Properties

Label 882.4.a.q.1.1
Level $882$
Weight $4$
Character 882.1
Self dual yes
Analytic conductor $52.040$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,4,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(52.0396846251\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 882.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +8.00000 q^{5} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +8.00000 q^{5} +8.00000 q^{8} +16.0000 q^{10} -40.0000 q^{11} -4.00000 q^{13} +16.0000 q^{16} -84.0000 q^{17} -148.000 q^{19} +32.0000 q^{20} -80.0000 q^{22} -84.0000 q^{23} -61.0000 q^{25} -8.00000 q^{26} -58.0000 q^{29} +136.000 q^{31} +32.0000 q^{32} -168.000 q^{34} -222.000 q^{37} -296.000 q^{38} +64.0000 q^{40} +420.000 q^{41} -164.000 q^{43} -160.000 q^{44} -168.000 q^{46} +488.000 q^{47} -122.000 q^{50} -16.0000 q^{52} -478.000 q^{53} -320.000 q^{55} -116.000 q^{58} +548.000 q^{59} -692.000 q^{61} +272.000 q^{62} +64.0000 q^{64} -32.0000 q^{65} -908.000 q^{67} -336.000 q^{68} +524.000 q^{71} -440.000 q^{73} -444.000 q^{74} -592.000 q^{76} +1216.00 q^{79} +128.000 q^{80} +840.000 q^{82} -684.000 q^{83} -672.000 q^{85} -328.000 q^{86} -320.000 q^{88} +604.000 q^{89} -336.000 q^{92} +976.000 q^{94} -1184.00 q^{95} +832.000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 0.707107
\(3\) 0 0
\(4\) 4.00000 0.500000
\(5\) 8.00000 0.715542 0.357771 0.933809i \(-0.383537\pi\)
0.357771 + 0.933809i \(0.383537\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 8.00000 0.353553
\(9\) 0 0
\(10\) 16.0000 0.505964
\(11\) −40.0000 −1.09640 −0.548202 0.836346i \(-0.684688\pi\)
−0.548202 + 0.836346i \(0.684688\pi\)
\(12\) 0 0
\(13\) −4.00000 −0.0853385 −0.0426692 0.999089i \(-0.513586\pi\)
−0.0426692 + 0.999089i \(0.513586\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 0.250000
\(17\) −84.0000 −1.19841 −0.599206 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) 0 0
\(19\) −148.000 −1.78703 −0.893514 0.449036i \(-0.851768\pi\)
−0.893514 + 0.449036i \(0.851768\pi\)
\(20\) 32.0000 0.357771
\(21\) 0 0
\(22\) −80.0000 −0.775275
\(23\) −84.0000 −0.761531 −0.380765 0.924672i \(-0.624339\pi\)
−0.380765 + 0.924672i \(0.624339\pi\)
\(24\) 0 0
\(25\) −61.0000 −0.488000
\(26\) −8.00000 −0.0603434
\(27\) 0 0
\(28\) 0 0
\(29\) −58.0000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 136.000 0.787946 0.393973 0.919122i \(-0.371100\pi\)
0.393973 + 0.919122i \(0.371100\pi\)
\(32\) 32.0000 0.176777
\(33\) 0 0
\(34\) −168.000 −0.847405
\(35\) 0 0
\(36\) 0 0
\(37\) −222.000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −296.000 −1.26362
\(39\) 0 0
\(40\) 64.0000 0.252982
\(41\) 420.000 1.59983 0.799914 0.600114i \(-0.204878\pi\)
0.799914 + 0.600114i \(0.204878\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) −160.000 −0.548202
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) 488.000 1.51451 0.757257 0.653118i \(-0.226539\pi\)
0.757257 + 0.653118i \(0.226539\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −122.000 −0.345068
\(51\) 0 0
\(52\) −16.0000 −0.0426692
\(53\) −478.000 −1.23884 −0.619418 0.785061i \(-0.712632\pi\)
−0.619418 + 0.785061i \(0.712632\pi\)
\(54\) 0 0
\(55\) −320.000 −0.784523
\(56\) 0 0
\(57\) 0 0
\(58\) −116.000 −0.262613
\(59\) 548.000 1.20921 0.604606 0.796525i \(-0.293331\pi\)
0.604606 + 0.796525i \(0.293331\pi\)
\(60\) 0 0
\(61\) −692.000 −1.45248 −0.726242 0.687439i \(-0.758735\pi\)
−0.726242 + 0.687439i \(0.758735\pi\)
\(62\) 272.000 0.557162
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) −32.0000 −0.0610633
\(66\) 0 0
\(67\) −908.000 −1.65567 −0.827835 0.560972i \(-0.810428\pi\)
−0.827835 + 0.560972i \(0.810428\pi\)
\(68\) −336.000 −0.599206
\(69\) 0 0
\(70\) 0 0
\(71\) 524.000 0.875878 0.437939 0.899005i \(-0.355709\pi\)
0.437939 + 0.899005i \(0.355709\pi\)
\(72\) 0 0
\(73\) −440.000 −0.705453 −0.352727 0.935726i \(-0.614745\pi\)
−0.352727 + 0.935726i \(0.614745\pi\)
\(74\) −444.000 −0.697486
\(75\) 0 0
\(76\) −592.000 −0.893514
\(77\) 0 0
\(78\) 0 0
\(79\) 1216.00 1.73178 0.865890 0.500234i \(-0.166753\pi\)
0.865890 + 0.500234i \(0.166753\pi\)
\(80\) 128.000 0.178885
\(81\) 0 0
\(82\) 840.000 1.13125
\(83\) −684.000 −0.904563 −0.452282 0.891875i \(-0.649390\pi\)
−0.452282 + 0.891875i \(0.649390\pi\)
\(84\) 0 0
\(85\) −672.000 −0.857513
\(86\) −328.000 −0.411269
\(87\) 0 0
\(88\) −320.000 −0.387638
\(89\) 604.000 0.719369 0.359685 0.933074i \(-0.382884\pi\)
0.359685 + 0.933074i \(0.382884\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −336.000 −0.380765
\(93\) 0 0
\(94\) 976.000 1.07092
\(95\) −1184.00 −1.27869
\(96\) 0 0
\(97\) 832.000 0.870895 0.435447 0.900214i \(-0.356590\pi\)
0.435447 + 0.900214i \(0.356590\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −244.000 −0.244000
\(101\) 464.000 0.457126 0.228563 0.973529i \(-0.426597\pi\)
0.228563 + 0.973529i \(0.426597\pi\)
\(102\) 0 0
\(103\) 632.000 0.604590 0.302295 0.953214i \(-0.402247\pi\)
0.302295 + 0.953214i \(0.402247\pi\)
\(104\) −32.0000 −0.0301717
\(105\) 0 0
\(106\) −956.000 −0.875990
\(107\) 160.000 0.144559 0.0722794 0.997384i \(-0.476973\pi\)
0.0722794 + 0.997384i \(0.476973\pi\)
\(108\) 0 0
\(109\) −2198.00 −1.93147 −0.965735 0.259530i \(-0.916432\pi\)
−0.965735 + 0.259530i \(0.916432\pi\)
\(110\) −640.000 −0.554742
\(111\) 0 0
\(112\) 0 0
\(113\) −770.000 −0.641022 −0.320511 0.947245i \(-0.603855\pi\)
−0.320511 + 0.947245i \(0.603855\pi\)
\(114\) 0 0
\(115\) −672.000 −0.544907
\(116\) −232.000 −0.185695
\(117\) 0 0
\(118\) 1096.00 0.855042
\(119\) 0 0
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) −1384.00 −1.02706
\(123\) 0 0
\(124\) 544.000 0.393973
\(125\) −1488.00 −1.06473
\(126\) 0 0
\(127\) −184.000 −0.128562 −0.0642809 0.997932i \(-0.520475\pi\)
−0.0642809 + 0.997932i \(0.520475\pi\)
\(128\) 128.000 0.0883883
\(129\) 0 0
\(130\) −64.0000 −0.0431782
\(131\) −1452.00 −0.968411 −0.484205 0.874954i \(-0.660891\pi\)
−0.484205 + 0.874954i \(0.660891\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −1816.00 −1.17074
\(135\) 0 0
\(136\) −672.000 −0.423702
\(137\) −646.000 −0.402858 −0.201429 0.979503i \(-0.564558\pi\)
−0.201429 + 0.979503i \(0.564558\pi\)
\(138\) 0 0
\(139\) 3012.00 1.83795 0.918973 0.394320i \(-0.129020\pi\)
0.918973 + 0.394320i \(0.129020\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1048.00 0.619339
\(143\) 160.000 0.0935655
\(144\) 0 0
\(145\) −464.000 −0.265746
\(146\) −880.000 −0.498831
\(147\) 0 0
\(148\) −888.000 −0.493197
\(149\) 3170.00 1.74293 0.871465 0.490458i \(-0.163170\pi\)
0.871465 + 0.490458i \(0.163170\pi\)
\(150\) 0 0
\(151\) −1880.00 −1.01319 −0.506597 0.862183i \(-0.669097\pi\)
−0.506597 + 0.862183i \(0.669097\pi\)
\(152\) −1184.00 −0.631810
\(153\) 0 0
\(154\) 0 0
\(155\) 1088.00 0.563808
\(156\) 0 0
\(157\) −604.000 −0.307035 −0.153517 0.988146i \(-0.549060\pi\)
−0.153517 + 0.988146i \(0.549060\pi\)
\(158\) 2432.00 1.22455
\(159\) 0 0
\(160\) 256.000 0.126491
\(161\) 0 0
\(162\) 0 0
\(163\) 1116.00 0.536269 0.268135 0.963381i \(-0.413593\pi\)
0.268135 + 0.963381i \(0.413593\pi\)
\(164\) 1680.00 0.799914
\(165\) 0 0
\(166\) −1368.00 −0.639623
\(167\) −1784.00 −0.826647 −0.413324 0.910584i \(-0.635632\pi\)
−0.413324 + 0.910584i \(0.635632\pi\)
\(168\) 0 0
\(169\) −2181.00 −0.992717
\(170\) −1344.00 −0.606353
\(171\) 0 0
\(172\) −656.000 −0.290811
\(173\) −344.000 −0.151178 −0.0755891 0.997139i \(-0.524084\pi\)
−0.0755891 + 0.997139i \(0.524084\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −640.000 −0.274101
\(177\) 0 0
\(178\) 1208.00 0.508671
\(179\) −1392.00 −0.581246 −0.290623 0.956838i \(-0.593862\pi\)
−0.290623 + 0.956838i \(0.593862\pi\)
\(180\) 0 0
\(181\) −4052.00 −1.66399 −0.831997 0.554781i \(-0.812802\pi\)
−0.831997 + 0.554781i \(0.812802\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −672.000 −0.269242
\(185\) −1776.00 −0.705806
\(186\) 0 0
\(187\) 3360.00 1.31394
\(188\) 1952.00 0.757257
\(189\) 0 0
\(190\) −2368.00 −0.904173
\(191\) 3108.00 1.17742 0.588709 0.808345i \(-0.299636\pi\)
0.588709 + 0.808345i \(0.299636\pi\)
\(192\) 0 0
\(193\) 50.0000 0.0186481 0.00932404 0.999957i \(-0.497032\pi\)
0.00932404 + 0.999957i \(0.497032\pi\)
\(194\) 1664.00 0.615816
\(195\) 0 0
\(196\) 0 0
\(197\) 162.000 0.0585889 0.0292945 0.999571i \(-0.490674\pi\)
0.0292945 + 0.999571i \(0.490674\pi\)
\(198\) 0 0
\(199\) −1544.00 −0.550006 −0.275003 0.961443i \(-0.588679\pi\)
−0.275003 + 0.961443i \(0.588679\pi\)
\(200\) −488.000 −0.172534
\(201\) 0 0
\(202\) 928.000 0.323237
\(203\) 0 0
\(204\) 0 0
\(205\) 3360.00 1.14474
\(206\) 1264.00 0.427510
\(207\) 0 0
\(208\) −64.0000 −0.0213346
\(209\) 5920.00 1.95931
\(210\) 0 0
\(211\) −1204.00 −0.392828 −0.196414 0.980521i \(-0.562930\pi\)
−0.196414 + 0.980521i \(0.562930\pi\)
\(212\) −1912.00 −0.619418
\(213\) 0 0
\(214\) 320.000 0.102218
\(215\) −1312.00 −0.416175
\(216\) 0 0
\(217\) 0 0
\(218\) −4396.00 −1.36576
\(219\) 0 0
\(220\) −1280.00 −0.392262
\(221\) 336.000 0.102271
\(222\) 0 0
\(223\) −2000.00 −0.600583 −0.300291 0.953848i \(-0.597084\pi\)
−0.300291 + 0.953848i \(0.597084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1540.00 −0.453271
\(227\) 388.000 0.113447 0.0567235 0.998390i \(-0.481935\pi\)
0.0567235 + 0.998390i \(0.481935\pi\)
\(228\) 0 0
\(229\) −4180.00 −1.20621 −0.603105 0.797662i \(-0.706070\pi\)
−0.603105 + 0.797662i \(0.706070\pi\)
\(230\) −1344.00 −0.385308
\(231\) 0 0
\(232\) −464.000 −0.131306
\(233\) 1322.00 0.371704 0.185852 0.982578i \(-0.440495\pi\)
0.185852 + 0.982578i \(0.440495\pi\)
\(234\) 0 0
\(235\) 3904.00 1.08370
\(236\) 2192.00 0.604606
\(237\) 0 0
\(238\) 0 0
\(239\) −2412.00 −0.652800 −0.326400 0.945232i \(-0.605836\pi\)
−0.326400 + 0.945232i \(0.605836\pi\)
\(240\) 0 0
\(241\) 4336.00 1.15895 0.579474 0.814991i \(-0.303258\pi\)
0.579474 + 0.814991i \(0.303258\pi\)
\(242\) 538.000 0.142909
\(243\) 0 0
\(244\) −2768.00 −0.726242
\(245\) 0 0
\(246\) 0 0
\(247\) 592.000 0.152502
\(248\) 1088.00 0.278581
\(249\) 0 0
\(250\) −2976.00 −0.752875
\(251\) 764.000 0.192125 0.0960623 0.995375i \(-0.469375\pi\)
0.0960623 + 0.995375i \(0.469375\pi\)
\(252\) 0 0
\(253\) 3360.00 0.834946
\(254\) −368.000 −0.0909070
\(255\) 0 0
\(256\) 256.000 0.0625000
\(257\) 4300.00 1.04368 0.521842 0.853042i \(-0.325245\pi\)
0.521842 + 0.853042i \(0.325245\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −128.000 −0.0305316
\(261\) 0 0
\(262\) −2904.00 −0.684770
\(263\) 3860.00 0.905011 0.452505 0.891762i \(-0.350530\pi\)
0.452505 + 0.891762i \(0.350530\pi\)
\(264\) 0 0
\(265\) −3824.00 −0.886439
\(266\) 0 0
\(267\) 0 0
\(268\) −3632.00 −0.827835
\(269\) −2800.00 −0.634643 −0.317322 0.948318i \(-0.602783\pi\)
−0.317322 + 0.948318i \(0.602783\pi\)
\(270\) 0 0
\(271\) 4880.00 1.09387 0.546935 0.837175i \(-0.315794\pi\)
0.546935 + 0.837175i \(0.315794\pi\)
\(272\) −1344.00 −0.299603
\(273\) 0 0
\(274\) −1292.00 −0.284863
\(275\) 2440.00 0.535046
\(276\) 0 0
\(277\) −6674.00 −1.44766 −0.723830 0.689978i \(-0.757620\pi\)
−0.723830 + 0.689978i \(0.757620\pi\)
\(278\) 6024.00 1.29962
\(279\) 0 0
\(280\) 0 0
\(281\) 9402.00 1.99600 0.998001 0.0632056i \(-0.0201324\pi\)
0.998001 + 0.0632056i \(0.0201324\pi\)
\(282\) 0 0
\(283\) 9100.00 1.91144 0.955722 0.294270i \(-0.0950765\pi\)
0.955722 + 0.294270i \(0.0950765\pi\)
\(284\) 2096.00 0.437939
\(285\) 0 0
\(286\) 320.000 0.0661608
\(287\) 0 0
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) −928.000 −0.187910
\(291\) 0 0
\(292\) −1760.00 −0.352727
\(293\) 5952.00 1.18676 0.593378 0.804924i \(-0.297794\pi\)
0.593378 + 0.804924i \(0.297794\pi\)
\(294\) 0 0
\(295\) 4384.00 0.865242
\(296\) −1776.00 −0.348743
\(297\) 0 0
\(298\) 6340.00 1.23244
\(299\) 336.000 0.0649879
\(300\) 0 0
\(301\) 0 0
\(302\) −3760.00 −0.716436
\(303\) 0 0
\(304\) −2368.00 −0.446757
\(305\) −5536.00 −1.03931
\(306\) 0 0
\(307\) 3004.00 0.558460 0.279230 0.960224i \(-0.409921\pi\)
0.279230 + 0.960224i \(0.409921\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2176.00 0.398673
\(311\) 688.000 0.125443 0.0627217 0.998031i \(-0.480022\pi\)
0.0627217 + 0.998031i \(0.480022\pi\)
\(312\) 0 0
\(313\) −5592.00 −1.00984 −0.504918 0.863167i \(-0.668477\pi\)
−0.504918 + 0.863167i \(0.668477\pi\)
\(314\) −1208.00 −0.217106
\(315\) 0 0
\(316\) 4864.00 0.865890
\(317\) 2922.00 0.517716 0.258858 0.965915i \(-0.416654\pi\)
0.258858 + 0.965915i \(0.416654\pi\)
\(318\) 0 0
\(319\) 2320.00 0.407195
\(320\) 512.000 0.0894427
\(321\) 0 0
\(322\) 0 0
\(323\) 12432.0 2.14159
\(324\) 0 0
\(325\) 244.000 0.0416452
\(326\) 2232.00 0.379200
\(327\) 0 0
\(328\) 3360.00 0.565625
\(329\) 0 0
\(330\) 0 0
\(331\) −7492.00 −1.24410 −0.622051 0.782977i \(-0.713700\pi\)
−0.622051 + 0.782977i \(0.713700\pi\)
\(332\) −2736.00 −0.452282
\(333\) 0 0
\(334\) −3568.00 −0.584528
\(335\) −7264.00 −1.18470
\(336\) 0 0
\(337\) 10766.0 1.74024 0.870121 0.492839i \(-0.164041\pi\)
0.870121 + 0.492839i \(0.164041\pi\)
\(338\) −4362.00 −0.701957
\(339\) 0 0
\(340\) −2688.00 −0.428757
\(341\) −5440.00 −0.863908
\(342\) 0 0
\(343\) 0 0
\(344\) −1312.00 −0.205635
\(345\) 0 0
\(346\) −688.000 −0.106899
\(347\) 3984.00 0.616347 0.308173 0.951330i \(-0.400282\pi\)
0.308173 + 0.951330i \(0.400282\pi\)
\(348\) 0 0
\(349\) 180.000 0.0276080 0.0138040 0.999905i \(-0.495606\pi\)
0.0138040 + 0.999905i \(0.495606\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1280.00 −0.193819
\(353\) −10428.0 −1.57231 −0.786156 0.618028i \(-0.787932\pi\)
−0.786156 + 0.618028i \(0.787932\pi\)
\(354\) 0 0
\(355\) 4192.00 0.626727
\(356\) 2416.00 0.359685
\(357\) 0 0
\(358\) −2784.00 −0.411003
\(359\) −8684.00 −1.27667 −0.638334 0.769759i \(-0.720376\pi\)
−0.638334 + 0.769759i \(0.720376\pi\)
\(360\) 0 0
\(361\) 15045.0 2.19347
\(362\) −8104.00 −1.17662
\(363\) 0 0
\(364\) 0 0
\(365\) −3520.00 −0.504781
\(366\) 0 0
\(367\) −5648.00 −0.803333 −0.401666 0.915786i \(-0.631569\pi\)
−0.401666 + 0.915786i \(0.631569\pi\)
\(368\) −1344.00 −0.190383
\(369\) 0 0
\(370\) −3552.00 −0.499080
\(371\) 0 0
\(372\) 0 0
\(373\) −2546.00 −0.353423 −0.176712 0.984263i \(-0.556546\pi\)
−0.176712 + 0.984263i \(0.556546\pi\)
\(374\) 6720.00 0.929099
\(375\) 0 0
\(376\) 3904.00 0.535461
\(377\) 232.000 0.0316939
\(378\) 0 0
\(379\) 8268.00 1.12058 0.560288 0.828298i \(-0.310690\pi\)
0.560288 + 0.828298i \(0.310690\pi\)
\(380\) −4736.00 −0.639347
\(381\) 0 0
\(382\) 6216.00 0.832561
\(383\) −10872.0 −1.45048 −0.725239 0.688497i \(-0.758271\pi\)
−0.725239 + 0.688497i \(0.758271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 100.000 0.0131862
\(387\) 0 0
\(388\) 3328.00 0.435447
\(389\) −10434.0 −1.35996 −0.679980 0.733230i \(-0.738012\pi\)
−0.679980 + 0.733230i \(0.738012\pi\)
\(390\) 0 0
\(391\) 7056.00 0.912627
\(392\) 0 0
\(393\) 0 0
\(394\) 324.000 0.0414286
\(395\) 9728.00 1.23916
\(396\) 0 0
\(397\) 3044.00 0.384821 0.192411 0.981315i \(-0.438369\pi\)
0.192411 + 0.981315i \(0.438369\pi\)
\(398\) −3088.00 −0.388913
\(399\) 0 0
\(400\) −976.000 −0.122000
\(401\) −8910.00 −1.10959 −0.554793 0.831988i \(-0.687203\pi\)
−0.554793 + 0.831988i \(0.687203\pi\)
\(402\) 0 0
\(403\) −544.000 −0.0672421
\(404\) 1856.00 0.228563
\(405\) 0 0
\(406\) 0 0
\(407\) 8880.00 1.08149
\(408\) 0 0
\(409\) −5616.00 −0.678957 −0.339478 0.940614i \(-0.610251\pi\)
−0.339478 + 0.940614i \(0.610251\pi\)
\(410\) 6720.00 0.809456
\(411\) 0 0
\(412\) 2528.00 0.302295
\(413\) 0 0
\(414\) 0 0
\(415\) −5472.00 −0.647253
\(416\) −128.000 −0.0150859
\(417\) 0 0
\(418\) 11840.0 1.38544
\(419\) −8932.00 −1.04142 −0.520712 0.853732i \(-0.674334\pi\)
−0.520712 + 0.853732i \(0.674334\pi\)
\(420\) 0 0
\(421\) −5538.00 −0.641106 −0.320553 0.947231i \(-0.603869\pi\)
−0.320553 + 0.947231i \(0.603869\pi\)
\(422\) −2408.00 −0.277772
\(423\) 0 0
\(424\) −3824.00 −0.437995
\(425\) 5124.00 0.584825
\(426\) 0 0
\(427\) 0 0
\(428\) 640.000 0.0722794
\(429\) 0 0
\(430\) −2624.00 −0.294280
\(431\) 6700.00 0.748788 0.374394 0.927270i \(-0.377851\pi\)
0.374394 + 0.927270i \(0.377851\pi\)
\(432\) 0 0
\(433\) 5048.00 0.560257 0.280129 0.959962i \(-0.409623\pi\)
0.280129 + 0.959962i \(0.409623\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −8792.00 −0.965735
\(437\) 12432.0 1.36088
\(438\) 0 0
\(439\) 1344.00 0.146118 0.0730588 0.997328i \(-0.476724\pi\)
0.0730588 + 0.997328i \(0.476724\pi\)
\(440\) −2560.00 −0.277371
\(441\) 0 0
\(442\) 672.000 0.0723162
\(443\) 4392.00 0.471039 0.235519 0.971870i \(-0.424321\pi\)
0.235519 + 0.971870i \(0.424321\pi\)
\(444\) 0 0
\(445\) 4832.00 0.514739
\(446\) −4000.00 −0.424676
\(447\) 0 0
\(448\) 0 0
\(449\) −3666.00 −0.385321 −0.192661 0.981265i \(-0.561712\pi\)
−0.192661 + 0.981265i \(0.561712\pi\)
\(450\) 0 0
\(451\) −16800.0 −1.75406
\(452\) −3080.00 −0.320511
\(453\) 0 0
\(454\) 776.000 0.0802191
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 0.00266133 0.00133067 0.999999i \(-0.499576\pi\)
0.00133067 + 0.999999i \(0.499576\pi\)
\(458\) −8360.00 −0.852920
\(459\) 0 0
\(460\) −2688.00 −0.272454
\(461\) 7656.00 0.773483 0.386741 0.922188i \(-0.373601\pi\)
0.386741 + 0.922188i \(0.373601\pi\)
\(462\) 0 0
\(463\) 12608.0 1.26554 0.632768 0.774341i \(-0.281919\pi\)
0.632768 + 0.774341i \(0.281919\pi\)
\(464\) −928.000 −0.0928477
\(465\) 0 0
\(466\) 2644.00 0.262835
\(467\) −3068.00 −0.304005 −0.152002 0.988380i \(-0.548572\pi\)
−0.152002 + 0.988380i \(0.548572\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 7808.00 0.766290
\(471\) 0 0
\(472\) 4384.00 0.427521
\(473\) 6560.00 0.637694
\(474\) 0 0
\(475\) 9028.00 0.872070
\(476\) 0 0
\(477\) 0 0
\(478\) −4824.00 −0.461600
\(479\) 6456.00 0.615829 0.307915 0.951414i \(-0.400369\pi\)
0.307915 + 0.951414i \(0.400369\pi\)
\(480\) 0 0
\(481\) 888.000 0.0841774
\(482\) 8672.00 0.819500
\(483\) 0 0
\(484\) 1076.00 0.101052
\(485\) 6656.00 0.623162
\(486\) 0 0
\(487\) 11896.0 1.10690 0.553449 0.832883i \(-0.313311\pi\)
0.553449 + 0.832883i \(0.313311\pi\)
\(488\) −5536.00 −0.513531
\(489\) 0 0
\(490\) 0 0
\(491\) 264.000 0.0242651 0.0121325 0.999926i \(-0.496138\pi\)
0.0121325 + 0.999926i \(0.496138\pi\)
\(492\) 0 0
\(493\) 4872.00 0.445079
\(494\) 1184.00 0.107835
\(495\) 0 0
\(496\) 2176.00 0.196986
\(497\) 0 0
\(498\) 0 0
\(499\) −2628.00 −0.235762 −0.117881 0.993028i \(-0.537610\pi\)
−0.117881 + 0.993028i \(0.537610\pi\)
\(500\) −5952.00 −0.532363
\(501\) 0 0
\(502\) 1528.00 0.135853
\(503\) −13568.0 −1.20272 −0.601359 0.798979i \(-0.705374\pi\)
−0.601359 + 0.798979i \(0.705374\pi\)
\(504\) 0 0
\(505\) 3712.00 0.327093
\(506\) 6720.00 0.590396
\(507\) 0 0
\(508\) −736.000 −0.0642809
\(509\) 20656.0 1.79874 0.899372 0.437183i \(-0.144024\pi\)
0.899372 + 0.437183i \(0.144024\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000 0.0441942
\(513\) 0 0
\(514\) 8600.00 0.737996
\(515\) 5056.00 0.432610
\(516\) 0 0
\(517\) −19520.0 −1.66052
\(518\) 0 0
\(519\) 0 0
\(520\) −256.000 −0.0215891
\(521\) 3628.00 0.305078 0.152539 0.988297i \(-0.451255\pi\)
0.152539 + 0.988297i \(0.451255\pi\)
\(522\) 0 0
\(523\) −4852.00 −0.405666 −0.202833 0.979213i \(-0.565015\pi\)
−0.202833 + 0.979213i \(0.565015\pi\)
\(524\) −5808.00 −0.484205
\(525\) 0 0
\(526\) 7720.00 0.639939
\(527\) −11424.0 −0.944283
\(528\) 0 0
\(529\) −5111.00 −0.420071
\(530\) −7648.00 −0.626807
\(531\) 0 0
\(532\) 0 0
\(533\) −1680.00 −0.136527
\(534\) 0 0
\(535\) 1280.00 0.103438
\(536\) −7264.00 −0.585368
\(537\) 0 0
\(538\) −5600.00 −0.448760
\(539\) 0 0
\(540\) 0 0
\(541\) −7130.00 −0.566622 −0.283311 0.959028i \(-0.591433\pi\)
−0.283311 + 0.959028i \(0.591433\pi\)
\(542\) 9760.00 0.773483
\(543\) 0 0
\(544\) −2688.00 −0.211851
\(545\) −17584.0 −1.38205
\(546\) 0 0
\(547\) −12788.0 −0.999589 −0.499795 0.866144i \(-0.666591\pi\)
−0.499795 + 0.866144i \(0.666591\pi\)
\(548\) −2584.00 −0.201429
\(549\) 0 0
\(550\) 4880.00 0.378334
\(551\) 8584.00 0.663685
\(552\) 0 0
\(553\) 0 0
\(554\) −13348.0 −1.02365
\(555\) 0 0
\(556\) 12048.0 0.918973
\(557\) −2406.00 −0.183026 −0.0915130 0.995804i \(-0.529170\pi\)
−0.0915130 + 0.995804i \(0.529170\pi\)
\(558\) 0 0
\(559\) 656.000 0.0496348
\(560\) 0 0
\(561\) 0 0
\(562\) 18804.0 1.41139
\(563\) −25412.0 −1.90229 −0.951144 0.308748i \(-0.900090\pi\)
−0.951144 + 0.308748i \(0.900090\pi\)
\(564\) 0 0
\(565\) −6160.00 −0.458678
\(566\) 18200.0 1.35160
\(567\) 0 0
\(568\) 4192.00 0.309670
\(569\) 9690.00 0.713930 0.356965 0.934118i \(-0.383812\pi\)
0.356965 + 0.934118i \(0.383812\pi\)
\(570\) 0 0
\(571\) 5604.00 0.410718 0.205359 0.978687i \(-0.434164\pi\)
0.205359 + 0.978687i \(0.434164\pi\)
\(572\) 640.000 0.0467828
\(573\) 0 0
\(574\) 0 0
\(575\) 5124.00 0.371627
\(576\) 0 0
\(577\) 21568.0 1.55613 0.778066 0.628183i \(-0.216201\pi\)
0.778066 + 0.628183i \(0.216201\pi\)
\(578\) 4286.00 0.308433
\(579\) 0 0
\(580\) −1856.00 −0.132873
\(581\) 0 0
\(582\) 0 0
\(583\) 19120.0 1.35827
\(584\) −3520.00 −0.249415
\(585\) 0 0
\(586\) 11904.0 0.839163
\(587\) 20300.0 1.42738 0.713689 0.700463i \(-0.247023\pi\)
0.713689 + 0.700463i \(0.247023\pi\)
\(588\) 0 0
\(589\) −20128.0 −1.40808
\(590\) 8768.00 0.611818
\(591\) 0 0
\(592\) −3552.00 −0.246598
\(593\) 13812.0 0.956477 0.478238 0.878230i \(-0.341275\pi\)
0.478238 + 0.878230i \(0.341275\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 12680.0 0.871465
\(597\) 0 0
\(598\) 672.000 0.0459534
\(599\) 21996.0 1.50039 0.750194 0.661218i \(-0.229960\pi\)
0.750194 + 0.661218i \(0.229960\pi\)
\(600\) 0 0
\(601\) −8368.00 −0.567950 −0.283975 0.958832i \(-0.591653\pi\)
−0.283975 + 0.958832i \(0.591653\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −7520.00 −0.506597
\(605\) 2152.00 0.144614
\(606\) 0 0
\(607\) −21504.0 −1.43792 −0.718962 0.695049i \(-0.755383\pi\)
−0.718962 + 0.695049i \(0.755383\pi\)
\(608\) −4736.00 −0.315905
\(609\) 0 0
\(610\) −11072.0 −0.734905
\(611\) −1952.00 −0.129246
\(612\) 0 0
\(613\) −10270.0 −0.676674 −0.338337 0.941025i \(-0.609864\pi\)
−0.338337 + 0.941025i \(0.609864\pi\)
\(614\) 6008.00 0.394891
\(615\) 0 0
\(616\) 0 0
\(617\) −28358.0 −1.85032 −0.925162 0.379572i \(-0.876071\pi\)
−0.925162 + 0.379572i \(0.876071\pi\)
\(618\) 0 0
\(619\) −16292.0 −1.05788 −0.528942 0.848658i \(-0.677411\pi\)
−0.528942 + 0.848658i \(0.677411\pi\)
\(620\) 4352.00 0.281904
\(621\) 0 0
\(622\) 1376.00 0.0887019
\(623\) 0 0
\(624\) 0 0
\(625\) −4279.00 −0.273856
\(626\) −11184.0 −0.714062
\(627\) 0 0
\(628\) −2416.00 −0.153517
\(629\) 18648.0 1.18211
\(630\) 0 0
\(631\) 11256.0 0.710134 0.355067 0.934841i \(-0.384458\pi\)
0.355067 + 0.934841i \(0.384458\pi\)
\(632\) 9728.00 0.612277
\(633\) 0 0
\(634\) 5844.00 0.366080
\(635\) −1472.00 −0.0919914
\(636\) 0 0
\(637\) 0 0
\(638\) 4640.00 0.287930
\(639\) 0 0
\(640\) 1024.00 0.0632456
\(641\) −15518.0 −0.956200 −0.478100 0.878305i \(-0.658674\pi\)
−0.478100 + 0.878305i \(0.658674\pi\)
\(642\) 0 0
\(643\) 10452.0 0.641037 0.320518 0.947242i \(-0.396143\pi\)
0.320518 + 0.947242i \(0.396143\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24864.0 1.51434
\(647\) 72.0000 0.00437498 0.00218749 0.999998i \(-0.499304\pi\)
0.00218749 + 0.999998i \(0.499304\pi\)
\(648\) 0 0
\(649\) −21920.0 −1.32579
\(650\) 488.000 0.0294476
\(651\) 0 0
\(652\) 4464.00 0.268135
\(653\) −11962.0 −0.716859 −0.358430 0.933557i \(-0.616688\pi\)
−0.358430 + 0.933557i \(0.616688\pi\)
\(654\) 0 0
\(655\) −11616.0 −0.692938
\(656\) 6720.00 0.399957
\(657\) 0 0
\(658\) 0 0
\(659\) 6016.00 0.355615 0.177807 0.984065i \(-0.443100\pi\)
0.177807 + 0.984065i \(0.443100\pi\)
\(660\) 0 0
\(661\) −26068.0 −1.53393 −0.766965 0.641689i \(-0.778234\pi\)
−0.766965 + 0.641689i \(0.778234\pi\)
\(662\) −14984.0 −0.879713
\(663\) 0 0
\(664\) −5472.00 −0.319811
\(665\) 0 0
\(666\) 0 0
\(667\) 4872.00 0.282825
\(668\) −7136.00 −0.413324
\(669\) 0 0
\(670\) −14528.0 −0.837710
\(671\) 27680.0 1.59251
\(672\) 0 0
\(673\) −20530.0 −1.17589 −0.587945 0.808901i \(-0.700063\pi\)
−0.587945 + 0.808901i \(0.700063\pi\)
\(674\) 21532.0 1.23054
\(675\) 0 0
\(676\) −8724.00 −0.496359
\(677\) −10056.0 −0.570877 −0.285438 0.958397i \(-0.592139\pi\)
−0.285438 + 0.958397i \(0.592139\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5376.00 −0.303177
\(681\) 0 0
\(682\) −10880.0 −0.610875
\(683\) −6152.00 −0.344656 −0.172328 0.985040i \(-0.555129\pi\)
−0.172328 + 0.985040i \(0.555129\pi\)
\(684\) 0 0
\(685\) −5168.00 −0.288262
\(686\) 0 0
\(687\) 0 0
\(688\) −2624.00 −0.145406
\(689\) 1912.00 0.105720
\(690\) 0 0
\(691\) 14716.0 0.810164 0.405082 0.914280i \(-0.367243\pi\)
0.405082 + 0.914280i \(0.367243\pi\)
\(692\) −1376.00 −0.0755891
\(693\) 0 0
\(694\) 7968.00 0.435823
\(695\) 24096.0 1.31513
\(696\) 0 0
\(697\) −35280.0 −1.91725
\(698\) 360.000 0.0195218
\(699\) 0 0
\(700\) 0 0
\(701\) −28202.0 −1.51951 −0.759754 0.650211i \(-0.774681\pi\)
−0.759754 + 0.650211i \(0.774681\pi\)
\(702\) 0 0
\(703\) 32856.0 1.76271
\(704\) −2560.00 −0.137051
\(705\) 0 0
\(706\) −20856.0 −1.11179
\(707\) 0 0
\(708\) 0 0
\(709\) 22114.0 1.17138 0.585690 0.810535i \(-0.300824\pi\)
0.585690 + 0.810535i \(0.300824\pi\)
\(710\) 8384.00 0.443163
\(711\) 0 0
\(712\) 4832.00 0.254335
\(713\) −11424.0 −0.600045
\(714\) 0 0
\(715\) 1280.00 0.0669501
\(716\) −5568.00 −0.290623
\(717\) 0 0
\(718\) −17368.0 −0.902741
\(719\) −9288.00 −0.481758 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 30090.0 1.55102
\(723\) 0 0
\(724\) −16208.0 −0.831997
\(725\) 3538.00 0.181239
\(726\) 0 0
\(727\) −23848.0 −1.21661 −0.608304 0.793704i \(-0.708150\pi\)
−0.608304 + 0.793704i \(0.708150\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7040.00 −0.356934
\(731\) 13776.0 0.697023
\(732\) 0 0
\(733\) 34756.0 1.75135 0.875677 0.482898i \(-0.160416\pi\)
0.875677 + 0.482898i \(0.160416\pi\)
\(734\) −11296.0 −0.568042
\(735\) 0 0
\(736\) −2688.00 −0.134621
\(737\) 36320.0 1.81528
\(738\) 0 0
\(739\) 26044.0 1.29641 0.648203 0.761468i \(-0.275521\pi\)
0.648203 + 0.761468i \(0.275521\pi\)
\(740\) −7104.00 −0.352903
\(741\) 0 0
\(742\) 0 0
\(743\) −36204.0 −1.78761 −0.893806 0.448454i \(-0.851975\pi\)
−0.893806 + 0.448454i \(0.851975\pi\)
\(744\) 0 0
\(745\) 25360.0 1.24714
\(746\) −5092.00 −0.249908
\(747\) 0 0
\(748\) 13440.0 0.656972
\(749\) 0 0
\(750\) 0 0
\(751\) −11424.0 −0.555083 −0.277542 0.960714i \(-0.589520\pi\)
−0.277542 + 0.960714i \(0.589520\pi\)
\(752\) 7808.00 0.378628
\(753\) 0 0
\(754\) 464.000 0.0224110
\(755\) −15040.0 −0.724982
\(756\) 0 0
\(757\) −16622.0 −0.798067 −0.399034 0.916936i \(-0.630654\pi\)
−0.399034 + 0.916936i \(0.630654\pi\)
\(758\) 16536.0 0.792368
\(759\) 0 0
\(760\) −9472.00 −0.452086
\(761\) −38524.0 −1.83508 −0.917539 0.397646i \(-0.869827\pi\)
−0.917539 + 0.397646i \(0.869827\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 12432.0 0.588709
\(765\) 0 0
\(766\) −21744.0 −1.02564
\(767\) −2192.00 −0.103192
\(768\) 0 0
\(769\) 18440.0 0.864712 0.432356 0.901703i \(-0.357682\pi\)
0.432356 + 0.901703i \(0.357682\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 200.000 0.00932404
\(773\) −13968.0 −0.649928 −0.324964 0.945726i \(-0.605352\pi\)
−0.324964 + 0.945726i \(0.605352\pi\)
\(774\) 0 0
\(775\) −8296.00 −0.384518
\(776\) 6656.00 0.307908
\(777\) 0 0
\(778\) −20868.0 −0.961638
\(779\) −62160.0 −2.85894
\(780\) 0 0
\(781\) −20960.0 −0.960317
\(782\) 14112.0 0.645325
\(783\) 0 0
\(784\) 0 0
\(785\) −4832.00 −0.219696
\(786\) 0 0
\(787\) −10916.0 −0.494426 −0.247213 0.968961i \(-0.579515\pi\)
−0.247213 + 0.968961i \(0.579515\pi\)
\(788\) 648.000 0.0292945
\(789\) 0 0
\(790\) 19456.0 0.876220
\(791\) 0 0
\(792\) 0 0
\(793\) 2768.00 0.123953
\(794\) 6088.00 0.272110
\(795\) 0 0
\(796\) −6176.00 −0.275003
\(797\) 12360.0 0.549327 0.274664 0.961540i \(-0.411434\pi\)
0.274664 + 0.961540i \(0.411434\pi\)
\(798\) 0 0
\(799\) −40992.0 −1.81501
\(800\) −1952.00 −0.0862670
\(801\) 0 0
\(802\) −17820.0 −0.784596
\(803\) 17600.0 0.773463
\(804\) 0 0
\(805\) 0 0
\(806\) −1088.00 −0.0475474
\(807\) 0 0
\(808\) 3712.00 0.161618
\(809\) −3402.00 −0.147847 −0.0739233 0.997264i \(-0.523552\pi\)
−0.0739233 + 0.997264i \(0.523552\pi\)
\(810\) 0 0
\(811\) −292.000 −0.0126430 −0.00632152 0.999980i \(-0.502012\pi\)
−0.00632152 + 0.999980i \(0.502012\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 17760.0 0.764727
\(815\) 8928.00 0.383723
\(816\) 0 0
\(817\) 24272.0 1.03938
\(818\) −11232.0 −0.480095
\(819\) 0 0
\(820\) 13440.0 0.572372
\(821\) −6910.00 −0.293740 −0.146870 0.989156i \(-0.546920\pi\)
−0.146870 + 0.989156i \(0.546920\pi\)
\(822\) 0 0
\(823\) 568.000 0.0240574 0.0120287 0.999928i \(-0.496171\pi\)
0.0120287 + 0.999928i \(0.496171\pi\)
\(824\) 5056.00 0.213755
\(825\) 0 0
\(826\) 0 0
\(827\) 12144.0 0.510627 0.255313 0.966858i \(-0.417821\pi\)
0.255313 + 0.966858i \(0.417821\pi\)
\(828\) 0 0
\(829\) −14828.0 −0.621228 −0.310614 0.950536i \(-0.600535\pi\)
−0.310614 + 0.950536i \(0.600535\pi\)
\(830\) −10944.0 −0.457677
\(831\) 0 0
\(832\) −256.000 −0.0106673
\(833\) 0 0
\(834\) 0 0
\(835\) −14272.0 −0.591501
\(836\) 23680.0 0.979653
\(837\) 0 0
\(838\) −17864.0 −0.736398
\(839\) −22824.0 −0.939180 −0.469590 0.882885i \(-0.655598\pi\)
−0.469590 + 0.882885i \(0.655598\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) −11076.0 −0.453330
\(843\) 0 0
\(844\) −4816.00 −0.196414
\(845\) −17448.0 −0.710331
\(846\) 0 0
\(847\) 0 0
\(848\) −7648.00 −0.309709
\(849\) 0 0
\(850\) 10248.0 0.413534
\(851\) 18648.0 0.751169
\(852\) 0 0
\(853\) 41780.0 1.67705 0.838523 0.544866i \(-0.183420\pi\)
0.838523 + 0.544866i \(0.183420\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1280.00 0.0511092
\(857\) −21420.0 −0.853784 −0.426892 0.904303i \(-0.640392\pi\)
−0.426892 + 0.904303i \(0.640392\pi\)
\(858\) 0 0
\(859\) 18132.0 0.720205 0.360102 0.932913i \(-0.382742\pi\)
0.360102 + 0.932913i \(0.382742\pi\)
\(860\) −5248.00 −0.208088
\(861\) 0 0
\(862\) 13400.0 0.529473
\(863\) −24036.0 −0.948082 −0.474041 0.880503i \(-0.657205\pi\)
−0.474041 + 0.880503i \(0.657205\pi\)
\(864\) 0 0
\(865\) −2752.00 −0.108174
\(866\) 10096.0 0.396162
\(867\) 0 0
\(868\) 0 0
\(869\) −48640.0 −1.89873
\(870\) 0 0
\(871\) 3632.00 0.141292
\(872\) −17584.0 −0.682878
\(873\) 0 0
\(874\) 24864.0 0.962285
\(875\) 0 0
\(876\) 0 0
\(877\) −4374.00 −0.168414 −0.0842072 0.996448i \(-0.526836\pi\)
−0.0842072 + 0.996448i \(0.526836\pi\)
\(878\) 2688.00 0.103321
\(879\) 0 0
\(880\) −5120.00 −0.196131
\(881\) −46348.0 −1.77242 −0.886211 0.463282i \(-0.846672\pi\)
−0.886211 + 0.463282i \(0.846672\pi\)
\(882\) 0 0
\(883\) −20660.0 −0.787389 −0.393694 0.919241i \(-0.628803\pi\)
−0.393694 + 0.919241i \(0.628803\pi\)
\(884\) 1344.00 0.0511353
\(885\) 0 0
\(886\) 8784.00 0.333075
\(887\) −1800.00 −0.0681376 −0.0340688 0.999419i \(-0.510847\pi\)
−0.0340688 + 0.999419i \(0.510847\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 9664.00 0.363975
\(891\) 0 0
\(892\) −8000.00 −0.300291
\(893\) −72224.0 −2.70648
\(894\) 0 0
\(895\) −11136.0 −0.415906
\(896\) 0 0
\(897\) 0 0
\(898\) −7332.00 −0.272463
\(899\) −7888.00 −0.292636
\(900\) 0 0
\(901\) 40152.0 1.48464
\(902\) −33600.0 −1.24031
\(903\) 0 0
\(904\) −6160.00 −0.226636
\(905\) −32416.0 −1.19066
\(906\) 0 0
\(907\) −41996.0 −1.53744 −0.768718 0.639588i \(-0.779105\pi\)
−0.768718 + 0.639588i \(0.779105\pi\)
\(908\) 1552.00 0.0567235
\(909\) 0 0
\(910\) 0 0
\(911\) 41308.0 1.50230 0.751150 0.660132i \(-0.229500\pi\)
0.751150 + 0.660132i \(0.229500\pi\)
\(912\) 0 0
\(913\) 27360.0 0.991768
\(914\) 52.0000 0.00188185
\(915\) 0 0
\(916\) −16720.0 −0.603105
\(917\) 0 0
\(918\) 0 0
\(919\) 3936.00 0.141280 0.0706402 0.997502i \(-0.477496\pi\)
0.0706402 + 0.997502i \(0.477496\pi\)
\(920\) −5376.00 −0.192654
\(921\) 0 0
\(922\) 15312.0 0.546935
\(923\) −2096.00 −0.0747461
\(924\) 0 0
\(925\) 13542.0 0.481360
\(926\) 25216.0 0.894870
\(927\) 0 0
\(928\) −1856.00 −0.0656532
\(929\) −7212.00 −0.254702 −0.127351 0.991858i \(-0.540647\pi\)
−0.127351 + 0.991858i \(0.540647\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 5288.00 0.185852
\(933\) 0 0
\(934\) −6136.00 −0.214964
\(935\) 26880.0 0.940182
\(936\) 0 0
\(937\) 38976.0 1.35890 0.679451 0.733721i \(-0.262218\pi\)
0.679451 + 0.733721i \(0.262218\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 15616.0 0.541849
\(941\) 53544.0 1.85493 0.927463 0.373916i \(-0.121985\pi\)
0.927463 + 0.373916i \(0.121985\pi\)
\(942\) 0 0
\(943\) −35280.0 −1.21832
\(944\) 8768.00 0.302303
\(945\) 0 0
\(946\) 13120.0 0.450918
\(947\) 21392.0 0.734051 0.367026 0.930211i \(-0.380376\pi\)
0.367026 + 0.930211i \(0.380376\pi\)
\(948\) 0 0
\(949\) 1760.00 0.0602023
\(950\) 18056.0 0.616646
\(951\) 0 0
\(952\) 0 0
\(953\) −21162.0 −0.719312 −0.359656 0.933085i \(-0.617106\pi\)
−0.359656 + 0.933085i \(0.617106\pi\)
\(954\) 0 0
\(955\) 24864.0 0.842492
\(956\) −9648.00 −0.326400
\(957\) 0 0
\(958\) 12912.0 0.435457
\(959\) 0 0
\(960\) 0 0
\(961\) −11295.0 −0.379141
\(962\) 1776.00 0.0595224
\(963\) 0 0
\(964\) 17344.0 0.579474
\(965\) 400.000 0.0133435
\(966\) 0 0
\(967\) 8224.00 0.273491 0.136746 0.990606i \(-0.456336\pi\)
0.136746 + 0.990606i \(0.456336\pi\)
\(968\) 2152.00 0.0714544
\(969\) 0 0
\(970\) 13312.0 0.440642
\(971\) −8140.00 −0.269027 −0.134513 0.990912i \(-0.542947\pi\)
−0.134513 + 0.990912i \(0.542947\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 23792.0 0.782695
\(975\) 0 0
\(976\) −11072.0 −0.363121
\(977\) −32158.0 −1.05305 −0.526523 0.850161i \(-0.676505\pi\)
−0.526523 + 0.850161i \(0.676505\pi\)
\(978\) 0 0
\(979\) −24160.0 −0.788720
\(980\) 0 0
\(981\) 0 0
\(982\) 528.000 0.0171580
\(983\) −41416.0 −1.34381 −0.671905 0.740637i \(-0.734524\pi\)
−0.671905 + 0.740637i \(0.734524\pi\)
\(984\) 0 0
\(985\) 1296.00 0.0419228
\(986\) 9744.00 0.314718
\(987\) 0 0
\(988\) 2368.00 0.0762511
\(989\) 13776.0 0.442923
\(990\) 0 0
\(991\) 12296.0 0.394143 0.197071 0.980389i \(-0.436857\pi\)
0.197071 + 0.980389i \(0.436857\pi\)
\(992\) 4352.00 0.139290
\(993\) 0 0
\(994\) 0 0
\(995\) −12352.0 −0.393552
\(996\) 0 0
\(997\) −57652.0 −1.83135 −0.915676 0.401918i \(-0.868344\pi\)
−0.915676 + 0.401918i \(0.868344\pi\)
\(998\) −5256.00 −0.166709
\(999\) 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 882.4.a.q.1.1 1
3.2 odd 2 294.4.a.b.1.1 1
7.2 even 3 882.4.g.c.361.1 2
7.3 odd 6 882.4.g.j.667.1 2
7.4 even 3 882.4.g.c.667.1 2
7.5 odd 6 882.4.g.j.361.1 2
7.6 odd 2 882.4.a.j.1.1 1
12.11 even 2 2352.4.a.z.1.1 1
21.2 odd 6 294.4.e.j.67.1 2
21.5 even 6 294.4.e.f.67.1 2
21.11 odd 6 294.4.e.j.79.1 2
21.17 even 6 294.4.e.f.79.1 2
21.20 even 2 294.4.a.f.1.1 yes 1
84.83 odd 2 2352.4.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.b.1.1 1 3.2 odd 2
294.4.a.f.1.1 yes 1 21.20 even 2
294.4.e.f.67.1 2 21.5 even 6
294.4.e.f.79.1 2 21.17 even 6
294.4.e.j.67.1 2 21.2 odd 6
294.4.e.j.79.1 2 21.11 odd 6
882.4.a.j.1.1 1 7.6 odd 2
882.4.a.q.1.1 1 1.1 even 1 trivial
882.4.g.c.361.1 2 7.2 even 3
882.4.g.c.667.1 2 7.4 even 3
882.4.g.j.361.1 2 7.5 odd 6
882.4.g.j.667.1 2 7.3 odd 6
2352.4.a.m.1.1 1 84.83 odd 2
2352.4.a.z.1.1 1 12.11 even 2