Properties

Label 5586.2.a.ce.1.8
Level $5586$
Weight $2$
Character 5586.1
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 60x^{5} + 87x^{4} - 176x^{3} - 40x^{2} + 64x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-0.514116\) of defining polynomial
Character \(\chi\) \(=\) 5586.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.93854 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.93854 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +3.93854 q^{10} +1.27293 q^{11} -1.00000 q^{12} +5.94914 q^{13} -3.93854 q^{15} +1.00000 q^{16} -3.05310 q^{17} +1.00000 q^{18} -1.00000 q^{19} +3.93854 q^{20} +1.27293 q^{22} -0.148774 q^{23} -1.00000 q^{24} +10.5121 q^{25} +5.94914 q^{26} -1.00000 q^{27} -6.60877 q^{29} -3.93854 q^{30} +8.01988 q^{31} +1.00000 q^{32} -1.27293 q^{33} -3.05310 q^{34} +1.00000 q^{36} +6.14824 q^{37} -1.00000 q^{38} -5.94914 q^{39} +3.93854 q^{40} -6.28274 q^{41} +7.91174 q^{43} +1.27293 q^{44} +3.93854 q^{45} -0.148774 q^{46} -8.22182 q^{47} -1.00000 q^{48} +10.5121 q^{50} +3.05310 q^{51} +5.94914 q^{52} +11.3312 q^{53} -1.00000 q^{54} +5.01349 q^{55} +1.00000 q^{57} -6.60877 q^{58} -1.84029 q^{59} -3.93854 q^{60} -12.1881 q^{61} +8.01988 q^{62} +1.00000 q^{64} +23.4309 q^{65} -1.27293 q^{66} +2.61205 q^{67} -3.05310 q^{68} +0.148774 q^{69} -12.8117 q^{71} +1.00000 q^{72} +16.2859 q^{73} +6.14824 q^{74} -10.5121 q^{75} -1.00000 q^{76} -5.94914 q^{78} +7.44674 q^{79} +3.93854 q^{80} +1.00000 q^{81} -6.28274 q^{82} -12.1746 q^{83} -12.0248 q^{85} +7.91174 q^{86} +6.60877 q^{87} +1.27293 q^{88} +5.40934 q^{89} +3.93854 q^{90} -0.148774 q^{92} -8.01988 q^{93} -8.22182 q^{94} -3.93854 q^{95} -1.00000 q^{96} -11.2277 q^{97} +1.27293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} - 4 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} - 4 q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} - 4 q^{10} + 8 q^{11} - 8 q^{12} + 4 q^{13} + 4 q^{15} + 8 q^{16} - 8 q^{17} + 8 q^{18} - 8 q^{19} - 4 q^{20} + 8 q^{22} + 4 q^{23} - 8 q^{24} + 24 q^{25} + 4 q^{26} - 8 q^{27} + 16 q^{29} + 4 q^{30} - 4 q^{31} + 8 q^{32} - 8 q^{33} - 8 q^{34} + 8 q^{36} + 12 q^{37} - 8 q^{38} - 4 q^{39} - 4 q^{40} + 16 q^{43} + 8 q^{44} - 4 q^{45} + 4 q^{46} - 12 q^{47} - 8 q^{48} + 24 q^{50} + 8 q^{51} + 4 q^{52} + 24 q^{53} - 8 q^{54} + 8 q^{55} + 8 q^{57} + 16 q^{58} + 8 q^{59} + 4 q^{60} - 4 q^{62} + 8 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} - 8 q^{68} - 4 q^{69} + 24 q^{71} + 8 q^{72} + 16 q^{73} + 12 q^{74} - 24 q^{75} - 8 q^{76} - 4 q^{78} + 20 q^{79} - 4 q^{80} + 8 q^{81} - 16 q^{83} + 48 q^{85} + 16 q^{86} - 16 q^{87} + 8 q^{88} - 4 q^{90} + 4 q^{92} + 4 q^{93} - 12 q^{94} + 4 q^{95} - 8 q^{96} + 8 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.93854 1.76137 0.880685 0.473702i \(-0.157083\pi\)
0.880685 + 0.473702i \(0.157083\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.93854 1.24548
\(11\) 1.27293 0.383803 0.191901 0.981414i \(-0.438535\pi\)
0.191901 + 0.981414i \(0.438535\pi\)
\(12\) −1.00000 −0.288675
\(13\) 5.94914 1.64999 0.824997 0.565137i \(-0.191177\pi\)
0.824997 + 0.565137i \(0.191177\pi\)
\(14\) 0 0
\(15\) −3.93854 −1.01693
\(16\) 1.00000 0.250000
\(17\) −3.05310 −0.740486 −0.370243 0.928935i \(-0.620726\pi\)
−0.370243 + 0.928935i \(0.620726\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 3.93854 0.880685
\(21\) 0 0
\(22\) 1.27293 0.271390
\(23\) −0.148774 −0.0310216 −0.0155108 0.999880i \(-0.504937\pi\)
−0.0155108 + 0.999880i \(0.504937\pi\)
\(24\) −1.00000 −0.204124
\(25\) 10.5121 2.10242
\(26\) 5.94914 1.16672
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.60877 −1.22722 −0.613609 0.789610i \(-0.710283\pi\)
−0.613609 + 0.789610i \(0.710283\pi\)
\(30\) −3.93854 −0.719076
\(31\) 8.01988 1.44041 0.720206 0.693760i \(-0.244047\pi\)
0.720206 + 0.693760i \(0.244047\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.27293 −0.221589
\(34\) −3.05310 −0.523603
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.14824 1.01076 0.505382 0.862896i \(-0.331352\pi\)
0.505382 + 0.862896i \(0.331352\pi\)
\(38\) −1.00000 −0.162221
\(39\) −5.94914 −0.952624
\(40\) 3.93854 0.622738
\(41\) −6.28274 −0.981198 −0.490599 0.871385i \(-0.663222\pi\)
−0.490599 + 0.871385i \(0.663222\pi\)
\(42\) 0 0
\(43\) 7.91174 1.20653 0.603264 0.797541i \(-0.293866\pi\)
0.603264 + 0.797541i \(0.293866\pi\)
\(44\) 1.27293 0.191901
\(45\) 3.93854 0.587123
\(46\) −0.148774 −0.0219356
\(47\) −8.22182 −1.19928 −0.599638 0.800272i \(-0.704689\pi\)
−0.599638 + 0.800272i \(0.704689\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 10.5121 1.48664
\(51\) 3.05310 0.427520
\(52\) 5.94914 0.824997
\(53\) 11.3312 1.55646 0.778232 0.627977i \(-0.216117\pi\)
0.778232 + 0.627977i \(0.216117\pi\)
\(54\) −1.00000 −0.136083
\(55\) 5.01349 0.676019
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −6.60877 −0.867774
\(59\) −1.84029 −0.239585 −0.119793 0.992799i \(-0.538223\pi\)
−0.119793 + 0.992799i \(0.538223\pi\)
\(60\) −3.93854 −0.508464
\(61\) −12.1881 −1.56053 −0.780266 0.625448i \(-0.784916\pi\)
−0.780266 + 0.625448i \(0.784916\pi\)
\(62\) 8.01988 1.01853
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 23.4309 2.90625
\(66\) −1.27293 −0.156687
\(67\) 2.61205 0.319112 0.159556 0.987189i \(-0.448994\pi\)
0.159556 + 0.987189i \(0.448994\pi\)
\(68\) −3.05310 −0.370243
\(69\) 0.148774 0.0179103
\(70\) 0 0
\(71\) −12.8117 −1.52047 −0.760236 0.649647i \(-0.774917\pi\)
−0.760236 + 0.649647i \(0.774917\pi\)
\(72\) 1.00000 0.117851
\(73\) 16.2859 1.90612 0.953059 0.302784i \(-0.0979162\pi\)
0.953059 + 0.302784i \(0.0979162\pi\)
\(74\) 6.14824 0.714718
\(75\) −10.5121 −1.21384
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −5.94914 −0.673607
\(79\) 7.44674 0.837824 0.418912 0.908027i \(-0.362412\pi\)
0.418912 + 0.908027i \(0.362412\pi\)
\(80\) 3.93854 0.440343
\(81\) 1.00000 0.111111
\(82\) −6.28274 −0.693812
\(83\) −12.1746 −1.33634 −0.668170 0.744009i \(-0.732922\pi\)
−0.668170 + 0.744009i \(0.732922\pi\)
\(84\) 0 0
\(85\) −12.0248 −1.30427
\(86\) 7.91174 0.853144
\(87\) 6.60877 0.708534
\(88\) 1.27293 0.135695
\(89\) 5.40934 0.573389 0.286694 0.958022i \(-0.407444\pi\)
0.286694 + 0.958022i \(0.407444\pi\)
\(90\) 3.93854 0.415159
\(91\) 0 0
\(92\) −0.148774 −0.0155108
\(93\) −8.01988 −0.831623
\(94\) −8.22182 −0.848016
\(95\) −3.93854 −0.404086
\(96\) −1.00000 −0.102062
\(97\) −11.2277 −1.14000 −0.570002 0.821643i \(-0.693058\pi\)
−0.570002 + 0.821643i \(0.693058\pi\)
\(98\) 0 0
\(99\) 1.27293 0.127934
\(100\) 10.5121 1.05121
\(101\) −4.38789 −0.436612 −0.218306 0.975880i \(-0.570053\pi\)
−0.218306 + 0.975880i \(0.570053\pi\)
\(102\) 3.05310 0.302302
\(103\) −6.48370 −0.638858 −0.319429 0.947610i \(-0.603491\pi\)
−0.319429 + 0.947610i \(0.603491\pi\)
\(104\) 5.94914 0.583361
\(105\) 0 0
\(106\) 11.3312 1.10059
\(107\) 4.09675 0.396047 0.198024 0.980197i \(-0.436548\pi\)
0.198024 + 0.980197i \(0.436548\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 5.93860 0.568815 0.284407 0.958704i \(-0.408203\pi\)
0.284407 + 0.958704i \(0.408203\pi\)
\(110\) 5.01349 0.478017
\(111\) −6.14824 −0.583565
\(112\) 0 0
\(113\) 3.83385 0.360658 0.180329 0.983606i \(-0.442284\pi\)
0.180329 + 0.983606i \(0.442284\pi\)
\(114\) 1.00000 0.0936586
\(115\) −0.585954 −0.0546405
\(116\) −6.60877 −0.613609
\(117\) 5.94914 0.549998
\(118\) −1.84029 −0.169412
\(119\) 0 0
\(120\) −3.93854 −0.359538
\(121\) −9.37965 −0.852695
\(122\) −12.1881 −1.10346
\(123\) 6.28274 0.566495
\(124\) 8.01988 0.720206
\(125\) 21.7097 1.94178
\(126\) 0 0
\(127\) −15.4576 −1.37164 −0.685821 0.727771i \(-0.740557\pi\)
−0.685821 + 0.727771i \(0.740557\pi\)
\(128\) 1.00000 0.0883883
\(129\) −7.91174 −0.696589
\(130\) 23.4309 2.05503
\(131\) 5.76015 0.503267 0.251633 0.967823i \(-0.419032\pi\)
0.251633 + 0.967823i \(0.419032\pi\)
\(132\) −1.27293 −0.110794
\(133\) 0 0
\(134\) 2.61205 0.225646
\(135\) −3.93854 −0.338976
\(136\) −3.05310 −0.261801
\(137\) 16.2798 1.39088 0.695440 0.718584i \(-0.255209\pi\)
0.695440 + 0.718584i \(0.255209\pi\)
\(138\) 0.148774 0.0126645
\(139\) −8.24825 −0.699608 −0.349804 0.936823i \(-0.613752\pi\)
−0.349804 + 0.936823i \(0.613752\pi\)
\(140\) 0 0
\(141\) 8.22182 0.692402
\(142\) −12.8117 −1.07514
\(143\) 7.57283 0.633272
\(144\) 1.00000 0.0833333
\(145\) −26.0289 −2.16158
\(146\) 16.2859 1.34783
\(147\) 0 0
\(148\) 6.14824 0.505382
\(149\) −5.92450 −0.485354 −0.242677 0.970107i \(-0.578026\pi\)
−0.242677 + 0.970107i \(0.578026\pi\)
\(150\) −10.5121 −0.858311
\(151\) −1.09856 −0.0893992 −0.0446996 0.999000i \(-0.514233\pi\)
−0.0446996 + 0.999000i \(0.514233\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −3.05310 −0.246829
\(154\) 0 0
\(155\) 31.5866 2.53710
\(156\) −5.94914 −0.476312
\(157\) 8.29607 0.662098 0.331049 0.943613i \(-0.392597\pi\)
0.331049 + 0.943613i \(0.392597\pi\)
\(158\) 7.44674 0.592431
\(159\) −11.3312 −0.898625
\(160\) 3.93854 0.311369
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 12.5954 0.986550 0.493275 0.869873i \(-0.335800\pi\)
0.493275 + 0.869873i \(0.335800\pi\)
\(164\) −6.28274 −0.490599
\(165\) −5.01349 −0.390300
\(166\) −12.1746 −0.944935
\(167\) −9.65057 −0.746784 −0.373392 0.927674i \(-0.621805\pi\)
−0.373392 + 0.927674i \(0.621805\pi\)
\(168\) 0 0
\(169\) 22.3922 1.72248
\(170\) −12.0248 −0.922258
\(171\) −1.00000 −0.0764719
\(172\) 7.91174 0.603264
\(173\) −14.1088 −1.07267 −0.536335 0.844005i \(-0.680192\pi\)
−0.536335 + 0.844005i \(0.680192\pi\)
\(174\) 6.60877 0.501009
\(175\) 0 0
\(176\) 1.27293 0.0959507
\(177\) 1.84029 0.138325
\(178\) 5.40934 0.405447
\(179\) −12.4256 −0.928731 −0.464365 0.885644i \(-0.653718\pi\)
−0.464365 + 0.885644i \(0.653718\pi\)
\(180\) 3.93854 0.293562
\(181\) 12.3729 0.919671 0.459836 0.888004i \(-0.347908\pi\)
0.459836 + 0.888004i \(0.347908\pi\)
\(182\) 0 0
\(183\) 12.1881 0.900973
\(184\) −0.148774 −0.0109678
\(185\) 24.2151 1.78033
\(186\) −8.01988 −0.588046
\(187\) −3.88638 −0.284201
\(188\) −8.22182 −0.599638
\(189\) 0 0
\(190\) −3.93854 −0.285732
\(191\) −4.07435 −0.294810 −0.147405 0.989076i \(-0.547092\pi\)
−0.147405 + 0.989076i \(0.547092\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −4.50280 −0.324119 −0.162059 0.986781i \(-0.551814\pi\)
−0.162059 + 0.986781i \(0.551814\pi\)
\(194\) −11.2277 −0.806105
\(195\) −23.4309 −1.67792
\(196\) 0 0
\(197\) 0.322407 0.0229706 0.0114853 0.999934i \(-0.496344\pi\)
0.0114853 + 0.999934i \(0.496344\pi\)
\(198\) 1.27293 0.0904632
\(199\) −22.6125 −1.60296 −0.801478 0.598024i \(-0.795953\pi\)
−0.801478 + 0.598024i \(0.795953\pi\)
\(200\) 10.5121 0.743319
\(201\) −2.61205 −0.184239
\(202\) −4.38789 −0.308731
\(203\) 0 0
\(204\) 3.05310 0.213760
\(205\) −24.7448 −1.72825
\(206\) −6.48370 −0.451741
\(207\) −0.148774 −0.0103405
\(208\) 5.94914 0.412498
\(209\) −1.27293 −0.0880504
\(210\) 0 0
\(211\) 0.346997 0.0238882 0.0119441 0.999929i \(-0.496198\pi\)
0.0119441 + 0.999929i \(0.496198\pi\)
\(212\) 11.3312 0.778232
\(213\) 12.8117 0.877844
\(214\) 4.09675 0.280048
\(215\) 31.1607 2.12514
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 5.93860 0.402213
\(219\) −16.2859 −1.10050
\(220\) 5.01349 0.338009
\(221\) −18.1633 −1.22180
\(222\) −6.14824 −0.412643
\(223\) 2.29672 0.153800 0.0769000 0.997039i \(-0.475498\pi\)
0.0769000 + 0.997039i \(0.475498\pi\)
\(224\) 0 0
\(225\) 10.5121 0.700808
\(226\) 3.83385 0.255024
\(227\) −15.3728 −1.02033 −0.510166 0.860076i \(-0.670416\pi\)
−0.510166 + 0.860076i \(0.670416\pi\)
\(228\) 1.00000 0.0662266
\(229\) 11.7178 0.774336 0.387168 0.922009i \(-0.373453\pi\)
0.387168 + 0.922009i \(0.373453\pi\)
\(230\) −0.585954 −0.0386367
\(231\) 0 0
\(232\) −6.60877 −0.433887
\(233\) 13.7683 0.901989 0.450995 0.892527i \(-0.351069\pi\)
0.450995 + 0.892527i \(0.351069\pi\)
\(234\) 5.94914 0.388907
\(235\) −32.3820 −2.11237
\(236\) −1.84029 −0.119793
\(237\) −7.44674 −0.483718
\(238\) 0 0
\(239\) 27.3226 1.76735 0.883676 0.468100i \(-0.155061\pi\)
0.883676 + 0.468100i \(0.155061\pi\)
\(240\) −3.93854 −0.254232
\(241\) 29.4781 1.89885 0.949427 0.313988i \(-0.101665\pi\)
0.949427 + 0.313988i \(0.101665\pi\)
\(242\) −9.37965 −0.602947
\(243\) −1.00000 −0.0641500
\(244\) −12.1881 −0.780266
\(245\) 0 0
\(246\) 6.28274 0.400573
\(247\) −5.94914 −0.378534
\(248\) 8.01988 0.509263
\(249\) 12.1746 0.771536
\(250\) 21.7097 1.37304
\(251\) −22.6060 −1.42688 −0.713440 0.700716i \(-0.752864\pi\)
−0.713440 + 0.700716i \(0.752864\pi\)
\(252\) 0 0
\(253\) −0.189379 −0.0119062
\(254\) −15.4576 −0.969897
\(255\) 12.0248 0.753021
\(256\) 1.00000 0.0625000
\(257\) −25.9880 −1.62109 −0.810543 0.585679i \(-0.800828\pi\)
−0.810543 + 0.585679i \(0.800828\pi\)
\(258\) −7.91174 −0.492563
\(259\) 0 0
\(260\) 23.4309 1.45312
\(261\) −6.60877 −0.409072
\(262\) 5.76015 0.355863
\(263\) 6.16848 0.380365 0.190182 0.981749i \(-0.439092\pi\)
0.190182 + 0.981749i \(0.439092\pi\)
\(264\) −1.27293 −0.0783434
\(265\) 44.6285 2.74151
\(266\) 0 0
\(267\) −5.40934 −0.331046
\(268\) 2.61205 0.159556
\(269\) −10.1268 −0.617442 −0.308721 0.951153i \(-0.599901\pi\)
−0.308721 + 0.951153i \(0.599901\pi\)
\(270\) −3.93854 −0.239692
\(271\) −10.9731 −0.666568 −0.333284 0.942826i \(-0.608157\pi\)
−0.333284 + 0.942826i \(0.608157\pi\)
\(272\) −3.05310 −0.185122
\(273\) 0 0
\(274\) 16.2798 0.983501
\(275\) 13.3812 0.806916
\(276\) 0.148774 0.00895516
\(277\) 2.90574 0.174589 0.0872946 0.996183i \(-0.472178\pi\)
0.0872946 + 0.996183i \(0.472178\pi\)
\(278\) −8.24825 −0.494697
\(279\) 8.01988 0.480138
\(280\) 0 0
\(281\) 13.5310 0.807191 0.403595 0.914938i \(-0.367760\pi\)
0.403595 + 0.914938i \(0.367760\pi\)
\(282\) 8.22182 0.489602
\(283\) −15.3969 −0.915252 −0.457626 0.889145i \(-0.651300\pi\)
−0.457626 + 0.889145i \(0.651300\pi\)
\(284\) −12.8117 −0.760236
\(285\) 3.93854 0.233299
\(286\) 7.57283 0.447791
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −7.67857 −0.451680
\(290\) −26.0289 −1.52847
\(291\) 11.2277 0.658182
\(292\) 16.2859 0.953059
\(293\) 14.5720 0.851307 0.425654 0.904886i \(-0.360044\pi\)
0.425654 + 0.904886i \(0.360044\pi\)
\(294\) 0 0
\(295\) −7.24806 −0.421998
\(296\) 6.14824 0.357359
\(297\) −1.27293 −0.0738629
\(298\) −5.92450 −0.343197
\(299\) −0.885079 −0.0511854
\(300\) −10.5121 −0.606918
\(301\) 0 0
\(302\) −1.09856 −0.0632148
\(303\) 4.38789 0.252078
\(304\) −1.00000 −0.0573539
\(305\) −48.0035 −2.74867
\(306\) −3.05310 −0.174534
\(307\) 26.2382 1.49749 0.748745 0.662858i \(-0.230657\pi\)
0.748745 + 0.662858i \(0.230657\pi\)
\(308\) 0 0
\(309\) 6.48370 0.368845
\(310\) 31.5866 1.79400
\(311\) −7.89074 −0.447443 −0.223721 0.974653i \(-0.571821\pi\)
−0.223721 + 0.974653i \(0.571821\pi\)
\(312\) −5.94914 −0.336804
\(313\) 17.3496 0.980655 0.490327 0.871538i \(-0.336877\pi\)
0.490327 + 0.871538i \(0.336877\pi\)
\(314\) 8.29607 0.468174
\(315\) 0 0
\(316\) 7.44674 0.418912
\(317\) 31.0193 1.74222 0.871108 0.491092i \(-0.163402\pi\)
0.871108 + 0.491092i \(0.163402\pi\)
\(318\) −11.3312 −0.635424
\(319\) −8.41250 −0.471009
\(320\) 3.93854 0.220171
\(321\) −4.09675 −0.228658
\(322\) 0 0
\(323\) 3.05310 0.169879
\(324\) 1.00000 0.0555556
\(325\) 62.5381 3.46899
\(326\) 12.5954 0.697597
\(327\) −5.93860 −0.328405
\(328\) −6.28274 −0.346906
\(329\) 0 0
\(330\) −5.01349 −0.275983
\(331\) −27.7615 −1.52591 −0.762954 0.646453i \(-0.776252\pi\)
−0.762954 + 0.646453i \(0.776252\pi\)
\(332\) −12.1746 −0.668170
\(333\) 6.14824 0.336921
\(334\) −9.65057 −0.528056
\(335\) 10.2877 0.562075
\(336\) 0 0
\(337\) −11.9053 −0.648523 −0.324261 0.945968i \(-0.605116\pi\)
−0.324261 + 0.945968i \(0.605116\pi\)
\(338\) 22.3922 1.21798
\(339\) −3.83385 −0.208226
\(340\) −12.0248 −0.652135
\(341\) 10.2087 0.552834
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) 7.91174 0.426572
\(345\) 0.585954 0.0315467
\(346\) −14.1088 −0.758493
\(347\) 18.8103 1.00979 0.504895 0.863181i \(-0.331531\pi\)
0.504895 + 0.863181i \(0.331531\pi\)
\(348\) 6.60877 0.354267
\(349\) 8.76894 0.469391 0.234695 0.972069i \(-0.424591\pi\)
0.234695 + 0.972069i \(0.424591\pi\)
\(350\) 0 0
\(351\) −5.94914 −0.317541
\(352\) 1.27293 0.0678474
\(353\) 29.8162 1.58696 0.793478 0.608599i \(-0.208268\pi\)
0.793478 + 0.608599i \(0.208268\pi\)
\(354\) 1.84029 0.0978103
\(355\) −50.4595 −2.67811
\(356\) 5.40934 0.286694
\(357\) 0 0
\(358\) −12.4256 −0.656712
\(359\) 17.1263 0.903890 0.451945 0.892046i \(-0.350730\pi\)
0.451945 + 0.892046i \(0.350730\pi\)
\(360\) 3.93854 0.207579
\(361\) 1.00000 0.0526316
\(362\) 12.3729 0.650306
\(363\) 9.37965 0.492304
\(364\) 0 0
\(365\) 64.1426 3.35738
\(366\) 12.1881 0.637084
\(367\) −24.3662 −1.27190 −0.635952 0.771728i \(-0.719393\pi\)
−0.635952 + 0.771728i \(0.719393\pi\)
\(368\) −0.148774 −0.00775540
\(369\) −6.28274 −0.327066
\(370\) 24.2151 1.25888
\(371\) 0 0
\(372\) −8.01988 −0.415811
\(373\) 24.1414 1.24999 0.624997 0.780627i \(-0.285100\pi\)
0.624997 + 0.780627i \(0.285100\pi\)
\(374\) −3.88638 −0.200960
\(375\) −21.7097 −1.12109
\(376\) −8.22182 −0.424008
\(377\) −39.3165 −2.02490
\(378\) 0 0
\(379\) 8.51695 0.437486 0.218743 0.975782i \(-0.429804\pi\)
0.218743 + 0.975782i \(0.429804\pi\)
\(380\) −3.93854 −0.202043
\(381\) 15.4576 0.791917
\(382\) −4.07435 −0.208462
\(383\) 17.3654 0.887331 0.443665 0.896193i \(-0.353678\pi\)
0.443665 + 0.896193i \(0.353678\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −4.50280 −0.229187
\(387\) 7.91174 0.402176
\(388\) −11.2277 −0.570002
\(389\) −22.5117 −1.14139 −0.570694 0.821163i \(-0.693326\pi\)
−0.570694 + 0.821163i \(0.693326\pi\)
\(390\) −23.4309 −1.18647
\(391\) 0.454223 0.0229711
\(392\) 0 0
\(393\) −5.76015 −0.290561
\(394\) 0.322407 0.0162426
\(395\) 29.3293 1.47572
\(396\) 1.27293 0.0639671
\(397\) −33.8779 −1.70028 −0.850140 0.526556i \(-0.823483\pi\)
−0.850140 + 0.526556i \(0.823483\pi\)
\(398\) −22.6125 −1.13346
\(399\) 0 0
\(400\) 10.5121 0.525606
\(401\) 33.1403 1.65495 0.827474 0.561504i \(-0.189777\pi\)
0.827474 + 0.561504i \(0.189777\pi\)
\(402\) −2.61205 −0.130277
\(403\) 47.7114 2.37667
\(404\) −4.38789 −0.218306
\(405\) 3.93854 0.195708
\(406\) 0 0
\(407\) 7.82627 0.387934
\(408\) 3.05310 0.151151
\(409\) 28.2927 1.39899 0.699493 0.714639i \(-0.253409\pi\)
0.699493 + 0.714639i \(0.253409\pi\)
\(410\) −24.7448 −1.22206
\(411\) −16.2798 −0.803025
\(412\) −6.48370 −0.319429
\(413\) 0 0
\(414\) −0.148774 −0.00731186
\(415\) −47.9503 −2.35379
\(416\) 5.94914 0.291680
\(417\) 8.24825 0.403919
\(418\) −1.27293 −0.0622610
\(419\) −24.5157 −1.19767 −0.598836 0.800872i \(-0.704370\pi\)
−0.598836 + 0.800872i \(0.704370\pi\)
\(420\) 0 0
\(421\) −36.4678 −1.77733 −0.888667 0.458553i \(-0.848368\pi\)
−0.888667 + 0.458553i \(0.848368\pi\)
\(422\) 0.346997 0.0168915
\(423\) −8.22182 −0.399758
\(424\) 11.3312 0.550293
\(425\) −32.0946 −1.55682
\(426\) 12.8117 0.620730
\(427\) 0 0
\(428\) 4.09675 0.198024
\(429\) −7.57283 −0.365620
\(430\) 31.1607 1.50270
\(431\) 7.38080 0.355521 0.177760 0.984074i \(-0.443115\pi\)
0.177760 + 0.984074i \(0.443115\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 11.4859 0.551975 0.275988 0.961161i \(-0.410995\pi\)
0.275988 + 0.961161i \(0.410995\pi\)
\(434\) 0 0
\(435\) 26.0289 1.24799
\(436\) 5.93860 0.284407
\(437\) 0.148774 0.00711684
\(438\) −16.2859 −0.778169
\(439\) 22.2521 1.06204 0.531018 0.847360i \(-0.321809\pi\)
0.531018 + 0.847360i \(0.321809\pi\)
\(440\) 5.01349 0.239009
\(441\) 0 0
\(442\) −18.1633 −0.863941
\(443\) −22.7804 −1.08233 −0.541165 0.840917i \(-0.682016\pi\)
−0.541165 + 0.840917i \(0.682016\pi\)
\(444\) −6.14824 −0.291782
\(445\) 21.3049 1.00995
\(446\) 2.29672 0.108753
\(447\) 5.92450 0.280219
\(448\) 0 0
\(449\) −8.19585 −0.386786 −0.193393 0.981121i \(-0.561949\pi\)
−0.193393 + 0.981121i \(0.561949\pi\)
\(450\) 10.5121 0.495546
\(451\) −7.99748 −0.376587
\(452\) 3.83385 0.180329
\(453\) 1.09856 0.0516146
\(454\) −15.3728 −0.721483
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −17.9078 −0.837691 −0.418846 0.908058i \(-0.637565\pi\)
−0.418846 + 0.908058i \(0.637565\pi\)
\(458\) 11.7178 0.547539
\(459\) 3.05310 0.142507
\(460\) −0.585954 −0.0273203
\(461\) 10.8937 0.507372 0.253686 0.967287i \(-0.418357\pi\)
0.253686 + 0.967287i \(0.418357\pi\)
\(462\) 0 0
\(463\) −39.6911 −1.84460 −0.922300 0.386475i \(-0.873693\pi\)
−0.922300 + 0.386475i \(0.873693\pi\)
\(464\) −6.60877 −0.306804
\(465\) −31.5866 −1.46480
\(466\) 13.7683 0.637803
\(467\) −32.0868 −1.48480 −0.742400 0.669956i \(-0.766313\pi\)
−0.742400 + 0.669956i \(0.766313\pi\)
\(468\) 5.94914 0.274999
\(469\) 0 0
\(470\) −32.3820 −1.49367
\(471\) −8.29607 −0.382263
\(472\) −1.84029 −0.0847062
\(473\) 10.0711 0.463069
\(474\) −7.44674 −0.342040
\(475\) −10.5121 −0.482329
\(476\) 0 0
\(477\) 11.3312 0.518821
\(478\) 27.3226 1.24971
\(479\) 21.5125 0.982933 0.491466 0.870897i \(-0.336461\pi\)
0.491466 + 0.870897i \(0.336461\pi\)
\(480\) −3.93854 −0.179769
\(481\) 36.5767 1.66775
\(482\) 29.4781 1.34269
\(483\) 0 0
\(484\) −9.37965 −0.426348
\(485\) −44.2209 −2.00797
\(486\) −1.00000 −0.0453609
\(487\) −22.7496 −1.03088 −0.515441 0.856925i \(-0.672372\pi\)
−0.515441 + 0.856925i \(0.672372\pi\)
\(488\) −12.1881 −0.551731
\(489\) −12.5954 −0.569585
\(490\) 0 0
\(491\) −26.4966 −1.19577 −0.597887 0.801581i \(-0.703993\pi\)
−0.597887 + 0.801581i \(0.703993\pi\)
\(492\) 6.28274 0.283248
\(493\) 20.1772 0.908737
\(494\) −5.94914 −0.267664
\(495\) 5.01349 0.225340
\(496\) 8.01988 0.360103
\(497\) 0 0
\(498\) 12.1746 0.545558
\(499\) 20.4302 0.914580 0.457290 0.889318i \(-0.348820\pi\)
0.457290 + 0.889318i \(0.348820\pi\)
\(500\) 21.7097 0.970889
\(501\) 9.65057 0.431156
\(502\) −22.6060 −1.00896
\(503\) 0.0114171 0.000509063 0 0.000254532 1.00000i \(-0.499919\pi\)
0.000254532 1.00000i \(0.499919\pi\)
\(504\) 0 0
\(505\) −17.2819 −0.769035
\(506\) −0.189379 −0.00841894
\(507\) −22.3922 −0.994474
\(508\) −15.4576 −0.685821
\(509\) −17.0678 −0.756518 −0.378259 0.925700i \(-0.623477\pi\)
−0.378259 + 0.925700i \(0.623477\pi\)
\(510\) 12.0248 0.532466
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −25.9880 −1.14628
\(515\) −25.5363 −1.12527
\(516\) −7.91174 −0.348295
\(517\) −10.4658 −0.460285
\(518\) 0 0
\(519\) 14.1088 0.619307
\(520\) 23.4309 1.02751
\(521\) −6.84370 −0.299828 −0.149914 0.988699i \(-0.547900\pi\)
−0.149914 + 0.988699i \(0.547900\pi\)
\(522\) −6.60877 −0.289258
\(523\) 10.9826 0.480237 0.240119 0.970744i \(-0.422814\pi\)
0.240119 + 0.970744i \(0.422814\pi\)
\(524\) 5.76015 0.251633
\(525\) 0 0
\(526\) 6.16848 0.268958
\(527\) −24.4855 −1.06661
\(528\) −1.27293 −0.0553972
\(529\) −22.9779 −0.999038
\(530\) 44.6285 1.93854
\(531\) −1.84029 −0.0798618
\(532\) 0 0
\(533\) −37.3769 −1.61897
\(534\) −5.40934 −0.234085
\(535\) 16.1352 0.697586
\(536\) 2.61205 0.112823
\(537\) 12.4256 0.536203
\(538\) −10.1268 −0.436597
\(539\) 0 0
\(540\) −3.93854 −0.169488
\(541\) 18.7473 0.806011 0.403006 0.915197i \(-0.367965\pi\)
0.403006 + 0.915197i \(0.367965\pi\)
\(542\) −10.9731 −0.471335
\(543\) −12.3729 −0.530973
\(544\) −3.05310 −0.130901
\(545\) 23.3894 1.00189
\(546\) 0 0
\(547\) −5.12794 −0.219255 −0.109628 0.993973i \(-0.534966\pi\)
−0.109628 + 0.993973i \(0.534966\pi\)
\(548\) 16.2798 0.695440
\(549\) −12.1881 −0.520177
\(550\) 13.3812 0.570576
\(551\) 6.60877 0.281543
\(552\) 0.148774 0.00633226
\(553\) 0 0
\(554\) 2.90574 0.123453
\(555\) −24.2151 −1.02787
\(556\) −8.24825 −0.349804
\(557\) −19.8712 −0.841970 −0.420985 0.907068i \(-0.638316\pi\)
−0.420985 + 0.907068i \(0.638316\pi\)
\(558\) 8.01988 0.339509
\(559\) 47.0680 1.99076
\(560\) 0 0
\(561\) 3.88638 0.164083
\(562\) 13.5310 0.570770
\(563\) 5.45760 0.230010 0.115005 0.993365i \(-0.463312\pi\)
0.115005 + 0.993365i \(0.463312\pi\)
\(564\) 8.22182 0.346201
\(565\) 15.0998 0.635252
\(566\) −15.3969 −0.647181
\(567\) 0 0
\(568\) −12.8117 −0.537568
\(569\) 5.92409 0.248351 0.124175 0.992260i \(-0.460371\pi\)
0.124175 + 0.992260i \(0.460371\pi\)
\(570\) 3.93854 0.164967
\(571\) 22.8266 0.955265 0.477633 0.878560i \(-0.341495\pi\)
0.477633 + 0.878560i \(0.341495\pi\)
\(572\) 7.57283 0.316636
\(573\) 4.07435 0.170208
\(574\) 0 0
\(575\) −1.56393 −0.0652206
\(576\) 1.00000 0.0416667
\(577\) −12.2583 −0.510320 −0.255160 0.966899i \(-0.582128\pi\)
−0.255160 + 0.966899i \(0.582128\pi\)
\(578\) −7.67857 −0.319386
\(579\) 4.50280 0.187130
\(580\) −26.0289 −1.08079
\(581\) 0 0
\(582\) 11.2277 0.465405
\(583\) 14.4239 0.597375
\(584\) 16.2859 0.673915
\(585\) 23.4309 0.968750
\(586\) 14.5720 0.601965
\(587\) 43.2598 1.78552 0.892762 0.450528i \(-0.148764\pi\)
0.892762 + 0.450528i \(0.148764\pi\)
\(588\) 0 0
\(589\) −8.01988 −0.330453
\(590\) −7.24806 −0.298398
\(591\) −0.322407 −0.0132621
\(592\) 6.14824 0.252691
\(593\) −18.4196 −0.756403 −0.378201 0.925723i \(-0.623457\pi\)
−0.378201 + 0.925723i \(0.623457\pi\)
\(594\) −1.27293 −0.0522289
\(595\) 0 0
\(596\) −5.92450 −0.242677
\(597\) 22.6125 0.925467
\(598\) −0.885079 −0.0361936
\(599\) −6.36854 −0.260212 −0.130106 0.991500i \(-0.541532\pi\)
−0.130106 + 0.991500i \(0.541532\pi\)
\(600\) −10.5121 −0.429156
\(601\) 26.8167 1.09388 0.546939 0.837173i \(-0.315793\pi\)
0.546939 + 0.837173i \(0.315793\pi\)
\(602\) 0 0
\(603\) 2.61205 0.106371
\(604\) −1.09856 −0.0446996
\(605\) −36.9422 −1.50191
\(606\) 4.38789 0.178246
\(607\) 45.5076 1.84710 0.923549 0.383482i \(-0.125275\pi\)
0.923549 + 0.383482i \(0.125275\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −48.0035 −1.94361
\(611\) −48.9127 −1.97880
\(612\) −3.05310 −0.123414
\(613\) −4.94269 −0.199633 −0.0998167 0.995006i \(-0.531826\pi\)
−0.0998167 + 0.995006i \(0.531826\pi\)
\(614\) 26.2382 1.05889
\(615\) 24.7448 0.997808
\(616\) 0 0
\(617\) −41.7247 −1.67977 −0.839887 0.542762i \(-0.817379\pi\)
−0.839887 + 0.542762i \(0.817379\pi\)
\(618\) 6.48370 0.260813
\(619\) −28.8296 −1.15876 −0.579380 0.815058i \(-0.696705\pi\)
−0.579380 + 0.815058i \(0.696705\pi\)
\(620\) 31.5866 1.26855
\(621\) 0.148774 0.00597011
\(622\) −7.89074 −0.316390
\(623\) 0 0
\(624\) −5.94914 −0.238156
\(625\) 32.9441 1.31777
\(626\) 17.3496 0.693428
\(627\) 1.27293 0.0508359
\(628\) 8.29607 0.331049
\(629\) −18.7712 −0.748457
\(630\) 0 0
\(631\) −18.8512 −0.750455 −0.375228 0.926933i \(-0.622435\pi\)
−0.375228 + 0.926933i \(0.622435\pi\)
\(632\) 7.44674 0.296215
\(633\) −0.346997 −0.0137919
\(634\) 31.0193 1.23193
\(635\) −60.8805 −2.41597
\(636\) −11.3312 −0.449312
\(637\) 0 0
\(638\) −8.41250 −0.333054
\(639\) −12.8117 −0.506824
\(640\) 3.93854 0.155685
\(641\) −12.1065 −0.478178 −0.239089 0.970998i \(-0.576849\pi\)
−0.239089 + 0.970998i \(0.576849\pi\)
\(642\) −4.09675 −0.161686
\(643\) −40.2074 −1.58563 −0.792813 0.609465i \(-0.791384\pi\)
−0.792813 + 0.609465i \(0.791384\pi\)
\(644\) 0 0
\(645\) −31.1607 −1.22695
\(646\) 3.05310 0.120123
\(647\) 18.0855 0.711016 0.355508 0.934673i \(-0.384308\pi\)
0.355508 + 0.934673i \(0.384308\pi\)
\(648\) 1.00000 0.0392837
\(649\) −2.34256 −0.0919535
\(650\) 62.5381 2.45294
\(651\) 0 0
\(652\) 12.5954 0.493275
\(653\) −42.2280 −1.65251 −0.826255 0.563296i \(-0.809533\pi\)
−0.826255 + 0.563296i \(0.809533\pi\)
\(654\) −5.93860 −0.232218
\(655\) 22.6866 0.886439
\(656\) −6.28274 −0.245300
\(657\) 16.2859 0.635373
\(658\) 0 0
\(659\) 43.2743 1.68573 0.842864 0.538126i \(-0.180868\pi\)
0.842864 + 0.538126i \(0.180868\pi\)
\(660\) −5.01349 −0.195150
\(661\) 38.8777 1.51217 0.756083 0.654476i \(-0.227111\pi\)
0.756083 + 0.654476i \(0.227111\pi\)
\(662\) −27.7615 −1.07898
\(663\) 18.1633 0.705405
\(664\) −12.1746 −0.472467
\(665\) 0 0
\(666\) 6.14824 0.238239
\(667\) 0.983215 0.0380702
\(668\) −9.65057 −0.373392
\(669\) −2.29672 −0.0887965
\(670\) 10.2877 0.397447
\(671\) −15.5146 −0.598936
\(672\) 0 0
\(673\) −13.3534 −0.514737 −0.257369 0.966313i \(-0.582855\pi\)
−0.257369 + 0.966313i \(0.582855\pi\)
\(674\) −11.9053 −0.458575
\(675\) −10.5121 −0.404612
\(676\) 22.3922 0.861239
\(677\) 12.3254 0.473704 0.236852 0.971546i \(-0.423884\pi\)
0.236852 + 0.971546i \(0.423884\pi\)
\(678\) −3.83385 −0.147238
\(679\) 0 0
\(680\) −12.0248 −0.461129
\(681\) 15.3728 0.589089
\(682\) 10.2087 0.390913
\(683\) −14.1727 −0.542305 −0.271152 0.962536i \(-0.587405\pi\)
−0.271152 + 0.962536i \(0.587405\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 64.1189 2.44986
\(686\) 0 0
\(687\) −11.7178 −0.447063
\(688\) 7.91174 0.301632
\(689\) 67.4110 2.56816
\(690\) 0.585954 0.0223069
\(691\) −41.6858 −1.58580 −0.792902 0.609350i \(-0.791431\pi\)
−0.792902 + 0.609350i \(0.791431\pi\)
\(692\) −14.1088 −0.536335
\(693\) 0 0
\(694\) 18.8103 0.714029
\(695\) −32.4861 −1.23227
\(696\) 6.60877 0.250505
\(697\) 19.1818 0.726564
\(698\) 8.76894 0.331909
\(699\) −13.7683 −0.520764
\(700\) 0 0
\(701\) −17.0286 −0.643162 −0.321581 0.946882i \(-0.604214\pi\)
−0.321581 + 0.946882i \(0.604214\pi\)
\(702\) −5.94914 −0.224536
\(703\) −6.14824 −0.231885
\(704\) 1.27293 0.0479753
\(705\) 32.3820 1.21958
\(706\) 29.8162 1.12215
\(707\) 0 0
\(708\) 1.84029 0.0691623
\(709\) 32.3459 1.21478 0.607388 0.794405i \(-0.292217\pi\)
0.607388 + 0.794405i \(0.292217\pi\)
\(710\) −50.4595 −1.89371
\(711\) 7.44674 0.279275
\(712\) 5.40934 0.202724
\(713\) −1.19315 −0.0446839
\(714\) 0 0
\(715\) 29.8259 1.11543
\(716\) −12.4256 −0.464365
\(717\) −27.3226 −1.02038
\(718\) 17.1263 0.639146
\(719\) 6.79180 0.253291 0.126646 0.991948i \(-0.459579\pi\)
0.126646 + 0.991948i \(0.459579\pi\)
\(720\) 3.93854 0.146781
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −29.4781 −1.09630
\(724\) 12.3729 0.459836
\(725\) −69.4722 −2.58013
\(726\) 9.37965 0.348111
\(727\) 35.1033 1.30191 0.650955 0.759116i \(-0.274368\pi\)
0.650955 + 0.759116i \(0.274368\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 64.1426 2.37403
\(731\) −24.1553 −0.893417
\(732\) 12.1881 0.450487
\(733\) −18.2408 −0.673737 −0.336869 0.941552i \(-0.609368\pi\)
−0.336869 + 0.941552i \(0.609368\pi\)
\(734\) −24.3662 −0.899373
\(735\) 0 0
\(736\) −0.148774 −0.00548390
\(737\) 3.32495 0.122476
\(738\) −6.28274 −0.231271
\(739\) 2.59627 0.0955054 0.0477527 0.998859i \(-0.484794\pi\)
0.0477527 + 0.998859i \(0.484794\pi\)
\(740\) 24.2151 0.890165
\(741\) 5.94914 0.218547
\(742\) 0 0
\(743\) 5.38432 0.197531 0.0987657 0.995111i \(-0.468511\pi\)
0.0987657 + 0.995111i \(0.468511\pi\)
\(744\) −8.01988 −0.294023
\(745\) −23.3339 −0.854889
\(746\) 24.1414 0.883879
\(747\) −12.1746 −0.445447
\(748\) −3.88638 −0.142100
\(749\) 0 0
\(750\) −21.7097 −0.792728
\(751\) 52.2522 1.90671 0.953356 0.301850i \(-0.0976041\pi\)
0.953356 + 0.301850i \(0.0976041\pi\)
\(752\) −8.22182 −0.299819
\(753\) 22.6060 0.823810
\(754\) −39.3165 −1.43182
\(755\) −4.32671 −0.157465
\(756\) 0 0
\(757\) −14.2585 −0.518233 −0.259116 0.965846i \(-0.583431\pi\)
−0.259116 + 0.965846i \(0.583431\pi\)
\(758\) 8.51695 0.309349
\(759\) 0.189379 0.00687403
\(760\) −3.93854 −0.142866
\(761\) −4.51447 −0.163649 −0.0818247 0.996647i \(-0.526075\pi\)
−0.0818247 + 0.996647i \(0.526075\pi\)
\(762\) 15.4576 0.559970
\(763\) 0 0
\(764\) −4.07435 −0.147405
\(765\) −12.0248 −0.434757
\(766\) 17.3654 0.627437
\(767\) −10.9481 −0.395314
\(768\) −1.00000 −0.0360844
\(769\) −16.9779 −0.612239 −0.306120 0.951993i \(-0.599031\pi\)
−0.306120 + 0.951993i \(0.599031\pi\)
\(770\) 0 0
\(771\) 25.9880 0.935935
\(772\) −4.50280 −0.162059
\(773\) −39.7267 −1.42887 −0.714435 0.699702i \(-0.753316\pi\)
−0.714435 + 0.699702i \(0.753316\pi\)
\(774\) 7.91174 0.284381
\(775\) 84.3060 3.02836
\(776\) −11.2277 −0.403052
\(777\) 0 0
\(778\) −22.5117 −0.807083
\(779\) 6.28274 0.225102
\(780\) −23.4309 −0.838962
\(781\) −16.3084 −0.583561
\(782\) 0.454223 0.0162430
\(783\) 6.60877 0.236178
\(784\) 0 0
\(785\) 32.6744 1.16620
\(786\) −5.76015 −0.205458
\(787\) 54.3699 1.93808 0.969039 0.246908i \(-0.0794144\pi\)
0.969039 + 0.246908i \(0.0794144\pi\)
\(788\) 0.322407 0.0114853
\(789\) −6.16848 −0.219604
\(790\) 29.3293 1.04349
\(791\) 0 0
\(792\) 1.27293 0.0452316
\(793\) −72.5089 −2.57487
\(794\) −33.8779 −1.20228
\(795\) −44.6285 −1.58281
\(796\) −22.6125 −0.801478
\(797\) −17.4243 −0.617201 −0.308600 0.951192i \(-0.599861\pi\)
−0.308600 + 0.951192i \(0.599861\pi\)
\(798\) 0 0
\(799\) 25.1020 0.888046
\(800\) 10.5121 0.371660
\(801\) 5.40934 0.191130
\(802\) 33.1403 1.17022
\(803\) 20.7308 0.731573
\(804\) −2.61205 −0.0921197
\(805\) 0 0
\(806\) 47.7114 1.68056
\(807\) 10.1268 0.356480
\(808\) −4.38789 −0.154366
\(809\) 31.2970 1.10034 0.550172 0.835051i \(-0.314562\pi\)
0.550172 + 0.835051i \(0.314562\pi\)
\(810\) 3.93854 0.138386
\(811\) −12.7410 −0.447398 −0.223699 0.974658i \(-0.571813\pi\)
−0.223699 + 0.974658i \(0.571813\pi\)
\(812\) 0 0
\(813\) 10.9731 0.384843
\(814\) 7.82627 0.274311
\(815\) 49.6077 1.73768
\(816\) 3.05310 0.106880
\(817\) −7.91174 −0.276797
\(818\) 28.2927 0.989233
\(819\) 0 0
\(820\) −24.7448 −0.864127
\(821\) −0.210997 −0.00736386 −0.00368193 0.999993i \(-0.501172\pi\)
−0.00368193 + 0.999993i \(0.501172\pi\)
\(822\) −16.2798 −0.567825
\(823\) −26.1195 −0.910468 −0.455234 0.890372i \(-0.650444\pi\)
−0.455234 + 0.890372i \(0.650444\pi\)
\(824\) −6.48370 −0.225870
\(825\) −13.3812 −0.465873
\(826\) 0 0
\(827\) −18.9617 −0.659364 −0.329682 0.944092i \(-0.606942\pi\)
−0.329682 + 0.944092i \(0.606942\pi\)
\(828\) −0.148774 −0.00517027
\(829\) −7.91930 −0.275049 −0.137524 0.990498i \(-0.543915\pi\)
−0.137524 + 0.990498i \(0.543915\pi\)
\(830\) −47.9503 −1.66438
\(831\) −2.90574 −0.100799
\(832\) 5.94914 0.206249
\(833\) 0 0
\(834\) 8.24825 0.285614
\(835\) −38.0092 −1.31536
\(836\) −1.27293 −0.0440252
\(837\) −8.01988 −0.277208
\(838\) −24.5157 −0.846882
\(839\) −1.11414 −0.0384644 −0.0192322 0.999815i \(-0.506122\pi\)
−0.0192322 + 0.999815i \(0.506122\pi\)
\(840\) 0 0
\(841\) 14.6758 0.506062
\(842\) −36.4678 −1.25676
\(843\) −13.5310 −0.466032
\(844\) 0.346997 0.0119441
\(845\) 88.1928 3.03392
\(846\) −8.22182 −0.282672
\(847\) 0 0
\(848\) 11.3312 0.389116
\(849\) 15.3969 0.528421
\(850\) −32.0946 −1.10084
\(851\) −0.914700 −0.0313555
\(852\) 12.8117 0.438922
\(853\) −32.7675 −1.12194 −0.560969 0.827837i \(-0.689571\pi\)
−0.560969 + 0.827837i \(0.689571\pi\)
\(854\) 0 0
\(855\) −3.93854 −0.134695
\(856\) 4.09675 0.140024
\(857\) 19.0262 0.649921 0.324960 0.945728i \(-0.394649\pi\)
0.324960 + 0.945728i \(0.394649\pi\)
\(858\) −7.57283 −0.258532
\(859\) −4.70887 −0.160664 −0.0803322 0.996768i \(-0.525598\pi\)
−0.0803322 + 0.996768i \(0.525598\pi\)
\(860\) 31.1607 1.06257
\(861\) 0 0
\(862\) 7.38080 0.251391
\(863\) 3.17462 0.108065 0.0540326 0.998539i \(-0.482793\pi\)
0.0540326 + 0.998539i \(0.482793\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −55.5680 −1.88937
\(866\) 11.4859 0.390306
\(867\) 7.67857 0.260778
\(868\) 0 0
\(869\) 9.47918 0.321559
\(870\) 26.0289 0.882463
\(871\) 15.5394 0.526533
\(872\) 5.93860 0.201106
\(873\) −11.2277 −0.380001
\(874\) 0.148774 0.00503237
\(875\) 0 0
\(876\) −16.2859 −0.550249
\(877\) −1.38358 −0.0467201 −0.0233601 0.999727i \(-0.507436\pi\)
−0.0233601 + 0.999727i \(0.507436\pi\)
\(878\) 22.2521 0.750974
\(879\) −14.5720 −0.491503
\(880\) 5.01349 0.169005
\(881\) −0.922545 −0.0310813 −0.0155407 0.999879i \(-0.504947\pi\)
−0.0155407 + 0.999879i \(0.504947\pi\)
\(882\) 0 0
\(883\) −57.2691 −1.92726 −0.963630 0.267241i \(-0.913888\pi\)
−0.963630 + 0.267241i \(0.913888\pi\)
\(884\) −18.1633 −0.610899
\(885\) 7.24806 0.243641
\(886\) −22.7804 −0.765322
\(887\) −52.8038 −1.77298 −0.886488 0.462751i \(-0.846862\pi\)
−0.886488 + 0.462751i \(0.846862\pi\)
\(888\) −6.14824 −0.206321
\(889\) 0 0
\(890\) 21.3049 0.714143
\(891\) 1.27293 0.0426447
\(892\) 2.29672 0.0769000
\(893\) 8.22182 0.275133
\(894\) 5.92450 0.198145
\(895\) −48.9386 −1.63584
\(896\) 0 0
\(897\) 0.885079 0.0295519
\(898\) −8.19585 −0.273499
\(899\) −53.0015 −1.76770
\(900\) 10.5121 0.350404
\(901\) −34.5954 −1.15254
\(902\) −7.99748 −0.266287
\(903\) 0 0
\(904\) 3.83385 0.127512
\(905\) 48.7313 1.61988
\(906\) 1.09856 0.0364971
\(907\) 35.0561 1.16402 0.582009 0.813182i \(-0.302267\pi\)
0.582009 + 0.813182i \(0.302267\pi\)
\(908\) −15.3728 −0.510166
\(909\) −4.38789 −0.145537
\(910\) 0 0
\(911\) 23.3448 0.773449 0.386724 0.922195i \(-0.373606\pi\)
0.386724 + 0.922195i \(0.373606\pi\)
\(912\) 1.00000 0.0331133
\(913\) −15.4975 −0.512891
\(914\) −17.9078 −0.592337
\(915\) 48.0035 1.58695
\(916\) 11.7178 0.387168
\(917\) 0 0
\(918\) 3.05310 0.100767
\(919\) 2.68302 0.0885045 0.0442523 0.999020i \(-0.485909\pi\)
0.0442523 + 0.999020i \(0.485909\pi\)
\(920\) −0.585954 −0.0193183
\(921\) −26.2382 −0.864577
\(922\) 10.8937 0.358766
\(923\) −76.2187 −2.50877
\(924\) 0 0
\(925\) 64.6310 2.12506
\(926\) −39.6911 −1.30433
\(927\) −6.48370 −0.212953
\(928\) −6.60877 −0.216943
\(929\) −16.0004 −0.524955 −0.262478 0.964938i \(-0.584540\pi\)
−0.262478 + 0.964938i \(0.584540\pi\)
\(930\) −31.5866 −1.03577
\(931\) 0 0
\(932\) 13.7683 0.450995
\(933\) 7.89074 0.258331
\(934\) −32.0868 −1.04991
\(935\) −15.3067 −0.500582
\(936\) 5.94914 0.194454
\(937\) −49.2753 −1.60975 −0.804876 0.593443i \(-0.797768\pi\)
−0.804876 + 0.593443i \(0.797768\pi\)
\(938\) 0 0
\(939\) −17.3496 −0.566181
\(940\) −32.3820 −1.05618
\(941\) −12.9474 −0.422074 −0.211037 0.977478i \(-0.567684\pi\)
−0.211037 + 0.977478i \(0.567684\pi\)
\(942\) −8.29607 −0.270301
\(943\) 0.934710 0.0304383
\(944\) −1.84029 −0.0598963
\(945\) 0 0
\(946\) 10.0711 0.327439
\(947\) −24.6036 −0.799510 −0.399755 0.916622i \(-0.630905\pi\)
−0.399755 + 0.916622i \(0.630905\pi\)
\(948\) −7.44674 −0.241859
\(949\) 96.8869 3.14508
\(950\) −10.5121 −0.341058
\(951\) −31.0193 −1.00587
\(952\) 0 0
\(953\) −33.2606 −1.07742 −0.538708 0.842493i \(-0.681087\pi\)
−0.538708 + 0.842493i \(0.681087\pi\)
\(954\) 11.3312 0.366862
\(955\) −16.0470 −0.519269
\(956\) 27.3226 0.883676
\(957\) 8.41250 0.271937
\(958\) 21.5125 0.695038
\(959\) 0 0
\(960\) −3.93854 −0.127116
\(961\) 33.3184 1.07479
\(962\) 36.5767 1.17928
\(963\) 4.09675 0.132016
\(964\) 29.4781 0.949427
\(965\) −17.7345 −0.570893
\(966\) 0 0
\(967\) −2.48135 −0.0797949 −0.0398974 0.999204i \(-0.512703\pi\)
−0.0398974 + 0.999204i \(0.512703\pi\)
\(968\) −9.37965 −0.301473
\(969\) −3.05310 −0.0980798
\(970\) −44.2209 −1.41985
\(971\) −9.65888 −0.309968 −0.154984 0.987917i \(-0.549533\pi\)
−0.154984 + 0.987917i \(0.549533\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −22.7496 −0.728944
\(975\) −62.5381 −2.00282
\(976\) −12.1881 −0.390133
\(977\) −34.6842 −1.10965 −0.554823 0.831968i \(-0.687214\pi\)
−0.554823 + 0.831968i \(0.687214\pi\)
\(978\) −12.5954 −0.402758
\(979\) 6.88571 0.220068
\(980\) 0 0
\(981\) 5.93860 0.189605
\(982\) −26.4966 −0.845539
\(983\) 2.51987 0.0803715 0.0401857 0.999192i \(-0.487205\pi\)
0.0401857 + 0.999192i \(0.487205\pi\)
\(984\) 6.28274 0.200286
\(985\) 1.26981 0.0404597
\(986\) 20.1772 0.642574
\(987\) 0 0
\(988\) −5.94914 −0.189267
\(989\) −1.17706 −0.0374284
\(990\) 5.01349 0.159339
\(991\) −40.5602 −1.28844 −0.644219 0.764841i \(-0.722817\pi\)
−0.644219 + 0.764841i \(0.722817\pi\)
\(992\) 8.01988 0.254631
\(993\) 27.7615 0.880983
\(994\) 0 0
\(995\) −89.0603 −2.82340
\(996\) 12.1746 0.385768
\(997\) 0.253401 0.00802530 0.00401265 0.999992i \(-0.498723\pi\)
0.00401265 + 0.999992i \(0.498723\pi\)
\(998\) 20.4302 0.646706
\(999\) −6.14824 −0.194522
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 5586.2.a.ce.1.8 8
7.6 odd 2 5586.2.a.cf.1.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5586.2.a.ce.1.8 8 1.1 even 1 trivial
5586.2.a.cf.1.1 yes 8 7.6 odd 2