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Properties

Label 900.4.a.d.1.1
Level $900$
Weight $4$
Character 900.1
Self dual yes
Analytic conductor $53.102$
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] = [900,4,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("900.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(53.1017190052\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 900.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-22.0000 q^{7} +O(q^{10})\) \(q-22.0000 q^{7} +14.0000 q^{11} +30.0000 q^{13} +62.0000 q^{17} -120.000 q^{19} +188.000 q^{23} -96.0000 q^{29} +184.000 q^{31} -406.000 q^{37} -130.000 q^{41} -148.000 q^{43} +448.000 q^{47} +141.000 q^{49} -414.000 q^{53} -266.000 q^{59} -838.000 q^{61} -248.000 q^{67} -1020.00 q^{71} -484.000 q^{73} -308.000 q^{77} -48.0000 q^{79} +548.000 q^{83} +650.000 q^{89} -660.000 q^{91} +1816.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −22.0000 −1.18789 −0.593944 − 0.804506i \(-0.702430\pi\)
−0.593944 + 0.804506i \(0.702430\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.0000 0.383742 0.191871 − 0.981420i \(-0.438545\pi\)
0.191871 + 0.981420i \(0.438545\pi\)
\(12\) 0 0
\(13\) 30.0000 0.640039 0.320019 − 0.947411i \(-0.396311\pi\)
0.320019 + 0.947411i \(0.396311\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 62.0000 0.884542 0.442271 − 0.896882i \(-0.354173\pi\)
0.442271 + 0.896882i \(0.354173\pi\)
\(18\) 0 0
\(19\) −120.000 −1.44894 −0.724471 − 0.689306i \(-0.757916\pi\)
−0.724471 + 0.689306i \(0.757916\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 188.000 1.70438 0.852189 − 0.523234i \(-0.175274\pi\)
0.852189 + 0.523234i \(0.175274\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −96.0000 −0.614716 −0.307358 − 0.951594i \(-0.599445\pi\)
−0.307358 + 0.951594i \(0.599445\pi\)
\(30\) 0 0
\(31\) 184.000 1.06604 0.533022 − 0.846101i \(-0.321056\pi\)
0.533022 + 0.846101i \(0.321056\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −406.000 −1.80395 −0.901973 − 0.431793i \(-0.857881\pi\)
−0.901973 + 0.431793i \(0.857881\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −130.000 −0.495185 −0.247593 − 0.968864i \(-0.579639\pi\)
−0.247593 + 0.968864i \(0.579639\pi\)
\(42\) 0 0
\(43\) −148.000 −0.524879 −0.262439 − 0.964948i \(-0.584527\pi\)
−0.262439 + 0.964948i \(0.584527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 448.000 1.39037 0.695186 − 0.718830i \(-0.255322\pi\)
0.695186 + 0.718830i \(0.255322\pi\)
\(48\) 0 0
\(49\) 141.000 0.411079
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −414.000 −1.07297 −0.536484 − 0.843911i \(-0.680248\pi\)
−0.536484 + 0.843911i \(0.680248\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −266.000 −0.586953 −0.293477 − 0.955966i \(-0.594812\pi\)
−0.293477 + 0.955966i \(0.594812\pi\)
\(60\) 0 0
\(61\) −838.000 −1.75893 −0.879466 − 0.475961i \(-0.842100\pi\)
−0.879466 + 0.475961i \(0.842100\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −248.000 −0.452209 −0.226105 − 0.974103i \(-0.572599\pi\)
−0.226105 + 0.974103i \(0.572599\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1020.00 −1.70495 −0.852477 − 0.522765i \(-0.824901\pi\)
−0.852477 + 0.522765i \(0.824901\pi\)
\(72\) 0 0
\(73\) −484.000 −0.775999 −0.387999 − 0.921660i \(-0.626834\pi\)
−0.387999 + 0.921660i \(0.626834\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −308.000 −0.455842
\(78\) 0 0
\(79\) −48.0000 −0.0683598 −0.0341799 − 0.999416i \(-0.510882\pi\)
−0.0341799 + 0.999416i \(0.510882\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 548.000 0.724709 0.362354 − 0.932040i \(-0.381973\pi\)
0.362354 + 0.932040i \(0.381973\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 650.000 0.774156 0.387078 − 0.922047i \(-0.373484\pi\)
0.387078 + 0.922047i \(0.373484\pi\)
\(90\) 0 0
\(91\) −660.000 −0.760294
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1816.00 1.90090 0.950448 − 0.310884i \(-0.100625\pi\)
0.950448 + 0.310884i \(0.100625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1688.00 −1.66299 −0.831496 − 0.555530i \(-0.812515\pi\)
−0.831496 + 0.555530i \(0.812515\pi\)
\(102\) 0 0
\(103\) 298.000 0.285076 0.142538 − 0.989789i \(-0.454474\pi\)
0.142538 + 0.989789i \(0.454474\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 276.000 0.249364 0.124682 − 0.992197i \(-0.460209\pi\)
0.124682 + 0.992197i \(0.460209\pi\)
\(108\) 0 0
\(109\) 322.000 0.282954 0.141477 − 0.989942i \(-0.454815\pi\)
0.141477 + 0.989942i \(0.454815\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −486.000 −0.404593 −0.202297 − 0.979324i \(-0.564841\pi\)
−0.202297 + 0.979324i \(0.564841\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1364.00 −1.05074
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −502.000 −0.350750 −0.175375 − 0.984502i \(-0.556114\pi\)
−0.175375 + 0.984502i \(0.556114\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.0000 0.0120051 0.00600255 − 0.999982i \(-0.498089\pi\)
0.00600255 + 0.999982i \(0.498089\pi\)
\(132\) 0 0
\(133\) 2640.00 1.72118
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2334.00 −1.45553 −0.727763 − 0.685829i \(-0.759440\pi\)
−0.727763 + 0.685829i \(0.759440\pi\)
\(138\) 0 0
\(139\) −1676.00 −1.02271 −0.511354 − 0.859370i \(-0.670856\pi\)
−0.511354 + 0.859370i \(0.670856\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 420.000 0.245610
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3452.00 −1.89798 −0.948989 − 0.315308i \(-0.897892\pi\)
−0.948989 + 0.315308i \(0.897892\pi\)
\(150\) 0 0
\(151\) 1720.00 0.926964 0.463482 − 0.886106i \(-0.346600\pi\)
0.463482 + 0.886106i \(0.346600\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1246.00 0.633386 0.316693 − 0.948528i \(-0.397427\pi\)
0.316693 + 0.948528i \(0.397427\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4136.00 −2.02461
\(162\) 0 0
\(163\) −1760.00 −0.845729 −0.422865 − 0.906193i \(-0.638975\pi\)
−0.422865 + 0.906193i \(0.638975\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2724.00 −1.26221 −0.631106 − 0.775696i \(-0.717399\pi\)
−0.631106 + 0.775696i \(0.717399\pi\)
\(168\) 0 0
\(169\) −1297.00 −0.590350
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2378.00 −1.04506 −0.522532 − 0.852620i \(-0.675012\pi\)
−0.522532 + 0.852620i \(0.675012\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3094.00 1.29194 0.645968 − 0.763365i \(-0.276454\pi\)
0.645968 + 0.763365i \(0.276454\pi\)
\(180\) 0 0
\(181\) −310.000 −0.127305 −0.0636523 − 0.997972i \(-0.520275\pi\)
−0.0636523 + 0.997972i \(0.520275\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 868.000 0.339436
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 516.000 0.195479 0.0977394 − 0.995212i \(-0.468839\pi\)
0.0977394 + 0.995212i \(0.468839\pi\)
\(192\) 0 0
\(193\) 188.000 0.0701168 0.0350584 − 0.999385i \(-0.488838\pi\)
0.0350584 + 0.999385i \(0.488838\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2578.00 0.932360 0.466180 − 0.884690i \(-0.345630\pi\)
0.466180 + 0.884690i \(0.345630\pi\)
\(198\) 0 0
\(199\) −528.000 −0.188085 −0.0940425 − 0.995568i \(-0.529979\pi\)
−0.0940425 + 0.995568i \(0.529979\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2112.00 0.730213
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1680.00 −0.556019
\(210\) 0 0
\(211\) −3660.00 −1.19415 −0.597073 − 0.802187i \(-0.703670\pi\)
−0.597073 + 0.802187i \(0.703670\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4048.00 −1.26634
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1860.00 0.566141
\(222\) 0 0
\(223\) −2350.00 −0.705684 −0.352842 − 0.935683i \(-0.614785\pi\)
−0.352842 + 0.935683i \(0.614785\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3260.00 0.953189 0.476594 − 0.879123i \(-0.341871\pi\)
0.476594 + 0.879123i \(0.341871\pi\)
\(228\) 0 0
\(229\) −466.000 −0.134472 −0.0672361 − 0.997737i \(-0.521418\pi\)
−0.0672361 + 0.997737i \(0.521418\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3170.00 −0.891303 −0.445652 − 0.895207i \(-0.647028\pi\)
−0.445652 + 0.895207i \(0.647028\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −292.000 −0.0790289 −0.0395145 − 0.999219i \(-0.512581\pi\)
−0.0395145 + 0.999219i \(0.512581\pi\)
\(240\) 0 0
\(241\) −842.000 −0.225054 −0.112527 − 0.993649i \(-0.535894\pi\)
−0.112527 + 0.993649i \(0.535894\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −3600.00 −0.927379
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5838.00 −1.46809 −0.734046 − 0.679099i \(-0.762371\pi\)
−0.734046 + 0.679099i \(0.762371\pi\)
\(252\) 0 0
\(253\) 2632.00 0.654041
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4626.00 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 8932.00 2.14289
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5468.00 1.28202 0.641010 − 0.767532i \(-0.278516\pi\)
0.641010 + 0.767532i \(0.278516\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2976.00 0.674535 0.337268 − 0.941409i \(-0.390497\pi\)
0.337268 + 0.941409i \(0.390497\pi\)
\(270\) 0 0
\(271\) −56.0000 −0.0125526 −0.00627631 − 0.999980i \(-0.501998\pi\)
−0.00627631 + 0.999980i \(0.501998\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4106.00 0.890634 0.445317 − 0.895373i \(-0.353091\pi\)
0.445317 + 0.895373i \(0.353091\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2274.00 0.482760 0.241380 − 0.970431i \(-0.422400\pi\)
0.241380 + 0.970431i \(0.422400\pi\)
\(282\) 0 0
\(283\) −5504.00 −1.15611 −0.578054 − 0.815998i \(-0.696188\pi\)
−0.578054 + 0.815998i \(0.696188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2860.00 0.588225
\(288\) 0 0
\(289\) −1069.00 −0.217586
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8034.00 1.60188 0.800941 − 0.598744i \(-0.204333\pi\)
0.800941 + 0.598744i \(0.204333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5640.00 1.09087
\(300\) 0 0
\(301\) 3256.00 0.623497
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −996.000 −0.185162 −0.0925810 − 0.995705i \(-0.529512\pi\)
−0.0925810 + 0.995705i \(0.529512\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8676.00 1.58190 0.790950 − 0.611881i \(-0.209587\pi\)
0.790950 + 0.611881i \(0.209587\pi\)
\(312\) 0 0
\(313\) −1732.00 −0.312775 −0.156387 − 0.987696i \(-0.549985\pi\)
−0.156387 + 0.987696i \(0.549985\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2938.00 −0.520551 −0.260275 − 0.965534i \(-0.583813\pi\)
−0.260275 + 0.965534i \(0.583813\pi\)
\(318\) 0 0
\(319\) −1344.00 −0.235892
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7440.00 −1.28165
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9856.00 −1.65161
\(330\) 0 0
\(331\) −128.000 −0.0212553 −0.0106277 − 0.999944i \(-0.503383\pi\)
−0.0106277 + 0.999944i \(0.503383\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5596.00 −0.904551 −0.452275 − 0.891878i \(-0.649388\pi\)
−0.452275 + 0.891878i \(0.649388\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2576.00 0.409086
\(342\) 0 0
\(343\) 4444.00 0.699573
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −924.000 −0.142948 −0.0714739 − 0.997442i \(-0.522770\pi\)
−0.0714739 + 0.997442i \(0.522770\pi\)
\(348\) 0 0
\(349\) −6206.00 −0.951861 −0.475931 − 0.879483i \(-0.657889\pi\)
−0.475931 + 0.879483i \(0.657889\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3506.00 −0.528628 −0.264314 − 0.964437i \(-0.585145\pi\)
−0.264314 + 0.964437i \(0.585145\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7104.00 1.04439 0.522193 − 0.852827i \(-0.325114\pi\)
0.522193 + 0.852827i \(0.325114\pi\)
\(360\) 0 0
\(361\) 7541.00 1.09943
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2902.00 −0.412761 −0.206380 − 0.978472i \(-0.566168\pi\)
−0.206380 + 0.978472i \(0.566168\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9108.00 1.27457
\(372\) 0 0
\(373\) 4542.00 0.630498 0.315249 − 0.949009i \(-0.397912\pi\)
0.315249 + 0.949009i \(0.397912\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2880.00 −0.393442
\(378\) 0 0
\(379\) 5852.00 0.793132 0.396566 − 0.918006i \(-0.370202\pi\)
0.396566 + 0.918006i \(0.370202\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8936.00 −1.19219 −0.596094 − 0.802914i \(-0.703282\pi\)
−0.596094 + 0.802914i \(0.703282\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1924.00 −0.250773 −0.125386 − 0.992108i \(-0.540017\pi\)
−0.125386 + 0.992108i \(0.540017\pi\)
\(390\) 0 0
\(391\) 11656.0 1.50759
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9194.00 1.16230 0.581151 − 0.813796i \(-0.302603\pi\)
0.581151 + 0.813796i \(0.302603\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10714.0 1.33424 0.667122 − 0.744949i \(-0.267526\pi\)
0.667122 + 0.744949i \(0.267526\pi\)
\(402\) 0 0
\(403\) 5520.00 0.682310
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5684.00 −0.692249
\(408\) 0 0
\(409\) −12346.0 −1.49259 −0.746296 − 0.665614i \(-0.768170\pi\)
−0.746296 + 0.665614i \(0.768170\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5852.00 0.697235
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13562.0 1.58126 0.790629 − 0.612296i \(-0.209754\pi\)
0.790629 + 0.612296i \(0.209754\pi\)
\(420\) 0 0
\(421\) 5230.00 0.605450 0.302725 − 0.953078i \(-0.402104\pi\)
0.302725 + 0.953078i \(0.402104\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 18436.0 2.08942
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11552.0 1.29104 0.645522 − 0.763741i \(-0.276640\pi\)
0.645522 + 0.763741i \(0.276640\pi\)
\(432\) 0 0
\(433\) −7972.00 −0.884780 −0.442390 − 0.896823i \(-0.645869\pi\)
−0.442390 + 0.896823i \(0.645869\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −22560.0 −2.46954
\(438\) 0 0
\(439\) 7152.00 0.777554 0.388777 − 0.921332i \(-0.372898\pi\)
0.388777 + 0.921332i \(0.372898\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2652.00 0.284425 0.142213 − 0.989836i \(-0.454578\pi\)
0.142213 + 0.989836i \(0.454578\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4542.00 −0.477395 −0.238697 − 0.971094i \(-0.576720\pi\)
−0.238697 + 0.971094i \(0.576720\pi\)
\(450\) 0 0
\(451\) −1820.00 −0.190023
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −16000.0 −1.63774 −0.818871 − 0.573977i \(-0.805400\pi\)
−0.818871 + 0.573977i \(0.805400\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2076.00 −0.209737 −0.104869 − 0.994486i \(-0.533442\pi\)
−0.104869 + 0.994486i \(0.533442\pi\)
\(462\) 0 0
\(463\) 5406.00 0.542631 0.271315 − 0.962490i \(-0.412541\pi\)
0.271315 + 0.962490i \(0.412541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −444.000 −0.0439954 −0.0219977 − 0.999758i \(-0.507003\pi\)
−0.0219977 + 0.999758i \(0.507003\pi\)
\(468\) 0 0
\(469\) 5456.00 0.537174
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2072.00 −0.201418
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1724.00 0.164450 0.0822250 − 0.996614i \(-0.473797\pi\)
0.0822250 + 0.996614i \(0.473797\pi\)
\(480\) 0 0
\(481\) −12180.0 −1.15460
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 17046.0 1.58609 0.793047 − 0.609160i \(-0.208493\pi\)
0.793047 + 0.609160i \(0.208493\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −8814.00 −0.810123 −0.405061 − 0.914290i \(-0.632750\pi\)
−0.405061 + 0.914290i \(0.632750\pi\)
\(492\) 0 0
\(493\) −5952.00 −0.543742
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22440.0 2.02529
\(498\) 0 0
\(499\) 5256.00 0.471525 0.235762 − 0.971811i \(-0.424241\pi\)
0.235762 + 0.971811i \(0.424241\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5232.00 0.463784 0.231892 − 0.972742i \(-0.425508\pi\)
0.231892 + 0.972742i \(0.425508\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4128.00 −0.359470 −0.179735 − 0.983715i \(-0.557524\pi\)
−0.179735 + 0.983715i \(0.557524\pi\)
\(510\) 0 0
\(511\) 10648.0 0.921800
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6272.00 0.533544
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 538.000 0.0452403 0.0226202 − 0.999744i \(-0.492799\pi\)
0.0226202 + 0.999744i \(0.492799\pi\)
\(522\) 0 0
\(523\) 10336.0 0.864172 0.432086 − 0.901833i \(-0.357778\pi\)
0.432086 + 0.901833i \(0.357778\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11408.0 0.942961
\(528\) 0 0
\(529\) 23177.0 1.90491
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3900.00 −0.316938
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1974.00 0.157748
\(540\) 0 0
\(541\) −8942.00 −0.710622 −0.355311 − 0.934748i \(-0.615625\pi\)
−0.355311 + 0.934748i \(0.615625\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −11404.0 −0.891407 −0.445704 − 0.895181i \(-0.647046\pi\)
−0.445704 + 0.895181i \(0.647046\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11520.0 0.890687
\(552\) 0 0
\(553\) 1056.00 0.0812038
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3114.00 0.236884 0.118442 − 0.992961i \(-0.462210\pi\)
0.118442 + 0.992961i \(0.462210\pi\)
\(558\) 0 0
\(559\) −4440.00 −0.335943
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3620.00 −0.270985 −0.135493 − 0.990778i \(-0.543262\pi\)
−0.135493 + 0.990778i \(0.543262\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6506.00 0.479342 0.239671 − 0.970854i \(-0.422960\pi\)
0.239671 + 0.970854i \(0.422960\pi\)
\(570\) 0 0
\(571\) −17600.0 −1.28991 −0.644954 − 0.764222i \(-0.723123\pi\)
−0.644954 + 0.764222i \(0.723123\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4864.00 −0.350938 −0.175469 − 0.984485i \(-0.556144\pi\)
−0.175469 + 0.984485i \(0.556144\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12056.0 −0.860873
\(582\) 0 0
\(583\) −5796.00 −0.411742
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18348.0 −1.29012 −0.645062 − 0.764130i \(-0.723169\pi\)
−0.645062 + 0.764130i \(0.723169\pi\)
\(588\) 0 0
\(589\) −22080.0 −1.54464
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16218.0 1.12309 0.561546 − 0.827446i \(-0.310207\pi\)
0.561546 + 0.827446i \(0.310207\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24576.0 1.67637 0.838187 − 0.545383i \(-0.183616\pi\)
0.838187 + 0.545383i \(0.183616\pi\)
\(600\) 0 0
\(601\) 6578.00 0.446460 0.223230 − 0.974766i \(-0.428340\pi\)
0.223230 + 0.974766i \(0.428340\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −25346.0 −1.69483 −0.847415 − 0.530930i \(-0.821843\pi\)
−0.847415 + 0.530930i \(0.821843\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13440.0 0.889892
\(612\) 0 0
\(613\) −27214.0 −1.79309 −0.896544 − 0.442954i \(-0.853930\pi\)
−0.896544 + 0.442954i \(0.853930\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 22478.0 1.46666 0.733331 − 0.679872i \(-0.237965\pi\)
0.733331 + 0.679872i \(0.237965\pi\)
\(618\) 0 0
\(619\) 3356.00 0.217914 0.108957 − 0.994046i \(-0.465249\pi\)
0.108957 + 0.994046i \(0.465249\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −14300.0 −0.919611
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25172.0 −1.59567
\(630\) 0 0
\(631\) 19952.0 1.25876 0.629379 − 0.777098i \(-0.283309\pi\)
0.629379 + 0.777098i \(0.283309\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4230.00 0.263106
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −26434.0 −1.62883 −0.814415 − 0.580283i \(-0.802942\pi\)
−0.814415 + 0.580283i \(0.802942\pi\)
\(642\) 0 0
\(643\) −10632.0 −0.652076 −0.326038 − 0.945357i \(-0.605714\pi\)
−0.326038 + 0.945357i \(0.605714\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9384.00 −0.570206 −0.285103 − 0.958497i \(-0.592028\pi\)
−0.285103 + 0.958497i \(0.592028\pi\)
\(648\) 0 0
\(649\) −3724.00 −0.225239
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3922.00 −0.235038 −0.117519 − 0.993071i \(-0.537494\pi\)
−0.117519 + 0.993071i \(0.537494\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12542.0 −0.741376 −0.370688 − 0.928757i \(-0.620878\pi\)
−0.370688 + 0.928757i \(0.620878\pi\)
\(660\) 0 0
\(661\) −8662.00 −0.509702 −0.254851 − 0.966980i \(-0.582026\pi\)
−0.254851 + 0.966980i \(0.582026\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −18048.0 −1.04771
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11732.0 −0.674976
\(672\) 0 0
\(673\) −5820.00 −0.333350 −0.166675 − 0.986012i \(-0.553303\pi\)
−0.166675 + 0.986012i \(0.553303\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −778.000 −0.0441669 −0.0220834 − 0.999756i \(-0.507030\pi\)
−0.0220834 + 0.999756i \(0.507030\pi\)
\(678\) 0 0
\(679\) −39952.0 −2.25805
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5548.00 −0.310817 −0.155409 − 0.987850i \(-0.549669\pi\)
−0.155409 + 0.987850i \(0.549669\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12420.0 −0.686741
\(690\) 0 0
\(691\) 5488.00 0.302132 0.151066 − 0.988524i \(-0.451729\pi\)
0.151066 + 0.988524i \(0.451729\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8060.00 −0.438012
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1216.00 −0.0655174 −0.0327587 − 0.999463i \(-0.510429\pi\)
−0.0327587 + 0.999463i \(0.510429\pi\)
\(702\) 0 0
\(703\) 48720.0 2.61381
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 37136.0 1.97545
\(708\) 0 0
\(709\) 20406.0 1.08091 0.540454 − 0.841374i \(-0.318253\pi\)
0.540454 + 0.841374i \(0.318253\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 34592.0 1.81694
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19672.0 −1.02036 −0.510182 − 0.860066i \(-0.670422\pi\)
−0.510182 + 0.860066i \(0.670422\pi\)
\(720\) 0 0
\(721\) −6556.00 −0.338638
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1194.00 −0.0609120 −0.0304560 − 0.999536i \(-0.509696\pi\)
−0.0304560 + 0.999536i \(0.509696\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9176.00 −0.464277
\(732\) 0 0
\(733\) 21802.0 1.09860 0.549301 − 0.835625i \(-0.314894\pi\)
0.549301 + 0.835625i \(0.314894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3472.00 −0.173532
\(738\) 0 0
\(739\) 15280.0 0.760601 0.380300 − 0.924863i \(-0.375821\pi\)
0.380300 + 0.924863i \(0.375821\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6672.00 0.329437 0.164719 − 0.986341i \(-0.447328\pi\)
0.164719 + 0.986341i \(0.447328\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6072.00 −0.296216
\(750\) 0 0
\(751\) −11008.0 −0.534870 −0.267435 − 0.963576i \(-0.586176\pi\)
−0.267435 + 0.963576i \(0.586176\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3242.00 0.155657 0.0778286 − 0.996967i \(-0.475201\pi\)
0.0778286 + 0.996967i \(0.475201\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −17982.0 −0.856566 −0.428283 − 0.903645i \(-0.640881\pi\)
−0.428283 + 0.903645i \(0.640881\pi\)
\(762\) 0 0
\(763\) −7084.00 −0.336118
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7980.00 −0.375673
\(768\) 0 0
\(769\) −30462.0 −1.42846 −0.714231 − 0.699910i \(-0.753224\pi\)
−0.714231 + 0.699910i \(0.753224\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −17394.0 −0.809339 −0.404669 − 0.914463i \(-0.632613\pi\)
−0.404669 + 0.914463i \(0.632613\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 15600.0 0.717494
\(780\) 0 0
\(781\) −14280.0 −0.654262
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 20336.0 0.921093 0.460546 − 0.887636i \(-0.347653\pi\)
0.460546 + 0.887636i \(0.347653\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 10692.0 0.480612
\(792\) 0 0
\(793\) −25140.0 −1.12579
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19546.0 −0.868701 −0.434351 − 0.900744i \(-0.643022\pi\)
−0.434351 + 0.900744i \(0.643022\pi\)
\(798\) 0 0
\(799\) 27776.0 1.22984
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6776.00 −0.297783
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3954.00 −0.171836 −0.0859179 − 0.996302i \(-0.527382\pi\)
−0.0859179 + 0.996302i \(0.527382\pi\)
\(810\) 0 0
\(811\) −3860.00 −0.167131 −0.0835653 − 0.996502i \(-0.526631\pi\)
−0.0835653 + 0.996502i \(0.526631\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 17760.0 0.760519
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −37016.0 −1.57353 −0.786764 − 0.617253i \(-0.788245\pi\)
−0.786764 + 0.617253i \(0.788245\pi\)
\(822\) 0 0
\(823\) −25794.0 −1.09249 −0.546247 − 0.837624i \(-0.683944\pi\)
−0.546247 + 0.837624i \(0.683944\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −44116.0 −1.85497 −0.927487 − 0.373855i \(-0.878036\pi\)
−0.927487 + 0.373855i \(0.878036\pi\)
\(828\) 0 0
\(829\) −11622.0 −0.486910 −0.243455 − 0.969912i \(-0.578281\pi\)
−0.243455 + 0.969912i \(0.578281\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8742.00 0.363616
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2584.00 0.106328 0.0531642 − 0.998586i \(-0.483069\pi\)
0.0531642 + 0.998586i \(0.483069\pi\)
\(840\) 0 0
\(841\) −15173.0 −0.622125
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 24970.0 1.01296
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −76328.0 −3.07461
\(852\) 0 0
\(853\) 39754.0 1.59572 0.797861 − 0.602841i \(-0.205965\pi\)
0.797861 + 0.602841i \(0.205965\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18534.0 0.738751 0.369375 − 0.929280i \(-0.379572\pi\)
0.369375 + 0.929280i \(0.379572\pi\)
\(858\) 0 0
\(859\) −11140.0 −0.442482 −0.221241 − 0.975219i \(-0.571011\pi\)
−0.221241 + 0.975219i \(0.571011\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28356.0 1.11848 0.559241 − 0.829005i \(-0.311093\pi\)
0.559241 + 0.829005i \(0.311093\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −672.000 −0.0262325
\(870\) 0 0
\(871\) −7440.00 −0.289431
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 34942.0 1.34539 0.672695 − 0.739920i \(-0.265137\pi\)
0.672695 + 0.739920i \(0.265137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 13890.0 0.531176 0.265588 − 0.964087i \(-0.414434\pi\)
0.265588 + 0.964087i \(0.414434\pi\)
\(882\) 0 0
\(883\) −11496.0 −0.438133 −0.219066 − 0.975710i \(-0.570301\pi\)
−0.219066 + 0.975710i \(0.570301\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29988.0 −1.13517 −0.567587 − 0.823314i \(-0.692123\pi\)
−0.567587 + 0.823314i \(0.692123\pi\)
\(888\) 0 0
\(889\) 11044.0 0.416652
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −53760.0 −2.01457
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −17664.0 −0.655314
\(900\) 0 0
\(901\) −25668.0 −0.949084
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 16764.0 0.613715 0.306857 − 0.951755i \(-0.400723\pi\)
0.306857 + 0.951755i \(0.400723\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 19152.0 0.696525 0.348262 − 0.937397i \(-0.386772\pi\)
0.348262 + 0.937397i \(0.386772\pi\)
\(912\) 0 0
\(913\) 7672.00 0.278101
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −396.000 −0.0142607
\(918\) 0 0
\(919\) 35712.0 1.28186 0.640930 − 0.767599i \(-0.278549\pi\)
0.640930 + 0.767599i \(0.278549\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −30600.0 −1.09124
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16098.0 0.568523 0.284262 − 0.958747i \(-0.408252\pi\)
0.284262 + 0.958747i \(0.408252\pi\)
\(930\) 0 0
\(931\) −16920.0 −0.595629
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24776.0 −0.863817 −0.431909 − 0.901917i \(-0.642160\pi\)
−0.431909 + 0.901917i \(0.642160\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 39500.0 1.36840 0.684199 − 0.729295i \(-0.260152\pi\)
0.684199 + 0.729295i \(0.260152\pi\)
\(942\) 0 0
\(943\) −24440.0 −0.843983
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 22836.0 0.783601 0.391801 − 0.920050i \(-0.371852\pi\)
0.391801 + 0.920050i \(0.371852\pi\)
\(948\) 0 0
\(949\) −14520.0 −0.496669
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −28478.0 −0.967988 −0.483994 − 0.875071i \(-0.660814\pi\)
−0.483994 + 0.875071i \(0.660814\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 51348.0 1.72900
\(960\) 0 0
\(961\) 4065.00 0.136451
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24830.0 −0.825728 −0.412864 − 0.910793i \(-0.635472\pi\)
−0.412864 + 0.910793i \(0.635472\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −37038.0 −1.22411 −0.612053 − 0.790817i \(-0.709656\pi\)
−0.612053 + 0.790817i \(0.709656\pi\)
\(972\) 0 0
\(973\) 36872.0 1.21486
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −26346.0 −0.862726 −0.431363 − 0.902178i \(-0.641967\pi\)
−0.431363 + 0.902178i \(0.641967\pi\)
\(978\) 0 0
\(979\) 9100.00 0.297076
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11464.0 0.371968 0.185984 − 0.982553i \(-0.440453\pi\)
0.185984 + 0.982553i \(0.440453\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −27824.0 −0.894592
\(990\) 0 0
\(991\) −28952.0 −0.928043 −0.464021 − 0.885824i \(-0.653594\pi\)
−0.464021 + 0.885824i \(0.653594\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8022.00 0.254824 0.127412 − 0.991850i \(-0.459333\pi\)
0.127412 + 0.991850i \(0.459333\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 900.4.a.d.1.1 1
3.2 odd 2 300.4.a.f.1.1 1
5.2 odd 4 180.4.d.b.109.2 2
5.3 odd 4 180.4.d.b.109.1 2
5.4 even 2 900.4.a.o.1.1 1
12.11 even 2 1200.4.a.q.1.1 1
15.2 even 4 60.4.d.a.49.1 ✓ 2
15.8 even 4 60.4.d.a.49.2 yes 2
15.14 odd 2 300.4.a.d.1.1 1
20.3 even 4 720.4.f.h.289.1 2
20.7 even 4 720.4.f.h.289.2 2
60.23 odd 4 240.4.f.a.49.1 2
60.47 odd 4 240.4.f.a.49.2 2
60.59 even 2 1200.4.a.w.1.1 1
120.53 even 4 960.4.f.j.769.1 2
120.77 even 4 960.4.f.j.769.2 2
120.83 odd 4 960.4.f.i.769.2 2
120.107 odd 4 960.4.f.i.769.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.d.a.49.1 ✓ 2 15.2 even 4
60.4.d.a.49.2 yes 2 15.8 even 4
180.4.d.b.109.1 2 5.3 odd 4
180.4.d.b.109.2 2 5.2 odd 4
240.4.f.a.49.1 2 60.23 odd 4
240.4.f.a.49.2 2 60.47 odd 4
300.4.a.d.1.1 1 15.14 odd 2
300.4.a.f.1.1 1 3.2 odd 2
720.4.f.h.289.1 2 20.3 even 4
720.4.f.h.289.2 2 20.7 even 4
900.4.a.d.1.1 1 1.1 even 1 trivial
900.4.a.o.1.1 1 5.4 even 2
960.4.f.i.769.1 2 120.107 odd 4
960.4.f.i.769.2 2 120.83 odd 4
960.4.f.j.769.1 2 120.53 even 4
960.4.f.j.769.2 2 120.77 even 4
1200.4.a.q.1.1 1 12.11 even 2
1200.4.a.w.1.1 1 60.59 even 2