Properties

Label 2352.4.a.z.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,4,Mod(1,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2352.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(138.772492334\)
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 \(\chi\) \(=\) 2352.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -8.00000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -8.00000 q^{5} +9.00000 q^{9} -40.0000 q^{11} -4.00000 q^{13} -24.0000 q^{15} +84.0000 q^{17} +148.000 q^{19} -84.0000 q^{23} -61.0000 q^{25} +27.0000 q^{27} +58.0000 q^{29} -136.000 q^{31} -120.000 q^{33} -222.000 q^{37} -12.0000 q^{39} -420.000 q^{41} +164.000 q^{43} -72.0000 q^{45} +488.000 q^{47} +252.000 q^{51} +478.000 q^{53} +320.000 q^{55} +444.000 q^{57} +548.000 q^{59} -692.000 q^{61} +32.0000 q^{65} +908.000 q^{67} -252.000 q^{69} +524.000 q^{71} -440.000 q^{73} -183.000 q^{75} -1216.00 q^{79} +81.0000 q^{81} -684.000 q^{83} -672.000 q^{85} +174.000 q^{87} -604.000 q^{89} -408.000 q^{93} -1184.00 q^{95} +832.000 q^{97} -360.000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −8.00000 −0.715542 −0.357771 0.933809i \(-0.616463\pi\)
−0.357771 + 0.933809i \(0.616463\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −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
\(15\) −24.0000 −0.413118
\(16\) 0 0
\(17\) 84.0000 1.19841 0.599206 0.800595i \(-0.295483\pi\)
0.599206 + 0.800595i \(0.295483\pi\)
\(18\) 0 0
\(19\) 148.000 1.78703 0.893514 0.449036i \(-0.148232\pi\)
0.893514 + 0.449036i \(0.148232\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −136.000 −0.787946 −0.393973 0.919122i \(-0.628900\pi\)
−0.393973 + 0.919122i \(0.628900\pi\)
\(32\) 0 0
\(33\) −120.000 −0.633010
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −222.000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) −12.0000 −0.0492702
\(40\) 0 0
\(41\) −420.000 −1.59983 −0.799914 0.600114i \(-0.795122\pi\)
−0.799914 + 0.600114i \(0.795122\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) 0 0
\(45\) −72.0000 −0.238514
\(46\) 0 0
\(47\) 488.000 1.51451 0.757257 0.653118i \(-0.226539\pi\)
0.757257 + 0.653118i \(0.226539\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 252.000 0.691903
\(52\) 0 0
\(53\) 478.000 1.23884 0.619418 0.785061i \(-0.287368\pi\)
0.619418 + 0.785061i \(0.287368\pi\)
\(54\) 0 0
\(55\) 320.000 0.784523
\(56\) 0 0
\(57\) 444.000 1.03174
\(58\) 0 0
\(59\) 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 \(0.758735\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 32.0000 0.0610633
\(66\) 0 0
\(67\) 908.000 1.65567 0.827835 0.560972i \(-0.189572\pi\)
0.827835 + 0.560972i \(0.189572\pi\)
\(68\) 0 0
\(69\) −252.000 −0.439670
\(70\) 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 \(0.614745\pi\)
\(74\) 0 0
\(75\) −183.000 −0.281747
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1216.00 −1.73178 −0.865890 0.500234i \(-0.833247\pi\)
−0.865890 + 0.500234i \(0.833247\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) 174.000 0.214423
\(88\) 0 0
\(89\) −604.000 −0.719369 −0.359685 0.933074i \(-0.617116\pi\)
−0.359685 + 0.933074i \(0.617116\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −408.000 −0.454921
\(94\) 0 0
\(95\) −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
\(99\) −360.000 −0.365468
\(100\) 0 0
\(101\) −464.000 −0.457126 −0.228563 0.973529i \(-0.573403\pi\)
−0.228563 + 0.973529i \(0.573403\pi\)
\(102\) 0 0
\(103\) −632.000 −0.604590 −0.302295 0.953214i \(-0.597753\pi\)
−0.302295 + 0.953214i \(0.597753\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 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 \(0.916432\pi\)
\(110\) 0 0
\(111\) −666.000 −0.569495
\(112\) 0 0
\(113\) 770.000 0.641022 0.320511 0.947245i \(-0.396145\pi\)
0.320511 + 0.947245i \(0.396145\pi\)
\(114\) 0 0
\(115\) 672.000 0.544907
\(116\) 0 0
\(117\) −36.0000 −0.0284462
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 269.000 0.202104
\(122\) 0 0
\(123\) −1260.00 −0.923662
\(124\) 0 0
\(125\) 1488.00 1.06473
\(126\) 0 0
\(127\) 184.000 0.128562 0.0642809 0.997932i \(-0.479525\pi\)
0.0642809 + 0.997932i \(0.479525\pi\)
\(128\) 0 0
\(129\) 492.000 0.335800
\(130\) 0 0
\(131\) −1452.00 −0.968411 −0.484205 0.874954i \(-0.660891\pi\)
−0.484205 + 0.874954i \(0.660891\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −216.000 −0.137706
\(136\) 0 0
\(137\) 646.000 0.402858 0.201429 0.979503i \(-0.435442\pi\)
0.201429 + 0.979503i \(0.435442\pi\)
\(138\) 0 0
\(139\) −3012.00 −1.83795 −0.918973 0.394320i \(-0.870980\pi\)
−0.918973 + 0.394320i \(0.870980\pi\)
\(140\) 0 0
\(141\) 1464.00 0.874405
\(142\) 0 0
\(143\) 160.000 0.0935655
\(144\) 0 0
\(145\) −464.000 −0.265746
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3170.00 −1.74293 −0.871465 0.490458i \(-0.836830\pi\)
−0.871465 + 0.490458i \(0.836830\pi\)
\(150\) 0 0
\(151\) 1880.00 1.01319 0.506597 0.862183i \(-0.330903\pi\)
0.506597 + 0.862183i \(0.330903\pi\)
\(152\) 0 0
\(153\) 756.000 0.399470
\(154\) 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 \(0.549060\pi\)
\(158\) 0 0
\(159\) 1434.00 0.715243
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1116.00 −0.536269 −0.268135 0.963381i \(-0.586407\pi\)
−0.268135 + 0.963381i \(0.586407\pi\)
\(164\) 0 0
\(165\) 960.000 0.452945
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) 1332.00 0.595676
\(172\) 0 0
\(173\) 344.000 0.151178 0.0755891 0.997139i \(-0.475916\pi\)
0.0755891 + 0.997139i \(0.475916\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1644.00 0.698139
\(178\) 0 0
\(179\) −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
\(183\) −2076.00 −0.838592
\(184\) 0 0
\(185\) 1776.00 0.705806
\(186\) 0 0
\(187\) −3360.00 −1.31394
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(0.497032\pi\)
\(194\) 0 0
\(195\) 96.0000 0.0352549
\(196\) 0 0
\(197\) −162.000 −0.0585889 −0.0292945 0.999571i \(-0.509326\pi\)
−0.0292945 + 0.999571i \(0.509326\pi\)
\(198\) 0 0
\(199\) 1544.00 0.550006 0.275003 0.961443i \(-0.411321\pi\)
0.275003 + 0.961443i \(0.411321\pi\)
\(200\) 0 0
\(201\) 2724.00 0.955901
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3360.00 1.14474
\(206\) 0 0
\(207\) −756.000 −0.253844
\(208\) 0 0
\(209\) −5920.00 −1.95931
\(210\) 0 0
\(211\) 1204.00 0.392828 0.196414 0.980521i \(-0.437070\pi\)
0.196414 + 0.980521i \(0.437070\pi\)
\(212\) 0 0
\(213\) 1572.00 0.505689
\(214\) 0 0
\(215\) −1312.00 −0.416175
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1320.00 −0.407294
\(220\) 0 0
\(221\) −336.000 −0.102271
\(222\) 0 0
\(223\) 2000.00 0.600583 0.300291 0.953848i \(-0.402916\pi\)
0.300291 + 0.953848i \(0.402916\pi\)
\(224\) 0 0
\(225\) −549.000 −0.162667
\(226\) 0 0
\(227\) 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 \(0.706070\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1322.00 −0.371704 −0.185852 0.982578i \(-0.559505\pi\)
−0.185852 + 0.982578i \(0.559505\pi\)
\(234\) 0 0
\(235\) −3904.00 −1.08370
\(236\) 0 0
\(237\) −3648.00 −0.999844
\(238\) 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 \(0.303258\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −592.000 −0.152502
\(248\) 0 0
\(249\) −2052.00 −0.522250
\(250\) 0 0
\(251\) 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
\(254\) 0 0
\(255\) −2016.00 −0.495086
\(256\) 0 0
\(257\) −4300.00 −1.04368 −0.521842 0.853042i \(-0.674755\pi\)
−0.521842 + 0.853042i \(0.674755\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 522.000 0.123797
\(262\) 0 0
\(263\) 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
\(267\) −1812.00 −0.415328
\(268\) 0 0
\(269\) 2800.00 0.634643 0.317322 0.948318i \(-0.397217\pi\)
0.317322 + 0.948318i \(0.397217\pi\)
\(270\) 0 0
\(271\) −4880.00 −1.09387 −0.546935 0.837175i \(-0.684206\pi\)
−0.546935 + 0.837175i \(0.684206\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2440.00 0.535046
\(276\) 0 0
\(277\) −6674.00 −1.44766 −0.723830 0.689978i \(-0.757620\pi\)
−0.723830 + 0.689978i \(0.757620\pi\)
\(278\) 0 0
\(279\) −1224.00 −0.262649
\(280\) 0 0
\(281\) −9402.00 −1.99600 −0.998001 0.0632056i \(-0.979868\pi\)
−0.998001 + 0.0632056i \(0.979868\pi\)
\(282\) 0 0
\(283\) −9100.00 −1.91144 −0.955722 0.294270i \(-0.904924\pi\)
−0.955722 + 0.294270i \(0.904924\pi\)
\(284\) 0 0
\(285\) −3552.00 −0.738254
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) 0 0
\(291\) 2496.00 0.502811
\(292\) 0 0
\(293\) −5952.00 −1.18676 −0.593378 0.804924i \(-0.702206\pi\)
−0.593378 + 0.804924i \(0.702206\pi\)
\(294\) 0 0
\(295\) −4384.00 −0.865242
\(296\) 0 0
\(297\) −1080.00 −0.211003
\(298\) 0 0
\(299\) 336.000 0.0649879
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1392.00 −0.263922
\(304\) 0 0
\(305\) 5536.00 1.03931
\(306\) 0 0
\(307\) −3004.00 −0.558460 −0.279230 0.960224i \(-0.590079\pi\)
−0.279230 + 0.960224i \(0.590079\pi\)
\(308\) 0 0
\(309\) −1896.00 −0.349060
\(310\) 0 0
\(311\) 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 \(0.668477\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2922.00 −0.517716 −0.258858 0.965915i \(-0.583346\pi\)
−0.258858 + 0.965915i \(0.583346\pi\)
\(318\) 0 0
\(319\) −2320.00 −0.407195
\(320\) 0 0
\(321\) 480.000 0.0834610
\(322\) 0 0
\(323\) 12432.0 2.14159
\(324\) 0 0
\(325\) 244.000 0.0416452
\(326\) 0 0
\(327\) −6594.00 −1.11513
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7492.00 1.24410 0.622051 0.782977i \(-0.286300\pi\)
0.622051 + 0.782977i \(0.286300\pi\)
\(332\) 0 0
\(333\) −1998.00 −0.328798
\(334\) 0 0
\(335\) −7264.00 −1.18470
\(336\) 0 0
\(337\) 10766.0 1.74024 0.870121 0.492839i \(-0.164041\pi\)
0.870121 + 0.492839i \(0.164041\pi\)
\(338\) 0 0
\(339\) 2310.00 0.370094
\(340\) 0 0
\(341\) 5440.00 0.863908
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2016.00 0.314602
\(346\) 0 0
\(347\) 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
\(351\) −108.000 −0.0164234
\(352\) 0 0
\(353\) 10428.0 1.57231 0.786156 0.618028i \(-0.212068\pi\)
0.786156 + 0.618028i \(0.212068\pi\)
\(354\) 0 0
\(355\) −4192.00 −0.626727
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 807.000 0.116685
\(364\) 0 0
\(365\) 3520.00 0.504781
\(366\) 0 0
\(367\) 5648.00 0.803333 0.401666 0.915786i \(-0.368431\pi\)
0.401666 + 0.915786i \(0.368431\pi\)
\(368\) 0 0
\(369\) −3780.00 −0.533276
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2546.00 −0.353423 −0.176712 0.984263i \(-0.556546\pi\)
−0.176712 + 0.984263i \(0.556546\pi\)
\(374\) 0 0
\(375\) 4464.00 0.614720
\(376\) 0 0
\(377\) −232.000 −0.0316939
\(378\) 0 0
\(379\) −8268.00 −1.12058 −0.560288 0.828298i \(-0.689310\pi\)
−0.560288 + 0.828298i \(0.689310\pi\)
\(380\) 0 0
\(381\) 552.000 0.0742252
\(382\) 0 0
\(383\) −10872.0 −1.45048 −0.725239 0.688497i \(-0.758271\pi\)
−0.725239 + 0.688497i \(0.758271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1476.00 0.193874
\(388\) 0 0
\(389\) 10434.0 1.35996 0.679980 0.733230i \(-0.261988\pi\)
0.679980 + 0.733230i \(0.261988\pi\)
\(390\) 0 0
\(391\) −7056.00 −0.912627
\(392\) 0 0
\(393\) −4356.00 −0.559112
\(394\) 0 0
\(395\) 9728.00 1.23916
\(396\) 0 0
\(397\) 3044.00 0.384821 0.192411 0.981315i \(-0.438369\pi\)
0.192411 + 0.981315i \(0.438369\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8910.00 1.10959 0.554793 0.831988i \(-0.312797\pi\)
0.554793 + 0.831988i \(0.312797\pi\)
\(402\) 0 0
\(403\) 544.000 0.0672421
\(404\) 0 0
\(405\) −648.000 −0.0795046
\(406\) 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 \(0.610251\pi\)
\(410\) 0 0
\(411\) 1938.00 0.232590
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 5472.00 0.647253
\(416\) 0 0
\(417\) −9036.00 −1.06114
\(418\) 0 0
\(419\) −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 \(0.603869\pi\)
\(422\) 0 0
\(423\) 4392.00 0.504838
\(424\) 0 0
\(425\) −5124.00 −0.584825
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 480.000 0.0540201
\(430\) 0 0
\(431\) 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
\(435\) −1392.00 −0.153428
\(436\) 0 0
\(437\) −12432.0 −1.36088
\(438\) 0 0
\(439\) −1344.00 −0.146118 −0.0730588 0.997328i \(-0.523276\pi\)
−0.0730588 + 0.997328i \(0.523276\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 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
\(446\) 0 0
\(447\) −9510.00 −1.00628
\(448\) 0 0
\(449\) 3666.00 0.385321 0.192661 0.981265i \(-0.438288\pi\)
0.192661 + 0.981265i \(0.438288\pi\)
\(450\) 0 0
\(451\) 16800.0 1.75406
\(452\) 0 0
\(453\) 5640.00 0.584968
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.0000 0.00266133 0.00133067 0.999999i \(-0.499576\pi\)
0.00133067 + 0.999999i \(0.499576\pi\)
\(458\) 0 0
\(459\) 2268.00 0.230634
\(460\) 0 0
\(461\) −7656.00 −0.773483 −0.386741 0.922188i \(-0.626399\pi\)
−0.386741 + 0.922188i \(0.626399\pi\)
\(462\) 0 0
\(463\) −12608.0 −1.26554 −0.632768 0.774341i \(-0.718081\pi\)
−0.632768 + 0.774341i \(0.718081\pi\)
\(464\) 0 0
\(465\) 3264.00 0.325515
\(466\) 0 0
\(467\) −3068.00 −0.304005 −0.152002 0.988380i \(-0.548572\pi\)
−0.152002 + 0.988380i \(0.548572\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1812.00 −0.177267
\(472\) 0 0
\(473\) −6560.00 −0.637694
\(474\) 0 0
\(475\) −9028.00 −0.872070
\(476\) 0 0
\(477\) 4302.00 0.412946
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6656.00 −0.623162
\(486\) 0 0
\(487\) −11896.0 −1.10690 −0.553449 0.832883i \(-0.686689\pi\)
−0.553449 + 0.832883i \(0.686689\pi\)
\(488\) 0 0
\(489\) −3348.00 −0.309615
\(490\) 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
\(494\) 0 0
\(495\) 2880.00 0.261508
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2628.00 0.235762 0.117881 0.993028i \(-0.462390\pi\)
0.117881 + 0.993028i \(0.462390\pi\)
\(500\) 0 0
\(501\) −5352.00 −0.477265
\(502\) 0 0
\(503\) −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
\(506\) 0 0
\(507\) −6543.00 −0.573146
\(508\) 0 0
\(509\) −20656.0 −1.79874 −0.899372 0.437183i \(-0.855976\pi\)
−0.899372 + 0.437183i \(0.855976\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3996.00 0.343914
\(514\) 0 0
\(515\) 5056.00 0.432610
\(516\) 0 0
\(517\) −19520.0 −1.66052
\(518\) 0 0
\(519\) 1032.00 0.0872828
\(520\) 0 0
\(521\) −3628.00 −0.305078 −0.152539 0.988297i \(-0.548745\pi\)
−0.152539 + 0.988297i \(0.548745\pi\)
\(522\) 0 0
\(523\) 4852.00 0.405666 0.202833 0.979213i \(-0.434985\pi\)
0.202833 + 0.979213i \(0.434985\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11424.0 −0.944283
\(528\) 0 0
\(529\) −5111.00 −0.420071
\(530\) 0 0
\(531\) 4932.00 0.403071
\(532\) 0 0
\(533\) 1680.00 0.136527
\(534\) 0 0
\(535\) −1280.00 −0.103438
\(536\) 0 0
\(537\) −4176.00 −0.335582
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −7130.00 −0.566622 −0.283311 0.959028i \(-0.591433\pi\)
−0.283311 + 0.959028i \(0.591433\pi\)
\(542\) 0 0
\(543\) −12156.0 −0.960707
\(544\) 0 0
\(545\) 17584.0 1.38205
\(546\) 0 0
\(547\) 12788.0 0.999589 0.499795 0.866144i \(-0.333409\pi\)
0.499795 + 0.866144i \(0.333409\pi\)
\(548\) 0 0
\(549\) −6228.00 −0.484161
\(550\) 0 0
\(551\) 8584.00 0.663685
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 5328.00 0.407497
\(556\) 0 0
\(557\) 2406.00 0.183026 0.0915130 0.995804i \(-0.470830\pi\)
0.0915130 + 0.995804i \(0.470830\pi\)
\(558\) 0 0
\(559\) −656.000 −0.0496348
\(560\) 0 0
\(561\) −10080.0 −0.758606
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9690.00 −0.713930 −0.356965 0.934118i \(-0.616188\pi\)
−0.356965 + 0.934118i \(0.616188\pi\)
\(570\) 0 0
\(571\) −5604.00 −0.410718 −0.205359 0.978687i \(-0.565836\pi\)
−0.205359 + 0.978687i \(0.565836\pi\)
\(572\) 0 0
\(573\) 9324.00 0.679783
\(574\) 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 \(0.216201\pi\)
\(578\) 0 0
\(579\) 150.000 0.0107665
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −19120.0 −1.35827
\(584\) 0 0
\(585\) 288.000 0.0203544
\(586\) 0 0
\(587\) 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
\(590\) 0 0
\(591\) −486.000 −0.0338263
\(592\) 0 0
\(593\) −13812.0 −0.956477 −0.478238 0.878230i \(-0.658725\pi\)
−0.478238 + 0.878230i \(0.658725\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4632.00 0.317546
\(598\) 0 0
\(599\) 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
\(603\) 8172.00 0.551890
\(604\) 0 0
\(605\) −2152.00 −0.144614
\(606\) 0 0
\(607\) 21504.0 1.43792 0.718962 0.695049i \(-0.244617\pi\)
0.718962 + 0.695049i \(0.244617\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1952.00 −0.129246
\(612\) 0 0
\(613\) −10270.0 −0.676674 −0.338337 0.941025i \(-0.609864\pi\)
−0.338337 + 0.941025i \(0.609864\pi\)
\(614\) 0 0
\(615\) 10080.0 0.660918
\(616\) 0 0
\(617\) 28358.0 1.85032 0.925162 0.379572i \(-0.123929\pi\)
0.925162 + 0.379572i \(0.123929\pi\)
\(618\) 0 0
\(619\) 16292.0 1.05788 0.528942 0.848658i \(-0.322589\pi\)
0.528942 + 0.848658i \(0.322589\pi\)
\(620\) 0 0
\(621\) −2268.00 −0.146557
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4279.00 −0.273856
\(626\) 0 0
\(627\) −17760.0 −1.13121
\(628\) 0 0
\(629\) −18648.0 −1.18211
\(630\) 0 0
\(631\) −11256.0 −0.710134 −0.355067 0.934841i \(-0.615542\pi\)
−0.355067 + 0.934841i \(0.615542\pi\)
\(632\) 0 0
\(633\) 3612.00 0.226800
\(634\) 0 0
\(635\) −1472.00 −0.0919914
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4716.00 0.291959
\(640\) 0 0
\(641\) 15518.0 0.956200 0.478100 0.878305i \(-0.341326\pi\)
0.478100 + 0.878305i \(0.341326\pi\)
\(642\) 0 0
\(643\) −10452.0 −0.641037 −0.320518 0.947242i \(-0.603857\pi\)
−0.320518 + 0.947242i \(0.603857\pi\)
\(644\) 0 0
\(645\) −3936.00 −0.240279
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 11962.0 0.716859 0.358430 0.933557i \(-0.383312\pi\)
0.358430 + 0.933557i \(0.383312\pi\)
\(654\) 0 0
\(655\) 11616.0 0.692938
\(656\) 0 0
\(657\) −3960.00 −0.235151
\(658\) 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 \(0.778234\pi\)
\(662\) 0 0
\(663\) −1008.00 −0.0590460
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4872.00 −0.282825
\(668\) 0 0
\(669\) 6000.00 0.346746
\(670\) 0 0
\(671\) 27680.0 1.59251
\(672\) 0 0
\(673\) −20530.0 −1.17589 −0.587945 0.808901i \(-0.700063\pi\)
−0.587945 + 0.808901i \(0.700063\pi\)
\(674\) 0 0
\(675\) −1647.00 −0.0939156
\(676\) 0 0
\(677\) 10056.0 0.570877 0.285438 0.958397i \(-0.407861\pi\)
0.285438 + 0.958397i \(0.407861\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 1164.00 0.0654986
\(682\) 0 0
\(683\) −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
\(687\) −12540.0 −0.696406
\(688\) 0 0
\(689\) −1912.00 −0.105720
\(690\) 0 0
\(691\) −14716.0 −0.810164 −0.405082 0.914280i \(-0.632757\pi\)
−0.405082 + 0.914280i \(0.632757\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24096.0 1.31513
\(696\) 0 0
\(697\) −35280.0 −1.91725
\(698\) 0 0
\(699\) −3966.00 −0.214604
\(700\) 0 0
\(701\) 28202.0 1.51951 0.759754 0.650211i \(-0.225319\pi\)
0.759754 + 0.650211i \(0.225319\pi\)
\(702\) 0 0
\(703\) −32856.0 −1.76271
\(704\) 0 0
\(705\) −11712.0 −0.625673
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 22114.0 1.17138 0.585690 0.810535i \(-0.300824\pi\)
0.585690 + 0.810535i \(0.300824\pi\)
\(710\) 0 0
\(711\) −10944.0 −0.577260
\(712\) 0 0
\(713\) 11424.0 0.600045
\(714\) 0 0
\(715\) −1280.00 −0.0669501
\(716\) 0 0
\(717\) −7236.00 −0.376895
\(718\) 0 0
\(719\) −9288.00 −0.481758 −0.240879 0.970555i \(-0.577436\pi\)
−0.240879 + 0.970555i \(0.577436\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 13008.0 0.669119
\(724\) 0 0
\(725\) −3538.00 −0.181239
\(726\) 0 0
\(727\) 23848.0 1.21661 0.608304 0.793704i \(-0.291850\pi\)
0.608304 + 0.793704i \(0.291850\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 13776.0 0.697023
\(732\) 0 0
\(733\) 34756.0 1.75135 0.875677 0.482898i \(-0.160416\pi\)
0.875677 + 0.482898i \(0.160416\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −36320.0 −1.81528
\(738\) 0 0
\(739\) −26044.0 −1.29641 −0.648203 0.761468i \(-0.724479\pi\)
−0.648203 + 0.761468i \(0.724479\pi\)
\(740\) 0 0
\(741\) −1776.00 −0.0880472
\(742\) 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
\(746\) 0 0
\(747\) −6156.00 −0.301521
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11424.0 0.555083 0.277542 0.960714i \(-0.410480\pi\)
0.277542 + 0.960714i \(0.410480\pi\)
\(752\) 0 0
\(753\) 2292.00 0.110923
\(754\) 0 0
\(755\) −15040.0 −0.724982
\(756\) 0 0
\(757\) −16622.0 −0.798067 −0.399034 0.916936i \(-0.630654\pi\)
−0.399034 + 0.916936i \(0.630654\pi\)
\(758\) 0 0
\(759\) 10080.0 0.482056
\(760\) 0 0
\(761\) 38524.0 1.83508 0.917539 0.397646i \(-0.130173\pi\)
0.917539 + 0.397646i \(0.130173\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6048.00 −0.285838
\(766\) 0 0
\(767\) −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
\(771\) −12900.0 −0.602571
\(772\) 0 0
\(773\) 13968.0 0.649928 0.324964 0.945726i \(-0.394648\pi\)
0.324964 + 0.945726i \(0.394648\pi\)
\(774\) 0 0
\(775\) 8296.00 0.384518
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −62160.0 −2.85894
\(780\) 0 0
\(781\) −20960.0 −0.960317
\(782\) 0 0
\(783\) 1566.00 0.0714742
\(784\) 0 0
\(785\) 4832.00 0.219696
\(786\) 0 0
\(787\) 10916.0 0.494426 0.247213 0.968961i \(-0.420485\pi\)
0.247213 + 0.968961i \(0.420485\pi\)
\(788\) 0 0
\(789\) 11580.0 0.522508
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2768.00 0.123953
\(794\) 0 0
\(795\) −11472.0 −0.511786
\(796\) 0 0
\(797\) −12360.0 −0.549327 −0.274664 0.961540i \(-0.588566\pi\)
−0.274664 + 0.961540i \(0.588566\pi\)
\(798\) 0 0
\(799\) 40992.0 1.81501
\(800\) 0 0
\(801\) −5436.00 −0.239790
\(802\) 0 0
\(803\) 17600.0 0.773463
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8400.00 0.366411
\(808\) 0 0
\(809\) 3402.00 0.147847 0.0739233 0.997264i \(-0.476448\pi\)
0.0739233 + 0.997264i \(0.476448\pi\)
\(810\) 0 0
\(811\) 292.000 0.0126430 0.00632152 0.999980i \(-0.497988\pi\)
0.00632152 + 0.999980i \(0.497988\pi\)
\(812\) 0 0
\(813\) −14640.0 −0.631546
\(814\) 0 0
\(815\) 8928.00 0.383723
\(816\) 0 0
\(817\) 24272.0 1.03938
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6910.00 0.293740 0.146870 0.989156i \(-0.453080\pi\)
0.146870 + 0.989156i \(0.453080\pi\)
\(822\) 0 0
\(823\) −568.000 −0.0240574 −0.0120287 0.999928i \(-0.503829\pi\)
−0.0120287 + 0.999928i \(0.503829\pi\)
\(824\) 0 0
\(825\) 7320.00 0.308909
\(826\) 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 \(0.600535\pi\)
\(830\) 0 0
\(831\) −20022.0 −0.835807
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 14272.0 0.591501
\(836\) 0 0
\(837\) −3672.00 −0.151640
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) −28206.0 −1.15239
\(844\) 0 0
\(845\) 17448.0 0.710331
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −27300.0 −1.10357
\(850\) 0 0
\(851\) 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
\(855\) −10656.0 −0.426231
\(856\) 0 0
\(857\) 21420.0 0.853784 0.426892 0.904303i \(-0.359608\pi\)
0.426892 + 0.904303i \(0.359608\pi\)
\(858\) 0 0
\(859\) −18132.0 −0.720205 −0.360102 0.932913i \(-0.617258\pi\)
−0.360102 + 0.932913i \(0.617258\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −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
\(866\) 0 0
\(867\) 6429.00 0.251834
\(868\) 0 0
\(869\) 48640.0 1.89873
\(870\) 0 0
\(871\) −3632.00 −0.141292
\(872\) 0 0
\(873\) 7488.00 0.290298
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4374.00 −0.168414 −0.0842072 0.996448i \(-0.526836\pi\)
−0.0842072 + 0.996448i \(0.526836\pi\)
\(878\) 0 0
\(879\) −17856.0 −0.685174
\(880\) 0 0
\(881\) 46348.0 1.77242 0.886211 0.463282i \(-0.153328\pi\)
0.886211 + 0.463282i \(0.153328\pi\)
\(882\) 0 0
\(883\) 20660.0 0.787389 0.393694 0.919241i \(-0.371197\pi\)
0.393694 + 0.919241i \(0.371197\pi\)
\(884\) 0 0
\(885\) −13152.0 −0.499548
\(886\) 0 0
\(887\) −1800.00 −0.0681376 −0.0340688 0.999419i \(-0.510847\pi\)
−0.0340688 + 0.999419i \(0.510847\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3240.00 −0.121823
\(892\) 0 0
\(893\) 72224.0 2.70648
\(894\) 0 0
\(895\) 11136.0 0.415906
\(896\) 0 0
\(897\) 1008.00 0.0375208
\(898\) 0 0
\(899\) −7888.00 −0.292636
\(900\) 0 0
\(901\) 40152.0 1.48464
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 32416.0 1.19066
\(906\) 0 0
\(907\) 41996.0 1.53744 0.768718 0.639588i \(-0.220895\pi\)
0.768718 + 0.639588i \(0.220895\pi\)
\(908\) 0 0
\(909\) −4176.00 −0.152375
\(910\) 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
\(914\) 0 0
\(915\) 16608.0 0.600048
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −3936.00 −0.141280 −0.0706402 0.997502i \(-0.522504\pi\)
−0.0706402 + 0.997502i \(0.522504\pi\)
\(920\) 0 0
\(921\) −9012.00 −0.322427
\(922\) 0 0
\(923\) −2096.00 −0.0747461
\(924\) 0 0
\(925\) 13542.0 0.481360
\(926\) 0 0
\(927\) −5688.00 −0.201530
\(928\) 0 0
\(929\) 7212.00 0.254702 0.127351 0.991858i \(-0.459353\pi\)
0.127351 + 0.991858i \(0.459353\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2064.00 0.0724248
\(934\) 0 0
\(935\) 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
\(939\) −16776.0 −0.583029
\(940\) 0 0
\(941\) −53544.0 −1.85493 −0.927463 0.373916i \(-0.878015\pi\)
−0.927463 + 0.373916i \(0.878015\pi\)
\(942\) 0 0
\(943\) 35280.0 1.21832
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 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
\(950\) 0 0
\(951\) −8766.00 −0.298903
\(952\) 0 0
\(953\) 21162.0 0.719312 0.359656 0.933085i \(-0.382894\pi\)
0.359656 + 0.933085i \(0.382894\pi\)
\(954\) 0 0
\(955\) −24864.0 −0.842492
\(956\) 0 0
\(957\) −6960.00 −0.235094
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −11295.0 −0.379141
\(962\) 0 0
\(963\) 1440.00 0.0481862
\(964\) 0 0
\(965\) −400.000 −0.0133435
\(966\) 0 0
\(967\) −8224.00 −0.273491 −0.136746 0.990606i \(-0.543664\pi\)
−0.136746 + 0.990606i \(0.543664\pi\)
\(968\) 0 0
\(969\) 37296.0 1.23645
\(970\) 0 0
\(971\) −8140.00 −0.269027 −0.134513 0.990912i \(-0.542947\pi\)
−0.134513 + 0.990912i \(0.542947\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 732.000 0.0240439
\(976\) 0 0
\(977\) 32158.0 1.05305 0.526523 0.850161i \(-0.323495\pi\)
0.526523 + 0.850161i \(0.323495\pi\)
\(978\) 0 0
\(979\) 24160.0 0.788720
\(980\) 0 0
\(981\) −19782.0 −0.643823
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −13776.0 −0.442923
\(990\) 0 0
\(991\) −12296.0 −0.394143 −0.197071 0.980389i \(-0.563143\pi\)
−0.197071 + 0.980389i \(0.563143\pi\)
\(992\) 0 0
\(993\) 22476.0 0.718282
\(994\) 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 \(0.868344\pi\)
\(998\) 0 0
\(999\) −5994.00 −0.189832
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 2352.4.a.z.1.1 1
4.3 odd 2 294.4.a.b.1.1 1
7.6 odd 2 2352.4.a.m.1.1 1
12.11 even 2 882.4.a.q.1.1 1
28.3 even 6 294.4.e.f.79.1 2
28.11 odd 6 294.4.e.j.79.1 2
28.19 even 6 294.4.e.f.67.1 2
28.23 odd 6 294.4.e.j.67.1 2
28.27 even 2 294.4.a.f.1.1 yes 1
84.11 even 6 882.4.g.c.667.1 2
84.23 even 6 882.4.g.c.361.1 2
84.47 odd 6 882.4.g.j.361.1 2
84.59 odd 6 882.4.g.j.667.1 2
84.83 odd 2 882.4.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.4.a.b.1.1 1 4.3 odd 2
294.4.a.f.1.1 yes 1 28.27 even 2
294.4.e.f.67.1 2 28.19 even 6
294.4.e.f.79.1 2 28.3 even 6
294.4.e.j.67.1 2 28.23 odd 6
294.4.e.j.79.1 2 28.11 odd 6
882.4.a.j.1.1 1 84.83 odd 2
882.4.a.q.1.1 1 12.11 even 2
882.4.g.c.361.1 2 84.23 even 6
882.4.g.c.667.1 2 84.11 even 6
882.4.g.j.361.1 2 84.47 odd 6
882.4.g.j.667.1 2 84.59 odd 6
2352.4.a.m.1.1 1 7.6 odd 2
2352.4.a.z.1.1 1 1.1 even 1 trivial