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Properties

Label 4002.2.a.h.1.1
Level $4002$
Weight $2$
Character 4002.1
Self dual yes
Analytic conductor $31.956$
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] = [4002,2,Mod(1,4002)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, 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("4002.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.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: \(31.9561308889\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4002.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} +2.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} -1.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{40} +6.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} +2.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} -2.00000 q^{56} -4.00000 q^{57} -1.00000 q^{58} +2.00000 q^{60} -4.00000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -2.00000 q^{66} -8.00000 q^{67} -1.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{78} -10.0000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -14.0000 q^{83} +2.00000 q^{84} -8.00000 q^{86} +1.00000 q^{87} +2.00000 q^{88} -2.00000 q^{90} +4.00000 q^{91} +1.00000 q^{92} -4.00000 q^{93} -8.00000 q^{94} -8.00000 q^{95} -1.00000 q^{96} +10.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.00000 −0.534522
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 2.00000 0.436436
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −1.00000 −0.185695
\(30\) 2.00000 0.365148
\(31\) 4.00000 0.718421 0.359211 − 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) −2.00000 −0.316228
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 2.00000 0.301511
\(45\) −2.00000 −0.298142
\(46\) 1.00000 0.147442
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) −2.00000 −0.267261
\(57\) −4.00000 −0.529813
\(58\) −1.00000 −0.131306
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −2.00000 −0.246183
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 4.00000 0.478091
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) −4.00000 −0.455842
\(78\) 2.00000 0.226455
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 1.00000 0.107211
\(88\) 2.00000 0.213201
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.00000 0.419314
\(92\) 1.00000 0.104257
\(93\) −4.00000 −0.414781
\(94\) −8.00000 −0.825137
\(95\) −8.00000 −0.820783
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −3.00000 −0.303046
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 14.0000 1.39305 0.696526 − 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 − 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −2.00000 −0.196116
\(105\) −4.00000 −0.390360
\(106\) −6.00000 −0.582772
\(107\) 10.0000 0.966736 0.483368 − 0.875417i \(-0.339413\pi\)
0.483368 + 0.875417i \(0.339413\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.00000 0.383131 0.191565 − 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) −4.00000 −0.381385
\(111\) −8.00000 −0.759326
\(112\) −2.00000 −0.188982
\(113\) −16.0000 −1.50515 −0.752577 − 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −4.00000 −0.374634
\(115\) −2.00000 −0.186501
\(116\) −1.00000 −0.0928477
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −7.00000 −0.636364
\(122\) −4.00000 −0.362143
\(123\) −6.00000 −0.541002
\(124\) 4.00000 0.359211
\(125\) 12.0000 1.07331
\(126\) −2.00000 −0.178174
\(127\) −16.0000 −1.41977 −0.709885 − 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 4.00000 0.350823
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −2.00000 −0.174078
\(133\) −8.00000 −0.693688
\(134\) −8.00000 −0.691095
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 4.00000 0.338062
\(141\) 8.00000 0.673722
\(142\) −8.00000 −0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) −10.0000 −0.827606
\(147\) 3.00000 0.247436
\(148\) 8.00000 0.657596
\(149\) −18.0000 −1.47462 −0.737309 − 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) −4.00000 −0.322329
\(155\) −8.00000 −0.642575
\(156\) 2.00000 0.160128
\(157\) −8.00000 −0.638470 −0.319235 − 0.947676i \(-0.603426\pi\)
−0.319235 + 0.947676i \(0.603426\pi\)
\(158\) −10.0000 −0.795557
\(159\) 6.00000 0.475831
\(160\) −2.00000 −0.158114
\(161\) −2.00000 −0.157622
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 − 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 6.00000 0.468521
\(165\) 4.00000 0.311400
\(166\) −14.0000 −1.08661
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 2.00000 0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −8.00000 −0.609994
\(173\) 18.0000 1.36851 0.684257 − 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 1.00000 0.0758098
\(175\) 2.00000 0.151186
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 − 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −2.00000 −0.149071
\(181\) −12.0000 −0.891953 −0.445976 − 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 4.00000 0.296500
\(183\) 4.00000 0.295689
\(184\) 1.00000 0.0737210
\(185\) −16.0000 −1.17634
\(186\) −4.00000 −0.293294
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 2.00000 0.145479
\(190\) −8.00000 −0.580381
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.0000 −1.29567 −0.647834 − 0.761781i \(-0.724325\pi\)
−0.647834 + 0.761781i \(0.724325\pi\)
\(194\) 10.0000 0.717958
\(195\) −4.00000 −0.286446
\(196\) −3.00000 −0.214286
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 2.00000 0.142134
\(199\) 10.0000 0.708881 0.354441 − 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 14.0000 0.985037
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 14.0000 0.975426
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 8.00000 0.553372
\(210\) −4.00000 −0.276026
\(211\) −12.0000 −0.826114 −0.413057 − 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −6.00000 −0.412082
\(213\) 8.00000 0.548151
\(214\) 10.0000 0.683586
\(215\) 16.0000 1.09119
\(216\) −1.00000 −0.0680414
\(217\) −8.00000 −0.543075
\(218\) 4.00000 0.270914
\(219\) 10.0000 0.675737
\(220\) −4.00000 −0.269680
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −2.00000 −0.133631
\(225\) −1.00000 −0.0666667
\(226\) −16.0000 −1.06430
\(227\) 10.0000 0.663723 0.331862 − 0.943328i \(-0.392323\pi\)
0.331862 + 0.943328i \(0.392323\pi\)
\(228\) −4.00000 −0.264906
\(229\) 8.00000 0.528655 0.264327 − 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) −2.00000 −0.131876
\(231\) 4.00000 0.263181
\(232\) −1.00000 −0.0656532
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −2.00000 −0.130744
\(235\) 16.0000 1.04372
\(236\) 0 0
\(237\) 10.0000 0.649570
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 − 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 2.00000 0.129099
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −7.00000 −0.449977
\(243\) −1.00000 −0.0641500
\(244\) −4.00000 −0.256074
\(245\) 6.00000 0.383326
\(246\) −6.00000 −0.382546
\(247\) −8.00000 −0.509028
\(248\) 4.00000 0.254000
\(249\) 14.0000 0.887214
\(250\) 12.0000 0.758947
\(251\) 2.00000 0.126239 0.0631194 − 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) −2.00000 −0.125988
\(253\) 2.00000 0.125739
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 − 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 8.00000 0.498058
\(259\) −16.0000 −0.994192
\(260\) 4.00000 0.248069
\(261\) −1.00000 −0.0618984
\(262\) 0 0
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) −2.00000 −0.123091
\(265\) 12.0000 0.737154
\(266\) −8.00000 −0.490511
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −18.0000 −1.09748 −0.548740 − 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 2.00000 0.121716
\(271\) 16.0000 0.971931 0.485965 − 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) −12.0000 −0.724947
\(275\) −2.00000 −0.120605
\(276\) −1.00000 −0.0601929
\(277\) −30.0000 −1.80253 −0.901263 − 0.433273i \(-0.857359\pi\)
−0.901263 + 0.433273i \(0.857359\pi\)
\(278\) −12.0000 −0.719712
\(279\) 4.00000 0.239474
\(280\) 4.00000 0.239046
\(281\) 20.0000 1.19310 0.596550 − 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 8.00000 0.476393
\(283\) −12.0000 −0.713326 −0.356663 − 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −8.00000 −0.474713
\(285\) 8.00000 0.473879
\(286\) −4.00000 −0.236525
\(287\) −12.0000 −0.708338
\(288\) 1.00000 0.0589256
\(289\) −17.0000 −1.00000
\(290\) 2.00000 0.117444
\(291\) −10.0000 −0.586210
\(292\) −10.0000 −0.585206
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −2.00000 −0.116052
\(298\) −18.0000 −1.04271
\(299\) −2.00000 −0.115663
\(300\) 1.00000 0.0577350
\(301\) 16.0000 0.922225
\(302\) 20.0000 1.15087
\(303\) −14.0000 −0.804279
\(304\) 4.00000 0.229416
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 − 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −4.00000 −0.227921
\(309\) −14.0000 −0.796432
\(310\) −8.00000 −0.454369
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −8.00000 −0.451466
\(315\) 4.00000 0.225374
\(316\) −10.0000 −0.562544
\(317\) −30.0000 −1.68497 −0.842484 − 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 6.00000 0.336463
\(319\) −2.00000 −0.111979
\(320\) −2.00000 −0.111803
\(321\) −10.0000 −0.558146
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −20.0000 −1.10770
\(327\) −4.00000 −0.221201
\(328\) 6.00000 0.331295
\(329\) 16.0000 0.882109
\(330\) 4.00000 0.220193
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −14.0000 −0.768350
\(333\) 8.00000 0.438397
\(334\) −8.00000 −0.437741
\(335\) 16.0000 0.874173
\(336\) 2.00000 0.109109
\(337\) −18.0000 −0.980522 −0.490261 − 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −9.00000 −0.489535
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 4.00000 0.216295
\(343\) 20.0000 1.07990
\(344\) −8.00000 −0.431331
\(345\) 2.00000 0.107676
\(346\) 18.0000 0.967686
\(347\) 4.00000 0.214731 0.107366 − 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 1.00000 0.0536056
\(349\) −30.0000 −1.60586 −0.802932 − 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 2.00000 0.106904
\(351\) 2.00000 0.106752
\(352\) 2.00000 0.106600
\(353\) 6.00000 0.319348 0.159674 − 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) −2.00000 −0.105409
\(361\) −3.00000 −0.157895
\(362\) −12.0000 −0.630706
\(363\) 7.00000 0.367405
\(364\) 4.00000 0.209657
\(365\) 20.0000 1.04685
\(366\) 4.00000 0.209083
\(367\) −10.0000 −0.521996 −0.260998 − 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) −16.0000 −0.831800
\(371\) 12.0000 0.623009
\(372\) −4.00000 −0.207390
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) −8.00000 −0.412568
\(377\) 2.00000 0.103005
\(378\) 2.00000 0.102869
\(379\) 4.00000 0.205466 0.102733 − 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) −8.00000 −0.410391
\(381\) 16.0000 0.819705
\(382\) −8.00000 −0.409316
\(383\) 20.0000 1.02195 0.510976 − 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.00000 0.407718
\(386\) −18.0000 −0.916176
\(387\) −8.00000 −0.406663
\(388\) 10.0000 0.507673
\(389\) 30.0000 1.52106 0.760530 − 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 0 0
\(394\) −2.00000 −0.100759
\(395\) 20.0000 1.00631
\(396\) 2.00000 0.100504
\(397\) 22.0000 1.10415 0.552074 − 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 10.0000 0.501255
\(399\) 8.00000 0.400501
\(400\) −1.00000 −0.0500000
\(401\) −12.0000 −0.599251 −0.299626 − 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 8.00000 0.399004
\(403\) −8.00000 −0.398508
\(404\) 14.0000 0.696526
\(405\) −2.00000 −0.0993808
\(406\) 2.00000 0.0992583
\(407\) 16.0000 0.793091
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) −12.0000 −0.592638
\(411\) 12.0000 0.591916
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 28.0000 1.37447
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) 8.00000 0.391293
\(419\) 34.0000 1.66101 0.830504 − 0.557012i \(-0.188052\pi\)
0.830504 + 0.557012i \(0.188052\pi\)
\(420\) −4.00000 −0.195180
\(421\) −20.0000 −0.974740 −0.487370 − 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) −12.0000 −0.584151
\(423\) −8.00000 −0.388973
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 8.00000 0.387147
\(428\) 10.0000 0.483368
\(429\) 4.00000 0.193122
\(430\) 16.0000 0.771589
\(431\) −36.0000 −1.73406 −0.867029 − 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 6.00000 0.288342 0.144171 − 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) −8.00000 −0.384012
\(435\) −2.00000 −0.0958927
\(436\) 4.00000 0.191565
\(437\) 4.00000 0.191346
\(438\) 10.0000 0.477818
\(439\) −24.0000 −1.14546 −0.572729 − 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) −4.00000 −0.190693
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 − 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 18.0000 0.851371
\(448\) −2.00000 −0.0944911
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) −16.0000 −0.752577
\(453\) −20.0000 −0.939682
\(454\) 10.0000 0.469323
\(455\) −8.00000 −0.375046
\(456\) −4.00000 −0.187317
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 8.00000 0.373815
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) 14.0000 0.652045 0.326023 − 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 4.00000 0.186097
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 8.00000 0.370991
\(466\) 10.0000 0.463241
\(467\) 10.0000 0.462745 0.231372 − 0.972865i \(-0.425678\pi\)
0.231372 + 0.972865i \(0.425678\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) 16.0000 0.738025
\(471\) 8.00000 0.368621
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 10.0000 0.459315
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 2.00000 0.0912871
\(481\) −16.0000 −0.729537
\(482\) −6.00000 −0.273293
\(483\) 2.00000 0.0910032
\(484\) −7.00000 −0.318182
\(485\) −20.0000 −0.908153
\(486\) −1.00000 −0.0453609
\(487\) 24.0000 1.08754 0.543772 − 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −4.00000 −0.181071
\(489\) 20.0000 0.904431
\(490\) 6.00000 0.271052
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) −4.00000 −0.179787
\(496\) 4.00000 0.179605
\(497\) 16.0000 0.717698
\(498\) 14.0000 0.627355
\(499\) 12.0000 0.537194 0.268597 − 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 12.0000 0.536656
\(501\) 8.00000 0.357414
\(502\) 2.00000 0.0892644
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −2.00000 −0.0890871
\(505\) −28.0000 −1.24598
\(506\) 2.00000 0.0889108
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 10.0000 0.441081
\(515\) −28.0000 −1.23383
\(516\) 8.00000 0.352180
\(517\) −16.0000 −0.703679
\(518\) −16.0000 −0.703000
\(519\) −18.0000 −0.790112
\(520\) 4.00000 0.175412
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 12.0000 0.523225
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) −8.00000 −0.346844
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) −8.00000 −0.345547
\(537\) 4.00000 0.172613
\(538\) −18.0000 −0.776035
\(539\) −6.00000 −0.258438
\(540\) 2.00000 0.0860663
\(541\) 30.0000 1.28980 0.644900 − 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 16.0000 0.687259
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) −4.00000 −0.171184
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −12.0000 −0.512615
\(549\) −4.00000 −0.170716
\(550\) −2.00000 −0.0852803
\(551\) −4.00000 −0.170406
\(552\) −1.00000 −0.0425628
\(553\) 20.0000 0.850487
\(554\) −30.0000 −1.27458
\(555\) 16.0000 0.679162
\(556\) −12.0000 −0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 4.00000 0.169334
\(559\) 16.0000 0.676728
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) 20.0000 0.843649
\(563\) 14.0000 0.590030 0.295015 − 0.955493i \(-0.404675\pi\)
0.295015 + 0.955493i \(0.404675\pi\)
\(564\) 8.00000 0.336861
\(565\) 32.0000 1.34625
\(566\) −12.0000 −0.504398
\(567\) −2.00000 −0.0839921
\(568\) −8.00000 −0.335673
\(569\) −24.0000 −1.00613 −0.503066 − 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 8.00000 0.335083
\(571\) 24.0000 1.00437 0.502184 − 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) −4.00000 −0.167248
\(573\) 8.00000 0.334205
\(574\) −12.0000 −0.500870
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −46.0000 −1.91501 −0.957503 − 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) −17.0000 −0.707107
\(579\) 18.0000 0.748054
\(580\) 2.00000 0.0830455
\(581\) 28.0000 1.16164
\(582\) −10.0000 −0.414513
\(583\) −12.0000 −0.496989
\(584\) −10.0000 −0.413803
\(585\) 4.00000 0.165380
\(586\) −14.0000 −0.578335
\(587\) −4.00000 −0.165098 −0.0825488 − 0.996587i \(-0.526306\pi\)
−0.0825488 + 0.996587i \(0.526306\pi\)
\(588\) 3.00000 0.123718
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) 8.00000 0.328798
\(593\) −30.0000 −1.23195 −0.615976 − 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −10.0000 −0.409273
\(598\) −2.00000 −0.0817861
\(599\) 16.0000 0.653742 0.326871 − 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000 0.0408248
\(601\) −38.0000 −1.55005 −0.775026 − 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 16.0000 0.652111
\(603\) −8.00000 −0.325785
\(604\) 20.0000 0.813788
\(605\) 14.0000 0.569181
\(606\) −14.0000 −0.568711
\(607\) 40.0000 1.62355 0.811775 − 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 4.00000 0.162221
\(609\) −2.00000 −0.0810441
\(610\) 8.00000 0.323911
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 − 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 20.0000 0.807134
\(615\) 12.0000 0.483887
\(616\) −4.00000 −0.161165
\(617\) 24.0000 0.966204 0.483102 − 0.875564i \(-0.339510\pi\)
0.483102 + 0.875564i \(0.339510\pi\)
\(618\) −14.0000 −0.563163
\(619\) −20.0000 −0.803868 −0.401934 − 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) −8.00000 −0.321288
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) −8.00000 −0.319489
\(628\) −8.00000 −0.319235
\(629\) 0 0
\(630\) 4.00000 0.159364
\(631\) 6.00000 0.238856 0.119428 − 0.992843i \(-0.461894\pi\)
0.119428 + 0.992843i \(0.461894\pi\)
\(632\) −10.0000 −0.397779
\(633\) 12.0000 0.476957
\(634\) −30.0000 −1.19145
\(635\) 32.0000 1.26988
\(636\) 6.00000 0.237915
\(637\) 6.00000 0.237729
\(638\) −2.00000 −0.0791808
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) 40.0000 1.57991 0.789953 − 0.613168i \(-0.210105\pi\)
0.789953 + 0.613168i \(0.210105\pi\)
\(642\) −10.0000 −0.394669
\(643\) 40.0000 1.57745 0.788723 − 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) −2.00000 −0.0788110
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) −20.0000 −0.783260
\(653\) −18.0000 −0.704394 −0.352197 − 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −10.0000 −0.390137
\(658\) 16.0000 0.623745
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 4.00000 0.155700
\(661\) 24.0000 0.933492 0.466746 − 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) 16.0000 0.620453
\(666\) 8.00000 0.309994
\(667\) −1.00000 −0.0387202
\(668\) −8.00000 −0.309529
\(669\) −4.00000 −0.154649
\(670\) 16.0000 0.618134
\(671\) −8.00000 −0.308837
\(672\) 2.00000 0.0771517
\(673\) 18.0000 0.693849 0.346925 − 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −18.0000 −0.693334
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 16.0000 0.614476
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) −10.0000 −0.383201
\(682\) 8.00000 0.306336
\(683\) −28.0000 −1.07139 −0.535695 − 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 4.00000 0.152944
\(685\) 24.0000 0.916993
\(686\) 20.0000 0.763604
\(687\) −8.00000 −0.305219
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) 2.00000 0.0761387
\(691\) −44.0000 −1.67384 −0.836919 − 0.547326i \(-0.815646\pi\)
−0.836919 + 0.547326i \(0.815646\pi\)
\(692\) 18.0000 0.684257
\(693\) −4.00000 −0.151947
\(694\) 4.00000 0.151838
\(695\) 24.0000 0.910372
\(696\) 1.00000 0.0379049
\(697\) 0 0
\(698\) −30.0000 −1.13552
\(699\) −10.0000 −0.378235
\(700\) 2.00000 0.0755929
\(701\) 42.0000 1.58632 0.793159 − 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 2.00000 0.0754851
\(703\) 32.0000 1.20690
\(704\) 2.00000 0.0753778
\(705\) −16.0000 −0.602595
\(706\) 6.00000 0.225813
\(707\) −28.0000 −1.05305
\(708\) 0 0
\(709\) −40.0000 −1.50223 −0.751116 − 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 16.0000 0.600469
\(711\) −10.0000 −0.375029
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) −4.00000 −0.149487
\(717\) −24.0000 −0.896296
\(718\) 32.0000 1.19423
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −28.0000 −1.04277
\(722\) −3.00000 −0.111648
\(723\) 6.00000 0.223142
\(724\) −12.0000 −0.445976
\(725\) 1.00000 0.0371391
\(726\) 7.00000 0.259794
\(727\) −46.0000 −1.70605 −0.853023 − 0.521874i \(-0.825233\pi\)
−0.853023 + 0.521874i \(0.825233\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) 0 0
\(732\) 4.00000 0.147844
\(733\) 36.0000 1.32969 0.664845 − 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −10.0000 −0.369107
\(735\) −6.00000 −0.221313
\(736\) 1.00000 0.0368605
\(737\) −16.0000 −0.589368
\(738\) 6.00000 0.220863
\(739\) 4.00000 0.147142 0.0735712 − 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) −16.0000 −0.588172
\(741\) 8.00000 0.293887
\(742\) 12.0000 0.440534
\(743\) 12.0000 0.440237 0.220119 − 0.975473i \(-0.429356\pi\)
0.220119 + 0.975473i \(0.429356\pi\)
\(744\) −4.00000 −0.146647
\(745\) 36.0000 1.31894
\(746\) −4.00000 −0.146450
\(747\) −14.0000 −0.512233
\(748\) 0 0
\(749\) −20.0000 −0.730784
\(750\) −12.0000 −0.438178
\(751\) 38.0000 1.38664 0.693320 − 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) −8.00000 −0.291730
\(753\) −2.00000 −0.0728841
\(754\) 2.00000 0.0728357
\(755\) −40.0000 −1.45575
\(756\) 2.00000 0.0727393
\(757\) 16.0000 0.581530 0.290765 − 0.956795i \(-0.406090\pi\)
0.290765 + 0.956795i \(0.406090\pi\)
\(758\) 4.00000 0.145287
\(759\) −2.00000 −0.0725954
\(760\) −8.00000 −0.290191
\(761\) −10.0000 −0.362500 −0.181250 − 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 16.0000 0.579619
\(763\) −8.00000 −0.289619
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −26.0000 −0.937584 −0.468792 − 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 8.00000 0.288300
\(771\) −10.0000 −0.360141
\(772\) −18.0000 −0.647834
\(773\) 2.00000 0.0719350 0.0359675 − 0.999353i \(-0.488549\pi\)
0.0359675 + 0.999353i \(0.488549\pi\)
\(774\) −8.00000 −0.287554
\(775\) −4.00000 −0.143684
\(776\) 10.0000 0.358979
\(777\) 16.0000 0.573997
\(778\) 30.0000 1.07555
\(779\) 24.0000 0.859889
\(780\) −4.00000 −0.143223
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 1.00000 0.0357371
\(784\) −3.00000 −0.107143
\(785\) 16.0000 0.571064
\(786\) 0 0
\(787\) 48.0000 1.71102 0.855508 − 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) −2.00000 −0.0712470
\(789\) −12.0000 −0.427211
\(790\) 20.0000 0.711568
\(791\) 32.0000 1.13779
\(792\) 2.00000 0.0710669
\(793\) 8.00000 0.284088
\(794\) 22.0000 0.780751
\(795\) −12.0000 −0.425596
\(796\) 10.0000 0.354441
\(797\) 30.0000 1.06265 0.531327 − 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 8.00000 0.283197
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −12.0000 −0.423735
\(803\) −20.0000 −0.705785
\(804\) 8.00000 0.282138
\(805\) 4.00000 0.140981
\(806\) −8.00000 −0.281788
\(807\) 18.0000 0.633630
\(808\) 14.0000 0.492518
\(809\) 26.0000 0.914111 0.457056 − 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 2.00000 0.0701862
\(813\) −16.0000 −0.561144
\(814\) 16.0000 0.560800
\(815\) 40.0000 1.40114
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) −18.0000 −0.629355
\(819\) 4.00000 0.139771
\(820\) −12.0000 −0.419058
\(821\) 38.0000 1.32621 0.663105 − 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 12.0000 0.418548
\(823\) −52.0000 −1.81261 −0.906303 − 0.422628i \(-0.861108\pi\)
−0.906303 + 0.422628i \(0.861108\pi\)
\(824\) 14.0000 0.487713
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 − 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 1.00000 0.0347524
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 28.0000 0.971894
\(831\) 30.0000 1.04069
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) 16.0000 0.553703
\(836\) 8.00000 0.276686
\(837\) −4.00000 −0.138260
\(838\) 34.0000 1.17451
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) −4.00000 −0.138013
\(841\) 1.00000 0.0344828
\(842\) −20.0000 −0.689246
\(843\) −20.0000 −0.688837
\(844\) −12.0000 −0.413057
\(845\) 18.0000 0.619219
\(846\) −8.00000 −0.275046
\(847\) 14.0000 0.481046
\(848\) −6.00000 −0.206041
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 8.00000 0.274075
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 8.00000 0.273754
\(855\) −8.00000 −0.273594
\(856\) 10.0000 0.341793
\(857\) −46.0000 −1.57133 −0.785665 − 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 4.00000 0.136558
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 16.0000 0.545595
\(861\) 12.0000 0.408959
\(862\) −36.0000 −1.22616
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −36.0000 −1.22404
\(866\) 6.00000 0.203888
\(867\) 17.0000 0.577350
\(868\) −8.00000 −0.271538
\(869\) −20.0000 −0.678454
\(870\) −2.00000 −0.0678064
\(871\) 16.0000 0.542139
\(872\) 4.00000 0.135457
\(873\) 10.0000 0.338449
\(874\) 4.00000 0.135302
\(875\) −24.0000 −0.811348
\(876\) 10.0000 0.337869
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −24.0000 −0.809961
\(879\) 14.0000 0.472208
\(880\) −4.00000 −0.134840
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) −3.00000 −0.101015
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −8.00000 −0.268462
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 4.00000 0.133930
\(893\) −32.0000 −1.07084
\(894\) 18.0000 0.602010
\(895\) 8.00000 0.267411
\(896\) −2.00000 −0.0668153
\(897\) 2.00000 0.0667781
\(898\) 30.0000 1.00111
\(899\) −4.00000 −0.133407
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) 12.0000 0.399556
\(903\) −16.0000 −0.532447
\(904\) −16.0000 −0.532152
\(905\) 24.0000 0.797787
\(906\) −20.0000 −0.664455
\(907\) −20.0000 −0.664089 −0.332045 − 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 10.0000 0.331862
\(909\) 14.0000 0.464351
\(910\) −8.00000 −0.265197
\(911\) 12.0000 0.397578 0.198789 − 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −4.00000 −0.132453
\(913\) −28.0000 −0.926665
\(914\) 10.0000 0.330771
\(915\) −8.00000 −0.264472
\(916\) 8.00000 0.264327
\(917\) 0 0
\(918\) 0 0
\(919\) 50.0000 1.64935 0.824674 − 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −20.0000 −0.659022
\(922\) 14.0000 0.461065
\(923\) 16.0000 0.526646
\(924\) 4.00000 0.131590
\(925\) −8.00000 −0.263038
\(926\) 24.0000 0.788689
\(927\) 14.0000 0.459820
\(928\) −1.00000 −0.0328266
\(929\) −14.0000 −0.459325 −0.229663 − 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 8.00000 0.262330
\(931\) −12.0000 −0.393284
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) 10.0000 0.327210
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 26.0000 0.849383 0.424691 − 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 16.0000 0.522419
\(939\) 10.0000 0.326338
\(940\) 16.0000 0.521862
\(941\) 22.0000 0.717180 0.358590 − 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 8.00000 0.260654
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) −16.0000 −0.520205
\(947\) 24.0000 0.779895 0.389948 − 0.920837i \(-0.372493\pi\)
0.389948 + 0.920837i \(0.372493\pi\)
\(948\) 10.0000 0.324785
\(949\) 20.0000 0.649227
\(950\) −4.00000 −0.129777
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) −6.00000 −0.194257
\(955\) 16.0000 0.517748
\(956\) 24.0000 0.776215
\(957\) 2.00000 0.0646508
\(958\) 24.0000 0.775405
\(959\) 24.0000 0.775000
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) −16.0000 −0.515861
\(963\) 10.0000 0.322245
\(964\) −6.00000 −0.193247
\(965\) 36.0000 1.15888
\(966\) 2.00000 0.0643489
\(967\) 12.0000 0.385894 0.192947 − 0.981209i \(-0.438195\pi\)
0.192947 + 0.981209i \(0.438195\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) −20.0000 −0.642161
\(971\) −42.0000 −1.34784 −0.673922 − 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 24.0000 0.769405
\(974\) 24.0000 0.769010
\(975\) −2.00000 −0.0640513
\(976\) −4.00000 −0.128037
\(977\) 32.0000 1.02377 0.511885 − 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 20.0000 0.639529
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) 4.00000 0.127710
\(982\) −24.0000 −0.765871
\(983\) −28.0000 −0.893061 −0.446531 − 0.894768i \(-0.647341\pi\)
−0.446531 + 0.894768i \(0.647341\pi\)
\(984\) −6.00000 −0.191273
\(985\) 4.00000 0.127451
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) −8.00000 −0.254514
\(989\) −8.00000 −0.254385
\(990\) −4.00000 −0.127128
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 4.00000 0.127000
\(993\) 4.00000 0.126936
\(994\) 16.0000 0.507489
\(995\) −20.0000 −0.634043
\(996\) 14.0000 0.443607
\(997\) 46.0000 1.45683 0.728417 − 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 12.0000 0.379853
\(999\) −8.00000 −0.253109
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 4002.2.a.h.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.h.1.1 ✓ 1 1.1 even 1 trivial