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Properties

Label 252.12.a.a.1.1
Level $252$
Weight $12$
Character 252.1
Self dual yes
Analytic conductor $193.622$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,12,Mod(1,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(193.622481501\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 252.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2130.00 q^{5} +16807.0 q^{7} +O(q^{10})\) \(q+2130.00 q^{5} +16807.0 q^{7} +704196. q^{11} +952286. q^{13} -5.10551e6 q^{17} -1.39051e7 q^{19} +1.89454e7 q^{23} -4.42912e7 q^{25} -1.48937e8 q^{29} -1.59226e8 q^{31} +3.57989e7 q^{35} -8.25482e7 q^{37} +7.29417e8 q^{41} +1.18514e9 q^{43} -2.86059e8 q^{47} +2.82475e8 q^{49} -3.85354e9 q^{53} +1.49994e9 q^{55} -5.28827e9 q^{59} -8.15633e9 q^{61} +2.02837e9 q^{65} +9.25005e9 q^{67} -2.00517e10 q^{71} +2.38532e10 q^{73} +1.18354e10 q^{77} +3.51368e9 q^{79} +2.14974e10 q^{83} -1.08747e10 q^{85} -6.68399e10 q^{89} +1.60051e10 q^{91} -2.96180e10 q^{95} +1.46493e11 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) 2130.00 0.304821 0.152410 − 0.988317i \(-0.451296\pi\)
0.152410 + 0.988317i \(0.451296\pi\)
\(6\) 0 0
\(7\) 16807.0 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 704196. 1.31836 0.659180 − 0.751986i \(-0.270904\pi\)
0.659180 + 0.751986i \(0.270904\pi\)
\(12\) 0 0
\(13\) 952286. 0.711343 0.355671 − 0.934611i \(-0.384252\pi\)
0.355671 + 0.934611i \(0.384252\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.10551e6 −0.872108 −0.436054 − 0.899920i \(-0.643624\pi\)
−0.436054 + 0.899920i \(0.643624\pi\)
\(18\) 0 0
\(19\) −1.39051e7 −1.28834 −0.644170 − 0.764882i \(-0.722797\pi\)
−0.644170 + 0.764882i \(0.722797\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.89454e7 0.613763 0.306882 − 0.951748i \(-0.400714\pi\)
0.306882 + 0.951748i \(0.400714\pi\)
\(24\) 0 0
\(25\) −4.42912e7 −0.907084
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.48937e8 −1.34838 −0.674192 − 0.738556i \(-0.735508\pi\)
−0.674192 + 0.738556i \(0.735508\pi\)
\(30\) 0 0
\(31\) −1.59226e8 −0.998907 −0.499454 − 0.866341i \(-0.666466\pi\)
−0.499454 + 0.866341i \(0.666466\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.57989e7 0.115211
\(36\) 0 0
\(37\) −8.25482e7 −0.195703 −0.0978516 − 0.995201i \(-0.531197\pi\)
−0.0978516 + 0.995201i \(0.531197\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.29417e8 0.983252 0.491626 − 0.870807i \(-0.336403\pi\)
0.491626 + 0.870807i \(0.336403\pi\)
\(42\) 0 0
\(43\) 1.18514e9 1.22940 0.614699 − 0.788761i \(-0.289277\pi\)
0.614699 + 0.788761i \(0.289277\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.86059e8 −0.181935 −0.0909677 − 0.995854i \(-0.528996\pi\)
−0.0909677 + 0.995854i \(0.528996\pi\)
\(48\) 0 0
\(49\) 2.82475e8 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.85354e9 −1.26573 −0.632867 − 0.774260i \(-0.718122\pi\)
−0.632867 + 0.774260i \(0.718122\pi\)
\(54\) 0 0
\(55\) 1.49994e9 0.401863
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.28827e9 −0.963002 −0.481501 − 0.876445i \(-0.659908\pi\)
−0.481501 + 0.876445i \(0.659908\pi\)
\(60\) 0 0
\(61\) −8.15633e9 −1.23646 −0.618231 − 0.785997i \(-0.712150\pi\)
−0.618231 + 0.785997i \(0.712150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.02837e9 0.216832
\(66\) 0 0
\(67\) 9.25005e9 0.837014 0.418507 − 0.908214i \(-0.362554\pi\)
0.418507 + 0.908214i \(0.362554\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00517e10 −1.31895 −0.659477 − 0.751725i \(-0.729222\pi\)
−0.659477 + 0.751725i \(0.729222\pi\)
\(72\) 0 0
\(73\) 2.38532e10 1.34670 0.673350 − 0.739324i \(-0.264855\pi\)
0.673350 + 0.739324i \(0.264855\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.18354e10 0.498293
\(78\) 0 0
\(79\) 3.51368e9 0.128473 0.0642366 − 0.997935i \(-0.479539\pi\)
0.0642366 + 0.997935i \(0.479539\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.14974e10 0.599039 0.299520 − 0.954090i \(-0.403174\pi\)
0.299520 + 0.954090i \(0.403174\pi\)
\(84\) 0 0
\(85\) −1.08747e10 −0.265837
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.68399e10 −1.26879 −0.634397 − 0.773007i \(-0.718751\pi\)
−0.634397 + 0.773007i \(0.718751\pi\)
\(90\) 0 0
\(91\) 1.60051e10 0.268862
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.96180e10 −0.392713
\(96\) 0 0
\(97\) 1.46493e11 1.73209 0.866047 − 0.499962i \(-0.166653\pi\)
0.866047 + 0.499962i \(0.166653\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.17073e11 −1.10838 −0.554192 − 0.832389i \(-0.686973\pi\)
−0.554192 + 0.832389i \(0.686973\pi\)
\(102\) 0 0
\(103\) 1.67202e11 1.42114 0.710571 − 0.703625i \(-0.248437\pi\)
0.710571 + 0.703625i \(0.248437\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.83513e10 −0.126490 −0.0632451 − 0.997998i \(-0.520145\pi\)
−0.0632451 + 0.997998i \(0.520145\pi\)
\(108\) 0 0
\(109\) −1.57815e11 −0.982432 −0.491216 − 0.871038i \(-0.663447\pi\)
−0.491216 + 0.871038i \(0.663447\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.39425e11 −0.711886 −0.355943 − 0.934508i \(-0.615840\pi\)
−0.355943 + 0.934508i \(0.615840\pi\)
\(114\) 0 0
\(115\) 4.03537e10 0.187088
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.58084e10 −0.329626
\(120\) 0 0
\(121\) 2.10580e11 0.738071
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.98344e11 −0.581319
\(126\) 0 0
\(127\) −1.36528e11 −0.366692 −0.183346 − 0.983048i \(-0.558693\pi\)
−0.183346 + 0.983048i \(0.558693\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.25591e11 0.737361 0.368680 − 0.929556i \(-0.379810\pi\)
0.368680 + 0.929556i \(0.379810\pi\)
\(132\) 0 0
\(133\) −2.33704e11 −0.486947
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.83345e11 −1.38672 −0.693362 − 0.720589i \(-0.743871\pi\)
−0.693362 + 0.720589i \(0.743871\pi\)
\(138\) 0 0
\(139\) 9.40330e10 0.153709 0.0768544 − 0.997042i \(-0.475512\pi\)
0.0768544 + 0.997042i \(0.475512\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.70596e11 0.937805
\(144\) 0 0
\(145\) −3.17236e11 −0.411016
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.67149e12 1.86457 0.932287 − 0.361719i \(-0.117810\pi\)
0.932287 + 0.361719i \(0.117810\pi\)
\(150\) 0 0
\(151\) −1.55944e12 −1.61657 −0.808286 − 0.588791i \(-0.799604\pi\)
−0.808286 + 0.588791i \(0.799604\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.39152e11 −0.304488
\(156\) 0 0
\(157\) 1.14021e12 0.953973 0.476986 − 0.878911i \(-0.341729\pi\)
0.476986 + 0.878911i \(0.341729\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.18415e11 0.231981
\(162\) 0 0
\(163\) −6.68894e11 −0.455329 −0.227665 − 0.973740i \(-0.573109\pi\)
−0.227665 + 0.973740i \(0.573109\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.16683e11 −0.188662 −0.0943311 − 0.995541i \(-0.530071\pi\)
−0.0943311 + 0.995541i \(0.530071\pi\)
\(168\) 0 0
\(169\) −8.85312e11 −0.493991
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.35593e12 −1.15587 −0.577936 − 0.816082i \(-0.696142\pi\)
−0.577936 + 0.816082i \(0.696142\pi\)
\(174\) 0 0
\(175\) −7.44403e11 −0.342846
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.20397e11 0.0896424 0.0448212 − 0.998995i \(-0.485728\pi\)
0.0448212 + 0.998995i \(0.485728\pi\)
\(180\) 0 0
\(181\) −4.11490e12 −1.57444 −0.787222 − 0.616670i \(-0.788481\pi\)
−0.787222 + 0.616670i \(0.788481\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.75828e11 −0.0596544
\(186\) 0 0
\(187\) −3.59528e12 −1.14975
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.26480e11 −0.121399 −0.0606995 − 0.998156i \(-0.519333\pi\)
−0.0606995 + 0.998156i \(0.519333\pi\)
\(192\) 0 0
\(193\) 2.34283e12 0.629759 0.314880 − 0.949132i \(-0.398036\pi\)
0.314880 + 0.949132i \(0.398036\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.60709e12 1.10627 0.553136 − 0.833091i \(-0.313431\pi\)
0.553136 + 0.833091i \(0.313431\pi\)
\(198\) 0 0
\(199\) 3.54693e12 0.805676 0.402838 − 0.915271i \(-0.368024\pi\)
0.402838 + 0.915271i \(0.368024\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.50319e12 −0.509642
\(204\) 0 0
\(205\) 1.55366e12 0.299716
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.79195e12 −1.69850
\(210\) 0 0
\(211\) −1.08382e13 −1.78404 −0.892021 − 0.451993i \(-0.850713\pi\)
−0.892021 + 0.451993i \(0.850713\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.52435e12 0.374746
\(216\) 0 0
\(217\) −2.67611e12 −0.377551
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.86191e12 −0.620368
\(222\) 0 0
\(223\) 1.16866e13 1.41909 0.709547 − 0.704658i \(-0.248900\pi\)
0.709547 + 0.704658i \(0.248900\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.92194e12 −0.762230 −0.381115 − 0.924528i \(-0.624460\pi\)
−0.381115 + 0.924528i \(0.624460\pi\)
\(228\) 0 0
\(229\) 7.62155e12 0.799738 0.399869 − 0.916572i \(-0.369056\pi\)
0.399869 + 0.916572i \(0.369056\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.55874e12 −0.530297 −0.265148 − 0.964208i \(-0.585421\pi\)
−0.265148 + 0.964208i \(0.585421\pi\)
\(234\) 0 0
\(235\) −6.09306e11 −0.0554577
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.02086e13 −0.846795 −0.423398 − 0.905944i \(-0.639163\pi\)
−0.423398 + 0.905944i \(0.639163\pi\)
\(240\) 0 0
\(241\) −6.04974e12 −0.479339 −0.239669 − 0.970855i \(-0.577039\pi\)
−0.239669 + 0.970855i \(0.577039\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.01672e11 0.0435458
\(246\) 0 0
\(247\) −1.32417e13 −0.916452
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.88411e13 −1.19371 −0.596857 − 0.802347i \(-0.703584\pi\)
−0.596857 + 0.802347i \(0.703584\pi\)
\(252\) 0 0
\(253\) 1.33413e13 0.809160
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.12183e12 −0.451878 −0.225939 − 0.974141i \(-0.572545\pi\)
−0.225939 + 0.974141i \(0.572545\pi\)
\(258\) 0 0
\(259\) −1.38739e12 −0.0739689
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.94680e13 −0.954034 −0.477017 − 0.878894i \(-0.658282\pi\)
−0.477017 + 0.878894i \(0.658282\pi\)
\(264\) 0 0
\(265\) −8.20804e12 −0.385822
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.47779e13 −1.07257 −0.536287 − 0.844035i \(-0.680174\pi\)
−0.536287 + 0.844035i \(0.680174\pi\)
\(270\) 0 0
\(271\) −2.10568e13 −0.875108 −0.437554 − 0.899192i \(-0.644155\pi\)
−0.437554 + 0.899192i \(0.644155\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.11897e13 −1.19586
\(276\) 0 0
\(277\) 1.27183e13 0.468587 0.234294 − 0.972166i \(-0.424722\pi\)
0.234294 + 0.972166i \(0.424722\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.78054e13 −1.96827 −0.984133 − 0.177432i \(-0.943221\pi\)
−0.984133 + 0.177432i \(0.943221\pi\)
\(282\) 0 0
\(283\) 1.54561e13 0.506145 0.253072 − 0.967447i \(-0.418559\pi\)
0.253072 + 0.967447i \(0.418559\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.22593e13 0.371634
\(288\) 0 0
\(289\) −8.20562e12 −0.239427
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.24618e13 −0.337140 −0.168570 − 0.985690i \(-0.553915\pi\)
−0.168570 + 0.985690i \(0.553915\pi\)
\(294\) 0 0
\(295\) −1.12640e13 −0.293543
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.80414e13 0.436596
\(300\) 0 0
\(301\) 1.99186e13 0.464669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.73730e13 −0.376899
\(306\) 0 0
\(307\) 2.52342e11 0.00528116 0.00264058 − 0.999997i \(-0.499159\pi\)
0.00264058 + 0.999997i \(0.499159\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.66226e12 0.0908687 0.0454344 − 0.998967i \(-0.485533\pi\)
0.0454344 + 0.998967i \(0.485533\pi\)
\(312\) 0 0
\(313\) −4.79752e13 −0.902657 −0.451329 − 0.892358i \(-0.649050\pi\)
−0.451329 + 0.892358i \(0.649050\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −8.79391e12 −0.154296 −0.0771482 − 0.997020i \(-0.524581\pi\)
−0.0771482 + 0.997020i \(0.524581\pi\)
\(318\) 0 0
\(319\) −1.04881e14 −1.77766
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.09929e13 1.12357
\(324\) 0 0
\(325\) −4.21779e13 −0.645248
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.80779e12 −0.0687651
\(330\) 0 0
\(331\) −8.64638e13 −1.19614 −0.598068 − 0.801445i \(-0.704065\pi\)
−0.598068 + 0.801445i \(0.704065\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.97026e13 0.255139
\(336\) 0 0
\(337\) 5.28130e13 0.661875 0.330938 − 0.943653i \(-0.392635\pi\)
0.330938 + 0.943653i \(0.392635\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.12126e14 −1.31692
\(342\) 0 0
\(343\) 4.74756e12 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.98101e13 0.851619 0.425810 − 0.904813i \(-0.359989\pi\)
0.425810 + 0.904813i \(0.359989\pi\)
\(348\) 0 0
\(349\) 2.27529e12 0.0235233 0.0117616 − 0.999931i \(-0.496256\pi\)
0.0117616 + 0.999931i \(0.496256\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.51217e13 −0.729465 −0.364732 − 0.931112i \(-0.618840\pi\)
−0.364732 + 0.931112i \(0.618840\pi\)
\(354\) 0 0
\(355\) −4.27100e13 −0.402044
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.71592e13 0.240379 0.120190 − 0.992751i \(-0.461650\pi\)
0.120190 + 0.992751i \(0.461650\pi\)
\(360\) 0 0
\(361\) 7.68629e13 0.659822
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.08073e13 0.410502
\(366\) 0 0
\(367\) −2.32864e14 −1.82574 −0.912869 − 0.408254i \(-0.866138\pi\)
−0.912869 + 0.408254i \(0.866138\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.47664e13 −0.478403
\(372\) 0 0
\(373\) −9.73302e12 −0.0697989 −0.0348995 − 0.999391i \(-0.511111\pi\)
−0.0348995 + 0.999391i \(0.511111\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.41831e14 −0.959164
\(378\) 0 0
\(379\) 1.66886e14 1.09623 0.548117 − 0.836402i \(-0.315345\pi\)
0.548117 + 0.836402i \(0.315345\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.05302e13 −0.499305 −0.249652 − 0.968336i \(-0.580316\pi\)
−0.249652 + 0.968336i \(0.580316\pi\)
\(384\) 0 0
\(385\) 2.52094e13 0.151890
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.36876e14 1.91755 0.958776 − 0.284163i \(-0.0917157\pi\)
0.958776 + 0.284163i \(0.0917157\pi\)
\(390\) 0 0
\(391\) −9.67260e13 −0.535268
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.48413e12 0.0391613
\(396\) 0 0
\(397\) −5.52099e12 −0.0280976 −0.0140488 − 0.999901i \(-0.504472\pi\)
−0.0140488 + 0.999901i \(0.504472\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.05783e14 −0.991094 −0.495547 − 0.868581i \(-0.665032\pi\)
−0.495547 + 0.868581i \(0.665032\pi\)
\(402\) 0 0
\(403\) −1.51629e14 −0.710565
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.81301e13 −0.258007
\(408\) 0 0
\(409\) 3.86125e14 1.66821 0.834103 − 0.551609i \(-0.185986\pi\)
0.834103 + 0.551609i \(0.185986\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.88799e13 −0.363981
\(414\) 0 0
\(415\) 4.57894e13 0.182600
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.95554e14 1.11805 0.559023 − 0.829152i \(-0.311176\pi\)
0.559023 + 0.829152i \(0.311176\pi\)
\(420\) 0 0
\(421\) 1.27109e14 0.468409 0.234205 − 0.972187i \(-0.424751\pi\)
0.234205 + 0.972187i \(0.424751\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.26129e14 0.791076
\(426\) 0 0
\(427\) −1.37083e14 −0.467338
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −3.76840e14 −1.22048 −0.610242 − 0.792215i \(-0.708928\pi\)
−0.610242 + 0.792215i \(0.708928\pi\)
\(432\) 0 0
\(433\) −2.46767e13 −0.0779118 −0.0389559 − 0.999241i \(-0.512403\pi\)
−0.0389559 + 0.999241i \(0.512403\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.63439e14 −0.790736
\(438\) 0 0
\(439\) 6.02655e13 0.176406 0.0882031 − 0.996103i \(-0.471888\pi\)
0.0882031 + 0.996103i \(0.471888\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.14750e14 1.15496 0.577478 − 0.816406i \(-0.304037\pi\)
0.577478 + 0.816406i \(0.304037\pi\)
\(444\) 0 0
\(445\) −1.42369e14 −0.386755
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −5.30260e14 −1.37131 −0.685653 − 0.727929i \(-0.740483\pi\)
−0.685653 + 0.727929i \(0.740483\pi\)
\(450\) 0 0
\(451\) 5.13653e14 1.29628
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.40908e13 0.0819548
\(456\) 0 0
\(457\) 3.36550e14 0.789787 0.394894 − 0.918727i \(-0.370781\pi\)
0.394894 + 0.918727i \(0.370781\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.11970e14 −0.921533 −0.460766 − 0.887521i \(-0.652425\pi\)
−0.460766 + 0.887521i \(0.652425\pi\)
\(462\) 0 0
\(463\) 1.52755e14 0.333657 0.166829 − 0.985986i \(-0.446647\pi\)
0.166829 + 0.985986i \(0.446647\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.38546e14 0.705301 0.352651 − 0.935755i \(-0.385280\pi\)
0.352651 + 0.935755i \(0.385280\pi\)
\(468\) 0 0
\(469\) 1.55466e14 0.316361
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.34570e14 1.62079
\(474\) 0 0
\(475\) 6.15876e14 1.16863
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.76632e14 0.863650 0.431825 − 0.901957i \(-0.357870\pi\)
0.431825 + 0.901957i \(0.357870\pi\)
\(480\) 0 0
\(481\) −7.86095e13 −0.139212
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.12030e14 0.527978
\(486\) 0 0
\(487\) −5.22106e14 −0.863674 −0.431837 − 0.901952i \(-0.642134\pi\)
−0.431837 + 0.901952i \(0.642134\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.59493e14 0.410372 0.205186 − 0.978723i \(-0.434220\pi\)
0.205186 + 0.978723i \(0.434220\pi\)
\(492\) 0 0
\(493\) 7.60400e14 1.17594
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −3.37008e14 −0.498517
\(498\) 0 0
\(499\) 9.20473e14 1.33186 0.665929 − 0.746015i \(-0.268035\pi\)
0.665929 + 0.746015i \(0.268035\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7.73641e13 0.107131 0.0535656 − 0.998564i \(-0.482941\pi\)
0.0535656 + 0.998564i \(0.482941\pi\)
\(504\) 0 0
\(505\) −2.49366e14 −0.337859
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.20522e14 0.675291 0.337646 − 0.941273i \(-0.390369\pi\)
0.337646 + 0.941273i \(0.390369\pi\)
\(510\) 0 0
\(511\) 4.00901e14 0.509005
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.56141e14 0.433194
\(516\) 0 0
\(517\) −2.01442e14 −0.239856
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.45616e14 0.850957 0.425478 − 0.904969i \(-0.360106\pi\)
0.425478 + 0.904969i \(0.360106\pi\)
\(522\) 0 0
\(523\) −5.21902e14 −0.583216 −0.291608 − 0.956538i \(-0.594190\pi\)
−0.291608 + 0.956538i \(0.594190\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.12931e14 0.871155
\(528\) 0 0
\(529\) −5.93881e14 −0.623295
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.94614e14 0.699429
\(534\) 0 0
\(535\) −3.90883e13 −0.0385568
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.98918e14 0.188337
\(540\) 0 0
\(541\) −1.89890e15 −1.76164 −0.880819 − 0.473453i \(-0.843007\pi\)
−0.880819 + 0.473453i \(0.843007\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.36146e14 −0.299466
\(546\) 0 0
\(547\) −5.04656e14 −0.440621 −0.220310 − 0.975430i \(-0.570707\pi\)
−0.220310 + 0.975430i \(0.570707\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.07099e15 1.73718
\(552\) 0 0
\(553\) 5.90543e13 0.0485583
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.51067e15 1.98420 0.992100 − 0.125446i \(-0.0400362\pi\)
0.992100 + 0.125446i \(0.0400362\pi\)
\(558\) 0 0
\(559\) 1.12859e15 0.874524
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.89266e15 −1.41018 −0.705091 − 0.709116i \(-0.749094\pi\)
−0.705091 + 0.709116i \(0.749094\pi\)
\(564\) 0 0
\(565\) −2.96976e14 −0.216998
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.68489e14 0.188716 0.0943580 − 0.995538i \(-0.469920\pi\)
0.0943580 + 0.995538i \(0.469920\pi\)
\(570\) 0 0
\(571\) −2.54494e15 −1.75461 −0.877303 − 0.479936i \(-0.840660\pi\)
−0.877303 + 0.479936i \(0.840660\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.39115e14 −0.556735
\(576\) 0 0
\(577\) 2.22726e15 1.44978 0.724892 − 0.688862i \(-0.241889\pi\)
0.724892 + 0.688862i \(0.241889\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.61306e14 0.226416
\(582\) 0 0
\(583\) −2.71365e15 −1.66869
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.09890e15 −1.24303 −0.621517 − 0.783401i \(-0.713483\pi\)
−0.621517 + 0.783401i \(0.713483\pi\)
\(588\) 0 0
\(589\) 2.21406e15 1.28693
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.94829e14 0.333113 0.166556 − 0.986032i \(-0.446735\pi\)
0.166556 + 0.986032i \(0.446735\pi\)
\(594\) 0 0
\(595\) −1.82772e14 −0.100477
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.41645e15 1.28035 0.640177 − 0.768228i \(-0.278861\pi\)
0.640177 + 0.768228i \(0.278861\pi\)
\(600\) 0 0
\(601\) 2.99329e15 1.55718 0.778590 − 0.627532i \(-0.215935\pi\)
0.778590 + 0.627532i \(0.215935\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.48536e14 0.224979
\(606\) 0 0
\(607\) −6.02897e14 −0.296965 −0.148483 − 0.988915i \(-0.547439\pi\)
−0.148483 + 0.988915i \(0.547439\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.72410e14 −0.129418
\(612\) 0 0
\(613\) −3.42265e14 −0.159709 −0.0798545 − 0.996807i \(-0.525446\pi\)
−0.0798545 + 0.996807i \(0.525446\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.20429e15 1.44266 0.721329 − 0.692592i \(-0.243531\pi\)
0.721329 + 0.692592i \(0.243531\pi\)
\(618\) 0 0
\(619\) −7.12161e14 −0.314978 −0.157489 − 0.987521i \(-0.550340\pi\)
−0.157489 + 0.987521i \(0.550340\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.12338e15 −0.479559
\(624\) 0 0
\(625\) 1.74018e15 0.729886
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.21451e14 0.170674
\(630\) 0 0
\(631\) 4.42200e15 1.75978 0.879888 − 0.475181i \(-0.157618\pi\)
0.879888 + 0.475181i \(0.157618\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.90805e14 −0.111775
\(636\) 0 0
\(637\) 2.68997e14 0.101620
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.94982e14 −0.217162 −0.108581 − 0.994088i \(-0.534631\pi\)
−0.108581 + 0.994088i \(0.534631\pi\)
\(642\) 0 0
\(643\) 3.84803e15 1.38063 0.690317 − 0.723507i \(-0.257471\pi\)
0.690317 + 0.723507i \(0.257471\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.56818e15 −1.23730 −0.618648 − 0.785668i \(-0.712319\pi\)
−0.618648 + 0.785668i \(0.712319\pi\)
\(648\) 0 0
\(649\) −3.72398e15 −1.26958
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.93910e13 0.0228708 0.0114354 − 0.999935i \(-0.496360\pi\)
0.0114354 + 0.999935i \(0.496360\pi\)
\(654\) 0 0
\(655\) 6.93508e14 0.224763
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.99239e15 −1.56473 −0.782364 − 0.622822i \(-0.785986\pi\)
−0.782364 + 0.622822i \(0.785986\pi\)
\(660\) 0 0
\(661\) −2.29399e15 −0.707105 −0.353553 − 0.935415i \(-0.615026\pi\)
−0.353553 + 0.935415i \(0.615026\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.97789e14 −0.148432
\(666\) 0 0
\(667\) −2.82167e15 −0.827589
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −5.74365e15 −1.63010
\(672\) 0 0
\(673\) −5.03980e15 −1.40712 −0.703559 − 0.710637i \(-0.748407\pi\)
−0.703559 + 0.710637i \(0.748407\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.33043e15 −0.359547 −0.179774 − 0.983708i \(-0.557537\pi\)
−0.179774 + 0.983708i \(0.557537\pi\)
\(678\) 0 0
\(679\) 2.46210e15 0.654670
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.60565e15 0.670815 0.335408 − 0.942073i \(-0.391126\pi\)
0.335408 + 0.942073i \(0.391126\pi\)
\(684\) 0 0
\(685\) −1.66853e15 −0.422702
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.66967e15 −0.900371
\(690\) 0 0
\(691\) 2.70082e15 0.652179 0.326090 − 0.945339i \(-0.394269\pi\)
0.326090 + 0.945339i \(0.394269\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00290e14 0.0468537
\(696\) 0 0
\(697\) −3.72405e15 −0.857502
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.94816e15 −1.10407 −0.552033 − 0.833822i \(-0.686148\pi\)
−0.552033 + 0.833822i \(0.686148\pi\)
\(702\) 0 0
\(703\) 1.14784e15 0.252132
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.96765e15 −0.418930
\(708\) 0 0
\(709\) 5.94161e15 1.24552 0.622759 − 0.782414i \(-0.286012\pi\)
0.622759 + 0.782414i \(0.286012\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.01660e15 −0.613092
\(714\) 0 0
\(715\) 1.42837e15 0.285863
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.41393e14 0.163302 0.0816508 − 0.996661i \(-0.473981\pi\)
0.0816508 + 0.996661i \(0.473981\pi\)
\(720\) 0 0
\(721\) 2.81017e15 0.537142
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.59661e15 1.22310
\(726\) 0 0
\(727\) 6.16528e15 1.12593 0.562967 − 0.826479i \(-0.309660\pi\)
0.562967 + 0.826479i \(0.309660\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.05074e15 −1.07217
\(732\) 0 0
\(733\) −4.80682e15 −0.839046 −0.419523 − 0.907745i \(-0.637803\pi\)
−0.419523 + 0.907745i \(0.637803\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.51385e15 1.10348
\(738\) 0 0
\(739\) −6.55617e15 −1.09422 −0.547112 − 0.837060i \(-0.684273\pi\)
−0.547112 + 0.837060i \(0.684273\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.06202e15 −0.820135 −0.410068 − 0.912055i \(-0.634495\pi\)
−0.410068 + 0.912055i \(0.634495\pi\)
\(744\) 0 0
\(745\) 3.56028e15 0.568361
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.08431e14 −0.0478088
\(750\) 0 0
\(751\) −4.17267e15 −0.637374 −0.318687 − 0.947860i \(-0.603242\pi\)
−0.318687 + 0.947860i \(0.603242\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.32160e15 −0.492765
\(756\) 0 0
\(757\) −2.25217e15 −0.329287 −0.164644 − 0.986353i \(-0.552647\pi\)
−0.164644 + 0.986353i \(0.552647\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.72899e15 0.245570 0.122785 − 0.992433i \(-0.460817\pi\)
0.122785 + 0.992433i \(0.460817\pi\)
\(762\) 0 0
\(763\) −2.65240e15 −0.371324
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.03594e15 −0.685025
\(768\) 0 0
\(769\) −7.07369e15 −0.948530 −0.474265 − 0.880382i \(-0.657286\pi\)
−0.474265 + 0.880382i \(0.657286\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.06373e16 −1.38626 −0.693132 − 0.720810i \(-0.743770\pi\)
−0.693132 + 0.720810i \(0.743770\pi\)
\(774\) 0 0
\(775\) 7.05232e15 0.906093
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.01427e16 −1.26676
\(780\) 0 0
\(781\) −1.41203e16 −1.73885
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.42864e15 0.290791
\(786\) 0 0
\(787\) −9.36381e15 −1.10558 −0.552792 − 0.833319i \(-0.686438\pi\)
−0.552792 + 0.833319i \(0.686438\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.34332e15 −0.269068
\(792\) 0 0
\(793\) −7.76716e15 −0.879548
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.30839e16 1.44118 0.720588 − 0.693364i \(-0.243872\pi\)
0.720588 + 0.693364i \(0.243872\pi\)
\(798\) 0 0
\(799\) 1.46048e15 0.158667
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.67973e16 1.77543
\(804\) 0 0
\(805\) 6.78225e14 0.0707125
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.35450e15 0.644710 0.322355 − 0.946619i \(-0.395526\pi\)
0.322355 + 0.946619i \(0.395526\pi\)
\(810\) 0 0
\(811\) −4.23471e15 −0.423847 −0.211924 − 0.977286i \(-0.567973\pi\)
−0.211924 + 0.977286i \(0.567973\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.42474e15 −0.138794
\(816\) 0 0
\(817\) −1.64795e16 −1.58388
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.94663e15 −0.182136 −0.0910682 − 0.995845i \(-0.529028\pi\)
−0.0910682 + 0.995845i \(0.529028\pi\)
\(822\) 0 0
\(823\) −1.54905e16 −1.43010 −0.715051 − 0.699072i \(-0.753597\pi\)
−0.715051 + 0.699072i \(0.753597\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1.28411e16 −1.15431 −0.577154 − 0.816636i \(-0.695837\pi\)
−0.577154 + 0.816636i \(0.695837\pi\)
\(828\) 0 0
\(829\) 1.70504e16 1.51246 0.756229 − 0.654307i \(-0.227040\pi\)
0.756229 + 0.654307i \(0.227040\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.44218e15 −0.124587
\(834\) 0 0
\(835\) −6.74536e14 −0.0575081
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3.71947e15 0.308881 0.154440 − 0.988002i \(-0.450643\pi\)
0.154440 + 0.988002i \(0.450643\pi\)
\(840\) 0 0
\(841\) 9.98175e15 0.818142
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.88571e15 −0.150579
\(846\) 0 0
\(847\) 3.53922e15 0.278965
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.56391e15 −0.120115
\(852\) 0 0
\(853\) 7.37445e15 0.559126 0.279563 − 0.960127i \(-0.409810\pi\)
0.279563 + 0.960127i \(0.409810\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.88812e16 1.39520 0.697598 − 0.716490i \(-0.254252\pi\)
0.697598 + 0.716490i \(0.254252\pi\)
\(858\) 0 0
\(859\) −4.46083e15 −0.325427 −0.162713 − 0.986673i \(-0.552025\pi\)
−0.162713 + 0.986673i \(0.552025\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.11445e15 −0.0792501 −0.0396250 − 0.999215i \(-0.512616\pi\)
−0.0396250 + 0.999215i \(0.512616\pi\)
\(864\) 0 0
\(865\) −5.01814e15 −0.352334
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.47432e15 0.169374
\(870\) 0 0
\(871\) 8.80869e15 0.595404
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.33357e15 −0.219718
\(876\) 0 0
\(877\) 2.39112e16 1.55634 0.778169 − 0.628055i \(-0.216149\pi\)
0.778169 + 0.628055i \(0.216149\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.58603e16 −1.64160 −0.820799 − 0.571217i \(-0.806472\pi\)
−0.820799 + 0.571217i \(0.806472\pi\)
\(882\) 0 0
\(883\) −1.93296e16 −1.21182 −0.605911 − 0.795532i \(-0.707191\pi\)
−0.605911 + 0.795532i \(0.707191\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.69053e16 −1.64535 −0.822675 − 0.568513i \(-0.807519\pi\)
−0.822675 + 0.568513i \(0.807519\pi\)
\(888\) 0 0
\(889\) −2.29463e15 −0.138597
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.97769e15 0.234395
\(894\) 0 0
\(895\) 4.69445e14 0.0273249
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.37147e16 1.34691
\(900\) 0 0
\(901\) 1.96743e16 1.10386
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.76473e15 −0.479923
\(906\) 0 0
\(907\) 8.41876e15 0.455416 0.227708 − 0.973729i \(-0.426877\pi\)
0.227708 + 0.973729i \(0.426877\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.56863e16 0.828266 0.414133 − 0.910216i \(-0.364085\pi\)
0.414133 + 0.910216i \(0.364085\pi\)
\(912\) 0 0
\(913\) 1.51383e16 0.789749
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.47220e15 0.278696
\(918\) 0 0
\(919\) −2.60562e16 −1.31122 −0.655610 − 0.755100i \(-0.727589\pi\)
−0.655610 + 0.755100i \(0.727589\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.90949e16 −0.938228
\(924\) 0 0
\(925\) 3.65616e15 0.177519
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.44058e15 0.115720 0.0578598 − 0.998325i \(-0.481572\pi\)
0.0578598 + 0.998325i \(0.481572\pi\)
\(930\) 0 0
\(931\) −3.92786e15 −0.184049
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.65795e15 −0.350468
\(936\) 0 0
\(937\) 1.80843e16 0.817963 0.408982 − 0.912543i \(-0.365884\pi\)
0.408982 + 0.912543i \(0.365884\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.48874e13 0.00375059 0.00187530 − 0.999998i \(-0.499403\pi\)
0.00187530 + 0.999998i \(0.499403\pi\)
\(942\) 0 0
\(943\) 1.38191e16 0.603484
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.64390e15 −0.0701373 −0.0350687 − 0.999385i \(-0.511165\pi\)
−0.0350687 + 0.999385i \(0.511165\pi\)
\(948\) 0 0
\(949\) 2.27151e16 0.957965
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.80712e15 0.0744690 0.0372345 − 0.999307i \(-0.488145\pi\)
0.0372345 + 0.999307i \(0.488145\pi\)
\(954\) 0 0
\(955\) −9.08402e14 −0.0370049
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.31657e16 −0.524133
\(960\) 0 0
\(961\) −5.55120e13 −0.00218478
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.99022e15 0.191964
\(966\) 0 0
\(967\) 1.80376e16 0.686013 0.343006 − 0.939333i \(-0.388555\pi\)
0.343006 + 0.939333i \(0.388555\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.47922e16 1.66532 0.832658 − 0.553787i \(-0.186818\pi\)
0.832658 + 0.553787i \(0.186818\pi\)
\(972\) 0 0
\(973\) 1.58041e15 0.0580965
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.38403e16 0.856825 0.428413 − 0.903583i \(-0.359073\pi\)
0.428413 + 0.903583i \(0.359073\pi\)
\(978\) 0 0
\(979\) −4.70684e16 −1.67273
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.03980e16 −1.75134 −0.875668 − 0.482914i \(-0.839578\pi\)
−0.875668 + 0.482914i \(0.839578\pi\)
\(984\) 0 0
\(985\) 9.81309e15 0.337215
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.24529e16 0.754560
\(990\) 0 0
\(991\) 2.04972e16 0.681224 0.340612 − 0.940204i \(-0.389366\pi\)
0.340612 + 0.940204i \(0.389366\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.55495e15 0.245587
\(996\) 0 0
\(997\) −2.97428e16 −0.956220 −0.478110 − 0.878300i \(-0.658678\pi\)
−0.478110 + 0.878300i \(0.658678\pi\)
\(998\) 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 252.12.a.a.1.1 1
3.2 odd 2 84.12.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.12.a.a.1.1 ✓ 1 3.2 odd 2
252.12.a.a.1.1 1 1.1 even 1 trivial