Properties

Label 624.4.a.e.1.1
Level $624$
Weight $4$
Character 624.1
Self dual yes
Analytic conductor $36.817$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
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)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -20.0000 q^{5} +32.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -20.0000 q^{5} +32.0000 q^{7} +9.00000 q^{9} -50.0000 q^{11} -13.0000 q^{13} -60.0000 q^{15} -30.0000 q^{17} +120.000 q^{19} +96.0000 q^{21} +20.0000 q^{23} +275.000 q^{25} +27.0000 q^{27} +82.0000 q^{29} +44.0000 q^{31} -150.000 q^{33} -640.000 q^{35} -306.000 q^{37} -39.0000 q^{39} +108.000 q^{41} +356.000 q^{43} -180.000 q^{45} +178.000 q^{47} +681.000 q^{49} -90.0000 q^{51} +198.000 q^{53} +1000.00 q^{55} +360.000 q^{57} -94.0000 q^{59} -62.0000 q^{61} +288.000 q^{63} +260.000 q^{65} +140.000 q^{67} +60.0000 q^{69} +778.000 q^{71} +62.0000 q^{73} +825.000 q^{75} -1600.00 q^{77} +1096.00 q^{79} +81.0000 q^{81} +462.000 q^{83} +600.000 q^{85} +246.000 q^{87} +1224.00 q^{89} -416.000 q^{91} +132.000 q^{93} -2400.00 q^{95} +614.000 q^{97} -450.000 q^{99} +O(q^{100})\)

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
\(5\) −20.0000 −1.78885 −0.894427 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 0 0
\(7\) 32.0000 1.72784 0.863919 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −50.0000 −1.37051 −0.685253 0.728305i \(-0.740308\pi\)
−0.685253 + 0.728305i \(0.740308\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) −60.0000 −1.03280
\(16\) 0 0
\(17\) −30.0000 −0.428004 −0.214002 0.976833i \(-0.568650\pi\)
−0.214002 + 0.976833i \(0.568650\pi\)
\(18\) 0 0
\(19\) 120.000 1.44894 0.724471 0.689306i \(-0.242084\pi\)
0.724471 + 0.689306i \(0.242084\pi\)
\(20\) 0 0
\(21\) 96.0000 0.997567
\(22\) 0 0
\(23\) 20.0000 0.181317 0.0906584 0.995882i \(-0.471103\pi\)
0.0906584 + 0.995882i \(0.471103\pi\)
\(24\) 0 0
\(25\) 275.000 2.20000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 82.0000 0.525070 0.262535 0.964923i \(-0.415442\pi\)
0.262535 + 0.964923i \(0.415442\pi\)
\(30\) 0 0
\(31\) 44.0000 0.254924 0.127462 0.991843i \(-0.459317\pi\)
0.127462 + 0.991843i \(0.459317\pi\)
\(32\) 0 0
\(33\) −150.000 −0.791262
\(34\) 0 0
\(35\) −640.000 −3.09085
\(36\) 0 0
\(37\) −306.000 −1.35962 −0.679812 0.733386i \(-0.737939\pi\)
−0.679812 + 0.733386i \(0.737939\pi\)
\(38\) 0 0
\(39\) −39.0000 −0.160128
\(40\) 0 0
\(41\) 108.000 0.411385 0.205692 0.978617i \(-0.434055\pi\)
0.205692 + 0.978617i \(0.434055\pi\)
\(42\) 0 0
\(43\) 356.000 1.26255 0.631273 0.775561i \(-0.282533\pi\)
0.631273 + 0.775561i \(0.282533\pi\)
\(44\) 0 0
\(45\) −180.000 −0.596285
\(46\) 0 0
\(47\) 178.000 0.552425 0.276212 0.961097i \(-0.410921\pi\)
0.276212 + 0.961097i \(0.410921\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) −90.0000 −0.247108
\(52\) 0 0
\(53\) 198.000 0.513158 0.256579 0.966523i \(-0.417405\pi\)
0.256579 + 0.966523i \(0.417405\pi\)
\(54\) 0 0
\(55\) 1000.00 2.45164
\(56\) 0 0
\(57\) 360.000 0.836547
\(58\) 0 0
\(59\) −94.0000 −0.207420 −0.103710 0.994608i \(-0.533071\pi\)
−0.103710 + 0.994608i \(0.533071\pi\)
\(60\) 0 0
\(61\) −62.0000 −0.130136 −0.0650679 0.997881i \(-0.520726\pi\)
−0.0650679 + 0.997881i \(0.520726\pi\)
\(62\) 0 0
\(63\) 288.000 0.575946
\(64\) 0 0
\(65\) 260.000 0.496139
\(66\) 0 0
\(67\) 140.000 0.255279 0.127640 0.991821i \(-0.459260\pi\)
0.127640 + 0.991821i \(0.459260\pi\)
\(68\) 0 0
\(69\) 60.0000 0.104683
\(70\) 0 0
\(71\) 778.000 1.30045 0.650223 0.759744i \(-0.274676\pi\)
0.650223 + 0.759744i \(0.274676\pi\)
\(72\) 0 0
\(73\) 62.0000 0.0994048 0.0497024 0.998764i \(-0.484173\pi\)
0.0497024 + 0.998764i \(0.484173\pi\)
\(74\) 0 0
\(75\) 825.000 1.27017
\(76\) 0 0
\(77\) −1600.00 −2.36801
\(78\) 0 0
\(79\) 1096.00 1.56088 0.780441 0.625230i \(-0.214995\pi\)
0.780441 + 0.625230i \(0.214995\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 462.000 0.610977 0.305488 0.952196i \(-0.401180\pi\)
0.305488 + 0.952196i \(0.401180\pi\)
\(84\) 0 0
\(85\) 600.000 0.765637
\(86\) 0 0
\(87\) 246.000 0.303149
\(88\) 0 0
\(89\) 1224.00 1.45779 0.728897 0.684623i \(-0.240033\pi\)
0.728897 + 0.684623i \(0.240033\pi\)
\(90\) 0 0
\(91\) −416.000 −0.479216
\(92\) 0 0
\(93\) 132.000 0.147180
\(94\) 0 0
\(95\) −2400.00 −2.59195
\(96\) 0 0
\(97\) 614.000 0.642704 0.321352 0.946960i \(-0.395863\pi\)
0.321352 + 0.946960i \(0.395863\pi\)
\(98\) 0 0
\(99\) −450.000 −0.456835
\(100\) 0 0
\(101\) 1058.00 1.04233 0.521163 0.853457i \(-0.325498\pi\)
0.521163 + 0.853457i \(0.325498\pi\)
\(102\) 0 0
\(103\) −1768.00 −1.69132 −0.845661 0.533720i \(-0.820794\pi\)
−0.845661 + 0.533720i \(0.820794\pi\)
\(104\) 0 0
\(105\) −1920.00 −1.78450
\(106\) 0 0
\(107\) 1808.00 1.63351 0.816757 0.576982i \(-0.195770\pi\)
0.816757 + 0.576982i \(0.195770\pi\)
\(108\) 0 0
\(109\) −1886.00 −1.65730 −0.828652 0.559765i \(-0.810891\pi\)
−0.828652 + 0.559765i \(0.810891\pi\)
\(110\) 0 0
\(111\) −918.000 −0.784979
\(112\) 0 0
\(113\) 1246.00 1.03729 0.518645 0.854990i \(-0.326437\pi\)
0.518645 + 0.854990i \(0.326437\pi\)
\(114\) 0 0
\(115\) −400.000 −0.324349
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) −960.000 −0.739521
\(120\) 0 0
\(121\) 1169.00 0.878287
\(122\) 0 0
\(123\) 324.000 0.237513
\(124\) 0 0
\(125\) −3000.00 −2.14663
\(126\) 0 0
\(127\) −1624.00 −1.13470 −0.567349 0.823477i \(-0.692031\pi\)
−0.567349 + 0.823477i \(0.692031\pi\)
\(128\) 0 0
\(129\) 1068.00 0.728931
\(130\) 0 0
\(131\) 2072.00 1.38192 0.690960 0.722893i \(-0.257188\pi\)
0.690960 + 0.722893i \(0.257188\pi\)
\(132\) 0 0
\(133\) 3840.00 2.50354
\(134\) 0 0
\(135\) −540.000 −0.344265
\(136\) 0 0
\(137\) −756.000 −0.471456 −0.235728 0.971819i \(-0.575747\pi\)
−0.235728 + 0.971819i \(0.575747\pi\)
\(138\) 0 0
\(139\) −172.000 −0.104956 −0.0524779 0.998622i \(-0.516712\pi\)
−0.0524779 + 0.998622i \(0.516712\pi\)
\(140\) 0 0
\(141\) 534.000 0.318943
\(142\) 0 0
\(143\) 650.000 0.380110
\(144\) 0 0
\(145\) −1640.00 −0.939273
\(146\) 0 0
\(147\) 2043.00 1.14628
\(148\) 0 0
\(149\) 1272.00 0.699371 0.349686 0.936867i \(-0.386288\pi\)
0.349686 + 0.936867i \(0.386288\pi\)
\(150\) 0 0
\(151\) −1404.00 −0.756662 −0.378331 0.925670i \(-0.623502\pi\)
−0.378331 + 0.925670i \(0.623502\pi\)
\(152\) 0 0
\(153\) −270.000 −0.142668
\(154\) 0 0
\(155\) −880.000 −0.456021
\(156\) 0 0
\(157\) −2170.00 −1.10309 −0.551544 0.834146i \(-0.685961\pi\)
−0.551544 + 0.834146i \(0.685961\pi\)
\(158\) 0 0
\(159\) 594.000 0.296272
\(160\) 0 0
\(161\) 640.000 0.313286
\(162\) 0 0
\(163\) −248.000 −0.119171 −0.0595855 0.998223i \(-0.518978\pi\)
−0.0595855 + 0.998223i \(0.518978\pi\)
\(164\) 0 0
\(165\) 3000.00 1.41545
\(166\) 0 0
\(167\) −102.000 −0.0472635 −0.0236317 0.999721i \(-0.507523\pi\)
−0.0236317 + 0.999721i \(0.507523\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 1080.00 0.482980
\(172\) 0 0
\(173\) 682.000 0.299720 0.149860 0.988707i \(-0.452118\pi\)
0.149860 + 0.988707i \(0.452118\pi\)
\(174\) 0 0
\(175\) 8800.00 3.80124
\(176\) 0 0
\(177\) −282.000 −0.119754
\(178\) 0 0
\(179\) 612.000 0.255548 0.127774 0.991803i \(-0.459217\pi\)
0.127774 + 0.991803i \(0.459217\pi\)
\(180\) 0 0
\(181\) −66.0000 −0.0271035 −0.0135518 0.999908i \(-0.504314\pi\)
−0.0135518 + 0.999908i \(0.504314\pi\)
\(182\) 0 0
\(183\) −186.000 −0.0751340
\(184\) 0 0
\(185\) 6120.00 2.43217
\(186\) 0 0
\(187\) 1500.00 0.586582
\(188\) 0 0
\(189\) 864.000 0.332522
\(190\) 0 0
\(191\) −608.000 −0.230332 −0.115166 0.993346i \(-0.536740\pi\)
−0.115166 + 0.993346i \(0.536740\pi\)
\(192\) 0 0
\(193\) 1370.00 0.510957 0.255479 0.966815i \(-0.417767\pi\)
0.255479 + 0.966815i \(0.417767\pi\)
\(194\) 0 0
\(195\) 780.000 0.286446
\(196\) 0 0
\(197\) −4908.00 −1.77503 −0.887514 0.460781i \(-0.847569\pi\)
−0.887514 + 0.460781i \(0.847569\pi\)
\(198\) 0 0
\(199\) 328.000 0.116841 0.0584204 0.998292i \(-0.481394\pi\)
0.0584204 + 0.998292i \(0.481394\pi\)
\(200\) 0 0
\(201\) 420.000 0.147386
\(202\) 0 0
\(203\) 2624.00 0.907235
\(204\) 0 0
\(205\) −2160.00 −0.735907
\(206\) 0 0
\(207\) 180.000 0.0604390
\(208\) 0 0
\(209\) −6000.00 −1.98578
\(210\) 0 0
\(211\) −1316.00 −0.429371 −0.214685 0.976683i \(-0.568873\pi\)
−0.214685 + 0.976683i \(0.568873\pi\)
\(212\) 0 0
\(213\) 2334.00 0.750812
\(214\) 0 0
\(215\) −7120.00 −2.25851
\(216\) 0 0
\(217\) 1408.00 0.440467
\(218\) 0 0
\(219\) 186.000 0.0573914
\(220\) 0 0
\(221\) 390.000 0.118707
\(222\) 0 0
\(223\) 1932.00 0.580163 0.290081 0.957002i \(-0.406318\pi\)
0.290081 + 0.957002i \(0.406318\pi\)
\(224\) 0 0
\(225\) 2475.00 0.733333
\(226\) 0 0
\(227\) −4998.00 −1.46136 −0.730680 0.682720i \(-0.760797\pi\)
−0.730680 + 0.682720i \(0.760797\pi\)
\(228\) 0 0
\(229\) −78.0000 −0.0225082 −0.0112541 0.999937i \(-0.503582\pi\)
−0.0112541 + 0.999937i \(0.503582\pi\)
\(230\) 0 0
\(231\) −4800.00 −1.36717
\(232\) 0 0
\(233\) −1282.00 −0.360458 −0.180229 0.983625i \(-0.557684\pi\)
−0.180229 + 0.983625i \(0.557684\pi\)
\(234\) 0 0
\(235\) −3560.00 −0.988208
\(236\) 0 0
\(237\) 3288.00 0.901175
\(238\) 0 0
\(239\) −294.000 −0.0795702 −0.0397851 0.999208i \(-0.512667\pi\)
−0.0397851 + 0.999208i \(0.512667\pi\)
\(240\) 0 0
\(241\) −4962.00 −1.32627 −0.663134 0.748501i \(-0.730774\pi\)
−0.663134 + 0.748501i \(0.730774\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) −13620.0 −3.55163
\(246\) 0 0
\(247\) −1560.00 −0.401864
\(248\) 0 0
\(249\) 1386.00 0.352748
\(250\) 0 0
\(251\) −744.000 −0.187095 −0.0935475 0.995615i \(-0.529821\pi\)
−0.0935475 + 0.995615i \(0.529821\pi\)
\(252\) 0 0
\(253\) −1000.00 −0.248496
\(254\) 0 0
\(255\) 1800.00 0.442041
\(256\) 0 0
\(257\) −1026.00 −0.249028 −0.124514 0.992218i \(-0.539737\pi\)
−0.124514 + 0.992218i \(0.539737\pi\)
\(258\) 0 0
\(259\) −9792.00 −2.34921
\(260\) 0 0
\(261\) 738.000 0.175023
\(262\) 0 0
\(263\) 5532.00 1.29703 0.648513 0.761204i \(-0.275391\pi\)
0.648513 + 0.761204i \(0.275391\pi\)
\(264\) 0 0
\(265\) −3960.00 −0.917966
\(266\) 0 0
\(267\) 3672.00 0.841658
\(268\) 0 0
\(269\) −3534.00 −0.801010 −0.400505 0.916294i \(-0.631165\pi\)
−0.400505 + 0.916294i \(0.631165\pi\)
\(270\) 0 0
\(271\) −2392.00 −0.536176 −0.268088 0.963394i \(-0.586392\pi\)
−0.268088 + 0.963394i \(0.586392\pi\)
\(272\) 0 0
\(273\) −1248.00 −0.276675
\(274\) 0 0
\(275\) −13750.0 −3.01511
\(276\) 0 0
\(277\) 6102.00 1.32359 0.661794 0.749686i \(-0.269796\pi\)
0.661794 + 0.749686i \(0.269796\pi\)
\(278\) 0 0
\(279\) 396.000 0.0849746
\(280\) 0 0
\(281\) −7540.00 −1.60071 −0.800354 0.599528i \(-0.795355\pi\)
−0.800354 + 0.599528i \(0.795355\pi\)
\(282\) 0 0
\(283\) 2756.00 0.578895 0.289447 0.957194i \(-0.406528\pi\)
0.289447 + 0.957194i \(0.406528\pi\)
\(284\) 0 0
\(285\) −7200.00 −1.49646
\(286\) 0 0
\(287\) 3456.00 0.710806
\(288\) 0 0
\(289\) −4013.00 −0.816813
\(290\) 0 0
\(291\) 1842.00 0.371065
\(292\) 0 0
\(293\) 968.000 0.193007 0.0965037 0.995333i \(-0.469234\pi\)
0.0965037 + 0.995333i \(0.469234\pi\)
\(294\) 0 0
\(295\) 1880.00 0.371043
\(296\) 0 0
\(297\) −1350.00 −0.263754
\(298\) 0 0
\(299\) −260.000 −0.0502883
\(300\) 0 0
\(301\) 11392.0 2.18147
\(302\) 0 0
\(303\) 3174.00 0.601787
\(304\) 0 0
\(305\) 1240.00 0.232794
\(306\) 0 0
\(307\) 6436.00 1.19649 0.598244 0.801314i \(-0.295865\pi\)
0.598244 + 0.801314i \(0.295865\pi\)
\(308\) 0 0
\(309\) −5304.00 −0.976485
\(310\) 0 0
\(311\) −7932.00 −1.44625 −0.723123 0.690719i \(-0.757294\pi\)
−0.723123 + 0.690719i \(0.757294\pi\)
\(312\) 0 0
\(313\) 10358.0 1.87051 0.935254 0.353978i \(-0.115171\pi\)
0.935254 + 0.353978i \(0.115171\pi\)
\(314\) 0 0
\(315\) −5760.00 −1.03028
\(316\) 0 0
\(317\) −2820.00 −0.499643 −0.249822 0.968292i \(-0.580372\pi\)
−0.249822 + 0.968292i \(0.580372\pi\)
\(318\) 0 0
\(319\) −4100.00 −0.719611
\(320\) 0 0
\(321\) 5424.00 0.943110
\(322\) 0 0
\(323\) −3600.00 −0.620153
\(324\) 0 0
\(325\) −3575.00 −0.610170
\(326\) 0 0
\(327\) −5658.00 −0.956844
\(328\) 0 0
\(329\) 5696.00 0.954500
\(330\) 0 0
\(331\) 4180.00 0.694120 0.347060 0.937843i \(-0.387180\pi\)
0.347060 + 0.937843i \(0.387180\pi\)
\(332\) 0 0
\(333\) −2754.00 −0.453208
\(334\) 0 0
\(335\) −2800.00 −0.456658
\(336\) 0 0
\(337\) −5026.00 −0.812414 −0.406207 0.913781i \(-0.633149\pi\)
−0.406207 + 0.913781i \(0.633149\pi\)
\(338\) 0 0
\(339\) 3738.00 0.598880
\(340\) 0 0
\(341\) −2200.00 −0.349374
\(342\) 0 0
\(343\) 10816.0 1.70265
\(344\) 0 0
\(345\) −1200.00 −0.187263
\(346\) 0 0
\(347\) 7332.00 1.13430 0.567150 0.823614i \(-0.308046\pi\)
0.567150 + 0.823614i \(0.308046\pi\)
\(348\) 0 0
\(349\) −8162.00 −1.25187 −0.625934 0.779876i \(-0.715282\pi\)
−0.625934 + 0.779876i \(0.715282\pi\)
\(350\) 0 0
\(351\) −351.000 −0.0533761
\(352\) 0 0
\(353\) 1244.00 0.187568 0.0937839 0.995593i \(-0.470104\pi\)
0.0937839 + 0.995593i \(0.470104\pi\)
\(354\) 0 0
\(355\) −15560.0 −2.32631
\(356\) 0 0
\(357\) −2880.00 −0.426963
\(358\) 0 0
\(359\) −9558.00 −1.40516 −0.702579 0.711605i \(-0.747968\pi\)
−0.702579 + 0.711605i \(0.747968\pi\)
\(360\) 0 0
\(361\) 7541.00 1.09943
\(362\) 0 0
\(363\) 3507.00 0.507079
\(364\) 0 0
\(365\) −1240.00 −0.177821
\(366\) 0 0
\(367\) 11032.0 1.56912 0.784558 0.620055i \(-0.212890\pi\)
0.784558 + 0.620055i \(0.212890\pi\)
\(368\) 0 0
\(369\) 972.000 0.137128
\(370\) 0 0
\(371\) 6336.00 0.886654
\(372\) 0 0
\(373\) 5474.00 0.759874 0.379937 0.925012i \(-0.375946\pi\)
0.379937 + 0.925012i \(0.375946\pi\)
\(374\) 0 0
\(375\) −9000.00 −1.23935
\(376\) 0 0
\(377\) −1066.00 −0.145628
\(378\) 0 0
\(379\) 7040.00 0.954144 0.477072 0.878864i \(-0.341698\pi\)
0.477072 + 0.878864i \(0.341698\pi\)
\(380\) 0 0
\(381\) −4872.00 −0.655118
\(382\) 0 0
\(383\) 1830.00 0.244148 0.122074 0.992521i \(-0.461045\pi\)
0.122074 + 0.992521i \(0.461045\pi\)
\(384\) 0 0
\(385\) 32000.0 4.23603
\(386\) 0 0
\(387\) 3204.00 0.420849
\(388\) 0 0
\(389\) 10158.0 1.32399 0.661994 0.749509i \(-0.269711\pi\)
0.661994 + 0.749509i \(0.269711\pi\)
\(390\) 0 0
\(391\) −600.000 −0.0776044
\(392\) 0 0
\(393\) 6216.00 0.797852
\(394\) 0 0
\(395\) −21920.0 −2.79219
\(396\) 0 0
\(397\) −12658.0 −1.60022 −0.800109 0.599854i \(-0.795225\pi\)
−0.800109 + 0.599854i \(0.795225\pi\)
\(398\) 0 0
\(399\) 11520.0 1.44542
\(400\) 0 0
\(401\) 15720.0 1.95765 0.978827 0.204689i \(-0.0656182\pi\)
0.978827 + 0.204689i \(0.0656182\pi\)
\(402\) 0 0
\(403\) −572.000 −0.0707031
\(404\) 0 0
\(405\) −1620.00 −0.198762
\(406\) 0 0
\(407\) 15300.0 1.86337
\(408\) 0 0
\(409\) 7654.00 0.925345 0.462672 0.886529i \(-0.346891\pi\)
0.462672 + 0.886529i \(0.346891\pi\)
\(410\) 0 0
\(411\) −2268.00 −0.272195
\(412\) 0 0
\(413\) −3008.00 −0.358387
\(414\) 0 0
\(415\) −9240.00 −1.09295
\(416\) 0 0
\(417\) −516.000 −0.0605962
\(418\) 0 0
\(419\) 1848.00 0.215467 0.107734 0.994180i \(-0.465641\pi\)
0.107734 + 0.994180i \(0.465641\pi\)
\(420\) 0 0
\(421\) −12542.0 −1.45192 −0.725962 0.687735i \(-0.758605\pi\)
−0.725962 + 0.687735i \(0.758605\pi\)
\(422\) 0 0
\(423\) 1602.00 0.184142
\(424\) 0 0
\(425\) −8250.00 −0.941609
\(426\) 0 0
\(427\) −1984.00 −0.224854
\(428\) 0 0
\(429\) 1950.00 0.219457
\(430\) 0 0
\(431\) 5238.00 0.585396 0.292698 0.956205i \(-0.405447\pi\)
0.292698 + 0.956205i \(0.405447\pi\)
\(432\) 0 0
\(433\) −8258.00 −0.916522 −0.458261 0.888818i \(-0.651528\pi\)
−0.458261 + 0.888818i \(0.651528\pi\)
\(434\) 0 0
\(435\) −4920.00 −0.542290
\(436\) 0 0
\(437\) 2400.00 0.262718
\(438\) 0 0
\(439\) 6304.00 0.685361 0.342681 0.939452i \(-0.388665\pi\)
0.342681 + 0.939452i \(0.388665\pi\)
\(440\) 0 0
\(441\) 6129.00 0.661808
\(442\) 0 0
\(443\) −12744.0 −1.36678 −0.683392 0.730051i \(-0.739496\pi\)
−0.683392 + 0.730051i \(0.739496\pi\)
\(444\) 0 0
\(445\) −24480.0 −2.60778
\(446\) 0 0
\(447\) 3816.00 0.403782
\(448\) 0 0
\(449\) −11776.0 −1.23774 −0.618868 0.785495i \(-0.712409\pi\)
−0.618868 + 0.785495i \(0.712409\pi\)
\(450\) 0 0
\(451\) −5400.00 −0.563805
\(452\) 0 0
\(453\) −4212.00 −0.436859
\(454\) 0 0
\(455\) 8320.00 0.857248
\(456\) 0 0
\(457\) 2134.00 0.218434 0.109217 0.994018i \(-0.465166\pi\)
0.109217 + 0.994018i \(0.465166\pi\)
\(458\) 0 0
\(459\) −810.000 −0.0823694
\(460\) 0 0
\(461\) 2724.00 0.275205 0.137602 0.990488i \(-0.456060\pi\)
0.137602 + 0.990488i \(0.456060\pi\)
\(462\) 0 0
\(463\) 5648.00 0.566922 0.283461 0.958984i \(-0.408517\pi\)
0.283461 + 0.958984i \(0.408517\pi\)
\(464\) 0 0
\(465\) −2640.00 −0.263284
\(466\) 0 0
\(467\) 18224.0 1.80579 0.902897 0.429856i \(-0.141436\pi\)
0.902897 + 0.429856i \(0.141436\pi\)
\(468\) 0 0
\(469\) 4480.00 0.441081
\(470\) 0 0
\(471\) −6510.00 −0.636868
\(472\) 0 0
\(473\) −17800.0 −1.73033
\(474\) 0 0
\(475\) 33000.0 3.18767
\(476\) 0 0
\(477\) 1782.00 0.171053
\(478\) 0 0
\(479\) −9066.00 −0.864794 −0.432397 0.901683i \(-0.642332\pi\)
−0.432397 + 0.901683i \(0.642332\pi\)
\(480\) 0 0
\(481\) 3978.00 0.377092
\(482\) 0 0
\(483\) 1920.00 0.180876
\(484\) 0 0
\(485\) −12280.0 −1.14970
\(486\) 0 0
\(487\) −8948.00 −0.832593 −0.416296 0.909229i \(-0.636672\pi\)
−0.416296 + 0.909229i \(0.636672\pi\)
\(488\) 0 0
\(489\) −744.000 −0.0688034
\(490\) 0 0
\(491\) −8720.00 −0.801483 −0.400741 0.916191i \(-0.631247\pi\)
−0.400741 + 0.916191i \(0.631247\pi\)
\(492\) 0 0
\(493\) −2460.00 −0.224732
\(494\) 0 0
\(495\) 9000.00 0.817212
\(496\) 0 0
\(497\) 24896.0 2.24696
\(498\) 0 0
\(499\) −6604.00 −0.592456 −0.296228 0.955117i \(-0.595729\pi\)
−0.296228 + 0.955117i \(0.595729\pi\)
\(500\) 0 0
\(501\) −306.000 −0.0272876
\(502\) 0 0
\(503\) −3404.00 −0.301743 −0.150872 0.988553i \(-0.548208\pi\)
−0.150872 + 0.988553i \(0.548208\pi\)
\(504\) 0 0
\(505\) −21160.0 −1.86457
\(506\) 0 0
\(507\) 507.000 0.0444116
\(508\) 0 0
\(509\) −76.0000 −0.00661815 −0.00330908 0.999995i \(-0.501053\pi\)
−0.00330908 + 0.999995i \(0.501053\pi\)
\(510\) 0 0
\(511\) 1984.00 0.171755
\(512\) 0 0
\(513\) 3240.00 0.278849
\(514\) 0 0
\(515\) 35360.0 3.02553
\(516\) 0 0
\(517\) −8900.00 −0.757102
\(518\) 0 0
\(519\) 2046.00 0.173043
\(520\) 0 0
\(521\) 12054.0 1.01362 0.506809 0.862058i \(-0.330825\pi\)
0.506809 + 0.862058i \(0.330825\pi\)
\(522\) 0 0
\(523\) −276.000 −0.0230758 −0.0115379 0.999933i \(-0.503673\pi\)
−0.0115379 + 0.999933i \(0.503673\pi\)
\(524\) 0 0
\(525\) 26400.0 2.19465
\(526\) 0 0
\(527\) −1320.00 −0.109108
\(528\) 0 0
\(529\) −11767.0 −0.967124
\(530\) 0 0
\(531\) −846.000 −0.0691399
\(532\) 0 0
\(533\) −1404.00 −0.114098
\(534\) 0 0
\(535\) −36160.0 −2.92212
\(536\) 0 0
\(537\) 1836.00 0.147540
\(538\) 0 0
\(539\) −34050.0 −2.72103
\(540\) 0 0
\(541\) 13778.0 1.09494 0.547470 0.836825i \(-0.315591\pi\)
0.547470 + 0.836825i \(0.315591\pi\)
\(542\) 0 0
\(543\) −198.000 −0.0156482
\(544\) 0 0
\(545\) 37720.0 2.96467
\(546\) 0 0
\(547\) 10844.0 0.847634 0.423817 0.905748i \(-0.360690\pi\)
0.423817 + 0.905748i \(0.360690\pi\)
\(548\) 0 0
\(549\) −558.000 −0.0433786
\(550\) 0 0
\(551\) 9840.00 0.760795
\(552\) 0 0
\(553\) 35072.0 2.69695
\(554\) 0 0
\(555\) 18360.0 1.40421
\(556\) 0 0
\(557\) 20544.0 1.56280 0.781398 0.624033i \(-0.214507\pi\)
0.781398 + 0.624033i \(0.214507\pi\)
\(558\) 0 0
\(559\) −4628.00 −0.350167
\(560\) 0 0
\(561\) 4500.00 0.338663
\(562\) 0 0
\(563\) −6988.00 −0.523107 −0.261553 0.965189i \(-0.584235\pi\)
−0.261553 + 0.965189i \(0.584235\pi\)
\(564\) 0 0
\(565\) −24920.0 −1.85556
\(566\) 0 0
\(567\) 2592.00 0.191982
\(568\) 0 0
\(569\) −706.000 −0.0520159 −0.0260080 0.999662i \(-0.508280\pi\)
−0.0260080 + 0.999662i \(0.508280\pi\)
\(570\) 0 0
\(571\) 17532.0 1.28492 0.642462 0.766318i \(-0.277913\pi\)
0.642462 + 0.766318i \(0.277913\pi\)
\(572\) 0 0
\(573\) −1824.00 −0.132982
\(574\) 0 0
\(575\) 5500.00 0.398897
\(576\) 0 0
\(577\) −14814.0 −1.06883 −0.534415 0.845222i \(-0.679468\pi\)
−0.534415 + 0.845222i \(0.679468\pi\)
\(578\) 0 0
\(579\) 4110.00 0.295001
\(580\) 0 0
\(581\) 14784.0 1.05567
\(582\) 0 0
\(583\) −9900.00 −0.703287
\(584\) 0 0
\(585\) 2340.00 0.165380
\(586\) 0 0
\(587\) −14170.0 −0.996352 −0.498176 0.867076i \(-0.665997\pi\)
−0.498176 + 0.867076i \(0.665997\pi\)
\(588\) 0 0
\(589\) 5280.00 0.369369
\(590\) 0 0
\(591\) −14724.0 −1.02481
\(592\) 0 0
\(593\) −11744.0 −0.813269 −0.406634 0.913591i \(-0.633298\pi\)
−0.406634 + 0.913591i \(0.633298\pi\)
\(594\) 0 0
\(595\) 19200.0 1.32290
\(596\) 0 0
\(597\) 984.000 0.0674580
\(598\) 0 0
\(599\) 15076.0 1.02836 0.514181 0.857682i \(-0.328096\pi\)
0.514181 + 0.857682i \(0.328096\pi\)
\(600\) 0 0
\(601\) 20230.0 1.37304 0.686522 0.727109i \(-0.259137\pi\)
0.686522 + 0.727109i \(0.259137\pi\)
\(602\) 0 0
\(603\) 1260.00 0.0850931
\(604\) 0 0
\(605\) −23380.0 −1.57113
\(606\) 0 0
\(607\) 28056.0 1.87604 0.938021 0.346577i \(-0.112656\pi\)
0.938021 + 0.346577i \(0.112656\pi\)
\(608\) 0 0
\(609\) 7872.00 0.523792
\(610\) 0 0
\(611\) −2314.00 −0.153215
\(612\) 0 0
\(613\) 27446.0 1.80837 0.904187 0.427136i \(-0.140478\pi\)
0.904187 + 0.427136i \(0.140478\pi\)
\(614\) 0 0
\(615\) −6480.00 −0.424876
\(616\) 0 0
\(617\) 8804.00 0.574450 0.287225 0.957863i \(-0.407267\pi\)
0.287225 + 0.957863i \(0.407267\pi\)
\(618\) 0 0
\(619\) −3508.00 −0.227784 −0.113892 0.993493i \(-0.536332\pi\)
−0.113892 + 0.993493i \(0.536332\pi\)
\(620\) 0 0
\(621\) 540.000 0.0348945
\(622\) 0 0
\(623\) 39168.0 2.51883
\(624\) 0 0
\(625\) 25625.0 1.64000
\(626\) 0 0
\(627\) −18000.0 −1.14649
\(628\) 0 0
\(629\) 9180.00 0.581925
\(630\) 0 0
\(631\) −22084.0 −1.39326 −0.696632 0.717428i \(-0.745319\pi\)
−0.696632 + 0.717428i \(0.745319\pi\)
\(632\) 0 0
\(633\) −3948.00 −0.247897
\(634\) 0 0
\(635\) 32480.0 2.02981
\(636\) 0 0
\(637\) −8853.00 −0.550657
\(638\) 0 0
\(639\) 7002.00 0.433482
\(640\) 0 0
\(641\) −7342.00 −0.452405 −0.226202 0.974080i \(-0.572631\pi\)
−0.226202 + 0.974080i \(0.572631\pi\)
\(642\) 0 0
\(643\) −2996.00 −0.183749 −0.0918746 0.995771i \(-0.529286\pi\)
−0.0918746 + 0.995771i \(0.529286\pi\)
\(644\) 0 0
\(645\) −21360.0 −1.30395
\(646\) 0 0
\(647\) −9344.00 −0.567775 −0.283888 0.958858i \(-0.591624\pi\)
−0.283888 + 0.958858i \(0.591624\pi\)
\(648\) 0 0
\(649\) 4700.00 0.284270
\(650\) 0 0
\(651\) 4224.00 0.254304
\(652\) 0 0
\(653\) −16686.0 −0.999960 −0.499980 0.866037i \(-0.666659\pi\)
−0.499980 + 0.866037i \(0.666659\pi\)
\(654\) 0 0
\(655\) −41440.0 −2.47205
\(656\) 0 0
\(657\) 558.000 0.0331349
\(658\) 0 0
\(659\) −31356.0 −1.85350 −0.926750 0.375679i \(-0.877410\pi\)
−0.926750 + 0.375679i \(0.877410\pi\)
\(660\) 0 0
\(661\) 590.000 0.0347176 0.0173588 0.999849i \(-0.494474\pi\)
0.0173588 + 0.999849i \(0.494474\pi\)
\(662\) 0 0
\(663\) 1170.00 0.0685355
\(664\) 0 0
\(665\) −76800.0 −4.47846
\(666\) 0 0
\(667\) 1640.00 0.0952040
\(668\) 0 0
\(669\) 5796.00 0.334957
\(670\) 0 0
\(671\) 3100.00 0.178352
\(672\) 0 0
\(673\) 5938.00 0.340109 0.170054 0.985435i \(-0.445606\pi\)
0.170054 + 0.985435i \(0.445606\pi\)
\(674\) 0 0
\(675\) 7425.00 0.423390
\(676\) 0 0
\(677\) 9486.00 0.538518 0.269259 0.963068i \(-0.413221\pi\)
0.269259 + 0.963068i \(0.413221\pi\)
\(678\) 0 0
\(679\) 19648.0 1.11049
\(680\) 0 0
\(681\) −14994.0 −0.843717
\(682\) 0 0
\(683\) −26162.0 −1.46568 −0.732841 0.680400i \(-0.761806\pi\)
−0.732841 + 0.680400i \(0.761806\pi\)
\(684\) 0 0
\(685\) 15120.0 0.843366
\(686\) 0 0
\(687\) −234.000 −0.0129951
\(688\) 0 0
\(689\) −2574.00 −0.142325
\(690\) 0 0
\(691\) 17348.0 0.955064 0.477532 0.878614i \(-0.341532\pi\)
0.477532 + 0.878614i \(0.341532\pi\)
\(692\) 0 0
\(693\) −14400.0 −0.789337
\(694\) 0 0
\(695\) 3440.00 0.187751
\(696\) 0 0
\(697\) −3240.00 −0.176074
\(698\) 0 0
\(699\) −3846.00 −0.208110
\(700\) 0 0
\(701\) 30.0000 0.00161638 0.000808191 1.00000i \(-0.499743\pi\)
0.000808191 1.00000i \(0.499743\pi\)
\(702\) 0 0
\(703\) −36720.0 −1.97002
\(704\) 0 0
\(705\) −10680.0 −0.570542
\(706\) 0 0
\(707\) 33856.0 1.80097
\(708\) 0 0
\(709\) 31466.0 1.66676 0.833378 0.552703i \(-0.186404\pi\)
0.833378 + 0.552703i \(0.186404\pi\)
\(710\) 0 0
\(711\) 9864.00 0.520294
\(712\) 0 0
\(713\) 880.000 0.0462220
\(714\) 0 0
\(715\) −13000.0 −0.679961
\(716\) 0 0
\(717\) −882.000 −0.0459399
\(718\) 0 0
\(719\) 28892.0 1.49859 0.749297 0.662234i \(-0.230391\pi\)
0.749297 + 0.662234i \(0.230391\pi\)
\(720\) 0 0
\(721\) −56576.0 −2.92233
\(722\) 0 0
\(723\) −14886.0 −0.765721
\(724\) 0 0
\(725\) 22550.0 1.15515
\(726\) 0 0
\(727\) −13384.0 −0.682786 −0.341393 0.939921i \(-0.610899\pi\)
−0.341393 + 0.939921i \(0.610899\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10680.0 −0.540375
\(732\) 0 0
\(733\) 7130.00 0.359280 0.179640 0.983732i \(-0.442507\pi\)
0.179640 + 0.983732i \(0.442507\pi\)
\(734\) 0 0
\(735\) −40860.0 −2.05054
\(736\) 0 0
\(737\) −7000.00 −0.349862
\(738\) 0 0
\(739\) 29268.0 1.45689 0.728444 0.685105i \(-0.240244\pi\)
0.728444 + 0.685105i \(0.240244\pi\)
\(740\) 0 0
\(741\) −4680.00 −0.232016
\(742\) 0 0
\(743\) 9898.00 0.488725 0.244362 0.969684i \(-0.421421\pi\)
0.244362 + 0.969684i \(0.421421\pi\)
\(744\) 0 0
\(745\) −25440.0 −1.25107
\(746\) 0 0
\(747\) 4158.00 0.203659
\(748\) 0 0
\(749\) 57856.0 2.82245
\(750\) 0 0
\(751\) 15120.0 0.734669 0.367335 0.930089i \(-0.380270\pi\)
0.367335 + 0.930089i \(0.380270\pi\)
\(752\) 0 0
\(753\) −2232.00 −0.108019
\(754\) 0 0
\(755\) 28080.0 1.35356
\(756\) 0 0
\(757\) −5454.00 −0.261861 −0.130931 0.991392i \(-0.541797\pi\)
−0.130931 + 0.991392i \(0.541797\pi\)
\(758\) 0 0
\(759\) −3000.00 −0.143469
\(760\) 0 0
\(761\) −11988.0 −0.571044 −0.285522 0.958372i \(-0.592167\pi\)
−0.285522 + 0.958372i \(0.592167\pi\)
\(762\) 0 0
\(763\) −60352.0 −2.86355
\(764\) 0 0
\(765\) 5400.00 0.255212
\(766\) 0 0
\(767\) 1222.00 0.0575279
\(768\) 0 0
\(769\) 1338.00 0.0627432 0.0313716 0.999508i \(-0.490012\pi\)
0.0313716 + 0.999508i \(0.490012\pi\)
\(770\) 0 0
\(771\) −3078.00 −0.143776
\(772\) 0 0
\(773\) −14408.0 −0.670401 −0.335200 0.942147i \(-0.608804\pi\)
−0.335200 + 0.942147i \(0.608804\pi\)
\(774\) 0 0
\(775\) 12100.0 0.560832
\(776\) 0 0
\(777\) −29376.0 −1.35632
\(778\) 0 0
\(779\) 12960.0 0.596072
\(780\) 0 0
\(781\) −38900.0 −1.78227
\(782\) 0 0
\(783\) 2214.00 0.101050
\(784\) 0 0
\(785\) 43400.0 1.97326
\(786\) 0 0
\(787\) 10660.0 0.482831 0.241415 0.970422i \(-0.422388\pi\)
0.241415 + 0.970422i \(0.422388\pi\)
\(788\) 0 0
\(789\) 16596.0 0.748838
\(790\) 0 0
\(791\) 39872.0 1.79227
\(792\) 0 0
\(793\) 806.000 0.0360932
\(794\) 0 0
\(795\) −11880.0 −0.529988
\(796\) 0 0
\(797\) 1974.00 0.0877323 0.0438662 0.999037i \(-0.486032\pi\)
0.0438662 + 0.999037i \(0.486032\pi\)
\(798\) 0 0
\(799\) −5340.00 −0.236440
\(800\) 0 0
\(801\) 11016.0 0.485932
\(802\) 0 0
\(803\) −3100.00 −0.136235
\(804\) 0 0
\(805\) −12800.0 −0.560423
\(806\) 0 0
\(807\) −10602.0 −0.462464
\(808\) 0 0
\(809\) 31734.0 1.37912 0.689560 0.724229i \(-0.257804\pi\)
0.689560 + 0.724229i \(0.257804\pi\)
\(810\) 0 0
\(811\) 38824.0 1.68100 0.840502 0.541808i \(-0.182260\pi\)
0.840502 + 0.541808i \(0.182260\pi\)
\(812\) 0 0
\(813\) −7176.00 −0.309561
\(814\) 0 0
\(815\) 4960.00 0.213179
\(816\) 0 0
\(817\) 42720.0 1.82936
\(818\) 0 0
\(819\) −3744.00 −0.159739
\(820\) 0 0
\(821\) −16736.0 −0.711438 −0.355719 0.934593i \(-0.615764\pi\)
−0.355719 + 0.934593i \(0.615764\pi\)
\(822\) 0 0
\(823\) −42096.0 −1.78296 −0.891479 0.453062i \(-0.850332\pi\)
−0.891479 + 0.453062i \(0.850332\pi\)
\(824\) 0 0
\(825\) −41250.0 −1.74078
\(826\) 0 0
\(827\) −24858.0 −1.04522 −0.522610 0.852572i \(-0.675042\pi\)
−0.522610 + 0.852572i \(0.675042\pi\)
\(828\) 0 0
\(829\) 922.000 0.0386277 0.0193139 0.999813i \(-0.493852\pi\)
0.0193139 + 0.999813i \(0.493852\pi\)
\(830\) 0 0
\(831\) 18306.0 0.764173
\(832\) 0 0
\(833\) −20430.0 −0.849769
\(834\) 0 0
\(835\) 2040.00 0.0845474
\(836\) 0 0
\(837\) 1188.00 0.0490601
\(838\) 0 0
\(839\) 14294.0 0.588181 0.294090 0.955778i \(-0.404983\pi\)
0.294090 + 0.955778i \(0.404983\pi\)
\(840\) 0 0
\(841\) −17665.0 −0.724302
\(842\) 0 0
\(843\) −22620.0 −0.924169
\(844\) 0 0
\(845\) −3380.00 −0.137604
\(846\) 0 0
\(847\) 37408.0 1.51754
\(848\) 0 0
\(849\) 8268.00 0.334225
\(850\) 0 0
\(851\) −6120.00 −0.246523
\(852\) 0 0
\(853\) 37966.0 1.52395 0.761976 0.647605i \(-0.224229\pi\)
0.761976 + 0.647605i \(0.224229\pi\)
\(854\) 0 0
\(855\) −21600.0 −0.863982
\(856\) 0 0
\(857\) 39038.0 1.55602 0.778012 0.628249i \(-0.216228\pi\)
0.778012 + 0.628249i \(0.216228\pi\)
\(858\) 0 0
\(859\) −20564.0 −0.816804 −0.408402 0.912802i \(-0.633914\pi\)
−0.408402 + 0.912802i \(0.633914\pi\)
\(860\) 0 0
\(861\) 10368.0 0.410384
\(862\) 0 0
\(863\) −39866.0 −1.57248 −0.786242 0.617918i \(-0.787976\pi\)
−0.786242 + 0.617918i \(0.787976\pi\)
\(864\) 0 0
\(865\) −13640.0 −0.536155
\(866\) 0 0
\(867\) −12039.0 −0.471587
\(868\) 0 0
\(869\) −54800.0 −2.13920
\(870\) 0 0
\(871\) −1820.00 −0.0708018
\(872\) 0 0
\(873\) 5526.00 0.214235
\(874\) 0 0
\(875\) −96000.0 −3.70902
\(876\) 0 0
\(877\) 30990.0 1.19322 0.596612 0.802530i \(-0.296513\pi\)
0.596612 + 0.802530i \(0.296513\pi\)
\(878\) 0 0
\(879\) 2904.00 0.111433
\(880\) 0 0
\(881\) −4458.00 −0.170481 −0.0852405 0.996360i \(-0.527166\pi\)
−0.0852405 + 0.996360i \(0.527166\pi\)
\(882\) 0 0
\(883\) 3164.00 0.120586 0.0602928 0.998181i \(-0.480797\pi\)
0.0602928 + 0.998181i \(0.480797\pi\)
\(884\) 0 0
\(885\) 5640.00 0.214222
\(886\) 0 0
\(887\) 32512.0 1.23072 0.615359 0.788247i \(-0.289011\pi\)
0.615359 + 0.788247i \(0.289011\pi\)
\(888\) 0 0
\(889\) −51968.0 −1.96057
\(890\) 0 0
\(891\) −4050.00 −0.152278
\(892\) 0 0
\(893\) 21360.0 0.800431
\(894\) 0 0
\(895\) −12240.0 −0.457138
\(896\) 0 0
\(897\) −780.000 −0.0290339
\(898\) 0 0
\(899\) 3608.00 0.133853
\(900\) 0 0
\(901\) −5940.00 −0.219634
\(902\) 0 0
\(903\) 34176.0 1.25948
\(904\) 0 0
\(905\) 1320.00 0.0484843
\(906\) 0 0
\(907\) −10500.0 −0.384396 −0.192198 0.981356i \(-0.561562\pi\)
−0.192198 + 0.981356i \(0.561562\pi\)
\(908\) 0 0
\(909\) 9522.00 0.347442
\(910\) 0 0
\(911\) −9840.00 −0.357864 −0.178932 0.983861i \(-0.557264\pi\)
−0.178932 + 0.983861i \(0.557264\pi\)
\(912\) 0 0
\(913\) −23100.0 −0.837348
\(914\) 0 0
\(915\) 3720.00 0.134404
\(916\) 0 0
\(917\) 66304.0 2.38773
\(918\) 0 0
\(919\) 35040.0 1.25774 0.628870 0.777511i \(-0.283518\pi\)
0.628870 + 0.777511i \(0.283518\pi\)
\(920\) 0 0
\(921\) 19308.0 0.690793
\(922\) 0 0
\(923\) −10114.0 −0.360679
\(924\) 0 0
\(925\) −84150.0 −2.99117
\(926\) 0 0
\(927\) −15912.0 −0.563774
\(928\) 0 0
\(929\) 44172.0 1.56000 0.779998 0.625782i \(-0.215220\pi\)
0.779998 + 0.625782i \(0.215220\pi\)
\(930\) 0 0
\(931\) 81720.0 2.87676
\(932\) 0 0
\(933\) −23796.0 −0.834990
\(934\) 0 0
\(935\) −30000.0 −1.04931
\(936\) 0 0
\(937\) −54018.0 −1.88334 −0.941671 0.336535i \(-0.890745\pi\)
−0.941671 + 0.336535i \(0.890745\pi\)
\(938\) 0 0
\(939\) 31074.0 1.07994
\(940\) 0 0
\(941\) −1672.00 −0.0579231 −0.0289616 0.999581i \(-0.509220\pi\)
−0.0289616 + 0.999581i \(0.509220\pi\)
\(942\) 0 0
\(943\) 2160.00 0.0745910
\(944\) 0 0
\(945\) −17280.0 −0.594834
\(946\) 0 0
\(947\) −5238.00 −0.179738 −0.0898691 0.995954i \(-0.528645\pi\)
−0.0898691 + 0.995954i \(0.528645\pi\)
\(948\) 0 0
\(949\) −806.000 −0.0275699
\(950\) 0 0
\(951\) −8460.00 −0.288469
\(952\) 0 0
\(953\) −50042.0 −1.70096 −0.850482 0.526004i \(-0.823690\pi\)
−0.850482 + 0.526004i \(0.823690\pi\)
\(954\) 0 0
\(955\) 12160.0 0.412030
\(956\) 0 0
\(957\) −12300.0 −0.415468
\(958\) 0 0
\(959\) −24192.0 −0.814599
\(960\) 0 0
\(961\) −27855.0 −0.935014
\(962\) 0 0
\(963\) 16272.0 0.544505
\(964\) 0 0
\(965\) −27400.0 −0.914028
\(966\) 0 0
\(967\) −37676.0 −1.25293 −0.626463 0.779452i \(-0.715498\pi\)
−0.626463 + 0.779452i \(0.715498\pi\)
\(968\) 0 0
\(969\) −10800.0 −0.358045
\(970\) 0 0
\(971\) 17364.0 0.573880 0.286940 0.957949i \(-0.407362\pi\)
0.286940 + 0.957949i \(0.407362\pi\)
\(972\) 0 0
\(973\) −5504.00 −0.181346
\(974\) 0 0
\(975\) −10725.0 −0.352282
\(976\) 0 0
\(977\) −14904.0 −0.488046 −0.244023 0.969769i \(-0.578467\pi\)
−0.244023 + 0.969769i \(0.578467\pi\)
\(978\) 0 0
\(979\) −61200.0 −1.99792
\(980\) 0 0
\(981\) −16974.0 −0.552434
\(982\) 0 0
\(983\) 18038.0 0.585272 0.292636 0.956224i \(-0.405467\pi\)
0.292636 + 0.956224i \(0.405467\pi\)
\(984\) 0 0
\(985\) 98160.0 3.17527
\(986\) 0 0
\(987\) 17088.0 0.551081
\(988\) 0 0
\(989\) 7120.00 0.228921
\(990\) 0 0
\(991\) −46176.0 −1.48015 −0.740075 0.672524i \(-0.765210\pi\)
−0.740075 + 0.672524i \(0.765210\pi\)
\(992\) 0 0
\(993\) 12540.0 0.400750
\(994\) 0 0
\(995\) −6560.00 −0.209011
\(996\) 0 0
\(997\) 55838.0 1.77373 0.886864 0.462030i \(-0.152879\pi\)
0.886864 + 0.462030i \(0.152879\pi\)
\(998\) 0 0
\(999\) −8262.00 −0.261660
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.e.1.1 1
3.2 odd 2 1872.4.a.r.1.1 1
4.3 odd 2 78.4.a.d.1.1 1
8.3 odd 2 2496.4.a.r.1.1 1
8.5 even 2 2496.4.a.i.1.1 1
12.11 even 2 234.4.a.f.1.1 1
20.19 odd 2 1950.4.a.h.1.1 1
52.31 even 4 1014.4.b.e.337.1 2
52.47 even 4 1014.4.b.e.337.2 2
52.51 odd 2 1014.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.4.a.d.1.1 1 4.3 odd 2
234.4.a.f.1.1 1 12.11 even 2
624.4.a.e.1.1 1 1.1 even 1 trivial
1014.4.a.d.1.1 1 52.51 odd 2
1014.4.b.e.337.1 2 52.31 even 4
1014.4.b.e.337.2 2 52.47 even 4
1872.4.a.r.1.1 1 3.2 odd 2
1950.4.a.h.1.1 1 20.19 odd 2
2496.4.a.i.1.1 1 8.5 even 2
2496.4.a.r.1.1 1 8.3 odd 2