Properties

Label 624.4.a.h.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
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] = [624,4,Mod(1,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, 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("624.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -6.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -6.00000 q^{5} +4.00000 q^{7} +9.00000 q^{9} -36.0000 q^{11} +13.0000 q^{13} -18.0000 q^{15} +66.0000 q^{17} -56.0000 q^{19} +12.0000 q^{21} -96.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +222.000 q^{29} -260.000 q^{31} -108.000 q^{33} -24.0000 q^{35} -106.000 q^{37} +39.0000 q^{39} -90.0000 q^{41} -44.0000 q^{43} -54.0000 q^{45} -168.000 q^{47} -327.000 q^{49} +198.000 q^{51} +30.0000 q^{53} +216.000 q^{55} -168.000 q^{57} -348.000 q^{59} -346.000 q^{61} +36.0000 q^{63} -78.0000 q^{65} +256.000 q^{67} -288.000 q^{69} +168.000 q^{71} -814.000 q^{73} -267.000 q^{75} -144.000 q^{77} -200.000 q^{79} +81.0000 q^{81} -1236.00 q^{83} -396.000 q^{85} +666.000 q^{87} +318.000 q^{89} +52.0000 q^{91} -780.000 q^{93} +336.000 q^{95} -502.000 q^{97} -324.000 q^{99} +O(q^{100})\)

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
\(5\) −6.00000 −0.536656 −0.268328 0.963328i \(-0.586471\pi\)
−0.268328 + 0.963328i \(0.586471\pi\)
\(6\) 0 0
\(7\) 4.00000 0.215980 0.107990 0.994152i \(-0.465559\pi\)
0.107990 + 0.994152i \(0.465559\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −36.0000 −0.986764 −0.493382 0.869813i \(-0.664240\pi\)
−0.493382 + 0.869813i \(0.664240\pi\)
\(12\) 0 0
\(13\) 13.0000 0.277350
\(14\) 0 0
\(15\) −18.0000 −0.309839
\(16\) 0 0
\(17\) 66.0000 0.941609 0.470804 0.882238i \(-0.343964\pi\)
0.470804 + 0.882238i \(0.343964\pi\)
\(18\) 0 0
\(19\) −56.0000 −0.676173 −0.338086 0.941115i \(-0.609780\pi\)
−0.338086 + 0.941115i \(0.609780\pi\)
\(20\) 0 0
\(21\) 12.0000 0.124696
\(22\) 0 0
\(23\) −96.0000 −0.870321 −0.435161 0.900353i \(-0.643308\pi\)
−0.435161 + 0.900353i \(0.643308\pi\)
\(24\) 0 0
\(25\) −89.0000 −0.712000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 222.000 1.42153 0.710765 0.703430i \(-0.248349\pi\)
0.710765 + 0.703430i \(0.248349\pi\)
\(30\) 0 0
\(31\) −260.000 −1.50637 −0.753184 0.657810i \(-0.771483\pi\)
−0.753184 + 0.657810i \(0.771483\pi\)
\(32\) 0 0
\(33\) −108.000 −0.569709
\(34\) 0 0
\(35\) −24.0000 −0.115907
\(36\) 0 0
\(37\) −106.000 −0.470981 −0.235490 0.971877i \(-0.575670\pi\)
−0.235490 + 0.971877i \(0.575670\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −90.0000 −0.342820 −0.171410 0.985200i \(-0.554832\pi\)
−0.171410 + 0.985200i \(0.554832\pi\)
\(42\) 0 0
\(43\) −44.0000 −0.156045 −0.0780225 0.996952i \(-0.524861\pi\)
−0.0780225 + 0.996952i \(0.524861\pi\)
\(44\) 0 0
\(45\) −54.0000 −0.178885
\(46\) 0 0
\(47\) −168.000 −0.521390 −0.260695 0.965421i \(-0.583952\pi\)
−0.260695 + 0.965421i \(0.583952\pi\)
\(48\) 0 0
\(49\) −327.000 −0.953353
\(50\) 0 0
\(51\) 198.000 0.543638
\(52\) 0 0
\(53\) 30.0000 0.0777513 0.0388756 0.999244i \(-0.487622\pi\)
0.0388756 + 0.999244i \(0.487622\pi\)
\(54\) 0 0
\(55\) 216.000 0.529553
\(56\) 0 0
\(57\) −168.000 −0.390388
\(58\) 0 0
\(59\) −348.000 −0.767894 −0.383947 0.923355i \(-0.625435\pi\)
−0.383947 + 0.923355i \(0.625435\pi\)
\(60\) 0 0
\(61\) −346.000 −0.726242 −0.363121 0.931742i \(-0.618289\pi\)
−0.363121 + 0.931742i \(0.618289\pi\)
\(62\) 0 0
\(63\) 36.0000 0.0719932
\(64\) 0 0
\(65\) −78.0000 −0.148842
\(66\) 0 0
\(67\) 256.000 0.466797 0.233398 0.972381i \(-0.425015\pi\)
0.233398 + 0.972381i \(0.425015\pi\)
\(68\) 0 0
\(69\) −288.000 −0.502480
\(70\) 0 0
\(71\) 168.000 0.280816 0.140408 0.990094i \(-0.455159\pi\)
0.140408 + 0.990094i \(0.455159\pi\)
\(72\) 0 0
\(73\) −814.000 −1.30509 −0.652544 0.757750i \(-0.726298\pi\)
−0.652544 + 0.757750i \(0.726298\pi\)
\(74\) 0 0
\(75\) −267.000 −0.411073
\(76\) 0 0
\(77\) −144.000 −0.213121
\(78\) 0 0
\(79\) −200.000 −0.284832 −0.142416 0.989807i \(-0.545487\pi\)
−0.142416 + 0.989807i \(0.545487\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −1236.00 −1.63456 −0.817281 0.576240i \(-0.804520\pi\)
−0.817281 + 0.576240i \(0.804520\pi\)
\(84\) 0 0
\(85\) −396.000 −0.505320
\(86\) 0 0
\(87\) 666.000 0.820721
\(88\) 0 0
\(89\) 318.000 0.378741 0.189370 0.981906i \(-0.439355\pi\)
0.189370 + 0.981906i \(0.439355\pi\)
\(90\) 0 0
\(91\) 52.0000 0.0599020
\(92\) 0 0
\(93\) −780.000 −0.869701
\(94\) 0 0
\(95\) 336.000 0.362872
\(96\) 0 0
\(97\) −502.000 −0.525468 −0.262734 0.964868i \(-0.584624\pi\)
−0.262734 + 0.964868i \(0.584624\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 1062.00 1.04627 0.523133 0.852251i \(-0.324763\pi\)
0.523133 + 0.852251i \(0.324763\pi\)
\(102\) 0 0
\(103\) 64.0000 0.0612243 0.0306122 0.999531i \(-0.490254\pi\)
0.0306122 + 0.999531i \(0.490254\pi\)
\(104\) 0 0
\(105\) −72.0000 −0.0669189
\(106\) 0 0
\(107\) 444.000 0.401150 0.200575 0.979678i \(-0.435719\pi\)
0.200575 + 0.979678i \(0.435719\pi\)
\(108\) 0 0
\(109\) 1382.00 1.21442 0.607209 0.794542i \(-0.292289\pi\)
0.607209 + 0.794542i \(0.292289\pi\)
\(110\) 0 0
\(111\) −318.000 −0.271921
\(112\) 0 0
\(113\) −870.000 −0.724272 −0.362136 0.932125i \(-0.617952\pi\)
−0.362136 + 0.932125i \(0.617952\pi\)
\(114\) 0 0
\(115\) 576.000 0.467063
\(116\) 0 0
\(117\) 117.000 0.0924500
\(118\) 0 0
\(119\) 264.000 0.203368
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) −270.000 −0.197927
\(124\) 0 0
\(125\) 1284.00 0.918756
\(126\) 0 0
\(127\) −464.000 −0.324200 −0.162100 0.986774i \(-0.551827\pi\)
−0.162100 + 0.986774i \(0.551827\pi\)
\(128\) 0 0
\(129\) −132.000 −0.0900927
\(130\) 0 0
\(131\) −1548.00 −1.03244 −0.516219 0.856457i \(-0.672661\pi\)
−0.516219 + 0.856457i \(0.672661\pi\)
\(132\) 0 0
\(133\) −224.000 −0.146040
\(134\) 0 0
\(135\) −162.000 −0.103280
\(136\) 0 0
\(137\) 294.000 0.183344 0.0916720 0.995789i \(-0.470779\pi\)
0.0916720 + 0.995789i \(0.470779\pi\)
\(138\) 0 0
\(139\) −2564.00 −1.56457 −0.782286 0.622919i \(-0.785947\pi\)
−0.782286 + 0.622919i \(0.785947\pi\)
\(140\) 0 0
\(141\) −504.000 −0.301025
\(142\) 0 0
\(143\) −468.000 −0.273679
\(144\) 0 0
\(145\) −1332.00 −0.762873
\(146\) 0 0
\(147\) −981.000 −0.550418
\(148\) 0 0
\(149\) 114.000 0.0626795 0.0313397 0.999509i \(-0.490023\pi\)
0.0313397 + 0.999509i \(0.490023\pi\)
\(150\) 0 0
\(151\) −2036.00 −1.09727 −0.548634 0.836063i \(-0.684852\pi\)
−0.548634 + 0.836063i \(0.684852\pi\)
\(152\) 0 0
\(153\) 594.000 0.313870
\(154\) 0 0
\(155\) 1560.00 0.808401
\(156\) 0 0
\(157\) 2870.00 1.45892 0.729462 0.684022i \(-0.239771\pi\)
0.729462 + 0.684022i \(0.239771\pi\)
\(158\) 0 0
\(159\) 90.0000 0.0448897
\(160\) 0 0
\(161\) −384.000 −0.187972
\(162\) 0 0
\(163\) −1472.00 −0.707337 −0.353669 0.935371i \(-0.615066\pi\)
−0.353669 + 0.935371i \(0.615066\pi\)
\(164\) 0 0
\(165\) 648.000 0.305738
\(166\) 0 0
\(167\) 240.000 0.111208 0.0556041 0.998453i \(-0.482292\pi\)
0.0556041 + 0.998453i \(0.482292\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −504.000 −0.225391
\(172\) 0 0
\(173\) −306.000 −0.134478 −0.0672392 0.997737i \(-0.521419\pi\)
−0.0672392 + 0.997737i \(0.521419\pi\)
\(174\) 0 0
\(175\) −356.000 −0.153778
\(176\) 0 0
\(177\) −1044.00 −0.443344
\(178\) 0 0
\(179\) 2052.00 0.856836 0.428418 0.903581i \(-0.359071\pi\)
0.428418 + 0.903581i \(0.359071\pi\)
\(180\) 0 0
\(181\) −4498.00 −1.84715 −0.923574 0.383421i \(-0.874746\pi\)
−0.923574 + 0.383421i \(0.874746\pi\)
\(182\) 0 0
\(183\) −1038.00 −0.419296
\(184\) 0 0
\(185\) 636.000 0.252755
\(186\) 0 0
\(187\) −2376.00 −0.929146
\(188\) 0 0
\(189\) 108.000 0.0415653
\(190\) 0 0
\(191\) 4056.00 1.53655 0.768277 0.640117i \(-0.221114\pi\)
0.768277 + 0.640117i \(0.221114\pi\)
\(192\) 0 0
\(193\) −2062.00 −0.769047 −0.384523 0.923115i \(-0.625634\pi\)
−0.384523 + 0.923115i \(0.625634\pi\)
\(194\) 0 0
\(195\) −234.000 −0.0859338
\(196\) 0 0
\(197\) −4374.00 −1.58190 −0.790951 0.611880i \(-0.790414\pi\)
−0.790951 + 0.611880i \(0.790414\pi\)
\(198\) 0 0
\(199\) 2536.00 0.903378 0.451689 0.892175i \(-0.350822\pi\)
0.451689 + 0.892175i \(0.350822\pi\)
\(200\) 0 0
\(201\) 768.000 0.269505
\(202\) 0 0
\(203\) 888.000 0.307022
\(204\) 0 0
\(205\) 540.000 0.183977
\(206\) 0 0
\(207\) −864.000 −0.290107
\(208\) 0 0
\(209\) 2016.00 0.667223
\(210\) 0 0
\(211\) 4444.00 1.44994 0.724971 0.688780i \(-0.241853\pi\)
0.724971 + 0.688780i \(0.241853\pi\)
\(212\) 0 0
\(213\) 504.000 0.162129
\(214\) 0 0
\(215\) 264.000 0.0837426
\(216\) 0 0
\(217\) −1040.00 −0.325345
\(218\) 0 0
\(219\) −2442.00 −0.753493
\(220\) 0 0
\(221\) 858.000 0.261155
\(222\) 0 0
\(223\) 2716.00 0.815591 0.407796 0.913073i \(-0.366298\pi\)
0.407796 + 0.913073i \(0.366298\pi\)
\(224\) 0 0
\(225\) −801.000 −0.237333
\(226\) 0 0
\(227\) 4692.00 1.37189 0.685945 0.727653i \(-0.259389\pi\)
0.685945 + 0.727653i \(0.259389\pi\)
\(228\) 0 0
\(229\) 6446.00 1.86010 0.930052 0.367429i \(-0.119762\pi\)
0.930052 + 0.367429i \(0.119762\pi\)
\(230\) 0 0
\(231\) −432.000 −0.123046
\(232\) 0 0
\(233\) −3102.00 −0.872184 −0.436092 0.899902i \(-0.643638\pi\)
−0.436092 + 0.899902i \(0.643638\pi\)
\(234\) 0 0
\(235\) 1008.00 0.279807
\(236\) 0 0
\(237\) −600.000 −0.164448
\(238\) 0 0
\(239\) −816.000 −0.220848 −0.110424 0.993885i \(-0.535221\pi\)
−0.110424 + 0.993885i \(0.535221\pi\)
\(240\) 0 0
\(241\) 3818.00 1.02049 0.510247 0.860028i \(-0.329554\pi\)
0.510247 + 0.860028i \(0.329554\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 1962.00 0.511623
\(246\) 0 0
\(247\) −728.000 −0.187537
\(248\) 0 0
\(249\) −3708.00 −0.943715
\(250\) 0 0
\(251\) 6612.00 1.66273 0.831366 0.555725i \(-0.187559\pi\)
0.831366 + 0.555725i \(0.187559\pi\)
\(252\) 0 0
\(253\) 3456.00 0.858802
\(254\) 0 0
\(255\) −1188.00 −0.291747
\(256\) 0 0
\(257\) −4806.00 −1.16650 −0.583249 0.812293i \(-0.698219\pi\)
−0.583249 + 0.812293i \(0.698219\pi\)
\(258\) 0 0
\(259\) −424.000 −0.101722
\(260\) 0 0
\(261\) 1998.00 0.473843
\(262\) 0 0
\(263\) 4584.00 1.07476 0.537379 0.843341i \(-0.319414\pi\)
0.537379 + 0.843341i \(0.319414\pi\)
\(264\) 0 0
\(265\) −180.000 −0.0417257
\(266\) 0 0
\(267\) 954.000 0.218666
\(268\) 0 0
\(269\) 7134.00 1.61698 0.808490 0.588510i \(-0.200285\pi\)
0.808490 + 0.588510i \(0.200285\pi\)
\(270\) 0 0
\(271\) −3140.00 −0.703843 −0.351921 0.936030i \(-0.614472\pi\)
−0.351921 + 0.936030i \(0.614472\pi\)
\(272\) 0 0
\(273\) 156.000 0.0345844
\(274\) 0 0
\(275\) 3204.00 0.702576
\(276\) 0 0
\(277\) −4786.00 −1.03813 −0.519067 0.854734i \(-0.673720\pi\)
−0.519067 + 0.854734i \(0.673720\pi\)
\(278\) 0 0
\(279\) −2340.00 −0.502122
\(280\) 0 0
\(281\) 3798.00 0.806298 0.403149 0.915134i \(-0.367916\pi\)
0.403149 + 0.915134i \(0.367916\pi\)
\(282\) 0 0
\(283\) −3572.00 −0.750295 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(284\) 0 0
\(285\) 1008.00 0.209504
\(286\) 0 0
\(287\) −360.000 −0.0740423
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) −1506.00 −0.303379
\(292\) 0 0
\(293\) 7122.00 1.42004 0.710020 0.704182i \(-0.248686\pi\)
0.710020 + 0.704182i \(0.248686\pi\)
\(294\) 0 0
\(295\) 2088.00 0.412095
\(296\) 0 0
\(297\) −972.000 −0.189903
\(298\) 0 0
\(299\) −1248.00 −0.241384
\(300\) 0 0
\(301\) −176.000 −0.0337026
\(302\) 0 0
\(303\) 3186.00 0.604062
\(304\) 0 0
\(305\) 2076.00 0.389742
\(306\) 0 0
\(307\) 6856.00 1.27457 0.637284 0.770629i \(-0.280058\pi\)
0.637284 + 0.770629i \(0.280058\pi\)
\(308\) 0 0
\(309\) 192.000 0.0353479
\(310\) 0 0
\(311\) 8832.00 1.61034 0.805172 0.593042i \(-0.202073\pi\)
0.805172 + 0.593042i \(0.202073\pi\)
\(312\) 0 0
\(313\) 3626.00 0.654804 0.327402 0.944885i \(-0.393827\pi\)
0.327402 + 0.944885i \(0.393827\pi\)
\(314\) 0 0
\(315\) −216.000 −0.0386356
\(316\) 0 0
\(317\) 10146.0 1.79765 0.898827 0.438304i \(-0.144421\pi\)
0.898827 + 0.438304i \(0.144421\pi\)
\(318\) 0 0
\(319\) −7992.00 −1.40272
\(320\) 0 0
\(321\) 1332.00 0.231604
\(322\) 0 0
\(323\) −3696.00 −0.636690
\(324\) 0 0
\(325\) −1157.00 −0.197473
\(326\) 0 0
\(327\) 4146.00 0.701145
\(328\) 0 0
\(329\) −672.000 −0.112610
\(330\) 0 0
\(331\) −6536.00 −1.08535 −0.542675 0.839943i \(-0.682589\pi\)
−0.542675 + 0.839943i \(0.682589\pi\)
\(332\) 0 0
\(333\) −954.000 −0.156994
\(334\) 0 0
\(335\) −1536.00 −0.250509
\(336\) 0 0
\(337\) −6094.00 −0.985048 −0.492524 0.870299i \(-0.663926\pi\)
−0.492524 + 0.870299i \(0.663926\pi\)
\(338\) 0 0
\(339\) −2610.00 −0.418159
\(340\) 0 0
\(341\) 9360.00 1.48643
\(342\) 0 0
\(343\) −2680.00 −0.421885
\(344\) 0 0
\(345\) 1728.00 0.269659
\(346\) 0 0
\(347\) 2724.00 0.421418 0.210709 0.977549i \(-0.432423\pi\)
0.210709 + 0.977549i \(0.432423\pi\)
\(348\) 0 0
\(349\) −1522.00 −0.233441 −0.116720 0.993165i \(-0.537238\pi\)
−0.116720 + 0.993165i \(0.537238\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) −1362.00 −0.205360 −0.102680 0.994714i \(-0.532742\pi\)
−0.102680 + 0.994714i \(0.532742\pi\)
\(354\) 0 0
\(355\) −1008.00 −0.150702
\(356\) 0 0
\(357\) 792.000 0.117415
\(358\) 0 0
\(359\) −8880.00 −1.30548 −0.652742 0.757581i \(-0.726381\pi\)
−0.652742 + 0.757581i \(0.726381\pi\)
\(360\) 0 0
\(361\) −3723.00 −0.542790
\(362\) 0 0
\(363\) −105.000 −0.0151820
\(364\) 0 0
\(365\) 4884.00 0.700384
\(366\) 0 0
\(367\) 3712.00 0.527970 0.263985 0.964527i \(-0.414963\pi\)
0.263985 + 0.964527i \(0.414963\pi\)
\(368\) 0 0
\(369\) −810.000 −0.114273
\(370\) 0 0
\(371\) 120.000 0.0167927
\(372\) 0 0
\(373\) 5726.00 0.794855 0.397428 0.917634i \(-0.369903\pi\)
0.397428 + 0.917634i \(0.369903\pi\)
\(374\) 0 0
\(375\) 3852.00 0.530444
\(376\) 0 0
\(377\) 2886.00 0.394261
\(378\) 0 0
\(379\) 13168.0 1.78468 0.892341 0.451361i \(-0.149061\pi\)
0.892341 + 0.451361i \(0.149061\pi\)
\(380\) 0 0
\(381\) −1392.00 −0.187177
\(382\) 0 0
\(383\) −4872.00 −0.649994 −0.324997 0.945715i \(-0.605363\pi\)
−0.324997 + 0.945715i \(0.605363\pi\)
\(384\) 0 0
\(385\) 864.000 0.114373
\(386\) 0 0
\(387\) −396.000 −0.0520150
\(388\) 0 0
\(389\) −1266.00 −0.165010 −0.0825048 0.996591i \(-0.526292\pi\)
−0.0825048 + 0.996591i \(0.526292\pi\)
\(390\) 0 0
\(391\) −6336.00 −0.819502
\(392\) 0 0
\(393\) −4644.00 −0.596078
\(394\) 0 0
\(395\) 1200.00 0.152857
\(396\) 0 0
\(397\) −4882.00 −0.617180 −0.308590 0.951195i \(-0.599857\pi\)
−0.308590 + 0.951195i \(0.599857\pi\)
\(398\) 0 0
\(399\) −672.000 −0.0843160
\(400\) 0 0
\(401\) −90.0000 −0.0112079 −0.00560397 0.999984i \(-0.501784\pi\)
−0.00560397 + 0.999984i \(0.501784\pi\)
\(402\) 0 0
\(403\) −3380.00 −0.417791
\(404\) 0 0
\(405\) −486.000 −0.0596285
\(406\) 0 0
\(407\) 3816.00 0.464747
\(408\) 0 0
\(409\) 2354.00 0.284591 0.142296 0.989824i \(-0.454552\pi\)
0.142296 + 0.989824i \(0.454552\pi\)
\(410\) 0 0
\(411\) 882.000 0.105854
\(412\) 0 0
\(413\) −1392.00 −0.165849
\(414\) 0 0
\(415\) 7416.00 0.877198
\(416\) 0 0
\(417\) −7692.00 −0.903307
\(418\) 0 0
\(419\) −7020.00 −0.818495 −0.409248 0.912423i \(-0.634209\pi\)
−0.409248 + 0.912423i \(0.634209\pi\)
\(420\) 0 0
\(421\) 302.000 0.0349610 0.0174805 0.999847i \(-0.494436\pi\)
0.0174805 + 0.999847i \(0.494436\pi\)
\(422\) 0 0
\(423\) −1512.00 −0.173797
\(424\) 0 0
\(425\) −5874.00 −0.670426
\(426\) 0 0
\(427\) −1384.00 −0.156854
\(428\) 0 0
\(429\) −1404.00 −0.158009
\(430\) 0 0
\(431\) −9816.00 −1.09703 −0.548515 0.836141i \(-0.684807\pi\)
−0.548515 + 0.836141i \(0.684807\pi\)
\(432\) 0 0
\(433\) −14782.0 −1.64059 −0.820297 0.571937i \(-0.806192\pi\)
−0.820297 + 0.571937i \(0.806192\pi\)
\(434\) 0 0
\(435\) −3996.00 −0.440445
\(436\) 0 0
\(437\) 5376.00 0.588487
\(438\) 0 0
\(439\) −3584.00 −0.389647 −0.194823 0.980838i \(-0.562413\pi\)
−0.194823 + 0.980838i \(0.562413\pi\)
\(440\) 0 0
\(441\) −2943.00 −0.317784
\(442\) 0 0
\(443\) −180.000 −0.0193049 −0.00965244 0.999953i \(-0.503073\pi\)
−0.00965244 + 0.999953i \(0.503073\pi\)
\(444\) 0 0
\(445\) −1908.00 −0.203254
\(446\) 0 0
\(447\) 342.000 0.0361880
\(448\) 0 0
\(449\) −3450.00 −0.362618 −0.181309 0.983426i \(-0.558033\pi\)
−0.181309 + 0.983426i \(0.558033\pi\)
\(450\) 0 0
\(451\) 3240.00 0.338283
\(452\) 0 0
\(453\) −6108.00 −0.633507
\(454\) 0 0
\(455\) −312.000 −0.0321468
\(456\) 0 0
\(457\) −16654.0 −1.70469 −0.852343 0.522984i \(-0.824819\pi\)
−0.852343 + 0.522984i \(0.824819\pi\)
\(458\) 0 0
\(459\) 1782.00 0.181213
\(460\) 0 0
\(461\) −14046.0 −1.41906 −0.709531 0.704674i \(-0.751093\pi\)
−0.709531 + 0.704674i \(0.751093\pi\)
\(462\) 0 0
\(463\) 4588.00 0.460524 0.230262 0.973129i \(-0.426042\pi\)
0.230262 + 0.973129i \(0.426042\pi\)
\(464\) 0 0
\(465\) 4680.00 0.466731
\(466\) 0 0
\(467\) 15372.0 1.52319 0.761597 0.648051i \(-0.224416\pi\)
0.761597 + 0.648051i \(0.224416\pi\)
\(468\) 0 0
\(469\) 1024.00 0.100819
\(470\) 0 0
\(471\) 8610.00 0.842310
\(472\) 0 0
\(473\) 1584.00 0.153980
\(474\) 0 0
\(475\) 4984.00 0.481435
\(476\) 0 0
\(477\) 270.000 0.0259171
\(478\) 0 0
\(479\) −12864.0 −1.22708 −0.613540 0.789664i \(-0.710255\pi\)
−0.613540 + 0.789664i \(0.710255\pi\)
\(480\) 0 0
\(481\) −1378.00 −0.130627
\(482\) 0 0
\(483\) −1152.00 −0.108525
\(484\) 0 0
\(485\) 3012.00 0.281996
\(486\) 0 0
\(487\) 10276.0 0.956160 0.478080 0.878316i \(-0.341333\pi\)
0.478080 + 0.878316i \(0.341333\pi\)
\(488\) 0 0
\(489\) −4416.00 −0.408381
\(490\) 0 0
\(491\) 11220.0 1.03127 0.515633 0.856810i \(-0.327557\pi\)
0.515633 + 0.856810i \(0.327557\pi\)
\(492\) 0 0
\(493\) 14652.0 1.33853
\(494\) 0 0
\(495\) 1944.00 0.176518
\(496\) 0 0
\(497\) 672.000 0.0606505
\(498\) 0 0
\(499\) −17264.0 −1.54878 −0.774392 0.632707i \(-0.781944\pi\)
−0.774392 + 0.632707i \(0.781944\pi\)
\(500\) 0 0
\(501\) 720.000 0.0642060
\(502\) 0 0
\(503\) 1896.00 0.168069 0.0840343 0.996463i \(-0.473219\pi\)
0.0840343 + 0.996463i \(0.473219\pi\)
\(504\) 0 0
\(505\) −6372.00 −0.561486
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) 5010.00 0.436276 0.218138 0.975918i \(-0.430002\pi\)
0.218138 + 0.975918i \(0.430002\pi\)
\(510\) 0 0
\(511\) −3256.00 −0.281873
\(512\) 0 0
\(513\) −1512.00 −0.130129
\(514\) 0 0
\(515\) −384.000 −0.0328564
\(516\) 0 0
\(517\) 6048.00 0.514489
\(518\) 0 0
\(519\) −918.000 −0.0776411
\(520\) 0 0
\(521\) 8610.00 0.724013 0.362007 0.932176i \(-0.382092\pi\)
0.362007 + 0.932176i \(0.382092\pi\)
\(522\) 0 0
\(523\) 5308.00 0.443791 0.221895 0.975070i \(-0.428776\pi\)
0.221895 + 0.975070i \(0.428776\pi\)
\(524\) 0 0
\(525\) −1068.00 −0.0887835
\(526\) 0 0
\(527\) −17160.0 −1.41841
\(528\) 0 0
\(529\) −2951.00 −0.242541
\(530\) 0 0
\(531\) −3132.00 −0.255965
\(532\) 0 0
\(533\) −1170.00 −0.0950813
\(534\) 0 0
\(535\) −2664.00 −0.215280
\(536\) 0 0
\(537\) 6156.00 0.494695
\(538\) 0 0
\(539\) 11772.0 0.940735
\(540\) 0 0
\(541\) 6182.00 0.491285 0.245642 0.969361i \(-0.421001\pi\)
0.245642 + 0.969361i \(0.421001\pi\)
\(542\) 0 0
\(543\) −13494.0 −1.06645
\(544\) 0 0
\(545\) −8292.00 −0.651725
\(546\) 0 0
\(547\) −1292.00 −0.100991 −0.0504954 0.998724i \(-0.516080\pi\)
−0.0504954 + 0.998724i \(0.516080\pi\)
\(548\) 0 0
\(549\) −3114.00 −0.242081
\(550\) 0 0
\(551\) −12432.0 −0.961200
\(552\) 0 0
\(553\) −800.000 −0.0615180
\(554\) 0 0
\(555\) 1908.00 0.145928
\(556\) 0 0
\(557\) −12774.0 −0.971727 −0.485863 0.874035i \(-0.661495\pi\)
−0.485863 + 0.874035i \(0.661495\pi\)
\(558\) 0 0
\(559\) −572.000 −0.0432791
\(560\) 0 0
\(561\) −7128.00 −0.536443
\(562\) 0 0
\(563\) −16908.0 −1.26570 −0.632848 0.774276i \(-0.718114\pi\)
−0.632848 + 0.774276i \(0.718114\pi\)
\(564\) 0 0
\(565\) 5220.00 0.388685
\(566\) 0 0
\(567\) 324.000 0.0239977
\(568\) 0 0
\(569\) −11214.0 −0.826213 −0.413107 0.910683i \(-0.635556\pi\)
−0.413107 + 0.910683i \(0.635556\pi\)
\(570\) 0 0
\(571\) −25220.0 −1.84838 −0.924189 0.381935i \(-0.875258\pi\)
−0.924189 + 0.381935i \(0.875258\pi\)
\(572\) 0 0
\(573\) 12168.0 0.887130
\(574\) 0 0
\(575\) 8544.00 0.619669
\(576\) 0 0
\(577\) −17710.0 −1.27778 −0.638888 0.769300i \(-0.720605\pi\)
−0.638888 + 0.769300i \(0.720605\pi\)
\(578\) 0 0
\(579\) −6186.00 −0.444009
\(580\) 0 0
\(581\) −4944.00 −0.353032
\(582\) 0 0
\(583\) −1080.00 −0.0767222
\(584\) 0 0
\(585\) −702.000 −0.0496139
\(586\) 0 0
\(587\) −20028.0 −1.40825 −0.704126 0.710075i \(-0.748661\pi\)
−0.704126 + 0.710075i \(0.748661\pi\)
\(588\) 0 0
\(589\) 14560.0 1.01856
\(590\) 0 0
\(591\) −13122.0 −0.913311
\(592\) 0 0
\(593\) 19926.0 1.37987 0.689935 0.723871i \(-0.257639\pi\)
0.689935 + 0.723871i \(0.257639\pi\)
\(594\) 0 0
\(595\) −1584.00 −0.109139
\(596\) 0 0
\(597\) 7608.00 0.521566
\(598\) 0 0
\(599\) 1704.00 0.116233 0.0581165 0.998310i \(-0.481491\pi\)
0.0581165 + 0.998310i \(0.481491\pi\)
\(600\) 0 0
\(601\) 11018.0 0.747810 0.373905 0.927467i \(-0.378019\pi\)
0.373905 + 0.927467i \(0.378019\pi\)
\(602\) 0 0
\(603\) 2304.00 0.155599
\(604\) 0 0
\(605\) 210.000 0.0141119
\(606\) 0 0
\(607\) 448.000 0.0299568 0.0149784 0.999888i \(-0.495232\pi\)
0.0149784 + 0.999888i \(0.495232\pi\)
\(608\) 0 0
\(609\) 2664.00 0.177259
\(610\) 0 0
\(611\) −2184.00 −0.144608
\(612\) 0 0
\(613\) −12586.0 −0.829272 −0.414636 0.909987i \(-0.636091\pi\)
−0.414636 + 0.909987i \(0.636091\pi\)
\(614\) 0 0
\(615\) 1620.00 0.106219
\(616\) 0 0
\(617\) −29610.0 −1.93202 −0.966008 0.258513i \(-0.916768\pi\)
−0.966008 + 0.258513i \(0.916768\pi\)
\(618\) 0 0
\(619\) 7120.00 0.462321 0.231161 0.972916i \(-0.425748\pi\)
0.231161 + 0.972916i \(0.425748\pi\)
\(620\) 0 0
\(621\) −2592.00 −0.167493
\(622\) 0 0
\(623\) 1272.00 0.0818003
\(624\) 0 0
\(625\) 3421.00 0.218944
\(626\) 0 0
\(627\) 6048.00 0.385221
\(628\) 0 0
\(629\) −6996.00 −0.443480
\(630\) 0 0
\(631\) 15580.0 0.982932 0.491466 0.870897i \(-0.336461\pi\)
0.491466 + 0.870897i \(0.336461\pi\)
\(632\) 0 0
\(633\) 13332.0 0.837124
\(634\) 0 0
\(635\) 2784.00 0.173984
\(636\) 0 0
\(637\) −4251.00 −0.264412
\(638\) 0 0
\(639\) 1512.00 0.0936053
\(640\) 0 0
\(641\) −19806.0 −1.22042 −0.610211 0.792239i \(-0.708915\pi\)
−0.610211 + 0.792239i \(0.708915\pi\)
\(642\) 0 0
\(643\) −24032.0 −1.47392 −0.736959 0.675937i \(-0.763739\pi\)
−0.736959 + 0.675937i \(0.763739\pi\)
\(644\) 0 0
\(645\) 792.000 0.0483488
\(646\) 0 0
\(647\) 2808.00 0.170624 0.0853121 0.996354i \(-0.472811\pi\)
0.0853121 + 0.996354i \(0.472811\pi\)
\(648\) 0 0
\(649\) 12528.0 0.757730
\(650\) 0 0
\(651\) −3120.00 −0.187838
\(652\) 0 0
\(653\) 23886.0 1.43144 0.715721 0.698386i \(-0.246098\pi\)
0.715721 + 0.698386i \(0.246098\pi\)
\(654\) 0 0
\(655\) 9288.00 0.554064
\(656\) 0 0
\(657\) −7326.00 −0.435030
\(658\) 0 0
\(659\) 3948.00 0.233372 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(660\) 0 0
\(661\) 5750.00 0.338350 0.169175 0.985586i \(-0.445890\pi\)
0.169175 + 0.985586i \(0.445890\pi\)
\(662\) 0 0
\(663\) 2574.00 0.150778
\(664\) 0 0
\(665\) 1344.00 0.0783731
\(666\) 0 0
\(667\) −21312.0 −1.23719
\(668\) 0 0
\(669\) 8148.00 0.470882
\(670\) 0 0
\(671\) 12456.0 0.716630
\(672\) 0 0
\(673\) 28082.0 1.60844 0.804221 0.594330i \(-0.202583\pi\)
0.804221 + 0.594330i \(0.202583\pi\)
\(674\) 0 0
\(675\) −2403.00 −0.137024
\(676\) 0 0
\(677\) −27954.0 −1.58694 −0.793471 0.608608i \(-0.791728\pi\)
−0.793471 + 0.608608i \(0.791728\pi\)
\(678\) 0 0
\(679\) −2008.00 −0.113490
\(680\) 0 0
\(681\) 14076.0 0.792061
\(682\) 0 0
\(683\) 28428.0 1.59263 0.796316 0.604881i \(-0.206779\pi\)
0.796316 + 0.604881i \(0.206779\pi\)
\(684\) 0 0
\(685\) −1764.00 −0.0983927
\(686\) 0 0
\(687\) 19338.0 1.07393
\(688\) 0 0
\(689\) 390.000 0.0215643
\(690\) 0 0
\(691\) −21680.0 −1.19355 −0.596777 0.802407i \(-0.703552\pi\)
−0.596777 + 0.802407i \(0.703552\pi\)
\(692\) 0 0
\(693\) −1296.00 −0.0710404
\(694\) 0 0
\(695\) 15384.0 0.839638
\(696\) 0 0
\(697\) −5940.00 −0.322803
\(698\) 0 0
\(699\) −9306.00 −0.503555
\(700\) 0 0
\(701\) −7482.00 −0.403126 −0.201563 0.979476i \(-0.564602\pi\)
−0.201563 + 0.979476i \(0.564602\pi\)
\(702\) 0 0
\(703\) 5936.00 0.318464
\(704\) 0 0
\(705\) 3024.00 0.161547
\(706\) 0 0
\(707\) 4248.00 0.225972
\(708\) 0 0
\(709\) 2270.00 0.120242 0.0601210 0.998191i \(-0.480851\pi\)
0.0601210 + 0.998191i \(0.480851\pi\)
\(710\) 0 0
\(711\) −1800.00 −0.0949441
\(712\) 0 0
\(713\) 24960.0 1.31102
\(714\) 0 0
\(715\) 2808.00 0.146872
\(716\) 0 0
\(717\) −2448.00 −0.127507
\(718\) 0 0
\(719\) −36024.0 −1.86852 −0.934262 0.356588i \(-0.883940\pi\)
−0.934262 + 0.356588i \(0.883940\pi\)
\(720\) 0 0
\(721\) 256.000 0.0132232
\(722\) 0 0
\(723\) 11454.0 0.589182
\(724\) 0 0
\(725\) −19758.0 −1.01213
\(726\) 0 0
\(727\) 21544.0 1.09907 0.549534 0.835471i \(-0.314805\pi\)
0.549534 + 0.835471i \(0.314805\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −2904.00 −0.146933
\(732\) 0 0
\(733\) −1018.00 −0.0512970 −0.0256485 0.999671i \(-0.508165\pi\)
−0.0256485 + 0.999671i \(0.508165\pi\)
\(734\) 0 0
\(735\) 5886.00 0.295386
\(736\) 0 0
\(737\) −9216.00 −0.460618
\(738\) 0 0
\(739\) 24568.0 1.22293 0.611467 0.791270i \(-0.290580\pi\)
0.611467 + 0.791270i \(0.290580\pi\)
\(740\) 0 0
\(741\) −2184.00 −0.108274
\(742\) 0 0
\(743\) −16968.0 −0.837814 −0.418907 0.908029i \(-0.637587\pi\)
−0.418907 + 0.908029i \(0.637587\pi\)
\(744\) 0 0
\(745\) −684.000 −0.0336373
\(746\) 0 0
\(747\) −11124.0 −0.544854
\(748\) 0 0
\(749\) 1776.00 0.0866404
\(750\) 0 0
\(751\) −3224.00 −0.156652 −0.0783259 0.996928i \(-0.524957\pi\)
−0.0783259 + 0.996928i \(0.524957\pi\)
\(752\) 0 0
\(753\) 19836.0 0.959979
\(754\) 0 0
\(755\) 12216.0 0.588855
\(756\) 0 0
\(757\) −31570.0 −1.51576 −0.757881 0.652393i \(-0.773765\pi\)
−0.757881 + 0.652393i \(0.773765\pi\)
\(758\) 0 0
\(759\) 10368.0 0.495829
\(760\) 0 0
\(761\) −34890.0 −1.66197 −0.830987 0.556293i \(-0.812223\pi\)
−0.830987 + 0.556293i \(0.812223\pi\)
\(762\) 0 0
\(763\) 5528.00 0.262290
\(764\) 0 0
\(765\) −3564.00 −0.168440
\(766\) 0 0
\(767\) −4524.00 −0.212975
\(768\) 0 0
\(769\) 11522.0 0.540304 0.270152 0.962818i \(-0.412926\pi\)
0.270152 + 0.962818i \(0.412926\pi\)
\(770\) 0 0
\(771\) −14418.0 −0.673478
\(772\) 0 0
\(773\) −28158.0 −1.31018 −0.655092 0.755549i \(-0.727370\pi\)
−0.655092 + 0.755549i \(0.727370\pi\)
\(774\) 0 0
\(775\) 23140.0 1.07253
\(776\) 0 0
\(777\) −1272.00 −0.0587294
\(778\) 0 0
\(779\) 5040.00 0.231806
\(780\) 0 0
\(781\) −6048.00 −0.277099
\(782\) 0 0
\(783\) 5994.00 0.273574
\(784\) 0 0
\(785\) −17220.0 −0.782940
\(786\) 0 0
\(787\) −14504.0 −0.656940 −0.328470 0.944514i \(-0.606533\pi\)
−0.328470 + 0.944514i \(0.606533\pi\)
\(788\) 0 0
\(789\) 13752.0 0.620512
\(790\) 0 0
\(791\) −3480.00 −0.156428
\(792\) 0 0
\(793\) −4498.00 −0.201423
\(794\) 0 0
\(795\) −540.000 −0.0240903
\(796\) 0 0
\(797\) −18090.0 −0.803991 −0.401995 0.915642i \(-0.631683\pi\)
−0.401995 + 0.915642i \(0.631683\pi\)
\(798\) 0 0
\(799\) −11088.0 −0.490945
\(800\) 0 0
\(801\) 2862.00 0.126247
\(802\) 0 0
\(803\) 29304.0 1.28782
\(804\) 0 0
\(805\) 2304.00 0.100876
\(806\) 0 0
\(807\) 21402.0 0.933564
\(808\) 0 0
\(809\) 36402.0 1.58199 0.790993 0.611826i \(-0.209565\pi\)
0.790993 + 0.611826i \(0.209565\pi\)
\(810\) 0 0
\(811\) 32368.0 1.40147 0.700736 0.713420i \(-0.252855\pi\)
0.700736 + 0.713420i \(0.252855\pi\)
\(812\) 0 0
\(813\) −9420.00 −0.406364
\(814\) 0 0
\(815\) 8832.00 0.379597
\(816\) 0 0
\(817\) 2464.00 0.105513
\(818\) 0 0
\(819\) 468.000 0.0199673
\(820\) 0 0
\(821\) 35778.0 1.52090 0.760451 0.649395i \(-0.224978\pi\)
0.760451 + 0.649395i \(0.224978\pi\)
\(822\) 0 0
\(823\) 10240.0 0.433711 0.216855 0.976204i \(-0.430420\pi\)
0.216855 + 0.976204i \(0.430420\pi\)
\(824\) 0 0
\(825\) 9612.00 0.405633
\(826\) 0 0
\(827\) 16284.0 0.684704 0.342352 0.939572i \(-0.388776\pi\)
0.342352 + 0.939572i \(0.388776\pi\)
\(828\) 0 0
\(829\) 14150.0 0.592822 0.296411 0.955060i \(-0.404210\pi\)
0.296411 + 0.955060i \(0.404210\pi\)
\(830\) 0 0
\(831\) −14358.0 −0.599366
\(832\) 0 0
\(833\) −21582.0 −0.897685
\(834\) 0 0
\(835\) −1440.00 −0.0596805
\(836\) 0 0
\(837\) −7020.00 −0.289900
\(838\) 0 0
\(839\) −39576.0 −1.62850 −0.814252 0.580511i \(-0.802853\pi\)
−0.814252 + 0.580511i \(0.802853\pi\)
\(840\) 0 0
\(841\) 24895.0 1.02075
\(842\) 0 0
\(843\) 11394.0 0.465516
\(844\) 0 0
\(845\) −1014.00 −0.0412813
\(846\) 0 0
\(847\) −140.000 −0.00567941
\(848\) 0 0
\(849\) −10716.0 −0.433183
\(850\) 0 0
\(851\) 10176.0 0.409905
\(852\) 0 0
\(853\) −6922.00 −0.277848 −0.138924 0.990303i \(-0.544364\pi\)
−0.138924 + 0.990303i \(0.544364\pi\)
\(854\) 0 0
\(855\) 3024.00 0.120957
\(856\) 0 0
\(857\) 48162.0 1.91970 0.959850 0.280514i \(-0.0905050\pi\)
0.959850 + 0.280514i \(0.0905050\pi\)
\(858\) 0 0
\(859\) 27652.0 1.09834 0.549170 0.835711i \(-0.314944\pi\)
0.549170 + 0.835711i \(0.314944\pi\)
\(860\) 0 0
\(861\) −1080.00 −0.0427483
\(862\) 0 0
\(863\) −648.000 −0.0255599 −0.0127799 0.999918i \(-0.504068\pi\)
−0.0127799 + 0.999918i \(0.504068\pi\)
\(864\) 0 0
\(865\) 1836.00 0.0721686
\(866\) 0 0
\(867\) −1671.00 −0.0654558
\(868\) 0 0
\(869\) 7200.00 0.281062
\(870\) 0 0
\(871\) 3328.00 0.129466
\(872\) 0 0
\(873\) −4518.00 −0.175156
\(874\) 0 0
\(875\) 5136.00 0.198433
\(876\) 0 0
\(877\) 7166.00 0.275916 0.137958 0.990438i \(-0.455946\pi\)
0.137958 + 0.990438i \(0.455946\pi\)
\(878\) 0 0
\(879\) 21366.0 0.819860
\(880\) 0 0
\(881\) −37062.0 −1.41731 −0.708655 0.705555i \(-0.750698\pi\)
−0.708655 + 0.705555i \(0.750698\pi\)
\(882\) 0 0
\(883\) −24716.0 −0.941970 −0.470985 0.882141i \(-0.656101\pi\)
−0.470985 + 0.882141i \(0.656101\pi\)
\(884\) 0 0
\(885\) 6264.00 0.237923
\(886\) 0 0
\(887\) −48672.0 −1.84244 −0.921221 0.389040i \(-0.872807\pi\)
−0.921221 + 0.389040i \(0.872807\pi\)
\(888\) 0 0
\(889\) −1856.00 −0.0700205
\(890\) 0 0
\(891\) −2916.00 −0.109640
\(892\) 0 0
\(893\) 9408.00 0.352550
\(894\) 0 0
\(895\) −12312.0 −0.459827
\(896\) 0 0
\(897\) −3744.00 −0.139363
\(898\) 0 0
\(899\) −57720.0 −2.14135
\(900\) 0 0
\(901\) 1980.00 0.0732113
\(902\) 0 0
\(903\) −528.000 −0.0194582
\(904\) 0 0
\(905\) 26988.0 0.991283
\(906\) 0 0
\(907\) 9484.00 0.347201 0.173600 0.984816i \(-0.444460\pi\)
0.173600 + 0.984816i \(0.444460\pi\)
\(908\) 0 0
\(909\) 9558.00 0.348756
\(910\) 0 0
\(911\) −12792.0 −0.465223 −0.232611 0.972570i \(-0.574727\pi\)
−0.232611 + 0.972570i \(0.574727\pi\)
\(912\) 0 0
\(913\) 44496.0 1.61293
\(914\) 0 0
\(915\) 6228.00 0.225018
\(916\) 0 0
\(917\) −6192.00 −0.222986
\(918\) 0 0
\(919\) 18592.0 0.667349 0.333674 0.942688i \(-0.391711\pi\)
0.333674 + 0.942688i \(0.391711\pi\)
\(920\) 0 0
\(921\) 20568.0 0.735873
\(922\) 0 0
\(923\) 2184.00 0.0778843
\(924\) 0 0
\(925\) 9434.00 0.335338
\(926\) 0 0
\(927\) 576.000 0.0204081
\(928\) 0 0
\(929\) −15378.0 −0.543096 −0.271548 0.962425i \(-0.587536\pi\)
−0.271548 + 0.962425i \(0.587536\pi\)
\(930\) 0 0
\(931\) 18312.0 0.644631
\(932\) 0 0
\(933\) 26496.0 0.929732
\(934\) 0 0
\(935\) 14256.0 0.498632
\(936\) 0 0
\(937\) −37078.0 −1.29273 −0.646364 0.763030i \(-0.723711\pi\)
−0.646364 + 0.763030i \(0.723711\pi\)
\(938\) 0 0
\(939\) 10878.0 0.378051
\(940\) 0 0
\(941\) 10842.0 0.375599 0.187800 0.982207i \(-0.439864\pi\)
0.187800 + 0.982207i \(0.439864\pi\)
\(942\) 0 0
\(943\) 8640.00 0.298364
\(944\) 0 0
\(945\) −648.000 −0.0223063
\(946\) 0 0
\(947\) −41508.0 −1.42432 −0.712159 0.702018i \(-0.752282\pi\)
−0.712159 + 0.702018i \(0.752282\pi\)
\(948\) 0 0
\(949\) −10582.0 −0.361967
\(950\) 0 0
\(951\) 30438.0 1.03788
\(952\) 0 0
\(953\) 38706.0 1.31565 0.657823 0.753173i \(-0.271478\pi\)
0.657823 + 0.753173i \(0.271478\pi\)
\(954\) 0 0
\(955\) −24336.0 −0.824602
\(956\) 0 0
\(957\) −23976.0 −0.809858
\(958\) 0 0
\(959\) 1176.00 0.0395986
\(960\) 0 0
\(961\) 37809.0 1.26914
\(962\) 0 0
\(963\) 3996.00 0.133717
\(964\) 0 0
\(965\) 12372.0 0.412714
\(966\) 0 0
\(967\) 24388.0 0.811029 0.405515 0.914089i \(-0.367092\pi\)
0.405515 + 0.914089i \(0.367092\pi\)
\(968\) 0 0
\(969\) −11088.0 −0.367593
\(970\) 0 0
\(971\) 14100.0 0.466005 0.233002 0.972476i \(-0.425145\pi\)
0.233002 + 0.972476i \(0.425145\pi\)
\(972\) 0 0
\(973\) −10256.0 −0.337916
\(974\) 0 0
\(975\) −3471.00 −0.114011
\(976\) 0 0
\(977\) 44838.0 1.46826 0.734132 0.679006i \(-0.237589\pi\)
0.734132 + 0.679006i \(0.237589\pi\)
\(978\) 0 0
\(979\) −11448.0 −0.373728
\(980\) 0 0
\(981\) 12438.0 0.404806
\(982\) 0 0
\(983\) −13176.0 −0.427517 −0.213758 0.976887i \(-0.568571\pi\)
−0.213758 + 0.976887i \(0.568571\pi\)
\(984\) 0 0
\(985\) 26244.0 0.848937
\(986\) 0 0
\(987\) −2016.00 −0.0650152
\(988\) 0 0
\(989\) 4224.00 0.135809
\(990\) 0 0
\(991\) 43648.0 1.39912 0.699558 0.714576i \(-0.253380\pi\)
0.699558 + 0.714576i \(0.253380\pi\)
\(992\) 0 0
\(993\) −19608.0 −0.626627
\(994\) 0 0
\(995\) −15216.0 −0.484804
\(996\) 0 0
\(997\) 62750.0 1.99329 0.996646 0.0818317i \(-0.0260770\pi\)
0.996646 + 0.0818317i \(0.0260770\pi\)
\(998\) 0 0
\(999\) −2862.00 −0.0906403
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 624.4.a.h.1.1 1
3.2 odd 2 1872.4.a.j.1.1 1
4.3 odd 2 156.4.a.a.1.1 1
8.3 odd 2 2496.4.a.n.1.1 1
8.5 even 2 2496.4.a.e.1.1 1
12.11 even 2 468.4.a.b.1.1 1
52.31 even 4 2028.4.b.a.337.2 2
52.47 even 4 2028.4.b.a.337.1 2
52.51 odd 2 2028.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.4.a.a.1.1 1 4.3 odd 2
468.4.a.b.1.1 1 12.11 even 2
624.4.a.h.1.1 1 1.1 even 1 trivial
1872.4.a.j.1.1 1 3.2 odd 2
2028.4.a.a.1.1 1 52.51 odd 2
2028.4.b.a.337.1 2 52.47 even 4
2028.4.b.a.337.2 2 52.31 even 4
2496.4.a.e.1.1 1 8.5 even 2
2496.4.a.n.1.1 1 8.3 odd 2