Properties

Label 576.8.a.bd.1.1
Level $576$
Weight $8$
Character 576.1
Self dual yes
Analytic conductor $179.934$
Analytic rank $0$
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] = [576,8,Mod(1,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("576.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(179.933774679\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{46}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-6.78233\) of defining polynomial
Character \(\chi\) \(=\) 576.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-437.552 q^{5} +1722.21 q^{7} +O(q^{10})\) \(q-437.552 q^{5} +1722.21 q^{7} +4226.62 q^{11} +6638.83 q^{13} +34165.6 q^{17} +20358.1 q^{19} +45985.9 q^{23} +113327. q^{25} -51462.3 q^{29} -15184.8 q^{31} -753555. q^{35} +384588. q^{37} +602282. q^{41} +248405. q^{43} -718924. q^{47} +2.14246e6 q^{49} -1.39480e6 q^{53} -1.84937e6 q^{55} -1.19713e6 q^{59} -764413. q^{61} -2.90483e6 q^{65} +2.54127e6 q^{67} -4.66313e6 q^{71} -339311. q^{73} +7.27912e6 q^{77} +328624. q^{79} -1.77607e6 q^{83} -1.49492e7 q^{85} -582340. q^{89} +1.14334e7 q^{91} -8.90771e6 q^{95} -3.28652e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 224 q^{5} + 840 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 224 q^{5} + 840 q^{7} + 640 q^{11} + 2860 q^{13} + 35776 q^{17} + 14672 q^{19} + 118016 q^{23} + 80806 q^{25} - 306720 q^{29} - 95480 q^{31} - 941952 q^{35} - 53820 q^{37} + 1143360 q^{41} + 679120 q^{43} + 567552 q^{47} + 2097202 q^{49} - 1165088 q^{53} - 2615296 q^{55} + 121600 q^{59} - 3143564 q^{61} - 3711808 q^{65} + 5468000 q^{67} - 6763520 q^{71} + 5009420 q^{73} + 10443264 q^{77} - 5450104 q^{79} + 6071168 q^{83} - 14605312 q^{85} - 7063680 q^{89} + 14767152 q^{91} - 10121984 q^{95} + 5219740 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −437.552 −1.56543 −0.782717 0.622378i \(-0.786167\pi\)
−0.782717 + 0.622378i \(0.786167\pi\)
\(6\) 0 0
\(7\) 1722.21 1.89776 0.948882 0.315630i \(-0.102216\pi\)
0.948882 + 0.315630i \(0.102216\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4226.62 0.957456 0.478728 0.877963i \(-0.341098\pi\)
0.478728 + 0.877963i \(0.341098\pi\)
\(12\) 0 0
\(13\) 6638.83 0.838088 0.419044 0.907966i \(-0.362365\pi\)
0.419044 + 0.907966i \(0.362365\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 34165.6 1.68662 0.843311 0.537426i \(-0.180603\pi\)
0.843311 + 0.537426i \(0.180603\pi\)
\(18\) 0 0
\(19\) 20358.1 0.680925 0.340462 0.940258i \(-0.389416\pi\)
0.340462 + 0.940258i \(0.389416\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 45985.9 0.788093 0.394047 0.919090i \(-0.371075\pi\)
0.394047 + 0.919090i \(0.371075\pi\)
\(24\) 0 0
\(25\) 113327. 1.45058
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −51462.3 −0.391828 −0.195914 0.980621i \(-0.562767\pi\)
−0.195914 + 0.980621i \(0.562767\pi\)
\(30\) 0 0
\(31\) −15184.8 −0.0915469 −0.0457734 0.998952i \(-0.514575\pi\)
−0.0457734 + 0.998952i \(0.514575\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −753555. −2.97082
\(36\) 0 0
\(37\) 384588. 1.24821 0.624107 0.781339i \(-0.285463\pi\)
0.624107 + 0.781339i \(0.285463\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 602282. 1.36476 0.682380 0.730998i \(-0.260945\pi\)
0.682380 + 0.730998i \(0.260945\pi\)
\(42\) 0 0
\(43\) 248405. 0.476455 0.238227 0.971209i \(-0.423434\pi\)
0.238227 + 0.971209i \(0.423434\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −718924. −1.01004 −0.505022 0.863106i \(-0.668516\pi\)
−0.505022 + 0.863106i \(0.668516\pi\)
\(48\) 0 0
\(49\) 2.14246e6 2.60151
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.39480e6 −1.28690 −0.643450 0.765488i \(-0.722498\pi\)
−0.643450 + 0.765488i \(0.722498\pi\)
\(54\) 0 0
\(55\) −1.84937e6 −1.49883
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.19713e6 −0.758857 −0.379429 0.925221i \(-0.623879\pi\)
−0.379429 + 0.925221i \(0.623879\pi\)
\(60\) 0 0
\(61\) −764413. −0.431195 −0.215598 0.976482i \(-0.569170\pi\)
−0.215598 + 0.976482i \(0.569170\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.90483e6 −1.31197
\(66\) 0 0
\(67\) 2.54127e6 1.03226 0.516131 0.856510i \(-0.327372\pi\)
0.516131 + 0.856510i \(0.327372\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.66313e6 −1.54623 −0.773114 0.634267i \(-0.781302\pi\)
−0.773114 + 0.634267i \(0.781302\pi\)
\(72\) 0 0
\(73\) −339311. −0.102086 −0.0510432 0.998696i \(-0.516255\pi\)
−0.0510432 + 0.998696i \(0.516255\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.27912e6 1.81703
\(78\) 0 0
\(79\) 328624. 0.0749902 0.0374951 0.999297i \(-0.488062\pi\)
0.0374951 + 0.999297i \(0.488062\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.77607e6 −0.340947 −0.170474 0.985362i \(-0.554530\pi\)
−0.170474 + 0.985362i \(0.554530\pi\)
\(84\) 0 0
\(85\) −1.49492e7 −2.64029
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −582340. −0.0875612 −0.0437806 0.999041i \(-0.513940\pi\)
−0.0437806 + 0.999041i \(0.513940\pi\)
\(90\) 0 0
\(91\) 1.14334e7 1.59049
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.90771e6 −1.06594
\(96\) 0 0
\(97\) −3.28652e6 −0.365625 −0.182812 0.983148i \(-0.558520\pi\)
−0.182812 + 0.983148i \(0.558520\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.54549e6 −0.342414 −0.171207 0.985235i \(-0.554767\pi\)
−0.171207 + 0.985235i \(0.554767\pi\)
\(102\) 0 0
\(103\) −1.24694e7 −1.12438 −0.562192 0.827007i \(-0.690042\pi\)
−0.562192 + 0.827007i \(0.690042\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.77904e7 1.40392 0.701962 0.712214i \(-0.252308\pi\)
0.701962 + 0.712214i \(0.252308\pi\)
\(108\) 0 0
\(109\) −668188. −0.0494204 −0.0247102 0.999695i \(-0.507866\pi\)
−0.0247102 + 0.999695i \(0.507866\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.59643e7 1.04082 0.520408 0.853918i \(-0.325780\pi\)
0.520408 + 0.853918i \(0.325780\pi\)
\(114\) 0 0
\(115\) −2.01212e7 −1.23371
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.88402e7 3.20081
\(120\) 0 0
\(121\) −1.62284e6 −0.0832772
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.54025e7 −0.705354
\(126\) 0 0
\(127\) 3.97232e7 1.72081 0.860403 0.509615i \(-0.170212\pi\)
0.860403 + 0.509615i \(0.170212\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −6.07841e6 −0.236233 −0.118116 0.993000i \(-0.537686\pi\)
−0.118116 + 0.993000i \(0.537686\pi\)
\(132\) 0 0
\(133\) 3.50608e7 1.29223
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.87491e7 −0.955216 −0.477608 0.878573i \(-0.658496\pi\)
−0.477608 + 0.878573i \(0.658496\pi\)
\(138\) 0 0
\(139\) 4.65405e7 1.46987 0.734935 0.678138i \(-0.237213\pi\)
0.734935 + 0.678138i \(0.237213\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.80598e7 0.802433
\(144\) 0 0
\(145\) 2.25174e7 0.613381
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.10645e7 −0.769330 −0.384665 0.923056i \(-0.625683\pi\)
−0.384665 + 0.923056i \(0.625683\pi\)
\(150\) 0 0
\(151\) 1.42377e7 0.336528 0.168264 0.985742i \(-0.446184\pi\)
0.168264 + 0.985742i \(0.446184\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.64414e6 0.143311
\(156\) 0 0
\(157\) 7.35878e7 1.51760 0.758800 0.651324i \(-0.225786\pi\)
0.758800 + 0.651324i \(0.225786\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.91973e7 1.49562
\(162\) 0 0
\(163\) −8.10928e7 −1.46665 −0.733324 0.679880i \(-0.762032\pi\)
−0.733324 + 0.679880i \(0.762032\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.11778e8 −1.85716 −0.928581 0.371130i \(-0.878971\pi\)
−0.928581 + 0.371130i \(0.878971\pi\)
\(168\) 0 0
\(169\) −1.86745e7 −0.297608
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.97788e7 1.31829 0.659147 0.752014i \(-0.270917\pi\)
0.659147 + 0.752014i \(0.270917\pi\)
\(174\) 0 0
\(175\) 1.95172e8 2.75286
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.65667e7 0.606862 0.303431 0.952853i \(-0.401868\pi\)
0.303431 + 0.952853i \(0.401868\pi\)
\(180\) 0 0
\(181\) 6.73507e7 0.844242 0.422121 0.906539i \(-0.361286\pi\)
0.422121 + 0.906539i \(0.361286\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.68277e8 −1.95400
\(186\) 0 0
\(187\) 1.44405e8 1.61487
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.95557e7 −0.306920 −0.153460 0.988155i \(-0.549042\pi\)
−0.153460 + 0.988155i \(0.549042\pi\)
\(192\) 0 0
\(193\) −1.23751e8 −1.23908 −0.619540 0.784965i \(-0.712681\pi\)
−0.619540 + 0.784965i \(0.712681\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.28755e7 0.213177 0.106588 0.994303i \(-0.466007\pi\)
0.106588 + 0.994303i \(0.466007\pi\)
\(198\) 0 0
\(199\) −8.69164e7 −0.781837 −0.390918 0.920425i \(-0.627842\pi\)
−0.390918 + 0.920425i \(0.627842\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.86287e7 −0.743598
\(204\) 0 0
\(205\) −2.63530e8 −2.13644
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.60459e7 0.651956
\(210\) 0 0
\(211\) 3.82685e7 0.280448 0.140224 0.990120i \(-0.455218\pi\)
0.140224 + 0.990120i \(0.455218\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.08690e8 −0.745858
\(216\) 0 0
\(217\) −2.61514e7 −0.173734
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.26820e8 1.41354
\(222\) 0 0
\(223\) 2.07224e8 1.25133 0.625667 0.780090i \(-0.284827\pi\)
0.625667 + 0.780090i \(0.284827\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.07779e8 −1.17899 −0.589497 0.807770i \(-0.700674\pi\)
−0.589497 + 0.807770i \(0.700674\pi\)
\(228\) 0 0
\(229\) −1.86435e8 −1.02589 −0.512947 0.858420i \(-0.671446\pi\)
−0.512947 + 0.858420i \(0.671446\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.55106e7 −0.0803309 −0.0401654 0.999193i \(-0.512788\pi\)
−0.0401654 + 0.999193i \(0.512788\pi\)
\(234\) 0 0
\(235\) 3.14566e8 1.58116
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.74511e8 −0.826856 −0.413428 0.910537i \(-0.635669\pi\)
−0.413428 + 0.910537i \(0.635669\pi\)
\(240\) 0 0
\(241\) −7.69055e7 −0.353914 −0.176957 0.984219i \(-0.556625\pi\)
−0.176957 + 0.984219i \(0.556625\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.37435e8 −4.07249
\(246\) 0 0
\(247\) 1.35154e8 0.570675
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.99888e8 −0.797865 −0.398933 0.916980i \(-0.630619\pi\)
−0.398933 + 0.916980i \(0.630619\pi\)
\(252\) 0 0
\(253\) 1.94365e8 0.754565
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.96410e8 0.721769 0.360884 0.932611i \(-0.382475\pi\)
0.360884 + 0.932611i \(0.382475\pi\)
\(258\) 0 0
\(259\) 6.62339e8 2.36882
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 3.80764e8 1.29066 0.645328 0.763905i \(-0.276721\pi\)
0.645328 + 0.763905i \(0.276721\pi\)
\(264\) 0 0
\(265\) 6.10295e8 2.01456
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.47962e8 1.40316 0.701581 0.712590i \(-0.252478\pi\)
0.701581 + 0.712590i \(0.252478\pi\)
\(270\) 0 0
\(271\) 1.62567e8 0.496180 0.248090 0.968737i \(-0.420197\pi\)
0.248090 + 0.968737i \(0.420197\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.78989e8 1.38887
\(276\) 0 0
\(277\) −2.52029e8 −0.712477 −0.356239 0.934395i \(-0.615941\pi\)
−0.356239 + 0.934395i \(0.615941\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.40349e7 −0.145279 −0.0726393 0.997358i \(-0.523142\pi\)
−0.0726393 + 0.997358i \(0.523142\pi\)
\(282\) 0 0
\(283\) 5.76782e8 1.51272 0.756361 0.654155i \(-0.226976\pi\)
0.756361 + 0.654155i \(0.226976\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.03725e9 2.58999
\(288\) 0 0
\(289\) 7.56949e8 1.84469
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.67850e8 1.31885 0.659427 0.751769i \(-0.270799\pi\)
0.659427 + 0.751769i \(0.270799\pi\)
\(294\) 0 0
\(295\) 5.23807e8 1.18794
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.05293e8 0.660492
\(300\) 0 0
\(301\) 4.27806e8 0.904199
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.34471e8 0.675007
\(306\) 0 0
\(307\) 2.12785e8 0.419718 0.209859 0.977732i \(-0.432699\pi\)
0.209859 + 0.977732i \(0.432699\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.52178e8 −0.852408 −0.426204 0.904627i \(-0.640149\pi\)
−0.426204 + 0.904627i \(0.640149\pi\)
\(312\) 0 0
\(313\) −8.42763e8 −1.55346 −0.776730 0.629834i \(-0.783123\pi\)
−0.776730 + 0.629834i \(0.783123\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.02806e8 −0.181264 −0.0906318 0.995884i \(-0.528889\pi\)
−0.0906318 + 0.995884i \(0.528889\pi\)
\(318\) 0 0
\(319\) −2.17512e8 −0.375159
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.95546e8 1.14846
\(324\) 0 0
\(325\) 7.52356e8 1.21571
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.23814e9 −1.91683
\(330\) 0 0
\(331\) −4.55210e8 −0.689944 −0.344972 0.938613i \(-0.612112\pi\)
−0.344972 + 0.938613i \(0.612112\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.11194e9 −1.61594
\(336\) 0 0
\(337\) 1.38644e8 0.197331 0.0986655 0.995121i \(-0.468543\pi\)
0.0986655 + 0.995121i \(0.468543\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.41805e7 −0.0876522
\(342\) 0 0
\(343\) 2.27144e9 3.03929
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.99476e8 1.02719 0.513597 0.858032i \(-0.328313\pi\)
0.513597 + 0.858032i \(0.328313\pi\)
\(348\) 0 0
\(349\) −9.15205e8 −1.15247 −0.576234 0.817284i \(-0.695478\pi\)
−0.576234 + 0.817284i \(0.695478\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.76398e8 0.818448 0.409224 0.912434i \(-0.365799\pi\)
0.409224 + 0.912434i \(0.365799\pi\)
\(354\) 0 0
\(355\) 2.04036e9 2.42052
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.99641e8 0.455869 0.227934 0.973676i \(-0.426803\pi\)
0.227934 + 0.973676i \(0.426803\pi\)
\(360\) 0 0
\(361\) −4.79421e8 −0.536342
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.48466e8 0.159809
\(366\) 0 0
\(367\) 4.15569e8 0.438846 0.219423 0.975630i \(-0.429582\pi\)
0.219423 + 0.975630i \(0.429582\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.40213e9 −2.44223
\(372\) 0 0
\(373\) −6.54100e8 −0.652624 −0.326312 0.945262i \(-0.605806\pi\)
−0.326312 + 0.945262i \(0.605806\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.41649e8 −0.328387
\(378\) 0 0
\(379\) −6.68523e8 −0.630782 −0.315391 0.948962i \(-0.602136\pi\)
−0.315391 + 0.948962i \(0.602136\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.38983e8 −0.672107 −0.336054 0.941843i \(-0.609092\pi\)
−0.336054 + 0.941843i \(0.609092\pi\)
\(384\) 0 0
\(385\) −3.18499e9 −2.84443
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.04715e7 −0.0348598 −0.0174299 0.999848i \(-0.505548\pi\)
−0.0174299 + 0.999848i \(0.505548\pi\)
\(390\) 0 0
\(391\) 1.57114e9 1.32922
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.43790e8 −0.117392
\(396\) 0 0
\(397\) 1.87564e9 1.50447 0.752233 0.658897i \(-0.228977\pi\)
0.752233 + 0.658897i \(0.228977\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −9.96483e8 −0.771728 −0.385864 0.922556i \(-0.626097\pi\)
−0.385864 + 0.922556i \(0.626097\pi\)
\(402\) 0 0
\(403\) −1.00809e8 −0.0767244
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.62551e9 1.19511
\(408\) 0 0
\(409\) 1.01806e8 0.0735772 0.0367886 0.999323i \(-0.488287\pi\)
0.0367886 + 0.999323i \(0.488287\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.06171e9 −1.44013
\(414\) 0 0
\(415\) 7.77124e8 0.533730
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.61886e9 −1.07513 −0.537564 0.843223i \(-0.680655\pi\)
−0.537564 + 0.843223i \(0.680655\pi\)
\(420\) 0 0
\(421\) −1.76813e9 −1.15486 −0.577428 0.816441i \(-0.695944\pi\)
−0.577428 + 0.816441i \(0.695944\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.87187e9 2.44658
\(426\) 0 0
\(427\) −1.31648e9 −0.818307
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.37116e8 −0.262982 −0.131491 0.991317i \(-0.541976\pi\)
−0.131491 + 0.991317i \(0.541976\pi\)
\(432\) 0 0
\(433\) 9.69007e8 0.573613 0.286807 0.957989i \(-0.407406\pi\)
0.286807 + 0.957989i \(0.407406\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.36185e8 0.536632
\(438\) 0 0
\(439\) −8.86918e8 −0.500331 −0.250166 0.968203i \(-0.580485\pi\)
−0.250166 + 0.968203i \(0.580485\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.47780e9 −1.35411 −0.677053 0.735934i \(-0.736743\pi\)
−0.677053 + 0.735934i \(0.736743\pi\)
\(444\) 0 0
\(445\) 2.54804e8 0.137071
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.68169e7 −0.0504765 −0.0252382 0.999681i \(-0.508034\pi\)
−0.0252382 + 0.999681i \(0.508034\pi\)
\(450\) 0 0
\(451\) 2.54562e9 1.30670
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.00272e9 −2.48981
\(456\) 0 0
\(457\) 3.53320e9 1.73166 0.865828 0.500341i \(-0.166792\pi\)
0.865828 + 0.500341i \(0.166792\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.88532e7 0.0279780 0.0139890 0.999902i \(-0.495547\pi\)
0.0139890 + 0.999902i \(0.495547\pi\)
\(462\) 0 0
\(463\) −2.28948e9 −1.07202 −0.536011 0.844211i \(-0.680069\pi\)
−0.536011 + 0.844211i \(0.680069\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.76508e9 1.71067 0.855333 0.518079i \(-0.173353\pi\)
0.855333 + 0.518079i \(0.173353\pi\)
\(468\) 0 0
\(469\) 4.37660e9 1.95899
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.04992e9 0.456185
\(474\) 0 0
\(475\) 2.30711e9 0.987736
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.15354e9 0.895320 0.447660 0.894204i \(-0.352257\pi\)
0.447660 + 0.894204i \(0.352257\pi\)
\(480\) 0 0
\(481\) 2.55321e9 1.04611
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.43803e9 0.572361
\(486\) 0 0
\(487\) −4.64683e8 −0.182308 −0.0911540 0.995837i \(-0.529056\pi\)
−0.0911540 + 0.995837i \(0.529056\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.88216e8 −0.300511 −0.150255 0.988647i \(-0.548010\pi\)
−0.150255 + 0.988647i \(0.548010\pi\)
\(492\) 0 0
\(493\) −1.75824e9 −0.660866
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −8.03088e9 −2.93438
\(498\) 0 0
\(499\) −4.30040e9 −1.54938 −0.774688 0.632344i \(-0.782093\pi\)
−0.774688 + 0.632344i \(0.782093\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.47920e9 1.91968 0.959840 0.280547i \(-0.0905157\pi\)
0.959840 + 0.280547i \(0.0905157\pi\)
\(504\) 0 0
\(505\) 1.55134e9 0.536027
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.25720e9 0.422562 0.211281 0.977425i \(-0.432236\pi\)
0.211281 + 0.977425i \(0.432236\pi\)
\(510\) 0 0
\(511\) −5.84364e8 −0.193736
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.45600e9 1.76015
\(516\) 0 0
\(517\) −3.03862e9 −0.967073
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.69281e8 0.269295 0.134648 0.990894i \(-0.457010\pi\)
0.134648 + 0.990894i \(0.457010\pi\)
\(522\) 0 0
\(523\) −8.30165e7 −0.0253752 −0.0126876 0.999920i \(-0.504039\pi\)
−0.0126876 + 0.999920i \(0.504039\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.18798e8 −0.154405
\(528\) 0 0
\(529\) −1.29012e9 −0.378909
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.99845e9 1.14379
\(534\) 0 0
\(535\) −7.78424e9 −2.19775
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.05535e9 2.49083
\(540\) 0 0
\(541\) −9.05014e8 −0.245734 −0.122867 0.992423i \(-0.539209\pi\)
−0.122867 + 0.992423i \(0.539209\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.92367e8 0.0773643
\(546\) 0 0
\(547\) 2.61326e9 0.682696 0.341348 0.939937i \(-0.389116\pi\)
0.341348 + 0.939937i \(0.389116\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.04767e9 −0.266806
\(552\) 0 0
\(553\) 5.65959e8 0.142314
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −8.06824e9 −1.97827 −0.989136 0.147006i \(-0.953036\pi\)
−0.989136 + 0.147006i \(0.953036\pi\)
\(558\) 0 0
\(559\) 1.64912e9 0.399311
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.55851e9 −1.31274 −0.656371 0.754439i \(-0.727909\pi\)
−0.656371 + 0.754439i \(0.727909\pi\)
\(564\) 0 0
\(565\) −6.98519e9 −1.62933
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.82207e9 0.642207 0.321104 0.947044i \(-0.395946\pi\)
0.321104 + 0.947044i \(0.395946\pi\)
\(570\) 0 0
\(571\) 6.20632e9 1.39511 0.697554 0.716533i \(-0.254272\pi\)
0.697554 + 0.716533i \(0.254272\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.21143e9 1.14319
\(576\) 0 0
\(577\) −5.43205e9 −1.17720 −0.588598 0.808426i \(-0.700320\pi\)
−0.588598 + 0.808426i \(0.700320\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.05876e9 −0.647038
\(582\) 0 0
\(583\) −5.89527e9 −1.23215
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.08447e9 −0.629431 −0.314715 0.949186i \(-0.601909\pi\)
−0.314715 + 0.949186i \(0.601909\pi\)
\(588\) 0 0
\(589\) −3.09134e8 −0.0623365
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.83257e8 0.134553 0.0672764 0.997734i \(-0.478569\pi\)
0.0672764 + 0.997734i \(0.478569\pi\)
\(594\) 0 0
\(595\) −2.57457e10 −5.01066
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.24846e9 −0.997787 −0.498893 0.866663i \(-0.666260\pi\)
−0.498893 + 0.866663i \(0.666260\pi\)
\(600\) 0 0
\(601\) −4.00032e9 −0.751682 −0.375841 0.926684i \(-0.622646\pi\)
−0.375841 + 0.926684i \(0.622646\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.10075e8 0.130365
\(606\) 0 0
\(607\) −2.52401e9 −0.458069 −0.229034 0.973418i \(-0.573557\pi\)
−0.229034 + 0.973418i \(0.573557\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.77281e9 −0.846506
\(612\) 0 0
\(613\) 1.17565e9 0.206142 0.103071 0.994674i \(-0.467133\pi\)
0.103071 + 0.994674i \(0.467133\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.05146e9 1.03720 0.518599 0.855017i \(-0.326454\pi\)
0.518599 + 0.855017i \(0.326454\pi\)
\(618\) 0 0
\(619\) −9.38449e9 −1.59035 −0.795176 0.606378i \(-0.792622\pi\)
−0.795176 + 0.606378i \(0.792622\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.00291e9 −0.166171
\(624\) 0 0
\(625\) −2.11424e9 −0.346396
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.31397e10 2.10527
\(630\) 0 0
\(631\) −4.21509e9 −0.667889 −0.333945 0.942593i \(-0.608380\pi\)
−0.333945 + 0.942593i \(0.608380\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.73810e10 −2.69381
\(636\) 0 0
\(637\) 1.42234e10 2.18029
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.96210e9 1.49399 0.746995 0.664829i \(-0.231496\pi\)
0.746995 + 0.664829i \(0.231496\pi\)
\(642\) 0 0
\(643\) 6.98296e9 1.03586 0.517930 0.855423i \(-0.326703\pi\)
0.517930 + 0.855423i \(0.326703\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.65100e9 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(648\) 0 0
\(649\) −5.05983e9 −0.726573
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.57145e9 −1.34518 −0.672592 0.740014i \(-0.734819\pi\)
−0.672592 + 0.740014i \(0.734819\pi\)
\(654\) 0 0
\(655\) 2.65962e9 0.369807
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.13794e9 1.10768 0.553842 0.832622i \(-0.313161\pi\)
0.553842 + 0.832622i \(0.313161\pi\)
\(660\) 0 0
\(661\) −9.82041e9 −1.32259 −0.661294 0.750127i \(-0.729992\pi\)
−0.661294 + 0.750127i \(0.729992\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.53409e10 −2.02291
\(666\) 0 0
\(667\) −2.36654e9 −0.308797
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.23089e9 −0.412851
\(672\) 0 0
\(673\) 3.93826e9 0.498026 0.249013 0.968500i \(-0.419894\pi\)
0.249013 + 0.968500i \(0.419894\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.66884e9 −0.949882 −0.474941 0.880018i \(-0.657531\pi\)
−0.474941 + 0.880018i \(0.657531\pi\)
\(678\) 0 0
\(679\) −5.66008e9 −0.693870
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.65167e9 0.918933 0.459466 0.888195i \(-0.348041\pi\)
0.459466 + 0.888195i \(0.348041\pi\)
\(684\) 0 0
\(685\) 1.25792e10 1.49533
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.25981e9 −1.07854
\(690\) 0 0
\(691\) −1.58561e10 −1.82820 −0.914098 0.405493i \(-0.867100\pi\)
−0.914098 + 0.405493i \(0.867100\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.03639e10 −2.30098
\(696\) 0 0
\(697\) 2.05773e10 2.30183
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.61910e9 0.616104 0.308052 0.951370i \(-0.400323\pi\)
0.308052 + 0.951370i \(0.400323\pi\)
\(702\) 0 0
\(703\) 7.82946e9 0.849940
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.10608e9 −0.649822
\(708\) 0 0
\(709\) 1.34254e10 1.41471 0.707353 0.706860i \(-0.249889\pi\)
0.707353 + 0.706860i \(0.249889\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.98288e8 −0.0721475
\(714\) 0 0
\(715\) −1.22776e10 −1.25616
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.63313e10 −1.63858 −0.819292 0.573377i \(-0.805633\pi\)
−0.819292 + 0.573377i \(0.805633\pi\)
\(720\) 0 0
\(721\) −2.14749e10 −2.13382
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.83205e9 −0.568379
\(726\) 0 0
\(727\) −4.33823e9 −0.418738 −0.209369 0.977837i \(-0.567141\pi\)
−0.209369 + 0.977837i \(0.567141\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.48692e9 0.803599
\(732\) 0 0
\(733\) 7.43119e9 0.696939 0.348469 0.937320i \(-0.386702\pi\)
0.348469 + 0.937320i \(0.386702\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.07410e10 0.988345
\(738\) 0 0
\(739\) 5.03186e9 0.458641 0.229321 0.973351i \(-0.426350\pi\)
0.229321 + 0.973351i \(0.426350\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.63121e10 1.45898 0.729491 0.683990i \(-0.239757\pi\)
0.729491 + 0.683990i \(0.239757\pi\)
\(744\) 0 0
\(745\) 1.35923e10 1.20433
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.06388e10 2.66432
\(750\) 0 0
\(751\) −2.07632e10 −1.78877 −0.894385 0.447299i \(-0.852386\pi\)
−0.894385 + 0.447299i \(0.852386\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −6.22975e9 −0.526813
\(756\) 0 0
\(757\) −3.85677e9 −0.323138 −0.161569 0.986861i \(-0.551655\pi\)
−0.161569 + 0.986861i \(0.551655\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.80368e9 −0.148358 −0.0741792 0.997245i \(-0.523634\pi\)
−0.0741792 + 0.997245i \(0.523634\pi\)
\(762\) 0 0
\(763\) −1.15076e9 −0.0937882
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.94756e9 −0.635989
\(768\) 0 0
\(769\) 3.33349e9 0.264336 0.132168 0.991227i \(-0.457806\pi\)
0.132168 + 0.991227i \(0.457806\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.11920e10 −0.871526 −0.435763 0.900062i \(-0.643521\pi\)
−0.435763 + 0.900062i \(0.643521\pi\)
\(774\) 0 0
\(775\) −1.72084e9 −0.132796
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.22613e10 0.929299
\(780\) 0 0
\(781\) −1.97093e10 −1.48045
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.21985e10 −2.37570
\(786\) 0 0
\(787\) 2.19525e9 0.160536 0.0802679 0.996773i \(-0.474422\pi\)
0.0802679 + 0.996773i \(0.474422\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.74938e10 1.97522
\(792\) 0 0
\(793\) −5.07481e9 −0.361380
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.43541e9 0.380302 0.190151 0.981755i \(-0.439102\pi\)
0.190151 + 0.981755i \(0.439102\pi\)
\(798\) 0 0
\(799\) −2.45625e10 −1.70356
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.43414e9 −0.0977433
\(804\) 0 0
\(805\) −3.46529e10 −2.34129
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9.89544e9 0.657075 0.328538 0.944491i \(-0.393444\pi\)
0.328538 + 0.944491i \(0.393444\pi\)
\(810\) 0 0
\(811\) −1.00426e10 −0.661111 −0.330556 0.943787i \(-0.607236\pi\)
−0.330556 + 0.943787i \(0.607236\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.54823e10 2.29594
\(816\) 0 0
\(817\) 5.05706e9 0.324430
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.07209e10 −1.93746 −0.968731 0.248112i \(-0.920190\pi\)
−0.968731 + 0.248112i \(0.920190\pi\)
\(822\) 0 0
\(823\) 2.01255e10 1.25848 0.629241 0.777210i \(-0.283366\pi\)
0.629241 + 0.777210i \(0.283366\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.55472e10 0.955832 0.477916 0.878405i \(-0.341392\pi\)
0.477916 + 0.878405i \(0.341392\pi\)
\(828\) 0 0
\(829\) −1.42396e10 −0.868077 −0.434038 0.900894i \(-0.642912\pi\)
−0.434038 + 0.900894i \(0.642912\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.31982e10 4.38776
\(834\) 0 0
\(835\) 4.89088e10 2.90726
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.17959e9 −0.185868 −0.0929338 0.995672i \(-0.529625\pi\)
−0.0929338 + 0.995672i \(0.529625\pi\)
\(840\) 0 0
\(841\) −1.46015e10 −0.846470
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.17104e9 0.465885
\(846\) 0 0
\(847\) −2.79486e9 −0.158040
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.76856e10 0.983709
\(852\) 0 0
\(853\) 4.62326e9 0.255051 0.127525 0.991835i \(-0.459297\pi\)
0.127525 + 0.991835i \(0.459297\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.13062e10 −1.15631 −0.578153 0.815929i \(-0.696226\pi\)
−0.578153 + 0.815929i \(0.696226\pi\)
\(858\) 0 0
\(859\) 7.86988e9 0.423635 0.211818 0.977309i \(-0.432062\pi\)
0.211818 + 0.977309i \(0.432062\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.55483e9 0.0823465 0.0411732 0.999152i \(-0.486890\pi\)
0.0411732 + 0.999152i \(0.486890\pi\)
\(864\) 0 0
\(865\) −3.92829e10 −2.06370
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.38897e9 0.0717999
\(870\) 0 0
\(871\) 1.68711e10 0.865126
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.65264e10 −1.33860
\(876\) 0 0
\(877\) 2.34828e10 1.17558 0.587790 0.809014i \(-0.299998\pi\)
0.587790 + 0.809014i \(0.299998\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.10488e9 −0.0544378 −0.0272189 0.999629i \(-0.508665\pi\)
−0.0272189 + 0.999629i \(0.508665\pi\)
\(882\) 0 0
\(883\) −1.52780e10 −0.746800 −0.373400 0.927670i \(-0.621808\pi\)
−0.373400 + 0.927670i \(0.621808\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.13521e10 −0.546189 −0.273094 0.961987i \(-0.588047\pi\)
−0.273094 + 0.961987i \(0.588047\pi\)
\(888\) 0 0
\(889\) 6.84117e10 3.26568
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.46359e10 −0.687764
\(894\) 0 0
\(895\) −2.03754e10 −0.950002
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7.81445e8 0.0358707
\(900\) 0 0
\(901\) −4.76540e10 −2.17052
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.94694e10 −1.32160
\(906\) 0 0
\(907\) 2.41214e9 0.107344 0.0536718 0.998559i \(-0.482908\pi\)
0.0536718 + 0.998559i \(0.482908\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.63158e10 1.15319 0.576597 0.817029i \(-0.304380\pi\)
0.576597 + 0.817029i \(0.304380\pi\)
\(912\) 0 0
\(913\) −7.50679e9 −0.326442
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.04683e10 −0.448314
\(918\) 0 0
\(919\) 2.35629e10 1.00144 0.500719 0.865610i \(-0.333069\pi\)
0.500719 + 0.865610i \(0.333069\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3.09577e10 −1.29588
\(924\) 0 0
\(925\) 4.35840e10 1.81064
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.08846e9 −0.0854618 −0.0427309 0.999087i \(-0.513606\pi\)
−0.0427309 + 0.999087i \(0.513606\pi\)
\(930\) 0 0
\(931\) 4.36163e10 1.77143
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.31847e10 −2.52797
\(936\) 0 0
\(937\) 3.67845e10 1.46075 0.730376 0.683045i \(-0.239345\pi\)
0.730376 + 0.683045i \(0.239345\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.19271e9 0.359650 0.179825 0.983699i \(-0.442447\pi\)
0.179825 + 0.983699i \(0.442447\pi\)
\(942\) 0 0
\(943\) 2.76965e10 1.07556
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.30566e10 −0.499581 −0.249791 0.968300i \(-0.580362\pi\)
−0.249791 + 0.968300i \(0.580362\pi\)
\(948\) 0 0
\(949\) −2.25263e9 −0.0855574
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.42599e10 −0.907954 −0.453977 0.891013i \(-0.649995\pi\)
−0.453977 + 0.891013i \(0.649995\pi\)
\(954\) 0 0
\(955\) 1.29322e10 0.480462
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −4.95118e10 −1.81278
\(960\) 0 0
\(961\) −2.72820e10 −0.991619
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.41476e10 1.93970
\(966\) 0 0
\(967\) 1.37434e10 0.488766 0.244383 0.969679i \(-0.421415\pi\)
0.244383 + 0.969679i \(0.421415\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.37656e9 −0.223521 −0.111761 0.993735i \(-0.535649\pi\)
−0.111761 + 0.993735i \(0.535649\pi\)
\(972\) 0 0
\(973\) 8.01523e10 2.78947
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.35347e10 −1.49350 −0.746748 0.665107i \(-0.768386\pi\)
−0.746748 + 0.665107i \(0.768386\pi\)
\(978\) 0 0
\(979\) −2.46133e9 −0.0838361
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.03387e10 0.682946 0.341473 0.939892i \(-0.389074\pi\)
0.341473 + 0.939892i \(0.389074\pi\)
\(984\) 0 0
\(985\) −1.00092e10 −0.333714
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.14232e10 0.375491
\(990\) 0 0
\(991\) 5.03245e10 1.64256 0.821280 0.570525i \(-0.193260\pi\)
0.821280 + 0.570525i \(0.193260\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.80304e10 1.22391
\(996\) 0 0
\(997\) 2.65066e10 0.847073 0.423537 0.905879i \(-0.360789\pi\)
0.423537 + 0.905879i \(0.360789\pi\)
\(998\) 0 0
\(999\) 0 0
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 576.8.a.bd.1.1 2
3.2 odd 2 576.8.a.bq.1.2 2
4.3 odd 2 576.8.a.bc.1.1 2
8.3 odd 2 144.8.a.n.1.2 2
8.5 even 2 72.8.a.g.1.2 yes 2
12.11 even 2 576.8.a.bp.1.2 2
24.5 odd 2 72.8.a.f.1.1 2
24.11 even 2 144.8.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.a.f.1.1 2 24.5 odd 2
72.8.a.g.1.2 yes 2 8.5 even 2
144.8.a.l.1.1 2 24.11 even 2
144.8.a.n.1.2 2 8.3 odd 2
576.8.a.bc.1.1 2 4.3 odd 2
576.8.a.bd.1.1 2 1.1 even 1 trivial
576.8.a.bp.1.2 2 12.11 even 2
576.8.a.bq.1.2 2 3.2 odd 2