Properties

Label 6012.2.h.a.3005.8
Level 6012
Weight 2
Character 6012.3005
Analytic conductor 48.006
Analytic rank 0
Dimension 56
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3005.8
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.7

$q$-expansion

\(f(q)\) \(=\) \(q-3.30258 q^{5} -4.47594 q^{7} +O(q^{10})\) \(q-3.30258 q^{5} -4.47594 q^{7} +1.74620i q^{11} +2.94627i q^{13} +2.90566 q^{17} +3.77245 q^{19} +0.201252 q^{23} +5.90705 q^{25} -6.23069i q^{29} +4.31658 q^{31} +14.7822 q^{35} +5.99423i q^{37} +6.64706 q^{41} -6.12658i q^{43} +5.02820i q^{47} +13.0341 q^{49} -2.29277 q^{53} -5.76699i q^{55} -0.540391 q^{59} -8.87576 q^{61} -9.73030i q^{65} +12.8297i q^{67} +5.03751 q^{71} +4.72315i q^{73} -7.81591i q^{77} +0.282217i q^{79} +3.56680 q^{83} -9.59617 q^{85} +3.81680i q^{89} -13.1873i q^{91} -12.4588 q^{95} -8.03056 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times\).

\(n\) \(3007\) \(3341\) \(4681\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 0
\(4\) 0 0
\(5\) −3.30258 −1.47696 −0.738480 0.674276i \(-0.764456\pi\)
−0.738480 + 0.674276i \(0.764456\pi\)
\(6\) 0 0
\(7\) −4.47594 −1.69175 −0.845874 0.533383i \(-0.820920\pi\)
−0.845874 + 0.533383i \(0.820920\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.74620i 0.526501i 0.964728 + 0.263250i \(0.0847945\pi\)
−0.964728 + 0.263250i \(0.915206\pi\)
\(12\) 0 0
\(13\) 2.94627i 0.817148i 0.912725 + 0.408574i \(0.133974\pi\)
−0.912725 + 0.408574i \(0.866026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.90566 0.704725 0.352362 0.935864i \(-0.385378\pi\)
0.352362 + 0.935864i \(0.385378\pi\)
\(18\) 0 0
\(19\) 3.77245 0.865458 0.432729 0.901524i \(-0.357551\pi\)
0.432729 + 0.901524i \(0.357551\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.201252 0.0419640 0.0209820 0.999780i \(-0.493321\pi\)
0.0209820 + 0.999780i \(0.493321\pi\)
\(24\) 0 0
\(25\) 5.90705 1.18141
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.23069i 1.15701i −0.815679 0.578505i \(-0.803636\pi\)
0.815679 0.578505i \(-0.196364\pi\)
\(30\) 0 0
\(31\) 4.31658 0.775281 0.387640 0.921811i \(-0.373290\pi\)
0.387640 + 0.921811i \(0.373290\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 14.7822 2.49864
\(36\) 0 0
\(37\) 5.99423i 0.985445i 0.870187 + 0.492722i \(0.163998\pi\)
−0.870187 + 0.492722i \(0.836002\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.64706 1.03810 0.519048 0.854745i \(-0.326286\pi\)
0.519048 + 0.854745i \(0.326286\pi\)
\(42\) 0 0
\(43\) 6.12658i 0.934294i −0.884180 0.467147i \(-0.845282\pi\)
0.884180 0.467147i \(-0.154718\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.02820i 0.733438i 0.930332 + 0.366719i \(0.119519\pi\)
−0.930332 + 0.366719i \(0.880481\pi\)
\(48\) 0 0
\(49\) 13.0341 1.86201
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.29277 −0.314936 −0.157468 0.987524i \(-0.550333\pi\)
−0.157468 + 0.987524i \(0.550333\pi\)
\(54\) 0 0
\(55\) 5.76699i 0.777620i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.540391 −0.0703530 −0.0351765 0.999381i \(-0.511199\pi\)
−0.0351765 + 0.999381i \(0.511199\pi\)
\(60\) 0 0
\(61\) −8.87576 −1.13642 −0.568212 0.822882i \(-0.692365\pi\)
−0.568212 + 0.822882i \(0.692365\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.73030i 1.20690i
\(66\) 0 0
\(67\) 12.8297i 1.56740i 0.621138 + 0.783701i \(0.286671\pi\)
−0.621138 + 0.783701i \(0.713329\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.03751 0.597842 0.298921 0.954278i \(-0.403373\pi\)
0.298921 + 0.954278i \(0.403373\pi\)
\(72\) 0 0
\(73\) 4.72315i 0.552802i 0.961042 + 0.276401i \(0.0891419\pi\)
−0.961042 + 0.276401i \(0.910858\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.81591i 0.890706i
\(78\) 0 0
\(79\) 0.282217i 0.0317519i 0.999874 + 0.0158759i \(0.00505368\pi\)
−0.999874 + 0.0158759i \(0.994946\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.56680 0.391507 0.195754 0.980653i \(-0.437285\pi\)
0.195754 + 0.980653i \(0.437285\pi\)
\(84\) 0 0
\(85\) −9.59617 −1.04085
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.81680i 0.404580i 0.979326 + 0.202290i \(0.0648383\pi\)
−0.979326 + 0.202290i \(0.935162\pi\)
\(90\) 0 0
\(91\) 13.1873i 1.38241i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −12.4588 −1.27825
\(96\) 0 0
\(97\) −8.03056 −0.815380 −0.407690 0.913120i \(-0.633666\pi\)
−0.407690 + 0.913120i \(0.633666\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.01447 −0.299951 −0.149976 0.988690i \(-0.547920\pi\)
−0.149976 + 0.988690i \(0.547920\pi\)
\(102\) 0 0
\(103\) 1.10909i 0.109282i −0.998506 0.0546408i \(-0.982599\pi\)
0.998506 0.0546408i \(-0.0174014\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0758i 1.74745i −0.486416 0.873727i \(-0.661696\pi\)
0.486416 0.873727i \(-0.338304\pi\)
\(108\) 0 0
\(109\) 15.8681i 1.51989i 0.649990 + 0.759943i \(0.274773\pi\)
−0.649990 + 0.759943i \(0.725227\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.3715 −1.06974 −0.534869 0.844935i \(-0.679639\pi\)
−0.534869 + 0.844935i \(0.679639\pi\)
\(114\) 0 0
\(115\) −0.664653 −0.0619792
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −13.0055 −1.19222
\(120\) 0 0
\(121\) 7.95077 0.722797
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.99561 −0.267936
\(126\) 0 0
\(127\) −2.90908 −0.258139 −0.129070 0.991636i \(-0.541199\pi\)
−0.129070 + 0.991636i \(0.541199\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.21247 0.193305 0.0966523 0.995318i \(-0.469187\pi\)
0.0966523 + 0.995318i \(0.469187\pi\)
\(132\) 0 0
\(133\) −16.8852 −1.46414
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.2993i 1.73429i −0.498056 0.867145i \(-0.665953\pi\)
0.498056 0.867145i \(-0.334047\pi\)
\(138\) 0 0
\(139\) 13.0422i 1.10623i 0.833105 + 0.553115i \(0.186561\pi\)
−0.833105 + 0.553115i \(0.813439\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.14479 −0.430229
\(144\) 0 0
\(145\) 20.5774i 1.70886i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.97767 −0.162017 −0.0810086 0.996713i \(-0.525814\pi\)
−0.0810086 + 0.996713i \(0.525814\pi\)
\(150\) 0 0
\(151\) 7.45460i 0.606646i 0.952888 + 0.303323i \(0.0980962\pi\)
−0.952888 + 0.303323i \(0.901904\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −14.2559 −1.14506
\(156\) 0 0
\(157\) −7.89114 −0.629782 −0.314891 0.949128i \(-0.601968\pi\)
−0.314891 + 0.949128i \(0.601968\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −0.900794 −0.0709925
\(162\) 0 0
\(163\) 2.36362i 0.185133i −0.995707 0.0925664i \(-0.970493\pi\)
0.995707 0.0925664i \(-0.0295070\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.47390 8.78893i 0.733112 0.680108i
\(168\) 0 0
\(169\) 4.31949 0.332269
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.7117i 0.890421i −0.895426 0.445211i \(-0.853129\pi\)
0.895426 0.445211i \(-0.146871\pi\)
\(174\) 0 0
\(175\) −26.4396 −1.99865
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.05482i 0.0788411i −0.999223 0.0394205i \(-0.987449\pi\)
0.999223 0.0394205i \(-0.0125512\pi\)
\(180\) 0 0
\(181\) −19.3368 −1.43729 −0.718647 0.695375i \(-0.755238\pi\)
−0.718647 + 0.695375i \(0.755238\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 19.7964i 1.45546i
\(186\) 0 0
\(187\) 5.07387i 0.371038i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.18869i 0.520155i 0.965588 + 0.260078i \(0.0837482\pi\)
−0.965588 + 0.260078i \(0.916252\pi\)
\(192\) 0 0
\(193\) 24.5942i 1.77033i 0.465279 + 0.885164i \(0.345954\pi\)
−0.465279 + 0.885164i \(0.654046\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.7355 −1.04986 −0.524932 0.851144i \(-0.675909\pi\)
−0.524932 + 0.851144i \(0.675909\pi\)
\(198\) 0 0
\(199\) −22.1485 −1.57007 −0.785034 0.619453i \(-0.787354\pi\)
−0.785034 + 0.619453i \(0.787354\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 27.8882i 1.95737i
\(204\) 0 0
\(205\) −21.9525 −1.53323
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.58746i 0.455664i
\(210\) 0 0
\(211\) 16.9623 1.16773 0.583867 0.811849i \(-0.301539\pi\)
0.583867 + 0.811849i \(0.301539\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 20.2335i 1.37991i
\(216\) 0 0
\(217\) −19.3208 −1.31158
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.56084i 0.575865i
\(222\) 0 0
\(223\) 0.960326 0.0643082 0.0321541 0.999483i \(-0.489763\pi\)
0.0321541 + 0.999483i \(0.489763\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −26.3838 −1.75115 −0.875576 0.483081i \(-0.839518\pi\)
−0.875576 + 0.483081i \(0.839518\pi\)
\(228\) 0 0
\(229\) 3.90353 0.257953 0.128976 0.991648i \(-0.458831\pi\)
0.128976 + 0.991648i \(0.458831\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.848301i 0.0555741i 0.999614 + 0.0277870i \(0.00884602\pi\)
−0.999614 + 0.0277870i \(0.991154\pi\)
\(234\) 0 0
\(235\) 16.6060i 1.08326i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.93714i 0.254672i 0.991860 + 0.127336i \(0.0406427\pi\)
−0.991860 + 0.127336i \(0.959357\pi\)
\(240\) 0 0
\(241\) 13.3587i 0.860510i 0.902707 + 0.430255i \(0.141576\pi\)
−0.902707 + 0.430255i \(0.858424\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −43.0461 −2.75011
\(246\) 0 0
\(247\) 11.1146i 0.707208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.1441i 1.14524i 0.819820 + 0.572621i \(0.194073\pi\)
−0.819820 + 0.572621i \(0.805927\pi\)
\(252\) 0 0
\(253\) 0.351428i 0.0220941i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.1437 −1.38129 −0.690644 0.723195i \(-0.742673\pi\)
−0.690644 + 0.723195i \(0.742673\pi\)
\(258\) 0 0
\(259\) 26.8298i 1.66712i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.8406i 0.791785i −0.918297 0.395893i \(-0.870435\pi\)
0.918297 0.395893i \(-0.129565\pi\)
\(264\) 0 0
\(265\) 7.57205 0.465147
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 11.4969 0.700976 0.350488 0.936567i \(-0.386016\pi\)
0.350488 + 0.936567i \(0.386016\pi\)
\(270\) 0 0
\(271\) 12.6014i 0.765482i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.3149i 0.622013i
\(276\) 0 0
\(277\) 11.4112i 0.685632i 0.939403 + 0.342816i \(0.111381\pi\)
−0.939403 + 0.342816i \(0.888619\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.4111i 0.859694i −0.902902 0.429847i \(-0.858568\pi\)
0.902902 0.429847i \(-0.141432\pi\)
\(282\) 0 0
\(283\) 1.27067 0.0755334 0.0377667 0.999287i \(-0.487976\pi\)
0.0377667 + 0.999287i \(0.487976\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −29.7519 −1.75620
\(288\) 0 0
\(289\) −8.55717 −0.503363
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.6014i 0.736183i 0.929790 + 0.368092i \(0.119989\pi\)
−0.929790 + 0.368092i \(0.880011\pi\)
\(294\) 0 0
\(295\) 1.78469 0.103909
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.592944i 0.0342908i
\(300\) 0 0
\(301\) 27.4222i 1.58059i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 29.3129 1.67845
\(306\) 0 0
\(307\) 30.1036i 1.71810i −0.511888 0.859052i \(-0.671054\pi\)
0.511888 0.859052i \(-0.328946\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.4913i 1.27536i 0.770300 + 0.637681i \(0.220106\pi\)
−0.770300 + 0.637681i \(0.779894\pi\)
\(312\) 0 0
\(313\) 1.74957i 0.0988917i −0.998777 0.0494459i \(-0.984254\pi\)
0.998777 0.0494459i \(-0.0157455\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.1405i 0.962708i −0.876526 0.481354i \(-0.840145\pi\)
0.876526 0.481354i \(-0.159855\pi\)
\(318\) 0 0
\(319\) 10.8801 0.609166
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.9614 0.609910
\(324\) 0 0
\(325\) 17.4038i 0.965387i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 22.5059i 1.24079i
\(330\) 0 0
\(331\) 12.7118i 0.698704i −0.936991 0.349352i \(-0.886402\pi\)
0.936991 0.349352i \(-0.113598\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 42.3713i 2.31499i
\(336\) 0 0
\(337\) −3.34572 −0.182253 −0.0911266 0.995839i \(-0.529047\pi\)
−0.0911266 + 0.995839i \(0.529047\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.53763i 0.408186i
\(342\) 0 0
\(343\) −27.0081 −1.45830
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.3087 0.607085 0.303543 0.952818i \(-0.401831\pi\)
0.303543 + 0.952818i \(0.401831\pi\)
\(348\) 0 0
\(349\) 6.81106i 0.364588i −0.983244 0.182294i \(-0.941648\pi\)
0.983244 0.182294i \(-0.0583522\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.02447i 0.107752i 0.998548 + 0.0538759i \(0.0171575\pi\)
−0.998548 + 0.0538759i \(0.982842\pi\)
\(354\) 0 0
\(355\) −16.6368 −0.882989
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.7973i 0.622640i 0.950305 + 0.311320i \(0.100771\pi\)
−0.950305 + 0.311320i \(0.899229\pi\)
\(360\) 0 0
\(361\) −4.76866 −0.250982
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.5986i 0.816467i
\(366\) 0 0
\(367\) −4.29029 −0.223952 −0.111976 0.993711i \(-0.535718\pi\)
−0.111976 + 0.993711i \(0.535718\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.2623 0.532792
\(372\) 0 0
\(373\) 0.419097i 0.0217000i 0.999941 + 0.0108500i \(0.00345373\pi\)
−0.999941 + 0.0108500i \(0.996546\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 18.3573 0.945449
\(378\) 0 0
\(379\) 0.455203i 0.0233822i −0.999932 0.0116911i \(-0.996279\pi\)
0.999932 0.0116911i \(-0.00372148\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.3395i 0.783812i −0.920005 0.391906i \(-0.871816\pi\)
0.920005 0.391906i \(-0.128184\pi\)
\(384\) 0 0
\(385\) 25.8127i 1.31554i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.4909 0.633315 0.316658 0.948540i \(-0.397439\pi\)
0.316658 + 0.948540i \(0.397439\pi\)
\(390\) 0 0
\(391\) 0.584770 0.0295731
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.932044i 0.0468962i
\(396\) 0 0
\(397\) −34.6895 −1.74102 −0.870509 0.492152i \(-0.836210\pi\)
−0.870509 + 0.492152i \(0.836210\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −38.4417 −1.91969 −0.959844 0.280534i \(-0.909489\pi\)
−0.959844 + 0.280534i \(0.909489\pi\)
\(402\) 0 0
\(403\) 12.7178i 0.633519i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −10.4671 −0.518837
\(408\) 0 0
\(409\) 24.0857 1.19096 0.595481 0.803370i \(-0.296962\pi\)
0.595481 + 0.803370i \(0.296962\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.41876 0.119019
\(414\) 0 0
\(415\) −11.7797 −0.578241
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.72636i 0.279751i 0.990169 + 0.139875i \(0.0446702\pi\)
−0.990169 + 0.139875i \(0.955330\pi\)
\(420\) 0 0
\(421\) 15.0108 0.731584 0.365792 0.930697i \(-0.380798\pi\)
0.365792 + 0.930697i \(0.380798\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.1639 0.832569
\(426\) 0 0
\(427\) 39.7274 1.92254
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.1945i 0.731891i 0.930636 + 0.365946i \(0.119254\pi\)
−0.930636 + 0.365946i \(0.880746\pi\)
\(432\) 0 0
\(433\) 13.1293 0.630954 0.315477 0.948933i \(-0.397835\pi\)
0.315477 + 0.948933i \(0.397835\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.759214 0.0363181
\(438\) 0 0
\(439\) 0.946508i 0.0451743i −0.999745 0.0225872i \(-0.992810\pi\)
0.999745 0.0225872i \(-0.00719033\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.4579 −0.876961 −0.438481 0.898741i \(-0.644483\pi\)
−0.438481 + 0.898741i \(0.644483\pi\)
\(444\) 0 0
\(445\) 12.6053i 0.597548i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.2668i 0.767677i 0.923400 + 0.383838i \(0.125398\pi\)
−0.923400 + 0.383838i \(0.874602\pi\)
\(450\) 0 0
\(451\) 11.6071i 0.546559i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 43.5523i 2.04176i
\(456\) 0 0
\(457\) 11.8957i 0.556460i 0.960515 + 0.278230i \(0.0897477\pi\)
−0.960515 + 0.278230i \(0.910252\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 31.3806i 1.46154i −0.682624 0.730770i \(-0.739161\pi\)
0.682624 0.730770i \(-0.260839\pi\)
\(462\) 0 0
\(463\) 35.6282i 1.65578i −0.560888 0.827891i \(-0.689540\pi\)
0.560888 0.827891i \(-0.310460\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.27776i 0.383049i −0.981488 0.191524i \(-0.938657\pi\)
0.981488 0.191524i \(-0.0613431\pi\)
\(468\) 0 0
\(469\) 57.4252i 2.65165i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.6983 0.491906
\(474\) 0 0
\(475\) 22.2840 1.02246
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.39416 0.292157 0.146078 0.989273i \(-0.453335\pi\)
0.146078 + 0.989273i \(0.453335\pi\)
\(480\) 0 0
\(481\) −17.6606 −0.805254
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 26.5216 1.20428
\(486\) 0 0
\(487\) 23.5822i 1.06861i 0.845292 + 0.534305i \(0.179426\pi\)
−0.845292 + 0.534305i \(0.820574\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.84530i 0.263795i 0.991263 + 0.131897i \(0.0421069\pi\)
−0.991263 + 0.131897i \(0.957893\pi\)
\(492\) 0 0
\(493\) 18.1042i 0.815374i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −22.5476 −1.01140
\(498\) 0 0
\(499\) 33.4438i 1.49715i −0.663051 0.748574i \(-0.730739\pi\)
0.663051 0.748574i \(-0.269261\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 12.0827i 0.538741i −0.963037 0.269371i \(-0.913184\pi\)
0.963037 0.269371i \(-0.0868157\pi\)
\(504\) 0 0
\(505\) 9.95555 0.443016
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 17.0315i 0.754908i 0.926028 + 0.377454i \(0.123200\pi\)
−0.926028 + 0.377454i \(0.876800\pi\)
\(510\) 0 0
\(511\) 21.1405i 0.935202i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.66285i 0.161405i
\(516\) 0 0
\(517\) −8.78026 −0.386155
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 41.9883 1.83954 0.919770 0.392457i \(-0.128375\pi\)
0.919770 + 0.392457i \(0.128375\pi\)
\(522\) 0 0
\(523\) −16.1413 −0.705810 −0.352905 0.935659i \(-0.614806\pi\)
−0.352905 + 0.935659i \(0.614806\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.5425 0.546360
\(528\) 0 0
\(529\) −22.9595 −0.998239
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 19.5840i 0.848279i
\(534\) 0 0
\(535\) 59.6968i 2.58092i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 22.7601i 0.980348i
\(540\) 0 0
\(541\) 10.0272i 0.431105i −0.976492 0.215552i \(-0.930845\pi\)
0.976492 0.215552i \(-0.0691552\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 52.4056i 2.24481i
\(546\) 0 0
\(547\) 41.1458i 1.75927i 0.475650 + 0.879635i \(0.342213\pi\)
−0.475650 + 0.879635i \(0.657787\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 23.5049i 1.00134i
\(552\) 0 0
\(553\) 1.26319i 0.0537162i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.92681i 0.378241i −0.981954 0.189120i \(-0.939436\pi\)
0.981954 0.189120i \(-0.0605637\pi\)
\(558\) 0 0
\(559\) 18.0505 0.763457
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.92539i 0.207581i −0.994599 0.103790i \(-0.966903\pi\)
0.994599 0.103790i \(-0.0330971\pi\)
\(564\) 0 0
\(565\) 37.5552 1.57996
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −36.2533 −1.51982 −0.759908 0.650030i \(-0.774756\pi\)
−0.759908 + 0.650030i \(0.774756\pi\)
\(570\) 0 0
\(571\) 7.17787i 0.300384i 0.988657 + 0.150192i \(0.0479892\pi\)
−0.988657 + 0.150192i \(0.952011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.18881 0.0495767
\(576\) 0 0
\(577\) 28.7645 1.19748 0.598741 0.800943i \(-0.295668\pi\)
0.598741 + 0.800943i \(0.295668\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −15.9648 −0.662331
\(582\) 0 0
\(583\) 4.00364i 0.165814i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.99976 0.0825390 0.0412695 0.999148i \(-0.486860\pi\)
0.0412695 + 0.999148i \(0.486860\pi\)
\(588\) 0 0
\(589\) 16.2841 0.670973
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.97200 0.163110 0.0815552 0.996669i \(-0.474011\pi\)
0.0815552 + 0.996669i \(0.474011\pi\)
\(594\) 0 0
\(595\) 42.9519 1.76086
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 48.6269i 1.98684i −0.114524 0.993420i \(-0.536534\pi\)
0.114524 0.993420i \(-0.463466\pi\)
\(600\) 0 0
\(601\) −17.9517 −0.732264 −0.366132 0.930563i \(-0.619318\pi\)
−0.366132 + 0.930563i \(0.619318\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −26.2581 −1.06754
\(606\) 0 0
\(607\) 32.2026i 1.30706i 0.756900 + 0.653531i \(0.226713\pi\)
−0.756900 + 0.653531i \(0.773287\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −14.8144 −0.599328
\(612\) 0 0
\(613\) −47.7854 −1.93003 −0.965017 0.262189i \(-0.915556\pi\)
−0.965017 + 0.262189i \(0.915556\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.53524i 0.102065i −0.998697 0.0510325i \(-0.983749\pi\)
0.998697 0.0510325i \(-0.0162512\pi\)
\(618\) 0 0
\(619\) 48.3473i 1.94324i −0.236542 0.971621i \(-0.576014\pi\)
0.236542 0.971621i \(-0.423986\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.0838i 0.684447i
\(624\) 0 0
\(625\) −19.6420 −0.785680
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 17.4172i 0.694467i
\(630\) 0 0
\(631\) 10.4366 0.415474 0.207737 0.978185i \(-0.433390\pi\)
0.207737 + 0.978185i \(0.433390\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 9.60748 0.381261
\(636\) 0 0
\(637\) 38.4019i 1.52154i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.677992 −0.0267791 −0.0133895 0.999910i \(-0.504262\pi\)
−0.0133895 + 0.999910i \(0.504262\pi\)
\(642\) 0 0
\(643\) 31.9191i 1.25876i 0.777096 + 0.629382i \(0.216692\pi\)
−0.777096 + 0.629382i \(0.783308\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.1454 1.14583 0.572913 0.819616i \(-0.305813\pi\)
0.572913 + 0.819616i \(0.305813\pi\)
\(648\) 0 0
\(649\) 0.943634i 0.0370409i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 38.1069i 1.49124i −0.666372 0.745619i \(-0.732154\pi\)
0.666372 0.745619i \(-0.267846\pi\)
\(654\) 0 0
\(655\) −7.30687 −0.285503
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 19.0667 0.742732 0.371366 0.928487i \(-0.378890\pi\)
0.371366 + 0.928487i \(0.378890\pi\)
\(660\) 0 0
\(661\) 16.2058i 0.630331i −0.949037 0.315166i \(-0.897940\pi\)
0.949037 0.315166i \(-0.102060\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 55.7649 2.16247
\(666\) 0 0
\(667\) 1.25394i 0.0485528i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 15.4989i 0.598328i
\(672\) 0 0
\(673\) 4.76689i 0.183750i 0.995771 + 0.0918750i \(0.0292860\pi\)
−0.995771 + 0.0918750i \(0.970714\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.5383i 1.09682i 0.836211 + 0.548408i \(0.184766\pi\)
−0.836211 + 0.548408i \(0.815234\pi\)
\(678\) 0 0
\(679\) 35.9443 1.37942
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.6812 1.02093 0.510464 0.859899i \(-0.329474\pi\)
0.510464 + 0.859899i \(0.329474\pi\)
\(684\) 0 0
\(685\) 67.0402i 2.56148i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 6.75511i 0.257349i
\(690\) 0 0
\(691\) 29.2893i 1.11422i −0.830440 0.557109i \(-0.811911\pi\)
0.830440 0.557109i \(-0.188089\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 43.0731i 1.63386i
\(696\) 0 0
\(697\) 19.3141 0.731573
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21.2439i 0.802370i 0.915997 + 0.401185i \(0.131402\pi\)
−0.915997 + 0.401185i \(0.868598\pi\)
\(702\) 0 0
\(703\) 22.6129i 0.852861i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.4926 0.507442
\(708\) 0 0
\(709\) 2.54912i 0.0957342i −0.998854 0.0478671i \(-0.984758\pi\)
0.998854 0.0478671i \(-0.0152424\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0.868722 0.0325339
\(714\) 0 0
\(715\) 16.9911 0.635431
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28.7150 −1.07089 −0.535445 0.844570i \(-0.679856\pi\)
−0.535445 + 0.844570i \(0.679856\pi\)
\(720\) 0 0
\(721\) 4.96421i 0.184877i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 36.8050i 1.36690i
\(726\) 0 0
\(727\) 36.0147i 1.33571i 0.744290 + 0.667856i \(0.232788\pi\)
−0.744290 + 0.667856i \(0.767212\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 17.8017i 0.658420i
\(732\) 0 0
\(733\) −9.00576 −0.332635 −0.166318 0.986072i \(-0.553188\pi\)
−0.166318 + 0.986072i \(0.553188\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −22.4034 −0.825238
\(738\) 0 0
\(739\) 41.7652i 1.53636i −0.640236 0.768179i \(-0.721163\pi\)
0.640236 0.768179i \(-0.278837\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 17.2569i 0.633093i 0.948577 + 0.316547i \(0.102523\pi\)
−0.948577 + 0.316547i \(0.897477\pi\)
\(744\) 0 0
\(745\) 6.53143 0.239293
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 80.9063i 2.95625i
\(750\) 0 0
\(751\) 18.6483i 0.680487i 0.940337 + 0.340244i \(0.110510\pi\)
−0.940337 + 0.340244i \(0.889490\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.6194i 0.895992i
\(756\) 0 0
\(757\) −17.8340 −0.648186 −0.324093 0.946025i \(-0.605059\pi\)
−0.324093 + 0.946025i \(0.605059\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 25.4205i 0.921492i 0.887532 + 0.460746i \(0.152418\pi\)
−0.887532 + 0.460746i \(0.847582\pi\)
\(762\) 0 0
\(763\) 71.0246i 2.57126i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.59214i 0.0574888i
\(768\) 0 0
\(769\) 13.8535i 0.499569i 0.968301 + 0.249784i \(0.0803598\pi\)
−0.968301 + 0.249784i \(0.919640\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −26.0966 −0.938631 −0.469315 0.883031i \(-0.655499\pi\)
−0.469315 + 0.883031i \(0.655499\pi\)
\(774\) 0 0
\(775\) 25.4983 0.915924
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 25.0757 0.898429
\(780\) 0 0
\(781\) 8.79652i 0.314764i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 26.0611 0.930162
\(786\) 0 0
\(787\) 5.19066i 0.185027i 0.995711 + 0.0925136i \(0.0294902\pi\)
−0.995711 + 0.0925136i \(0.970510\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 50.8981 1.80973
\(792\) 0 0
\(793\) 26.1504i 0.928627i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.5481 −1.04665 −0.523324 0.852134i \(-0.675308\pi\)
−0.523324 + 0.852134i \(0.675308\pi\)
\(798\) 0 0