Properties

Label 4012.2.b.b.237.19
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.19
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.28

$q$-expansion

\(f(q)\) \(=\) \(q-0.593296i q^{3} +3.87533i q^{5} +2.92807i q^{7} +2.64800 q^{9} +O(q^{10})\) \(q-0.593296i q^{3} +3.87533i q^{5} +2.92807i q^{7} +2.64800 q^{9} -4.95085i q^{11} -3.80221 q^{13} +2.29922 q^{15} +(2.88587 + 2.94479i) q^{17} +1.82016 q^{19} +1.73721 q^{21} +8.40097i q^{23} -10.0182 q^{25} -3.35094i q^{27} -1.64389i q^{29} +7.62019i q^{31} -2.93732 q^{33} -11.3473 q^{35} +0.0884485i q^{37} +2.25584i q^{39} +2.48344i q^{41} +10.2661 q^{43} +10.2619i q^{45} -1.02570 q^{47} -1.57361 q^{49} +(1.74713 - 1.71217i) q^{51} +0.471127 q^{53} +19.1862 q^{55} -1.07989i q^{57} +1.00000 q^{59} -6.85661i q^{61} +7.75354i q^{63} -14.7348i q^{65} -12.9279 q^{67} +4.98426 q^{69} +8.36046i q^{71} -7.16000i q^{73} +5.94375i q^{75} +14.4964 q^{77} +8.40731i q^{79} +5.95590 q^{81} +2.81612 q^{83} +(-11.4120 + 11.1837i) q^{85} -0.975310 q^{87} -12.6865 q^{89} -11.1331i q^{91} +4.52103 q^{93} +7.05372i q^{95} +2.93724i q^{97} -13.1098i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.593296i 0.342540i −0.985224 0.171270i \(-0.945213\pi\)
0.985224 0.171270i \(-0.0547870\pi\)
\(4\) 0 0
\(5\) 3.87533i 1.73310i 0.499090 + 0.866550i \(0.333668\pi\)
−0.499090 + 0.866550i \(0.666332\pi\)
\(6\) 0 0
\(7\) 2.92807i 1.10671i 0.832946 + 0.553354i \(0.186652\pi\)
−0.832946 + 0.553354i \(0.813348\pi\)
\(8\) 0 0
\(9\) 2.64800 0.882667
\(10\) 0 0
\(11\) 4.95085i 1.49274i −0.665533 0.746368i \(-0.731796\pi\)
0.665533 0.746368i \(-0.268204\pi\)
\(12\) 0 0
\(13\) −3.80221 −1.05454 −0.527272 0.849697i \(-0.676785\pi\)
−0.527272 + 0.849697i \(0.676785\pi\)
\(14\) 0 0
\(15\) 2.29922 0.593656
\(16\) 0 0
\(17\) 2.88587 + 2.94479i 0.699926 + 0.714216i
\(18\) 0 0
\(19\) 1.82016 0.417573 0.208787 0.977961i \(-0.433049\pi\)
0.208787 + 0.977961i \(0.433049\pi\)
\(20\) 0 0
\(21\) 1.73721 0.379091
\(22\) 0 0
\(23\) 8.40097i 1.75172i 0.482562 + 0.875862i \(0.339706\pi\)
−0.482562 + 0.875862i \(0.660294\pi\)
\(24\) 0 0
\(25\) −10.0182 −2.00364
\(26\) 0 0
\(27\) 3.35094i 0.644888i
\(28\) 0 0
\(29\) 1.64389i 0.305262i −0.988283 0.152631i \(-0.951225\pi\)
0.988283 0.152631i \(-0.0487746\pi\)
\(30\) 0 0
\(31\) 7.62019i 1.36863i 0.729188 + 0.684313i \(0.239898\pi\)
−0.729188 + 0.684313i \(0.760102\pi\)
\(32\) 0 0
\(33\) −2.93732 −0.511321
\(34\) 0 0
\(35\) −11.3473 −1.91804
\(36\) 0 0
\(37\) 0.0884485i 0.0145408i 0.999974 + 0.00727042i \(0.00231427\pi\)
−0.999974 + 0.00727042i \(0.997686\pi\)
\(38\) 0 0
\(39\) 2.25584i 0.361223i
\(40\) 0 0
\(41\) 2.48344i 0.387849i 0.981016 + 0.193924i \(0.0621217\pi\)
−0.981016 + 0.193924i \(0.937878\pi\)
\(42\) 0 0
\(43\) 10.2661 1.56557 0.782785 0.622292i \(-0.213798\pi\)
0.782785 + 0.622292i \(0.213798\pi\)
\(44\) 0 0
\(45\) 10.2619i 1.52975i
\(46\) 0 0
\(47\) −1.02570 −0.149614 −0.0748070 0.997198i \(-0.523834\pi\)
−0.0748070 + 0.997198i \(0.523834\pi\)
\(48\) 0 0
\(49\) −1.57361 −0.224802
\(50\) 0 0
\(51\) 1.74713 1.71217i 0.244647 0.239752i
\(52\) 0 0
\(53\) 0.471127 0.0647143 0.0323571 0.999476i \(-0.489699\pi\)
0.0323571 + 0.999476i \(0.489699\pi\)
\(54\) 0 0
\(55\) 19.1862 2.58706
\(56\) 0 0
\(57\) 1.07989i 0.143035i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 6.85661i 0.877898i −0.898512 0.438949i \(-0.855351\pi\)
0.898512 0.438949i \(-0.144649\pi\)
\(62\) 0 0
\(63\) 7.75354i 0.976854i
\(64\) 0 0
\(65\) 14.7348i 1.82763i
\(66\) 0 0
\(67\) −12.9279 −1.57939 −0.789696 0.613499i \(-0.789762\pi\)
−0.789696 + 0.613499i \(0.789762\pi\)
\(68\) 0 0
\(69\) 4.98426 0.600035
\(70\) 0 0
\(71\) 8.36046i 0.992204i 0.868264 + 0.496102i \(0.165236\pi\)
−0.868264 + 0.496102i \(0.834764\pi\)
\(72\) 0 0
\(73\) 7.16000i 0.838014i −0.907983 0.419007i \(-0.862378\pi\)
0.907983 0.419007i \(-0.137622\pi\)
\(74\) 0 0
\(75\) 5.94375i 0.686325i
\(76\) 0 0
\(77\) 14.4964 1.65202
\(78\) 0 0
\(79\) 8.40731i 0.945896i 0.881090 + 0.472948i \(0.156810\pi\)
−0.881090 + 0.472948i \(0.843190\pi\)
\(80\) 0 0
\(81\) 5.95590 0.661767
\(82\) 0 0
\(83\) 2.81612 0.309110 0.154555 0.987984i \(-0.450606\pi\)
0.154555 + 0.987984i \(0.450606\pi\)
\(84\) 0 0
\(85\) −11.4120 + 11.1837i −1.23781 + 1.21304i
\(86\) 0 0
\(87\) −0.975310 −0.104564
\(88\) 0 0
\(89\) −12.6865 −1.34476 −0.672381 0.740206i \(-0.734728\pi\)
−0.672381 + 0.740206i \(0.734728\pi\)
\(90\) 0 0
\(91\) 11.1331i 1.16707i
\(92\) 0 0
\(93\) 4.52103 0.468809
\(94\) 0 0
\(95\) 7.05372i 0.723697i
\(96\) 0 0
\(97\) 2.93724i 0.298232i 0.988820 + 0.149116i \(0.0476427\pi\)
−0.988820 + 0.149116i \(0.952357\pi\)
\(98\) 0 0
\(99\) 13.1098i 1.31759i
\(100\) 0 0
\(101\) −12.5736 −1.25112 −0.625562 0.780174i \(-0.715130\pi\)
−0.625562 + 0.780174i \(0.715130\pi\)
\(102\) 0 0
\(103\) −0.918766 −0.0905287 −0.0452644 0.998975i \(-0.514413\pi\)
−0.0452644 + 0.998975i \(0.514413\pi\)
\(104\) 0 0
\(105\) 6.73228i 0.657003i
\(106\) 0 0
\(107\) 7.47339i 0.722480i 0.932473 + 0.361240i \(0.117646\pi\)
−0.932473 + 0.361240i \(0.882354\pi\)
\(108\) 0 0
\(109\) 3.87296i 0.370963i −0.982648 0.185481i \(-0.940616\pi\)
0.982648 0.185481i \(-0.0593844\pi\)
\(110\) 0 0
\(111\) 0.0524761 0.00498082
\(112\) 0 0
\(113\) 5.26006i 0.494825i −0.968910 0.247413i \(-0.920420\pi\)
0.968910 0.247413i \(-0.0795803\pi\)
\(114\) 0 0
\(115\) −32.5566 −3.03591
\(116\) 0 0
\(117\) −10.0683 −0.930810
\(118\) 0 0
\(119\) −8.62255 + 8.45003i −0.790428 + 0.774613i
\(120\) 0 0
\(121\) −13.5109 −1.22826
\(122\) 0 0
\(123\) 1.47342 0.132854
\(124\) 0 0
\(125\) 19.4472i 1.73941i
\(126\) 0 0
\(127\) 8.46239 0.750915 0.375458 0.926840i \(-0.377486\pi\)
0.375458 + 0.926840i \(0.377486\pi\)
\(128\) 0 0
\(129\) 6.09085i 0.536270i
\(130\) 0 0
\(131\) 17.4727i 1.52660i 0.646046 + 0.763298i \(0.276421\pi\)
−0.646046 + 0.763298i \(0.723579\pi\)
\(132\) 0 0
\(133\) 5.32956i 0.462132i
\(134\) 0 0
\(135\) 12.9860 1.11766
\(136\) 0 0
\(137\) 2.30666 0.197071 0.0985357 0.995134i \(-0.468584\pi\)
0.0985357 + 0.995134i \(0.468584\pi\)
\(138\) 0 0
\(139\) 10.7926i 0.915413i 0.889103 + 0.457707i \(0.151329\pi\)
−0.889103 + 0.457707i \(0.848671\pi\)
\(140\) 0 0
\(141\) 0.608545i 0.0512487i
\(142\) 0 0
\(143\) 18.8242i 1.57415i
\(144\) 0 0
\(145\) 6.37060 0.529049
\(146\) 0 0
\(147\) 0.933619i 0.0770036i
\(148\) 0 0
\(149\) −13.8742 −1.13662 −0.568311 0.822814i \(-0.692403\pi\)
−0.568311 + 0.822814i \(0.692403\pi\)
\(150\) 0 0
\(151\) −11.1809 −0.909891 −0.454945 0.890519i \(-0.650341\pi\)
−0.454945 + 0.890519i \(0.650341\pi\)
\(152\) 0 0
\(153\) 7.64178 + 7.79780i 0.617801 + 0.630415i
\(154\) 0 0
\(155\) −29.5308 −2.37197
\(156\) 0 0
\(157\) 2.01139 0.160527 0.0802633 0.996774i \(-0.474424\pi\)
0.0802633 + 0.996774i \(0.474424\pi\)
\(158\) 0 0
\(159\) 0.279518i 0.0221672i
\(160\) 0 0
\(161\) −24.5987 −1.93865
\(162\) 0 0
\(163\) 12.6552i 0.991235i −0.868541 0.495618i \(-0.834942\pi\)
0.868541 0.495618i \(-0.165058\pi\)
\(164\) 0 0
\(165\) 11.3831i 0.886171i
\(166\) 0 0
\(167\) 18.5213i 1.43322i −0.697475 0.716609i \(-0.745693\pi\)
0.697475 0.716609i \(-0.254307\pi\)
\(168\) 0 0
\(169\) 1.45680 0.112061
\(170\) 0 0
\(171\) 4.81978 0.368578
\(172\) 0 0
\(173\) 22.2040i 1.68814i −0.536236 0.844068i \(-0.680154\pi\)
0.536236 0.844068i \(-0.319846\pi\)
\(174\) 0 0
\(175\) 29.3340i 2.21744i
\(176\) 0 0
\(177\) 0.593296i 0.0445949i
\(178\) 0 0
\(179\) −18.3235 −1.36957 −0.684783 0.728747i \(-0.740103\pi\)
−0.684783 + 0.728747i \(0.740103\pi\)
\(180\) 0 0
\(181\) 5.00327i 0.371890i 0.982560 + 0.185945i \(0.0595347\pi\)
−0.982560 + 0.185945i \(0.940465\pi\)
\(182\) 0 0
\(183\) −4.06800 −0.300715
\(184\) 0 0
\(185\) −0.342767 −0.0252008
\(186\) 0 0
\(187\) 14.5792 14.2875i 1.06614 1.04480i
\(188\) 0 0
\(189\) 9.81179 0.713702
\(190\) 0 0
\(191\) 3.52982 0.255409 0.127704 0.991812i \(-0.459239\pi\)
0.127704 + 0.991812i \(0.459239\pi\)
\(192\) 0 0
\(193\) 17.0812i 1.22953i −0.788711 0.614764i \(-0.789251\pi\)
0.788711 0.614764i \(-0.210749\pi\)
\(194\) 0 0
\(195\) −8.74211 −0.626035
\(196\) 0 0
\(197\) 9.16663i 0.653095i −0.945181 0.326548i \(-0.894115\pi\)
0.945181 0.326548i \(-0.105885\pi\)
\(198\) 0 0
\(199\) 4.96742i 0.352131i 0.984378 + 0.176066i \(0.0563371\pi\)
−0.984378 + 0.176066i \(0.943663\pi\)
\(200\) 0 0
\(201\) 7.67005i 0.541004i
\(202\) 0 0
\(203\) 4.81342 0.337836
\(204\) 0 0
\(205\) −9.62417 −0.672181
\(206\) 0 0
\(207\) 22.2458i 1.54619i
\(208\) 0 0
\(209\) 9.01133i 0.623327i
\(210\) 0 0
\(211\) 16.3353i 1.12457i −0.826944 0.562284i \(-0.809923\pi\)
0.826944 0.562284i \(-0.190077\pi\)
\(212\) 0 0
\(213\) 4.96023 0.339869
\(214\) 0 0
\(215\) 39.7847i 2.71329i
\(216\) 0 0
\(217\) −22.3125 −1.51467
\(218\) 0 0
\(219\) −4.24800 −0.287053
\(220\) 0 0
\(221\) −10.9727 11.1967i −0.738102 0.753171i
\(222\) 0 0
\(223\) −6.02994 −0.403795 −0.201897 0.979407i \(-0.564711\pi\)
−0.201897 + 0.979407i \(0.564711\pi\)
\(224\) 0 0
\(225\) −26.5282 −1.76854
\(226\) 0 0
\(227\) 19.0807i 1.26643i 0.773975 + 0.633217i \(0.218266\pi\)
−0.773975 + 0.633217i \(0.781734\pi\)
\(228\) 0 0
\(229\) −9.92975 −0.656176 −0.328088 0.944647i \(-0.606404\pi\)
−0.328088 + 0.944647i \(0.606404\pi\)
\(230\) 0 0
\(231\) 8.60068i 0.565883i
\(232\) 0 0
\(233\) 17.1706i 1.12488i 0.826838 + 0.562441i \(0.190138\pi\)
−0.826838 + 0.562441i \(0.809862\pi\)
\(234\) 0 0
\(235\) 3.97493i 0.259296i
\(236\) 0 0
\(237\) 4.98802 0.324007
\(238\) 0 0
\(239\) 4.30844 0.278690 0.139345 0.990244i \(-0.455500\pi\)
0.139345 + 0.990244i \(0.455500\pi\)
\(240\) 0 0
\(241\) 27.4140i 1.76589i 0.469473 + 0.882947i \(0.344444\pi\)
−0.469473 + 0.882947i \(0.655556\pi\)
\(242\) 0 0
\(243\) 13.5864i 0.871569i
\(244\) 0 0
\(245\) 6.09828i 0.389605i
\(246\) 0 0
\(247\) −6.92063 −0.440349
\(248\) 0 0
\(249\) 1.67080i 0.105882i
\(250\) 0 0
\(251\) 6.84664 0.432156 0.216078 0.976376i \(-0.430673\pi\)
0.216078 + 0.976376i \(0.430673\pi\)
\(252\) 0 0
\(253\) 41.5919 2.61486
\(254\) 0 0
\(255\) 6.63524 + 6.77071i 0.415515 + 0.423998i
\(256\) 0 0
\(257\) 20.2578 1.26364 0.631822 0.775113i \(-0.282307\pi\)
0.631822 + 0.775113i \(0.282307\pi\)
\(258\) 0 0
\(259\) −0.258984 −0.0160925
\(260\) 0 0
\(261\) 4.35301i 0.269444i
\(262\) 0 0
\(263\) −1.14819 −0.0708001 −0.0354001 0.999373i \(-0.511271\pi\)
−0.0354001 + 0.999373i \(0.511271\pi\)
\(264\) 0 0
\(265\) 1.82577i 0.112156i
\(266\) 0 0
\(267\) 7.52682i 0.460634i
\(268\) 0 0
\(269\) 17.4737i 1.06539i −0.846307 0.532695i \(-0.821179\pi\)
0.846307 0.532695i \(-0.178821\pi\)
\(270\) 0 0
\(271\) −0.871087 −0.0529148 −0.0264574 0.999650i \(-0.508423\pi\)
−0.0264574 + 0.999650i \(0.508423\pi\)
\(272\) 0 0
\(273\) −6.60525 −0.399768
\(274\) 0 0
\(275\) 49.5985i 2.99090i
\(276\) 0 0
\(277\) 13.8936i 0.834787i 0.908726 + 0.417394i \(0.137056\pi\)
−0.908726 + 0.417394i \(0.862944\pi\)
\(278\) 0 0
\(279\) 20.1783i 1.20804i
\(280\) 0 0
\(281\) 6.08777 0.363166 0.181583 0.983376i \(-0.441878\pi\)
0.181583 + 0.983376i \(0.441878\pi\)
\(282\) 0 0
\(283\) 27.5681i 1.63876i 0.573254 + 0.819378i \(0.305681\pi\)
−0.573254 + 0.819378i \(0.694319\pi\)
\(284\) 0 0
\(285\) 4.18495 0.247895
\(286\) 0 0
\(287\) −7.27171 −0.429235
\(288\) 0 0
\(289\) −0.343546 + 16.9965i −0.0202086 + 0.999796i
\(290\) 0 0
\(291\) 1.74265 0.102156
\(292\) 0 0
\(293\) −5.36853 −0.313633 −0.156816 0.987628i \(-0.550123\pi\)
−0.156816 + 0.987628i \(0.550123\pi\)
\(294\) 0 0
\(295\) 3.87533i 0.225631i
\(296\) 0 0
\(297\) −16.5900 −0.962647
\(298\) 0 0
\(299\) 31.9423i 1.84727i
\(300\) 0 0
\(301\) 30.0600i 1.73263i
\(302\) 0 0
\(303\) 7.45989i 0.428560i
\(304\) 0 0
\(305\) 26.5716 1.52149
\(306\) 0 0
\(307\) 15.5224 0.885910 0.442955 0.896544i \(-0.353930\pi\)
0.442955 + 0.896544i \(0.353930\pi\)
\(308\) 0 0
\(309\) 0.545100i 0.0310097i
\(310\) 0 0
\(311\) 27.5865i 1.56428i 0.623100 + 0.782142i \(0.285873\pi\)
−0.623100 + 0.782142i \(0.714127\pi\)
\(312\) 0 0
\(313\) 26.2054i 1.48122i −0.671935 0.740610i \(-0.734537\pi\)
0.671935 0.740610i \(-0.265463\pi\)
\(314\) 0 0
\(315\) −30.0475 −1.69299
\(316\) 0 0
\(317\) 11.8975i 0.668227i 0.942533 + 0.334114i \(0.108437\pi\)
−0.942533 + 0.334114i \(0.891563\pi\)
\(318\) 0 0
\(319\) −8.13862 −0.455675
\(320\) 0 0
\(321\) 4.43393 0.247478
\(322\) 0 0
\(323\) 5.25274 + 5.35999i 0.292270 + 0.298238i
\(324\) 0 0
\(325\) 38.0913 2.11292
\(326\) 0 0
\(327\) −2.29781 −0.127069
\(328\) 0 0
\(329\) 3.00333i 0.165579i
\(330\) 0 0
\(331\) −21.4502 −1.17901 −0.589505 0.807764i \(-0.700677\pi\)
−0.589505 + 0.807764i \(0.700677\pi\)
\(332\) 0 0
\(333\) 0.234212i 0.0128347i
\(334\) 0 0
\(335\) 50.0998i 2.73724i
\(336\) 0 0
\(337\) 35.6109i 1.93985i 0.243408 + 0.969924i \(0.421735\pi\)
−0.243408 + 0.969924i \(0.578265\pi\)
\(338\) 0 0
\(339\) −3.12077 −0.169497
\(340\) 0 0
\(341\) 37.7264 2.04300
\(342\) 0 0
\(343\) 15.8889i 0.857918i
\(344\) 0 0
\(345\) 19.3157i 1.03992i
\(346\) 0 0
\(347\) 13.8045i 0.741064i −0.928820 0.370532i \(-0.879175\pi\)
0.928820 0.370532i \(-0.120825\pi\)
\(348\) 0 0
\(349\) −3.43255 −0.183740 −0.0918701 0.995771i \(-0.529284\pi\)
−0.0918701 + 0.995771i \(0.529284\pi\)
\(350\) 0 0
\(351\) 12.7410i 0.680062i
\(352\) 0 0
\(353\) −4.17223 −0.222065 −0.111033 0.993817i \(-0.535416\pi\)
−0.111033 + 0.993817i \(0.535416\pi\)
\(354\) 0 0
\(355\) −32.3996 −1.71959
\(356\) 0 0
\(357\) 5.01337 + 5.11573i 0.265336 + 0.270753i
\(358\) 0 0
\(359\) 26.5274 1.40006 0.700032 0.714112i \(-0.253169\pi\)
0.700032 + 0.714112i \(0.253169\pi\)
\(360\) 0 0
\(361\) −15.6870 −0.825632
\(362\) 0 0
\(363\) 8.01595i 0.420728i
\(364\) 0 0
\(365\) 27.7474 1.45236
\(366\) 0 0
\(367\) 27.2314i 1.42147i −0.703462 0.710733i \(-0.748363\pi\)
0.703462 0.710733i \(-0.251637\pi\)
\(368\) 0 0
\(369\) 6.57616i 0.342341i
\(370\) 0 0
\(371\) 1.37949i 0.0716198i
\(372\) 0 0
\(373\) 15.7377 0.814869 0.407434 0.913234i \(-0.366424\pi\)
0.407434 + 0.913234i \(0.366424\pi\)
\(374\) 0 0
\(375\) −11.5379 −0.595815
\(376\) 0 0
\(377\) 6.25040i 0.321912i
\(378\) 0 0
\(379\) 12.7567i 0.655267i 0.944805 + 0.327633i \(0.106251\pi\)
−0.944805 + 0.327633i \(0.893749\pi\)
\(380\) 0 0
\(381\) 5.02070i 0.257218i
\(382\) 0 0
\(383\) 37.1510 1.89833 0.949164 0.314782i \(-0.101931\pi\)
0.949164 + 0.314782i \(0.101931\pi\)
\(384\) 0 0
\(385\) 56.1785i 2.86312i
\(386\) 0 0
\(387\) 27.1847 1.38188
\(388\) 0 0
\(389\) 31.1313 1.57842 0.789209 0.614125i \(-0.210491\pi\)
0.789209 + 0.614125i \(0.210491\pi\)
\(390\) 0 0
\(391\) −24.7391 + 24.2441i −1.25111 + 1.22608i
\(392\) 0 0
\(393\) 10.3665 0.522920
\(394\) 0 0
\(395\) −32.5811 −1.63933
\(396\) 0 0
\(397\) 23.8999i 1.19950i 0.800187 + 0.599750i \(0.204733\pi\)
−0.800187 + 0.599750i \(0.795267\pi\)
\(398\) 0 0
\(399\) 3.16201 0.158298
\(400\) 0 0
\(401\) 16.5266i 0.825297i −0.910890 0.412648i \(-0.864604\pi\)
0.910890 0.412648i \(-0.135396\pi\)
\(402\) 0 0
\(403\) 28.9736i 1.44328i
\(404\) 0 0
\(405\) 23.0811i 1.14691i
\(406\) 0 0
\(407\) 0.437895 0.0217057
\(408\) 0 0
\(409\) −1.42032 −0.0702305 −0.0351152 0.999383i \(-0.511180\pi\)
−0.0351152 + 0.999383i \(0.511180\pi\)
\(410\) 0 0
\(411\) 1.36853i 0.0675047i
\(412\) 0 0
\(413\) 2.92807i 0.144081i
\(414\) 0 0
\(415\) 10.9134i 0.535719i
\(416\) 0 0
\(417\) 6.40318 0.313565
\(418\) 0 0
\(419\) 20.5811i 1.00545i −0.864445 0.502727i \(-0.832330\pi\)
0.864445 0.502727i \(-0.167670\pi\)
\(420\) 0 0
\(421\) −26.0994 −1.27201 −0.636003 0.771687i \(-0.719413\pi\)
−0.636003 + 0.771687i \(0.719413\pi\)
\(422\) 0 0
\(423\) −2.71606 −0.132059
\(424\) 0 0
\(425\) −28.9112 29.5014i −1.40240 1.43103i
\(426\) 0 0
\(427\) 20.0766 0.971577
\(428\) 0 0
\(429\) 11.1683 0.539210
\(430\) 0 0
\(431\) 8.97767i 0.432439i −0.976345 0.216219i \(-0.930627\pi\)
0.976345 0.216219i \(-0.0693727\pi\)
\(432\) 0 0
\(433\) 6.45697 0.310302 0.155151 0.987891i \(-0.450414\pi\)
0.155151 + 0.987891i \(0.450414\pi\)
\(434\) 0 0
\(435\) 3.77965i 0.181220i
\(436\) 0 0
\(437\) 15.2911i 0.731473i
\(438\) 0 0
\(439\) 1.97734i 0.0943735i −0.998886 0.0471868i \(-0.984974\pi\)
0.998886 0.0471868i \(-0.0150256\pi\)
\(440\) 0 0
\(441\) −4.16693 −0.198425
\(442\) 0 0
\(443\) −30.5708 −1.45246 −0.726230 0.687452i \(-0.758729\pi\)
−0.726230 + 0.687452i \(0.758729\pi\)
\(444\) 0 0
\(445\) 49.1642i 2.33061i
\(446\) 0 0
\(447\) 8.23153i 0.389338i
\(448\) 0 0
\(449\) 39.4623i 1.86234i −0.364583 0.931171i \(-0.618788\pi\)
0.364583 0.931171i \(-0.381212\pi\)
\(450\) 0 0
\(451\) 12.2952 0.578956
\(452\) 0 0
\(453\) 6.63360i 0.311674i
\(454\) 0 0
\(455\) 43.1446 2.02265
\(456\) 0 0
\(457\) −15.5916 −0.729344 −0.364672 0.931136i \(-0.618819\pi\)
−0.364672 + 0.931136i \(0.618819\pi\)
\(458\) 0 0
\(459\) 9.86779 9.67035i 0.460589 0.451373i
\(460\) 0 0
\(461\) 23.7521 1.10625 0.553123 0.833100i \(-0.313436\pi\)
0.553123 + 0.833100i \(0.313436\pi\)
\(462\) 0 0
\(463\) 30.8449 1.43349 0.716743 0.697338i \(-0.245632\pi\)
0.716743 + 0.697338i \(0.245632\pi\)
\(464\) 0 0
\(465\) 17.5205i 0.812493i
\(466\) 0 0
\(467\) 42.2503 1.95511 0.977555 0.210683i \(-0.0675687\pi\)
0.977555 + 0.210683i \(0.0675687\pi\)
\(468\) 0 0
\(469\) 37.8538i 1.74792i
\(470\) 0 0
\(471\) 1.19335i 0.0549867i
\(472\) 0 0
\(473\) 50.8260i 2.33698i
\(474\) 0 0
\(475\) −18.2347 −0.836666
\(476\) 0 0
\(477\) 1.24754 0.0571211
\(478\) 0 0
\(479\) 5.24436i 0.239621i 0.992797 + 0.119810i \(0.0382287\pi\)
−0.992797 + 0.119810i \(0.961771\pi\)
\(480\) 0 0
\(481\) 0.336300i 0.0153340i
\(482\) 0 0
\(483\) 14.5943i 0.664063i
\(484\) 0 0
\(485\) −11.3828 −0.516866
\(486\) 0 0
\(487\) 32.3556i 1.46617i −0.680137 0.733085i \(-0.738080\pi\)
0.680137 0.733085i \(-0.261920\pi\)
\(488\) 0 0
\(489\) −7.50831 −0.339537
\(490\) 0 0
\(491\) 23.9506 1.08088 0.540438 0.841384i \(-0.318258\pi\)
0.540438 + 0.841384i \(0.318258\pi\)
\(492\) 0 0
\(493\) 4.84089 4.74403i 0.218023 0.213661i
\(494\) 0 0
\(495\) 50.8050 2.28351
\(496\) 0 0
\(497\) −24.4800 −1.09808
\(498\) 0 0
\(499\) 12.4420i 0.556979i 0.960439 + 0.278489i \(0.0898337\pi\)
−0.960439 + 0.278489i \(0.910166\pi\)
\(500\) 0 0
\(501\) −10.9886 −0.490934
\(502\) 0 0
\(503\) 31.5232i 1.40555i 0.711413 + 0.702774i \(0.248056\pi\)
−0.711413 + 0.702774i \(0.751944\pi\)
\(504\) 0 0
\(505\) 48.7270i 2.16833i
\(506\) 0 0
\(507\) 0.864311i 0.0383854i
\(508\) 0 0
\(509\) −19.7227 −0.874195 −0.437097 0.899414i \(-0.643993\pi\)
−0.437097 + 0.899414i \(0.643993\pi\)
\(510\) 0 0
\(511\) 20.9650 0.927437
\(512\) 0 0
\(513\) 6.09924i 0.269288i
\(514\) 0 0
\(515\) 3.56052i 0.156895i
\(516\) 0 0
\(517\) 5.07809i 0.223334i
\(518\) 0 0
\(519\) −13.1735 −0.578254
\(520\) 0 0
\(521\) 13.4265i 0.588228i 0.955770 + 0.294114i \(0.0950245\pi\)
−0.955770 + 0.294114i \(0.904975\pi\)
\(522\) 0 0
\(523\) 3.84227 0.168011 0.0840054 0.996465i \(-0.473229\pi\)
0.0840054 + 0.996465i \(0.473229\pi\)
\(524\) 0 0
\(525\) −17.4037 −0.759562
\(526\) 0 0
\(527\) −22.4398 + 21.9909i −0.977495 + 0.957937i
\(528\) 0 0
\(529\) −47.5764 −2.06854
\(530\) 0 0
\(531\) 2.64800 0.114913
\(532\) 0 0
\(533\) 9.44258i 0.409003i
\(534\) 0 0
\(535\) −28.9618 −1.25213
\(536\) 0 0
\(537\) 10.8713i 0.469131i
\(538\) 0 0
\(539\) 7.79072i 0.335570i
\(540\) 0 0
\(541\) 19.6791i 0.846071i 0.906113 + 0.423036i \(0.139035\pi\)
−0.906113 + 0.423036i \(0.860965\pi\)
\(542\) 0 0
\(543\) 2.96842 0.127387
\(544\) 0 0
\(545\) 15.0090 0.642915
\(546\) 0 0
\(547\) 10.3727i 0.443505i 0.975103 + 0.221753i \(0.0711777\pi\)
−0.975103 + 0.221753i \(0.928822\pi\)
\(548\) 0 0
\(549\) 18.1563i 0.774892i
\(550\) 0 0
\(551\) 2.99213i 0.127469i
\(552\) 0 0
\(553\) −24.6172 −1.04683
\(554\) 0 0
\(555\) 0.203362i 0.00863226i
\(556\) 0 0
\(557\) 30.8749 1.30821 0.654105 0.756404i \(-0.273045\pi\)
0.654105 + 0.756404i \(0.273045\pi\)
\(558\) 0 0
\(559\) −39.0340 −1.65096
\(560\) 0 0
\(561\) −8.47670 8.64977i −0.357887 0.365194i
\(562\) 0 0
\(563\) 40.0945 1.68978 0.844890 0.534940i \(-0.179666\pi\)
0.844890 + 0.534940i \(0.179666\pi\)
\(564\) 0 0
\(565\) 20.3845 0.857582
\(566\) 0 0
\(567\) 17.4393i 0.732383i
\(568\) 0 0
\(569\) −42.7366 −1.79161 −0.895806 0.444446i \(-0.853400\pi\)
−0.895806 + 0.444446i \(0.853400\pi\)
\(570\) 0 0
\(571\) 24.8026i 1.03796i −0.854788 0.518978i \(-0.826313\pi\)
0.854788 0.518978i \(-0.173687\pi\)
\(572\) 0 0
\(573\) 2.09423i 0.0874876i
\(574\) 0 0
\(575\) 84.1626i 3.50982i
\(576\) 0 0
\(577\) 25.7144 1.07051 0.535253 0.844692i \(-0.320216\pi\)
0.535253 + 0.844692i \(0.320216\pi\)
\(578\) 0 0
\(579\) −10.1342 −0.421162
\(580\) 0 0
\(581\) 8.24582i 0.342094i
\(582\) 0 0
\(583\) 2.33248i 0.0966014i
\(584\) 0 0
\(585\) 39.0178i 1.61319i
\(586\) 0 0
\(587\) 29.9337 1.23550 0.617748 0.786376i \(-0.288045\pi\)
0.617748 + 0.786376i \(0.288045\pi\)
\(588\) 0 0
\(589\) 13.8700i 0.571502i
\(590\) 0 0
\(591\) −5.43852 −0.223711
\(592\) 0 0
\(593\) 13.7668 0.565336 0.282668 0.959218i \(-0.408781\pi\)
0.282668 + 0.959218i \(0.408781\pi\)
\(594\) 0 0
\(595\) −32.7467 33.4153i −1.34248 1.36989i
\(596\) 0 0
\(597\) 2.94715 0.120619
\(598\) 0 0
\(599\) −0.413974 −0.0169145 −0.00845727 0.999964i \(-0.502692\pi\)
−0.00845727 + 0.999964i \(0.502692\pi\)
\(600\) 0 0
\(601\) 9.76784i 0.398438i 0.979955 + 0.199219i \(0.0638406\pi\)
−0.979955 + 0.199219i \(0.936159\pi\)
\(602\) 0 0
\(603\) −34.2330 −1.39408
\(604\) 0 0
\(605\) 52.3591i 2.12870i
\(606\) 0 0
\(607\) 20.5906i 0.835746i 0.908505 + 0.417873i \(0.137224\pi\)
−0.908505 + 0.417873i \(0.862776\pi\)
\(608\) 0 0
\(609\) 2.85578i 0.115722i
\(610\) 0 0
\(611\) 3.89993 0.157774
\(612\) 0 0
\(613\) 19.2067 0.775749 0.387875 0.921712i \(-0.373209\pi\)
0.387875 + 0.921712i \(0.373209\pi\)
\(614\) 0 0
\(615\) 5.70998i 0.230249i
\(616\) 0 0
\(617\) 19.2322i 0.774257i 0.922026 + 0.387129i \(0.126533\pi\)
−0.922026 + 0.387129i \(0.873467\pi\)
\(618\) 0 0
\(619\) 2.18242i 0.0877188i 0.999038 + 0.0438594i \(0.0139654\pi\)
−0.999038 + 0.0438594i \(0.986035\pi\)
\(620\) 0 0
\(621\) 28.1511 1.12967
\(622\) 0 0
\(623\) 37.1469i 1.48826i
\(624\) 0 0
\(625\) 25.2732 1.01093
\(626\) 0 0
\(627\) −5.34639 −0.213514
\(628\) 0 0
\(629\) −0.260462 + 0.255251i −0.0103853 + 0.0101775i
\(630\) 0 0
\(631\) −49.7845 −1.98189 −0.990944 0.134277i \(-0.957129\pi\)
−0.990944 + 0.134277i \(0.957129\pi\)
\(632\) 0 0
\(633\) −9.69167 −0.385209
\(634\) 0 0
\(635\) 32.7945i 1.30141i
\(636\) 0 0
\(637\) 5.98321 0.237064
\(638\) 0 0
\(639\) 22.1385i 0.875786i
\(640\) 0 0
\(641\) 28.1843i 1.11321i −0.830776 0.556607i \(-0.812103\pi\)
0.830776 0.556607i \(-0.187897\pi\)
\(642\) 0 0
\(643\) 13.8432i 0.545924i −0.962025 0.272962i \(-0.911997\pi\)
0.962025 0.272962i \(-0.0880033\pi\)
\(644\) 0 0
\(645\) 23.6041 0.929410
\(646\) 0 0
\(647\) 14.5492 0.571989 0.285995 0.958231i \(-0.407676\pi\)
0.285995 + 0.958231i \(0.407676\pi\)
\(648\) 0 0
\(649\) 4.95085i 0.194338i
\(650\) 0 0
\(651\) 13.2379i 0.518834i
\(652\) 0 0
\(653\) 5.49220i 0.214926i −0.994209 0.107463i \(-0.965727\pi\)
0.994209 0.107463i \(-0.0342728\pi\)
\(654\) 0 0
\(655\) −67.7125 −2.64575
\(656\) 0 0
\(657\) 18.9597i 0.739687i
\(658\) 0 0
\(659\) −16.2140 −0.631608 −0.315804 0.948824i \(-0.602274\pi\)
−0.315804 + 0.948824i \(0.602274\pi\)
\(660\) 0 0
\(661\) 9.42581 0.366621 0.183311 0.983055i \(-0.441319\pi\)
0.183311 + 0.983055i \(0.441319\pi\)
\(662\) 0 0
\(663\) −6.64296 + 6.51004i −0.257991 + 0.252829i
\(664\) 0 0
\(665\) −20.6538 −0.800921
\(666\) 0 0
\(667\) 13.8102 0.534734
\(668\) 0 0
\(669\) 3.57754i 0.138316i
\(670\) 0 0
\(671\) −33.9460 −1.31047
\(672\) 0 0
\(673\) 4.71120i 0.181603i −0.995869 0.0908016i \(-0.971057\pi\)
0.995869 0.0908016i \(-0.0289429\pi\)
\(674\) 0 0
\(675\) 33.5703i 1.29212i
\(676\) 0 0
\(677\) 1.39320i 0.0535450i −0.999642 0.0267725i \(-0.991477\pi\)
0.999642 0.0267725i \(-0.00852297\pi\)
\(678\) 0 0
\(679\) −8.60046 −0.330055
\(680\) 0 0
\(681\) 11.3205 0.433803
\(682\) 0 0
\(683\) 37.0195i 1.41651i 0.705956 + 0.708256i \(0.250518\pi\)
−0.705956 + 0.708256i \(0.749482\pi\)
\(684\) 0 0
\(685\) 8.93908i 0.341545i
\(686\) 0 0
\(687\) 5.89128i 0.224766i
\(688\) 0 0
\(689\) −1.79132 −0.0682440
\(690\) 0 0
\(691\) 34.7332i 1.32131i 0.750688 + 0.660657i \(0.229722\pi\)
−0.750688 + 0.660657i \(0.770278\pi\)
\(692\) 0 0
\(693\) 38.3866 1.45819
\(694\) 0 0
\(695\) −41.8248 −1.58650
\(696\) 0 0
\(697\) −7.31322 + 7.16689i −0.277008 + 0.271465i
\(698\) 0 0
\(699\) 10.1872 0.385316
\(700\) 0 0
\(701\) 4.88392 0.184463 0.0922316 0.995738i \(-0.470600\pi\)
0.0922316 + 0.995738i \(0.470600\pi\)
\(702\) 0 0
\(703\) 0.160991i 0.00607187i
\(704\) 0 0
\(705\) −2.35831 −0.0888192
\(706\) 0 0
\(707\) 36.8166i 1.38463i
\(708\) 0 0
\(709\) 48.3839i 1.81710i −0.417781 0.908548i \(-0.637192\pi\)
0.417781 0.908548i \(-0.362808\pi\)
\(710\) 0 0
\(711\) 22.2626i 0.834911i
\(712\) 0 0
\(713\) −64.0170 −2.39746
\(714\) 0 0
\(715\) −72.9498 −2.72817
\(716\) 0 0
\(717\) 2.55618i 0.0954623i
\(718\) 0 0
\(719\) 9.80772i 0.365766i −0.983135 0.182883i \(-0.941457\pi\)
0.983135 0.182883i \(-0.0585430\pi\)
\(720\) 0 0
\(721\) 2.69022i 0.100189i
\(722\) 0 0
\(723\) 16.2646 0.604888
\(724\) 0 0
\(725\) 16.4688i 0.611634i
\(726\) 0 0
\(727\) 14.5244 0.538680 0.269340 0.963045i \(-0.413194\pi\)
0.269340 + 0.963045i \(0.413194\pi\)
\(728\) 0 0
\(729\) 9.80694 0.363220
\(730\) 0 0
\(731\) 29.6267 + 30.2316i 1.09578 + 1.11816i
\(732\) 0 0
\(733\) 19.1472 0.707216 0.353608 0.935394i \(-0.384955\pi\)
0.353608 + 0.935394i \(0.384955\pi\)
\(734\) 0 0
\(735\) −3.61808 −0.133455
\(736\) 0 0
\(737\) 64.0039i 2.35761i
\(738\) 0 0
\(739\) −36.7662 −1.35247 −0.676233 0.736688i \(-0.736389\pi\)
−0.676233 + 0.736688i \(0.736389\pi\)
\(740\) 0 0
\(741\) 4.10598i 0.150837i
\(742\) 0 0
\(743\) 0.481977i 0.0176820i −0.999961 0.00884102i \(-0.997186\pi\)
0.999961 0.00884102i \(-0.00281422\pi\)
\(744\) 0 0
\(745\) 53.7673i 1.96988i
\(746\) 0 0
\(747\) 7.45710 0.272841
\(748\) 0 0
\(749\) −21.8826 −0.799574
\(750\) 0 0
\(751\) 34.1238i 1.24519i 0.782543 + 0.622597i \(0.213922\pi\)
−0.782543 + 0.622597i \(0.786078\pi\)
\(752\) 0 0
\(753\) 4.06208i 0.148030i
\(754\) 0 0
\(755\) 43.3298i 1.57693i
\(756\) 0 0
\(757\) −15.2759 −0.555212 −0.277606 0.960695i \(-0.589541\pi\)
−0.277606 + 0.960695i \(0.589541\pi\)
\(758\) 0 0
\(759\) 24.6763i 0.895694i
\(760\) 0 0
\(761\) 38.2701 1.38729 0.693645 0.720317i \(-0.256004\pi\)
0.693645 + 0.720317i \(0.256004\pi\)
\(762\) 0 0
\(763\) 11.3403 0.410547
\(764\) 0 0
\(765\) −30.2190 + 29.6144i −1.09257 + 1.07071i
\(766\) 0 0
\(767\) −3.80221 −0.137290
\(768\) 0 0
\(769\) 39.8202 1.43595 0.717976 0.696068i \(-0.245069\pi\)
0.717976 + 0.696068i \(0.245069\pi\)
\(770\) 0 0
\(771\) 12.0189i 0.432848i
\(772\) 0 0
\(773\) 21.8048 0.784265 0.392133 0.919909i \(-0.371737\pi\)
0.392133 + 0.919909i \(0.371737\pi\)
\(774\) 0 0
\(775\) 76.3405i 2.74223i
\(776\) 0 0
\(777\) 0.153654i 0.00551231i
\(778\) 0 0
\(779\) 4.52027i 0.161955i
\(780\) 0 0
\(781\) 41.3914 1.48110
\(782\) 0 0
\(783\) −5.50855 −0.196860
\(784\) 0 0
\(785\) 7.79481i 0.278209i
\(786\) 0 0
\(787\) 36.9239i 1.31619i −0.752933 0.658097i \(-0.771362\pi\)
0.752933 0.658097i \(-0.228638\pi\)
\(788\) 0 0
\(789\) 0.681214i 0.0242518i
\(790\) 0 0
\(791\) 15.4018 0.547627
\(792\) 0 0
\(793\) 26.0702i 0.925782i
\(794\)