↑

Properties

Label 3840.2.a.m.1.1
Level $3840$
Weight $2$
Character 3840.1
Self dual yes
Analytic conductor $30.663$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3840,2,Mod(1,3840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3840.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3840 = 2^{8} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3840.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: \(30.6625543762\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3840.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{15} -6.00000 q^{17} -4.00000 q^{19} -2.00000 q^{21} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -10.0000 q^{31} -4.00000 q^{33} +2.00000 q^{35} -4.00000 q^{37} -10.0000 q^{41} -4.00000 q^{43} +1.00000 q^{45} +4.00000 q^{47} -3.00000 q^{49} +6.00000 q^{51} +10.0000 q^{53} +4.00000 q^{55} +4.00000 q^{57} +8.00000 q^{59} -8.00000 q^{61} +2.00000 q^{63} +12.0000 q^{67} +4.00000 q^{69} -4.00000 q^{71} -10.0000 q^{73} -1.00000 q^{75} +8.00000 q^{77} +14.0000 q^{79} +1.00000 q^{81} -6.00000 q^{85} +6.00000 q^{87} -14.0000 q^{89} +10.0000 q^{93} -4.00000 q^{95} -10.0000 q^{97} +4.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 − 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 − 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 − 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 − 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −14.0000 −1.48400 −0.741999 − 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 10.0000 1.03695
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 − 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 − 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 − 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.0000 −1.10004
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 − 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −8.00000 −0.693688
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 − 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 − 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −10.0000 −0.803219
\(156\) 0 0
\(157\) 20.0000 1.59617 0.798087 − 0.602542i \(-0.205846\pi\)
0.798087 + 0.602542i \(0.205846\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 − 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 − 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 − 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 4.00000 0.274075
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 − 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 − 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) −14.0000 −0.909398
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 − 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 − 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 0 0
\(255\) 6.00000 0.375735
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 − 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) 14.0000 0.856786
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) 16.0000 0.961347 0.480673 − 0.876900i \(-0.340392\pi\)
0.480673 + 0.876900i \(0.340392\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 − 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 0 0
\(293\) 2.00000 0.116841 0.0584206 − 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) 0 0
\(303\) 14.0000 0.804279
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 2.00000 0.113776
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 − 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −40.0000 −2.16612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 16.0000 0.858925 0.429463 − 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 − 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 12.0000 0.633336 0.316668 − 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 − 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) −20.0000 −1.03556 −0.517780 − 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 − 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) −6.00000 −0.307389
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.0000 0.704416
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 − 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 0 0
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 30.0000 1.49813 0.749064 − 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 0 0
\(423\) 4.00000 0.194487
\(424\) 0 0
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) −16.0000 −0.774294
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 − 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) 16.0000 0.765384
\(438\) 0 0
\(439\) 2.00000 0.0954548 0.0477274 − 0.998860i \(-0.484802\pi\)
0.0477274 + 0.998860i \(0.484802\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 16.0000 0.760183 0.380091 − 0.924949i \(-0.375893\pi\)
0.380091 + 0.924949i \(0.375893\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 − 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 0 0
\(453\) 2.00000 0.0939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 − 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 − 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 0 0
\(465\) 10.0000 0.463739
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 − 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 − 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) −30.0000 −1.35943 −0.679715 − 0.733476i \(-0.737896\pi\)
−0.679715 + 0.733476i \(0.737896\pi\)
\(488\) 0 0
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −40.0000 −1.80517 −0.902587 − 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 − 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) −14.0000 −0.622992
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) 18.0000 0.797836 0.398918 − 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 − 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 − 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 60.0000 2.61364
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 4.00000 0.172935
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) −44.0000 −1.89171 −0.945854 − 0.324593i \(-0.894773\pi\)
−0.945854 + 0.324593i \(0.894773\pi\)
\(542\) 0 0
\(543\) −20.0000 −0.858282
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 28.0000 1.19068
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 − 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −2.00000 −0.0838444 −0.0419222 − 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 0 0
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) 40.0000 1.64817
\(590\) 0 0
\(591\) −10.0000 −0.411345
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 − 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) 6.00000 0.245564
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 0 0
\(607\) −2.00000 −0.0811775 −0.0405887 − 0.999176i \(-0.512923\pi\)
−0.0405887 + 0.999176i \(0.512923\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 20.0000 0.807792 0.403896 − 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 − 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −28.0000 −1.12180
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 16.0000 0.638978
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −18.0000 −0.716569 −0.358284 − 0.933613i \(-0.616638\pi\)
−0.358284 + 0.933613i \(0.616638\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 6.00000 0.238103
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) 14.0000 0.552967 0.276483 − 0.961019i \(-0.410831\pi\)
0.276483 + 0.961019i \(0.410831\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 − 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 − 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 20.0000 0.783862
\(652\) 0 0
\(653\) 30.0000 1.17399 0.586995 − 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 − 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −32.0000 −1.23535
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 − 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 − 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 0 0
\(693\) 8.00000 0.303895
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 − 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) −4.00000 −0.150649
\(706\) 0 0
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) −44.0000 −1.65245 −0.826227 − 0.563337i \(-0.809517\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) 14.0000 0.525041
\(712\) 0 0
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16.0000 0.597531
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 − 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 0 0
\(723\) 30.0000 1.11571
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −14.0000 −0.519231 −0.259616 − 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −24.0000 −0.886460 −0.443230 − 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 − 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 − 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.00000 0.292314
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 − 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −2.00000 −0.0727875
\(756\) 0 0
\(757\) 8.00000 0.290765 0.145382 − 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.00000 −0.216366 −0.108183 − 0.994131i \(-0.534503\pi\)
−0.108183 + 0.994131i \(0.534503\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 40.0000 1.43315
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −10.0000 −0.354663
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 − 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 − 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) −20.0000 −0.700569
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 − 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 42.0000 1.46403 0.732014 − 0.681290i \(-0.238581\pi\)
0.732014 + 0.681290i \(0.238581\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 − 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −44.0000 −1.52818 −0.764092 − 0.645108i \(-0.776812\pi\)
−0.764092 + 0.645108i \(0.776812\pi\)
\(830\) 0 0
\(831\) −16.0000 −0.555034
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) 0 0
\(839\) 4.00000 0.138095 0.0690477 − 0.997613i \(-0.478004\pi\)
0.0690477 + 0.997613i \(0.478004\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) −13.0000 −0.447214
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 44.0000 1.50653 0.753266 − 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −6.00000 −0.204956 −0.102478 − 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 20.0000 0.681598
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 56.0000 1.89967
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 44.0000 1.48577 0.742887 − 0.669417i \(-0.233456\pi\)
0.742887 + 0.669417i \(0.233456\pi\)
\(878\) 0 0
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 − 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) −8.00000 −0.268917
\(886\) 0 0
\(887\) 16.0000 0.537227 0.268614 − 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 20.0000 0.664822
\(906\) 0 0
\(907\) −52.0000 −1.72663 −0.863316 − 0.504664i \(-0.831616\pi\)
−0.863316 + 0.504664i \(0.831616\pi\)
\(908\) 0 0
\(909\) −14.0000 −0.464351
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 − 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 8.00000 0.264472
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −26.0000 −0.857661 −0.428830 − 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) 0 0
\(921\) 28.0000 0.922631
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −2.00000 −0.0656886
\(928\) 0 0
\(929\) 46.0000 1.50921 0.754606 − 0.656179i \(-0.227828\pi\)
0.754606 + 0.656179i \(0.227828\pi\)
\(930\) 0 0
\(931\) 12.0000 0.393284
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 − 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 0 0
\(943\) 40.0000 1.30258
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 − 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 26.0000 0.842223 0.421111 − 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) 24.0000 0.775810
\(958\) 0 0
\(959\) 4.00000 0.129167
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 46.0000 1.47926 0.739630 − 0.673014i \(-0.235000\pi\)
0.739630 + 0.673014i \(0.235000\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 − 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −22.0000 −0.703842 −0.351921 − 0.936030i \(-0.614471\pi\)
−0.351921 + 0.936030i \(0.614471\pi\)
\(978\) 0 0
\(979\) −56.0000 −1.78977
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) −32.0000 −1.02064 −0.510321 − 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 − 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 − 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 0 0
\(999\) 4.00000 0.126554
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 3840.2.a.m.1.1 1
4.3 odd 2 3840.2.a.w.1.1 1
8.3 odd 2 3840.2.a.d.1.1 1
8.5 even 2 3840.2.a.r.1.1 1
16.3 odd 4 120.2.k.a.61.1 ✓ 2
16.5 even 4 480.2.k.a.241.2 2
16.11 odd 4 120.2.k.a.61.2 yes 2
16.13 even 4 480.2.k.a.241.1 2
48.5 odd 4 1440.2.k.a.721.1 2
48.11 even 4 360.2.k.b.181.1 2
48.29 odd 4 1440.2.k.a.721.2 2
48.35 even 4 360.2.k.b.181.2 2
80.3 even 4 600.2.d.a.349.1 2
80.13 odd 4 2400.2.d.d.49.2 2
80.19 odd 4 600.2.k.a.301.2 2
80.27 even 4 600.2.d.a.349.2 2
80.29 even 4 2400.2.k.b.1201.2 2
80.37 odd 4 2400.2.d.d.49.1 2
80.43 even 4 600.2.d.d.349.1 2
80.53 odd 4 2400.2.d.a.49.2 2
80.59 odd 4 600.2.k.a.301.1 2
80.67 even 4 600.2.d.d.349.2 2
80.69 even 4 2400.2.k.b.1201.1 2
80.77 odd 4 2400.2.d.a.49.1 2
240.29 odd 4 7200.2.k.f.3601.2 2
240.53 even 4 7200.2.d.e.2449.2 2
240.59 even 4 1800.2.k.g.901.2 2
240.77 even 4 7200.2.d.e.2449.1 2
240.83 odd 4 1800.2.d.h.1549.2 2
240.107 odd 4 1800.2.d.h.1549.1 2
240.149 odd 4 7200.2.k.f.3601.1 2
240.173 even 4 7200.2.d.f.2449.2 2
240.179 even 4 1800.2.k.g.901.1 2
240.197 even 4 7200.2.d.f.2449.1 2
240.203 odd 4 1800.2.d.c.1549.2 2
240.227 odd 4 1800.2.d.c.1549.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.a.61.1 ✓ 2 16.3 odd 4
120.2.k.a.61.2 yes 2 16.11 odd 4
360.2.k.b.181.1 2 48.11 even 4
360.2.k.b.181.2 2 48.35 even 4
480.2.k.a.241.1 2 16.13 even 4
480.2.k.a.241.2 2 16.5 even 4
600.2.d.a.349.1 2 80.3 even 4
600.2.d.a.349.2 2 80.27 even 4
600.2.d.d.349.1 2 80.43 even 4
600.2.d.d.349.2 2 80.67 even 4
600.2.k.a.301.1 2 80.59 odd 4
600.2.k.a.301.2 2 80.19 odd 4
1440.2.k.a.721.1 2 48.5 odd 4
1440.2.k.a.721.2 2 48.29 odd 4
1800.2.d.c.1549.1 2 240.227 odd 4
1800.2.d.c.1549.2 2 240.203 odd 4
1800.2.d.h.1549.1 2 240.107 odd 4
1800.2.d.h.1549.2 2 240.83 odd 4
1800.2.k.g.901.1 2 240.179 even 4
1800.2.k.g.901.2 2 240.59 even 4
2400.2.d.a.49.1 2 80.77 odd 4
2400.2.d.a.49.2 2 80.53 odd 4
2400.2.d.d.49.1 2 80.37 odd 4
2400.2.d.d.49.2 2 80.13 odd 4
2400.2.k.b.1201.1 2 80.69 even 4
2400.2.k.b.1201.2 2 80.29 even 4
3840.2.a.d.1.1 1 8.3 odd 2
3840.2.a.m.1.1 1 1.1 even 1 trivial
3840.2.a.r.1.1 1 8.5 even 2
3840.2.a.w.1.1 1 4.3 odd 2
7200.2.d.e.2449.1 2 240.77 even 4
7200.2.d.e.2449.2 2 240.53 even 4
7200.2.d.f.2449.1 2 240.197 even 4
7200.2.d.f.2449.2 2 240.173 even 4
7200.2.k.f.3601.1 2 240.149 odd 4
7200.2.k.f.3601.2 2 240.29 odd 4