Properties

Label 138.2.a.a.1.1
Level 138
Weight 2
Character 138.1
Self dual yes
Analytic conductor 1.102
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.10193554789\)
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
Root \(0\)
Character \(\chi\) = 138.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +2.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -2.00000 q^{20} +2.00000 q^{21} +6.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} -2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +4.00000 q^{35} +1.00000 q^{36} +2.00000 q^{39} +2.00000 q^{40} +10.0000 q^{41} -2.00000 q^{42} -12.0000 q^{43} -6.00000 q^{44} -2.00000 q^{45} +1.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} +2.00000 q^{56} -6.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} +4.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} -6.00000 q^{66} -12.0000 q^{67} +1.00000 q^{69} -4.00000 q^{70} -1.00000 q^{72} -10.0000 q^{73} +1.00000 q^{75} +12.0000 q^{77} -2.00000 q^{78} -6.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +14.0000 q^{83} +2.00000 q^{84} +12.0000 q^{86} -6.00000 q^{87} +6.00000 q^{88} +2.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} +1.00000 q^{96} -6.00000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(4\) 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
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 2.00000 0.436436
\(22\) 6.00000 1.27920
\(23\) −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
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) −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
\(33\) 6.00000 1.04447
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 2.00000 0.316228
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −6.00000 −0.904534
\(45\) −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
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 1.00000 0.136083
\(55\) 12.0000 1.61808
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −6.00000 −0.738549
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) −4.00000 −0.478091
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −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
\(76\) 0 0
\(77\) 12.0000 1.36753
\(78\) −2.00000 −0.226455
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) −6.00000 −0.643268
\(88\) 6.00000 0.639602
\(89\) 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
\(93\) −8.00000 −0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.00000 0.303046
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 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
\(106\) −2.00000 −0.194257
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −12.0000 −1.14416
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) 6.00000 0.557086
\(117\) −2.00000 −0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 25.0000 2.27273
\(122\) −4.00000 −0.362143
\(123\) −10.0000 −0.901670
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) 2.00000 0.178174
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) −4.00000 −0.350823
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 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 \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 4.00000 0.338062
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) 10.0000 0.827606
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −12.0000 −0.966988
\(155\) −16.0000 −1.28515
\(156\) 2.00000 0.160128
\(157\) 16.0000 1.27694 0.638470 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(158\) 6.00000 0.477334
\(159\) −2.00000 −0.158610
\(160\) 2.00000 0.158114
\(161\) 2.00000 0.157622
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 10.0000 0.780869
\(165\) −12.0000 −0.934199
\(166\) −14.0000 −1.08661
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.00000 0.454859
\(175\) 2.00000 0.151186
\(176\) −6.00000 −0.452267
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −2.00000 −0.149071
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) −4.00000 −0.296500
\(183\) −4.00000 −0.295689
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) −8.00000 −0.583460
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 6.00000 0.430775
\(195\) −4.00000 −0.286446
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 6.00000 0.426401
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) 6.00000 0.422159
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) −20.0000 −1.39686
\(206\) −14.0000 −0.975426
\(207\) −1.00000 −0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 4.00000 0.276026
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) −14.0000 −0.957020
\(215\) 24.0000 1.63679
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 16.0000 1.08366
\(219\) 10.0000 0.675737
\(220\) 12.0000 0.809040
\(221\) 0 0
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 2.00000 0.133631
\(225\) −1.00000 −0.0666667
\(226\) 8.00000 0.532152
\(227\) −10.0000 −0.663723 −0.331862 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) −24.0000 −1.58596 −0.792982 0.609245i \(-0.791473\pi\)
−0.792982 + 0.609245i \(0.791473\pi\)
\(230\) −2.00000 −0.131876
\(231\) −12.0000 −0.789542
\(232\) −6.00000 −0.393919
\(233\) 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
\(236\) −12.0000 −0.781133
\(237\) 6.00000 0.389742
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 2.00000 0.129099
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 6.00000 0.383326
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −14.0000 −0.887214
\(250\) −12.0000 −0.758947
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) −2.00000 −0.125988
\(253\) 6.00000 0.377217
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) 8.00000 0.494242
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) −6.00000 −0.369274
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −2.00000 −0.121716
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 12.0000 0.724947
\(275\) 6.00000 0.361814
\(276\) 1.00000 0.0601929
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −12.0000 −0.719712
\(279\) 8.00000 0.478947
\(280\) −4.00000 −0.239046
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) −8.00000 −0.476393
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) −20.0000 −1.18056
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 12.0000 0.704664
\(291\) 6.00000 0.351726
\(292\) −10.0000 −0.585206
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −3.00000 −0.174964
\(295\) 24.0000 1.39733
\(296\) 0 0
\(297\) 6.00000 0.348155
\(298\) 18.0000 1.04271
\(299\) 2.00000 0.115663
\(300\) 1.00000 0.0577350
\(301\) 24.0000 1.38334
\(302\) 12.0000 0.690522
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 12.0000 0.683763
\(309\) −14.0000 −0.796432
\(310\) 16.0000 0.908739
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −2.00000 −0.113228
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) −16.0000 −0.902932
\(315\) 4.00000 0.225374
\(316\) −6.00000 −0.337526
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 2.00000 0.112154
\(319\) −36.0000 −2.01561
\(320\) −2.00000 −0.111803
\(321\) −14.0000 −0.781404
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 12.0000 0.664619
\(327\) 16.0000 0.884802
\(328\) −10.0000 −0.552158
\(329\) 16.0000 0.882109
\(330\) 12.0000 0.660578
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 14.0000 0.768350
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 24.0000 1.31126
\(336\) 2.00000 0.109109
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 9.00000 0.489535
\(339\) 8.00000 0.434500
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 12.0000 0.646997
\(345\) −2.00000 −0.107676
\(346\) 18.0000 0.967686
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) −6.00000 −0.321634
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) −2.00000 −0.106904
\(351\) 2.00000 0.106752
\(352\) 6.00000 0.319801
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 2.00000 0.105409
\(361\) −19.0000 −1.00000
\(362\) −8.00000 −0.420471
\(363\) −25.0000 −1.31216
\(364\) 4.00000 0.209657
\(365\) 20.0000 1.04685
\(366\) 4.00000 0.209083
\(367\) −14.0000 −0.730794 −0.365397 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 10.0000 0.520579
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) −8.00000 −0.414781
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 8.00000 0.412568
\(377\) −12.0000 −0.618031
\(378\) −2.00000 −0.102869
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 4.00000 0.204390 0.102195 0.994764i \(-0.467413\pi\)
0.102195 + 0.994764i \(0.467413\pi\)
\(384\) 1.00000 0.0510310
\(385\) −24.0000 −1.22315
\(386\) −14.0000 −0.712581
\(387\) −12.0000 −0.609994
\(388\) −6.00000 −0.304604
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 8.00000 0.403547
\(394\) 6.00000 0.302276
\(395\) 12.0000 0.603786
\(396\) −6.00000 −0.301511
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) −2.00000 −0.100251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −32.0000 −1.59800 −0.799002 0.601329i \(-0.794638\pi\)
−0.799002 + 0.601329i \(0.794638\pi\)
\(402\) −12.0000 −0.598506
\(403\) −16.0000 −0.797017
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) 12.0000 0.595550
\(407\) 0 0
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 20.0000 0.987730
\(411\) 12.0000 0.591916
\(412\) 14.0000 0.689730
\(413\) 24.0000 1.18096
\(414\) 1.00000 0.0491473
\(415\) −28.0000 −1.37447
\(416\) 2.00000 0.0980581
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) −4.00000 −0.195180
\(421\) 4.00000 0.194948 0.0974740 0.995238i \(-0.468924\pi\)
0.0974740 + 0.995238i \(0.468924\pi\)
\(422\) −20.0000 −0.973585
\(423\) −8.00000 −0.388973
\(424\) −2.00000 −0.0971286
\(425\) 0 0
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 14.0000 0.676716
\(429\) −12.0000 −0.579365
\(430\) −24.0000 −1.15738
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 16.0000 0.768025
\(435\) 12.0000 0.575356
\(436\) −16.0000 −0.766261
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) −12.0000 −0.572078
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −12.0000 −0.568216
\(447\) 18.0000 0.851371
\(448\) −2.00000 −0.0944911
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 1.00000 0.0471405
\(451\) −60.0000 −2.82529
\(452\) −8.00000 −0.376288
\(453\) 12.0000 0.563809
\(454\) 10.0000 0.469323
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) 24.0000 1.12145
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 12.0000 0.558291
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 6.00000 0.278543
\(465\) 16.0000 0.741982
\(466\) −10.0000 −0.463241
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 24.0000 1.10822
\(470\) −16.0000 −0.738025
\(471\) −16.0000 −0.737241
\(472\) 12.0000 0.552345
\(473\) 72.0000 3.31056
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 0.0915737
\(478\) 16.0000 0.731823
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 0 0
\(482\) 6.00000 0.273293
\(483\) −2.00000 −0.0910032
\(484\) 25.0000 1.13636
\(485\) 12.0000 0.544892
\(486\) 1.00000 0.0453609
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −4.00000 −0.181071
\(489\) 12.0000 0.542659
\(490\) −6.00000 −0.271052
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 14.0000 0.627355
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) −8.00000 −0.357414
\(502\) 6.00000 0.267793
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 2.00000 0.0890871
\(505\) 12.0000 0.533993
\(506\) −6.00000 −0.266733
\(507\) 9.00000 0.399704
\(508\) 12.0000 0.532414
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) −28.0000 −1.23383
\(516\) 12.0000 0.528271
\(517\) 48.0000 2.11104
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) −4.00000 −0.175412
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) −6.00000 −0.262613
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −8.00000 −0.349482
\(525\) −2.00000 −0.0872872
\(526\) −4.00000 −0.174408
\(527\) 0 0
\(528\) 6.00000 0.261116
\(529\) 1.00000 0.0434783
\(530\) 4.00000 0.173749
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −20.0000 −0.866296
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 12.0000 0.518321
\(537\) 16.0000 0.690451
\(538\) −18.0000 −0.776035
\(539\) 18.0000 0.775315
\(540\) 2.00000 0.0860663
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −28.0000 −1.20270
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) 32.0000 1.37073
\(546\) 4.00000 0.171184
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −12.0000 −0.512615
\(549\) 4.00000 0.170716
\(550\) −6.00000 −0.255841
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 12.0000 0.510292
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) 6.00000 0.254228 0.127114 0.991888i \(-0.459429\pi\)
0.127114 + 0.991888i \(0.459429\pi\)
\(558\) −8.00000 −0.338667
\(559\) 24.0000 1.01509
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −16.0000 −0.674919
\(563\) 14.0000 0.590030 0.295015 0.955493i \(-0.404675\pi\)
0.295015 + 0.955493i \(0.404675\pi\)
\(564\) 8.00000 0.336861
\(565\) 16.0000 0.673125
\(566\) −24.0000 −1.00880
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) 16.0000 0.670755 0.335377 0.942084i \(-0.391136\pi\)
0.335377 + 0.942084i \(0.391136\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 12.0000 0.501745
\(573\) 0 0
\(574\) 20.0000 0.834784
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 17.0000 0.707107
\(579\) −14.0000 −0.581820
\(580\) −12.0000 −0.498273
\(581\) −28.0000 −1.16164
\(582\) −6.00000 −0.248708
\(583\) −12.0000 −0.496989
\(584\) 10.0000 0.413803
\(585\) 4.00000 0.165380
\(586\) 6.00000 0.247858
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) 6.00000 0.246807
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −2.00000 −0.0818546
\(598\) −2.00000 −0.0817861
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) −24.0000 −0.978167
\(603\) −12.0000 −0.488678
\(604\) −12.0000 −0.488273
\(605\) −50.0000 −2.03279
\(606\) −6.00000 −0.243733
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 8.00000 0.323911
\(611\) 16.0000 0.647291
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) 20.0000 0.807134
\(615\) 20.0000 0.806478
\(616\) −12.0000 −0.483494
\(617\) −32.0000 −1.28827 −0.644136 0.764911i \(-0.722783\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 14.0000 0.563163
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) −16.0000 −0.642575
\(621\) 1.00000 0.0401286
\(622\) −16.0000 −0.641542
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) −19.0000 −0.760000
\(626\) 26.0000 1.03917
\(627\) 0 0
\(628\) 16.0000 0.638470
\(629\) 0 0
\(630\) −4.00000 −0.159364
\(631\) −2.00000 −0.0796187 −0.0398094 0.999207i \(-0.512675\pi\)
−0.0398094 + 0.999207i \(0.512675\pi\)
\(632\) 6.00000 0.238667
\(633\) −20.0000 −0.794929
\(634\) −22.0000 −0.873732
\(635\) −24.0000 −0.952411
\(636\) −2.00000 −0.0793052
\(637\) 6.00000 0.237729
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) 2.00000 0.0790569
\(641\) 16.0000 0.631962 0.315981 0.948766i \(-0.397666\pi\)
0.315981 + 0.948766i \(0.397666\pi\)
\(642\) 14.0000 0.552536
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 2.00000 0.0788110
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 72.0000 2.82625
\(650\) −2.00000 −0.0784465
\(651\) 16.0000 0.627089
\(652\) −12.0000 −0.469956
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) −16.0000 −0.625650
\(655\) 16.0000 0.625172
\(656\) 10.0000 0.390434
\(657\) −10.0000 −0.390137
\(658\) −16.0000 −0.623745
\(659\) 42.0000 1.63609 0.818044 0.575156i \(-0.195059\pi\)
0.818044 + 0.575156i \(0.195059\pi\)
\(660\) −12.0000 −0.467099
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −14.0000 −0.543305
\(665\) 0 0
\(666\) 0 0
\(667\) −6.00000 −0.232321
\(668\) 8.00000 0.309529
\(669\) −12.0000 −0.463947
\(670\) −24.0000 −0.927201
\(671\) −24.0000 −0.926510
\(672\) −2.00000 −0.0771517
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 10.0000 0.385186
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −8.00000 −0.307238
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 48.0000 1.83801
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) −20.0000 −0.763604
\(687\) 24.0000 0.915657
\(688\) −12.0000 −0.457496
\(689\) −4.00000 −0.152388
\(690\) 2.00000 0.0761387
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −18.0000 −0.684257
\(693\) 12.0000 0.455842
\(694\) 8.00000 0.303676
\(695\) −24.0000 −0.910372
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −26.0000 −0.984115
\(699\) −10.0000 −0.378235
\(700\) 2.00000 0.0755929
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) −6.00000 −0.226134
\(705\) −16.0000 −0.602595
\(706\) 2.00000 0.0752710
\(707\) 12.0000 0.451306
\(708\) 12.0000 0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −24.0000 −0.897549
\(716\) −16.0000 −0.597948
\(717\) 16.0000 0.597531
\(718\) −24.0000 −0.895672
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −28.0000 −1.04277
\(722\) 19.0000 0.707107
\(723\) 6.00000 0.223142
\(724\) 8.00000 0.297318
\(725\) −6.00000 −0.222834
\(726\) 25.0000 0.927837
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) −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 \(0.268498\pi\)
\(734\) 14.0000 0.516749
\(735\) −6.00000 −0.221313
\(736\) 1.00000 0.0368605
\(737\) 72.0000 2.65215
\(738\) −10.0000 −0.368105
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.00000 0.146845
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 8.00000 0.293294
\(745\) 36.0000 1.31894
\(746\) 32.0000 1.17160
\(747\) 14.0000 0.512233
\(748\) 0 0
\(749\) −28.0000 −1.02310
\(750\) 12.0000 0.438178
\(751\) −38.0000 −1.38664 −0.693320 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) −8.00000 −0.291730
\(753\) 6.00000 0.218652
\(754\) 12.0000 0.437014
\(755\) 24.0000 0.873449
\(756\) 2.00000 0.0727393
\(757\) −32.0000 −1.16306 −0.581530 0.813525i \(-0.697546\pi\)
−0.581530 + 0.813525i \(0.697546\pi\)
\(758\) 0 0
\(759\) −6.00000 −0.217786
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 12.0000 0.434714
\(763\) 32.0000 1.15848
\(764\) 0 0
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 24.0000 0.866590
\(768\) −1.00000 −0.0360844
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 24.0000 0.864900
\(771\) 6.00000 0.216085
\(772\) 14.0000 0.503871
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 12.0000 0.431331
\(775\) −8.00000 −0.287368
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) −6.00000 −0.215110
\(779\) 0 0
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −3.00000 −0.107143
\(785\) −32.0000 −1.14213
\(786\) −8.00000 −0.285351
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −6.00000 −0.213741
\(789\) −4.00000 −0.142404
\(790\) −12.0000 −0.426941
\(791\) 16.0000 0.568895
\(792\) 6.00000 0.213201
\(793\) −8.00000 −0.284088
\(794\) −6.00000 −0.212932
\(795\) 4.00000 0.141865
\(796\) 2.00000 0.0708881
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 32.0000 1.12996
\(803\) 60.0000 2.11735
\(804\) 12.0000 0.423207
\(805\) −4.00000 −0.140981
\(806\) 16.0000 0.563576
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 2.00000 0.0702728
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) −12.0000 −0.421117
\(813\) −28.0000 −0.982003
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) 4.00000 0.139771
\(820\) −20.0000 −0.698430
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) −12.0000 −0.418548
\(823\) −24.0000 −0.836587 −0.418294 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) −14.0000 −0.487713
\(825\) −6.00000 −0.208893
\(826\) −24.0000 −0.835067
\(827\) 18.0000 0.625921 0.312961 0.949766i \(-0.398679\pi\)
0.312961 + 0.949766i \(0.398679\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 42.0000 1.45872 0.729360 0.684130i \(-0.239818\pi\)
0.729360 + 0.684130i \(0.239818\pi\)
\(830\) 28.0000 0.971894
\(831\) −26.0000 −0.901930
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) 12.0000 0.415526
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −8.00000 −0.276520
\(838\) 26.0000 0.898155
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 4.00000 0.138013
\(841\) 7.00000 0.241379
\(842\) −4.00000 −0.137849
\(843\) −16.0000 −0.551069
\(844\) 20.0000 0.688428
\(845\) 18.0000 0.619219
\(846\) 8.00000 0.275046
\(847\) −50.0000 −1.71802
\(848\) 2.00000 0.0686803
\(849\) −24.0000 −0.823678
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) 2.00000 0.0683187 0.0341593 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) 12.0000 0.409673
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 24.0000 0.818393
\(861\) 20.0000 0.681598
\(862\) 12.0000 0.408722
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 1.00000 0.0340207
\(865\) 36.0000 1.22404
\(866\) 2.00000 0.0679628
\(867\) 17.0000 0.577350
\(868\) −16.0000 −0.543075
\(869\) 36.0000 1.22122
\(870\) −12.0000 −0.406838
\(871\) 24.0000 0.813209
\(872\) 16.0000 0.541828
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) −24.0000 −0.811348
\(876\) 10.0000 0.337869
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) 16.0000 0.539974
\(879\) 6.00000 0.202375
\(880\) 12.0000 0.404520
\(881\) −32.0000 −1.07811 −0.539054 0.842271i \(-0.681218\pi\)
−0.539054 + 0.842271i \(0.681218\pi\)
\(882\) 3.00000 0.101015
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) −36.0000 −1.20944
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 12.0000 0.401790
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 32.0000 1.06964
\(896\) 2.00000 0.0668153
\(897\) −2.00000 −0.0667781
\(898\) −2.00000 −0.0667409
\(899\) 48.0000 1.60089
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) 60.0000 1.99778
\(903\) −24.0000 −0.798670
\(904\) 8.00000 0.266076
\(905\) −16.0000 −0.531858
\(906\) −12.0000 −0.398673
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −10.0000 −0.331862
\(909\) −6.00000 −0.199007
\(910\) 8.00000 0.265197
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) −84.0000 −2.77999
\(914\) −26.0000 −0.860004
\(915\) 8.00000 0.264472
\(916\) −24.0000 −0.792982
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 20.0000 0.659022
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) −12.0000 −0.394771
\(925\) 0 0
\(926\) 40.0000 1.31448
\(927\) 14.0000 0.459820
\(928\) −6.00000 −0.196960
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) −16.0000 −0.524661
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) −16.0000 −0.523816
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −24.0000 −0.783628
\(939\) 26.0000 0.848478
\(940\) 16.0000 0.521862
\(941\) 54.0000 1.76035 0.880175 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) 16.0000 0.521308
\(943\) −10.0000 −0.325645
\(944\) −12.0000 −0.390567
\(945\) −4.00000 −0.130120
\(946\) −72.0000 −2.34092
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 6.00000 0.194871
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 36.0000 1.16371
\(958\) −24.0000 −0.775405
\(959\) 24.0000 0.775000
\(960\) 2.00000 0.0645497
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 14.0000 0.451144
\(964\) −6.00000 −0.193247
\(965\) −28.0000 −0.901352
\(966\) 2.00000 0.0643489
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −25.0000 −0.803530
\(969\) 0 0
\(970\) −12.0000 −0.385297
\(971\) −10.0000 −0.320915 −0.160458 0.987043i \(-0.551297\pi\)
−0.160458 + 0.987043i \(0.551297\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −24.0000 −0.769405
\(974\) −8.00000 −0.256337
\(975\) −2.00000 −0.0640513
\(976\) 4.00000 0.128037
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) −12.0000 −0.383718
\(979\) 0 0
\(980\) 6.00000 0.191663
\(981\) −16.0000 −0.510841
\(982\) 32.0000 1.02116
\(983\) 52.0000 1.65854 0.829271 0.558846i \(-0.188756\pi\)
0.829271 + 0.558846i \(0.188756\pi\)
\(984\) 10.0000 0.318788
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) −12.0000 −0.381385
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −8.00000 −0.254000
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) −14.0000 −0.443607
\(997\) 10.0000 0.316703 0.158352 0.987383i \(-0.449382\pi\)
0.158352 + 0.987383i \(0.449382\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 138.2.a.a.1.1 1
3.2 odd 2 414.2.a.d.1.1 1
4.3 odd 2 1104.2.a.e.1.1 1
5.2 odd 4 3450.2.d.j.2899.1 2
5.3 odd 4 3450.2.d.j.2899.2 2
5.4 even 2 3450.2.a.y.1.1 1
7.6 odd 2 6762.2.a.q.1.1 1
8.3 odd 2 4416.2.a.m.1.1 1
8.5 even 2 4416.2.a.z.1.1 1
12.11 even 2 3312.2.a.n.1.1 1
23.22 odd 2 3174.2.a.b.1.1 1
69.68 even 2 9522.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.a.1.1 1 1.1 even 1 trivial
414.2.a.d.1.1 1 3.2 odd 2
1104.2.a.e.1.1 1 4.3 odd 2
3174.2.a.b.1.1 1 23.22 odd 2
3312.2.a.n.1.1 1 12.11 even 2
3450.2.a.y.1.1 1 5.4 even 2
3450.2.d.j.2899.1 2 5.2 odd 4
3450.2.d.j.2899.2 2 5.3 odd 4
4416.2.a.m.1.1 1 8.3 odd 2
4416.2.a.z.1.1 1 8.5 even 2
6762.2.a.q.1.1 1 7.6 odd 2
9522.2.a.i.1.1 1 69.68 even 2