Properties

Label 4026.2.a.f.1.1
Level 4026
Weight 2
Character 4026.1
Self dual yes
Analytic conductor 32.148
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4026.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1477718538\)
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
Root \(0\)
Character \(\chi\) = 4026.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -3.00000 q^{13} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -1.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} +2.00000 q^{28} -9.00000 q^{29} +1.00000 q^{30} -3.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +8.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +6.00000 q^{37} +4.00000 q^{38} +3.00000 q^{39} -1.00000 q^{40} +3.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} -8.00000 q^{51} -3.00000 q^{52} +8.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +2.00000 q^{56} -4.00000 q^{57} -9.00000 q^{58} -3.00000 q^{59} +1.00000 q^{60} -1.00000 q^{61} -3.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{65} +1.00000 q^{66} +4.00000 q^{67} +8.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} +1.00000 q^{72} +12.0000 q^{73} +6.00000 q^{74} +4.00000 q^{75} +4.00000 q^{76} -2.00000 q^{77} +3.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +3.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{85} +4.00000 q^{86} +9.00000 q^{87} -1.00000 q^{88} -17.0000 q^{89} -1.00000 q^{90} -6.00000 q^{91} +4.00000 q^{92} +3.00000 q^{93} -4.00000 q^{95} -1.00000 q^{96} +13.0000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 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 \(0.0779791\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) −1.00000 −0.213201
\(23\) 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
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 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
\(33\) 1.00000 0.174078
\(34\) 8.00000 1.37199
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 4.00000 0.648886
\(39\) 3.00000 0.480384
\(40\) −1.00000 −0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) −4.00000 −0.565685
\(51\) −8.00000 −1.12022
\(52\) −3.00000 −0.416025
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 2.00000 0.267261
\(57\) −4.00000 −0.529813
\(58\) −9.00000 −1.18176
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 1.00000 0.129099
\(61\) −1.00000 −0.128037
\(62\) −3.00000 −0.381000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 1.00000 0.123091
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 8.00000 0.970143
\(69\) −4.00000 −0.481543
\(70\) −2.00000 −0.239046
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 1.00000 0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 6.00000 0.697486
\(75\) 4.00000 0.461880
\(76\) 4.00000 0.458831
\(77\) −2.00000 −0.227921
\(78\) 3.00000 0.339683
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 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
\(86\) 4.00000 0.431331
\(87\) 9.00000 0.964901
\(88\) −1.00000 −0.106600
\(89\) −17.0000 −1.80200 −0.900998 0.433823i \(-0.857164\pi\)
−0.900998 + 0.433823i \(0.857164\pi\)
\(90\) −1.00000 −0.105409
\(91\) −6.00000 −0.628971
\(92\) 4.00000 0.417029
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) −1.00000 −0.102062
\(97\) 13.0000 1.31995 0.659975 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) −3.00000 −0.303046
\(99\) −1.00000 −0.100504
\(100\) −4.00000 −0.400000
\(101\) −1.00000 −0.0995037 −0.0497519 0.998762i \(-0.515843\pi\)
−0.0497519 + 0.998762i \(0.515843\pi\)
\(102\) −8.00000 −0.792118
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −3.00000 −0.294174
\(105\) 2.00000 0.195180
\(106\) 8.00000 0.777029
\(107\) 13.0000 1.25676 0.628379 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 1.00000 0.0953463
\(111\) −6.00000 −0.569495
\(112\) 2.00000 0.188982
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) −4.00000 −0.373002
\(116\) −9.00000 −0.835629
\(117\) −3.00000 −0.277350
\(118\) −3.00000 −0.276172
\(119\) 16.0000 1.46672
\(120\) 1.00000 0.0912871
\(121\) 1.00000 0.0909091
\(122\) −1.00000 −0.0905357
\(123\) −3.00000 −0.270501
\(124\) −3.00000 −0.269408
\(125\) 9.00000 0.804984
\(126\) 2.00000 0.178174
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 3.00000 0.263117
\(131\) 1.00000 0.0873704 0.0436852 0.999045i \(-0.486090\pi\)
0.0436852 + 0.999045i \(0.486090\pi\)
\(132\) 1.00000 0.0870388
\(133\) 8.00000 0.693688
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 8.00000 0.685994
\(137\) 14.0000 1.19610 0.598050 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) −4.00000 −0.340503
\(139\) −19.0000 −1.61156 −0.805779 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 3.00000 0.250873
\(144\) 1.00000 0.0833333
\(145\) 9.00000 0.747409
\(146\) 12.0000 0.993127
\(147\) 3.00000 0.247436
\(148\) 6.00000 0.493197
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 4.00000 0.326599
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 4.00000 0.324443
\(153\) 8.00000 0.646762
\(154\) −2.00000 −0.161165
\(155\) 3.00000 0.240966
\(156\) 3.00000 0.240192
\(157\) −7.00000 −0.558661 −0.279330 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −4.00000 −0.318223
\(159\) −8.00000 −0.634441
\(160\) −1.00000 −0.0790569
\(161\) 8.00000 0.630488
\(162\) 1.00000 0.0785674
\(163\) −5.00000 −0.391630 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) 3.00000 0.234261
\(165\) −1.00000 −0.0778499
\(166\) 4.00000 0.310460
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −2.00000 −0.154303
\(169\) −4.00000 −0.307692
\(170\) −8.00000 −0.613572
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 11.0000 0.836315 0.418157 0.908375i \(-0.362676\pi\)
0.418157 + 0.908375i \(0.362676\pi\)
\(174\) 9.00000 0.682288
\(175\) −8.00000 −0.604743
\(176\) −1.00000 −0.0753778
\(177\) 3.00000 0.225494
\(178\) −17.0000 −1.27420
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 1.00000 0.0743294 0.0371647 0.999309i \(-0.488167\pi\)
0.0371647 + 0.999309i \(0.488167\pi\)
\(182\) −6.00000 −0.444750
\(183\) 1.00000 0.0739221
\(184\) 4.00000 0.294884
\(185\) −6.00000 −0.441129
\(186\) 3.00000 0.219971
\(187\) −8.00000 −0.585018
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) −4.00000 −0.290191
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −17.0000 −1.22369 −0.611843 0.790979i \(-0.709572\pi\)
−0.611843 + 0.790979i \(0.709572\pi\)
\(194\) 13.0000 0.933346
\(195\) −3.00000 −0.214834
\(196\) −3.00000 −0.214286
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) −4.00000 −0.282843
\(201\) −4.00000 −0.282138
\(202\) −1.00000 −0.0703598
\(203\) −18.0000 −1.26335
\(204\) −8.00000 −0.560112
\(205\) −3.00000 −0.209529
\(206\) 12.0000 0.836080
\(207\) 4.00000 0.278019
\(208\) −3.00000 −0.208013
\(209\) −4.00000 −0.276686
\(210\) 2.00000 0.138013
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 8.00000 0.549442
\(213\) −6.00000 −0.411113
\(214\) 13.0000 0.888662
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) −6.00000 −0.407307
\(218\) 3.00000 0.203186
\(219\) −12.0000 −0.810885
\(220\) 1.00000 0.0674200
\(221\) −24.0000 −1.61441
\(222\) −6.00000 −0.402694
\(223\) −12.0000 −0.803579 −0.401790 0.915732i \(-0.631612\pi\)
−0.401790 + 0.915732i \(0.631612\pi\)
\(224\) 2.00000 0.133631
\(225\) −4.00000 −0.266667
\(226\) 2.00000 0.133038
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −4.00000 −0.264906
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) −4.00000 −0.263752
\(231\) 2.00000 0.131590
\(232\) −9.00000 −0.590879
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) −3.00000 −0.196116
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 4.00000 0.259828
\(238\) 16.0000 1.03713
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 1.00000 0.0645497
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −1.00000 −0.0640184
\(245\) 3.00000 0.191663
\(246\) −3.00000 −0.191273
\(247\) −12.0000 −0.763542
\(248\) −3.00000 −0.190500
\(249\) −4.00000 −0.253490
\(250\) 9.00000 0.569210
\(251\) −19.0000 −1.19927 −0.599635 0.800274i \(-0.704687\pi\)
−0.599635 + 0.800274i \(0.704687\pi\)
\(252\) 2.00000 0.125988
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 1.00000 0.0625000
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) −4.00000 −0.249029
\(259\) 12.0000 0.745644
\(260\) 3.00000 0.186052
\(261\) −9.00000 −0.557086
\(262\) 1.00000 0.0617802
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 1.00000 0.0615457
\(265\) −8.00000 −0.491436
\(266\) 8.00000 0.490511
\(267\) 17.0000 1.04038
\(268\) 4.00000 0.244339
\(269\) 23.0000 1.40233 0.701167 0.712997i \(-0.252663\pi\)
0.701167 + 0.712997i \(0.252663\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 8.00000 0.485071
\(273\) 6.00000 0.363137
\(274\) 14.0000 0.845771
\(275\) 4.00000 0.241209
\(276\) −4.00000 −0.240772
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −19.0000 −1.13954
\(279\) −3.00000 −0.179605
\(280\) −2.00000 −0.119523
\(281\) 20.0000 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(282\) 0 0
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 6.00000 0.356034
\(285\) 4.00000 0.236940
\(286\) 3.00000 0.177394
\(287\) 6.00000 0.354169
\(288\) 1.00000 0.0589256
\(289\) 47.0000 2.76471
\(290\) 9.00000 0.528498
\(291\) −13.0000 −0.762073
\(292\) 12.0000 0.702247
\(293\) −8.00000 −0.467365 −0.233682 0.972313i \(-0.575078\pi\)
−0.233682 + 0.972313i \(0.575078\pi\)
\(294\) 3.00000 0.174964
\(295\) 3.00000 0.174667
\(296\) 6.00000 0.348743
\(297\) 1.00000 0.0580259
\(298\) 20.0000 1.15857
\(299\) −12.0000 −0.693978
\(300\) 4.00000 0.230940
\(301\) 8.00000 0.461112
\(302\) 4.00000 0.230174
\(303\) 1.00000 0.0574485
\(304\) 4.00000 0.229416
\(305\) 1.00000 0.0572598
\(306\) 8.00000 0.457330
\(307\) −11.0000 −0.627803 −0.313902 0.949456i \(-0.601636\pi\)
−0.313902 + 0.949456i \(0.601636\pi\)
\(308\) −2.00000 −0.113961
\(309\) −12.0000 −0.682656
\(310\) 3.00000 0.170389
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 3.00000 0.169842
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −7.00000 −0.395033
\(315\) −2.00000 −0.112687
\(316\) −4.00000 −0.225018
\(317\) 3.00000 0.168497 0.0842484 0.996445i \(-0.473151\pi\)
0.0842484 + 0.996445i \(0.473151\pi\)
\(318\) −8.00000 −0.448618
\(319\) 9.00000 0.503903
\(320\) −1.00000 −0.0559017
\(321\) −13.0000 −0.725589
\(322\) 8.00000 0.445823
\(323\) 32.0000 1.78053
\(324\) 1.00000 0.0555556
\(325\) 12.0000 0.665640
\(326\) −5.00000 −0.276924
\(327\) −3.00000 −0.165900
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) 6.00000 0.328798
\(334\) 24.0000 1.31322
\(335\) −4.00000 −0.218543
\(336\) −2.00000 −0.109109
\(337\) 15.0000 0.817102 0.408551 0.912735i \(-0.366034\pi\)
0.408551 + 0.912735i \(0.366034\pi\)
\(338\) −4.00000 −0.217571
\(339\) −2.00000 −0.108625
\(340\) −8.00000 −0.433861
\(341\) 3.00000 0.162459
\(342\) 4.00000 0.216295
\(343\) −20.0000 −1.07990
\(344\) 4.00000 0.215666
\(345\) 4.00000 0.215353
\(346\) 11.0000 0.591364
\(347\) −25.0000 −1.34207 −0.671035 0.741426i \(-0.734150\pi\)
−0.671035 + 0.741426i \(0.734150\pi\)
\(348\) 9.00000 0.482451
\(349\) 36.0000 1.92704 0.963518 0.267644i \(-0.0862451\pi\)
0.963518 + 0.267644i \(0.0862451\pi\)
\(350\) −8.00000 −0.427618
\(351\) 3.00000 0.160128
\(352\) −1.00000 −0.0533002
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 3.00000 0.159448
\(355\) −6.00000 −0.318447
\(356\) −17.0000 −0.900998
\(357\) −16.0000 −0.846810
\(358\) 14.0000 0.739923
\(359\) −33.0000 −1.74167 −0.870837 0.491572i \(-0.836422\pi\)
−0.870837 + 0.491572i \(0.836422\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −3.00000 −0.157895
\(362\) 1.00000 0.0525588
\(363\) −1.00000 −0.0524864
\(364\) −6.00000 −0.314485
\(365\) −12.0000 −0.628109
\(366\) 1.00000 0.0522708
\(367\) −22.0000 −1.14839 −0.574195 0.818718i \(-0.694685\pi\)
−0.574195 + 0.818718i \(0.694685\pi\)
\(368\) 4.00000 0.208514
\(369\) 3.00000 0.156174
\(370\) −6.00000 −0.311925
\(371\) 16.0000 0.830679
\(372\) 3.00000 0.155543
\(373\) −20.0000 −1.03556 −0.517780 0.855514i \(-0.673242\pi\)
−0.517780 + 0.855514i \(0.673242\pi\)
\(374\) −8.00000 −0.413670
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 27.0000 1.39057
\(378\) −2.00000 −0.102869
\(379\) −13.0000 −0.667765 −0.333883 0.942615i \(-0.608359\pi\)
−0.333883 + 0.942615i \(0.608359\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 18.0000 0.920960
\(383\) 14.0000 0.715367 0.357683 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.00000 0.101929
\(386\) −17.0000 −0.865277
\(387\) 4.00000 0.203331
\(388\) 13.0000 0.659975
\(389\) 36.0000 1.82527 0.912636 0.408773i \(-0.134043\pi\)
0.912636 + 0.408773i \(0.134043\pi\)
\(390\) −3.00000 −0.151911
\(391\) 32.0000 1.61831
\(392\) −3.00000 −0.151523
\(393\) −1.00000 −0.0504433
\(394\) −14.0000 −0.705310
\(395\) 4.00000 0.201262
\(396\) −1.00000 −0.0502519
\(397\) −29.0000 −1.45547 −0.727734 0.685859i \(-0.759427\pi\)
−0.727734 + 0.685859i \(0.759427\pi\)
\(398\) 14.0000 0.701757
\(399\) −8.00000 −0.400501
\(400\) −4.00000 −0.200000
\(401\) 25.0000 1.24844 0.624220 0.781248i \(-0.285417\pi\)
0.624220 + 0.781248i \(0.285417\pi\)
\(402\) −4.00000 −0.199502
\(403\) 9.00000 0.448322
\(404\) −1.00000 −0.0497519
\(405\) −1.00000 −0.0496904
\(406\) −18.0000 −0.893325
\(407\) −6.00000 −0.297409
\(408\) −8.00000 −0.396059
\(409\) 31.0000 1.53285 0.766426 0.642333i \(-0.222033\pi\)
0.766426 + 0.642333i \(0.222033\pi\)
\(410\) −3.00000 −0.148159
\(411\) −14.0000 −0.690569
\(412\) 12.0000 0.591198
\(413\) −6.00000 −0.295241
\(414\) 4.00000 0.196589
\(415\) −4.00000 −0.196352
\(416\) −3.00000 −0.147087
\(417\) 19.0000 0.930434
\(418\) −4.00000 −0.195646
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 2.00000 0.0975900
\(421\) −31.0000 −1.51085 −0.755424 0.655237i \(-0.772569\pi\)
−0.755424 + 0.655237i \(0.772569\pi\)
\(422\) −25.0000 −1.21698
\(423\) 0 0
\(424\) 8.00000 0.388514
\(425\) −32.0000 −1.55223
\(426\) −6.00000 −0.290701
\(427\) −2.00000 −0.0967868
\(428\) 13.0000 0.628379
\(429\) −3.00000 −0.144841
\(430\) −4.00000 −0.192897
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) −6.00000 −0.288009
\(435\) −9.00000 −0.431517
\(436\) 3.00000 0.143674
\(437\) 16.0000 0.765384
\(438\) −12.0000 −0.573382
\(439\) 13.0000 0.620456 0.310228 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 1.00000 0.0476731
\(441\) −3.00000 −0.142857
\(442\) −24.0000 −1.14156
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) −6.00000 −0.284747
\(445\) 17.0000 0.805877
\(446\) −12.0000 −0.568216
\(447\) −20.0000 −0.945968
\(448\) 2.00000 0.0944911
\(449\) −40.0000 −1.88772 −0.943858 0.330350i \(-0.892833\pi\)
−0.943858 + 0.330350i \(0.892833\pi\)
\(450\) −4.00000 −0.188562
\(451\) −3.00000 −0.141264
\(452\) 2.00000 0.0940721
\(453\) −4.00000 −0.187936
\(454\) 24.0000 1.12638
\(455\) 6.00000 0.281284
\(456\) −4.00000 −0.187317
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −4.00000 −0.186908
\(459\) −8.00000 −0.373408
\(460\) −4.00000 −0.186501
\(461\) −22.0000 −1.02464 −0.512321 0.858794i \(-0.671214\pi\)
−0.512321 + 0.858794i \(0.671214\pi\)
\(462\) 2.00000 0.0930484
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −9.00000 −0.417815
\(465\) −3.00000 −0.139122
\(466\) −20.0000 −0.926482
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) −3.00000 −0.138675
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) −3.00000 −0.138086
\(473\) −4.00000 −0.183920
\(474\) 4.00000 0.183726
\(475\) −16.0000 −0.734130
\(476\) 16.0000 0.733359
\(477\) 8.00000 0.366295
\(478\) −16.0000 −0.731823
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 0.0456435
\(481\) −18.0000 −0.820729
\(482\) −8.00000 −0.364390
\(483\) −8.00000 −0.364013
\(484\) 1.00000 0.0454545
\(485\) −13.0000 −0.590300
\(486\) −1.00000 −0.0453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −1.00000 −0.0452679
\(489\) 5.00000 0.226108
\(490\) 3.00000 0.135526
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) −3.00000 −0.135250
\(493\) −72.0000 −3.24272
\(494\) −12.0000 −0.539906
\(495\) 1.00000 0.0449467
\(496\) −3.00000 −0.134704
\(497\) 12.0000 0.538274
\(498\) −4.00000 −0.179244
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 9.00000 0.402492
\(501\) −24.0000 −1.07224
\(502\) −19.0000 −0.848012
\(503\) 38.0000 1.69434 0.847168 0.531325i \(-0.178306\pi\)
0.847168 + 0.531325i \(0.178306\pi\)
\(504\) 2.00000 0.0890871
\(505\) 1.00000 0.0444994
\(506\) −4.00000 −0.177822
\(507\) 4.00000 0.177646
\(508\) 0 0
\(509\) −4.00000 −0.177297 −0.0886484 0.996063i \(-0.528255\pi\)
−0.0886484 + 0.996063i \(0.528255\pi\)
\(510\) 8.00000 0.354246
\(511\) 24.0000 1.06170
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) 26.0000 1.14681
\(515\) −12.0000 −0.528783
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 12.0000 0.527250
\(519\) −11.0000 −0.482846
\(520\) 3.00000 0.131559
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −9.00000 −0.393919
\(523\) −13.0000 −0.568450 −0.284225 0.958758i \(-0.591736\pi\)
−0.284225 + 0.958758i \(0.591736\pi\)
\(524\) 1.00000 0.0436852
\(525\) 8.00000 0.349149
\(526\) −6.00000 −0.261612
\(527\) −24.0000 −1.04546
\(528\) 1.00000 0.0435194
\(529\) −7.00000 −0.304348
\(530\) −8.00000 −0.347498
\(531\) −3.00000 −0.130189
\(532\) 8.00000 0.346844
\(533\) −9.00000 −0.389833
\(534\) 17.0000 0.735662
\(535\) −13.0000 −0.562039
\(536\) 4.00000 0.172774
\(537\) −14.0000 −0.604145
\(538\) 23.0000 0.991600
\(539\) 3.00000 0.129219
\(540\) 1.00000 0.0430331
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 20.0000 0.859074
\(543\) −1.00000 −0.0429141
\(544\) 8.00000 0.342997
\(545\) −3.00000 −0.128506
\(546\) 6.00000 0.256776
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 14.0000 0.598050
\(549\) −1.00000 −0.0426790
\(550\) 4.00000 0.170561
\(551\) −36.0000 −1.53365
\(552\) −4.00000 −0.170251
\(553\) −8.00000 −0.340195
\(554\) 26.0000 1.10463
\(555\) 6.00000 0.254686
\(556\) −19.0000 −0.805779
\(557\) 27.0000 1.14403 0.572013 0.820244i \(-0.306163\pi\)
0.572013 + 0.820244i \(0.306163\pi\)
\(558\) −3.00000 −0.127000
\(559\) −12.0000 −0.507546
\(560\) −2.00000 −0.0845154
\(561\) 8.00000 0.337760
\(562\) 20.0000 0.843649
\(563\) −32.0000 −1.34864 −0.674320 0.738440i \(-0.735563\pi\)
−0.674320 + 0.738440i \(0.735563\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −32.0000 −1.34506
\(567\) 2.00000 0.0839921
\(568\) 6.00000 0.251754
\(569\) −35.0000 −1.46728 −0.733638 0.679540i \(-0.762179\pi\)
−0.733638 + 0.679540i \(0.762179\pi\)
\(570\) 4.00000 0.167542
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 3.00000 0.125436
\(573\) −18.0000 −0.751961
\(574\) 6.00000 0.250435
\(575\) −16.0000 −0.667246
\(576\) 1.00000 0.0416667
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 47.0000 1.95494
\(579\) 17.0000 0.706496
\(580\) 9.00000 0.373705
\(581\) 8.00000 0.331896
\(582\) −13.0000 −0.538867
\(583\) −8.00000 −0.331326
\(584\) 12.0000 0.496564
\(585\) 3.00000 0.124035
\(586\) −8.00000 −0.330477
\(587\) 39.0000 1.60970 0.804851 0.593477i \(-0.202245\pi\)
0.804851 + 0.593477i \(0.202245\pi\)
\(588\) 3.00000 0.123718
\(589\) −12.0000 −0.494451
\(590\) 3.00000 0.123508
\(591\) 14.0000 0.575883
\(592\) 6.00000 0.246598
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 1.00000 0.0410305
\(595\) −16.0000 −0.655936
\(596\) 20.0000 0.819232
\(597\) −14.0000 −0.572982
\(598\) −12.0000 −0.490716
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 4.00000 0.163299
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 8.00000 0.326056
\(603\) 4.00000 0.162893
\(604\) 4.00000 0.162758
\(605\) −1.00000 −0.0406558
\(606\) 1.00000 0.0406222
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 4.00000 0.162221
\(609\) 18.0000 0.729397
\(610\) 1.00000 0.0404888
\(611\) 0 0
\(612\) 8.00000 0.323381
\(613\) −33.0000 −1.33286 −0.666429 0.745569i \(-0.732178\pi\)
−0.666429 + 0.745569i \(0.732178\pi\)
\(614\) −11.0000 −0.443924
\(615\) 3.00000 0.120972
\(616\) −2.00000 −0.0805823
\(617\) −19.0000 −0.764911 −0.382456 0.923974i \(-0.624922\pi\)
−0.382456 + 0.923974i \(0.624922\pi\)
\(618\) −12.0000 −0.482711
\(619\) −31.0000 −1.24600 −0.622998 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) 3.00000 0.120483
\(621\) −4.00000 −0.160514
\(622\) 16.0000 0.641542
\(623\) −34.0000 −1.36218
\(624\) 3.00000 0.120096
\(625\) 11.0000 0.440000
\(626\) −4.00000 −0.159872
\(627\) 4.00000 0.159745
\(628\) −7.00000 −0.279330
\(629\) 48.0000 1.91389
\(630\) −2.00000 −0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 25.0000 0.993661
\(634\) 3.00000 0.119145
\(635\) 0 0
\(636\) −8.00000 −0.317221
\(637\) 9.00000 0.356593
\(638\) 9.00000 0.356313
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −13.0000 −0.513069
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 8.00000 0.315244
\(645\) 4.00000 0.157500
\(646\) 32.0000 1.25902
\(647\) −10.0000 −0.393141 −0.196570 0.980490i \(-0.562980\pi\)
−0.196570 + 0.980490i \(0.562980\pi\)
\(648\) 1.00000 0.0392837
\(649\) 3.00000 0.117760
\(650\) 12.0000 0.470679
\(651\) 6.00000 0.235159
\(652\) −5.00000 −0.195815
\(653\) −12.0000 −0.469596 −0.234798 0.972044i \(-0.575443\pi\)
−0.234798 + 0.972044i \(0.575443\pi\)
\(654\) −3.00000 −0.117309
\(655\) −1.00000 −0.0390732
\(656\) 3.00000 0.117130
\(657\) 12.0000 0.468165
\(658\) 0 0
\(659\) −39.0000 −1.51922 −0.759612 0.650376i \(-0.774611\pi\)
−0.759612 + 0.650376i \(0.774611\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −21.0000 −0.816805 −0.408403 0.912802i \(-0.633914\pi\)
−0.408403 + 0.912802i \(0.633914\pi\)
\(662\) −4.00000 −0.155464
\(663\) 24.0000 0.932083
\(664\) 4.00000 0.155230
\(665\) −8.00000 −0.310227
\(666\) 6.00000 0.232495
\(667\) −36.0000 −1.39393
\(668\) 24.0000 0.928588
\(669\) 12.0000 0.463947
\(670\) −4.00000 −0.154533
\(671\) 1.00000 0.0386046
\(672\) −2.00000 −0.0771517
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 15.0000 0.577778
\(675\) 4.00000 0.153960
\(676\) −4.00000 −0.153846
\(677\) −50.0000 −1.92166 −0.960828 0.277145i \(-0.910612\pi\)
−0.960828 + 0.277145i \(0.910612\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 26.0000 0.997788
\(680\) −8.00000 −0.306786
\(681\) −24.0000 −0.919682
\(682\) 3.00000 0.114876
\(683\) −30.0000 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) 4.00000 0.152944
\(685\) −14.0000 −0.534913
\(686\) −20.0000 −0.763604
\(687\) 4.00000 0.152610
\(688\) 4.00000 0.152499
\(689\) −24.0000 −0.914327
\(690\) 4.00000 0.152277
\(691\) 33.0000 1.25538 0.627690 0.778464i \(-0.284001\pi\)
0.627690 + 0.778464i \(0.284001\pi\)
\(692\) 11.0000 0.418157
\(693\) −2.00000 −0.0759737
\(694\) −25.0000 −0.948987
\(695\) 19.0000 0.720711
\(696\) 9.00000 0.341144
\(697\) 24.0000 0.909065
\(698\) 36.0000 1.36262
\(699\) 20.0000 0.756469
\(700\) −8.00000 −0.302372
\(701\) 15.0000 0.566542 0.283271 0.959040i \(-0.408580\pi\)
0.283271 + 0.959040i \(0.408580\pi\)
\(702\) 3.00000 0.113228
\(703\) 24.0000 0.905177
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) −2.00000 −0.0752177
\(708\) 3.00000 0.112747
\(709\) 21.0000 0.788672 0.394336 0.918966i \(-0.370975\pi\)
0.394336 + 0.918966i \(0.370975\pi\)
\(710\) −6.00000 −0.225176
\(711\) −4.00000 −0.150012
\(712\) −17.0000 −0.637102
\(713\) −12.0000 −0.449404
\(714\) −16.0000 −0.598785
\(715\) −3.00000 −0.112194
\(716\) 14.0000 0.523205
\(717\) 16.0000 0.597531
\(718\) −33.0000 −1.23155
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 24.0000 0.893807
\(722\) −3.00000 −0.111648
\(723\) 8.00000 0.297523
\(724\) 1.00000 0.0371647
\(725\) 36.0000 1.33701
\(726\) −1.00000 −0.0371135
\(727\) −12.0000 −0.445055 −0.222528 0.974926i \(-0.571431\pi\)
−0.222528 + 0.974926i \(0.571431\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) 32.0000 1.18356
\(732\) 1.00000 0.0369611
\(733\) 14.0000 0.517102 0.258551 0.965998i \(-0.416755\pi\)
0.258551 + 0.965998i \(0.416755\pi\)
\(734\) −22.0000 −0.812035
\(735\) −3.00000 −0.110657
\(736\) 4.00000 0.147442
\(737\) −4.00000 −0.147342
\(738\) 3.00000 0.110432
\(739\) −31.0000 −1.14035 −0.570177 0.821522i \(-0.693125\pi\)
−0.570177 + 0.821522i \(0.693125\pi\)
\(740\) −6.00000 −0.220564
\(741\) 12.0000 0.440831
\(742\) 16.0000 0.587378
\(743\) −5.00000 −0.183432 −0.0917161 0.995785i \(-0.529235\pi\)
−0.0917161 + 0.995785i \(0.529235\pi\)
\(744\) 3.00000 0.109985
\(745\) −20.0000 −0.732743
\(746\) −20.0000 −0.732252
\(747\) 4.00000 0.146352
\(748\) −8.00000 −0.292509
\(749\) 26.0000 0.950019
\(750\) −9.00000 −0.328634
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) 19.0000 0.692398
\(754\) 27.0000 0.983282
\(755\) −4.00000 −0.145575
\(756\) −2.00000 −0.0727393
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) −13.0000 −0.472181
\(759\) 4.00000 0.145191
\(760\) −4.00000 −0.145095
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) 6.00000 0.217215
\(764\) 18.0000 0.651217
\(765\) −8.00000 −0.289241
\(766\) 14.0000 0.505841
\(767\) 9.00000 0.324971
\(768\) −1.00000 −0.0360844
\(769\) 13.0000 0.468792 0.234396 0.972141i \(-0.424689\pi\)
0.234396 + 0.972141i \(0.424689\pi\)
\(770\) 2.00000 0.0720750
\(771\) −26.0000 −0.936367
\(772\) −17.0000 −0.611843
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 4.00000 0.143777
\(775\) 12.0000 0.431053
\(776\) 13.0000 0.466673
\(777\) −12.0000 −0.430498
\(778\) 36.0000 1.29066
\(779\) 12.0000 0.429945
\(780\) −3.00000 −0.107417
\(781\) −6.00000 −0.214697
\(782\) 32.0000 1.14432
\(783\) 9.00000 0.321634
\(784\) −3.00000 −0.107143
\(785\) 7.00000 0.249841
\(786\) −1.00000 −0.0356688
\(787\) −35.0000 −1.24762 −0.623808 0.781578i \(-0.714415\pi\)
−0.623808 + 0.781578i \(0.714415\pi\)
\(788\) −14.0000 −0.498729
\(789\) 6.00000 0.213606
\(790\) 4.00000 0.142314
\(791\) 4.00000 0.142224
\(792\) −1.00000 −0.0355335
\(793\) 3.00000 0.106533
\(794\) −29.0000 −1.02917
\(795\) 8.00000 0.283731
\(796\) 14.0000 0.496217
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −8.00000 −0.283197
\(799\) 0 0
\(800\) −4.00000 −0.141421
\(801\) −17.0000 −0.600665
\(802\) 25.0000 0.882781
\(803\) −12.0000 −0.423471
\(804\) −4.00000 −0.141069
\(805\) −8.00000 −0.281963
\(806\) 9.00000 0.317011
\(807\) −23.0000 −0.809638
\(808\) −1.00000 −0.0351799
\(809\) 25.0000 0.878953 0.439477 0.898254i \(-0.355164\pi\)
0.439477 + 0.898254i \(0.355164\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −18.0000 −0.631676
\(813\) −20.0000 −0.701431
\(814\) −6.00000 −0.210300
\(815\) 5.00000 0.175142
\(816\) −8.00000 −0.280056
\(817\) 16.0000 0.559769
\(818\) 31.0000 1.08389
\(819\) −6.00000 −0.209657
\(820\) −3.00000 −0.104765
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −14.0000 −0.488306
\(823\) 11.0000 0.383436 0.191718 0.981450i \(-0.438594\pi\)
0.191718 + 0.981450i \(0.438594\pi\)
\(824\) 12.0000 0.418040
\(825\) −4.00000 −0.139262
\(826\) −6.00000 −0.208767
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 4.00000 0.139010
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) −4.00000 −0.138842
\(831\) −26.0000 −0.901930
\(832\) −3.00000 −0.104006
\(833\) −24.0000 −0.831551
\(834\) 19.0000 0.657916
\(835\) −24.0000 −0.830554
\(836\) −4.00000 −0.138343
\(837\) 3.00000 0.103695
\(838\) −36.0000 −1.24360
\(839\) 9.00000 0.310715 0.155357 0.987858i \(-0.450347\pi\)
0.155357 + 0.987858i \(0.450347\pi\)
\(840\) 2.00000 0.0690066
\(841\) 52.0000 1.79310
\(842\) −31.0000 −1.06833
\(843\) −20.0000 −0.688837
\(844\) −25.0000 −0.860535
\(845\) 4.00000 0.137604
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 8.00000 0.274721
\(849\) 32.0000 1.09824
\(850\) −32.0000 −1.09759
\(851\) 24.0000 0.822709
\(852\) −6.00000 −0.205557
\(853\) 3.00000 0.102718 0.0513590 0.998680i \(-0.483645\pi\)
0.0513590 + 0.998680i \(0.483645\pi\)
\(854\) −2.00000 −0.0684386
\(855\) −4.00000 −0.136797
\(856\) 13.0000 0.444331
\(857\) 17.0000 0.580709 0.290354 0.956919i \(-0.406227\pi\)
0.290354 + 0.956919i \(0.406227\pi\)
\(858\) −3.00000 −0.102418
\(859\) 31.0000 1.05771 0.528853 0.848713i \(-0.322622\pi\)
0.528853 + 0.848713i \(0.322622\pi\)
\(860\) −4.00000 −0.136399
\(861\) −6.00000 −0.204479
\(862\) 18.0000 0.613082
\(863\) 17.0000 0.578687 0.289343 0.957225i \(-0.406563\pi\)
0.289343 + 0.957225i \(0.406563\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −11.0000 −0.374011
\(866\) −4.00000 −0.135926
\(867\) −47.0000 −1.59620
\(868\) −6.00000 −0.203653
\(869\) 4.00000 0.135691
\(870\) −9.00000 −0.305129
\(871\) −12.0000 −0.406604
\(872\) 3.00000 0.101593
\(873\) 13.0000 0.439983
\(874\) 16.0000 0.541208
\(875\) 18.0000 0.608511
\(876\) −12.0000 −0.405442
\(877\) −36.0000 −1.21563 −0.607817 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(878\) 13.0000 0.438729
\(879\) 8.00000 0.269833
\(880\) 1.00000 0.0337100
\(881\) 52.0000 1.75192 0.875962 0.482380i \(-0.160227\pi\)
0.875962 + 0.482380i \(0.160227\pi\)
\(882\) −3.00000 −0.101015
\(883\) 4.00000 0.134611 0.0673054 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) −24.0000 −0.807207
\(885\) −3.00000 −0.100844
\(886\) −26.0000 −0.873487
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) −6.00000 −0.201347
\(889\) 0 0
\(890\) 17.0000 0.569841
\(891\) −1.00000 −0.0335013
\(892\) −12.0000 −0.401790
\(893\) 0 0
\(894\) −20.0000 −0.668900
\(895\) −14.0000 −0.467968
\(896\) 2.00000 0.0668153
\(897\) 12.0000 0.400668
\(898\) −40.0000 −1.33482
\(899\) 27.0000 0.900500
\(900\) −4.00000 −0.133333
\(901\) 64.0000 2.13215
\(902\) −3.00000 −0.0998891
\(903\) −8.00000 −0.266223
\(904\) 2.00000 0.0665190
\(905\) −1.00000 −0.0332411
\(906\) −4.00000 −0.132891
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 24.0000 0.796468
\(909\) −1.00000 −0.0331679
\(910\) 6.00000 0.198898
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) −4.00000 −0.132453
\(913\) −4.00000 −0.132381
\(914\) −14.0000 −0.463079
\(915\) −1.00000 −0.0330590
\(916\) −4.00000 −0.132164
\(917\) 2.00000 0.0660458
\(918\) −8.00000 −0.264039
\(919\) −49.0000 −1.61636 −0.808180 0.588935i \(-0.799547\pi\)
−0.808180 + 0.588935i \(0.799547\pi\)
\(920\) −4.00000 −0.131876
\(921\) 11.0000 0.362462
\(922\) −22.0000 −0.724531
\(923\) −18.0000 −0.592477
\(924\) 2.00000 0.0657952
\(925\) −24.0000 −0.789115
\(926\) 6.00000 0.197172
\(927\) 12.0000 0.394132
\(928\) −9.00000 −0.295439
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) −3.00000 −0.0983739
\(931\) −12.0000 −0.393284
\(932\) −20.0000 −0.655122
\(933\) −16.0000 −0.523816
\(934\) 33.0000 1.07979
\(935\) 8.00000 0.261628
\(936\) −3.00000 −0.0980581
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 8.00000 0.261209
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 27.0000 0.880175 0.440087 0.897955i \(-0.354947\pi\)
0.440087 + 0.897955i \(0.354947\pi\)
\(942\) 7.00000 0.228072
\(943\) 12.0000 0.390774
\(944\) −3.00000 −0.0976417
\(945\) 2.00000 0.0650600
\(946\) −4.00000 −0.130051
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 4.00000 0.129914
\(949\) −36.0000 −1.16861
\(950\) −16.0000 −0.519109
\(951\) −3.00000 −0.0972817
\(952\) 16.0000 0.518563
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 8.00000 0.259010
\(955\) −18.0000 −0.582466
\(956\) −16.0000 −0.517477
\(957\) −9.00000 −0.290929
\(958\) 0 0
\(959\) 28.0000 0.904167
\(960\) 1.00000 0.0322749
\(961\) −22.0000 −0.709677
\(962\) −18.0000 −0.580343
\(963\) 13.0000 0.418919
\(964\) −8.00000 −0.257663
\(965\) 17.0000 0.547249
\(966\) −8.00000 −0.257396
\(967\) −23.0000 −0.739630 −0.369815 0.929105i \(-0.620579\pi\)
−0.369815 + 0.929105i \(0.620579\pi\)
\(968\) 1.00000 0.0321412
\(969\) −32.0000 −1.02799
\(970\) −13.0000 −0.417405
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −38.0000 −1.21822
\(974\) 20.0000 0.640841
\(975\) −12.0000 −0.384308
\(976\) −1.00000 −0.0320092
\(977\) 44.0000 1.40768 0.703842 0.710356i \(-0.251466\pi\)
0.703842 + 0.710356i \(0.251466\pi\)
\(978\) 5.00000 0.159882
\(979\) 17.0000 0.543322
\(980\) 3.00000 0.0958315
\(981\) 3.00000 0.0957826
\(982\) −28.0000 −0.893516
\(983\) 6.00000 0.191370 0.0956851 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 14.0000 0.446077
\(986\) −72.0000 −2.29295
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 16.0000 0.508770
\(990\) 1.00000 0.0317821
\(991\) −10.0000 −0.317660 −0.158830 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 4.00000 0.126936
\(994\) 12.0000 0.380617
\(995\) −14.0000 −0.443830
\(996\) −4.00000 −0.126745
\(997\) 40.0000 1.26681 0.633406 0.773819i \(-0.281656\pi\)
0.633406 + 0.773819i \(0.281656\pi\)
\(998\) −22.0000 −0.696398
\(999\) −6.00000 −0.189832
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 4026.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4026.2.a.f.1.1 1 1.1 even 1 trivial