Properties

Label 8280.2.p.a.1241.39
Level $8280$
Weight $2$
Character 8280.1241
Analytic conductor $66.116$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8280.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(66.1161328736\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.39
Character \(\chi\) \(=\) 8280.1241
Dual form 8280.2.p.a.1241.10

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{5} +2.84853i q^{7} +O(q^{10})\) \(q-1.00000 q^{5} +2.84853i q^{7} -2.95839 q^{11} -0.451593 q^{13} +4.38605 q^{17} -7.84690i q^{19} +(3.10541 + 3.65464i) q^{23} +1.00000 q^{25} +3.01776i q^{29} -6.09850 q^{31} -2.84853i q^{35} -0.619475i q^{37} -5.35531i q^{41} +3.75365i q^{43} +5.23505i q^{47} -1.11412 q^{49} +14.0102 q^{53} +2.95839 q^{55} -11.8371i q^{59} +9.81751i q^{61} +0.451593 q^{65} +12.3590i q^{67} -13.6631i q^{71} +1.07832 q^{73} -8.42705i q^{77} +8.71942i q^{79} +3.20932 q^{83} -4.38605 q^{85} -3.64156 q^{89} -1.28638i q^{91} +7.84690i q^{95} +8.46148i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 48 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times\).

\(n\) \(1657\) \(2071\) \(3961\) \(4141\) \(4601\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(-1\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.84853i 1.07664i 0.842740 + 0.538322i \(0.180941\pi\)
−0.842740 + 0.538322i \(0.819059\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.95839 −0.891987 −0.445994 0.895036i \(-0.647150\pi\)
−0.445994 + 0.895036i \(0.647150\pi\)
\(12\) 0 0
\(13\) −0.451593 −0.125249 −0.0626246 0.998037i \(-0.519947\pi\)
−0.0626246 + 0.998037i \(0.519947\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.38605 1.06377 0.531887 0.846815i \(-0.321483\pi\)
0.531887 + 0.846815i \(0.321483\pi\)
\(18\) 0 0
\(19\) 7.84690i 1.80020i −0.435681 0.900101i \(-0.643492\pi\)
0.435681 0.900101i \(-0.356508\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.10541 + 3.65464i 0.647523 + 0.762046i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.01776i 0.560383i 0.959944 + 0.280192i \(0.0903980\pi\)
−0.959944 + 0.280192i \(0.909602\pi\)
\(30\) 0 0
\(31\) −6.09850 −1.09532 −0.547662 0.836700i \(-0.684482\pi\)
−0.547662 + 0.836700i \(0.684482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.84853i 0.481489i
\(36\) 0 0
\(37\) 0.619475i 0.101841i −0.998703 0.0509205i \(-0.983784\pi\)
0.998703 0.0509205i \(-0.0162155\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.35531i 0.836359i −0.908364 0.418180i \(-0.862668\pi\)
0.908364 0.418180i \(-0.137332\pi\)
\(42\) 0 0
\(43\) 3.75365i 0.572426i 0.958166 + 0.286213i \(0.0923965\pi\)
−0.958166 + 0.286213i \(0.907604\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.23505i 0.763610i 0.924243 + 0.381805i \(0.124697\pi\)
−0.924243 + 0.381805i \(0.875303\pi\)
\(48\) 0 0
\(49\) −1.11412 −0.159160
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 14.0102 1.92445 0.962227 0.272248i \(-0.0877671\pi\)
0.962227 + 0.272248i \(0.0877671\pi\)
\(54\) 0 0
\(55\) 2.95839 0.398909
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.8371i 1.54106i −0.637404 0.770530i \(-0.719992\pi\)
0.637404 0.770530i \(-0.280008\pi\)
\(60\) 0 0
\(61\) 9.81751i 1.25700i 0.777808 + 0.628502i \(0.216332\pi\)
−0.777808 + 0.628502i \(0.783668\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.451593 0.0560132
\(66\) 0 0
\(67\) 12.3590i 1.50990i 0.655785 + 0.754948i \(0.272338\pi\)
−0.655785 + 0.754948i \(0.727662\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.6631i 1.62151i −0.585385 0.810756i \(-0.699056\pi\)
0.585385 0.810756i \(-0.300944\pi\)
\(72\) 0 0
\(73\) 1.07832 0.126207 0.0631037 0.998007i \(-0.479900\pi\)
0.0631037 + 0.998007i \(0.479900\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.42705i 0.960352i
\(78\) 0 0
\(79\) 8.71942i 0.981011i 0.871438 + 0.490506i \(0.163188\pi\)
−0.871438 + 0.490506i \(0.836812\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.20932 0.352269 0.176134 0.984366i \(-0.443641\pi\)
0.176134 + 0.984366i \(0.443641\pi\)
\(84\) 0 0
\(85\) −4.38605 −0.475734
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.64156 −0.386004 −0.193002 0.981198i \(-0.561822\pi\)
−0.193002 + 0.981198i \(0.561822\pi\)
\(90\) 0 0
\(91\) 1.28638i 0.134849i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.84690i 0.805075i
\(96\) 0 0
\(97\) 8.46148i 0.859133i 0.903035 + 0.429567i \(0.141334\pi\)
−0.903035 + 0.429567i \(0.858666\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.2407i 1.01899i 0.860474 + 0.509495i \(0.170168\pi\)
−0.860474 + 0.509495i \(0.829832\pi\)
\(102\) 0 0
\(103\) 8.25352i 0.813244i −0.913597 0.406622i \(-0.866707\pi\)
0.913597 0.406622i \(-0.133293\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.4938 1.88453 0.942266 0.334865i \(-0.108691\pi\)
0.942266 + 0.334865i \(0.108691\pi\)
\(108\) 0 0
\(109\) 1.36398i 0.130646i −0.997864 0.0653228i \(-0.979192\pi\)
0.997864 0.0653228i \(-0.0208077\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.411889 −0.0387472 −0.0193736 0.999812i \(-0.506167\pi\)
−0.0193736 + 0.999812i \(0.506167\pi\)
\(114\) 0 0
\(115\) −3.10541 3.65464i −0.289581 0.340797i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.4938i 1.14531i
\(120\) 0 0
\(121\) −2.24795 −0.204359
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −17.8566 −1.58452 −0.792259 0.610185i \(-0.791095\pi\)
−0.792259 + 0.610185i \(0.791095\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.5945i 1.71198i −0.516996 0.855988i \(-0.672950\pi\)
0.516996 0.855988i \(-0.327050\pi\)
\(132\) 0 0
\(133\) 22.3521 1.93818
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.542882 −0.0463815 −0.0231908 0.999731i \(-0.507383\pi\)
−0.0231908 + 0.999731i \(0.507383\pi\)
\(138\) 0 0
\(139\) 8.44914 0.716647 0.358323 0.933598i \(-0.383349\pi\)
0.358323 + 0.933598i \(0.383349\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.33599 0.111721
\(144\) 0 0
\(145\) 3.01776i 0.250611i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.10938 0.664346 0.332173 0.943218i \(-0.392218\pi\)
0.332173 + 0.943218i \(0.392218\pi\)
\(150\) 0 0
\(151\) 1.74008 0.141606 0.0708030 0.997490i \(-0.477444\pi\)
0.0708030 + 0.997490i \(0.477444\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.09850 0.489844
\(156\) 0 0
\(157\) 1.69896i 0.135592i −0.997699 0.0677958i \(-0.978403\pi\)
0.997699 0.0677958i \(-0.0215966\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −10.4104 + 8.84586i −0.820451 + 0.697151i
\(162\) 0 0
\(163\) −21.8082 −1.70815 −0.854077 0.520147i \(-0.825877\pi\)
−0.854077 + 0.520147i \(0.825877\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.2292i 1.33324i 0.745399 + 0.666619i \(0.232259\pi\)
−0.745399 + 0.666619i \(0.767741\pi\)
\(168\) 0 0
\(169\) −12.7961 −0.984313
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 12.4800i 0.948838i 0.880299 + 0.474419i \(0.157342\pi\)
−0.880299 + 0.474419i \(0.842658\pi\)
\(174\) 0 0
\(175\) 2.84853i 0.215329i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.6646i 0.871850i −0.899983 0.435925i \(-0.856421\pi\)
0.899983 0.435925i \(-0.143579\pi\)
\(180\) 0 0
\(181\) 23.7617i 1.76619i 0.469194 + 0.883095i \(0.344544\pi\)
−0.469194 + 0.883095i \(0.655456\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.619475i 0.0455447i
\(186\) 0 0
\(187\) −12.9756 −0.948873
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −18.6185 −1.34719 −0.673594 0.739102i \(-0.735250\pi\)
−0.673594 + 0.739102i \(0.735250\pi\)
\(192\) 0 0
\(193\) −6.08797 −0.438221 −0.219111 0.975700i \(-0.570316\pi\)
−0.219111 + 0.975700i \(0.570316\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 14.5354i 1.03560i 0.855501 + 0.517801i \(0.173249\pi\)
−0.855501 + 0.517801i \(0.826751\pi\)
\(198\) 0 0
\(199\) 23.0638i 1.63495i 0.575965 + 0.817475i \(0.304627\pi\)
−0.575965 + 0.817475i \(0.695373\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.59617 −0.603333
\(204\) 0 0
\(205\) 5.35531i 0.374031i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 23.2142i 1.60576i
\(210\) 0 0
\(211\) 6.20881 0.427432 0.213716 0.976896i \(-0.431443\pi\)
0.213716 + 0.976896i \(0.431443\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.75365i 0.255997i
\(216\) 0 0
\(217\) 17.3718i 1.17927i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.98071 −0.133237
\(222\) 0 0
\(223\) 4.90111 0.328202 0.164101 0.986444i \(-0.447528\pi\)
0.164101 + 0.986444i \(0.447528\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.1700 1.60422 0.802109 0.597178i \(-0.203712\pi\)
0.802109 + 0.597178i \(0.203712\pi\)
\(228\) 0 0
\(229\) 6.21941i 0.410990i −0.978658 0.205495i \(-0.934120\pi\)
0.978658 0.205495i \(-0.0658805\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.7214i 1.68506i 0.538648 + 0.842531i \(0.318935\pi\)
−0.538648 + 0.842531i \(0.681065\pi\)
\(234\) 0 0
\(235\) 5.23505i 0.341497i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 26.9786i 1.74510i 0.488527 + 0.872549i \(0.337534\pi\)
−0.488527 + 0.872549i \(0.662466\pi\)
\(240\) 0 0
\(241\) 3.41127i 0.219739i −0.993946 0.109870i \(-0.964957\pi\)
0.993946 0.109870i \(-0.0350433\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.11412 0.0711787
\(246\) 0 0
\(247\) 3.54360i 0.225474i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.41740 −0.215704 −0.107852 0.994167i \(-0.534397\pi\)
−0.107852 + 0.994167i \(0.534397\pi\)
\(252\) 0 0
\(253\) −9.18701 10.8118i −0.577582 0.679735i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.7443i 1.41875i 0.704832 + 0.709375i \(0.251023\pi\)
−0.704832 + 0.709375i \(0.748977\pi\)
\(258\) 0 0
\(259\) 1.76459 0.109646
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −27.9026 −1.72055 −0.860274 0.509833i \(-0.829707\pi\)
−0.860274 + 0.509833i \(0.829707\pi\)
\(264\) 0 0
\(265\) −14.0102 −0.860642
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.4394i 1.12427i 0.827046 + 0.562134i \(0.190019\pi\)
−0.827046 + 0.562134i \(0.809981\pi\)
\(270\) 0 0
\(271\) −11.9202 −0.724100 −0.362050 0.932159i \(-0.617923\pi\)
−0.362050 + 0.932159i \(0.617923\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.95839 −0.178397
\(276\) 0 0
\(277\) −23.4021 −1.40609 −0.703046 0.711144i \(-0.748177\pi\)
−0.703046 + 0.711144i \(0.748177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.66269 −0.0991880 −0.0495940 0.998769i \(-0.515793\pi\)
−0.0495940 + 0.998769i \(0.515793\pi\)
\(282\) 0 0
\(283\) 30.1800i 1.79401i 0.442018 + 0.897006i \(0.354263\pi\)
−0.442018 + 0.897006i \(0.645737\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 15.2548 0.900461
\(288\) 0 0
\(289\) 2.23746 0.131615
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 16.6235 0.971158 0.485579 0.874193i \(-0.338609\pi\)
0.485579 + 0.874193i \(0.338609\pi\)
\(294\) 0 0
\(295\) 11.8371i 0.689183i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.40238 1.65041i −0.0811018 0.0954457i
\(300\) 0 0
\(301\) −10.6924 −0.616298
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.81751i 0.562149i
\(306\) 0 0
\(307\) −10.2948 −0.587556 −0.293778 0.955874i \(-0.594913\pi\)
−0.293778 + 0.955874i \(0.594913\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.06273i 0.287081i −0.989644 0.143540i \(-0.954151\pi\)
0.989644 0.143540i \(-0.0458487\pi\)
\(312\) 0 0
\(313\) 14.8335i 0.838438i 0.907885 + 0.419219i \(0.137696\pi\)
−0.907885 + 0.419219i \(0.862304\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.9847i 1.01012i −0.863084 0.505060i \(-0.831470\pi\)
0.863084 0.505060i \(-0.168530\pi\)
\(318\) 0 0
\(319\) 8.92769i 0.499855i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 34.4169i 1.91501i
\(324\) 0 0
\(325\) −0.451593 −0.0250499
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −14.9122 −0.822136
\(330\) 0 0
\(331\) −18.8433 −1.03572 −0.517859 0.855466i \(-0.673271\pi\)
−0.517859 + 0.855466i \(0.673271\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.3590i 0.675246i
\(336\) 0 0
\(337\) 20.4634i 1.11471i −0.830273 0.557356i \(-0.811816\pi\)
0.830273 0.557356i \(-0.188184\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.0417 0.977014
\(342\) 0 0
\(343\) 16.7661i 0.905284i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.6403i 1.10803i −0.832507 0.554014i \(-0.813095\pi\)
0.832507 0.554014i \(-0.186905\pi\)
\(348\) 0 0
\(349\) 36.6859 1.96375 0.981877 0.189521i \(-0.0606936\pi\)
0.981877 + 0.189521i \(0.0606936\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.82424i 0.150319i −0.997172 0.0751595i \(-0.976053\pi\)
0.997172 0.0751595i \(-0.0239466\pi\)
\(354\) 0 0
\(355\) 13.6631i 0.725162i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.383141 −0.0202214 −0.0101107 0.999949i \(-0.503218\pi\)
−0.0101107 + 0.999949i \(0.503218\pi\)
\(360\) 0 0
\(361\) −42.5738 −2.24073
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.07832 −0.0564417
\(366\) 0 0
\(367\) 3.34271i 0.174488i 0.996187 + 0.0872439i \(0.0278059\pi\)
−0.996187 + 0.0872439i \(0.972194\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 39.9086i 2.07195i
\(372\) 0 0
\(373\) 22.4610i 1.16299i −0.813550 0.581494i \(-0.802468\pi\)
0.813550 0.581494i \(-0.197532\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.36280i 0.0701876i
\(378\) 0 0
\(379\) 16.3221i 0.838411i 0.907891 + 0.419205i \(0.137691\pi\)
−0.907891 + 0.419205i \(0.862309\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.2685 −0.575794 −0.287897 0.957661i \(-0.592956\pi\)
−0.287897 + 0.957661i \(0.592956\pi\)
\(384\) 0 0
\(385\) 8.42705i 0.429482i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.17467 0.363770 0.181885 0.983320i \(-0.441780\pi\)
0.181885 + 0.983320i \(0.441780\pi\)
\(390\) 0 0
\(391\) 13.6205 + 16.0295i 0.688818 + 0.810645i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.71942i 0.438722i
\(396\) 0 0
\(397\) 34.9127 1.75222 0.876109 0.482113i \(-0.160130\pi\)
0.876109 + 0.482113i \(0.160130\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6127 1.12922 0.564611 0.825357i \(-0.309026\pi\)
0.564611 + 0.825357i \(0.309026\pi\)
\(402\) 0 0
\(403\) 2.75404 0.137188
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.83265i 0.0908409i
\(408\) 0 0
\(409\) −4.70826 −0.232808 −0.116404 0.993202i \(-0.537137\pi\)
−0.116404 + 0.993202i \(0.537137\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 33.7183 1.65917
\(414\) 0 0
\(415\) −3.20932 −0.157539
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.32551 0.260168 0.130084 0.991503i \(-0.458475\pi\)
0.130084 + 0.991503i \(0.458475\pi\)
\(420\) 0 0
\(421\) 11.4267i 0.556905i −0.960450 0.278452i \(-0.910179\pi\)
0.960450 0.278452i \(-0.0898214\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.38605 0.212755
\(426\) 0 0
\(427\) −27.9655 −1.35334
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.4968 −0.939129 −0.469565 0.882898i \(-0.655589\pi\)
−0.469565 + 0.882898i \(0.655589\pi\)
\(432\) 0 0
\(433\) 22.9884i 1.10475i 0.833595 + 0.552375i \(0.186279\pi\)
−0.833595 + 0.552375i \(0.813721\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.6776 24.3679i 1.37184 1.16567i
\(438\) 0 0
\(439\) 2.56767 0.122548 0.0612740 0.998121i \(-0.480484\pi\)
0.0612740 + 0.998121i \(0.480484\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 41.1892i 1.95696i 0.206347 + 0.978479i \(0.433842\pi\)
−0.206347 + 0.978479i \(0.566158\pi\)
\(444\) 0 0
\(445\) 3.64156 0.172626
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 24.9921i 1.17945i −0.807605 0.589724i \(-0.799237\pi\)
0.807605 0.589724i \(-0.200763\pi\)
\(450\) 0 0
\(451\) 15.8431i 0.746022i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.28638i 0.0603062i
\(456\) 0 0
\(457\) 9.01138i 0.421535i 0.977536 + 0.210767i \(0.0675962\pi\)
−0.977536 + 0.210767i \(0.932404\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.98035i 0.418257i 0.977888 + 0.209128i \(0.0670627\pi\)
−0.977888 + 0.209128i \(0.932937\pi\)
\(462\) 0 0
\(463\) 7.13347 0.331520 0.165760 0.986166i \(-0.446992\pi\)
0.165760 + 0.986166i \(0.446992\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.9368 1.20021 0.600107 0.799920i \(-0.295125\pi\)
0.600107 + 0.799920i \(0.295125\pi\)
\(468\) 0 0
\(469\) −35.2051 −1.62562
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 11.1047i 0.510596i
\(474\) 0 0
\(475\) 7.84690i 0.360040i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.278689 −0.0127336 −0.00636681 0.999980i \(-0.502027\pi\)
−0.00636681 + 0.999980i \(0.502027\pi\)
\(480\) 0 0
\(481\) 0.279750i 0.0127555i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.46148i 0.384216i
\(486\) 0 0
\(487\) 31.6247 1.43305 0.716527 0.697559i \(-0.245731\pi\)
0.716527 + 0.697559i \(0.245731\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.4654i 0.788202i 0.919067 + 0.394101i \(0.128944\pi\)
−0.919067 + 0.394101i \(0.871056\pi\)
\(492\) 0 0
\(493\) 13.2360i 0.596121i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 38.9197 1.74579
\(498\) 0 0
\(499\) 25.4931 1.14123 0.570613 0.821219i \(-0.306706\pi\)
0.570613 + 0.821219i \(0.306706\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −34.1952 −1.52469 −0.762345 0.647170i \(-0.775952\pi\)
−0.762345 + 0.647170i \(0.775952\pi\)
\(504\) 0 0
\(505\) 10.2407i 0.455706i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 35.9594i 1.59387i 0.604064 + 0.796936i \(0.293547\pi\)
−0.604064 + 0.796936i \(0.706453\pi\)
\(510\) 0 0
\(511\) 3.07162i 0.135880i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.25352i 0.363694i
\(516\) 0 0
\(517\) 15.4873i 0.681131i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.6426 1.12342 0.561710 0.827334i \(-0.310144\pi\)
0.561710 + 0.827334i \(0.310144\pi\)
\(522\) 0 0
\(523\) 14.5928i 0.638098i −0.947738 0.319049i \(-0.896637\pi\)
0.947738 0.319049i \(-0.103363\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.7484 −1.16518
\(528\) 0 0
\(529\) −3.71284 + 22.6983i −0.161428 + 0.986884i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.41842i 0.104753i
\(534\) 0 0
\(535\) −19.4938 −0.842789
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 3.29601 0.141969
\(540\) 0 0
\(541\) 20.6370 0.887253 0.443627 0.896212i \(-0.353692\pi\)
0.443627 + 0.896212i \(0.353692\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.36398i 0.0584264i
\(546\) 0 0
\(547\) 8.00106 0.342101 0.171050 0.985262i \(-0.445284\pi\)
0.171050 + 0.985262i \(0.445284\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 23.6800 1.00880
\(552\) 0 0
\(553\) −24.8375 −1.05620
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −23.8992 −1.01264 −0.506321 0.862345i \(-0.668995\pi\)
−0.506321 + 0.862345i \(0.668995\pi\)
\(558\) 0 0
\(559\) 1.69512i 0.0716959i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.6858 −0.618934 −0.309467 0.950910i \(-0.600151\pi\)
−0.309467 + 0.950910i \(0.600151\pi\)
\(564\) 0 0
\(565\) 0.411889 0.0173283
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 39.9436 1.67452 0.837261 0.546803i \(-0.184156\pi\)
0.837261 + 0.546803i \(0.184156\pi\)
\(570\) 0 0
\(571\) 25.0900i 1.04998i 0.851107 + 0.524992i \(0.175932\pi\)
−0.851107 + 0.524992i \(0.824068\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.10541 + 3.65464i 0.129505 + 0.152409i
\(576\) 0 0
\(577\) 0.519605 0.0216315 0.0108157 0.999942i \(-0.496557\pi\)
0.0108157 + 0.999942i \(0.496557\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.14185i 0.379268i
\(582\) 0 0
\(583\) −41.4477 −1.71659
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.43276i 0.100411i −0.998739 0.0502054i \(-0.984012\pi\)
0.998739 0.0502054i \(-0.0159876\pi\)
\(588\) 0 0
\(589\) 47.8543i 1.97180i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.69741i 0.151834i −0.997114 0.0759172i \(-0.975812\pi\)
0.997114 0.0759172i \(-0.0241885\pi\)
\(594\) 0 0
\(595\) 12.4938i 0.512196i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.88862i 0.199744i −0.995000 0.0998718i \(-0.968157\pi\)
0.995000 0.0998718i \(-0.0318433\pi\)
\(600\) 0 0
\(601\) 27.9886 1.14168 0.570838 0.821062i \(-0.306618\pi\)
0.570838 + 0.821062i \(0.306618\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.24795 0.0913921
\(606\) 0 0
\(607\) −22.5025 −0.913348 −0.456674 0.889634i \(-0.650959\pi\)
−0.456674 + 0.889634i \(0.650959\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.36411i 0.0956417i
\(612\) 0 0
\(613\) 26.4302i 1.06751i 0.845641 + 0.533753i \(0.179219\pi\)
−0.845641 + 0.533753i \(0.820781\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.74830 −0.110642 −0.0553211 0.998469i \(-0.517618\pi\)
−0.0553211 + 0.998469i \(0.517618\pi\)
\(618\) 0 0
\(619\) 0.463520i 0.0186304i −0.999957 0.00931522i \(-0.997035\pi\)
0.999957 0.00931522i \(-0.00296517\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 10.3731i 0.415589i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.71705i 0.108336i
\(630\) 0 0
\(631\) 0.514097i 0.0204659i 0.999948 + 0.0102329i \(0.00325731\pi\)
−0.999948 + 0.0102329i \(0.996743\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 17.8566 0.708618
\(636\) 0 0
\(637\) 0.503130 0.0199347
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.15293 −0.124533 −0.0622665 0.998060i \(-0.519833\pi\)
−0.0622665 + 0.998060i \(0.519833\pi\)
\(642\) 0 0
\(643\) 14.1082i 0.556374i 0.960527 + 0.278187i \(0.0897335\pi\)
−0.960527 + 0.278187i \(0.910266\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.62870i 0.181973i 0.995852 + 0.0909865i \(0.0290020\pi\)
−0.995852 + 0.0909865i \(0.970998\pi\)
\(648\) 0 0
\(649\) 35.0187i 1.37461i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.2118i 1.33881i 0.742897 + 0.669406i \(0.233451\pi\)
−0.742897 + 0.669406i \(0.766549\pi\)
\(654\) 0 0
\(655\) 19.5945i 0.765619i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −45.9387 −1.78952 −0.894760 0.446548i \(-0.852653\pi\)
−0.894760 + 0.446548i \(0.852653\pi\)
\(660\) 0 0
\(661\) 31.0803i 1.20888i −0.796650 0.604441i \(-0.793397\pi\)
0.796650 0.604441i \(-0.206603\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −22.3521 −0.866778
\(666\) 0 0
\(667\) −11.0288 + 9.37138i −0.427038 + 0.362861i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 29.0440i 1.12123i
\(672\) 0 0
\(673\) 38.3898 1.47982 0.739910 0.672706i \(-0.234868\pi\)
0.739910 + 0.672706i \(0.234868\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 37.4158 1.43801 0.719003 0.695007i \(-0.244599\pi\)
0.719003 + 0.695007i \(0.244599\pi\)
\(678\) 0 0
\(679\) −24.1028 −0.924980
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.21233i 0.275972i −0.990434 0.137986i \(-0.955937\pi\)
0.990434 0.137986i \(-0.0440629\pi\)
\(684\) 0 0
\(685\) 0.542882 0.0207424
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.32692 −0.241036
\(690\) 0 0
\(691\) −35.5977 −1.35420 −0.677101 0.735890i \(-0.736764\pi\)
−0.677101 + 0.735890i \(0.736764\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.44914 −0.320494
\(696\) 0 0
\(697\) 23.4887i 0.889698i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.5389 0.586894 0.293447 0.955975i \(-0.405198\pi\)
0.293447 + 0.955975i \(0.405198\pi\)
\(702\) 0 0
\(703\) −4.86096 −0.183335
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −29.1710 −1.09709
\(708\) 0 0
\(709\) 40.3904i 1.51689i −0.651735 0.758447i \(-0.725959\pi\)
0.651735 0.758447i \(-0.274041\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.9384 22.2879i −0.709247 0.834687i
\(714\) 0 0
\(715\) −1.33599 −0.0499630
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.35988i 0.0880086i −0.999031 0.0440043i \(-0.985988\pi\)
0.999031 0.0440043i \(-0.0140115\pi\)
\(720\) 0 0
\(721\) 23.5104 0.875573
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.01776i 0.112077i
\(726\) 0 0
\(727\) 11.8636i 0.439996i −0.975500 0.219998i \(-0.929395\pi\)
0.975500 0.219998i \(-0.0706051\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.4637i 0.608932i
\(732\) 0 0
\(733\) 7.17262i 0.264927i −0.991188 0.132463i \(-0.957711\pi\)
0.991188 0.132463i \(-0.0422887\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.5628i 1.34681i
\(738\) 0 0
\(739\) 49.6293 1.82564 0.912822 0.408357i \(-0.133898\pi\)
0.912822 + 0.408357i \(0.133898\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 30.8004 1.12996 0.564979 0.825105i \(-0.308884\pi\)
0.564979 + 0.825105i \(0.308884\pi\)
\(744\) 0 0
\(745\) −8.10938 −0.297105
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 55.5285i 2.02897i
\(750\) 0 0
\(751\) 19.3155i 0.704833i 0.935843 + 0.352416i \(0.114640\pi\)
−0.935843 + 0.352416i \(0.885360\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.74008 −0.0633281
\(756\) 0 0
\(757\) 44.4725i 1.61638i −0.588921 0.808190i \(-0.700447\pi\)
0.588921 0.808190i \(-0.299553\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35.6433i 1.29207i 0.763308 + 0.646035i \(0.223574\pi\)
−0.763308 + 0.646035i \(0.776426\pi\)
\(762\) 0 0
\(763\) 3.88534 0.140659
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.34555i 0.193017i
\(768\) 0 0
\(769\) 24.9184i 0.898580i 0.893386 + 0.449290i \(0.148323\pi\)
−0.893386 + 0.449290i \(0.851677\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −38.5019 −1.38482 −0.692408 0.721506i \(-0.743450\pi\)
−0.692408 + 0.721506i \(0.743450\pi\)
\(774\) 0 0
\(775\) −6.09850 −0.219065
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −42.0226 −1.50562
\(780\) 0 0
\(781\) 40.4207i 1.44637i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.69896i 0.0606384i
\(786\) 0 0
\(787\) 27.3157i 0.973701i 0.873485 + 0.486851i \(0.161854\pi\)
−0.873485 + 0.486851i \(0.838146\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.17328i 0.0417169i
\(792\) 0 0
\(793\) 4.43352i 0.157439i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −22.8131 −0.808080 −0.404040 0.914741i \(-0.632394\pi\)
−0.404040 + 0.914741i \(0.632394\pi\)
\(798\) 0 0
\(799\) 22.9612i 0.812309i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.19008 −0.112575
\(804\) 0 0
\(805\) 10.4104 8.84586i 0.366917 0.311775i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 49.7903i 1.75053i 0.483640 + 0.875267i \(0.339314\pi\)