Properties

Label 288.8.f.a.143.13
Level $288$
Weight $8$
Character 288.143
Analytic conductor $89.967$
Analytic rank $0$
Dimension $28$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,8,Mod(143,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.143");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.f (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 143.13
Character \(\chi\) \(=\) 288.143
Dual form 288.8.f.a.143.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-93.0830 q^{5} -1025.06i q^{7} +O(q^{10})\) \(q-93.0830 q^{5} -1025.06i q^{7} -1545.61i q^{11} +5677.41i q^{13} +15703.6i q^{17} -33939.0 q^{19} -30068.2 q^{23} -69460.6 q^{25} +99025.6 q^{29} -152785. i q^{31} +95416.0i q^{35} +336316. i q^{37} -663449. i q^{41} +707879. q^{43} -402012. q^{47} -227213. q^{49} +662469. q^{53} +143870. i q^{55} +1.74805e6i q^{59} +321803. i q^{61} -528470. i q^{65} +464913. q^{67} +3.53880e6 q^{71} -3.41741e6 q^{73} -1.58435e6 q^{77} +7.60430e6i q^{79} +8.11070e6i q^{83} -1.46174e6i q^{85} -1.00923e7i q^{89} +5.81971e6 q^{91} +3.15915e6 q^{95} -5.88229e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q+O(q^{10}) \) Copy content Toggle raw display \( 28 q + 121168 q^{19} + 437500 q^{25} - 1505696 q^{43} - 2272076 q^{49} + 776272 q^{67} - 2534128 q^{73} + 3406992 q^{91} - 26311456 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) −93.0830 −0.333024 −0.166512 0.986039i \(-0.553250\pi\)
−0.166512 + 0.986039i \(0.553250\pi\)
\(6\) 0 0
\(7\) − 1025.06i − 1.12956i −0.825243 0.564778i \(-0.808962\pi\)
0.825243 0.564778i \(-0.191038\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1545.61i − 0.350126i −0.984557 0.175063i \(-0.943987\pi\)
0.984557 0.175063i \(-0.0560130\pi\)
\(12\) 0 0
\(13\) 5677.41i 0.716719i 0.933584 + 0.358359i \(0.116664\pi\)
−0.933584 + 0.358359i \(0.883336\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15703.6i 0.775227i 0.921822 + 0.387613i \(0.126700\pi\)
−0.921822 + 0.387613i \(0.873300\pi\)
\(18\) 0 0
\(19\) −33939.0 −1.13517 −0.567586 0.823314i \(-0.692123\pi\)
−0.567586 + 0.823314i \(0.692123\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −30068.2 −0.515300 −0.257650 0.966238i \(-0.582948\pi\)
−0.257650 + 0.966238i \(0.582948\pi\)
\(24\) 0 0
\(25\) −69460.6 −0.889095
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 99025.6 0.753971 0.376985 0.926219i \(-0.376961\pi\)
0.376985 + 0.926219i \(0.376961\pi\)
\(30\) 0 0
\(31\) − 152785.i − 0.921115i −0.887630 0.460558i \(-0.847649\pi\)
0.887630 0.460558i \(-0.152351\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 95416.0i 0.376169i
\(36\) 0 0
\(37\) 336316.i 1.09154i 0.837933 + 0.545772i \(0.183764\pi\)
−0.837933 + 0.545772i \(0.816236\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 663449.i − 1.50336i −0.659527 0.751681i \(-0.729243\pi\)
0.659527 0.751681i \(-0.270757\pi\)
\(42\) 0 0
\(43\) 707879. 1.35775 0.678874 0.734255i \(-0.262468\pi\)
0.678874 + 0.734255i \(0.262468\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −402012. −0.564802 −0.282401 0.959296i \(-0.591131\pi\)
−0.282401 + 0.959296i \(0.591131\pi\)
\(48\) 0 0
\(49\) −227213. −0.275896
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 662469. 0.611223 0.305612 0.952156i \(-0.401139\pi\)
0.305612 + 0.952156i \(0.401139\pi\)
\(54\) 0 0
\(55\) 143870.i 0.116600i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.74805e6i 1.10808i 0.832489 + 0.554041i \(0.186915\pi\)
−0.832489 + 0.554041i \(0.813085\pi\)
\(60\) 0 0
\(61\) 321803.i 0.181525i 0.995873 + 0.0907624i \(0.0289304\pi\)
−0.995873 + 0.0907624i \(0.971070\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 528470.i − 0.238684i
\(66\) 0 0
\(67\) 464913. 0.188847 0.0944234 0.995532i \(-0.469899\pi\)
0.0944234 + 0.995532i \(0.469899\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.53880e6 1.17342 0.586708 0.809799i \(-0.300424\pi\)
0.586708 + 0.809799i \(0.300424\pi\)
\(72\) 0 0
\(73\) −3.41741e6 −1.02817 −0.514087 0.857738i \(-0.671869\pi\)
−0.514087 + 0.857738i \(0.671869\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.58435e6 −0.395487
\(78\) 0 0
\(79\) 7.60430e6i 1.73526i 0.497210 + 0.867630i \(0.334358\pi\)
−0.497210 + 0.867630i \(0.665642\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.11070e6i 1.55699i 0.627653 + 0.778494i \(0.284016\pi\)
−0.627653 + 0.778494i \(0.715984\pi\)
\(84\) 0 0
\(85\) − 1.46174e6i − 0.258169i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.00923e7i − 1.51749i −0.651387 0.758746i \(-0.725813\pi\)
0.651387 0.758746i \(-0.274187\pi\)
\(90\) 0 0
\(91\) 5.81971e6 0.809574
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.15915e6 0.378039
\(96\) 0 0
\(97\) −5.88229e6 −0.654403 −0.327201 0.944955i \(-0.606106\pi\)
−0.327201 + 0.944955i \(0.606106\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.32223e7 1.27697 0.638486 0.769633i \(-0.279561\pi\)
0.638486 + 0.769633i \(0.279561\pi\)
\(102\) 0 0
\(103\) 8.86665e6i 0.799520i 0.916620 + 0.399760i \(0.130907\pi\)
−0.916620 + 0.399760i \(0.869093\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 4.71318e6i − 0.371938i −0.982556 0.185969i \(-0.940458\pi\)
0.982556 0.185969i \(-0.0595424\pi\)
\(108\) 0 0
\(109\) 2.63965e7i 1.95233i 0.217027 + 0.976166i \(0.430364\pi\)
−0.217027 + 0.976166i \(0.569636\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 3.41021e6i − 0.222334i −0.993802 0.111167i \(-0.964541\pi\)
0.993802 0.111167i \(-0.0354589\pi\)
\(114\) 0 0
\(115\) 2.79884e6 0.171607
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.60972e7 0.875662
\(120\) 0 0
\(121\) 1.70983e7 0.877412
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.37377e7 0.629114
\(126\) 0 0
\(127\) 9.87499e6i 0.427783i 0.976857 + 0.213892i \(0.0686139\pi\)
−0.976857 + 0.213892i \(0.931386\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.16683e7i − 0.842124i −0.907032 0.421062i \(-0.861658\pi\)
0.907032 0.421062i \(-0.138342\pi\)
\(132\) 0 0
\(133\) 3.47897e7i 1.28224i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.12379e7i 1.70243i 0.524818 + 0.851214i \(0.324133\pi\)
−0.524818 + 0.851214i \(0.675867\pi\)
\(138\) 0 0
\(139\) −3.92028e7 −1.23813 −0.619063 0.785341i \(-0.712488\pi\)
−0.619063 + 0.785341i \(0.712488\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.77505e6 0.250942
\(144\) 0 0
\(145\) −9.21760e6 −0.251090
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.23566e7 1.79195 0.895975 0.444104i \(-0.146478\pi\)
0.895975 + 0.444104i \(0.146478\pi\)
\(150\) 0 0
\(151\) 5.61796e7i 1.32788i 0.747786 + 0.663940i \(0.231117\pi\)
−0.747786 + 0.663940i \(0.768883\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.42217e7i 0.306753i
\(156\) 0 0
\(157\) 8.29634e7i 1.71095i 0.517843 + 0.855476i \(0.326735\pi\)
−0.517843 + 0.855476i \(0.673265\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.08218e7i 0.582060i
\(162\) 0 0
\(163\) −1.32708e7 −0.240017 −0.120008 0.992773i \(-0.538292\pi\)
−0.120008 + 0.992773i \(0.538292\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.62945e7 1.10146 0.550732 0.834682i \(-0.314349\pi\)
0.550732 + 0.834682i \(0.314349\pi\)
\(168\) 0 0
\(169\) 3.05155e7 0.486314
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.99909e7 −0.880895 −0.440448 0.897778i \(-0.645180\pi\)
−0.440448 + 0.897778i \(0.645180\pi\)
\(174\) 0 0
\(175\) 7.12015e7i 1.00428i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 8.01095e7i − 1.04399i −0.852947 0.521997i \(-0.825187\pi\)
0.852947 0.521997i \(-0.174813\pi\)
\(180\) 0 0
\(181\) − 3.83641e7i − 0.480894i −0.970662 0.240447i \(-0.922706\pi\)
0.970662 0.240447i \(-0.0772941\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 3.13053e7i − 0.363510i
\(186\) 0 0
\(187\) 2.42716e7 0.271427
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.26246e8 1.31100 0.655499 0.755196i \(-0.272458\pi\)
0.655499 + 0.755196i \(0.272458\pi\)
\(192\) 0 0
\(193\) 1.20271e8 1.20423 0.602115 0.798409i \(-0.294325\pi\)
0.602115 + 0.798409i \(0.294325\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.23903e8 1.15465 0.577326 0.816514i \(-0.304096\pi\)
0.577326 + 0.816514i \(0.304096\pi\)
\(198\) 0 0
\(199\) 1.68027e8i 1.51145i 0.654888 + 0.755726i \(0.272716\pi\)
−0.654888 + 0.755726i \(0.727284\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.01508e8i − 0.851652i
\(204\) 0 0
\(205\) 6.17558e7i 0.500655i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.24564e7i 0.397454i
\(210\) 0 0
\(211\) −8.24919e7 −0.604537 −0.302268 0.953223i \(-0.597744\pi\)
−0.302268 + 0.953223i \(0.597744\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.58914e7 −0.452162
\(216\) 0 0
\(217\) −1.56614e8 −1.04045
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.91560e7 −0.555620
\(222\) 0 0
\(223\) − 1.10837e8i − 0.669296i −0.942343 0.334648i \(-0.891383\pi\)
0.942343 0.334648i \(-0.108617\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.44393e8i 1.38675i 0.720576 + 0.693376i \(0.243877\pi\)
−0.720576 + 0.693376i \(0.756123\pi\)
\(228\) 0 0
\(229\) 2.49235e7i 0.137146i 0.997646 + 0.0685732i \(0.0218447\pi\)
−0.997646 + 0.0685732i \(0.978155\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 8.59209e7i − 0.444993i −0.974934 0.222496i \(-0.928579\pi\)
0.974934 0.222496i \(-0.0714206\pi\)
\(234\) 0 0
\(235\) 3.74204e7 0.188092
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.03442e8 −0.490122 −0.245061 0.969508i \(-0.578808\pi\)
−0.245061 + 0.969508i \(0.578808\pi\)
\(240\) 0 0
\(241\) −2.12373e8 −0.977325 −0.488662 0.872473i \(-0.662515\pi\)
−0.488662 + 0.872473i \(0.662515\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.11496e7 0.0918801
\(246\) 0 0
\(247\) − 1.92686e8i − 0.813599i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.88318e8i 1.15084i 0.817859 + 0.575418i \(0.195161\pi\)
−0.817859 + 0.575418i \(0.804839\pi\)
\(252\) 0 0
\(253\) 4.64736e7i 0.180420i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3.06607e8i − 1.12672i −0.826211 0.563361i \(-0.809508\pi\)
0.826211 0.563361i \(-0.190492\pi\)
\(258\) 0 0
\(259\) 3.44745e8 1.23296
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.85030e7 −0.266098 −0.133049 0.991109i \(-0.542477\pi\)
−0.133049 + 0.991109i \(0.542477\pi\)
\(264\) 0 0
\(265\) −6.16645e7 −0.203552
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.64506e8 0.828519 0.414259 0.910159i \(-0.364041\pi\)
0.414259 + 0.910159i \(0.364041\pi\)
\(270\) 0 0
\(271\) − 3.60573e8i − 1.10053i −0.834991 0.550263i \(-0.814527\pi\)
0.834991 0.550263i \(-0.185473\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.07359e8i 0.311296i
\(276\) 0 0
\(277\) − 2.46978e8i − 0.698198i −0.937086 0.349099i \(-0.886488\pi\)
0.937086 0.349099i \(-0.113512\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.16347e8i − 0.312812i −0.987693 0.156406i \(-0.950009\pi\)
0.987693 0.156406i \(-0.0499908\pi\)
\(282\) 0 0
\(283\) 3.05548e8 0.801358 0.400679 0.916219i \(-0.368774\pi\)
0.400679 + 0.916219i \(0.368774\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.80077e8 −1.69813
\(288\) 0 0
\(289\) 1.63735e8 0.399023
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.14442e8 −1.65932 −0.829661 0.558268i \(-0.811466\pi\)
−0.829661 + 0.558268i \(0.811466\pi\)
\(294\) 0 0
\(295\) − 1.62714e8i − 0.369018i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.70709e8i − 0.369325i
\(300\) 0 0
\(301\) − 7.25621e8i − 1.53365i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2.99544e7i − 0.0604521i
\(306\) 0 0
\(307\) −3.95766e8 −0.780647 −0.390323 0.920678i \(-0.627637\pi\)
−0.390323 + 0.920678i \(0.627637\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.17018e8 0.786129 0.393064 0.919511i \(-0.371415\pi\)
0.393064 + 0.919511i \(0.371415\pi\)
\(312\) 0 0
\(313\) 2.16149e8 0.398426 0.199213 0.979956i \(-0.436161\pi\)
0.199213 + 0.979956i \(0.436161\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.69525e8 −1.18048 −0.590241 0.807227i \(-0.700967\pi\)
−0.590241 + 0.807227i \(0.700967\pi\)
\(318\) 0 0
\(319\) − 1.53055e8i − 0.263985i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 5.32966e8i − 0.880016i
\(324\) 0 0
\(325\) − 3.94356e8i − 0.637231i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.12088e8i 0.637975i
\(330\) 0 0
\(331\) −2.27775e8 −0.345230 −0.172615 0.984989i \(-0.555222\pi\)
−0.172615 + 0.984989i \(0.555222\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.32755e7 −0.0628905
\(336\) 0 0
\(337\) −3.09334e8 −0.440274 −0.220137 0.975469i \(-0.570650\pi\)
−0.220137 + 0.975469i \(0.570650\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.36145e8 −0.322507
\(342\) 0 0
\(343\) − 6.11277e8i − 0.817915i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.22835e8i 0.543272i 0.962400 + 0.271636i \(0.0875647\pi\)
−0.962400 + 0.271636i \(0.912435\pi\)
\(348\) 0 0
\(349\) − 1.23470e9i − 1.55479i −0.629011 0.777396i \(-0.716540\pi\)
0.629011 0.777396i \(-0.283460\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.17813e8i 0.142554i 0.997457 + 0.0712772i \(0.0227075\pi\)
−0.997457 + 0.0712772i \(0.977292\pi\)
\(354\) 0 0
\(355\) −3.29402e8 −0.390775
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.21083e9 −1.38119 −0.690595 0.723242i \(-0.742651\pi\)
−0.690595 + 0.723242i \(0.742651\pi\)
\(360\) 0 0
\(361\) 2.57987e8 0.288617
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.18102e8 0.342406
\(366\) 0 0
\(367\) − 2.02237e8i − 0.213565i −0.994282 0.106782i \(-0.965945\pi\)
0.994282 0.106782i \(-0.0340548\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 6.79072e8i − 0.690411i
\(372\) 0 0
\(373\) 1.43859e8i 0.143534i 0.997421 + 0.0717671i \(0.0228638\pi\)
−0.997421 + 0.0717671i \(0.977136\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.62209e8i 0.540385i
\(378\) 0 0
\(379\) −8.77228e8 −0.827705 −0.413852 0.910344i \(-0.635817\pi\)
−0.413852 + 0.910344i \(0.635817\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.04366e8 0.185871 0.0929356 0.995672i \(-0.470375\pi\)
0.0929356 + 0.995672i \(0.470375\pi\)
\(384\) 0 0
\(385\) 1.47476e8 0.131707
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.58576e9 −1.36589 −0.682943 0.730472i \(-0.739300\pi\)
−0.682943 + 0.730472i \(0.739300\pi\)
\(390\) 0 0
\(391\) − 4.72180e8i − 0.399474i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 7.07831e8i − 0.577883i
\(396\) 0 0
\(397\) 1.83002e8i 0.146787i 0.997303 + 0.0733935i \(0.0233829\pi\)
−0.997303 + 0.0733935i \(0.976617\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.57329e9i 1.21844i 0.793001 + 0.609220i \(0.208517\pi\)
−0.793001 + 0.609220i \(0.791483\pi\)
\(402\) 0 0
\(403\) 8.67422e8 0.660181
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.19813e8 0.382179
\(408\) 0 0
\(409\) 1.66645e9 1.20437 0.602187 0.798355i \(-0.294296\pi\)
0.602187 + 0.798355i \(0.294296\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.79186e9 1.25164
\(414\) 0 0
\(415\) − 7.54968e8i − 0.518514i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 8.92118e8i − 0.592479i −0.955114 0.296240i \(-0.904267\pi\)
0.955114 0.296240i \(-0.0957327\pi\)
\(420\) 0 0
\(421\) 1.98849e9i 1.29878i 0.760455 + 0.649391i \(0.224976\pi\)
−0.760455 + 0.649391i \(0.775024\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 1.09078e9i − 0.689250i
\(426\) 0 0
\(427\) 3.29869e8 0.205042
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.29190e8 −0.559029 −0.279514 0.960142i \(-0.590173\pi\)
−0.279514 + 0.960142i \(0.590173\pi\)
\(432\) 0 0
\(433\) 1.65923e9 0.982197 0.491099 0.871104i \(-0.336595\pi\)
0.491099 + 0.871104i \(0.336595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.02049e9 0.584954
\(438\) 0 0
\(439\) 9.17987e7i 0.0517858i 0.999665 + 0.0258929i \(0.00824288\pi\)
−0.999665 + 0.0258929i \(0.991757\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 7.47135e8i − 0.408306i −0.978939 0.204153i \(-0.934556\pi\)
0.978939 0.204153i \(-0.0654441\pi\)
\(444\) 0 0
\(445\) 9.39424e8i 0.505361i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.67451e8i 0.191574i 0.995402 + 0.0957871i \(0.0305368\pi\)
−0.995402 + 0.0957871i \(0.969463\pi\)
\(450\) 0 0
\(451\) −1.02543e9 −0.526367
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.41716e8 −0.269607
\(456\) 0 0
\(457\) −3.88980e9 −1.90643 −0.953215 0.302292i \(-0.902248\pi\)
−0.953215 + 0.302292i \(0.902248\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.45672e8 0.354482 0.177241 0.984167i \(-0.443283\pi\)
0.177241 + 0.984167i \(0.443283\pi\)
\(462\) 0 0
\(463\) 1.76294e9i 0.825473i 0.910850 + 0.412737i \(0.135427\pi\)
−0.910850 + 0.412737i \(0.864573\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.95161e9i 1.34106i 0.741881 + 0.670531i \(0.233934\pi\)
−0.741881 + 0.670531i \(0.766066\pi\)
\(468\) 0 0
\(469\) − 4.76565e8i − 0.213313i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 1.09410e9i − 0.475383i
\(474\) 0 0
\(475\) 2.35742e9 1.00928
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.12030e9 1.71299 0.856495 0.516156i \(-0.172637\pi\)
0.856495 + 0.516156i \(0.172637\pi\)
\(480\) 0 0
\(481\) −1.90941e9 −0.782331
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.47541e8 0.217932
\(486\) 0 0
\(487\) 4.42855e9i 1.73744i 0.495304 + 0.868720i \(0.335057\pi\)
−0.495304 + 0.868720i \(0.664943\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.43647e9i 1.69142i 0.533641 + 0.845711i \(0.320823\pi\)
−0.533641 + 0.845711i \(0.679177\pi\)
\(492\) 0 0
\(493\) 1.55506e9i 0.584498i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 3.62750e9i − 1.32544i
\(498\) 0 0
\(499\) −3.41428e9 −1.23012 −0.615060 0.788481i \(-0.710868\pi\)
−0.615060 + 0.788481i \(0.710868\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.56937e9 0.900201 0.450100 0.892978i \(-0.351388\pi\)
0.450100 + 0.892978i \(0.351388\pi\)
\(504\) 0 0
\(505\) −1.23077e9 −0.425262
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.06039e9 −0.356414 −0.178207 0.983993i \(-0.557030\pi\)
−0.178207 + 0.983993i \(0.557030\pi\)
\(510\) 0 0
\(511\) 3.50306e9i 1.16138i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 8.25334e8i − 0.266259i
\(516\) 0 0
\(517\) 6.21352e8i 0.197752i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 4.18471e8i − 0.129638i −0.997897 0.0648191i \(-0.979353\pi\)
0.997897 0.0648191i \(-0.0206470\pi\)
\(522\) 0 0
\(523\) 3.84414e9 1.17502 0.587508 0.809218i \(-0.300109\pi\)
0.587508 + 0.809218i \(0.300109\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.39927e9 0.714073
\(528\) 0 0
\(529\) −2.50073e9 −0.734466
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.76667e9 1.07749
\(534\) 0 0
\(535\) 4.38717e8i 0.123864i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.51181e8i 0.0965986i
\(540\) 0 0
\(541\) 3.23668e9i 0.878838i 0.898282 + 0.439419i \(0.144816\pi\)
−0.898282 + 0.439419i \(0.855184\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 2.45706e9i − 0.650173i
\(546\) 0 0
\(547\) 4.94425e9 1.29165 0.645824 0.763486i \(-0.276514\pi\)
0.645824 + 0.763486i \(0.276514\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.36083e9 −0.855887
\(552\) 0 0
\(553\) 7.79490e9 1.96007
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.48349e9 −0.363740 −0.181870 0.983323i \(-0.558215\pi\)
−0.181870 + 0.983323i \(0.558215\pi\)
\(558\) 0 0
\(559\) 4.01892e9i 0.973123i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 4.65548e9i − 1.09947i −0.835338 0.549737i \(-0.814728\pi\)
0.835338 0.549737i \(-0.185272\pi\)
\(564\) 0 0
\(565\) 3.17433e8i 0.0740426i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.03603e9i 0.918462i 0.888317 + 0.459231i \(0.151875\pi\)
−0.888317 + 0.459231i \(0.848125\pi\)
\(570\) 0 0
\(571\) −4.33338e9 −0.974094 −0.487047 0.873376i \(-0.661926\pi\)
−0.487047 + 0.873376i \(0.661926\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.08855e9 0.458150
\(576\) 0 0
\(577\) 1.65483e9 0.358623 0.179311 0.983792i \(-0.442613\pi\)
0.179311 + 0.983792i \(0.442613\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.31398e9 1.75870
\(582\) 0 0
\(583\) − 1.02392e9i − 0.214005i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.30362e9i 1.08228i 0.840933 + 0.541140i \(0.182007\pi\)
−0.840933 + 0.541140i \(0.817993\pi\)
\(588\) 0 0
\(589\) 5.18537e9i 1.04563i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.54656e7i 0.0148613i 0.999972 + 0.00743066i \(0.00236527\pi\)
−0.999972 + 0.00743066i \(0.997635\pi\)
\(594\) 0 0
\(595\) −1.49838e9 −0.291616
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7.13988e9 1.35737 0.678683 0.734431i \(-0.262551\pi\)
0.678683 + 0.734431i \(0.262551\pi\)
\(600\) 0 0
\(601\) −9.21234e9 −1.73105 −0.865524 0.500867i \(-0.833015\pi\)
−0.865524 + 0.500867i \(0.833015\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.59156e9 −0.292199
\(606\) 0 0
\(607\) − 2.53391e9i − 0.459866i −0.973206 0.229933i \(-0.926149\pi\)
0.973206 0.229933i \(-0.0738507\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.28239e9i − 0.404804i
\(612\) 0 0
\(613\) 2.83258e9i 0.496674i 0.968674 + 0.248337i \(0.0798840\pi\)
−0.968674 + 0.248337i \(0.920116\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.09456e9i 0.187604i 0.995591 + 0.0938018i \(0.0299020\pi\)
−0.995591 + 0.0938018i \(0.970098\pi\)
\(618\) 0 0
\(619\) 3.22342e9 0.546260 0.273130 0.961977i \(-0.411941\pi\)
0.273130 + 0.961977i \(0.411941\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.03453e10 −1.71409
\(624\) 0 0
\(625\) 4.14786e9 0.679585
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.28138e9 −0.846195
\(630\) 0 0
\(631\) − 8.31648e9i − 1.31776i −0.752248 0.658880i \(-0.771030\pi\)
0.752248 0.658880i \(-0.228970\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 9.19193e8i − 0.142462i
\(636\) 0 0
\(637\) − 1.28998e9i − 0.197740i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.71195e9i 1.30651i 0.757138 + 0.653255i \(0.226597\pi\)
−0.757138 + 0.653255i \(0.773403\pi\)
\(642\) 0 0
\(643\) −9.96578e9 −1.47834 −0.739168 0.673522i \(-0.764781\pi\)
−0.739168 + 0.673522i \(0.764781\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.53985e9 −0.658987 −0.329493 0.944158i \(-0.606878\pi\)
−0.329493 + 0.944158i \(0.606878\pi\)
\(648\) 0 0
\(649\) 2.70180e9 0.387969
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.90772e9 0.408655 0.204328 0.978903i \(-0.434499\pi\)
0.204328 + 0.978903i \(0.434499\pi\)
\(654\) 0 0
\(655\) 2.01695e9i 0.280447i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 3.96130e9i − 0.539186i −0.962974 0.269593i \(-0.913111\pi\)
0.962974 0.269593i \(-0.0868892\pi\)
\(660\) 0 0
\(661\) − 4.86484e9i − 0.655184i −0.944819 0.327592i \(-0.893763\pi\)
0.944819 0.327592i \(-0.106237\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.23833e9i − 0.427017i
\(666\) 0 0
\(667\) −2.97752e9 −0.388521
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.97381e8 0.0635566
\(672\) 0 0
\(673\) −2.43436e9 −0.307845 −0.153923 0.988083i \(-0.549191\pi\)
−0.153923 + 0.988083i \(0.549191\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.30568e9 0.161725 0.0808624 0.996725i \(-0.474233\pi\)
0.0808624 + 0.996725i \(0.474233\pi\)
\(678\) 0 0
\(679\) 6.02972e9i 0.739184i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 7.27394e9i − 0.873569i −0.899566 0.436785i \(-0.856117\pi\)
0.899566 0.436785i \(-0.143883\pi\)
\(684\) 0 0
\(685\) − 4.76937e9i − 0.566949i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.76111e9i 0.438075i
\(690\) 0 0
\(691\) 2.01649e9 0.232500 0.116250 0.993220i \(-0.462913\pi\)
0.116250 + 0.993220i \(0.462913\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.64911e9 0.412326
\(696\) 0 0
\(697\) 1.04185e10 1.16545
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.51344e10 −1.65940 −0.829700 0.558210i \(-0.811488\pi\)
−0.829700 + 0.558210i \(0.811488\pi\)
\(702\) 0 0
\(703\) − 1.14142e10i − 1.23909i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.35537e10i − 1.44241i
\(708\) 0 0
\(709\) − 1.47287e10i − 1.55204i −0.630708 0.776020i \(-0.717235\pi\)
0.630708 0.776020i \(-0.282765\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.59396e9i 0.474650i
\(714\) 0 0
\(715\) −8.16808e8 −0.0835697
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.99832e9 0.300834 0.150417 0.988623i \(-0.451938\pi\)
0.150417 + 0.988623i \(0.451938\pi\)
\(720\) 0 0
\(721\) 9.08888e9 0.903103
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.87837e9 −0.670352
\(726\) 0 0
\(727\) − 9.00906e9i − 0.869579i −0.900532 0.434789i \(-0.856823\pi\)
0.900532 0.434789i \(-0.143177\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.11163e10i 1.05256i
\(732\) 0 0
\(733\) − 8.89008e9i − 0.833761i −0.908961 0.416880i \(-0.863123\pi\)
0.908961 0.416880i \(-0.136877\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 7.18573e8i − 0.0661202i
\(738\) 0 0
\(739\) −8.53349e9 −0.777806 −0.388903 0.921279i \(-0.627146\pi\)
−0.388903 + 0.921279i \(0.627146\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.12936e10 −1.01011 −0.505057 0.863086i \(-0.668528\pi\)
−0.505057 + 0.863086i \(0.668528\pi\)
\(744\) 0 0
\(745\) −6.73516e9 −0.596762
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.83131e9 −0.420125
\(750\) 0 0
\(751\) − 6.47122e9i − 0.557502i −0.960363 0.278751i \(-0.910080\pi\)
0.960363 0.278751i \(-0.0899204\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 5.22936e9i − 0.442216i
\(756\) 0 0
\(757\) − 4.70372e9i − 0.394100i −0.980393 0.197050i \(-0.936864\pi\)
0.980393 0.197050i \(-0.0631361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 9.44372e9i − 0.776778i −0.921496 0.388389i \(-0.873032\pi\)
0.921496 0.388389i \(-0.126968\pi\)
\(762\) 0 0
\(763\) 2.70581e10 2.20527
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.92441e9 −0.794183
\(768\) 0 0
\(769\) −5.19374e9 −0.411849 −0.205925 0.978568i \(-0.566020\pi\)
−0.205925 + 0.978568i \(0.566020\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.43367e10 −1.11640 −0.558202 0.829705i \(-0.688509\pi\)
−0.558202 + 0.829705i \(0.688509\pi\)
\(774\) 0 0
\(775\) 1.06125e10i 0.818959i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.25168e10i 1.70658i
\(780\) 0 0
\(781\) − 5.46960e9i − 0.410844i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 7.72248e9i − 0.569788i
\(786\) 0 0
\(787\) −6.32430e9 −0.462489 −0.231244 0.972896i \(-0.574280\pi\)
−0.231244 + 0.972896i \(0.574280\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.49568e9 −0.251139
\(792\) 0 0
\(793\) −1.82701e9 −0.130102
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.99261e10 1.39418 0.697091 0.716983i \(-0.254477\pi\)
0.697091 + 0.716983i \(0.254477\pi\)
\(798\) 0 0
\(799\) − 6.31304e9i − 0.437850i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.28197e9i 0.359991i
\(804\) 0 0
\(805\) − 2.86899e9i − 0.193840i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 4.18784e9i − 0.278080i −0.990287 0.139040i \(-0.955598\pi\)
0.990287 0.139040i \(-0.0444017\pi\)
\(810\) 0 0
\(811\) 3.77264e9 0.248355 0.124177 0.992260i \(-0.460371\pi\)
0.124177 + 0.992260i \(0.460371\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.23529e9 0.0799314
\(816\) 0 0
\(817\) −2.40247e10 −1.54128
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.35665e9 −0.527025 −0.263512 0.964656i \(-0.584881\pi\)
−0.263512 + 0.964656i \(0.584881\pi\)
\(822\) 0 0
\(823\) − 2.54233e10i − 1.58977i −0.606763 0.794883i \(-0.707532\pi\)
0.606763 0.794883i \(-0.292468\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.71179e10i 1.66720i 0.552371 + 0.833599i \(0.313723\pi\)
−0.552371 + 0.833599i \(0.686277\pi\)
\(828\) 0 0
\(829\) − 1.95618e10i − 1.19252i −0.802790 0.596262i \(-0.796652\pi\)
0.802790 0.596262i \(-0.203348\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 3.56806e9i − 0.213882i
\(834\) 0 0
\(835\) −6.17089e9 −0.366813
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.86844e10 −1.09223 −0.546113 0.837711i \(-0.683893\pi\)
−0.546113 + 0.837711i \(0.683893\pi\)
\(840\) 0 0
\(841\) −7.44381e9 −0.431528
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.84047e9 −0.161954
\(846\) 0 0
\(847\) − 1.75268e10i − 0.991085i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1.01124e10i − 0.562473i
\(852\) 0 0
\(853\) 3.55557e10i 1.96150i 0.195273 + 0.980749i \(0.437441\pi\)
−0.195273 + 0.980749i \(0.562559\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 4.79883e6i 0 0.000260437i −1.00000 0.000130218i \(-0.999959\pi\)
1.00000 0.000130218i \(-4.14498e-5\pi\)
\(858\) 0 0
\(859\) 1.84728e10 0.994388 0.497194 0.867639i \(-0.334364\pi\)
0.497194 + 0.867639i \(0.334364\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.22171e9 −0.329513 −0.164756 0.986334i \(-0.552684\pi\)
−0.164756 + 0.986334i \(0.552684\pi\)
\(864\) 0 0
\(865\) 5.58414e9 0.293359
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.17533e10 0.607560
\(870\) 0 0
\(871\) 2.63950e9i 0.135350i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 1.40820e10i − 0.710619i
\(876\) 0 0
\(877\) 1.16634e10i 0.583884i 0.956436 + 0.291942i \(0.0943014\pi\)
−0.956436 + 0.291942i \(0.905699\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 9.09495e9i − 0.448110i −0.974577 0.224055i \(-0.928070\pi\)
0.974577 0.224055i \(-0.0719295\pi\)
\(882\) 0 0
\(883\) −1.53416e10 −0.749907 −0.374954 0.927044i \(-0.622341\pi\)
−0.374954 + 0.927044i \(0.622341\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.84449e10 1.36858 0.684291 0.729209i \(-0.260112\pi\)
0.684291 + 0.729209i \(0.260112\pi\)
\(888\) 0 0
\(889\) 1.01225e10 0.483205
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.36439e10 0.641148
\(894\) 0 0
\(895\) 7.45683e9i 0.347675i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1.51296e10i − 0.694494i
\(900\) 0 0
\(901\) 1.04032e10i 0.473837i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.57104e9i 0.160149i
\(906\) 0 0
\(907\) 1.63965e10 0.729666 0.364833 0.931073i \(-0.381126\pi\)
0.364833 + 0.931073i \(0.381126\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.21999e10 −0.534617 −0.267309 0.963611i \(-0.586134\pi\)
−0.267309 + 0.963611i \(0.586134\pi\)
\(912\) 0 0
\(913\) 1.25360e10 0.545142
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2.22114e10 −0.951226
\(918\) 0 0
\(919\) 3.87196e10i 1.64561i 0.568326 + 0.822803i \(0.307591\pi\)
−0.568326 + 0.822803i \(0.692409\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.00912e10i 0.841009i
\(924\) 0 0
\(925\) − 2.33607e10i − 0.970487i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 3.34318e10i − 1.36806i −0.729455 0.684029i \(-0.760226\pi\)
0.729455 0.684029i \(-0.239774\pi\)
\(930\) 0 0
\(931\) 7.71138e9 0.313190
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.25928e9 −0.0903917
\(936\) 0 0
\(937\) −1.13884e8 −0.00452247 −0.00226123 0.999997i \(-0.500720\pi\)
−0.00226123 + 0.999997i \(0.500720\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.30760e10 1.29404 0.647022 0.762472i \(-0.276014\pi\)
0.647022 + 0.762472i \(0.276014\pi\)
\(942\) 0 0
\(943\) 1.99487e10i 0.774682i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.30694e10i − 1.64795i −0.566627 0.823974i \(-0.691752\pi\)
0.566627 0.823974i \(-0.308248\pi\)
\(948\) 0 0
\(949\) − 1.94020e10i − 0.736912i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.45613e10i 0.919233i 0.888118 + 0.459616i \(0.152013\pi\)
−0.888118 + 0.459616i \(0.847987\pi\)
\(954\) 0 0
\(955\) −1.17514e10 −0.436594
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.25221e10 1.92299
\(960\) 0 0
\(961\) 4.16944e9 0.151546
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.11952e10 −0.401037
\(966\) 0 0
\(967\) 2.28479e10i 0.812556i 0.913750 + 0.406278i \(0.133174\pi\)
−0.913750 + 0.406278i \(0.866826\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.59295e10i 1.60999i 0.593279 + 0.804997i \(0.297833\pi\)
−0.593279 + 0.804997i \(0.702167\pi\)
\(972\) 0 0
\(973\) 4.01854e10i 1.39853i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.53672e10i 1.55636i 0.628038 + 0.778182i \(0.283858\pi\)
−0.628038 + 0.778182i \(0.716142\pi\)
\(978\) 0 0
\(979\) −1.55988e10 −0.531314
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.77122e10 0.930536 0.465268 0.885170i \(-0.345958\pi\)
0.465268 + 0.885170i \(0.345958\pi\)
\(984\) 0 0
\(985\) −1.15333e10 −0.384527
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.12846e10 −0.699647
\(990\) 0 0
\(991\) 7.33226e9i 0.239321i 0.992815 + 0.119660i \(0.0381806\pi\)
−0.992815 + 0.119660i \(0.961819\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.56405e10i − 0.503350i
\(996\) 0 0
\(997\) 2.27024e10i 0.725502i 0.931886 + 0.362751i \(0.118162\pi\)
−0.931886 + 0.362751i \(0.881838\pi\)
\(998\) 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 288.8.f.a.143.13 28
3.2 odd 2 inner 288.8.f.a.143.15 28
4.3 odd 2 72.8.f.a.35.27 yes 28
8.3 odd 2 inner 288.8.f.a.143.16 28
8.5 even 2 72.8.f.a.35.1 28
12.11 even 2 72.8.f.a.35.2 yes 28
24.5 odd 2 72.8.f.a.35.28 yes 28
24.11 even 2 inner 288.8.f.a.143.14 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
72.8.f.a.35.1 28 8.5 even 2
72.8.f.a.35.2 yes 28 12.11 even 2
72.8.f.a.35.27 yes 28 4.3 odd 2
72.8.f.a.35.28 yes 28 24.5 odd 2
288.8.f.a.143.13 28 1.1 even 1 trivial
288.8.f.a.143.14 28 24.11 even 2 inner
288.8.f.a.143.15 28 3.2 odd 2 inner
288.8.f.a.143.16 28 8.3 odd 2 inner