Properties

Label 58.2.a.b.1.1
Level $58$
Weight $2$
Character 58.1
Self dual yes
Analytic conductor $0.463$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 58.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.463132331723\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 58.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} -2.00000 q^{18} +1.00000 q^{20} +2.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +5.00000 q^{27} -2.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +3.00000 q^{33} +8.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} +1.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} +2.00000 q^{42} -11.0000 q^{43} -3.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} +13.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -8.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +5.00000 q^{54} -3.00000 q^{55} -2.00000 q^{56} -1.00000 q^{58} -1.00000 q^{60} -8.00000 q^{61} -3.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +3.00000 q^{66} -12.0000 q^{67} +8.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +2.00000 q^{71} -2.00000 q^{72} +4.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} +6.00000 q^{77} +1.00000 q^{78} +15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +2.00000 q^{84} +8.00000 q^{85} -11.0000 q^{86} +1.00000 q^{87} -3.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} +13.0000 q^{94} -1.00000 q^{96} -2.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −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
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −2.00000 −0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) −2.00000 −0.471405
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.00000 0.436436
\(22\) −3.00000 −0.639602
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) −1.00000 −0.196116
\(27\) 5.00000 0.962250
\(28\) −2.00000 −0.377964
\(29\) −1.00000 −0.185695
\(30\) −1.00000 −0.182574
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.00000 0.522233
\(34\) 8.00000 1.37199
\(35\) −2.00000 −0.338062
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 1.00000 0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 2.00000 0.308607
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) −3.00000 −0.452267
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) 13.0000 1.89624 0.948122 0.317905i \(-0.102979\pi\)
0.948122 + 0.317905i \(0.102979\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −4.00000 −0.565685
\(51\) −8.00000 −1.12022
\(52\) −1.00000 −0.138675
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 5.00000 0.680414
\(55\) −3.00000 −0.404520
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −1.00000 −0.129099
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −3.00000 −0.381000
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 3.00000 0.369274
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 8.00000 0.970143
\(69\) −4.00000 −0.481543
\(70\) −2.00000 −0.239046
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) −2.00000 −0.235702
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 8.00000 0.929981
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 1.00000 0.113228
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 2.00000 0.218218
\(85\) 8.00000 0.867722
\(86\) −11.0000 −1.18616
\(87\) 1.00000 0.107211
\(88\) −3.00000 −0.319801
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 2.00000 0.209657
\(92\) 4.00000 0.417029
\(93\) 3.00000 0.311086
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −3.00000 −0.303046
\(99\) 6.00000 0.603023
\(100\) −4.00000 −0.400000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) −8.00000 −0.792118
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.00000 0.195180
\(106\) −11.0000 −1.06841
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 5.00000 0.481125
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) −3.00000 −0.286039
\(111\) −8.00000 −0.759326
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −1.00000 −0.0928477
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −16.0000 −1.46672
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) −2.00000 −0.180334
\(124\) −3.00000 −0.269408
\(125\) −9.00000 −0.804984
\(126\) 4.00000 0.356348
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 11.0000 0.968496
\(130\) −1.00000 −0.0877058
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 3.00000 0.261116
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) 5.00000 0.430331
\(136\) 8.00000 0.685994
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −4.00000 −0.340503
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −2.00000 −0.169031
\(141\) −13.0000 −1.09480
\(142\) 2.00000 0.167836
\(143\) 3.00000 0.250873
\(144\) −2.00000 −0.166667
\(145\) −1.00000 −0.0830455
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) 8.00000 0.657596
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 4.00000 0.326599
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) −16.0000 −1.29352
\(154\) 6.00000 0.483494
\(155\) −3.00000 −0.240966
\(156\) 1.00000 0.0800641
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 15.0000 1.19334
\(159\) 11.0000 0.872357
\(160\) 1.00000 0.0790569
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) 9.00000 0.704934 0.352467 0.935824i \(-0.385343\pi\)
0.352467 + 0.935824i \(0.385343\pi\)
\(164\) 2.00000 0.156174
\(165\) 3.00000 0.233550
\(166\) 4.00000 0.310460
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 2.00000 0.154303
\(169\) −12.0000 −0.923077
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 1.00000 0.0758098
\(175\) 8.00000 0.604743
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) −2.00000 −0.149071
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 2.00000 0.148250
\(183\) 8.00000 0.591377
\(184\) 4.00000 0.294884
\(185\) 8.00000 0.588172
\(186\) 3.00000 0.219971
\(187\) −24.0000 −1.75505
\(188\) 13.0000 0.948122
\(189\) −10.0000 −0.727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −2.00000 −0.143592
\(195\) 1.00000 0.0716115
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 6.00000 0.426401
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) −4.00000 −0.282843
\(201\) 12.0000 0.846415
\(202\) −8.00000 −0.562878
\(203\) 2.00000 0.140372
\(204\) −8.00000 −0.560112
\(205\) 2.00000 0.139686
\(206\) 14.0000 0.975426
\(207\) −8.00000 −0.556038
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −11.0000 −0.755483
\(213\) −2.00000 −0.137038
\(214\) −2.00000 −0.136717
\(215\) −11.0000 −0.750194
\(216\) 5.00000 0.340207
\(217\) 6.00000 0.407307
\(218\) 5.00000 0.338643
\(219\) −4.00000 −0.270295
\(220\) −3.00000 −0.202260
\(221\) −8.00000 −0.538138
\(222\) −8.00000 −0.536925
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) −2.00000 −0.133631
\(225\) 8.00000 0.533333
\(226\) −6.00000 −0.399114
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 4.00000 0.263752
\(231\) −6.00000 −0.394771
\(232\) −1.00000 −0.0656532
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 2.00000 0.130744
\(235\) 13.0000 0.848026
\(236\) 0 0
\(237\) −15.0000 −0.974355
\(238\) −16.0000 −1.03713
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) −2.00000 −0.128565
\(243\) −16.0000 −1.02640
\(244\) −8.00000 −0.512148
\(245\) −3.00000 −0.191663
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) −4.00000 −0.253490
\(250\) −9.00000 −0.569210
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 4.00000 0.251976
\(253\) −12.0000 −0.754434
\(254\) 8.00000 0.501965
\(255\) −8.00000 −0.500979
\(256\) 1.00000 0.0625000
\(257\) 13.0000 0.810918 0.405459 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(258\) 11.0000 0.684830
\(259\) −16.0000 −0.994192
\(260\) −1.00000 −0.0620174
\(261\) 2.00000 0.123797
\(262\) 12.0000 0.741362
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 3.00000 0.184637
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −12.0000 −0.733017
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 5.00000 0.304290
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 8.00000 0.485071
\(273\) −2.00000 −0.121046
\(274\) −12.0000 −0.724947
\(275\) 12.0000 0.723627
\(276\) −4.00000 −0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −20.0000 −1.19952
\(279\) 6.00000 0.359211
\(280\) −2.00000 −0.119523
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) −13.0000 −0.774139
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) −4.00000 −0.236113
\(288\) −2.00000 −0.117851
\(289\) 47.0000 2.76471
\(290\) −1.00000 −0.0587220
\(291\) 2.00000 0.117242
\(292\) 4.00000 0.234082
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −15.0000 −0.870388
\(298\) 15.0000 0.868927
\(299\) −4.00000 −0.231326
\(300\) 4.00000 0.230940
\(301\) 22.0000 1.26806
\(302\) 2.00000 0.115087
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) −16.0000 −0.914659
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 6.00000 0.341882
\(309\) −14.0000 −0.796432
\(310\) −3.00000 −0.170389
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 1.00000 0.0566139
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) 18.0000 1.01580
\(315\) 4.00000 0.225374
\(316\) 15.0000 0.843816
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 11.0000 0.616849
\(319\) 3.00000 0.167968
\(320\) 1.00000 0.0559017
\(321\) 2.00000 0.111629
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 9.00000 0.498464
\(327\) −5.00000 −0.276501
\(328\) 2.00000 0.110432
\(329\) −26.0000 −1.43343
\(330\) 3.00000 0.165145
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) 4.00000 0.219529
\(333\) −16.0000 −0.876795
\(334\) −2.00000 −0.109435
\(335\) −12.0000 −0.655630
\(336\) 2.00000 0.109109
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) −12.0000 −0.652714
\(339\) 6.00000 0.325875
\(340\) 8.00000 0.433861
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) −11.0000 −0.593080
\(345\) −4.00000 −0.215353
\(346\) −6.00000 −0.322562
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 1.00000 0.0536056
\(349\) −15.0000 −0.802932 −0.401466 0.915874i \(-0.631499\pi\)
−0.401466 + 0.915874i \(0.631499\pi\)
\(350\) 8.00000 0.427618
\(351\) −5.00000 −0.266880
\(352\) −3.00000 −0.159901
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) −10.0000 −0.529999
\(357\) 16.0000 0.846810
\(358\) −10.0000 −0.528516
\(359\) −25.0000 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) −2.00000 −0.105409
\(361\) −19.0000 −1.00000
\(362\) 7.00000 0.367912
\(363\) 2.00000 0.104973
\(364\) 2.00000 0.104828
\(365\) 4.00000 0.209370
\(366\) 8.00000 0.418167
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 4.00000 0.208514
\(369\) −4.00000 −0.208232
\(370\) 8.00000 0.415900
\(371\) 22.0000 1.14218
\(372\) 3.00000 0.155543
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) −24.0000 −1.24101
\(375\) 9.00000 0.464758
\(376\) 13.0000 0.670424
\(377\) 1.00000 0.0515026
\(378\) −10.0000 −0.514344
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −8.00000 −0.409316
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 6.00000 0.305788
\(386\) 14.0000 0.712581
\(387\) 22.0000 1.11832
\(388\) −2.00000 −0.101535
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 1.00000 0.0506370
\(391\) 32.0000 1.61831
\(392\) −3.00000 −0.151523
\(393\) −12.0000 −0.605320
\(394\) 18.0000 0.906827
\(395\) 15.0000 0.754732
\(396\) 6.00000 0.301511
\(397\) −17.0000 −0.853206 −0.426603 0.904439i \(-0.640290\pi\)
−0.426603 + 0.904439i \(0.640290\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 12.0000 0.598506
\(403\) 3.00000 0.149441
\(404\) −8.00000 −0.398015
\(405\) 1.00000 0.0496904
\(406\) 2.00000 0.0992583
\(407\) −24.0000 −1.18964
\(408\) −8.00000 −0.396059
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 2.00000 0.0987730
\(411\) 12.0000 0.591916
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 4.00000 0.196352
\(416\) −1.00000 −0.0490290
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 2.00000 0.0975900
\(421\) 32.0000 1.55958 0.779792 0.626038i \(-0.215325\pi\)
0.779792 + 0.626038i \(0.215325\pi\)
\(422\) −3.00000 −0.146038
\(423\) −26.0000 −1.26416
\(424\) −11.0000 −0.534207
\(425\) −32.0000 −1.55223
\(426\) −2.00000 −0.0969003
\(427\) 16.0000 0.774294
\(428\) −2.00000 −0.0966736
\(429\) −3.00000 −0.144841
\(430\) −11.0000 −0.530467
\(431\) 32.0000 1.54139 0.770693 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) 5.00000 0.240563
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 6.00000 0.288009
\(435\) 1.00000 0.0479463
\(436\) 5.00000 0.239457
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −3.00000 −0.143019
\(441\) 6.00000 0.285714
\(442\) −8.00000 −0.380521
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −8.00000 −0.379663
\(445\) −10.0000 −0.474045
\(446\) −26.0000 −1.23114
\(447\) −15.0000 −0.709476
\(448\) −2.00000 −0.0944911
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 8.00000 0.377124
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) −2.00000 −0.0939682
\(454\) 18.0000 0.844782
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 10.0000 0.467269
\(459\) 40.0000 1.86704
\(460\) 4.00000 0.186501
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) −6.00000 −0.279145
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 3.00000 0.139122
\(466\) −1.00000 −0.0463241
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 2.00000 0.0924500
\(469\) 24.0000 1.10822
\(470\) 13.0000 0.599645
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) 33.0000 1.51734
\(474\) −15.0000 −0.688973
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) 22.0000 1.00731
\(478\) 0 0
\(479\) −5.00000 −0.228456 −0.114228 0.993455i \(-0.536439\pi\)
−0.114228 + 0.993455i \(0.536439\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −8.00000 −0.364769
\(482\) 17.0000 0.774329
\(483\) 8.00000 0.364013
\(484\) −2.00000 −0.0909091
\(485\) −2.00000 −0.0908153
\(486\) −16.0000 −0.725775
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −8.00000 −0.362143
\(489\) −9.00000 −0.406994
\(490\) −3.00000 −0.135526
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) −2.00000 −0.0901670
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) −3.00000 −0.134704
\(497\) −4.00000 −0.179425
\(498\) −4.00000 −0.179244
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −9.00000 −0.402492
\(501\) 2.00000 0.0893534
\(502\) 27.0000 1.20507
\(503\) 19.0000 0.847168 0.423584 0.905857i \(-0.360772\pi\)
0.423584 + 0.905857i \(0.360772\pi\)
\(504\) 4.00000 0.178174
\(505\) −8.00000 −0.355995
\(506\) −12.0000 −0.533465
\(507\) 12.0000 0.532939
\(508\) 8.00000 0.354943
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) −8.00000 −0.354246
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 13.0000 0.573405
\(515\) 14.0000 0.616914
\(516\) 11.0000 0.484248
\(517\) −39.0000 −1.71522
\(518\) −16.0000 −0.703000
\(519\) 6.00000 0.263371
\(520\) −1.00000 −0.0438529
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 2.00000 0.0875376
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 12.0000 0.524222
\(525\) −8.00000 −0.349149
\(526\) 9.00000 0.392419
\(527\) −24.0000 −1.04546
\(528\) 3.00000 0.130558
\(529\) −7.00000 −0.304348
\(530\) −11.0000 −0.477809
\(531\) 0 0
\(532\) 0 0
\(533\) −2.00000 −0.0866296
\(534\) 10.0000 0.432742
\(535\) −2.00000 −0.0864675
\(536\) −12.0000 −0.518321
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 5.00000 0.215166
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −13.0000 −0.558398
\(543\) −7.00000 −0.300399
\(544\) 8.00000 0.342997
\(545\) 5.00000 0.214176
\(546\) −2.00000 −0.0855921
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) −12.0000 −0.512615
\(549\) 16.0000 0.682863
\(550\) 12.0000 0.511682
\(551\) 0 0
\(552\) −4.00000 −0.170251
\(553\) −30.0000 −1.27573
\(554\) −2.00000 −0.0849719
\(555\) −8.00000 −0.339581
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 6.00000 0.254000
\(559\) 11.0000 0.465250
\(560\) −2.00000 −0.0845154
\(561\) 24.0000 1.01328
\(562\) 27.0000 1.13893
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) −13.0000 −0.547399
\(565\) −6.00000 −0.252422
\(566\) 4.00000 0.168133
\(567\) −2.00000 −0.0839921
\(568\) 2.00000 0.0839181
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 3.00000 0.125436
\(573\) 8.00000 0.334205
\(574\) −4.00000 −0.166957
\(575\) −16.0000 −0.667246
\(576\) −2.00000 −0.0833333
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 47.0000 1.95494
\(579\) −14.0000 −0.581820
\(580\) −1.00000 −0.0415227
\(581\) −8.00000 −0.331896
\(582\) 2.00000 0.0829027
\(583\) 33.0000 1.36672
\(584\) 4.00000 0.165521
\(585\) 2.00000 0.0826898
\(586\) 14.0000 0.578335
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) 8.00000 0.328798
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) −15.0000 −0.615457
\(595\) −16.0000 −0.655936
\(596\) 15.0000 0.614424
\(597\) 10.0000 0.409273
\(598\) −4.00000 −0.163572
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) 4.00000 0.163299
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 22.0000 0.896653
\(603\) 24.0000 0.977356
\(604\) 2.00000 0.0813788
\(605\) −2.00000 −0.0813116
\(606\) 8.00000 0.324978
\(607\) 3.00000 0.121766 0.0608831 0.998145i \(-0.480608\pi\)
0.0608831 + 0.998145i \(0.480608\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) −8.00000 −0.323911
\(611\) −13.0000 −0.525924
\(612\) −16.0000 −0.646762
\(613\) −31.0000 −1.25208 −0.626039 0.779792i \(-0.715325\pi\)
−0.626039 + 0.779792i \(0.715325\pi\)
\(614\) −7.00000 −0.282497
\(615\) −2.00000 −0.0806478
\(616\) 6.00000 0.241747
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −14.0000 −0.563163
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) −3.00000 −0.120483
\(621\) 20.0000 0.802572
\(622\) −8.00000 −0.320771
\(623\) 20.0000 0.801283
\(624\) 1.00000 0.0400320
\(625\) 11.0000 0.440000
\(626\) 9.00000 0.359712
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 64.0000 2.55185
\(630\) 4.00000 0.159364
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 15.0000 0.596668
\(633\) 3.00000 0.119239
\(634\) −12.0000 −0.476581
\(635\) 8.00000 0.317470
\(636\) 11.0000 0.436178
\(637\) 3.00000 0.118864
\(638\) 3.00000 0.118771
\(639\) −4.00000 −0.158238
\(640\) 1.00000 0.0395285
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 2.00000 0.0789337
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −8.00000 −0.315244
\(645\) 11.0000 0.433125
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) −6.00000 −0.235159
\(652\) 9.00000 0.352467
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −5.00000 −0.195515
\(655\) 12.0000 0.468879
\(656\) 2.00000 0.0780869
\(657\) −8.00000 −0.312110
\(658\) −26.0000 −1.01359
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 3.00000 0.116775
\(661\) −18.0000 −0.700119 −0.350059 0.936727i \(-0.613839\pi\)
−0.350059 + 0.936727i \(0.613839\pi\)
\(662\) −23.0000 −0.893920
\(663\) 8.00000 0.310694
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) −4.00000 −0.154881
\(668\) −2.00000 −0.0773823
\(669\) 26.0000 1.00522
\(670\) −12.0000 −0.463600
\(671\) 24.0000 0.926510
\(672\) 2.00000 0.0771517
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) −32.0000 −1.23259
\(675\) −20.0000 −0.769800
\(676\) −12.0000 −0.461538
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 6.00000 0.230429
\(679\) 4.00000 0.153506
\(680\) 8.00000 0.306786
\(681\) −18.0000 −0.689761
\(682\) 9.00000 0.344628
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 20.0000 0.763604
\(687\) −10.0000 −0.381524
\(688\) −11.0000 −0.419371
\(689\) 11.0000 0.419067
\(690\) −4.00000 −0.152277
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −6.00000 −0.228086
\(693\) −12.0000 −0.455842
\(694\) −2.00000 −0.0759190
\(695\) −20.0000 −0.758643
\(696\) 1.00000 0.0379049
\(697\) 16.0000 0.606043
\(698\) −15.0000 −0.567758
\(699\) 1.00000 0.0378235
\(700\) 8.00000 0.302372
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) −5.00000 −0.188713
\(703\) 0 0
\(704\) −3.00000 −0.113067
\(705\) −13.0000 −0.489608
\(706\) −26.0000 −0.978523
\(707\) 16.0000 0.601742
\(708\) 0 0
\(709\) 15.0000 0.563337 0.281668 0.959512i \(-0.409112\pi\)
0.281668 + 0.959512i \(0.409112\pi\)
\(710\) 2.00000 0.0750587
\(711\) −30.0000 −1.12509
\(712\) −10.0000 −0.374766
\(713\) −12.0000 −0.449404
\(714\) 16.0000 0.598785
\(715\) 3.00000 0.112194
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) −25.0000 −0.932992
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −28.0000 −1.04277
\(722\) −19.0000 −0.707107
\(723\) −17.0000 −0.632237
\(724\) 7.00000 0.260153
\(725\) 4.00000 0.148556
\(726\) 2.00000 0.0742270
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) −88.0000 −3.25480
\(732\) 8.00000 0.295689
\(733\) 24.0000 0.886460 0.443230 0.896408i \(-0.353832\pi\)
0.443230 + 0.896408i \(0.353832\pi\)
\(734\) −32.0000 −1.18114
\(735\) 3.00000 0.110657
\(736\) 4.00000 0.147442
\(737\) 36.0000 1.32608
\(738\) −4.00000 −0.147242
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 22.0000 0.807645
\(743\) 44.0000 1.61420 0.807102 0.590412i \(-0.201035\pi\)
0.807102 + 0.590412i \(0.201035\pi\)
\(744\) 3.00000 0.109985
\(745\) 15.0000 0.549557
\(746\) −21.0000 −0.768865
\(747\) −8.00000 −0.292705
\(748\) −24.0000 −0.877527
\(749\) 4.00000 0.146157
\(750\) 9.00000 0.328634
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 13.0000 0.474061
\(753\) −27.0000 −0.983935
\(754\) 1.00000 0.0364179
\(755\) 2.00000 0.0727875
\(756\) −10.0000 −0.363696
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 20.0000 0.726433
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) −8.00000 −0.289809
\(763\) −10.0000 −0.362024
\(764\) −8.00000 −0.289430
\(765\) −16.0000 −0.578481
\(766\) 14.0000 0.505841
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 6.00000 0.216225
\(771\) −13.0000 −0.468184
\(772\) 14.0000 0.503871
\(773\) 14.0000 0.503545 0.251773 0.967786i \(-0.418987\pi\)
0.251773 + 0.967786i \(0.418987\pi\)
\(774\) 22.0000 0.790774
\(775\) 12.0000 0.431053
\(776\) −2.00000 −0.0717958
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000 0.0358057
\(781\) −6.00000 −0.214697
\(782\) 32.0000 1.14432
\(783\) −5.00000 −0.178685
\(784\) −3.00000 −0.107143
\(785\) 18.0000 0.642448
\(786\) −12.0000 −0.428026
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 18.0000 0.641223
\(789\) −9.00000 −0.320408
\(790\) 15.0000 0.533676
\(791\) 12.0000 0.426671
\(792\) 6.00000 0.213201
\(793\) 8.00000 0.284088
\(794\) −17.0000 −0.603307
\(795\) 11.0000 0.390130
\(796\) −10.0000 −0.354441
\(797\) −32.0000 −1.13350 −0.566749 0.823890i \(-0.691799\pi\)
−0.566749 + 0.823890i \(0.691799\pi\)
\(798\) 0 0
\(799\) 104.000 3.67926
\(800\) −4.00000 −0.141421
\(801\) 20.0000 0.706665
\(802\) 27.0000 0.953403
\(803\) −12.0000 −0.423471
\(804\) 12.0000 0.423207
\(805\) −8.00000 −0.281963
\(806\) 3.00000 0.105670
\(807\) 0 0
\(808\) −8.00000 −0.281439
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 1.00000 0.0351364
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 2.00000 0.0701862
\(813\) 13.0000 0.455930
\(814\) −24.0000 −0.841200
\(815\) 9.00000 0.315256
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) −4.00000 −0.139771
\(820\) 2.00000 0.0698430
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 12.0000 0.418548
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 14.0000 0.487713
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 13.0000 0.452054 0.226027 0.974121i \(-0.427426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(828\) −8.00000 −0.278019
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) 4.00000 0.138842
\(831\) 2.00000 0.0693792
\(832\) −1.00000 −0.0346688
\(833\) −24.0000 −0.831551
\(834\) 20.0000 0.692543
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) −15.0000 −0.518476
\(838\) −10.0000 −0.345444
\(839\) 45.0000 1.55357 0.776786 0.629764i \(-0.216849\pi\)
0.776786 + 0.629764i \(0.216849\pi\)
\(840\) 2.00000 0.0690066
\(841\) 1.00000 0.0344828
\(842\) 32.0000 1.10279
\(843\) −27.0000 −0.929929
\(844\) −3.00000 −0.103264
\(845\) −12.0000 −0.412813
\(846\) −26.0000 −0.893898
\(847\) 4.00000 0.137442
\(848\) −11.0000 −0.377742
\(849\) −4.00000 −0.137280
\(850\) −32.0000 −1.09759
\(851\) 32.0000 1.09695
\(852\) −2.00000 −0.0685189
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) −3.00000 −0.102418
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) −11.0000 −0.375097
\(861\) 4.00000 0.136320
\(862\) 32.0000 1.08992
\(863\) −46.0000 −1.56586 −0.782929 0.622111i \(-0.786275\pi\)
−0.782929 + 0.622111i \(0.786275\pi\)
\(864\) 5.00000 0.170103
\(865\) −6.00000 −0.204006
\(866\) −16.0000 −0.543702
\(867\) −47.0000 −1.59620
\(868\) 6.00000 0.203653
\(869\) −45.0000 −1.52652
\(870\) 1.00000 0.0339032
\(871\) 12.0000 0.406604
\(872\) 5.00000 0.169321
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 18.0000 0.608511
\(876\) −4.00000 −0.135147
\(877\) 13.0000 0.438979 0.219489 0.975615i \(-0.429561\pi\)
0.219489 + 0.975615i \(0.429561\pi\)
\(878\) 20.0000 0.674967
\(879\) −14.0000 −0.472208
\(880\) −3.00000 −0.101130
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 6.00000 0.202031
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 33.0000 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) −8.00000 −0.268462
\(889\) −16.0000 −0.536623
\(890\) −10.0000 −0.335201
\(891\) −3.00000 −0.100504
\(892\) −26.0000 −0.870544
\(893\) 0 0
\(894\) −15.0000 −0.501675
\(895\) −10.0000 −0.334263
\(896\) −2.00000 −0.0668153
\(897\) 4.00000 0.133556
\(898\) −10.0000 −0.333704
\(899\) 3.00000 0.100056
\(900\) 8.00000 0.266667
\(901\) −88.0000 −2.93171
\(902\) −6.00000 −0.199778
\(903\) −22.0000 −0.732114
\(904\) −6.00000 −0.199557
\(905\) 7.00000 0.232688
\(906\) −2.00000 −0.0664455
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 18.0000 0.597351
\(909\) 16.0000 0.530687
\(910\) 2.00000 0.0662994
\(911\) −13.0000 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −2.00000 −0.0661541
\(915\) 8.00000 0.264472
\(916\) 10.0000 0.330409
\(917\) −24.0000 −0.792550
\(918\) 40.0000 1.32020
\(919\) 30.0000 0.989609 0.494804 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) 4.00000 0.131876
\(921\) 7.00000 0.230658
\(922\) 2.00000 0.0658665
\(923\) −2.00000 −0.0658308
\(924\) −6.00000 −0.197386
\(925\) −32.0000 −1.05215
\(926\) 4.00000 0.131448
\(927\) −28.0000 −0.919641
\(928\) −1.00000 −0.0328266
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 3.00000 0.0983739
\(931\) 0 0
\(932\) −1.00000 −0.0327561
\(933\) 8.00000 0.261908
\(934\) −27.0000 −0.883467
\(935\) −24.0000 −0.784884
\(936\) 2.00000 0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 24.0000 0.783628
\(939\) −9.00000 −0.293704
\(940\) 13.0000 0.424013
\(941\) 37.0000 1.20617 0.603083 0.797679i \(-0.293939\pi\)
0.603083 + 0.797679i \(0.293939\pi\)
\(942\) −18.0000 −0.586472
\(943\) 8.00000 0.260516
\(944\) 0 0
\(945\) −10.0000 −0.325300
\(946\) 33.0000 1.07292
\(947\) 33.0000 1.07236 0.536178 0.844105i \(-0.319868\pi\)
0.536178 + 0.844105i \(0.319868\pi\)
\(948\) −15.0000 −0.487177
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) −16.0000 −0.518563
\(953\) −1.00000 −0.0323932 −0.0161966 0.999869i \(-0.505156\pi\)
−0.0161966 + 0.999869i \(0.505156\pi\)
\(954\) 22.0000 0.712276
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) −3.00000 −0.0969762
\(958\) −5.00000 −0.161543
\(959\) 24.0000 0.775000
\(960\) −1.00000 −0.0322749
\(961\) −22.0000 −0.709677
\(962\) −8.00000 −0.257930
\(963\) 4.00000 0.128898
\(964\) 17.0000 0.547533
\(965\) 14.0000 0.450676
\(966\) 8.00000 0.257396
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) −2.00000 −0.0642161
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −16.0000 −0.513200
\(973\) 40.0000 1.28234
\(974\) −22.0000 −0.704925
\(975\) −4.00000 −0.128103
\(976\) −8.00000 −0.256074
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) −9.00000 −0.287788
\(979\) 30.0000 0.958804
\(980\) −3.00000 −0.0958315
\(981\) −10.0000 −0.319275
\(982\) −33.0000 −1.05307
\(983\) 49.0000 1.56286 0.781429 0.623995i \(-0.214491\pi\)
0.781429 + 0.623995i \(0.214491\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 18.0000 0.573528
\(986\) −8.00000 −0.254772
\(987\) 26.0000 0.827589
\(988\) 0 0
\(989\) −44.0000 −1.39912
\(990\) 6.00000 0.190693
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 23.0000 0.729883
\(994\) −4.00000 −0.126872
\(995\) −10.0000 −0.317021
\(996\) −4.00000 −0.126745
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −20.0000 −0.633089
\(999\) 40.0000 1.26554
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 58.2.a.b.1.1 1
3.2 odd 2 522.2.a.b.1.1 1
4.3 odd 2 464.2.a.e.1.1 1
5.2 odd 4 1450.2.b.b.349.2 2
5.3 odd 4 1450.2.b.b.349.1 2
5.4 even 2 1450.2.a.c.1.1 1
7.6 odd 2 2842.2.a.e.1.1 1
8.3 odd 2 1856.2.a.f.1.1 1
8.5 even 2 1856.2.a.k.1.1 1
11.10 odd 2 7018.2.a.a.1.1 1
12.11 even 2 4176.2.a.n.1.1 1
13.12 even 2 9802.2.a.a.1.1 1
29.12 odd 4 1682.2.b.a.1681.2 2
29.17 odd 4 1682.2.b.a.1681.1 2
29.28 even 2 1682.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 1.1 even 1 trivial
464.2.a.e.1.1 1 4.3 odd 2
522.2.a.b.1.1 1 3.2 odd 2
1450.2.a.c.1.1 1 5.4 even 2
1450.2.b.b.349.1 2 5.3 odd 4
1450.2.b.b.349.2 2 5.2 odd 4
1682.2.a.d.1.1 1 29.28 even 2
1682.2.b.a.1681.1 2 29.17 odd 4
1682.2.b.a.1681.2 2 29.12 odd 4
1856.2.a.f.1.1 1 8.3 odd 2
1856.2.a.k.1.1 1 8.5 even 2
2842.2.a.e.1.1 1 7.6 odd 2
4176.2.a.n.1.1 1 12.11 even 2
7018.2.a.a.1.1 1 11.10 odd 2
9802.2.a.a.1.1 1 13.12 even 2