322860 0.946447i \(-0.604644\pi\)
−0.322860 + 0.946447i \(0.604644\pi\)
\(948\) 5.50278e33 0.321578
\(949\) −9.15672e33 −0.528663
\(950\) −2.99808e33 −0.171010
\(951\) 1.01005e34 0.569204
\(952\) −5.98296e33 −0.333113
\(953\) 3.06171e34 1.68421 0.842103 0.539317i \(-0.181318\pi\)
0.842103 + 0.539317i \(0.181318\pi\)
\(954\) −1.85361e33 −0.100742
\(955\) −1.96203e33 −0.105358
\(956\) 8.90327e33 0.472372
\(957\) −2.47370e33 −0.129676
\(958\) −2.68325e34 −1.38982
\(959\) −1.83140e34 −0.937281
\(960\) 1.22181e33 0.0617854
\(961\) −7.70321e33 −0.384904
\(962\) −7.78672e33 −0.384451
\(963\) −7.18857e32 −0.0350704
\(964\) −8.12642e33 −0.391754
\(965\) 1.64826e34 0.785165
\(966\) 1.45528e34 0.685028
\(967\) −1.52535e34 −0.709519 −0.354759 0.934958i \(-0.615437\pi\)
−0.354759 + 0.934958i \(0.615437\pi\)
\(968\) −7.64603e33 −0.351455
\(969\) −2.55687e34 −1.16141
\(970\) −3.39080e32 −0.0152204
\(971\) −2.92889e34 −1.29922 −0.649610 0.760268i \(-0.725068\pi\)
−0.649610 + 0.760268i \(0.725068\pi\)
\(972\) 4.96773e33 0.217769
\(973\) −1.70740e33 −0.0739669
\(974\) −1.59491e34 −0.682822
\(975\) −6.07180e33 −0.256900
\(976\) 9.35003e33 0.390967
\(977\) −6.75647e33 −0.279211 −0.139605 0.990207i \(-0.544583\pi\)
−0.139605 + 0.990207i \(0.544583\pi\)
\(978\) 3.50052e33 0.142967
\(979\) −1.00731e33 −0.0406595
\(980\) −9.83938e32 −0.0392526
\(981\) −1.00150e34 −0.394874
\(982\) −6.07012e31 −0.00236546
\(983\) 2.01524e34 0.776178 0.388089 0.921622i \(-0.373135\pi\)
0.388089 + 0.921622i \(0.373135\pi\)
\(984\) −1.56500e33 −0.0595758
\(985\) −1.60614e34 −0.604319
\(986\) 2.51617e34 0.935741
\(987\) 5.46038e34 2.00713
\(988\) −1.93399e34 −0.702668
\(989\) 1.22062e34 0.438352
\(990\) 1.52083e32 0.00539852
\(991\) 6.56935e33 0.230502 0.115251 0.993336i \(-0.463233\pi\)
0.115251 + 0.993336i \(0.463233\pi\)
\(992\) −3.99743e33 −0.138642
\(993\) −1.77861e34 −0.609769
\(994\) 3.95691e34 1.34095
\(995\) −1.86647e34 −0.625254
\(996\) 2.72318e34 0.901767
\(997\) 8.17693e33 0.267668 0.133834 0.991004i \(-0.457271\pi\)
0.133834 + 0.991004i \(0.457271\pi\)
\(998\) −3.54917e34 −1.14848
\(999\) −1.25823e34 −0.402493
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 10.24.a.c.1.1 2
5.2 odd 4 50.24.b.c.49.4 4
5.3 odd 4 50.24.b.c.49.1 4
5.4 even 2 50.24.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.24.a.c.1.1 2 1.1 even 1 trivial
50.24.a.b.1.2 2 5.4 even 2
50.24.b.c.49.1 4 5.3 odd 4
50.24.b.c.49.4 4 5.2 odd 4