Properties

Label 6012.2.h.a.3005.11
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 3005.11
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.12

$q$-expansion

\(f(q)\) \(=\) \(q-2.76473 q^{5} +0.618008 q^{7} +O(q^{10})\) \(q-2.76473 q^{5} +0.618008 q^{7} -5.51365i q^{11} -2.58896i q^{13} +5.12683 q^{17} -4.09998 q^{19} +5.83852 q^{23} +2.64374 q^{25} -0.795676i q^{29} +4.82779 q^{31} -1.70862 q^{35} +1.19611i q^{37} +8.20721 q^{41} +1.84848i q^{43} -1.33447i q^{47} -6.61807 q^{49} +4.14070 q^{53} +15.2438i q^{55} +4.91196 q^{59} -14.6371 q^{61} +7.15778i q^{65} -1.39489i q^{67} -2.37783 q^{71} +5.69062i q^{73} -3.40748i q^{77} -7.74996i q^{79} -0.0266917 q^{83} -14.1743 q^{85} -8.65105i q^{89} -1.60000i q^{91} +11.3353 q^{95} -5.26085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) −2.76473 −1.23643 −0.618213 0.786011i \(-0.712143\pi\)
−0.618213 + 0.786011i \(0.712143\pi\)
\(6\) 0 0
\(7\) 0.618008 0.233585 0.116792 0.993156i \(-0.462739\pi\)
0.116792 + 0.993156i \(0.462739\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.51365i 1.66243i −0.555951 0.831215i \(-0.687646\pi\)
0.555951 0.831215i \(-0.312354\pi\)
\(12\) 0 0
\(13\) 2.58896i 0.718049i −0.933328 0.359024i \(-0.883110\pi\)
0.933328 0.359024i \(-0.116890\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.12683 1.24344 0.621720 0.783240i \(-0.286434\pi\)
0.621720 + 0.783240i \(0.286434\pi\)
\(18\) 0 0
\(19\) −4.09998 −0.940600 −0.470300 0.882507i \(-0.655854\pi\)
−0.470300 + 0.882507i \(0.655854\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.83852 1.21741 0.608707 0.793395i \(-0.291688\pi\)
0.608707 + 0.793395i \(0.291688\pi\)
\(24\) 0 0
\(25\) 2.64374 0.528747
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.795676i 0.147753i −0.997267 0.0738767i \(-0.976463\pi\)
0.997267 0.0738767i \(-0.0235371\pi\)
\(30\) 0 0
\(31\) 4.82779 0.867097 0.433549 0.901130i \(-0.357261\pi\)
0.433549 + 0.901130i \(0.357261\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.70862 −0.288810
\(36\) 0 0
\(37\) 1.19611i 0.196639i 0.995155 + 0.0983193i \(0.0313467\pi\)
−0.995155 + 0.0983193i \(0.968653\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.20721 1.28175 0.640875 0.767645i \(-0.278572\pi\)
0.640875 + 0.767645i \(0.278572\pi\)
\(42\) 0 0
\(43\) 1.84848i 0.281890i 0.990017 + 0.140945i \(0.0450141\pi\)
−0.990017 + 0.140945i \(0.954986\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.33447i 0.194653i −0.995253 0.0973263i \(-0.968971\pi\)
0.995253 0.0973263i \(-0.0310290\pi\)
\(48\) 0 0
\(49\) −6.61807 −0.945438
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.14070 0.568769 0.284385 0.958710i \(-0.408211\pi\)
0.284385 + 0.958710i \(0.408211\pi\)
\(54\) 0 0
\(55\) 15.2438i 2.05547i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.91196 0.639483 0.319741 0.947505i \(-0.396404\pi\)
0.319741 + 0.947505i \(0.396404\pi\)
\(60\) 0 0
\(61\) −14.6371 −1.87409 −0.937047 0.349202i \(-0.886453\pi\)
−0.937047 + 0.349202i \(0.886453\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.15778i 0.887813i
\(66\) 0 0
\(67\) 1.39489i 0.170412i −0.996363 0.0852062i \(-0.972845\pi\)
0.996363 0.0852062i \(-0.0271549\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.37783 −0.282196 −0.141098 0.989996i \(-0.545063\pi\)
−0.141098 + 0.989996i \(0.545063\pi\)
\(72\) 0 0
\(73\) 5.69062i 0.666037i 0.942920 + 0.333018i \(0.108067\pi\)
−0.942920 + 0.333018i \(0.891933\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.40748i 0.388318i
\(78\) 0 0
\(79\) 7.74996i 0.871939i −0.899961 0.435970i \(-0.856406\pi\)
0.899961 0.435970i \(-0.143594\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −0.0266917 −0.00292979 −0.00146490 0.999999i \(-0.500466\pi\)
−0.00146490 + 0.999999i \(0.500466\pi\)
\(84\) 0 0
\(85\) −14.1743 −1.53742
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.65105i 0.917009i −0.888692 0.458505i \(-0.848385\pi\)
0.888692 0.458505i \(-0.151615\pi\)
\(90\) 0 0
\(91\) 1.60000i 0.167725i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3353 1.16298
\(96\) 0 0
\(97\) −5.26085 −0.534158 −0.267079 0.963675i \(-0.586059\pi\)
−0.267079 + 0.963675i \(0.586059\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.49685 −0.347949 −0.173975 0.984750i \(-0.555661\pi\)
−0.173975 + 0.984750i \(0.555661\pi\)
\(102\) 0 0
\(103\) 0.704239i 0.0693907i 0.999398 + 0.0346954i \(0.0110461\pi\)
−0.999398 + 0.0346954i \(0.988954\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.27785i 0.413555i −0.978388 0.206778i \(-0.933702\pi\)
0.978388 0.206778i \(-0.0662977\pi\)
\(108\) 0 0
\(109\) 11.3752i 1.08955i −0.838583 0.544773i \(-0.816616\pi\)
0.838583 0.544773i \(-0.183384\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.2226 1.14980 0.574902 0.818222i \(-0.305040\pi\)
0.574902 + 0.818222i \(0.305040\pi\)
\(114\) 0 0
\(115\) −16.1419 −1.50524
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.16842 0.290449
\(120\) 0 0
\(121\) −19.4004 −1.76367
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.51444 0.582669
\(126\) 0 0
\(127\) 6.02121 0.534296 0.267148 0.963656i \(-0.413919\pi\)
0.267148 + 0.963656i \(0.413919\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −19.3564 −1.69118 −0.845588 0.533836i \(-0.820750\pi\)
−0.845588 + 0.533836i \(0.820750\pi\)
\(132\) 0 0
\(133\) −2.53382 −0.219710
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 21.6540i 1.85003i 0.379931 + 0.925015i \(0.375948\pi\)
−0.379931 + 0.925015i \(0.624052\pi\)
\(138\) 0 0
\(139\) 7.89202i 0.669393i 0.942326 + 0.334696i \(0.108634\pi\)
−0.942326 + 0.334696i \(0.891366\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −14.2746 −1.19371
\(144\) 0 0
\(145\) 2.19983i 0.182686i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.6114 −1.03316 −0.516582 0.856238i \(-0.672796\pi\)
−0.516582 + 0.856238i \(0.672796\pi\)
\(150\) 0 0
\(151\) 13.8446i 1.12666i −0.826233 0.563328i \(-0.809521\pi\)
0.826233 0.563328i \(-0.190479\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −13.3475 −1.07210
\(156\) 0 0
\(157\) 6.78446 0.541459 0.270730 0.962655i \(-0.412735\pi\)
0.270730 + 0.962655i \(0.412735\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.60825 0.284370
\(162\) 0 0
\(163\) 10.6704i 0.835771i −0.908500 0.417886i \(-0.862771\pi\)
0.908500 0.417886i \(-0.137229\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.30520 12.1846i −0.333147 0.942875i
\(168\) 0 0
\(169\) 6.29728 0.484406
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.1637i 1.15288i −0.817140 0.576439i \(-0.804442\pi\)
0.817140 0.576439i \(-0.195558\pi\)
\(174\) 0 0
\(175\) 1.63385 0.123507
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.202535i 0.0151382i 0.999971 + 0.00756908i \(0.00240934\pi\)
−0.999971 + 0.00756908i \(0.997591\pi\)
\(180\) 0 0
\(181\) 10.3582 0.769918 0.384959 0.922934i \(-0.374216\pi\)
0.384959 + 0.922934i \(0.374216\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.30691i 0.243129i
\(186\) 0 0
\(187\) 28.2676i 2.06713i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.7071i 0.991813i −0.868376 0.495906i \(-0.834836\pi\)
0.868376 0.495906i \(-0.165164\pi\)
\(192\) 0 0
\(193\) 9.11688i 0.656247i 0.944635 + 0.328124i \(0.106416\pi\)
−0.944635 + 0.328124i \(0.893584\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −12.2003 −0.869234 −0.434617 0.900615i \(-0.643116\pi\)
−0.434617 + 0.900615i \(0.643116\pi\)
\(198\) 0 0
\(199\) 2.79531 0.198154 0.0990770 0.995080i \(-0.468411\pi\)
0.0990770 + 0.995080i \(0.468411\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.491734i 0.0345129i
\(204\) 0 0
\(205\) −22.6907 −1.58479
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.6059i 1.56368i
\(210\) 0 0
\(211\) −10.0550 −0.692216 −0.346108 0.938195i \(-0.612497\pi\)
−0.346108 + 0.938195i \(0.612497\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.11055i 0.348536i
\(216\) 0 0
\(217\) 2.98361 0.202541
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.2732i 0.892850i
\(222\) 0 0
\(223\) 15.9618 1.06888 0.534442 0.845205i \(-0.320522\pi\)
0.534442 + 0.845205i \(0.320522\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.7386 1.17736 0.588678 0.808368i \(-0.299649\pi\)
0.588678 + 0.808368i \(0.299649\pi\)
\(228\) 0 0
\(229\) −3.06625 −0.202624 −0.101312 0.994855i \(-0.532304\pi\)
−0.101312 + 0.994855i \(0.532304\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 27.3615i 1.79251i −0.443535 0.896257i \(-0.646276\pi\)
0.443535 0.896257i \(-0.353724\pi\)
\(234\) 0 0
\(235\) 3.68945i 0.240673i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 11.6002i 0.750354i 0.926953 + 0.375177i \(0.122418\pi\)
−0.926953 + 0.375177i \(0.877582\pi\)
\(240\) 0 0
\(241\) 21.6048i 1.39169i −0.718192 0.695845i \(-0.755030\pi\)
0.718192 0.695845i \(-0.244970\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 18.2972 1.16896
\(246\) 0 0
\(247\) 10.6147i 0.675396i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.54576i 0.539404i −0.962944 0.269702i \(-0.913075\pi\)
0.962944 0.269702i \(-0.0869251\pi\)
\(252\) 0 0
\(253\) 32.1916i 2.02387i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.15043 0.383653 0.191827 0.981429i \(-0.438559\pi\)
0.191827 + 0.981429i \(0.438559\pi\)
\(258\) 0 0
\(259\) 0.739203i 0.0459318i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.6243i 1.39507i −0.716549 0.697537i \(-0.754279\pi\)
0.716549 0.697537i \(-0.245721\pi\)
\(264\) 0 0
\(265\) −11.4479 −0.703241
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.2318 −1.11161 −0.555807 0.831312i \(-0.687591\pi\)
−0.555807 + 0.831312i \(0.687591\pi\)
\(270\) 0 0
\(271\) 3.06817i 0.186378i −0.995648 0.0931891i \(-0.970294\pi\)
0.995648 0.0931891i \(-0.0297061\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.5766i 0.879005i
\(276\) 0 0
\(277\) 1.76542i 0.106074i −0.998593 0.0530368i \(-0.983110\pi\)
0.998593 0.0530368i \(-0.0168901\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 18.2786i 1.09041i −0.838303 0.545205i \(-0.816452\pi\)
0.838303 0.545205i \(-0.183548\pi\)
\(282\) 0 0
\(283\) −27.0531 −1.60814 −0.804070 0.594535i \(-0.797336\pi\)
−0.804070 + 0.594535i \(0.797336\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.07212 0.299398
\(288\) 0 0
\(289\) 9.28439 0.546141
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.2388i 1.00710i 0.863965 + 0.503552i \(0.167974\pi\)
−0.863965 + 0.503552i \(0.832026\pi\)
\(294\) 0 0
\(295\) −13.5802 −0.790673
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.1157i 0.874163i
\(300\) 0 0
\(301\) 1.14237i 0.0658453i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 40.4678 2.31718
\(306\) 0 0
\(307\) 16.6785i 0.951891i 0.879475 + 0.475945i \(0.157894\pi\)
−0.879475 + 0.475945i \(0.842106\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.38908i 0.475701i 0.971302 + 0.237851i \(0.0764429\pi\)
−0.971302 + 0.237851i \(0.923557\pi\)
\(312\) 0 0
\(313\) 25.7289i 1.45429i 0.686486 + 0.727143i \(0.259152\pi\)
−0.686486 + 0.727143i \(0.740848\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.35143i 0.469063i −0.972109 0.234532i \(-0.924644\pi\)
0.972109 0.234532i \(-0.0753556\pi\)
\(318\) 0 0
\(319\) −4.38708 −0.245630
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −21.0199 −1.16958
\(324\) 0 0
\(325\) 6.84453i 0.379666i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0.824713i 0.0454679i
\(330\) 0 0
\(331\) 28.1622i 1.54794i −0.633225 0.773968i \(-0.718269\pi\)
0.633225 0.773968i \(-0.281731\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.85648i 0.210702i
\(336\) 0 0
\(337\) −15.7525 −0.858092 −0.429046 0.903283i \(-0.641150\pi\)
−0.429046 + 0.903283i \(0.641150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 26.6188i 1.44149i
\(342\) 0 0
\(343\) −8.41607 −0.454425
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.6851 −0.680972 −0.340486 0.940250i \(-0.610592\pi\)
−0.340486 + 0.940250i \(0.610592\pi\)
\(348\) 0 0
\(349\) 7.51912i 0.402489i 0.979541 + 0.201245i \(0.0644986\pi\)
−0.979541 + 0.201245i \(0.935501\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.8355i 1.58798i −0.607928 0.793992i \(-0.707999\pi\)
0.607928 0.793992i \(-0.292001\pi\)
\(354\) 0 0
\(355\) 6.57405 0.348914
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.29188i 0.384851i 0.981312 + 0.192425i \(0.0616353\pi\)
−0.981312 + 0.192425i \(0.938365\pi\)
\(360\) 0 0
\(361\) −2.19016 −0.115272
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 15.7330i 0.823504i
\(366\) 0 0
\(367\) 1.45949 0.0761846 0.0380923 0.999274i \(-0.487872\pi\)
0.0380923 + 0.999274i \(0.487872\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.55899 0.132856
\(372\) 0 0
\(373\) 11.4525i 0.592986i 0.955035 + 0.296493i \(0.0958172\pi\)
−0.955035 + 0.296493i \(0.904183\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.05997 −0.106094
\(378\) 0 0
\(379\) 28.7140i 1.47494i −0.675379 0.737471i \(-0.736020\pi\)
0.675379 0.737471i \(-0.263980\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.96528i 0.458104i −0.973414 0.229052i \(-0.926437\pi\)
0.973414 0.229052i \(-0.0735626\pi\)
\(384\) 0 0
\(385\) 9.42077i 0.480127i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −23.0490 −1.16863 −0.584315 0.811527i \(-0.698637\pi\)
−0.584315 + 0.811527i \(0.698637\pi\)
\(390\) 0 0
\(391\) 29.9331 1.51378
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 21.4266i 1.07809i
\(396\) 0 0
\(397\) 6.18808 0.310571 0.155285 0.987870i \(-0.450370\pi\)
0.155285 + 0.987870i \(0.450370\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.7599 1.08664 0.543319 0.839527i \(-0.317168\pi\)
0.543319 + 0.839527i \(0.317168\pi\)
\(402\) 0 0
\(403\) 12.4990i 0.622618i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.59492 0.326898
\(408\) 0 0
\(409\) −31.0725 −1.53644 −0.768218 0.640189i \(-0.778856\pi\)
−0.768218 + 0.640189i \(0.778856\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.03563 0.149374
\(414\) 0 0
\(415\) 0.0737954 0.00362247
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 29.5123i 1.44177i 0.693054 + 0.720885i \(0.256265\pi\)
−0.693054 + 0.720885i \(0.743735\pi\)
\(420\) 0 0
\(421\) −22.4021 −1.09181 −0.545907 0.837846i \(-0.683815\pi\)
−0.545907 + 0.837846i \(0.683815\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.5540 0.657465
\(426\) 0 0
\(427\) −9.04587 −0.437760
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.69219i 0.322351i −0.986926 0.161176i \(-0.948471\pi\)
0.986926 0.161176i \(-0.0515286\pi\)
\(432\) 0 0
\(433\) −21.8508 −1.05008 −0.525041 0.851077i \(-0.675950\pi\)
−0.525041 + 0.851077i \(0.675950\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −23.9378 −1.14510
\(438\) 0 0
\(439\) 12.3049i 0.587283i −0.955916 0.293641i \(-0.905133\pi\)
0.955916 0.293641i \(-0.0948672\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.3204 −0.870429 −0.435215 0.900327i \(-0.643328\pi\)
−0.435215 + 0.900327i \(0.643328\pi\)
\(444\) 0 0
\(445\) 23.9178i 1.13381i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 15.1799i 0.716382i 0.933648 + 0.358191i \(0.116606\pi\)
−0.933648 + 0.358191i \(0.883394\pi\)
\(450\) 0 0
\(451\) 45.2517i 2.13082i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.42356i 0.207380i
\(456\) 0 0
\(457\) 4.65645i 0.217819i 0.994052 + 0.108910i \(0.0347359\pi\)
−0.994052 + 0.108910i \(0.965264\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 32.4248i 1.51017i −0.655626 0.755086i \(-0.727595\pi\)
0.655626 0.755086i \(-0.272405\pi\)
\(462\) 0 0
\(463\) 23.2195i 1.07910i 0.841953 + 0.539551i \(0.181406\pi\)
−0.841953 + 0.539551i \(0.818594\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.2354i 1.16776i 0.811841 + 0.583878i \(0.198465\pi\)
−0.811841 + 0.583878i \(0.801535\pi\)
\(468\) 0 0
\(469\) 0.862050i 0.0398058i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.1919 0.468623
\(474\) 0 0
\(475\) −10.8393 −0.497340
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −39.7086 −1.81433 −0.907166 0.420773i \(-0.861759\pi\)
−0.907166 + 0.420773i \(0.861759\pi\)
\(480\) 0 0
\(481\) 3.09667 0.141196
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.5448 0.660447
\(486\) 0 0
\(487\) 28.8653i 1.30801i 0.756490 + 0.654005i \(0.226913\pi\)
−0.756490 + 0.654005i \(0.773087\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 14.1667i 0.639336i 0.947530 + 0.319668i \(0.103571\pi\)
−0.947530 + 0.319668i \(0.896429\pi\)
\(492\) 0 0
\(493\) 4.07930i 0.183722i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.46951 −0.0659167
\(498\) 0 0
\(499\) 0.585757i 0.0262221i 0.999914 + 0.0131110i \(0.00417349\pi\)
−0.999914 + 0.0131110i \(0.995827\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.21259i 0.410769i 0.978681 + 0.205385i \(0.0658445\pi\)
−0.978681 + 0.205385i \(0.934155\pi\)
\(504\) 0 0
\(505\) 9.66784 0.430213
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 18.8442i 0.835253i 0.908619 + 0.417626i \(0.137138\pi\)
−0.908619 + 0.417626i \(0.862862\pi\)
\(510\) 0 0
\(511\) 3.51685i 0.155576i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.94703i 0.0857965i
\(516\) 0 0
\(517\) −7.35781 −0.323596
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18.7521 −0.821546 −0.410773 0.911738i \(-0.634741\pi\)
−0.410773 + 0.911738i \(0.634741\pi\)
\(522\) 0 0
\(523\) 4.82149 0.210829 0.105415 0.994428i \(-0.466383\pi\)
0.105415 + 0.994428i \(0.466383\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.7513 1.07818
\(528\) 0 0
\(529\) 11.0883 0.482099
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 21.2481i 0.920359i
\(534\) 0 0
\(535\) 11.8271i 0.511330i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.4897i 1.57172i
\(540\) 0 0
\(541\) 0.602251i 0.0258928i 0.999916 + 0.0129464i \(0.00412108\pi\)
−0.999916 + 0.0129464i \(0.995879\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 31.4494i 1.34714i
\(546\) 0 0
\(547\) 8.33181i 0.356242i −0.984009 0.178121i \(-0.942998\pi\)
0.984009 0.178121i \(-0.0570019\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.26226i 0.138977i
\(552\) 0 0
\(553\) 4.78954i 0.203672i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25.3948i 1.07601i −0.842941 0.538005i \(-0.819178\pi\)
0.842941 0.538005i \(-0.180822\pi\)
\(558\) 0 0
\(559\) 4.78564 0.202411
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.239030i 0.0100739i −0.999987 0.00503696i \(-0.998397\pi\)
0.999987 0.00503696i \(-0.00160332\pi\)
\(564\) 0 0
\(565\) −33.7922 −1.42165
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.16148 0.132536 0.0662680 0.997802i \(-0.478891\pi\)
0.0662680 + 0.997802i \(0.478891\pi\)
\(570\) 0 0
\(571\) 46.4685i 1.94465i 0.233640 + 0.972323i \(0.424936\pi\)
−0.233640 + 0.972323i \(0.575064\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 15.4355 0.643705
\(576\) 0 0
\(577\) 9.69066 0.403428 0.201714 0.979445i \(-0.435349\pi\)
0.201714 + 0.979445i \(0.435349\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −0.0164957 −0.000684356
\(582\) 0 0
\(583\) 22.8304i 0.945539i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 42.3954 1.74985 0.874924 0.484261i \(-0.160911\pi\)
0.874924 + 0.484261i \(0.160911\pi\)
\(588\) 0 0
\(589\) −19.7939 −0.815592
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 12.7221 0.522435 0.261218 0.965280i \(-0.415876\pi\)
0.261218 + 0.965280i \(0.415876\pi\)
\(594\) 0 0
\(595\) −8.75983 −0.359118
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 30.2589i 1.23635i −0.786042 0.618173i \(-0.787873\pi\)
0.786042 0.618173i \(-0.212127\pi\)
\(600\) 0 0
\(601\) −13.6377 −0.556293 −0.278146 0.960539i \(-0.589720\pi\)
−0.278146 + 0.960539i \(0.589720\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 53.6369 2.18065
\(606\) 0 0
\(607\) 21.7351i 0.882201i −0.897458 0.441101i \(-0.854588\pi\)
0.897458 0.441101i \(-0.145412\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.45489 −0.139770
\(612\) 0 0
\(613\) −20.4741 −0.826943 −0.413471 0.910517i \(-0.635684\pi\)
−0.413471 + 0.910517i \(0.635684\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.2770i 0.776061i 0.921646 + 0.388031i \(0.126845\pi\)
−0.921646 + 0.388031i \(0.873155\pi\)
\(618\) 0 0
\(619\) 25.9858i 1.04446i 0.852805 + 0.522229i \(0.174899\pi\)
−0.852805 + 0.522229i \(0.825101\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.34641i 0.214200i
\(624\) 0 0
\(625\) −31.2293 −1.24917
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.13223i 0.244508i
\(630\) 0 0
\(631\) −13.7377 −0.546891 −0.273445 0.961888i \(-0.588163\pi\)
−0.273445 + 0.961888i \(0.588163\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.6470 −0.660617
\(636\) 0 0
\(637\) 17.1339i 0.678870i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.9736 0.709915 0.354957 0.934882i \(-0.384495\pi\)
0.354957 + 0.934882i \(0.384495\pi\)
\(642\) 0 0
\(643\) 39.1122i 1.54243i 0.636573 + 0.771216i \(0.280351\pi\)
−0.636573 + 0.771216i \(0.719649\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −20.8207 −0.818547 −0.409274 0.912412i \(-0.634218\pi\)
−0.409274 + 0.912412i \(0.634218\pi\)
\(648\) 0 0
\(649\) 27.0829i 1.06310i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.1623i 0.867278i −0.901087 0.433639i \(-0.857229\pi\)
0.901087 0.433639i \(-0.142771\pi\)
\(654\) 0 0
\(655\) 53.5152 2.09101
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −6.16750 −0.240252 −0.120126 0.992759i \(-0.538330\pi\)
−0.120126 + 0.992759i \(0.538330\pi\)
\(660\) 0 0
\(661\) 35.0646i 1.36385i −0.731420 0.681927i \(-0.761142\pi\)
0.731420 0.681927i \(-0.238858\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.00533 0.271655
\(666\) 0 0
\(667\) 4.64557i 0.179877i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 80.7042i 3.11555i
\(672\) 0 0
\(673\) 19.9031i 0.767206i 0.923498 + 0.383603i \(0.125317\pi\)
−0.923498 + 0.383603i \(0.874683\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.4258i 0.592861i −0.955054 0.296431i \(-0.904204\pi\)
0.955054 0.296431i \(-0.0957963\pi\)
\(678\) 0 0
\(679\) −3.25124 −0.124771
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 35.6351 1.36354 0.681770 0.731567i \(-0.261211\pi\)
0.681770 + 0.731567i \(0.261211\pi\)
\(684\) 0 0
\(685\) 59.8676i 2.28742i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.7201i 0.408404i
\(690\) 0 0
\(691\) 19.3096i 0.734573i 0.930108 + 0.367287i \(0.119713\pi\)
−0.930108 + 0.367287i \(0.880287\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 21.8193i 0.827654i
\(696\) 0 0
\(697\) 42.0770 1.59378
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.57426i 0.210537i −0.994444 0.105269i \(-0.966430\pi\)
0.994444 0.105269i \(-0.0335702\pi\)
\(702\) 0 0
\(703\) 4.90401i 0.184958i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.16108 −0.0812757
\(708\) 0 0
\(709\) 7.79896i 0.292896i 0.989218 + 0.146448i \(0.0467841\pi\)
−0.989218 + 0.146448i \(0.953216\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 28.1871 1.05562
\(714\) 0 0
\(715\) 39.4655 1.47593
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.766739 0.0285946 0.0142973 0.999898i \(-0.495449\pi\)
0.0142973 + 0.999898i \(0.495449\pi\)
\(720\) 0 0
\(721\) 0.435225i 0.0162086i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.10356i 0.0781242i
\(726\) 0 0
\(727\) 19.4239i 0.720391i 0.932877 + 0.360196i \(0.117290\pi\)
−0.932877 + 0.360196i \(0.882710\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.47684i 0.350514i
\(732\) 0 0
\(733\) −3.26000 −0.120411 −0.0602053 0.998186i \(-0.519176\pi\)
−0.0602053 + 0.998186i \(0.519176\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.69092 −0.283299
\(738\) 0 0
\(739\) 29.4351i 1.08279i 0.840769 + 0.541394i \(0.182103\pi\)
−0.840769 + 0.541394i \(0.817897\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.35428i 0.0496838i −0.999691 0.0248419i \(-0.992092\pi\)
0.999691 0.0248419i \(-0.00790824\pi\)
\(744\) 0 0
\(745\) 34.8670 1.27743
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.64374i 0.0966003i
\(750\) 0 0
\(751\) 9.26564i 0.338108i −0.985607 0.169054i \(-0.945929\pi\)
0.985607 0.169054i \(-0.0540712\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 38.2766i 1.39303i
\(756\) 0 0
\(757\) −4.77283 −0.173471 −0.0867357 0.996231i \(-0.527644\pi\)
−0.0867357 + 0.996231i \(0.527644\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.4481i 1.46624i −0.680098 0.733121i \(-0.738063\pi\)
0.680098 0.733121i \(-0.261937\pi\)
\(762\) 0 0
\(763\) 7.02996i 0.254502i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.7169i 0.459180i
\(768\) 0 0
\(769\) 36.9631i 1.33292i −0.745540 0.666461i \(-0.767808\pi\)
0.745540 0.666461i \(-0.232192\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.7701 −1.53833 −0.769167 0.639048i \(-0.779328\pi\)
−0.769167 + 0.639048i \(0.779328\pi\)
\(774\) 0 0
\(775\) 12.7634 0.458475
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −33.6494 −1.20561
\(780\) 0 0
\(781\) 13.1105i 0.469131i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −18.7572 −0.669474
\(786\) 0 0
\(787\) 10.8862i 0.388052i 0.980996 + 0.194026i \(0.0621546\pi\)
−0.980996 + 0.194026i \(0.937845\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 7.55365 0.268577
\(792\) 0 0
\(793\) 37.8950i 1.34569i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.3384 1.35802 0.679008 0.734130i \(-0.262410\pi\)
0.679008 + 0.734130i \(0.262410\pi\)
\(798\) 0 0