Properties

Label 4056.2.a.r.1.1
Level $4056$
Weight $2$
Character 4056.1
Self dual yes
Analytic conductor $32.387$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4056,2,Mod(1,4056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4056, 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("4056.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4056.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: \(32.3873230598\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4056.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +4.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +4.00000 q^{15} +6.00000 q^{17} +2.00000 q^{19} +2.00000 q^{21} -8.00000 q^{23} +11.0000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -10.0000 q^{31} -2.00000 q^{33} +8.00000 q^{35} +4.00000 q^{37} +4.00000 q^{43} +4.00000 q^{45} +2.00000 q^{47} -3.00000 q^{49} +6.00000 q^{51} -10.0000 q^{53} -8.00000 q^{55} +2.00000 q^{57} +14.0000 q^{59} -2.00000 q^{61} +2.00000 q^{63} +2.00000 q^{67} -8.00000 q^{69} +6.00000 q^{71} +8.00000 q^{73} +11.0000 q^{75} -4.00000 q^{77} +1.00000 q^{81} -6.00000 q^{83} +24.0000 q^{85} +6.00000 q^{87} -10.0000 q^{93} +8.00000 q^{95} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 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
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 11.0000 2.20000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 8.00000 1.35225
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) 2.00000 0.291730 0.145865 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 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 \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −8.00000 −1.07872
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 14.0000 1.82264 0.911322 0.411693i \(-0.135063\pi\)
0.911322 + 0.411693i \(0.135063\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) 0 0
\(85\) 24.0000 2.60317
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 8.00000 0.780720
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 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 \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −32.0000 −2.98402
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 1.10004
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −16.0000 −1.36697 −0.683486 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 24.0000 1.99309
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) −40.0000 −3.21288
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) −8.00000 −0.622799
\(166\) 0 0
\(167\) 10.0000 0.773823 0.386912 0.922117i \(-0.373542\pi\)
0.386912 + 0.922117i \(0.373542\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 22.0000 1.66304
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) 16.0000 1.17634
\(186\) 0 0
\(187\) −12.0000 −0.877527
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −20.0000 −1.35769
\(218\) 0 0
\(219\) 8.00000 0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 10.0000 0.663723 0.331862 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) 0 0
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.0000 1.94054 0.970269 0.242028i \(-0.0778125\pi\)
0.970269 + 0.242028i \(0.0778125\pi\)
\(240\) 0 0
\(241\) 16.0000 1.03065 0.515325 0.856995i \(-0.327671\pi\)
0.515325 + 0.856995i \(0.327671\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 24.0000 1.50294
\(256\) 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
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) −40.0000 −2.45718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −22.0000 −1.32665
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 8.00000 0.473879
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 56.0000 3.26045
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 30.0000 1.71219 0.856095 0.516818i \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 8.00000 0.450749
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) −6.00000 −0.329790 −0.164895 0.986311i \(-0.552728\pi\)
−0.164895 + 0.986311i \(0.552728\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −32.0000 −1.72282
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.00000 0.425797 0.212899 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) 0 0
\(357\) 12.0000 0.635107
\(358\) 0 0
\(359\) 26.0000 1.37223 0.686114 0.727494i \(-0.259315\pi\)
0.686114 + 0.727494i \(0.259315\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 32.0000 1.67496
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) −2.00000 −0.102195 −0.0510976 0.998694i \(-0.516272\pi\)
−0.0510976 + 0.998694i \(0.516272\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −8.00000 −0.399501 −0.199750 0.979847i \(-0.564013\pi\)
−0.199750 + 0.979847i \(0.564013\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) −8.00000 −0.396545
\(408\) 0 0
\(409\) −8.00000 −0.395575 −0.197787 0.980245i \(-0.563376\pi\)
−0.197787 + 0.980245i \(0.563376\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) 0 0
\(413\) 28.0000 1.37779
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 0 0
\(423\) 2.00000 0.0972433
\(424\) 0 0
\(425\) 66.0000 3.20147
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.00000 −0.0963366 −0.0481683 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 24.0000 1.15071
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 20.0000 0.945968
\(448\) 0 0
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.00000 0.0939682
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 0 0
\(465\) −40.0000 −1.85496
\(466\) 0 0
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 22.0000 1.00943
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −16.0000 −0.728025
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 0 0
\(489\) −2.00000 −0.0904431
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 18.0000 0.805791 0.402895 0.915246i \(-0.368004\pi\)
0.402895 + 0.915246i \(0.368004\pi\)
\(500\) 0 0
\(501\) 10.0000 0.446767
\(502\) 0 0
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) −24.0000 −1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −32.0000 −1.41009
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 22.0000 0.960159
\(526\) 0 0
\(527\) −60.0000 −2.61364
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 14.0000 0.607548
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −16.0000 −0.691740
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −20.0000 −0.859867 −0.429934 0.902861i \(-0.641463\pi\)
−0.429934 + 0.902861i \(0.641463\pi\)
\(542\) 0 0
\(543\) −14.0000 −0.600798
\(544\) 0 0
\(545\) 16.0000 0.685365
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 0 0
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 16.0000 0.679162
\(556\) 0 0
\(557\) 20.0000 0.847427 0.423714 0.905796i \(-0.360726\pi\)
0.423714 + 0.905796i \(0.360726\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −24.0000 −1.00969
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) −88.0000 −3.66985
\(576\) 0 0
\(577\) −32.0000 −1.33218 −0.666089 0.745873i \(-0.732033\pi\)
−0.666089 + 0.745873i \(0.732033\pi\)
\(578\) 0 0
\(579\) −16.0000 −0.664937
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 20.0000 0.828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 0 0
\(593\) 8.00000 0.328521 0.164260 0.986417i \(-0.447476\pi\)
0.164260 + 0.986417i \(0.447476\pi\)
\(594\) 0 0
\(595\) 48.0000 1.96781
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −28.0000 −1.13836
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −48.0000 −1.93241 −0.966204 0.257780i \(-0.917009\pi\)
−0.966204 + 0.257780i \(0.917009\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 64.0000 2.53976
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 16.0000 0.629999
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) −20.0000 −0.783862
\(652\) 0 0
\(653\) 22.0000 0.860927 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(654\) 0 0
\(655\) −48.0000 −1.87552
\(656\) 0 0
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) 0 0
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) 0 0
\(675\) 11.0000 0.423390
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 30.0000 1.14792 0.573959 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) −64.0000 −2.44531
\(686\) 0 0
\(687\) −28.0000 −1.06827
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 50.0000 1.90209 0.951045 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) −12.0000 −0.451306
\(708\) 0 0
\(709\) −28.0000 −1.05156 −0.525781 0.850620i \(-0.676227\pi\)
−0.525781 + 0.850620i \(0.676227\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 80.0000 2.99602
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 30.0000 1.12037
\(718\) 0 0
\(719\) −32.0000 −1.19340 −0.596699 0.802465i \(-0.703521\pi\)
−0.596699 + 0.802465i \(0.703521\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 16.0000 0.595046
\(724\) 0 0
\(725\) 66.0000 2.45118
\(726\) 0 0
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 38.0000 1.39408 0.697042 0.717030i \(-0.254499\pi\)
0.697042 + 0.717030i \(0.254499\pi\)
\(744\) 0 0
\(745\) 80.0000 2.93097
\(746\) 0 0
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 4.00000 0.145768
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 24.0000 0.867722
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −48.0000 −1.73092 −0.865462 0.500974i \(-0.832975\pi\)
−0.865462 + 0.500974i \(0.832975\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) 0 0
\(775\) −110.000 −3.95132
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 88.0000 3.14085
\(786\) 0 0
\(787\) −2.00000 −0.0712923 −0.0356462 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) 0 0
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −40.0000 −1.41865
\(796\) 0 0
\(797\) −38.0000 −1.34603 −0.673015 0.739629i \(-0.735001\pi\)
−0.673015 + 0.739629i \(0.735001\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −16.0000 −0.564628
\(804\) 0 0
\(805\) −64.0000 −2.25570
\(806\) 0 0
\(807\) −10.0000 −0.352017
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 0 0
\(813\) −22.0000 −0.771574
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −28.0000 −0.977207 −0.488603 0.872506i \(-0.662493\pi\)
−0.488603 + 0.872506i \(0.662493\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) −22.0000 −0.765942
\(826\) 0 0
\(827\) −2.00000 −0.0695468 −0.0347734 0.999395i \(-0.511071\pi\)
−0.0347734 + 0.999395i \(0.511071\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 40.0000 1.38426
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 16.0000 0.551069
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) −28.0000 −0.958702 −0.479351 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 0 0
\(865\) −56.0000 −1.90406
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 48.0000 1.62270
\(876\) 0 0
\(877\) −28.0000 −0.945493 −0.472746 0.881199i \(-0.656737\pi\)
−0.472746 + 0.881199i \(0.656737\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) 0 0
\(885\) 56.0000 1.88242
\(886\) 0 0
\(887\) 40.0000 1.34307 0.671534 0.740973i \(-0.265636\pi\)
0.671534 + 0.740973i \(0.265636\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 4.00000 0.133855
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 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
\(905\) −56.0000 −1.86150
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) −8.00000 −0.264472
\(916\) 0 0
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 30.0000 0.988534
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 44.0000 1.44671
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 24.0000 0.787414 0.393707 0.919236i \(-0.371192\pi\)
0.393707 + 0.919236i \(0.371192\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −48.0000 −1.56977
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) −20.0000 −0.651981 −0.325991 0.945373i \(-0.605698\pi\)
−0.325991 + 0.945373i \(0.605698\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 8.00000 0.260240
\(946\) 0 0
\(947\) −42.0000 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) −64.0000 −2.07099
\(956\) 0 0
\(957\) −12.0000 −0.387905
\(958\) 0 0
\(959\) −32.0000 −1.03333
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −4.00000 −0.128898
\(964\) 0 0
\(965\) −64.0000 −2.06023
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24.0000 0.767828 0.383914 0.923369i \(-0.374576\pi\)
0.383914 + 0.923369i \(0.374576\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) −22.0000 −0.701691 −0.350846 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(984\) 0 0
\(985\) −48.0000 −1.52941
\(986\) 0 0
\(987\) 4.00000 0.127321
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) −6.00000 −0.190404
\(994\) 0 0
\(995\) −96.0000 −3.04340
\(996\) 0 0
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 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 4056.2.a.r.1.1 1
4.3 odd 2 8112.2.a.p.1.1 1
13.5 odd 4 312.2.c.c.25.1 2
13.8 odd 4 312.2.c.c.25.2 yes 2
13.12 even 2 4056.2.a.k.1.1 1
39.5 even 4 936.2.c.c.649.2 2
39.8 even 4 936.2.c.c.649.1 2
52.31 even 4 624.2.c.c.337.1 2
52.47 even 4 624.2.c.c.337.2 2
52.51 odd 2 8112.2.a.a.1.1 1
104.5 odd 4 2496.2.c.a.961.2 2
104.21 odd 4 2496.2.c.a.961.1 2
104.83 even 4 2496.2.c.h.961.2 2
104.99 even 4 2496.2.c.h.961.1 2
156.47 odd 4 1872.2.c.i.1585.1 2
156.83 odd 4 1872.2.c.i.1585.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.c.c.25.1 2 13.5 odd 4
312.2.c.c.25.2 yes 2 13.8 odd 4
624.2.c.c.337.1 2 52.31 even 4
624.2.c.c.337.2 2 52.47 even 4
936.2.c.c.649.1 2 39.8 even 4
936.2.c.c.649.2 2 39.5 even 4
1872.2.c.i.1585.1 2 156.47 odd 4
1872.2.c.i.1585.2 2 156.83 odd 4
2496.2.c.a.961.1 2 104.21 odd 4
2496.2.c.a.961.2 2 104.5 odd 4
2496.2.c.h.961.1 2 104.99 even 4
2496.2.c.h.961.2 2 104.83 even 4
4056.2.a.k.1.1 1 13.12 even 2
4056.2.a.r.1.1 1 1.1 even 1 trivial
8112.2.a.a.1.1 1 52.51 odd 2
8112.2.a.p.1.1 1 4.3 odd 2