Properties

Label 528.8.a.a.1.1
Level $528$
Weight $8$
Character 528.1
Self dual yes
Analytic conductor $164.939$
Analytic rank $0$
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] = [528,8,Mod(1,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 528.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: \(164.939293456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 528.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -410.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} -410.000 q^{5} +1028.00 q^{7} +729.000 q^{9} +1331.00 q^{11} +12958.0 q^{13} +11070.0 q^{15} +17062.0 q^{17} +54168.0 q^{19} -27756.0 q^{21} +11488.0 q^{23} +89975.0 q^{25} -19683.0 q^{27} -186654. q^{29} +188672. q^{31} -35937.0 q^{33} -421480. q^{35} +395886. q^{37} -349866. q^{39} -47546.0 q^{41} -602088. q^{43} -298890. q^{45} +647200. q^{47} +233241. q^{49} -460674. q^{51} -1.31272e6 q^{53} -545710. q^{55} -1.46254e6 q^{57} +2.68114e6 q^{59} +551190. q^{61} +749412. q^{63} -5.31278e6 q^{65} -459260. q^{67} -310176. q^{69} +18072.0 q^{71} -426062. q^{73} -2.42932e6 q^{75} +1.36827e6 q^{77} -297764. q^{79} +531441. q^{81} -5.68403e6 q^{83} -6.99542e6 q^{85} +5.03966e6 q^{87} -6.34297e6 q^{89} +1.33208e7 q^{91} -5.09414e6 q^{93} -2.22089e7 q^{95} +1.66516e7 q^{97} +970299. q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −410.000 −1.46686 −0.733430 0.679765i \(-0.762082\pi\)
−0.733430 + 0.679765i \(0.762082\pi\)
\(6\) 0 0
\(7\) 1028.00 1.13279 0.566396 0.824133i \(-0.308337\pi\)
0.566396 + 0.824133i \(0.308337\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 1331.00 0.301511
\(12\) 0 0
\(13\) 12958.0 1.63582 0.817911 0.575344i \(-0.195132\pi\)
0.817911 + 0.575344i \(0.195132\pi\)
\(14\) 0 0
\(15\) 11070.0 0.846892
\(16\) 0 0
\(17\) 17062.0 0.842284 0.421142 0.906995i \(-0.361629\pi\)
0.421142 + 0.906995i \(0.361629\pi\)
\(18\) 0 0
\(19\) 54168.0 1.81178 0.905889 0.423514i \(-0.139204\pi\)
0.905889 + 0.423514i \(0.139204\pi\)
\(20\) 0 0
\(21\) −27756.0 −0.654017
\(22\) 0 0
\(23\) 11488.0 0.196878 0.0984390 0.995143i \(-0.468615\pi\)
0.0984390 + 0.995143i \(0.468615\pi\)
\(24\) 0 0
\(25\) 89975.0 1.15168
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −186654. −1.42116 −0.710582 0.703614i \(-0.751568\pi\)
−0.710582 + 0.703614i \(0.751568\pi\)
\(30\) 0 0
\(31\) 188672. 1.13747 0.568737 0.822519i \(-0.307432\pi\)
0.568737 + 0.822519i \(0.307432\pi\)
\(32\) 0 0
\(33\) −35937.0 −0.174078
\(34\) 0 0
\(35\) −421480. −1.66165
\(36\) 0 0
\(37\) 395886. 1.28488 0.642442 0.766334i \(-0.277921\pi\)
0.642442 + 0.766334i \(0.277921\pi\)
\(38\) 0 0
\(39\) −349866. −0.944443
\(40\) 0 0
\(41\) −47546.0 −0.107738 −0.0538692 0.998548i \(-0.517155\pi\)
−0.0538692 + 0.998548i \(0.517155\pi\)
\(42\) 0 0
\(43\) −602088. −1.15484 −0.577418 0.816449i \(-0.695940\pi\)
−0.577418 + 0.816449i \(0.695940\pi\)
\(44\) 0 0
\(45\) −298890. −0.488954
\(46\) 0 0
\(47\) 647200. 0.909277 0.454638 0.890676i \(-0.349769\pi\)
0.454638 + 0.890676i \(0.349769\pi\)
\(48\) 0 0
\(49\) 233241. 0.283217
\(50\) 0 0
\(51\) −460674. −0.486293
\(52\) 0 0
\(53\) −1.31272e6 −1.21118 −0.605588 0.795778i \(-0.707062\pi\)
−0.605588 + 0.795778i \(0.707062\pi\)
\(54\) 0 0
\(55\) −545710. −0.442275
\(56\) 0 0
\(57\) −1.46254e6 −1.04603
\(58\) 0 0
\(59\) 2.68114e6 1.69956 0.849782 0.527135i \(-0.176734\pi\)
0.849782 + 0.527135i \(0.176734\pi\)
\(60\) 0 0
\(61\) 551190. 0.310919 0.155459 0.987842i \(-0.450314\pi\)
0.155459 + 0.987842i \(0.450314\pi\)
\(62\) 0 0
\(63\) 749412. 0.377597
\(64\) 0 0
\(65\) −5.31278e6 −2.39952
\(66\) 0 0
\(67\) −459260. −0.186551 −0.0932753 0.995640i \(-0.529734\pi\)
−0.0932753 + 0.995640i \(0.529734\pi\)
\(68\) 0 0
\(69\) −310176. −0.113668
\(70\) 0 0
\(71\) 18072.0 0.00599242 0.00299621 0.999996i \(-0.499046\pi\)
0.00299621 + 0.999996i \(0.499046\pi\)
\(72\) 0 0
\(73\) −426062. −0.128187 −0.0640933 0.997944i \(-0.520416\pi\)
−0.0640933 + 0.997944i \(0.520416\pi\)
\(74\) 0 0
\(75\) −2.42932e6 −0.664923
\(76\) 0 0
\(77\) 1.36827e6 0.341549
\(78\) 0 0
\(79\) −297764. −0.0679481 −0.0339741 0.999423i \(-0.510816\pi\)
−0.0339741 + 0.999423i \(0.510816\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −5.68403e6 −1.09115 −0.545573 0.838063i \(-0.683688\pi\)
−0.545573 + 0.838063i \(0.683688\pi\)
\(84\) 0 0
\(85\) −6.99542e6 −1.23551
\(86\) 0 0
\(87\) 5.03966e6 0.820510
\(88\) 0 0
\(89\) −6.34297e6 −0.953734 −0.476867 0.878975i \(-0.658228\pi\)
−0.476867 + 0.878975i \(0.658228\pi\)
\(90\) 0 0
\(91\) 1.33208e7 1.85305
\(92\) 0 0
\(93\) −5.09414e6 −0.656721
\(94\) 0 0
\(95\) −2.22089e7 −2.65763
\(96\) 0 0
\(97\) 1.66516e7 1.85248 0.926242 0.376929i \(-0.123020\pi\)
0.926242 + 0.376929i \(0.123020\pi\)
\(98\) 0 0
\(99\) 970299. 0.100504
\(100\) 0 0
\(101\) −2.08327e6 −0.201197 −0.100598 0.994927i \(-0.532076\pi\)
−0.100598 + 0.994927i \(0.532076\pi\)
\(102\) 0 0
\(103\) 2.39046e6 0.215552 0.107776 0.994175i \(-0.465627\pi\)
0.107776 + 0.994175i \(0.465627\pi\)
\(104\) 0 0
\(105\) 1.13800e7 0.959352
\(106\) 0 0
\(107\) 1.40615e7 1.10965 0.554827 0.831966i \(-0.312785\pi\)
0.554827 + 0.831966i \(0.312785\pi\)
\(108\) 0 0
\(109\) −1.11321e7 −0.823347 −0.411674 0.911331i \(-0.635056\pi\)
−0.411674 + 0.911331i \(0.635056\pi\)
\(110\) 0 0
\(111\) −1.06889e7 −0.741828
\(112\) 0 0
\(113\) 5.66903e6 0.369602 0.184801 0.982776i \(-0.440836\pi\)
0.184801 + 0.982776i \(0.440836\pi\)
\(114\) 0 0
\(115\) −4.71008e6 −0.288792
\(116\) 0 0
\(117\) 9.44638e6 0.545274
\(118\) 0 0
\(119\) 1.75397e7 0.954132
\(120\) 0 0
\(121\) 1.77156e6 0.0909091
\(122\) 0 0
\(123\) 1.28374e6 0.0622028
\(124\) 0 0
\(125\) −4.85850e6 −0.222493
\(126\) 0 0
\(127\) 2.09170e7 0.906123 0.453061 0.891479i \(-0.350332\pi\)
0.453061 + 0.891479i \(0.350332\pi\)
\(128\) 0 0
\(129\) 1.62564e7 0.666745
\(130\) 0 0
\(131\) 1.12649e7 0.437802 0.218901 0.975747i \(-0.429753\pi\)
0.218901 + 0.975747i \(0.429753\pi\)
\(132\) 0 0
\(133\) 5.56847e7 2.05237
\(134\) 0 0
\(135\) 8.07003e6 0.282297
\(136\) 0 0
\(137\) 444290. 0.0147620 0.00738099 0.999973i \(-0.497651\pi\)
0.00738099 + 0.999973i \(0.497651\pi\)
\(138\) 0 0
\(139\) −3.42613e7 −1.08206 −0.541030 0.841003i \(-0.681966\pi\)
−0.541030 + 0.841003i \(0.681966\pi\)
\(140\) 0 0
\(141\) −1.74744e7 −0.524971
\(142\) 0 0
\(143\) 1.72471e7 0.493219
\(144\) 0 0
\(145\) 7.65281e7 2.08465
\(146\) 0 0
\(147\) −6.29751e6 −0.163515
\(148\) 0 0
\(149\) −4.82211e7 −1.19422 −0.597112 0.802158i \(-0.703685\pi\)
−0.597112 + 0.802158i \(0.703685\pi\)
\(150\) 0 0
\(151\) 4.48693e7 1.06055 0.530273 0.847827i \(-0.322089\pi\)
0.530273 + 0.847827i \(0.322089\pi\)
\(152\) 0 0
\(153\) 1.24382e7 0.280761
\(154\) 0 0
\(155\) −7.73555e7 −1.66852
\(156\) 0 0
\(157\) −5.38907e6 −0.111139 −0.0555693 0.998455i \(-0.517697\pi\)
−0.0555693 + 0.998455i \(0.517697\pi\)
\(158\) 0 0
\(159\) 3.54435e7 0.699273
\(160\) 0 0
\(161\) 1.18097e7 0.223022
\(162\) 0 0
\(163\) −9.81674e7 −1.77546 −0.887730 0.460365i \(-0.847719\pi\)
−0.887730 + 0.460365i \(0.847719\pi\)
\(164\) 0 0
\(165\) 1.47342e7 0.255348
\(166\) 0 0
\(167\) 4.40611e7 0.732062 0.366031 0.930603i \(-0.380716\pi\)
0.366031 + 0.930603i \(0.380716\pi\)
\(168\) 0 0
\(169\) 1.05161e8 1.67592
\(170\) 0 0
\(171\) 3.94885e7 0.603926
\(172\) 0 0
\(173\) 6.71087e7 0.985411 0.492706 0.870196i \(-0.336008\pi\)
0.492706 + 0.870196i \(0.336008\pi\)
\(174\) 0 0
\(175\) 9.24943e7 1.30461
\(176\) 0 0
\(177\) −7.23908e7 −0.981244
\(178\) 0 0
\(179\) −4.34929e6 −0.0566804 −0.0283402 0.999598i \(-0.509022\pi\)
−0.0283402 + 0.999598i \(0.509022\pi\)
\(180\) 0 0
\(181\) −1.20238e7 −0.150719 −0.0753593 0.997156i \(-0.524010\pi\)
−0.0753593 + 0.997156i \(0.524010\pi\)
\(182\) 0 0
\(183\) −1.48821e7 −0.179509
\(184\) 0 0
\(185\) −1.62313e8 −1.88475
\(186\) 0 0
\(187\) 2.27095e7 0.253958
\(188\) 0 0
\(189\) −2.02341e7 −0.218006
\(190\) 0 0
\(191\) −5.96399e7 −0.619327 −0.309664 0.950846i \(-0.600216\pi\)
−0.309664 + 0.950846i \(0.600216\pi\)
\(192\) 0 0
\(193\) −9.81036e7 −0.982278 −0.491139 0.871081i \(-0.663419\pi\)
−0.491139 + 0.871081i \(0.663419\pi\)
\(194\) 0 0
\(195\) 1.43445e8 1.38537
\(196\) 0 0
\(197\) −1.09317e8 −1.01872 −0.509361 0.860553i \(-0.670118\pi\)
−0.509361 + 0.860553i \(0.670118\pi\)
\(198\) 0 0
\(199\) 3.64317e7 0.327713 0.163857 0.986484i \(-0.447607\pi\)
0.163857 + 0.986484i \(0.447607\pi\)
\(200\) 0 0
\(201\) 1.24000e7 0.107705
\(202\) 0 0
\(203\) −1.91880e8 −1.60988
\(204\) 0 0
\(205\) 1.94939e7 0.158037
\(206\) 0 0
\(207\) 8.37475e6 0.0656260
\(208\) 0 0
\(209\) 7.20976e7 0.546272
\(210\) 0 0
\(211\) −1.38637e7 −0.101599 −0.0507997 0.998709i \(-0.516177\pi\)
−0.0507997 + 0.998709i \(0.516177\pi\)
\(212\) 0 0
\(213\) −487944. −0.00345972
\(214\) 0 0
\(215\) 2.46856e8 1.69398
\(216\) 0 0
\(217\) 1.93955e8 1.28852
\(218\) 0 0
\(219\) 1.15037e7 0.0740086
\(220\) 0 0
\(221\) 2.21089e8 1.37783
\(222\) 0 0
\(223\) −1.35935e8 −0.820850 −0.410425 0.911894i \(-0.634620\pi\)
−0.410425 + 0.911894i \(0.634620\pi\)
\(224\) 0 0
\(225\) 6.55918e7 0.383893
\(226\) 0 0
\(227\) −2.82203e7 −0.160129 −0.0800646 0.996790i \(-0.525513\pi\)
−0.0800646 + 0.996790i \(0.525513\pi\)
\(228\) 0 0
\(229\) −5.31215e7 −0.292312 −0.146156 0.989262i \(-0.546690\pi\)
−0.146156 + 0.989262i \(0.546690\pi\)
\(230\) 0 0
\(231\) −3.69432e7 −0.197194
\(232\) 0 0
\(233\) 1.54589e8 0.800631 0.400316 0.916377i \(-0.368901\pi\)
0.400316 + 0.916377i \(0.368901\pi\)
\(234\) 0 0
\(235\) −2.65352e8 −1.33378
\(236\) 0 0
\(237\) 8.03963e6 0.0392299
\(238\) 0 0
\(239\) 1.86143e8 0.881972 0.440986 0.897514i \(-0.354629\pi\)
0.440986 + 0.897514i \(0.354629\pi\)
\(240\) 0 0
\(241\) 2.62107e8 1.20620 0.603100 0.797666i \(-0.293932\pi\)
0.603100 + 0.797666i \(0.293932\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) −9.56288e7 −0.415439
\(246\) 0 0
\(247\) 7.01909e8 2.96375
\(248\) 0 0
\(249\) 1.53469e8 0.629973
\(250\) 0 0
\(251\) 2.75827e8 1.10098 0.550489 0.834842i \(-0.314441\pi\)
0.550489 + 0.834842i \(0.314441\pi\)
\(252\) 0 0
\(253\) 1.52905e7 0.0593609
\(254\) 0 0
\(255\) 1.88876e8 0.713324
\(256\) 0 0
\(257\) 1.06856e6 0.00392675 0.00196338 0.999998i \(-0.499375\pi\)
0.00196338 + 0.999998i \(0.499375\pi\)
\(258\) 0 0
\(259\) 4.06971e8 1.45551
\(260\) 0 0
\(261\) −1.36071e8 −0.473721
\(262\) 0 0
\(263\) 7.92924e7 0.268774 0.134387 0.990929i \(-0.457094\pi\)
0.134387 + 0.990929i \(0.457094\pi\)
\(264\) 0 0
\(265\) 5.38216e8 1.77663
\(266\) 0 0
\(267\) 1.71260e8 0.550639
\(268\) 0 0
\(269\) 2.10170e8 0.658321 0.329160 0.944274i \(-0.393234\pi\)
0.329160 + 0.944274i \(0.393234\pi\)
\(270\) 0 0
\(271\) 2.65510e8 0.810378 0.405189 0.914233i \(-0.367206\pi\)
0.405189 + 0.914233i \(0.367206\pi\)
\(272\) 0 0
\(273\) −3.59662e8 −1.06986
\(274\) 0 0
\(275\) 1.19757e8 0.347245
\(276\) 0 0
\(277\) −6.23529e8 −1.76270 −0.881349 0.472466i \(-0.843364\pi\)
−0.881349 + 0.472466i \(0.843364\pi\)
\(278\) 0 0
\(279\) 1.37542e8 0.379158
\(280\) 0 0
\(281\) 1.30611e8 0.351162 0.175581 0.984465i \(-0.443820\pi\)
0.175581 + 0.984465i \(0.443820\pi\)
\(282\) 0 0
\(283\) 2.20874e7 0.0579283 0.0289642 0.999580i \(-0.490779\pi\)
0.0289642 + 0.999580i \(0.490779\pi\)
\(284\) 0 0
\(285\) 5.99640e8 1.53438
\(286\) 0 0
\(287\) −4.88773e7 −0.122045
\(288\) 0 0
\(289\) −1.19227e8 −0.290557
\(290\) 0 0
\(291\) −4.49593e8 −1.06953
\(292\) 0 0
\(293\) 2.00188e8 0.464944 0.232472 0.972603i \(-0.425319\pi\)
0.232472 + 0.972603i \(0.425319\pi\)
\(294\) 0 0
\(295\) −1.09927e9 −2.49302
\(296\) 0 0
\(297\) −2.61981e7 −0.0580259
\(298\) 0 0
\(299\) 1.48862e8 0.322057
\(300\) 0 0
\(301\) −6.18946e8 −1.30819
\(302\) 0 0
\(303\) 5.62483e7 0.116161
\(304\) 0 0
\(305\) −2.25988e8 −0.456074
\(306\) 0 0
\(307\) 4.79736e8 0.946276 0.473138 0.880988i \(-0.343121\pi\)
0.473138 + 0.880988i \(0.343121\pi\)
\(308\) 0 0
\(309\) −6.45425e7 −0.124449
\(310\) 0 0
\(311\) 5.19734e8 0.979761 0.489880 0.871790i \(-0.337040\pi\)
0.489880 + 0.871790i \(0.337040\pi\)
\(312\) 0 0
\(313\) 9.69759e8 1.78755 0.893776 0.448514i \(-0.148047\pi\)
0.893776 + 0.448514i \(0.148047\pi\)
\(314\) 0 0
\(315\) −3.07259e8 −0.553882
\(316\) 0 0
\(317\) 7.56875e8 1.33450 0.667248 0.744836i \(-0.267472\pi\)
0.667248 + 0.744836i \(0.267472\pi\)
\(318\) 0 0
\(319\) −2.48436e8 −0.428497
\(320\) 0 0
\(321\) −3.79660e8 −0.640659
\(322\) 0 0
\(323\) 9.24214e8 1.52603
\(324\) 0 0
\(325\) 1.16590e9 1.88394
\(326\) 0 0
\(327\) 3.00566e8 0.475360
\(328\) 0 0
\(329\) 6.65322e8 1.03002
\(330\) 0 0
\(331\) 1.79867e8 0.272618 0.136309 0.990666i \(-0.456476\pi\)
0.136309 + 0.990666i \(0.456476\pi\)
\(332\) 0 0
\(333\) 2.88601e8 0.428295
\(334\) 0 0
\(335\) 1.88297e8 0.273644
\(336\) 0 0
\(337\) −1.38092e9 −1.96546 −0.982728 0.185054i \(-0.940754\pi\)
−0.982728 + 0.185054i \(0.940754\pi\)
\(338\) 0 0
\(339\) −1.53064e8 −0.213390
\(340\) 0 0
\(341\) 2.51122e8 0.342961
\(342\) 0 0
\(343\) −6.06830e8 −0.811966
\(344\) 0 0
\(345\) 1.27172e8 0.166734
\(346\) 0 0
\(347\) −7.66253e8 −0.984507 −0.492254 0.870452i \(-0.663827\pi\)
−0.492254 + 0.870452i \(0.663827\pi\)
\(348\) 0 0
\(349\) −2.68852e8 −0.338552 −0.169276 0.985569i \(-0.554143\pi\)
−0.169276 + 0.985569i \(0.554143\pi\)
\(350\) 0 0
\(351\) −2.55052e8 −0.314814
\(352\) 0 0
\(353\) −3.95002e8 −0.477956 −0.238978 0.971025i \(-0.576812\pi\)
−0.238978 + 0.971025i \(0.576812\pi\)
\(354\) 0 0
\(355\) −7.40952e6 −0.00879004
\(356\) 0 0
\(357\) −4.73573e8 −0.550869
\(358\) 0 0
\(359\) 4.25768e7 0.0485671 0.0242836 0.999705i \(-0.492270\pi\)
0.0242836 + 0.999705i \(0.492270\pi\)
\(360\) 0 0
\(361\) 2.04030e9 2.28254
\(362\) 0 0
\(363\) −4.78321e7 −0.0524864
\(364\) 0 0
\(365\) 1.74685e8 0.188032
\(366\) 0 0
\(367\) −1.85295e9 −1.95673 −0.978366 0.206882i \(-0.933668\pi\)
−0.978366 + 0.206882i \(0.933668\pi\)
\(368\) 0 0
\(369\) −3.46610e7 −0.0359128
\(370\) 0 0
\(371\) −1.34948e9 −1.37201
\(372\) 0 0
\(373\) −4.83602e7 −0.0482511 −0.0241256 0.999709i \(-0.507680\pi\)
−0.0241256 + 0.999709i \(0.507680\pi\)
\(374\) 0 0
\(375\) 1.31180e8 0.128457
\(376\) 0 0
\(377\) −2.41866e9 −2.32477
\(378\) 0 0
\(379\) 2.26078e8 0.213315 0.106658 0.994296i \(-0.465985\pi\)
0.106658 + 0.994296i \(0.465985\pi\)
\(380\) 0 0
\(381\) −5.64760e8 −0.523150
\(382\) 0 0
\(383\) 1.35198e9 1.22963 0.614815 0.788671i \(-0.289231\pi\)
0.614815 + 0.788671i \(0.289231\pi\)
\(384\) 0 0
\(385\) −5.60990e8 −0.501005
\(386\) 0 0
\(387\) −4.38922e8 −0.384945
\(388\) 0 0
\(389\) −1.09107e9 −0.939789 −0.469894 0.882723i \(-0.655708\pi\)
−0.469894 + 0.882723i \(0.655708\pi\)
\(390\) 0 0
\(391\) 1.96008e8 0.165827
\(392\) 0 0
\(393\) −3.04152e8 −0.252765
\(394\) 0 0
\(395\) 1.22083e8 0.0996704
\(396\) 0 0
\(397\) −6.97868e8 −0.559766 −0.279883 0.960034i \(-0.590296\pi\)
−0.279883 + 0.960034i \(0.590296\pi\)
\(398\) 0 0
\(399\) −1.50349e9 −1.18494
\(400\) 0 0
\(401\) 1.74689e9 1.35288 0.676441 0.736497i \(-0.263521\pi\)
0.676441 + 0.736497i \(0.263521\pi\)
\(402\) 0 0
\(403\) 2.44481e9 1.86071
\(404\) 0 0
\(405\) −2.17891e8 −0.162985
\(406\) 0 0
\(407\) 5.26924e8 0.387407
\(408\) 0 0
\(409\) −1.30304e9 −0.941729 −0.470865 0.882205i \(-0.656058\pi\)
−0.470865 + 0.882205i \(0.656058\pi\)
\(410\) 0 0
\(411\) −1.19958e7 −0.00852283
\(412\) 0 0
\(413\) 2.75621e9 1.92525
\(414\) 0 0
\(415\) 2.33045e9 1.60056
\(416\) 0 0
\(417\) 9.25054e8 0.624728
\(418\) 0 0
\(419\) −2.87139e9 −1.90697 −0.953484 0.301443i \(-0.902532\pi\)
−0.953484 + 0.301443i \(0.902532\pi\)
\(420\) 0 0
\(421\) 1.15946e9 0.757299 0.378650 0.925540i \(-0.376389\pi\)
0.378650 + 0.925540i \(0.376389\pi\)
\(422\) 0 0
\(423\) 4.71809e8 0.303092
\(424\) 0 0
\(425\) 1.53515e9 0.970042
\(426\) 0 0
\(427\) 5.66623e8 0.352206
\(428\) 0 0
\(429\) −4.65672e8 −0.284760
\(430\) 0 0
\(431\) 1.66703e9 1.00294 0.501468 0.865176i \(-0.332793\pi\)
0.501468 + 0.865176i \(0.332793\pi\)
\(432\) 0 0
\(433\) 6.34094e8 0.375358 0.187679 0.982230i \(-0.439903\pi\)
0.187679 + 0.982230i \(0.439903\pi\)
\(434\) 0 0
\(435\) −2.06626e9 −1.20357
\(436\) 0 0
\(437\) 6.22282e8 0.356699
\(438\) 0 0
\(439\) 1.22368e9 0.690307 0.345154 0.938546i \(-0.387827\pi\)
0.345154 + 0.938546i \(0.387827\pi\)
\(440\) 0 0
\(441\) 1.70033e8 0.0944055
\(442\) 0 0
\(443\) 1.23213e9 0.673355 0.336677 0.941620i \(-0.390697\pi\)
0.336677 + 0.941620i \(0.390697\pi\)
\(444\) 0 0
\(445\) 2.60062e9 1.39900
\(446\) 0 0
\(447\) 1.30197e9 0.689485
\(448\) 0 0
\(449\) −3.07511e9 −1.60324 −0.801621 0.597833i \(-0.796029\pi\)
−0.801621 + 0.597833i \(0.796029\pi\)
\(450\) 0 0
\(451\) −6.32837e7 −0.0324843
\(452\) 0 0
\(453\) −1.21147e9 −0.612307
\(454\) 0 0
\(455\) −5.46154e9 −2.71816
\(456\) 0 0
\(457\) −2.44730e9 −1.19945 −0.599723 0.800207i \(-0.704723\pi\)
−0.599723 + 0.800207i \(0.704723\pi\)
\(458\) 0 0
\(459\) −3.35831e8 −0.162098
\(460\) 0 0
\(461\) 9.52419e8 0.452767 0.226383 0.974038i \(-0.427310\pi\)
0.226383 + 0.974038i \(0.427310\pi\)
\(462\) 0 0
\(463\) 6.05200e8 0.283378 0.141689 0.989911i \(-0.454747\pi\)
0.141689 + 0.989911i \(0.454747\pi\)
\(464\) 0 0
\(465\) 2.08860e9 0.963318
\(466\) 0 0
\(467\) 1.37708e9 0.625676 0.312838 0.949806i \(-0.398720\pi\)
0.312838 + 0.949806i \(0.398720\pi\)
\(468\) 0 0
\(469\) −4.72119e8 −0.211323
\(470\) 0 0
\(471\) 1.45505e8 0.0641659
\(472\) 0 0
\(473\) −8.01379e8 −0.348196
\(474\) 0 0
\(475\) 4.87377e9 2.08659
\(476\) 0 0
\(477\) −9.56974e8 −0.403725
\(478\) 0 0
\(479\) 4.00222e9 1.66390 0.831949 0.554851i \(-0.187225\pi\)
0.831949 + 0.554851i \(0.187225\pi\)
\(480\) 0 0
\(481\) 5.12989e9 2.10184
\(482\) 0 0
\(483\) −3.18861e8 −0.128762
\(484\) 0 0
\(485\) −6.82715e9 −2.71734
\(486\) 0 0
\(487\) 2.88677e9 1.13256 0.566279 0.824214i \(-0.308383\pi\)
0.566279 + 0.824214i \(0.308383\pi\)
\(488\) 0 0
\(489\) 2.65052e9 1.02506
\(490\) 0 0
\(491\) −1.19743e8 −0.0456525 −0.0228262 0.999739i \(-0.507266\pi\)
−0.0228262 + 0.999739i \(0.507266\pi\)
\(492\) 0 0
\(493\) −3.18469e9 −1.19702
\(494\) 0 0
\(495\) −3.97823e8 −0.147425
\(496\) 0 0
\(497\) 1.85780e7 0.00678816
\(498\) 0 0
\(499\) 4.78950e9 1.72559 0.862796 0.505552i \(-0.168711\pi\)
0.862796 + 0.505552i \(0.168711\pi\)
\(500\) 0 0
\(501\) −1.18965e9 −0.422656
\(502\) 0 0
\(503\) −3.83047e9 −1.34203 −0.671017 0.741442i \(-0.734142\pi\)
−0.671017 + 0.741442i \(0.734142\pi\)
\(504\) 0 0
\(505\) 8.54141e8 0.295127
\(506\) 0 0
\(507\) −2.83935e9 −0.967591
\(508\) 0 0
\(509\) −2.34385e9 −0.787803 −0.393902 0.919153i \(-0.628875\pi\)
−0.393902 + 0.919153i \(0.628875\pi\)
\(510\) 0 0
\(511\) −4.37992e8 −0.145209
\(512\) 0 0
\(513\) −1.06619e9 −0.348677
\(514\) 0 0
\(515\) −9.80090e8 −0.316185
\(516\) 0 0
\(517\) 8.61423e8 0.274157
\(518\) 0 0
\(519\) −1.81194e9 −0.568928
\(520\) 0 0
\(521\) 5.77085e9 1.78775 0.893877 0.448313i \(-0.147975\pi\)
0.893877 + 0.448313i \(0.147975\pi\)
\(522\) 0 0
\(523\) 3.49411e8 0.106802 0.0534012 0.998573i \(-0.482994\pi\)
0.0534012 + 0.998573i \(0.482994\pi\)
\(524\) 0 0
\(525\) −2.49735e9 −0.753219
\(526\) 0 0
\(527\) 3.21912e9 0.958077
\(528\) 0 0
\(529\) −3.27285e9 −0.961239
\(530\) 0 0
\(531\) 1.95455e9 0.566521
\(532\) 0 0
\(533\) −6.16101e8 −0.176241
\(534\) 0 0
\(535\) −5.76520e9 −1.62771
\(536\) 0 0
\(537\) 1.17431e8 0.0327244
\(538\) 0 0
\(539\) 3.10444e8 0.0853930
\(540\) 0 0
\(541\) −5.10025e9 −1.38484 −0.692422 0.721493i \(-0.743456\pi\)
−0.692422 + 0.721493i \(0.743456\pi\)
\(542\) 0 0
\(543\) 3.24643e8 0.0870175
\(544\) 0 0
\(545\) 4.56415e9 1.20774
\(546\) 0 0
\(547\) −4.96217e9 −1.29633 −0.648166 0.761499i \(-0.724464\pi\)
−0.648166 + 0.761499i \(0.724464\pi\)
\(548\) 0 0
\(549\) 4.01818e8 0.103640
\(550\) 0 0
\(551\) −1.01107e10 −2.57484
\(552\) 0 0
\(553\) −3.06101e8 −0.0769710
\(554\) 0 0
\(555\) 4.38246e9 1.08816
\(556\) 0 0
\(557\) 1.42590e9 0.349620 0.174810 0.984602i \(-0.444069\pi\)
0.174810 + 0.984602i \(0.444069\pi\)
\(558\) 0 0
\(559\) −7.80186e9 −1.88911
\(560\) 0 0
\(561\) −6.13157e8 −0.146623
\(562\) 0 0
\(563\) −5.96929e9 −1.40975 −0.704876 0.709330i \(-0.748998\pi\)
−0.704876 + 0.709330i \(0.748998\pi\)
\(564\) 0 0
\(565\) −2.32430e9 −0.542155
\(566\) 0 0
\(567\) 5.46321e8 0.125866
\(568\) 0 0
\(569\) −3.51616e9 −0.800158 −0.400079 0.916481i \(-0.631017\pi\)
−0.400079 + 0.916481i \(0.631017\pi\)
\(570\) 0 0
\(571\) −6.44706e8 −0.144922 −0.0724611 0.997371i \(-0.523085\pi\)
−0.0724611 + 0.997371i \(0.523085\pi\)
\(572\) 0 0
\(573\) 1.61028e9 0.357569
\(574\) 0 0
\(575\) 1.03363e9 0.226740
\(576\) 0 0
\(577\) −2.63322e9 −0.570652 −0.285326 0.958430i \(-0.592102\pi\)
−0.285326 + 0.958430i \(0.592102\pi\)
\(578\) 0 0
\(579\) 2.64880e9 0.567118
\(580\) 0 0
\(581\) −5.84318e9 −1.23604
\(582\) 0 0
\(583\) −1.74723e9 −0.365183
\(584\) 0 0
\(585\) −3.87302e9 −0.799841
\(586\) 0 0
\(587\) −6.76347e9 −1.38018 −0.690090 0.723723i \(-0.742429\pi\)
−0.690090 + 0.723723i \(0.742429\pi\)
\(588\) 0 0
\(589\) 1.02200e10 2.06085
\(590\) 0 0
\(591\) 2.95156e9 0.588159
\(592\) 0 0
\(593\) −4.22718e9 −0.832452 −0.416226 0.909261i \(-0.636648\pi\)
−0.416226 + 0.909261i \(0.636648\pi\)
\(594\) 0 0
\(595\) −7.19129e9 −1.39958
\(596\) 0 0
\(597\) −9.83657e8 −0.189205
\(598\) 0 0
\(599\) −4.00299e9 −0.761010 −0.380505 0.924779i \(-0.624250\pi\)
−0.380505 + 0.924779i \(0.624250\pi\)
\(600\) 0 0
\(601\) 6.67554e9 1.25437 0.627185 0.778870i \(-0.284207\pi\)
0.627185 + 0.778870i \(0.284207\pi\)
\(602\) 0 0
\(603\) −3.34801e8 −0.0621836
\(604\) 0 0
\(605\) −7.26340e8 −0.133351
\(606\) 0 0
\(607\) 5.30634e9 0.963018 0.481509 0.876441i \(-0.340089\pi\)
0.481509 + 0.876441i \(0.340089\pi\)
\(608\) 0 0
\(609\) 5.18077e9 0.929466
\(610\) 0 0
\(611\) 8.38642e9 1.48742
\(612\) 0 0
\(613\) −8.65802e9 −1.51812 −0.759061 0.651019i \(-0.774342\pi\)
−0.759061 + 0.651019i \(0.774342\pi\)
\(614\) 0 0
\(615\) −5.26334e8 −0.0912428
\(616\) 0 0
\(617\) −7.38891e9 −1.26643 −0.633217 0.773974i \(-0.718266\pi\)
−0.633217 + 0.773974i \(0.718266\pi\)
\(618\) 0 0
\(619\) −9.99141e9 −1.69321 −0.846603 0.532225i \(-0.821356\pi\)
−0.846603 + 0.532225i \(0.821356\pi\)
\(620\) 0 0
\(621\) −2.26118e8 −0.0378892
\(622\) 0 0
\(623\) −6.52057e9 −1.08038
\(624\) 0 0
\(625\) −5.03731e9 −0.825313
\(626\) 0 0
\(627\) −1.94664e9 −0.315390
\(628\) 0 0
\(629\) 6.75461e9 1.08224
\(630\) 0 0
\(631\) 3.29834e9 0.522628 0.261314 0.965254i \(-0.415844\pi\)
0.261314 + 0.965254i \(0.415844\pi\)
\(632\) 0 0
\(633\) 3.74320e8 0.0586584
\(634\) 0 0
\(635\) −8.57598e9 −1.32916
\(636\) 0 0
\(637\) 3.02234e9 0.463292
\(638\) 0 0
\(639\) 1.31745e7 0.00199747
\(640\) 0 0
\(641\) −9.76971e9 −1.46514 −0.732569 0.680692i \(-0.761679\pi\)
−0.732569 + 0.680692i \(0.761679\pi\)
\(642\) 0 0
\(643\) 4.18444e9 0.620724 0.310362 0.950618i \(-0.399550\pi\)
0.310362 + 0.950618i \(0.399550\pi\)
\(644\) 0 0
\(645\) −6.66511e9 −0.978022
\(646\) 0 0
\(647\) 6.96085e8 0.101041 0.0505204 0.998723i \(-0.483912\pi\)
0.0505204 + 0.998723i \(0.483912\pi\)
\(648\) 0 0
\(649\) 3.56860e9 0.512438
\(650\) 0 0
\(651\) −5.23678e9 −0.743928
\(652\) 0 0
\(653\) 6.20046e9 0.871420 0.435710 0.900087i \(-0.356497\pi\)
0.435710 + 0.900087i \(0.356497\pi\)
\(654\) 0 0
\(655\) −4.61861e9 −0.642194
\(656\) 0 0
\(657\) −3.10599e8 −0.0427289
\(658\) 0 0
\(659\) 1.11404e10 1.51636 0.758178 0.652047i \(-0.226090\pi\)
0.758178 + 0.652047i \(0.226090\pi\)
\(660\) 0 0
\(661\) 4.56096e9 0.614258 0.307129 0.951668i \(-0.400632\pi\)
0.307129 + 0.951668i \(0.400632\pi\)
\(662\) 0 0
\(663\) −5.96941e9 −0.795489
\(664\) 0 0
\(665\) −2.28307e10 −3.01054
\(666\) 0 0
\(667\) −2.14428e9 −0.279796
\(668\) 0 0
\(669\) 3.67024e9 0.473918
\(670\) 0 0
\(671\) 7.33634e8 0.0937455
\(672\) 0 0
\(673\) 5.82879e9 0.737099 0.368550 0.929608i \(-0.379854\pi\)
0.368550 + 0.929608i \(0.379854\pi\)
\(674\) 0 0
\(675\) −1.77098e9 −0.221641
\(676\) 0 0
\(677\) 4.99624e9 0.618846 0.309423 0.950924i \(-0.399864\pi\)
0.309423 + 0.950924i \(0.399864\pi\)
\(678\) 0 0
\(679\) 1.71178e10 2.09848
\(680\) 0 0
\(681\) 7.61947e8 0.0924507
\(682\) 0 0
\(683\) −1.21371e10 −1.45762 −0.728808 0.684718i \(-0.759925\pi\)
−0.728808 + 0.684718i \(0.759925\pi\)
\(684\) 0 0
\(685\) −1.82159e8 −0.0216538
\(686\) 0 0
\(687\) 1.43428e9 0.168766
\(688\) 0 0
\(689\) −1.70103e10 −1.98127
\(690\) 0 0
\(691\) 9.23403e9 1.06468 0.532339 0.846531i \(-0.321313\pi\)
0.532339 + 0.846531i \(0.321313\pi\)
\(692\) 0 0
\(693\) 9.97467e8 0.113850
\(694\) 0 0
\(695\) 1.40471e10 1.58723
\(696\) 0 0
\(697\) −8.11230e8 −0.0907464
\(698\) 0 0
\(699\) −4.17390e9 −0.462245
\(700\) 0 0
\(701\) 4.74530e9 0.520296 0.260148 0.965569i \(-0.416229\pi\)
0.260148 + 0.965569i \(0.416229\pi\)
\(702\) 0 0
\(703\) 2.14444e10 2.32793
\(704\) 0 0
\(705\) 7.16450e9 0.770059
\(706\) 0 0
\(707\) −2.14160e9 −0.227914
\(708\) 0 0
\(709\) 1.34547e10 1.41779 0.708894 0.705315i \(-0.249195\pi\)
0.708894 + 0.705315i \(0.249195\pi\)
\(710\) 0 0
\(711\) −2.17070e8 −0.0226494
\(712\) 0 0
\(713\) 2.16746e9 0.223944
\(714\) 0 0
\(715\) −7.07131e9 −0.723484
\(716\) 0 0
\(717\) −5.02587e9 −0.509207
\(718\) 0 0
\(719\) −2.63976e9 −0.264858 −0.132429 0.991192i \(-0.542278\pi\)
−0.132429 + 0.991192i \(0.542278\pi\)
\(720\) 0 0
\(721\) 2.45740e9 0.244175
\(722\) 0 0
\(723\) −7.07689e9 −0.696400
\(724\) 0 0
\(725\) −1.67942e10 −1.63673
\(726\) 0 0
\(727\) 3.52707e9 0.340442 0.170221 0.985406i \(-0.445552\pi\)
0.170221 + 0.985406i \(0.445552\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) −1.02728e10 −0.972700
\(732\) 0 0
\(733\) −1.03828e10 −0.973760 −0.486880 0.873469i \(-0.661865\pi\)
−0.486880 + 0.873469i \(0.661865\pi\)
\(734\) 0 0
\(735\) 2.58198e9 0.239854
\(736\) 0 0
\(737\) −6.11275e8 −0.0562471
\(738\) 0 0
\(739\) −2.05418e9 −0.187233 −0.0936164 0.995608i \(-0.529843\pi\)
−0.0936164 + 0.995608i \(0.529843\pi\)
\(740\) 0 0
\(741\) −1.89515e10 −1.71112
\(742\) 0 0
\(743\) −4.87476e9 −0.436006 −0.218003 0.975948i \(-0.569954\pi\)
−0.218003 + 0.975948i \(0.569954\pi\)
\(744\) 0 0
\(745\) 1.97707e10 1.75176
\(746\) 0 0
\(747\) −4.14366e9 −0.363715
\(748\) 0 0
\(749\) 1.44552e10 1.25701
\(750\) 0 0
\(751\) −1.15809e10 −0.997705 −0.498853 0.866687i \(-0.666245\pi\)
−0.498853 + 0.866687i \(0.666245\pi\)
\(752\) 0 0
\(753\) −7.44733e9 −0.635650
\(754\) 0 0
\(755\) −1.83964e10 −1.55567
\(756\) 0 0
\(757\) 3.46735e9 0.290511 0.145255 0.989394i \(-0.453600\pi\)
0.145255 + 0.989394i \(0.453600\pi\)
\(758\) 0 0
\(759\) −4.12844e8 −0.0342720
\(760\) 0 0
\(761\) −1.14023e10 −0.937877 −0.468938 0.883231i \(-0.655363\pi\)
−0.468938 + 0.883231i \(0.655363\pi\)
\(762\) 0 0
\(763\) −1.14438e10 −0.932681
\(764\) 0 0
\(765\) −5.09966e9 −0.411838
\(766\) 0 0
\(767\) 3.47422e10 2.78018
\(768\) 0 0
\(769\) 2.30715e10 1.82951 0.914754 0.404012i \(-0.132384\pi\)
0.914754 + 0.404012i \(0.132384\pi\)
\(770\) 0 0
\(771\) −2.88512e7 −0.00226711
\(772\) 0 0
\(773\) 2.15091e10 1.67492 0.837461 0.546497i \(-0.184039\pi\)
0.837461 + 0.546497i \(0.184039\pi\)
\(774\) 0 0
\(775\) 1.69758e10 1.31001
\(776\) 0 0
\(777\) −1.09882e10 −0.840337
\(778\) 0 0
\(779\) −2.57547e9 −0.195198
\(780\) 0 0
\(781\) 2.40538e7 0.00180678
\(782\) 0 0
\(783\) 3.67391e9 0.273503
\(784\) 0 0
\(785\) 2.20952e9 0.163025
\(786\) 0 0
\(787\) 4.46678e9 0.326651 0.163325 0.986572i \(-0.447778\pi\)
0.163325 + 0.986572i \(0.447778\pi\)
\(788\) 0 0
\(789\) −2.14090e9 −0.155176
\(790\) 0 0
\(791\) 5.82777e9 0.418682
\(792\) 0 0
\(793\) 7.14232e9 0.508608
\(794\) 0 0
\(795\) −1.45318e10 −1.02574
\(796\) 0 0
\(797\) 2.43899e10 1.70650 0.853248 0.521505i \(-0.174629\pi\)
0.853248 + 0.521505i \(0.174629\pi\)
\(798\) 0 0
\(799\) 1.10425e10 0.765869
\(800\) 0 0
\(801\) −4.62402e9 −0.317911
\(802\) 0 0
\(803\) −5.67089e8 −0.0386497
\(804\) 0 0
\(805\) −4.84196e9 −0.327142
\(806\) 0 0
\(807\) −5.67459e9 −0.380082
\(808\) 0 0
\(809\) 9.88857e9 0.656620 0.328310 0.944570i \(-0.393521\pi\)
0.328310 + 0.944570i \(0.393521\pi\)
\(810\) 0 0
\(811\) 1.15204e10 0.758395 0.379198 0.925316i \(-0.376200\pi\)
0.379198 + 0.925316i \(0.376200\pi\)
\(812\) 0 0
\(813\) −7.16876e9 −0.467872
\(814\) 0 0
\(815\) 4.02487e10 2.60435
\(816\) 0 0
\(817\) −3.26139e10 −2.09231
\(818\) 0 0
\(819\) 9.71088e9 0.617682
\(820\) 0 0
\(821\) 2.63516e9 0.166191 0.0830953 0.996542i \(-0.473519\pi\)
0.0830953 + 0.996542i \(0.473519\pi\)
\(822\) 0 0
\(823\) 1.27039e10 0.794400 0.397200 0.917732i \(-0.369982\pi\)
0.397200 + 0.917732i \(0.369982\pi\)
\(824\) 0 0
\(825\) −3.23343e9 −0.200482
\(826\) 0 0
\(827\) 1.11339e10 0.684504 0.342252 0.939608i \(-0.388810\pi\)
0.342252 + 0.939608i \(0.388810\pi\)
\(828\) 0 0
\(829\) 2.79852e10 1.70604 0.853018 0.521881i \(-0.174770\pi\)
0.853018 + 0.521881i \(0.174770\pi\)
\(830\) 0 0
\(831\) 1.68353e10 1.01769
\(832\) 0 0
\(833\) 3.97956e9 0.238549
\(834\) 0 0
\(835\) −1.80651e10 −1.07383
\(836\) 0 0
\(837\) −3.71363e9 −0.218907
\(838\) 0 0
\(839\) −2.71170e9 −0.158517 −0.0792583 0.996854i \(-0.525255\pi\)
−0.0792583 + 0.996854i \(0.525255\pi\)
\(840\) 0 0
\(841\) 1.75898e10 1.01971
\(842\) 0 0
\(843\) −3.52650e9 −0.202744
\(844\) 0 0
\(845\) −4.31161e10 −2.45834
\(846\) 0 0
\(847\) 1.82116e9 0.102981
\(848\) 0 0
\(849\) −5.96359e8 −0.0334449
\(850\) 0 0
\(851\) 4.54794e9 0.252965
\(852\) 0 0
\(853\) 1.97175e10 1.08775 0.543877 0.839165i \(-0.316956\pi\)
0.543877 + 0.839165i \(0.316956\pi\)
\(854\) 0 0
\(855\) −1.61903e10 −0.885876
\(856\) 0 0
\(857\) 1.89411e10 1.02795 0.513976 0.857804i \(-0.328172\pi\)
0.513976 + 0.857804i \(0.328172\pi\)
\(858\) 0 0
\(859\) 6.77637e9 0.364772 0.182386 0.983227i \(-0.441618\pi\)
0.182386 + 0.983227i \(0.441618\pi\)
\(860\) 0 0
\(861\) 1.31969e9 0.0704628
\(862\) 0 0
\(863\) −2.80635e10 −1.48629 −0.743146 0.669129i \(-0.766667\pi\)
−0.743146 + 0.669129i \(0.766667\pi\)
\(864\) 0 0
\(865\) −2.75146e10 −1.44546
\(866\) 0 0
\(867\) 3.21912e9 0.167753
\(868\) 0 0
\(869\) −3.96324e8 −0.0204871
\(870\) 0 0
\(871\) −5.95109e9 −0.305164
\(872\) 0 0
\(873\) 1.21390e10 0.617495
\(874\) 0 0
\(875\) −4.99454e9 −0.252039
\(876\) 0 0
\(877\) −1.01559e10 −0.508418 −0.254209 0.967149i \(-0.581815\pi\)
−0.254209 + 0.967149i \(0.581815\pi\)
\(878\) 0 0
\(879\) −5.40507e9 −0.268436
\(880\) 0 0
\(881\) −2.78023e10 −1.36982 −0.684912 0.728626i \(-0.740159\pi\)
−0.684912 + 0.728626i \(0.740159\pi\)
\(882\) 0 0
\(883\) −2.15199e10 −1.05191 −0.525954 0.850513i \(-0.676291\pi\)
−0.525954 + 0.850513i \(0.676291\pi\)
\(884\) 0 0
\(885\) 2.96802e10 1.43935
\(886\) 0 0
\(887\) 3.13376e10 1.50776 0.753881 0.657011i \(-0.228180\pi\)
0.753881 + 0.657011i \(0.228180\pi\)
\(888\) 0 0
\(889\) 2.15027e10 1.02645
\(890\) 0 0
\(891\) 7.07348e8 0.0335013
\(892\) 0 0
\(893\) 3.50575e10 1.64741
\(894\) 0 0
\(895\) 1.78321e9 0.0831423
\(896\) 0 0
\(897\) −4.01926e9 −0.185940
\(898\) 0 0
\(899\) −3.52164e10 −1.61654
\(900\) 0 0
\(901\) −2.23977e10 −1.02015
\(902\) 0 0
\(903\) 1.67116e10 0.755283
\(904\) 0 0
\(905\) 4.92976e9 0.221083
\(906\) 0 0
\(907\) 1.62459e10 0.722966 0.361483 0.932379i \(-0.382270\pi\)
0.361483 + 0.932379i \(0.382270\pi\)
\(908\) 0 0
\(909\) −1.51870e9 −0.0670656
\(910\) 0 0
\(911\) 3.15726e9 0.138356 0.0691778 0.997604i \(-0.477962\pi\)
0.0691778 + 0.997604i \(0.477962\pi\)
\(912\) 0 0
\(913\) −7.56544e9 −0.328993
\(914\) 0 0
\(915\) 6.10167e9 0.263315
\(916\) 0 0
\(917\) 1.15803e10 0.495938
\(918\) 0 0
\(919\) −2.53655e9 −0.107805 −0.0539026 0.998546i \(-0.517166\pi\)
−0.0539026 + 0.998546i \(0.517166\pi\)
\(920\) 0 0
\(921\) −1.29529e10 −0.546333
\(922\) 0 0
\(923\) 2.34177e8 0.00980253
\(924\) 0 0
\(925\) 3.56198e10 1.47978
\(926\) 0 0
\(927\) 1.74265e9 0.0718506
\(928\) 0 0
\(929\) −2.10282e10 −0.860494 −0.430247 0.902711i \(-0.641574\pi\)
−0.430247 + 0.902711i \(0.641574\pi\)
\(930\) 0 0
\(931\) 1.26342e10 0.513126
\(932\) 0 0
\(933\) −1.40328e10 −0.565665
\(934\) 0 0
\(935\) −9.31090e9 −0.372521
\(936\) 0 0
\(937\) 3.12893e10 1.24253 0.621265 0.783601i \(-0.286619\pi\)
0.621265 + 0.783601i \(0.286619\pi\)
\(938\) 0 0
\(939\) −2.61835e10 −1.03204
\(940\) 0 0
\(941\) 7.91706e9 0.309742 0.154871 0.987935i \(-0.450504\pi\)
0.154871 + 0.987935i \(0.450504\pi\)
\(942\) 0 0
\(943\) −5.46208e8 −0.0212113
\(944\) 0 0
\(945\) 8.29599e9 0.319784
\(946\) 0 0
\(947\) −8.55849e9 −0.327471 −0.163735 0.986504i \(-0.552354\pi\)
−0.163735 + 0.986504i \(0.552354\pi\)
\(948\) 0 0
\(949\) −5.52091e9 −0.209691
\(950\) 0 0
\(951\) −2.04356e10 −0.770471
\(952\) 0 0
\(953\) −8.49661e9 −0.317995 −0.158998 0.987279i \(-0.550826\pi\)
−0.158998 + 0.987279i \(0.550826\pi\)
\(954\) 0 0
\(955\) 2.44524e10 0.908466
\(956\) 0 0
\(957\) 6.70778e9 0.247393
\(958\) 0 0
\(959\) 4.56730e8 0.0167222
\(960\) 0 0
\(961\) 8.08451e9 0.293847
\(962\) 0 0
\(963\) 1.02508e10 0.369885
\(964\) 0 0
\(965\) 4.02225e10 1.44086
\(966\) 0 0
\(967\) 1.47988e10 0.526300 0.263150 0.964755i \(-0.415239\pi\)
0.263150 + 0.964755i \(0.415239\pi\)
\(968\) 0 0
\(969\) −2.49538e10 −0.881056
\(970\) 0 0
\(971\) 2.86157e10 1.00308 0.501542 0.865133i \(-0.332766\pi\)
0.501542 + 0.865133i \(0.332766\pi\)
\(972\) 0 0
\(973\) −3.52206e10 −1.22575
\(974\) 0 0
\(975\) −3.14792e10 −1.08770
\(976\) 0 0
\(977\) 3.37991e10 1.15951 0.579755 0.814791i \(-0.303148\pi\)
0.579755 + 0.814791i \(0.303148\pi\)
\(978\) 0 0
\(979\) −8.44249e9 −0.287562
\(980\) 0 0
\(981\) −8.11528e9 −0.274449
\(982\) 0 0
\(983\) −1.03134e9 −0.0346308 −0.0173154 0.999850i \(-0.505512\pi\)
−0.0173154 + 0.999850i \(0.505512\pi\)
\(984\) 0 0
\(985\) 4.48199e10 1.49432
\(986\) 0 0
\(987\) −1.79637e10 −0.594683
\(988\) 0 0
\(989\) −6.91679e9 −0.227362
\(990\) 0 0
\(991\) 5.63139e10 1.83805 0.919027 0.394195i \(-0.128977\pi\)
0.919027 + 0.394195i \(0.128977\pi\)
\(992\) 0 0
\(993\) −4.85642e9 −0.157396
\(994\) 0 0
\(995\) −1.49370e10 −0.480710
\(996\) 0 0
\(997\) 2.55531e10 0.816603 0.408301 0.912847i \(-0.366121\pi\)
0.408301 + 0.912847i \(0.366121\pi\)
\(998\) 0 0
\(999\) −7.79222e9 −0.247276
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 528.8.a.a.1.1 1
4.3 odd 2 33.8.a.a.1.1 1
12.11 even 2 99.8.a.a.1.1 1
44.43 even 2 363.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.8.a.a.1.1 1 4.3 odd 2
99.8.a.a.1.1 1 12.11 even 2
363.8.a.a.1.1 1 44.43 even 2
528.8.a.a.1.1 1 1.1 even 1 trivial