Properties

Label 8112.2.a.bi.1.1
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $1$
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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 156)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +4.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +4.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -4.00000 q^{11} +4.00000 q^{15} +2.00000 q^{17} -2.00000 q^{19} -2.00000 q^{21} +11.0000 q^{25} +1.00000 q^{27} -6.00000 q^{29} -10.0000 q^{31} -4.00000 q^{33} -8.00000 q^{35} -10.0000 q^{37} -8.00000 q^{41} -4.00000 q^{43} +4.00000 q^{45} -4.00000 q^{47} -3.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} -16.0000 q^{55} -2.00000 q^{57} -8.00000 q^{59} -14.0000 q^{61} -2.00000 q^{63} +2.00000 q^{67} +16.0000 q^{71} +10.0000 q^{73} +11.0000 q^{75} +8.00000 q^{77} +16.0000 q^{79} +1.00000 q^{81} +8.00000 q^{85} -6.00000 q^{87} +4.00000 q^{89} -10.0000 q^{93} -8.00000 q^{95} +2.00000 q^{97} -4.00000 q^{99} +O(q^{100})\)

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 \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 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
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 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 \(-0.688081\pi\)
−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
\(35\) −8.00000 −1.35225
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 4.00000 0.596285
\(46\) 0 0
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −16.0000 −2.15744
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 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
\(69\) 0 0
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 11.0000 1.27017
\(76\) 0 0
\(77\) 8.00000 0.911685
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 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
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 4.00000 0.423999 0.212000 0.977270i \(-0.432002\pi\)
0.212000 + 0.977270i \(0.432002\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 24.0000 2.14663
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 4.00000 0.344265
\(136\) 0 0
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −24.0000 −1.99309
\(146\) 0 0
\(147\) −3.00000 −0.247436
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −40.0000 −3.21288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) −16.0000 −1.24560
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −22.0000 −1.66304
\(176\) 0 0
\(177\) −8.00000 −0.601317
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −14.0000 −1.03491
\(184\) 0 0
\(185\) −40.0000 −2.94086
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 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 \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) 0 0
\(205\) −32.0000 −2.23498
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 16.0000 1.09630
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 20.0000 1.35769
\(218\) 0 0
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14.0000 0.937509 0.468755 0.883328i \(-0.344703\pi\)
0.468755 + 0.883328i \(0.344703\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 8.00000 0.526361
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 0 0
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −40.0000 −2.45718
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −44.0000 −2.65330
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) 0 0
\(287\) 16.0000 0.944450
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 0 0
\(295\) −32.0000 −1.86311
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −56.0000 −3.20655
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −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
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) 0 0
\(333\) −10.0000 −0.547997
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −30.0000 −1.63420 −0.817102 0.576493i \(-0.804421\pi\)
−0.817102 + 0.576493i \(0.804421\pi\)
\(338\) 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
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −18.0000 −0.963518 −0.481759 0.876304i \(-0.660002\pi\)
−0.481759 + 0.876304i \(0.660002\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.0000 0.851594 0.425797 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) 0 0
\(355\) 64.0000 3.39677
\(356\) 0 0
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 40.0000 2.09370
\(366\) 0 0
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 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
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 32.0000 1.63087
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 64.0000 3.22019
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.00000 0.198762
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 0 0
\(413\) 16.0000 0.787309
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) −4.00000 −0.194487
\(424\) 0 0
\(425\) 22.0000 1.06716
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 0 0
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 0 0
\(435\) −24.0000 −1.15071
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 32.0000 1.50682
\(452\) 0 0
\(453\) 18.0000 0.845714
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 22.0000 1.02243 0.511213 0.859454i \(-0.329196\pi\)
0.511213 + 0.859454i \(0.329196\pi\)
\(464\) 0 0
\(465\) −40.0000 −1.85496
\(466\) 0 0
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 0 0
\(475\) −22.0000 −1.00943
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 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
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) −16.0000 −0.719147
\(496\) 0 0
\(497\) −32.0000 −1.43540
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) 40.0000 1.77998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 32.0000 1.41009
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −2.00000 −0.0876216 −0.0438108 0.999040i \(-0.513950\pi\)
−0.0438108 + 0.999040i \(0.513950\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) −22.0000 −0.960159
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −48.0000 −2.07522
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) 0 0
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) −40.0000 −1.69791
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) 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
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 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
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) 0 0
\(593\) 36.0000 1.47834 0.739171 0.673517i \(-0.235217\pi\)
0.739171 + 0.673517i \(0.235217\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 20.0000 0.813116
\(606\) 0 0
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) −32.0000 −1.29036
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 0 0
\(619\) 14.0000 0.562708 0.281354 0.959604i \(-0.409217\pi\)
0.281354 + 0.959604i \(0.409217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 8.00000 0.319489
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) −30.0000 −1.19428 −0.597141 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) −48.0000 −1.90482
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) 26.0000 1.02534 0.512670 0.858586i \(-0.328656\pi\)
0.512670 + 0.858586i \(0.328656\pi\)
\(644\) 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
\(649\) 32.0000 1.25611
\(650\) 0 0
\(651\) 20.0000 0.783862
\(652\) 0 0
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) 10.0000 0.390137
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) 22.0000 0.848038 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(674\) 0 0
\(675\) 11.0000 0.423390
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 20.0000 0.765279 0.382639 0.923898i \(-0.375015\pi\)
0.382639 + 0.923898i \(0.375015\pi\)
\(684\) 0 0
\(685\) −32.0000 −1.22266
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 0 0
\(693\) 8.00000 0.303895
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) −16.0000 −0.602595
\(706\) 0 0
\(707\) −20.0000 −0.752177
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 0 0
\(725\) −66.0000 −2.45118
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) −12.0000 −0.442627
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 24.0000 0.876941
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 20.0000 0.728841
\(754\) 0 0
\(755\) 72.0000 2.62035
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −40.0000 −1.45000 −0.724999 0.688749i \(-0.758160\pi\)
−0.724999 + 0.688749i \(0.758160\pi\)
\(762\) 0 0
\(763\) −4.00000 −0.144810
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 0 0
\(773\) 48.0000 1.72644 0.863220 0.504828i \(-0.168444\pi\)
0.863220 + 0.504828i \(0.168444\pi\)
\(774\) 0 0
\(775\) −110.000 −3.95132
\(776\) 0 0
\(777\) 20.0000 0.717496
\(778\) 0 0
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 8.00000 0.285532
\(786\) 0 0
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) 0 0
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −40.0000 −1.41865
\(796\) 0 0
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 4.00000 0.141333
\(802\) 0 0
\(803\) −40.0000 −1.41157
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −14.0000 −0.492823
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) 0 0
\(813\) −10.0000 −0.350715
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 0 0
\(825\) −44.0000 −1.53188
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 22.0000 0.763172
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) −48.0000 −1.66111
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 8.00000 0.275535
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 0 0
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) −8.00000 −0.273594
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) 0 0
\(861\) 16.0000 0.545279
\(862\) 0 0
\(863\) 4.00000 0.136162 0.0680808 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 8.00000 0.272008
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) −48.0000 −1.62270
\(876\) 0 0
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 0 0
\(879\) −8.00000 −0.269833
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) 32.0000 1.07689 0.538443 0.842662i \(-0.319013\pi\)
0.538443 + 0.842662i \(0.319013\pi\)
\(884\) 0 0
\(885\) −32.0000 −1.07567
\(886\) 0 0
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) −48.0000 −1.60446
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) −20.0000 −0.666297
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) 0 0
\(905\) 8.00000 0.265929
\(906\) 0 0
\(907\) 56.0000 1.85945 0.929725 0.368255i \(-0.120045\pi\)
0.929725 + 0.368255i \(0.120045\pi\)
\(908\) 0 0
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −56.0000 −1.85130
\(916\) 0 0
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) 22.0000 0.724925
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −110.000 −3.61678
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −16.0000 −0.524943 −0.262471 0.964940i \(-0.584538\pi\)
−0.262471 + 0.964940i \(0.584538\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) −24.0000 −0.785725
\(934\) 0 0
\(935\) −32.0000 −1.04651
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 24.0000 0.775810
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 56.0000 1.80270
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) 0 0
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 48.0000 1.52941
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 0 0
\(993\) 2.00000 0.0634681
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 0 0
\(999\) −10.0000 −0.316386
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.bi.1.1 1
4.3 odd 2 2028.2.a.c.1.1 1
12.11 even 2 6084.2.a.b.1.1 1
13.12 even 2 624.2.a.e.1.1 1
39.38 odd 2 1872.2.a.s.1.1 1
52.3 odd 6 2028.2.i.g.529.1 2
52.7 even 12 2028.2.q.h.361.2 4
52.11 even 12 2028.2.q.h.1837.2 4
52.15 even 12 2028.2.q.h.1837.1 4
52.19 even 12 2028.2.q.h.361.1 4
52.23 odd 6 2028.2.i.e.529.1 2
52.31 even 4 2028.2.b.a.337.1 2
52.35 odd 6 2028.2.i.g.2005.1 2
52.43 odd 6 2028.2.i.e.2005.1 2
52.47 even 4 2028.2.b.a.337.2 2
52.51 odd 2 156.2.a.a.1.1 1
104.51 odd 2 2496.2.a.bc.1.1 1
104.77 even 2 2496.2.a.o.1.1 1
156.47 odd 4 6084.2.b.j.4393.1 2
156.83 odd 4 6084.2.b.j.4393.2 2
156.155 even 2 468.2.a.d.1.1 1
260.103 even 4 3900.2.h.b.1249.1 2
260.207 even 4 3900.2.h.b.1249.2 2
260.259 odd 2 3900.2.a.m.1.1 1
312.77 odd 2 7488.2.a.d.1.1 1
312.155 even 2 7488.2.a.c.1.1 1
364.363 even 2 7644.2.a.k.1.1 1
468.103 odd 6 4212.2.i.l.2809.1 2
468.155 even 6 4212.2.i.b.1405.1 2
468.259 odd 6 4212.2.i.l.1405.1 2
468.311 even 6 4212.2.i.b.2809.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.a.a.1.1 1 52.51 odd 2
468.2.a.d.1.1 1 156.155 even 2
624.2.a.e.1.1 1 13.12 even 2
1872.2.a.s.1.1 1 39.38 odd 2
2028.2.a.c.1.1 1 4.3 odd 2
2028.2.b.a.337.1 2 52.31 even 4
2028.2.b.a.337.2 2 52.47 even 4
2028.2.i.e.529.1 2 52.23 odd 6
2028.2.i.e.2005.1 2 52.43 odd 6
2028.2.i.g.529.1 2 52.3 odd 6
2028.2.i.g.2005.1 2 52.35 odd 6
2028.2.q.h.361.1 4 52.19 even 12
2028.2.q.h.361.2 4 52.7 even 12
2028.2.q.h.1837.1 4 52.15 even 12
2028.2.q.h.1837.2 4 52.11 even 12
2496.2.a.o.1.1 1 104.77 even 2
2496.2.a.bc.1.1 1 104.51 odd 2
3900.2.a.m.1.1 1 260.259 odd 2
3900.2.h.b.1249.1 2 260.103 even 4
3900.2.h.b.1249.2 2 260.207 even 4
4212.2.i.b.1405.1 2 468.155 even 6
4212.2.i.b.2809.1 2 468.311 even 6
4212.2.i.l.1405.1 2 468.259 odd 6
4212.2.i.l.2809.1 2 468.103 odd 6
6084.2.a.b.1.1 1 12.11 even 2
6084.2.b.j.4393.1 2 156.47 odd 4
6084.2.b.j.4393.2 2 156.83 odd 4
7488.2.a.c.1.1 1 312.155 even 2
7488.2.a.d.1.1 1 312.77 odd 2
7644.2.a.k.1.1 1 364.363 even 2
8112.2.a.bi.1.1 1 1.1 even 1 trivial