Properties

Label 1617.2.a.b.1.1
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
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] = [1617,2,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1617.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(12.9118100068\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} +1.00000 q^{9} -1.00000 q^{11} +2.00000 q^{12} +5.00000 q^{13} -4.00000 q^{16} -6.00000 q^{17} -2.00000 q^{18} -7.00000 q^{19} +2.00000 q^{22} -4.00000 q^{23} -5.00000 q^{25} -10.0000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +7.00000 q^{31} +8.00000 q^{32} -1.00000 q^{33} +12.0000 q^{34} +2.00000 q^{36} +7.00000 q^{37} +14.0000 q^{38} +5.00000 q^{39} -4.00000 q^{41} -9.00000 q^{43} -2.00000 q^{44} +8.00000 q^{46} -6.00000 q^{47} -4.00000 q^{48} +10.0000 q^{50} -6.00000 q^{51} +10.0000 q^{52} -2.00000 q^{53} -2.00000 q^{54} -7.00000 q^{57} +4.00000 q^{58} -12.0000 q^{59} +2.00000 q^{61} -14.0000 q^{62} -8.00000 q^{64} +2.00000 q^{66} +7.00000 q^{67} -12.0000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +5.00000 q^{73} -14.0000 q^{74} -5.00000 q^{75} -14.0000 q^{76} -10.0000 q^{78} -11.0000 q^{79} +1.00000 q^{81} +8.00000 q^{82} +4.00000 q^{83} +18.0000 q^{86} -2.00000 q^{87} -6.00000 q^{89} -8.00000 q^{92} +7.00000 q^{93} +12.0000 q^{94} +8.00000 q^{96} -2.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −2.00000 −0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.00000 0.577350
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −2.00000 −0.471405
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) −10.0000 −1.96116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 8.00000 1.41421
\(33\) −1.00000 −0.174078
\(34\) 12.0000 2.05798
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 7.00000 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 14.0000 2.27110
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) −4.00000 −0.577350
\(49\) 0 0
\(50\) 10.0000 1.41421
\(51\) −6.00000 −0.840168
\(52\) 10.0000 1.38675
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) −7.00000 −0.927173
\(58\) 4.00000 0.525226
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −14.0000 −1.77800
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) −12.0000 −1.45521
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 5.00000 0.585206 0.292603 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) −14.0000 −1.62747
\(75\) −5.00000 −0.577350
\(76\) −14.0000 −1.60591
\(77\) 0 0
\(78\) −10.0000 −1.13228
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 18.0000 1.94099
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 7.00000 0.725866
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −10.0000 −1.00000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 12.0000 1.18818
\(103\) 19.0000 1.87213 0.936063 0.351833i \(-0.114441\pi\)
0.936063 + 0.351833i \(0.114441\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 2.00000 0.192450
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 14.0000 1.31122
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 5.00000 0.462250
\(118\) 24.0000 2.20938
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) −4.00000 −0.360668
\(124\) 14.0000 1.25724
\(125\) 0 0
\(126\) 0 0
\(127\) −13.0000 −1.15356 −0.576782 0.816898i \(-0.695692\pi\)
−0.576782 + 0.816898i \(0.695692\pi\)
\(128\) 0 0
\(129\) −9.00000 −0.792406
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 8.00000 0.681005
\(139\) −11.0000 −0.933008 −0.466504 0.884519i \(-0.654487\pi\)
−0.466504 + 0.884519i \(0.654487\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) −16.0000 −1.34269
\(143\) −5.00000 −0.418121
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) 14.0000 1.15079
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 10.0000 0.816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 10.0000 0.800641
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 22.0000 1.75023
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) −2.00000 −0.157135
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −8.00000 −0.624695
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −7.00000 −0.535303
\(172\) −18.0000 −1.37249
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 4.00000 0.303239
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) −12.0000 −0.901975
\(178\) 12.0000 0.899438
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) −14.0000 −1.02653
\(187\) 6.00000 0.438763
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) 22.0000 1.59186 0.795932 0.605386i \(-0.206981\pi\)
0.795932 + 0.605386i \(0.206981\pi\)
\(192\) −8.00000 −0.577350
\(193\) −7.00000 −0.503871 −0.251936 0.967744i \(-0.581067\pi\)
−0.251936 + 0.967744i \(0.581067\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 0 0
\(197\) −4.00000 −0.284988 −0.142494 0.989796i \(-0.545512\pi\)
−0.142494 + 0.989796i \(0.545512\pi\)
\(198\) 2.00000 0.142134
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 7.00000 0.493742
\(202\) −4.00000 −0.281439
\(203\) 0 0
\(204\) −12.0000 −0.840168
\(205\) 0 0
\(206\) −38.0000 −2.64759
\(207\) −4.00000 −0.278019
\(208\) −20.0000 −1.38675
\(209\) 7.00000 0.484200
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −4.00000 −0.274721
\(213\) 8.00000 0.548151
\(214\) 40.0000 2.73434
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 5.00000 0.337869
\(220\) 0 0
\(221\) −30.0000 −2.01802
\(222\) −14.0000 −0.939618
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 12.0000 0.798228
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) −14.0000 −0.927173
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) −10.0000 −0.653720
\(235\) 0 0
\(236\) −24.0000 −1.56227
\(237\) −11.0000 −0.714527
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) −35.0000 −2.22700
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 26.0000 1.63139
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 18.0000 1.12063
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −8.00000 −0.494242
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 14.0000 0.855186
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 24.0000 1.45521
\(273\) 0 0
\(274\) 20.0000 1.20824
\(275\) 5.00000 0.301511
\(276\) −8.00000 −0.481543
\(277\) −1.00000 −0.0600842 −0.0300421 0.999549i \(-0.509564\pi\)
−0.0300421 + 0.999549i \(0.509564\pi\)
\(278\) 22.0000 1.31947
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 12.0000 0.714590
\(283\) −23.0000 −1.36721 −0.683604 0.729853i \(-0.739588\pi\)
−0.683604 + 0.729853i \(0.739588\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) 0 0
\(288\) 8.00000 0.471405
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 12.0000 0.695141
\(299\) −20.0000 −1.15663
\(300\) −10.0000 −0.577350
\(301\) 0 0
\(302\) 32.0000 1.84139
\(303\) 2.00000 0.114897
\(304\) 28.0000 1.60591
\(305\) 0 0
\(306\) 12.0000 0.685994
\(307\) 3.00000 0.171219 0.0856095 0.996329i \(-0.472716\pi\)
0.0856095 + 0.996329i \(0.472716\pi\)
\(308\) 0 0
\(309\) 19.0000 1.08087
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) −3.00000 −0.169570 −0.0847850 0.996399i \(-0.527020\pi\)
−0.0847850 + 0.996399i \(0.527020\pi\)
\(314\) −28.0000 −1.58013
\(315\) 0 0
\(316\) −22.0000 −1.23760
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 4.00000 0.224309
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) 42.0000 2.33694
\(324\) 2.00000 0.111111
\(325\) −25.0000 −1.38675
\(326\) 16.0000 0.886158
\(327\) −1.00000 −0.0553001
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −33.0000 −1.81384 −0.906922 0.421299i \(-0.861574\pi\)
−0.906922 + 0.421299i \(0.861574\pi\)
\(332\) 8.00000 0.439057
\(333\) 7.00000 0.383598
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) −24.0000 −1.30543
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) 14.0000 0.757033
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −20.0000 −1.07521
\(347\) 2.00000 0.107366 0.0536828 0.998558i \(-0.482904\pi\)
0.0536828 + 0.998558i \(0.482904\pi\)
\(348\) −4.00000 −0.214423
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) −8.00000 −0.426401
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 24.0000 1.27559
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 0 0
\(358\) 32.0000 1.69125
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) 10.0000 0.525588
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) 16.0000 0.834058
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 0 0
\(372\) 14.0000 0.725866
\(373\) 21.0000 1.08734 0.543669 0.839299i \(-0.317035\pi\)
0.543669 + 0.839299i \(0.317035\pi\)
\(374\) −12.0000 −0.620505
\(375\) 0 0
\(376\) 0 0
\(377\) −10.0000 −0.515026
\(378\) 0 0
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) −13.0000 −0.666010
\(382\) −44.0000 −2.25124
\(383\) 38.0000 1.94171 0.970855 0.239669i \(-0.0770389\pi\)
0.970855 + 0.239669i \(0.0770389\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) −9.00000 −0.457496
\(388\) −4.00000 −0.203069
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 8.00000 0.403034
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −27.0000 −1.35509 −0.677546 0.735481i \(-0.736956\pi\)
−0.677546 + 0.735481i \(0.736956\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 20.0000 1.00000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) −14.0000 −0.698257
\(403\) 35.0000 1.74347
\(404\) 4.00000 0.199007
\(405\) 0 0
\(406\) 0 0
\(407\) −7.00000 −0.346977
\(408\) 0 0
\(409\) −11.0000 −0.543915 −0.271957 0.962309i \(-0.587671\pi\)
−0.271957 + 0.962309i \(0.587671\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 38.0000 1.87213
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) 40.0000 1.96116
\(417\) −11.0000 −0.538672
\(418\) −14.0000 −0.684762
\(419\) 8.00000 0.390826 0.195413 0.980721i \(-0.437395\pi\)
0.195413 + 0.980721i \(0.437395\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 8.00000 0.389434
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 30.0000 1.45521
\(426\) −16.0000 −0.775203
\(427\) 0 0
\(428\) −40.0000 −1.93347
\(429\) −5.00000 −0.241402
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) −4.00000 −0.192450
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 28.0000 1.33942
\(438\) −10.0000 −0.477818
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 60.0000 2.85391
\(443\) 22.0000 1.04525 0.522626 0.852562i \(-0.324953\pi\)
0.522626 + 0.852562i \(0.324953\pi\)
\(444\) 14.0000 0.664411
\(445\) 0 0
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 10.0000 0.471405
\(451\) 4.00000 0.188353
\(452\) −12.0000 −0.564433
\(453\) −16.0000 −0.751746
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 36.0000 1.67669 0.838344 0.545142i \(-0.183524\pi\)
0.838344 + 0.545142i \(0.183524\pi\)
\(462\) 0 0
\(463\) −29.0000 −1.34774 −0.673872 0.738848i \(-0.735370\pi\)
−0.673872 + 0.738848i \(0.735370\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 40.0000 1.85296
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 10.0000 0.462250
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) 9.00000 0.413820
\(474\) 22.0000 1.01049
\(475\) 35.0000 1.60591
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) 48.0000 2.19547
\(479\) 38.0000 1.73626 0.868132 0.496333i \(-0.165321\pi\)
0.868132 + 0.496333i \(0.165321\pi\)
\(480\) 0 0
\(481\) 35.0000 1.59586
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) −2.00000 −0.0907218
\(487\) −29.0000 −1.31412 −0.657058 0.753840i \(-0.728199\pi\)
−0.657058 + 0.753840i \(0.728199\pi\)
\(488\) 0 0
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −30.0000 −1.35388 −0.676941 0.736038i \(-0.736695\pi\)
−0.676941 + 0.736038i \(0.736695\pi\)
\(492\) −8.00000 −0.360668
\(493\) 12.0000 0.540453
\(494\) 70.0000 3.14945
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) 13.0000 0.581960 0.290980 0.956729i \(-0.406019\pi\)
0.290980 + 0.956729i \(0.406019\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) −36.0000 −1.60676
\(503\) 44.0000 1.96186 0.980932 0.194354i \(-0.0622609\pi\)
0.980932 + 0.194354i \(0.0622609\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.00000 −0.355643
\(507\) 12.0000 0.532939
\(508\) −26.0000 −1.15356
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) −7.00000 −0.309058
\(514\) −16.0000 −0.705730
\(515\) 0 0
\(516\) −18.0000 −0.792406
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 4.00000 0.175075
\(523\) 31.0000 1.35554 0.677768 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 36.0000 1.56967
\(527\) −42.0000 −1.82955
\(528\) 4.00000 0.174078
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) −28.0000 −1.20717
\(539\) 0 0
\(540\) 0 0
\(541\) 15.0000 0.644900 0.322450 0.946586i \(-0.395494\pi\)
0.322450 + 0.946586i \(0.395494\pi\)
\(542\) −40.0000 −1.71815
\(543\) −5.00000 −0.214571
\(544\) −48.0000 −2.05798
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000 1.36822 0.684111 0.729378i \(-0.260191\pi\)
0.684111 + 0.729378i \(0.260191\pi\)
\(548\) −20.0000 −0.854358
\(549\) 2.00000 0.0853579
\(550\) −10.0000 −0.426401
\(551\) 14.0000 0.596420
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −22.0000 −0.933008
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) −14.0000 −0.592667
\(559\) −45.0000 −1.90330
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) −48.0000 −2.02476
\(563\) −18.0000 −0.758610 −0.379305 0.925272i \(-0.623837\pi\)
−0.379305 + 0.925272i \(0.623837\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 46.0000 1.93352
\(567\) 0 0
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) 35.0000 1.46470 0.732352 0.680926i \(-0.238422\pi\)
0.732352 + 0.680926i \(0.238422\pi\)
\(572\) −10.0000 −0.418121
\(573\) 22.0000 0.919063
\(574\) 0 0
\(575\) 20.0000 0.834058
\(576\) −8.00000 −0.333333
\(577\) 17.0000 0.707719 0.353860 0.935299i \(-0.384869\pi\)
0.353860 + 0.935299i \(0.384869\pi\)
\(578\) −38.0000 −1.58059
\(579\) −7.00000 −0.290910
\(580\) 0 0
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −49.0000 −2.01901
\(590\) 0 0
\(591\) −4.00000 −0.164538
\(592\) −28.0000 −1.15079
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) −4.00000 −0.163709
\(598\) 40.0000 1.63572
\(599\) 10.0000 0.408589 0.204294 0.978909i \(-0.434510\pi\)
0.204294 + 0.978909i \(0.434510\pi\)
\(600\) 0 0
\(601\) 15.0000 0.611863 0.305931 0.952054i \(-0.401032\pi\)
0.305931 + 0.952054i \(0.401032\pi\)
\(602\) 0 0
\(603\) 7.00000 0.285062
\(604\) −32.0000 −1.30206
\(605\) 0 0
\(606\) −4.00000 −0.162489
\(607\) −13.0000 −0.527654 −0.263827 0.964570i \(-0.584985\pi\)
−0.263827 + 0.964570i \(0.584985\pi\)
\(608\) −56.0000 −2.27110
\(609\) 0 0
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) −12.0000 −0.485071
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) −6.00000 −0.242140
\(615\) 0 0
\(616\) 0 0
\(617\) −24.0000 −0.966204 −0.483102 0.875564i \(-0.660490\pi\)
−0.483102 + 0.875564i \(0.660490\pi\)
\(618\) −38.0000 −1.52858
\(619\) 21.0000 0.844061 0.422031 0.906582i \(-0.361317\pi\)
0.422031 + 0.906582i \(0.361317\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −32.0000 −1.28308
\(623\) 0 0
\(624\) −20.0000 −0.800641
\(625\) 25.0000 1.00000
\(626\) 6.00000 0.239808
\(627\) 7.00000 0.279553
\(628\) 28.0000 1.11732
\(629\) −42.0000 −1.67465
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) −4.00000 −0.158610
\(637\) 0 0
\(638\) −4.00000 −0.158362
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 40.0000 1.57867
\(643\) −21.0000 −0.828159 −0.414080 0.910241i \(-0.635896\pi\)
−0.414080 + 0.910241i \(0.635896\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −84.0000 −3.30494
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 50.0000 1.96116
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 2.00000 0.0782062
\(655\) 0 0
\(656\) 16.0000 0.624695
\(657\) 5.00000 0.195069
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) 66.0000 2.56516
\(663\) −30.0000 −1.16510
\(664\) 0 0
\(665\) 0 0
\(666\) −14.0000 −0.542489
\(667\) 8.00000 0.309761
\(668\) 12.0000 0.464294
\(669\) 0 0
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 34.0000 1.30963
\(675\) −5.00000 −0.192450
\(676\) 24.0000 0.923077
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 14.0000 0.536088
\(683\) −40.0000 −1.53056 −0.765279 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) −14.0000 −0.535303
\(685\) 0 0
\(686\) 0 0
\(687\) −5.00000 −0.190762
\(688\) 36.0000 1.37249
\(689\) −10.0000 −0.380970
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 20.0000 0.760286
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 12.0000 0.454207
\(699\) −20.0000 −0.756469
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) −10.0000 −0.377426
\(703\) −49.0000 −1.84807
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 60.0000 2.25813
\(707\) 0 0
\(708\) −24.0000 −0.901975
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) 0 0
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) −32.0000 −1.19590
\(717\) −24.0000 −0.896296
\(718\) −72.0000 −2.68702
\(719\) 6.00000 0.223762 0.111881 0.993722i \(-0.464312\pi\)
0.111881 + 0.993722i \(0.464312\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −60.0000 −2.23297
\(723\) 10.0000 0.371904
\(724\) −10.0000 −0.371647
\(725\) 10.0000 0.371391
\(726\) −2.00000 −0.0742270
\(727\) 5.00000 0.185440 0.0927199 0.995692i \(-0.470444\pi\)
0.0927199 + 0.995692i \(0.470444\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 54.0000 1.99726
\(732\) 4.00000 0.147844
\(733\) 13.0000 0.480166 0.240083 0.970752i \(-0.422825\pi\)
0.240083 + 0.970752i \(0.422825\pi\)
\(734\) −34.0000 −1.25496
\(735\) 0 0
\(736\) −32.0000 −1.17954
\(737\) −7.00000 −0.257848
\(738\) 8.00000 0.294484
\(739\) −33.0000 −1.21392 −0.606962 0.794731i \(-0.707612\pi\)
−0.606962 + 0.794731i \(0.707612\pi\)
\(740\) 0 0
\(741\) −35.0000 −1.28576
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −42.0000 −1.53773
\(747\) 4.00000 0.146352
\(748\) 12.0000 0.438763
\(749\) 0 0
\(750\) 0 0
\(751\) −31.0000 −1.13121 −0.565603 0.824678i \(-0.691357\pi\)
−0.565603 + 0.824678i \(0.691357\pi\)
\(752\) 24.0000 0.875190
\(753\) 18.0000 0.655956
\(754\) 20.0000 0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 10.0000 0.363216
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 26.0000 0.941881
\(763\) 0 0
\(764\) 44.0000 1.59186
\(765\) 0 0
\(766\) −76.0000 −2.74599
\(767\) −60.0000 −2.16647
\(768\) 16.0000 0.577350
\(769\) 3.00000 0.108183 0.0540914 0.998536i \(-0.482774\pi\)
0.0540914 + 0.998536i \(0.482774\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) −14.0000 −0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 18.0000 0.646997
\(775\) −35.0000 −1.25724
\(776\) 0 0
\(777\) 0 0
\(778\) −44.0000 −1.57748
\(779\) 28.0000 1.00320
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) −48.0000 −1.71648
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 0 0
\(786\) −8.00000 −0.285351
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −8.00000 −0.284988
\(789\) −18.0000 −0.640817
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 54.0000 1.91639
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) 4.00000 0.141687 0.0708436 0.997487i \(-0.477431\pi\)
0.0708436 + 0.997487i \(0.477431\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) −40.0000 −1.41421
\(801\) −6.00000 −0.212000
\(802\) −72.0000 −2.54241
\(803\) −5.00000 −0.176446
\(804\) 14.0000 0.493742
\(805\) 0 0
\(806\) −70.0000 −2.46564
\(807\) 14.0000 0.492823
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 14.0000 0.490700
\(815\) 0 0
\(816\) 24.0000 0.840168
\(817\) 63.0000 2.20409
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 0 0
\(821\) 52.0000 1.81481 0.907406 0.420255i \(-0.138059\pi\)
0.907406 + 0.420255i \(0.138059\pi\)
\(822\) 20.0000 0.697580
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 5.00000 0.174078
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) −8.00000 −0.278019
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 0 0
\(831\) −1.00000 −0.0346896
\(832\) −40.0000 −1.38675
\(833\) 0 0
\(834\) 22.0000 0.761798
\(835\) 0 0
\(836\) 14.0000 0.484200
\(837\) 7.00000 0.241955
\(838\) −16.0000 −0.552711
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −18.0000 −0.620321
\(843\) 24.0000 0.826604
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 0 0
\(848\) 8.00000 0.274721
\(849\) −23.0000 −0.789358
\(850\) −60.0000 −2.05798
\(851\) −28.0000 −0.959828
\(852\) 16.0000 0.548151
\(853\) −17.0000 −0.582069 −0.291034 0.956713i \(-0.593999\pi\)
−0.291034 + 0.956713i \(0.593999\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 10.0000 0.341394
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −20.0000 −0.681203
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) 38.0000 1.29129
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 11.0000 0.373149
\(870\) 0 0
\(871\) 35.0000 1.18593
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) −56.0000 −1.89423
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 32.0000 1.07995
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −50.0000 −1.68454 −0.842271 0.539054i \(-0.818782\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 0 0
\(883\) −9.00000 −0.302874 −0.151437 0.988467i \(-0.548390\pi\)
−0.151437 + 0.988467i \(0.548390\pi\)
\(884\) −60.0000 −2.01802
\(885\) 0 0
\(886\) −44.0000 −1.47821
\(887\) −22.0000 −0.738688 −0.369344 0.929293i \(-0.620418\pi\)
−0.369344 + 0.929293i \(0.620418\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 42.0000 1.40548
\(894\) 12.0000 0.401340
\(895\) 0 0
\(896\) 0 0
\(897\) −20.0000 −0.667781
\(898\) 0 0
\(899\) −14.0000 −0.466926
\(900\) −10.0000 −0.333333
\(901\) 12.0000 0.399778
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 32.0000 1.06313
\(907\) 39.0000 1.29497 0.647487 0.762077i \(-0.275820\pi\)
0.647487 + 0.762077i \(0.275820\pi\)
\(908\) −28.0000 −0.929213
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 28.0000 0.927173
\(913\) −4.00000 −0.132381
\(914\) −6.00000 −0.198462
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 12.0000 0.396059
\(919\) −31.0000 −1.02260 −0.511298 0.859404i \(-0.670835\pi\)
−0.511298 + 0.859404i \(0.670835\pi\)
\(920\) 0 0
\(921\) 3.00000 0.0988534
\(922\) −72.0000 −2.37119
\(923\) 40.0000 1.31662
\(924\) 0 0
\(925\) −35.0000 −1.15079
\(926\) 58.0000 1.90600
\(927\) 19.0000 0.624042
\(928\) −16.0000 −0.525226
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −40.0000 −1.31024
\(933\) 16.0000 0.523816
\(934\) 16.0000 0.523536
\(935\) 0 0
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) 0 0
\(939\) −3.00000 −0.0979013
\(940\) 0 0
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) −28.0000 −0.912289
\(943\) 16.0000 0.521032
\(944\) 48.0000 1.56227
\(945\) 0 0
\(946\) −18.0000 −0.585230
\(947\) −54.0000 −1.75476 −0.877382 0.479792i \(-0.840712\pi\)
−0.877382 + 0.479792i \(0.840712\pi\)
\(948\) −22.0000 −0.714527
\(949\) 25.0000 0.811534
\(950\) −70.0000 −2.27110
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) 14.0000 0.453504 0.226752 0.973952i \(-0.427189\pi\)
0.226752 + 0.973952i \(0.427189\pi\)
\(954\) 4.00000 0.129505
\(955\) 0 0
\(956\) −48.0000 −1.55243
\(957\) 2.00000 0.0646508
\(958\) −76.0000 −2.45545
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −70.0000 −2.25689
\(963\) −20.0000 −0.644491
\(964\) 20.0000 0.644157
\(965\) 0 0
\(966\) 0 0
\(967\) 17.0000 0.546683 0.273342 0.961917i \(-0.411871\pi\)
0.273342 + 0.961917i \(0.411871\pi\)
\(968\) 0 0
\(969\) 42.0000 1.34923
\(970\) 0 0
\(971\) 8.00000 0.256732 0.128366 0.991727i \(-0.459027\pi\)
0.128366 + 0.991727i \(0.459027\pi\)
\(972\) 2.00000 0.0641500
\(973\) 0 0
\(974\) 58.0000 1.85844
\(975\) −25.0000 −0.800641
\(976\) −8.00000 −0.256074
\(977\) 52.0000 1.66363 0.831814 0.555055i \(-0.187303\pi\)
0.831814 + 0.555055i \(0.187303\pi\)
\(978\) 16.0000 0.511624
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) 60.0000 1.91468
\(983\) −34.0000 −1.08443 −0.542216 0.840239i \(-0.682414\pi\)
−0.542216 + 0.840239i \(0.682414\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) −70.0000 −2.22700
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 29.0000 0.921215 0.460608 0.887604i \(-0.347632\pi\)
0.460608 + 0.887604i \(0.347632\pi\)
\(992\) 56.0000 1.77800
\(993\) −33.0000 −1.04722
\(994\) 0 0
\(995\) 0 0
\(996\) 8.00000 0.253490
\(997\) 11.0000 0.348373 0.174187 0.984713i \(-0.444270\pi\)
0.174187 + 0.984713i \(0.444270\pi\)
\(998\) −26.0000 −0.823016
\(999\) 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 1617.2.a.b.1.1 1
3.2 odd 2 4851.2.a.s.1.1 1
7.3 odd 6 231.2.i.c.100.1 yes 2
7.5 odd 6 231.2.i.c.67.1 2
7.6 odd 2 1617.2.a.a.1.1 1
21.5 even 6 693.2.i.a.298.1 2
21.17 even 6 693.2.i.a.100.1 2
21.20 even 2 4851.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.i.c.67.1 2 7.5 odd 6
231.2.i.c.100.1 yes 2 7.3 odd 6
693.2.i.a.100.1 2 21.17 even 6
693.2.i.a.298.1 2 21.5 even 6
1617.2.a.a.1.1 1 7.6 odd 2
1617.2.a.b.1.1 1 1.1 even 1 trivial
4851.2.a.r.1.1 1 21.20 even 2
4851.2.a.s.1.1 1 3.2 odd 2