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Properties

Label 4002.2.a.f.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{18} -2.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -1.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +8.00000 q^{34} +1.00000 q^{36} +2.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} -6.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} -1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -8.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -2.00000 q^{57} +1.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} -8.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -2.00000 q^{66} -4.00000 q^{67} -8.00000 q^{68} +1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -1.00000 q^{75} -2.00000 q^{76} +6.00000 q^{78} -2.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -16.0000 q^{85} +6.00000 q^{86} -1.00000 q^{87} -2.00000 q^{88} +8.00000 q^{89} -2.00000 q^{90} +1.00000 q^{92} +8.00000 q^{93} -4.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} +12.0000 q^{97} +7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −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
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −8.00000 −1.94029 −0.970143 − 0.242536i \(-0.922021\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) −2.00000 −0.365148
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 8.00000 1.37199
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 2.00000 0.324443
\(39\) −6.00000 −0.960769
\(40\) −2.00000 −0.316228
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000 0.301511
\(45\) 2.00000 0.298142
\(46\) −1.00000 −0.147442
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −8.00000 −1.12022
\(52\) −6.00000 −0.832050
\(53\) −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
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 1.00000 0.131306
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 2.00000 0.258199
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) −2.00000 −0.246183
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −8.00000 −0.970143
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −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 \(0.698986\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) −2.00000 −0.225018 −0.112509 − 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 12.0000 1.31717 0.658586 − 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 0 0
\(85\) −16.0000 −1.73544
\(86\) 6.00000 0.646997
\(87\) −1.00000 −0.107211
\(88\) −2.00000 −0.213201
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) −4.00000 −0.412568
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 12.0000 1.21842 0.609208 − 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 7.00000 0.707107
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 8.00000 0.792118
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 − 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) −4.00000 −0.381385
\(111\) 0 0
\(112\) 0 0
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 2.00000 0.187317
\(115\) 2.00000 0.186501
\(116\) −1.00000 −0.0928477
\(117\) −6.00000 −0.554700
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.00000 −0.528271
\(130\) 12.0000 1.05247
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 2.00000 0.172133
\(136\) 8.00000 0.685994
\(137\) −16.0000 −1.36697 −0.683486 − 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 4.00000 0.339276 0.169638 − 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 8.00000 0.671345
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 10.0000 0.827606
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 1.00000 0.0816497
\(151\) 16.0000 1.30206 0.651031 − 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 2.00000 0.162221
\(153\) −8.00000 −0.646762
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) −6.00000 −0.480384
\(157\) 8.00000 0.638470 0.319235 − 0.947676i \(-0.396574\pi\)
0.319235 + 0.947676i \(0.396574\pi\)
\(158\) 2.00000 0.159111
\(159\) −6.00000 −0.475831
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 4.00000 0.311400
\(166\) −12.0000 −0.931381
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 16.0000 1.22714
\(171\) −2.00000 −0.152944
\(172\) −6.00000 −0.457496
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) −8.00000 −0.599625
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 − 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −16.0000 −1.17004
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) −6.00000 −0.434145 −0.217072 − 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −12.0000 −0.861550
\(195\) −12.0000 −0.859338
\(196\) −7.00000 −0.500000
\(197\) 6.00000 0.427482 0.213741 − 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −2.00000 −0.142134
\(199\) 12.0000 0.850657 0.425329 − 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) −8.00000 −0.560112
\(205\) −12.0000 −0.838116
\(206\) −4.00000 −0.278693
\(207\) 1.00000 0.0695048
\(208\) −6.00000 −0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −6.00000 −0.412082
\(213\) −8.00000 −0.548151
\(214\) 8.00000 0.546869
\(215\) −12.0000 −0.818393
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −10.0000 −0.675737
\(220\) 4.00000 0.269680
\(221\) 48.0000 3.22883
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 12.0000 0.798228
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −2.00000 −0.132453
\(229\) 20.0000 1.32164 0.660819 − 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) −2.00000 −0.131876
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) −18.0000 −1.17922 −0.589610 − 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 6.00000 0.392232
\(235\) 8.00000 0.521862
\(236\) −4.00000 −0.260378
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 − 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 2.00000 0.129099
\(241\) −10.0000 −0.644157 −0.322078 − 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) −8.00000 −0.512148
\(245\) −14.0000 −0.894427
\(246\) 6.00000 0.382546
\(247\) 12.0000 0.763542
\(248\) −8.00000 −0.508001
\(249\) 12.0000 0.760469
\(250\) 12.0000 0.758947
\(251\) −22.0000 −1.38863 −0.694314 − 0.719672i \(-0.744292\pi\)
−0.694314 + 0.719672i \(0.744292\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 6.00000 0.373544
\(259\) 0 0
\(260\) −12.0000 −0.744208
\(261\) −1.00000 −0.0618984
\(262\) 12.0000 0.741362
\(263\) 10.0000 0.616626 0.308313 − 0.951285i \(-0.400236\pi\)
0.308313 + 0.951285i \(0.400236\pi\)
\(264\) −2.00000 −0.123091
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 8.00000 0.489592
\(268\) −4.00000 −0.244339
\(269\) 10.0000 0.609711 0.304855 − 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) −2.00000 −0.120605
\(276\) 1.00000 0.0601929
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −4.00000 −0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 − 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −4.00000 −0.238197
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) −8.00000 −0.474713
\(285\) −4.00000 −0.236940
\(286\) 12.0000 0.709575
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 47.0000 2.76471
\(290\) 2.00000 0.117444
\(291\) 12.0000 0.703452
\(292\) −10.0000 −0.585206
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 7.00000 0.408248
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) −6.00000 −0.347571
\(299\) −6.00000 −0.346989
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 6.00000 0.344691
\(304\) −2.00000 −0.114708
\(305\) −16.0000 −0.916157
\(306\) 8.00000 0.457330
\(307\) −8.00000 −0.456584 −0.228292 − 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −16.0000 −0.908739
\(311\) −12.0000 −0.680458 −0.340229 − 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 6.00000 0.339683
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 6.00000 0.336463
\(319\) −2.00000 −0.111979
\(320\) 2.00000 0.111803
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 16.0000 0.890264
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −16.0000 −0.886158
\(327\) 10.0000 0.553001
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) 12.0000 0.659580 0.329790 − 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) 4.00000 0.217894 0.108947 − 0.994048i \(-0.465252\pi\)
0.108947 + 0.994048i \(0.465252\pi\)
\(338\) −23.0000 −1.25104
\(339\) −12.0000 −0.651751
\(340\) −16.0000 −0.867722
\(341\) 16.0000 0.866449
\(342\) 2.00000 0.108148
\(343\) 0 0
\(344\) 6.00000 0.323498
\(345\) 2.00000 0.107676
\(346\) 18.0000 0.967686
\(347\) −20.0000 −1.07366 −0.536828 − 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −1.00000 −0.0536056
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) −2.00000 −0.106600
\(353\) 30.0000 1.59674 0.798369 − 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 4.00000 0.212598
\(355\) −16.0000 −0.849192
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) −2.00000 −0.105409
\(361\) −15.0000 −0.789474
\(362\) 2.00000 0.105118
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) 8.00000 0.418167
\(367\) −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
\(370\) 0 0
\(371\) 0 0
\(372\) 8.00000 0.414781
\(373\) −34.0000 −1.76045 −0.880227 − 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 16.0000 0.827340
\(375\) −12.0000 −0.619677
\(376\) −4.00000 −0.206284
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 − 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) −28.0000 −1.43073 −0.715367 − 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −6.00000 −0.304997
\(388\) 12.0000 0.609208
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 12.0000 0.607644
\(391\) −8.00000 −0.404577
\(392\) 7.00000 0.353553
\(393\) −12.0000 −0.605320
\(394\) −6.00000 −0.302276
\(395\) −4.00000 −0.201262
\(396\) 2.00000 0.100504
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −12.0000 −0.601506
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 4.00000 0.199502
\(403\) −48.0000 −2.39105
\(404\) 6.00000 0.298511
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 8.00000 0.396059
\(409\) 14.0000 0.692255 0.346128 − 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 12.0000 0.592638
\(411\) −16.0000 −0.789222
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 24.0000 1.17811
\(416\) 6.00000 0.294174
\(417\) 4.00000 0.195881
\(418\) 4.00000 0.195646
\(419\) −12.0000 −0.586238 −0.293119 − 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 8.00000 0.389434
\(423\) 4.00000 0.194487
\(424\) 6.00000 0.291386
\(425\) 8.00000 0.388057
\(426\) 8.00000 0.387601
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) −12.0000 −0.579365
\(430\) 12.0000 0.578691
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 8.00000 0.384455 0.192228 − 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 10.0000 0.478913
\(437\) −2.00000 −0.0956730
\(438\) 10.0000 0.477818
\(439\) −16.0000 −0.763638 −0.381819 − 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) −48.0000 −2.28313
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 16.0000 0.758473
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 1.00000 0.0471405
\(451\) −12.0000 −0.565058
\(452\) −12.0000 −0.564433
\(453\) 16.0000 0.751746
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 26.0000 1.21623 0.608114 − 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −20.0000 −0.934539
\(459\) −8.00000 −0.373408
\(460\) 2.00000 0.0932505
\(461\) 26.0000 1.21094 0.605470 − 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 − 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 16.0000 0.741982
\(466\) 18.0000 0.833834
\(467\) −26.0000 −1.20314 −0.601568 − 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 8.00000 0.368621
\(472\) 4.00000 0.184115
\(473\) −12.0000 −0.551761
\(474\) 2.00000 0.0918630
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 8.00000 0.365911
\(479\) −18.0000 −0.822441 −0.411220 − 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 24.0000 1.08978
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 − 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) 8.00000 0.362143
\(489\) 16.0000 0.723545
\(490\) 14.0000 0.632456
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −6.00000 −0.270501
\(493\) 8.00000 0.360302
\(494\) −12.0000 −0.539906
\(495\) 4.00000 0.179787
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 4.00000 0.179065 0.0895323 − 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −12.0000 −0.536656
\(501\) −8.00000 −0.357414
\(502\) 22.0000 0.981908
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −2.00000 −0.0889108
\(507\) 23.0000 1.02147
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 − 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 16.0000 0.708492
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −2.00000 −0.0883022
\(514\) 14.0000 0.617514
\(515\) 8.00000 0.352522
\(516\) −6.00000 −0.264135
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 12.0000 0.526235
\(521\) −38.0000 −1.66481 −0.832405 − 0.554168i \(-0.813037\pi\)
−0.832405 + 0.554168i \(0.813037\pi\)
\(522\) 1.00000 0.0437688
\(523\) 36.0000 1.57417 0.787085 − 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) −64.0000 −2.78788
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 12.0000 0.521247
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) −8.00000 −0.346194
\(535\) −16.0000 −0.691740
\(536\) 4.00000 0.172774
\(537\) 12.0000 0.517838
\(538\) −10.0000 −0.431131
\(539\) −14.0000 −0.603023
\(540\) 2.00000 0.0860663
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −8.00000 −0.343629
\(543\) −2.00000 −0.0858282
\(544\) 8.00000 0.342997
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 − 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) −16.0000 −0.683486
\(549\) −8.00000 −0.341432
\(550\) 2.00000 0.0852803
\(551\) 2.00000 0.0852029
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) −38.0000 −1.61011 −0.805056 − 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) −8.00000 −0.338667
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) −18.0000 −0.759284
\(563\) 38.0000 1.60151 0.800755 − 0.598993i \(-0.204432\pi\)
0.800755 + 0.598993i \(0.204432\pi\)
\(564\) 4.00000 0.168430
\(565\) −24.0000 −1.00969
\(566\) −16.0000 −0.672530
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −20.0000 −0.838444 −0.419222 − 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 4.00000 0.167542
\(571\) 16.0000 0.669579 0.334790 − 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) −12.0000 −0.501745
\(573\) −6.00000 −0.250654
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) 22.0000 0.915872 0.457936 − 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) −47.0000 −1.95494
\(579\) 14.0000 0.581820
\(580\) −2.00000 −0.0830455
\(581\) 0 0
\(582\) −12.0000 −0.497416
\(583\) −12.0000 −0.496989
\(584\) 10.0000 0.413803
\(585\) −12.0000 −0.496139
\(586\) 0 0
\(587\) 36.0000 1.48588 0.742940 − 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −7.00000 −0.288675
\(589\) −16.0000 −0.659269
\(590\) 8.00000 0.329355
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 − 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 12.0000 0.491127
\(598\) 6.00000 0.245358
\(599\) 36.0000 1.47092 0.735460 − 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 1.00000 0.0408248
\(601\) 22.0000 0.897399 0.448699 − 0.893683i \(-0.351887\pi\)
0.448699 + 0.893683i \(0.351887\pi\)
\(602\) 0 0
\(603\) −4.00000 −0.162893
\(604\) 16.0000 0.651031
\(605\) −14.0000 −0.569181
\(606\) −6.00000 −0.243733
\(607\) 4.00000 0.162355 0.0811775 − 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 16.0000 0.647821
\(611\) −24.0000 −0.970936
\(612\) −8.00000 −0.323381
\(613\) 10.0000 0.403896 0.201948 − 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) 8.00000 0.322854
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −4.00000 −0.161034 −0.0805170 − 0.996753i \(-0.525657\pi\)
−0.0805170 + 0.996753i \(0.525657\pi\)
\(618\) −4.00000 −0.160904
\(619\) −10.0000 −0.401934 −0.200967 − 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 16.0000 0.642575
\(621\) 1.00000 0.0401286
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) −19.0000 −0.760000
\(626\) 10.0000 0.399680
\(627\) −4.00000 −0.159745
\(628\) 8.00000 0.319235
\(629\) 0 0
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 − 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 2.00000 0.0795557
\(633\) −8.00000 −0.317971
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 42.0000 1.66410
\(638\) 2.00000 0.0791808
\(639\) −8.00000 −0.316475
\(640\) −2.00000 −0.0790569
\(641\) 36.0000 1.42191 0.710957 − 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 8.00000 0.315735
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) −16.0000 −0.629512
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.00000 −0.314027
\(650\) −6.00000 −0.235339
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) 14.0000 0.547862 0.273931 − 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −10.0000 −0.391031
\(655\) −24.0000 −0.937758
\(656\) −6.00000 −0.234261
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −2.00000 −0.0779089 −0.0389545 − 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) 4.00000 0.155700
\(661\) −30.0000 −1.16686 −0.583432 − 0.812162i \(-0.698291\pi\)
−0.583432 + 0.812162i \(0.698291\pi\)
\(662\) −12.0000 −0.466393
\(663\) 48.0000 1.86417
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 − 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −4.00000 −0.154074
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) −8.00000 −0.307465 −0.153732 − 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(678\) 12.0000 0.460857
\(679\) 0 0
\(680\) 16.0000 0.613572
\(681\) −4.00000 −0.153280
\(682\) −16.0000 −0.612672
\(683\) 12.0000 0.459167 0.229584 − 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −32.0000 −1.22266
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) −6.00000 −0.228748
\(689\) 36.0000 1.37149
\(690\) −2.00000 −0.0761387
\(691\) −12.0000 −0.456502 −0.228251 − 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) 8.00000 0.303457
\(696\) 1.00000 0.0379049
\(697\) 48.0000 1.81813
\(698\) −22.0000 −0.832712
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 6.00000 0.226455
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 8.00000 0.301297
\(706\) −30.0000 −1.12906
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) −34.0000 −1.27690 −0.638448 − 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 16.0000 0.600469
\(711\) −2.00000 −0.0750059
\(712\) −8.00000 −0.299813
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) 12.0000 0.448461
\(717\) −8.00000 −0.298765
\(718\) −6.00000 −0.223918
\(719\) −8.00000 −0.298350 −0.149175 − 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) −10.0000 −0.371904
\(724\) −2.00000 −0.0743294
\(725\) 1.00000 0.0371391
\(726\) 7.00000 0.259794
\(727\) 2.00000 0.0741759 0.0370879 − 0.999312i \(-0.488192\pi\)
0.0370879 + 0.999312i \(0.488192\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 20.0000 0.740233
\(731\) 48.0000 1.77534
\(732\) −8.00000 −0.295689
\(733\) −20.0000 −0.738717 −0.369358 − 0.929287i \(-0.620423\pi\)
−0.369358 + 0.929287i \(0.620423\pi\)
\(734\) 10.0000 0.369107
\(735\) −14.0000 −0.516398
\(736\) −1.00000 −0.0368605
\(737\) −8.00000 −0.294684
\(738\) 6.00000 0.220863
\(739\) 16.0000 0.588570 0.294285 − 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 0 0
\(743\) 26.0000 0.953847 0.476924 − 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) −8.00000 −0.293294
\(745\) 12.0000 0.439646
\(746\) 34.0000 1.24483
\(747\) 12.0000 0.439057
\(748\) −16.0000 −0.585018
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 22.0000 0.802791 0.401396 − 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 4.00000 0.145865
\(753\) −22.0000 −0.801725
\(754\) −6.00000 −0.218507
\(755\) 32.0000 1.16460
\(756\) 0 0
\(757\) −4.00000 −0.145382 −0.0726912 − 0.997354i \(-0.523159\pi\)
−0.0726912 + 0.997354i \(0.523159\pi\)
\(758\) −6.00000 −0.217930
\(759\) 2.00000 0.0725954
\(760\) 4.00000 0.145095
\(761\) 6.00000 0.217500 0.108750 − 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) −16.0000 −0.578481
\(766\) 28.0000 1.01168
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) 32.0000 1.15395 0.576975 − 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 14.0000 0.503871
\(773\) 36.0000 1.29483 0.647415 − 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 6.00000 0.215666
\(775\) −8.00000 −0.287368
\(776\) −12.0000 −0.430775
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) 12.0000 0.429945
\(780\) −12.0000 −0.429669
\(781\) −16.0000 −0.572525
\(782\) 8.00000 0.286079
\(783\) −1.00000 −0.0357371
\(784\) −7.00000 −0.250000
\(785\) 16.0000 0.571064
\(786\) 12.0000 0.428026
\(787\) 12.0000 0.427754 0.213877 − 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 6.00000 0.213741
\(789\) 10.0000 0.356009
\(790\) 4.00000 0.142314
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) 48.0000 1.70453
\(794\) 22.0000 0.780751
\(795\) −12.0000 −0.425596
\(796\) 12.0000 0.425329
\(797\) −4.00000 −0.141687 −0.0708436 − 0.997487i \(-0.522569\pi\)
−0.0708436 + 0.997487i \(0.522569\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 1.00000 0.0353553
\(801\) 8.00000 0.282666
\(802\) −10.0000 −0.353112
\(803\) −20.0000 −0.705785
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) 10.0000 0.352017
\(808\) −6.00000 −0.211079
\(809\) −50.0000 −1.75791 −0.878953 − 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 52.0000 1.82597 0.912983 − 0.407997i \(-0.133772\pi\)
0.912983 + 0.407997i \(0.133772\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 32.0000 1.12091
\(816\) −8.00000 −0.280056
\(817\) 12.0000 0.419827
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) −12.0000 −0.419058
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 16.0000 0.558064
\(823\) 20.0000 0.697156 0.348578 − 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −4.00000 −0.139347
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −18.0000 −0.625921 −0.312961 − 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 1.00000 0.0347524
\(829\) −46.0000 −1.59765 −0.798823 − 0.601566i \(-0.794544\pi\)
−0.798823 + 0.601566i \(0.794544\pi\)
\(830\) −24.0000 −0.833052
\(831\) −2.00000 −0.0693792
\(832\) −6.00000 −0.208013
\(833\) 56.0000 1.94029
\(834\) −4.00000 −0.138509
\(835\) −16.0000 −0.553703
\(836\) −4.00000 −0.138343
\(837\) 8.00000 0.276520
\(838\) 12.0000 0.414533
\(839\) −2.00000 −0.0690477 −0.0345238 − 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) 18.0000 0.619953
\(844\) −8.00000 −0.275371
\(845\) 46.0000 1.58245
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 16.0000 0.549119
\(850\) −8.00000 −0.274398
\(851\) 0 0
\(852\) −8.00000 −0.274075
\(853\) −2.00000 −0.0684787 −0.0342393 − 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 8.00000 0.273434
\(857\) −26.0000 −0.888143 −0.444072 − 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 12.0000 0.409673
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −12.0000 −0.409197
\(861\) 0 0
\(862\) 0 0
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −36.0000 −1.22404
\(866\) −8.00000 −0.271851
\(867\) 47.0000 1.59620
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 2.00000 0.0678064
\(871\) 24.0000 0.813209
\(872\) −10.0000 −0.338643
\(873\) 12.0000 0.406138
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 18.0000 0.607817 0.303908 − 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 16.0000 0.539974
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) −40.0000 −1.34763 −0.673817 − 0.738898i \(-0.735346\pi\)
−0.673817 + 0.738898i \(0.735346\pi\)
\(882\) 7.00000 0.235702
\(883\) −12.0000 −0.403832 −0.201916 − 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 48.0000 1.61441
\(885\) −8.00000 −0.268917
\(886\) −24.0000 −0.806296
\(887\) −48.0000 −1.61168 −0.805841 − 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −16.0000 −0.536321
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) −8.00000 −0.267710
\(894\) −6.00000 −0.200670
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 6.00000 0.200223
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) 48.0000 1.59911
\(902\) 12.0000 0.399556
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) −4.00000 −0.132964
\(906\) −16.0000 −0.531564
\(907\) −6.00000 −0.199227 −0.0996134 − 0.995026i \(-0.531761\pi\)
−0.0996134 + 0.995026i \(0.531761\pi\)
\(908\) −4.00000 −0.132745
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −54.0000 −1.78910 −0.894550 − 0.446968i \(-0.852504\pi\)
−0.894550 + 0.446968i \(0.852504\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 24.0000 0.794284
\(914\) −26.0000 −0.860004
\(915\) −16.0000 −0.528944
\(916\) 20.0000 0.660819
\(917\) 0 0
\(918\) 8.00000 0.264039
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) −2.00000 −0.0659380
\(921\) −8.00000 −0.263609
\(922\) −26.0000 −0.856264
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) 4.00000 0.131377
\(928\) 1.00000 0.0328266
\(929\) −18.0000 −0.590561 −0.295280 − 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) −16.0000 −0.524661
\(931\) 14.0000 0.458831
\(932\) −18.0000 −0.589610
\(933\) −12.0000 −0.392862
\(934\) 26.0000 0.850746
\(935\) −32.0000 −1.04651
\(936\) 6.00000 0.196116
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −8.00000 −0.260654
\(943\) −6.00000 −0.195387
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −36.0000 −1.16984 −0.584921 − 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −2.00000 −0.0649570
\(949\) 60.0000 1.94768
\(950\) −2.00000 −0.0648886
\(951\) −6.00000 −0.194563
\(952\) 0 0
\(953\) −14.0000 −0.453504 −0.226752 − 0.973952i \(-0.572811\pi\)
−0.226752 + 0.973952i \(0.572811\pi\)
\(954\) 6.00000 0.194257
\(955\) −12.0000 −0.388311
\(956\) −8.00000 −0.258738
\(957\) −2.00000 −0.0646508
\(958\) 18.0000 0.581554
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) −10.0000 −0.322078
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 − 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 7.00000 0.224989
\(969\) 16.0000 0.513994
\(970\) −24.0000 −0.770594
\(971\) 22.0000 0.706014 0.353007 − 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 40.0000 1.28168
\(975\) 6.00000 0.192154
\(976\) −8.00000 −0.256074
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) −16.0000 −0.511624
\(979\) 16.0000 0.511362
\(980\) −14.0000 −0.447214
\(981\) 10.0000 0.319275
\(982\) 28.0000 0.893516
\(983\) 38.0000 1.21201 0.606006 − 0.795460i \(-0.292771\pi\)
0.606006 + 0.795460i \(0.292771\pi\)
\(984\) 6.00000 0.191273
\(985\) 12.0000 0.382352
\(986\) −8.00000 −0.254772
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −6.00000 −0.190789
\(990\) −4.00000 −0.127128
\(991\) 40.0000 1.27064 0.635321 − 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −8.00000 −0.254000
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 12.0000 0.380235
\(997\) −38.0000 −1.20347 −0.601736 − 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) −4.00000 −0.126618
\(999\) 0 0
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.f.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.f.1.1 ✓ 1 1.1 even 1 trivial