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Properties

Label 8112.2.a.m.1.1
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
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] = [8112,2,Mod(1,8112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8112, 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("8112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.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: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -6.00000 q^{11} -3.00000 q^{15} -3.00000 q^{17} -2.00000 q^{19} +2.00000 q^{21} +6.00000 q^{23} +4.00000 q^{25} -1.00000 q^{27} +3.00000 q^{29} +4.00000 q^{31} +6.00000 q^{33} -6.00000 q^{35} -7.00000 q^{37} -3.00000 q^{41} +10.0000 q^{43} +3.00000 q^{45} -6.00000 q^{47} -3.00000 q^{49} +3.00000 q^{51} +3.00000 q^{53} -18.0000 q^{55} +2.00000 q^{57} -7.00000 q^{61} -2.00000 q^{63} +10.0000 q^{67} -6.00000 q^{69} -6.00000 q^{71} -13.0000 q^{73} -4.00000 q^{75} +12.0000 q^{77} +4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -9.00000 q^{85} -3.00000 q^{87} +18.0000 q^{89} -4.00000 q^{93} -6.00000 q^{95} +14.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −3.00000 −0.774597
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 − 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 − 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 − 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) 0 0
\(55\) −18.0000 −2.42712
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 0 0
\(63\) −2.00000 −0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −13.0000 −1.52153 −0.760767 − 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 0 0
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 − 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 − 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 − 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 − 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −3.00000 −0.282216 −0.141108 − 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 0 0
\(115\) 18.0000 1.67851
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.00000 0.550019
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 0 0
\(123\) 3.00000 0.270501
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 4.00000 0.354943 0.177471 − 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 0 0
\(129\) −10.0000 −0.880451
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −3.00000 −0.258199
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −9.00000 −0.737309 −0.368654 − 0.929567i \(-0.620181\pi\)
−0.368654 + 0.929567i \(0.620181\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 − 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 0 0
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 18.0000 1.40130
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 − 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 − 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) −21.0000 −1.54395
\(186\) 0 0
\(187\) 18.0000 1.31629
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) 23.0000 1.65558 0.827788 − 0.561041i \(-0.189599\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −9.00000 −0.628587
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 − 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 30.0000 2.04598
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 13.0000 0.878459
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 − 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 − 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 − 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 − 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −9.00000 −0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 9.00000 0.563602
\(256\) 0 0
\(257\) −3.00000 −0.187135 −0.0935674 − 0.995613i \(-0.529827\pi\)
−0.0935674 + 0.995613i \(0.529827\pi\)
\(258\) 0 0
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 6.00000 0.369976 0.184988 − 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 0 0
\(269\) 18.0000 1.09748 0.548740 − 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.0000 −1.44725
\(276\) 0 0
\(277\) 17.0000 1.02143 0.510716 − 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) 9.00000 0.536895 0.268447 − 0.963294i \(-0.413489\pi\)
0.268447 + 0.963294i \(0.413489\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) −21.0000 −1.22683 −0.613417 − 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 6.00000 0.348155
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −20.0000 −1.15278
\(302\) 0 0
\(303\) −15.0000 −0.861727
\(304\) 0 0
\(305\) −21.0000 −1.20246
\(306\) 0 0
\(307\) 10.0000 0.570730 0.285365 − 0.958419i \(-0.407885\pi\)
0.285365 + 0.958419i \(0.407885\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 − 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) 0 0
\(317\) 3.00000 0.168497 0.0842484 − 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) 0 0
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 − 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −7.00000 −0.383598
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 − 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 3.00000 0.162938
\(340\) 0 0
\(341\) −24.0000 −1.29967
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −18.0000 −0.969087
\(346\) 0 0
\(347\) 30.0000 1.61048 0.805242 − 0.592946i \(-0.202035\pi\)
0.805242 + 0.592946i \(0.202035\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.0000 −0.798369 −0.399185 − 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) 0 0
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −39.0000 −2.04135
\(366\) 0 0
\(367\) −2.00000 −0.104399 −0.0521996 − 0.998637i \(-0.516623\pi\)
−0.0521996 + 0.998637i \(0.516623\pi\)
\(368\) 0 0
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 29.0000 1.50156 0.750782 − 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) 0 0
\(375\) 3.00000 0.154919
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) 0 0
\(383\) −24.0000 −1.22634 −0.613171 − 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) 0 0
\(385\) 36.0000 1.83473
\(386\) 0 0
\(387\) 10.0000 0.508329
\(388\) 0 0
\(389\) 39.0000 1.97738 0.988689 − 0.149979i \(-0.0479205\pi\)
0.988689 + 0.149979i \(0.0479205\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.00000 0.149071
\(406\) 0 0
\(407\) 42.0000 2.08186
\(408\) 0 0
\(409\) −1.00000 −0.0494468 −0.0247234 − 0.999694i \(-0.507871\pi\)
−0.0247234 + 0.999694i \(0.507871\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 − 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 29.0000 1.41337 0.706687 − 0.707527i \(-0.250189\pi\)
0.706687 + 0.707527i \(0.250189\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 14.0000 0.677507
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 − 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 0 0
\(433\) −13.0000 −0.624740 −0.312370 − 0.949960i \(-0.601123\pi\)
−0.312370 + 0.949960i \(0.601123\pi\)
\(434\) 0 0
\(435\) −9.00000 −0.431517
\(436\) 0 0
\(437\) −12.0000 −0.574038
\(438\) 0 0
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 54.0000 2.55985
\(446\) 0 0
\(447\) 9.00000 0.425685
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 − 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 18.0000 0.847587
\(452\) 0 0
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.0000 0.514558 0.257279 − 0.966337i \(-0.417174\pi\)
0.257279 + 0.966337i \(0.417174\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) 15.0000 0.698620 0.349310 − 0.937007i \(-0.386416\pi\)
0.349310 + 0.937007i \(0.386416\pi\)
\(462\) 0 0
\(463\) −38.0000 −1.76601 −0.883005 − 0.469364i \(-0.844483\pi\)
−0.883005 + 0.469364i \(0.844483\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) 18.0000 0.832941 0.416470 − 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) 0 0
\(471\) −5.00000 −0.230388
\(472\) 0 0
\(473\) −60.0000 −2.75880
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) 0 0
\(477\) 3.00000 0.137361
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 12.0000 0.546019
\(484\) 0 0
\(485\) 42.0000 1.90712
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 0 0
\(491\) 18.0000 0.812329 0.406164 − 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 0 0
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) −18.0000 −0.809040
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 − 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 − 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.00000 0.132973 0.0664863 − 0.997787i \(-0.478821\pi\)
0.0664863 + 0.997787i \(0.478821\pi\)
\(510\) 0 0
\(511\) 26.0000 1.15017
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −42.0000 −1.85074
\(516\) 0 0
\(517\) 36.0000 1.58328
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 − 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) 0 0
\(523\) 34.0000 1.48672 0.743358 − 0.668894i \(-0.233232\pi\)
0.743358 + 0.668894i \(0.233232\pi\)
\(524\) 0 0
\(525\) 8.00000 0.349149
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 29.0000 1.24681 0.623404 − 0.781900i \(-0.285749\pi\)
0.623404 + 0.781900i \(0.285749\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) 42.0000 1.79908
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 − 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) 0 0
\(549\) −7.00000 −0.298753
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 21.0000 0.891400
\(556\) 0 0
\(557\) 3.00000 0.127114 0.0635570 − 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 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
\(571\) 22.0000 0.920671 0.460336 − 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 0 0
\(575\) 24.0000 1.00087
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 − 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) −23.0000 −0.955847
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 9.00000 0.369586 0.184793 − 0.982777i \(-0.440839\pi\)
0.184793 + 0.982777i \(0.440839\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 − 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) 75.0000 3.04918
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 0 0
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −31.0000 −1.25208 −0.626039 − 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) −15.0000 −0.603877 −0.301939 − 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 − 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) −12.0000 −0.479234
\(628\) 0 0
\(629\) 21.0000 0.837325
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 − 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) −16.0000 −0.635943
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −3.00000 −0.118493 −0.0592464 − 0.998243i \(-0.518870\pi\)
−0.0592464 + 0.998243i \(0.518870\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) −30.0000 −1.18125
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −13.0000 −0.507178
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 − 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 5.00000 0.194477 0.0972387 − 0.995261i \(-0.468999\pi\)
0.0972387 + 0.995261i \(0.468999\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 42.0000 1.62139
\(672\) 0 0
\(673\) −13.0000 −0.501113 −0.250557 − 0.968102i \(-0.580614\pi\)
−0.250557 + 0.968102i \(0.580614\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 − 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) −28.0000 −1.07454
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 48.0000 1.83667 0.918334 − 0.395805i \(-0.129534\pi\)
0.918334 + 0.395805i \(0.129534\pi\)
\(684\) 0 0
\(685\) 27.0000 1.03162
\(686\) 0 0
\(687\) 22.0000 0.839352
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −26.0000 −0.989087 −0.494543 − 0.869153i \(-0.664665\pi\)
−0.494543 + 0.869153i \(0.664665\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 9.00000 0.340899
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 − 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 14.0000 0.528020
\(704\) 0 0
\(705\) 18.0000 0.677919
\(706\) 0 0
\(707\) −30.0000 −1.12827
\(708\) 0 0
\(709\) 5.00000 0.187779 0.0938895 − 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) 24.0000 0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 1.00000 0.0371904
\(724\) 0 0
\(725\) 12.0000 0.445669
\(726\) 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
\(731\) −30.0000 −1.10959
\(732\) 0 0
\(733\) −31.0000 −1.14501 −0.572506 − 0.819901i \(-0.694029\pi\)
−0.572506 + 0.819901i \(0.694029\pi\)
\(734\) 0 0
\(735\) 9.00000 0.331970
\(736\) 0 0
\(737\) −60.0000 −2.21013
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 36.0000 1.32071 0.660356 − 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 0 0
\(745\) −27.0000 −0.989203
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −14.0000 −0.510867 −0.255434 − 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 36.0000 1.30672
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) −28.0000 −1.01367
\(764\) 0 0
\(765\) −9.00000 −0.325396
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 3.00000 0.108042
\(772\) 0 0
\(773\) −30.0000 −1.07903 −0.539513 − 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) −14.0000 −0.502247
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) 15.0000 0.535373
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 − 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) 78.0000 2.75256
\(804\) 0 0
\(805\) −36.0000 −1.26883
\(806\) 0 0
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) −51.0000 −1.79306 −0.896532 − 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 − 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −20.0000 −0.699711
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) 24.0000 0.835573
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) 17.0000 0.590434 0.295217 − 0.955430i \(-0.404608\pi\)
0.295217 + 0.955430i \(0.404608\pi\)
\(830\) 0 0
\(831\) −17.0000 −0.589723
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −9.00000 −0.309976
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −50.0000 −1.71802
\(848\) 0 0
\(849\) 14.0000 0.480479
\(850\) 0 0
\(851\) −42.0000 −1.43974
\(852\) 0 0
\(853\) −19.0000 −0.650548 −0.325274 − 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) 21.0000 0.717346 0.358673 − 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 0 0
\(859\) −26.0000 −0.887109 −0.443554 − 0.896248i \(-0.646283\pi\)
−0.443554 + 0.896248i \(0.646283\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) −18.0000 −0.612727 −0.306364 − 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 6.00000 0.202837
\(876\) 0 0
\(877\) 41.0000 1.38447 0.692236 − 0.721671i \(-0.256626\pi\)
0.692236 + 0.721671i \(0.256626\pi\)
\(878\) 0 0
\(879\) 21.0000 0.708312
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 − 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 − 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) 20.0000 0.665558
\(904\) 0 0
\(905\) −21.0000 −0.698064
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 − 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) 15.0000 0.497519
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 − 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 21.0000 0.694239
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) 33.0000 1.08269 0.541347 − 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) −30.0000 −0.982156
\(934\) 0 0
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 − 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 − 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 0 0
\(945\) 6.00000 0.195180
\(946\) 0 0
\(947\) −24.0000 −0.779895 −0.389948 − 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) 54.0000 1.74923 0.874616 − 0.484817i \(-0.161114\pi\)
0.874616 + 0.484817i \(0.161114\pi\)
\(954\) 0 0
\(955\) 36.0000 1.16493
\(956\) 0 0
\(957\) 18.0000 0.581857
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) 69.0000 2.22119
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 − 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.00000 −0.0959785 −0.0479893 − 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) 0 0
\(979\) −108.000 −3.45169
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) −38.0000 −1.20711 −0.603555 − 0.797321i \(-0.706250\pi\)
−0.603555 + 0.797321i \(0.706250\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 30.0000 0.951064
\(996\) 0 0
\(997\) 5.00000 0.158352 0.0791758 − 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) 0 0
\(999\) 7.00000 0.221470
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 8112.2.a.m.1.1 1
4.3 odd 2 1014.2.a.c.1.1 1
12.11 even 2 3042.2.a.i.1.1 1
13.3 even 3 624.2.q.g.529.1 2
13.9 even 3 624.2.q.g.289.1 2
13.12 even 2 8112.2.a.c.1.1 1
39.29 odd 6 1872.2.t.c.1153.1 2
39.35 odd 6 1872.2.t.c.289.1 2
52.3 odd 6 78.2.e.a.61.1 yes 2
52.7 even 12 1014.2.i.b.361.1 4
52.11 even 12 1014.2.i.b.823.2 4
52.15 even 12 1014.2.i.b.823.1 4
52.19 even 12 1014.2.i.b.361.2 4
52.23 odd 6 1014.2.e.a.529.1 2
52.31 even 4 1014.2.b.c.337.2 2
52.35 odd 6 78.2.e.a.55.1 ✓ 2
52.43 odd 6 1014.2.e.a.991.1 2
52.47 even 4 1014.2.b.c.337.1 2
52.51 odd 2 1014.2.a.f.1.1 1
156.35 even 6 234.2.h.a.55.1 2
156.47 odd 4 3042.2.b.h.1351.2 2
156.83 odd 4 3042.2.b.h.1351.1 2
156.107 even 6 234.2.h.a.217.1 2
156.155 even 2 3042.2.a.h.1.1 1
260.3 even 12 1950.2.z.g.1699.1 4
260.87 even 12 1950.2.z.g.1849.1 4
260.107 even 12 1950.2.z.g.1699.2 4
260.139 odd 6 1950.2.i.m.601.1 2
260.159 odd 6 1950.2.i.m.451.1 2
260.243 even 12 1950.2.z.g.1849.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 ✓ 2 52.35 odd 6
78.2.e.a.61.1 yes 2 52.3 odd 6
234.2.h.a.55.1 2 156.35 even 6
234.2.h.a.217.1 2 156.107 even 6
624.2.q.g.289.1 2 13.9 even 3
624.2.q.g.529.1 2 13.3 even 3
1014.2.a.c.1.1 1 4.3 odd 2
1014.2.a.f.1.1 1 52.51 odd 2
1014.2.b.c.337.1 2 52.47 even 4
1014.2.b.c.337.2 2 52.31 even 4
1014.2.e.a.529.1 2 52.23 odd 6
1014.2.e.a.991.1 2 52.43 odd 6
1014.2.i.b.361.1 4 52.7 even 12
1014.2.i.b.361.2 4 52.19 even 12
1014.2.i.b.823.1 4 52.15 even 12
1014.2.i.b.823.2 4 52.11 even 12
1872.2.t.c.289.1 2 39.35 odd 6
1872.2.t.c.1153.1 2 39.29 odd 6
1950.2.i.m.451.1 2 260.159 odd 6
1950.2.i.m.601.1 2 260.139 odd 6
1950.2.z.g.1699.1 4 260.3 even 12
1950.2.z.g.1699.2 4 260.107 even 12
1950.2.z.g.1849.1 4 260.87 even 12
1950.2.z.g.1849.2 4 260.243 even 12
3042.2.a.h.1.1 1 156.155 even 2
3042.2.a.i.1.1 1 12.11 even 2
3042.2.b.h.1351.1 2 156.83 odd 4
3042.2.b.h.1351.2 2 156.47 odd 4
8112.2.a.c.1.1 1 13.12 even 2
8112.2.a.m.1.1 1 1.1 even 1 trivial