Properties

Label 6048.2.c.e.3025.11
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 20
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.11
Root \(-1.37874 + 0.314750i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.e.3025.10

$q$-expansion

\(f(q)\) \(=\) \(q+0.114591i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+0.114591i q^{5} +1.00000 q^{7} -0.412956i q^{11} -1.73584i q^{13} +2.50762 q^{17} -6.85261i q^{19} -4.42226 q^{23} +4.98687 q^{25} +1.85559i q^{29} -5.60373 q^{31} +0.114591i q^{35} +4.39099i q^{37} -2.39907 q^{41} -4.35614i q^{43} +7.23070 q^{47} +1.00000 q^{49} -11.2241i q^{53} +0.0473212 q^{55} +4.25900i q^{59} -7.35936i q^{61} +0.198912 q^{65} -6.25549i q^{67} +0.608276 q^{71} -14.1550 q^{73} -0.412956i q^{77} -8.19950 q^{79} +4.88023i q^{83} +0.287352i q^{85} -10.4217 q^{89} -1.73584i q^{91} +0.785249 q^{95} -3.42009 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{7} + O(q^{10}) \) \( 20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) 0.114591i 0.0512468i 0.999672 + 0.0256234i \(0.00815707\pi\)
−0.999672 + 0.0256234i \(0.991843\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 0.412956i − 0.124511i −0.998060 0.0622555i \(-0.980171\pi\)
0.998060 0.0622555i \(-0.0198294\pi\)
\(12\) 0 0
\(13\) − 1.73584i − 0.481435i −0.970595 0.240717i \(-0.922617\pi\)
0.970595 0.240717i \(-0.0773827\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.50762 0.608188 0.304094 0.952642i \(-0.401646\pi\)
0.304094 + 0.952642i \(0.401646\pi\)
\(18\) 0 0
\(19\) − 6.85261i − 1.57210i −0.618166 0.786048i \(-0.712124\pi\)
0.618166 0.786048i \(-0.287876\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.42226 −0.922106 −0.461053 0.887373i \(-0.652528\pi\)
−0.461053 + 0.887373i \(0.652528\pi\)
\(24\) 0 0
\(25\) 4.98687 0.997374
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.85559i 0.344575i 0.985047 + 0.172287i \(0.0551158\pi\)
−0.985047 + 0.172287i \(0.944884\pi\)
\(30\) 0 0
\(31\) −5.60373 −1.00646 −0.503230 0.864153i \(-0.667855\pi\)
−0.503230 + 0.864153i \(0.667855\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.114591i 0.0193695i
\(36\) 0 0
\(37\) 4.39099i 0.721875i 0.932590 + 0.360938i \(0.117543\pi\)
−0.932590 + 0.360938i \(0.882457\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.39907 −0.374672 −0.187336 0.982296i \(-0.559985\pi\)
−0.187336 + 0.982296i \(0.559985\pi\)
\(42\) 0 0
\(43\) − 4.35614i − 0.664306i −0.943226 0.332153i \(-0.892225\pi\)
0.943226 0.332153i \(-0.107775\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.23070 1.05471 0.527353 0.849646i \(-0.323185\pi\)
0.527353 + 0.849646i \(0.323185\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 11.2241i − 1.54175i −0.636983 0.770877i \(-0.719818\pi\)
0.636983 0.770877i \(-0.280182\pi\)
\(54\) 0 0
\(55\) 0.0473212 0.00638079
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.25900i 0.554475i 0.960801 + 0.277237i \(0.0894188\pi\)
−0.960801 + 0.277237i \(0.910581\pi\)
\(60\) 0 0
\(61\) − 7.35936i − 0.942269i −0.882061 0.471135i \(-0.843845\pi\)
0.882061 0.471135i \(-0.156155\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.198912 0.0246720
\(66\) 0 0
\(67\) − 6.25549i − 0.764230i −0.924115 0.382115i \(-0.875196\pi\)
0.924115 0.382115i \(-0.124804\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.608276 0.0721891 0.0360946 0.999348i \(-0.488508\pi\)
0.0360946 + 0.999348i \(0.488508\pi\)
\(72\) 0 0
\(73\) −14.1550 −1.65671 −0.828357 0.560201i \(-0.810724\pi\)
−0.828357 + 0.560201i \(0.810724\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.412956i − 0.0470607i
\(78\) 0 0
\(79\) −8.19950 −0.922516 −0.461258 0.887266i \(-0.652602\pi\)
−0.461258 + 0.887266i \(0.652602\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.88023i 0.535675i 0.963464 + 0.267838i \(0.0863091\pi\)
−0.963464 + 0.267838i \(0.913691\pi\)
\(84\) 0 0
\(85\) 0.287352i 0.0311677i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.4217 −1.10469 −0.552347 0.833614i \(-0.686268\pi\)
−0.552347 + 0.833614i \(0.686268\pi\)
\(90\) 0 0
\(91\) − 1.73584i − 0.181965i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.785249 0.0805649
\(96\) 0 0
\(97\) −3.42009 −0.347258 −0.173629 0.984811i \(-0.555549\pi\)
−0.173629 + 0.984811i \(0.555549\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 0.203905i − 0.0202893i −0.999949 0.0101446i \(-0.996771\pi\)
0.999949 0.0101446i \(-0.00322920\pi\)
\(102\) 0 0
\(103\) −7.16809 −0.706293 −0.353147 0.935568i \(-0.614888\pi\)
−0.353147 + 0.935568i \(0.614888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.46742i 0.721902i 0.932585 + 0.360951i \(0.117548\pi\)
−0.932585 + 0.360951i \(0.882452\pi\)
\(108\) 0 0
\(109\) − 2.11355i − 0.202442i −0.994864 0.101221i \(-0.967725\pi\)
0.994864 0.101221i \(-0.0322749\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.72972 0.444935 0.222467 0.974940i \(-0.428589\pi\)
0.222467 + 0.974940i \(0.428589\pi\)
\(114\) 0 0
\(115\) − 0.506753i − 0.0472550i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.50762 0.229873
\(120\) 0 0
\(121\) 10.8295 0.984497
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.14441i 0.102359i
\(126\) 0 0
\(127\) 9.55123 0.847535 0.423767 0.905771i \(-0.360707\pi\)
0.423767 + 0.905771i \(0.360707\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 18.5084i 1.61709i 0.588435 + 0.808544i \(0.299744\pi\)
−0.588435 + 0.808544i \(0.700256\pi\)
\(132\) 0 0
\(133\) − 6.85261i − 0.594196i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −8.42852 −0.720097 −0.360048 0.932934i \(-0.617240\pi\)
−0.360048 + 0.932934i \(0.617240\pi\)
\(138\) 0 0
\(139\) − 13.8639i − 1.17592i −0.808890 0.587961i \(-0.799931\pi\)
0.808890 0.587961i \(-0.200069\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.716825 −0.0599439
\(144\) 0 0
\(145\) −0.212635 −0.0176583
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 22.0358i − 1.80525i −0.430432 0.902623i \(-0.641639\pi\)
0.430432 0.902623i \(-0.358361\pi\)
\(150\) 0 0
\(151\) −5.40696 −0.440012 −0.220006 0.975498i \(-0.570608\pi\)
−0.220006 + 0.975498i \(0.570608\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 0.642139i − 0.0515778i
\(156\) 0 0
\(157\) − 13.1081i − 1.04614i −0.852290 0.523070i \(-0.824787\pi\)
0.852290 0.523070i \(-0.175213\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.42226 −0.348523
\(162\) 0 0
\(163\) − 23.1785i − 1.81548i −0.419533 0.907740i \(-0.637806\pi\)
0.419533 0.907740i \(-0.362194\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.12880 0.551643 0.275821 0.961209i \(-0.411050\pi\)
0.275821 + 0.961209i \(0.411050\pi\)
\(168\) 0 0
\(169\) 9.98687 0.768221
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 20.8217i − 1.58305i −0.611138 0.791524i \(-0.709288\pi\)
0.611138 0.791524i \(-0.290712\pi\)
\(174\) 0 0
\(175\) 4.98687 0.376972
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.51282i 0.561535i 0.959776 + 0.280767i \(0.0905890\pi\)
−0.959776 + 0.280767i \(0.909411\pi\)
\(180\) 0 0
\(181\) − 4.48480i − 0.333353i −0.986012 0.166676i \(-0.946697\pi\)
0.986012 0.166676i \(-0.0533035\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.503170 −0.0369938
\(186\) 0 0
\(187\) − 1.03554i − 0.0757260i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.6726 1.78525 0.892624 0.450802i \(-0.148862\pi\)
0.892624 + 0.450802i \(0.148862\pi\)
\(192\) 0 0
\(193\) −0.335818 −0.0241727 −0.0120863 0.999927i \(-0.503847\pi\)
−0.0120863 + 0.999927i \(0.503847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.3089i 0.948221i 0.880465 + 0.474111i \(0.157230\pi\)
−0.880465 + 0.474111i \(0.842770\pi\)
\(198\) 0 0
\(199\) 9.24682 0.655490 0.327745 0.944766i \(-0.393711\pi\)
0.327745 + 0.944766i \(0.393711\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.85559i 0.130237i
\(204\) 0 0
\(205\) − 0.274913i − 0.0192008i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.82983 −0.195743
\(210\) 0 0
\(211\) 4.73424i 0.325918i 0.986633 + 0.162959i \(0.0521039\pi\)
−0.986633 + 0.162959i \(0.947896\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.499176 0.0340435
\(216\) 0 0
\(217\) −5.60373 −0.380406
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 4.35282i − 0.292803i
\(222\) 0 0
\(223\) −9.68005 −0.648224 −0.324112 0.946019i \(-0.605065\pi\)
−0.324112 + 0.946019i \(0.605065\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 22.8566i − 1.51705i −0.651646 0.758523i \(-0.725921\pi\)
0.651646 0.758523i \(-0.274079\pi\)
\(228\) 0 0
\(229\) − 16.7944i − 1.10980i −0.831916 0.554901i \(-0.812756\pi\)
0.831916 0.554901i \(-0.187244\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.6756 −1.28899 −0.644494 0.764609i \(-0.722932\pi\)
−0.644494 + 0.764609i \(0.722932\pi\)
\(234\) 0 0
\(235\) 0.828576i 0.0540503i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.8874 −1.54515 −0.772573 0.634926i \(-0.781030\pi\)
−0.772573 + 0.634926i \(0.781030\pi\)
\(240\) 0 0
\(241\) −4.03141 −0.259686 −0.129843 0.991535i \(-0.541447\pi\)
−0.129843 + 0.991535i \(0.541447\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.114591i 0.00732097i
\(246\) 0 0
\(247\) −11.8950 −0.756861
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 8.40404i 0.530459i 0.964185 + 0.265229i \(0.0854476\pi\)
−0.964185 + 0.265229i \(0.914552\pi\)
\(252\) 0 0
\(253\) 1.82620i 0.114812i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.32955 0.207692 0.103846 0.994593i \(-0.466885\pi\)
0.103846 + 0.994593i \(0.466885\pi\)
\(258\) 0 0
\(259\) 4.39099i 0.272843i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 16.2375 1.00124 0.500622 0.865666i \(-0.333105\pi\)
0.500622 + 0.865666i \(0.333105\pi\)
\(264\) 0 0
\(265\) 1.28619 0.0790100
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5.74732i 0.350420i 0.984531 + 0.175210i \(0.0560605\pi\)
−0.984531 + 0.175210i \(0.943940\pi\)
\(270\) 0 0
\(271\) 11.6089 0.705189 0.352595 0.935776i \(-0.385300\pi\)
0.352595 + 0.935776i \(0.385300\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 2.05936i − 0.124184i
\(276\) 0 0
\(277\) − 19.1097i − 1.14819i −0.818788 0.574096i \(-0.805354\pi\)
0.818788 0.574096i \(-0.194646\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 19.9611 1.19078 0.595389 0.803438i \(-0.296998\pi\)
0.595389 + 0.803438i \(0.296998\pi\)
\(282\) 0 0
\(283\) 16.1413i 0.959503i 0.877404 + 0.479752i \(0.159273\pi\)
−0.877404 + 0.479752i \(0.840727\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.39907 −0.141613
\(288\) 0 0
\(289\) −10.7118 −0.630108
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 7.16969i − 0.418858i −0.977824 0.209429i \(-0.932840\pi\)
0.977824 0.209429i \(-0.0671604\pi\)
\(294\) 0 0
\(295\) −0.488044 −0.0284150
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.67633i 0.443934i
\(300\) 0 0
\(301\) − 4.35614i − 0.251084i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.843319 0.0482883
\(306\) 0 0
\(307\) − 15.4062i − 0.879278i −0.898175 0.439639i \(-0.855106\pi\)
0.898175 0.439639i \(-0.144894\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.3442 −1.15361 −0.576806 0.816881i \(-0.695701\pi\)
−0.576806 + 0.816881i \(0.695701\pi\)
\(312\) 0 0
\(313\) −26.5540 −1.50092 −0.750460 0.660916i \(-0.770168\pi\)
−0.750460 + 0.660916i \(0.770168\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.8006i 0.999782i 0.866088 + 0.499891i \(0.166627\pi\)
−0.866088 + 0.499891i \(0.833373\pi\)
\(318\) 0 0
\(319\) 0.766278 0.0429033
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 17.1837i − 0.956129i
\(324\) 0 0
\(325\) − 8.65639i − 0.480170i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 7.23070 0.398641
\(330\) 0 0
\(331\) − 27.1007i − 1.48959i −0.667295 0.744793i \(-0.732548\pi\)
0.667295 0.744793i \(-0.267452\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0.716825 0.0391643
\(336\) 0 0
\(337\) −9.19674 −0.500978 −0.250489 0.968119i \(-0.580591\pi\)
−0.250489 + 0.968119i \(0.580591\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.31409i 0.125315i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 33.4386i 1.79508i 0.440935 + 0.897539i \(0.354647\pi\)
−0.440935 + 0.897539i \(0.645353\pi\)
\(348\) 0 0
\(349\) − 17.5598i − 0.939954i −0.882679 0.469977i \(-0.844262\pi\)
0.882679 0.469977i \(-0.155738\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.4376 0.981335 0.490668 0.871347i \(-0.336753\pi\)
0.490668 + 0.871347i \(0.336753\pi\)
\(354\) 0 0
\(355\) 0.0697032i 0.00369946i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.80894 −0.412140 −0.206070 0.978537i \(-0.566067\pi\)
−0.206070 + 0.978537i \(0.566067\pi\)
\(360\) 0 0
\(361\) −27.9582 −1.47148
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.62204i − 0.0849012i
\(366\) 0 0
\(367\) 18.2964 0.955066 0.477533 0.878614i \(-0.341531\pi\)
0.477533 + 0.878614i \(0.341531\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 11.2241i − 0.582729i
\(372\) 0 0
\(373\) − 29.4005i − 1.52230i −0.648575 0.761150i \(-0.724635\pi\)
0.648575 0.761150i \(-0.275365\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.22100 0.165890
\(378\) 0 0
\(379\) 2.46161i 0.126444i 0.997999 + 0.0632222i \(0.0201377\pi\)
−0.997999 + 0.0632222i \(0.979862\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.59739 0.439306 0.219653 0.975578i \(-0.429507\pi\)
0.219653 + 0.975578i \(0.429507\pi\)
\(384\) 0 0
\(385\) 0.0473212 0.00241171
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 3.93550i − 0.199538i −0.995011 0.0997688i \(-0.968190\pi\)
0.995011 0.0997688i \(-0.0318103\pi\)
\(390\) 0 0
\(391\) −11.0894 −0.560813
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 0.939592i − 0.0472760i
\(396\) 0 0
\(397\) − 27.3549i − 1.37290i −0.727175 0.686452i \(-0.759167\pi\)
0.727175 0.686452i \(-0.240833\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.49913 −0.324551 −0.162276 0.986745i \(-0.551883\pi\)
−0.162276 + 0.986745i \(0.551883\pi\)
\(402\) 0 0
\(403\) 9.72716i 0.484545i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.81329 0.0898814
\(408\) 0 0
\(409\) 17.9897 0.889531 0.444765 0.895647i \(-0.353287\pi\)
0.444765 + 0.895647i \(0.353287\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.25900i 0.209572i
\(414\) 0 0
\(415\) −0.559232 −0.0274516
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 40.4247i − 1.97487i −0.158014 0.987437i \(-0.550509\pi\)
0.158014 0.987437i \(-0.449491\pi\)
\(420\) 0 0
\(421\) 11.3043i 0.550938i 0.961310 + 0.275469i \(0.0888332\pi\)
−0.961310 + 0.275469i \(0.911167\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.5052 0.606590
\(426\) 0 0
\(427\) − 7.35936i − 0.356144i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.2412 −1.02315 −0.511575 0.859238i \(-0.670938\pi\)
−0.511575 + 0.859238i \(0.670938\pi\)
\(432\) 0 0
\(433\) −5.34136 −0.256690 −0.128345 0.991730i \(-0.540966\pi\)
−0.128345 + 0.991730i \(0.540966\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.3040i 1.44964i
\(438\) 0 0
\(439\) 20.0290 0.955933 0.477966 0.878378i \(-0.341374\pi\)
0.477966 + 0.878378i \(0.341374\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.3027i 1.01212i 0.862497 + 0.506062i \(0.168899\pi\)
−0.862497 + 0.506062i \(0.831101\pi\)
\(444\) 0 0
\(445\) − 1.19423i − 0.0566121i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.8880 −0.844185 −0.422093 0.906553i \(-0.638704\pi\)
−0.422093 + 0.906553i \(0.638704\pi\)
\(450\) 0 0
\(451\) 0.990712i 0.0466508i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.198912 0.00932513
\(456\) 0 0
\(457\) 19.3625 0.905740 0.452870 0.891577i \(-0.350400\pi\)
0.452870 + 0.891577i \(0.350400\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 21.8751i 1.01882i 0.860523 + 0.509412i \(0.170137\pi\)
−0.860523 + 0.509412i \(0.829863\pi\)
\(462\) 0 0
\(463\) −26.0449 −1.21041 −0.605204 0.796070i \(-0.706909\pi\)
−0.605204 + 0.796070i \(0.706909\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 11.8214i − 0.547028i −0.961868 0.273514i \(-0.911814\pi\)
0.961868 0.273514i \(-0.0881860\pi\)
\(468\) 0 0
\(469\) − 6.25549i − 0.288852i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.79890 −0.0827133
\(474\) 0 0
\(475\) − 34.1730i − 1.56797i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.0868 1.32901 0.664504 0.747284i \(-0.268643\pi\)
0.664504 + 0.747284i \(0.268643\pi\)
\(480\) 0 0
\(481\) 7.62205 0.347536
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 0.391913i − 0.0177959i
\(486\) 0 0
\(487\) 21.3338 0.966728 0.483364 0.875419i \(-0.339415\pi\)
0.483364 + 0.875419i \(0.339415\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 20.2052i − 0.911846i −0.890019 0.455923i \(-0.849309\pi\)
0.890019 0.455923i \(-0.150691\pi\)
\(492\) 0 0
\(493\) 4.65312i 0.209566i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.608276 0.0272849
\(498\) 0 0
\(499\) − 25.2662i − 1.13107i −0.824724 0.565535i \(-0.808670\pi\)
0.824724 0.565535i \(-0.191330\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −30.2176 −1.34734 −0.673669 0.739033i \(-0.735283\pi\)
−0.673669 + 0.739033i \(0.735283\pi\)
\(504\) 0 0
\(505\) 0.0233657 0.00103976
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.78140i 0.0789592i 0.999220 + 0.0394796i \(0.0125700\pi\)
−0.999220 + 0.0394796i \(0.987430\pi\)
\(510\) 0 0
\(511\) −14.1550 −0.626179
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 0.821401i − 0.0361953i
\(516\) 0 0
\(517\) − 2.98596i − 0.131322i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.72198 0.250685 0.125342 0.992114i \(-0.459997\pi\)
0.125342 + 0.992114i \(0.459997\pi\)
\(522\) 0 0
\(523\) − 0.348058i − 0.0152195i −0.999971 0.00760976i \(-0.997578\pi\)
0.999971 0.00760976i \(-0.00242229\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.0520 −0.612116
\(528\) 0 0
\(529\) −3.44359 −0.149721
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 4.16440i 0.180380i
\(534\) 0 0
\(535\) −0.855701 −0.0369952
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 0.412956i − 0.0177873i
\(540\) 0 0
\(541\) 8.49464i 0.365213i 0.983186 + 0.182606i \(0.0584534\pi\)
−0.983186 + 0.182606i \(0.941547\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.242195 0.0103745
\(546\) 0 0
\(547\) 31.5175i 1.34759i 0.738918 + 0.673796i \(0.235337\pi\)
−0.738918 + 0.673796i \(0.764663\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.7156 0.541704
\(552\) 0 0
\(553\) −8.19950 −0.348678
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 25.5635i − 1.08316i −0.840649 0.541581i \(-0.817826\pi\)
0.840649 0.541581i \(-0.182174\pi\)
\(558\) 0 0
\(559\) −7.56156 −0.319820
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 18.6212i − 0.784791i −0.919796 0.392396i \(-0.871646\pi\)
0.919796 0.392396i \(-0.128354\pi\)
\(564\) 0 0
\(565\) 0.541985i 0.0228015i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 40.8350 1.71189 0.855946 0.517065i \(-0.172975\pi\)
0.855946 + 0.517065i \(0.172975\pi\)
\(570\) 0 0
\(571\) 20.7134i 0.866831i 0.901194 + 0.433415i \(0.142692\pi\)
−0.901194 + 0.433415i \(0.857308\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −22.0532 −0.919684
\(576\) 0 0
\(577\) 8.58023 0.357200 0.178600 0.983922i \(-0.442843\pi\)
0.178600 + 0.983922i \(0.442843\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.88023i 0.202466i
\(582\) 0 0
\(583\) −4.63508 −0.191965
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 28.7023i − 1.18467i −0.805692 0.592334i \(-0.798206\pi\)
0.805692 0.592334i \(-0.201794\pi\)
\(588\) 0 0
\(589\) 38.4001i 1.58225i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −42.2499 −1.73500 −0.867498 0.497440i \(-0.834273\pi\)
−0.867498 + 0.497440i \(0.834273\pi\)
\(594\) 0 0
\(595\) 0.287352i 0.0117803i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.3138 1.27945 0.639723 0.768605i \(-0.279049\pi\)
0.639723 + 0.768605i \(0.279049\pi\)
\(600\) 0 0
\(601\) 9.40650 0.383699 0.191850 0.981424i \(-0.438551\pi\)
0.191850 + 0.981424i \(0.438551\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.24096i 0.0504523i
\(606\) 0 0
\(607\) −4.95314 −0.201042 −0.100521 0.994935i \(-0.532051\pi\)
−0.100521 + 0.994935i \(0.532051\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 12.5513i − 0.507772i
\(612\) 0 0
\(613\) 6.61633i 0.267231i 0.991033 + 0.133616i \(0.0426587\pi\)
−0.991033 + 0.133616i \(0.957341\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.3584 0.859855 0.429928 0.902863i \(-0.358539\pi\)
0.429928 + 0.902863i \(0.358539\pi\)
\(618\) 0 0
\(619\) 0.0210409i 0 0.000845707i 1.00000 0.000422854i \(0.000134598\pi\)
−1.00000 0.000422854i \(0.999865\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −10.4217 −0.417535
\(624\) 0 0
\(625\) 24.8032 0.992128
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.0110i 0.439036i
\(630\) 0 0
\(631\) −19.6037 −0.780411 −0.390206 0.920728i \(-0.627596\pi\)
−0.390206 + 0.920728i \(0.627596\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.09449i 0.0434335i
\(636\) 0 0
\(637\) − 1.73584i − 0.0687764i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.84692 0.191442 0.0957210 0.995408i \(-0.469484\pi\)
0.0957210 + 0.995408i \(0.469484\pi\)
\(642\) 0 0
\(643\) − 3.72899i − 0.147057i −0.997293 0.0735285i \(-0.976574\pi\)
0.997293 0.0735285i \(-0.0234260\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −45.6179 −1.79342 −0.896712 0.442615i \(-0.854051\pi\)
−0.896712 + 0.442615i \(0.854051\pi\)
\(648\) 0 0
\(649\) 1.75878 0.0690382
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 6.34542i 0.248315i 0.992263 + 0.124158i \(0.0396229\pi\)
−0.992263 + 0.124158i \(0.960377\pi\)
\(654\) 0 0
\(655\) −2.12090 −0.0828706
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 12.6820i − 0.494022i −0.969013 0.247011i \(-0.920552\pi\)
0.969013 0.247011i \(-0.0794484\pi\)
\(660\) 0 0
\(661\) − 7.30385i − 0.284087i −0.989860 0.142043i \(-0.954633\pi\)
0.989860 0.142043i \(-0.0453672\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.785249 0.0304507
\(666\) 0 0
\(667\) − 8.20591i − 0.317734i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3.03909 −0.117323
\(672\) 0 0
\(673\) −4.21222 −0.162369 −0.0811846 0.996699i \(-0.525870\pi\)
−0.0811846 + 0.996699i \(0.525870\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 18.0658i − 0.694326i −0.937805 0.347163i \(-0.887145\pi\)
0.937805 0.347163i \(-0.112855\pi\)
\(678\) 0 0
\(679\) −3.42009 −0.131251
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 41.3196i 1.58105i 0.612429 + 0.790526i \(0.290192\pi\)
−0.612429 + 0.790526i \(0.709808\pi\)
\(684\) 0 0
\(685\) − 0.965835i − 0.0369027i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −19.4833 −0.742254
\(690\) 0 0
\(691\) − 44.2400i − 1.68297i −0.540281 0.841485i \(-0.681682\pi\)
0.540281 0.841485i \(-0.318318\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.58868 0.0602622
\(696\) 0 0
\(697\) −6.01597 −0.227871
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 33.0921i 1.24987i 0.780676 + 0.624936i \(0.214875\pi\)
−0.780676 + 0.624936i \(0.785125\pi\)
\(702\) 0 0
\(703\) 30.0898 1.13486
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 0.203905i − 0.00766863i
\(708\) 0 0
\(709\) 28.0152i 1.05213i 0.850444 + 0.526066i \(0.176333\pi\)
−0.850444 + 0.526066i \(0.823667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.7812 0.928062
\(714\) 0 0
\(715\) − 0.0821419i − 0.00307193i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 5.44311 0.202994 0.101497 0.994836i \(-0.467637\pi\)
0.101497 + 0.994836i \(0.467637\pi\)
\(720\) 0 0
\(721\) −7.16809 −0.266954
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.25359i 0.343670i
\(726\) 0 0
\(727\) −14.7480 −0.546972 −0.273486 0.961876i \(-0.588177\pi\)
−0.273486 + 0.961876i \(0.588177\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 10.9236i − 0.404022i
\(732\) 0 0
\(733\) − 45.4197i − 1.67762i −0.544428 0.838808i \(-0.683253\pi\)
0.544428 0.838808i \(-0.316747\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.58324 −0.0951550
\(738\) 0 0
\(739\) − 3.03287i − 0.111566i −0.998443 0.0557830i \(-0.982234\pi\)
0.998443 0.0557830i \(-0.0177655\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 41.3290 1.51621 0.758106 0.652131i \(-0.226125\pi\)
0.758106 + 0.652131i \(0.226125\pi\)
\(744\) 0 0
\(745\) 2.52512 0.0925131
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.46742i 0.272853i
\(750\) 0 0
\(751\) −18.6220 −0.679527 −0.339763 0.940511i \(-0.610347\pi\)
−0.339763 + 0.940511i \(0.610347\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 0.619591i − 0.0225492i
\(756\) 0 0
\(757\) 48.7279i 1.77104i 0.464597 + 0.885522i \(0.346199\pi\)
−0.464597 + 0.885522i \(0.653801\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −26.1095 −0.946468 −0.473234 0.880937i \(-0.656913\pi\)
−0.473234 + 0.880937i \(0.656913\pi\)
\(762\) 0 0
\(763\) − 2.11355i − 0.0765157i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.39293 0.266943
\(768\) 0 0
\(769\) 24.6590 0.889225 0.444612 0.895723i \(-0.353341\pi\)
0.444612 + 0.895723i \(0.353341\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 15.7370i 0.566019i 0.959117 + 0.283010i \(0.0913329\pi\)
−0.959117 + 0.283010i \(0.908667\pi\)
\(774\) 0 0
\(775\) −27.9451 −1.00382
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 16.4399i 0.589021i
\(780\) 0 0
\(781\) − 0.251191i − 0.00898834i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.50207 0.0536113
\(786\) 0 0
\(787\) 48.6354i 1.73367i 0.498599 + 0.866833i \(0.333848\pi\)
−0.498599 + 0.866833i \(0.666152\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.72972 0.168170
\(792\) 0 0
\(793\) −12.7746 −0.453641
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 53.4156i 1.89208i 0.324054 + 0.946038i \(0.394954\pi\)
−0.324054 + 0.946038i \(0.605046\pi\)
\(798\) 0 0
\(799\) 18.1319 0.641459
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.84538i 0.206279i
\(804\) 0 0
\(805\) − 0.506753i − 0.0178607i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −46.1176 −1.62141 −0.810704 0.585456i \(-0.800915\pi\)
−0.810704 + 0.585456i \(0.800915\pi\)
\(810\) 0 0
\(811\) − 13.2874i − 0.466585i −0.972407 0.233293i \(-0.925050\pi\)
0.972407 0.233293i \(-0.0749500\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.65606 0.0930376
\(816\) 0 0
\(817\) −29.8509 −1.04435
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 30.5087i − 1.06476i −0.846505 0.532380i \(-0.821298\pi\)
0.846505 0.532380i \(-0.178702\pi\)
\(822\) 0 0
\(823\) 19.6955 0.686542 0.343271 0.939236i \(-0.388465\pi\)
0.343271 + 0.939236i \(0.388465\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.77145i 0.200693i 0.994953 + 0.100347i \(0.0319951\pi\)
−0.994953 + 0.100347i \(0.968005\pi\)
\(828\) 0 0
\(829\) 50.3002i 1.74700i 0.486826 + 0.873499i \(0.338155\pi\)
−0.486826 + 0.873499i \(0.661845\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.50762 0.0868840
\(834\) 0 0
\(835\) 0.816898i 0.0282699i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 26.5035 0.915004 0.457502 0.889209i \(-0.348744\pi\)
0.457502 + 0.889209i \(0.348744\pi\)
\(840\) 0 0
\(841\) 25.5568 0.881268
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.14441i 0.0393688i
\(846\) 0 0
\(847\) 10.8295 0.372105
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 19.4181i − 0.665645i
\(852\) 0 0
\(853\) 9.22041i 0.315701i 0.987463 + 0.157850i \(0.0504564\pi\)
−0.987463 + 0.157850i \(0.949544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.1124 1.40437 0.702187 0.711993i \(-0.252207\pi\)
0.702187 + 0.711993i \(0.252207\pi\)
\(858\) 0 0
\(859\) − 7.51805i − 0.256513i −0.991741 0.128256i \(-0.959062\pi\)
0.991741 0.128256i \(-0.0409380\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.5122 0.698245 0.349122 0.937077i \(-0.386480\pi\)
0.349122 + 0.937077i \(0.386480\pi\)
\(864\) 0 0
\(865\) 2.38599 0.0811261
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.38604i 0.114863i
\(870\) 0 0
\(871\) −10.8585 −0.367927
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.14441i 0.0386881i
\(876\) 0 0
\(877\) 53.2820i 1.79921i 0.436710 + 0.899603i \(0.356144\pi\)
−0.436710 + 0.899603i \(0.643856\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.1591 1.25192 0.625960 0.779855i \(-0.284707\pi\)
0.625960 + 0.779855i \(0.284707\pi\)
\(882\) 0 0
\(883\) 30.5599i 1.02842i 0.857664 + 0.514211i \(0.171915\pi\)
−0.857664 + 0.514211i \(0.828085\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −47.5640 −1.59704 −0.798521 0.601967i \(-0.794384\pi\)
−0.798521 + 0.601967i \(0.794384\pi\)
\(888\) 0 0
\(889\) 9.55123 0.320338
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 49.5491i − 1.65810i
\(894\) 0 0
\(895\) −0.860905 −0.0287769
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 10.3982i − 0.346800i
\(900\) 0 0
\(901\) − 28.1459i − 0.937676i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.513919 0.0170833
\(906\) 0 0
\(907\) 45.9789i 1.52670i 0.645982 + 0.763352i \(0.276448\pi\)
−0.645982 + 0.763352i \(0.723552\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −9.42797 −0.312363 −0.156181 0.987728i \(-0.549918\pi\)
−0.156181 + 0.987728i \(0.549918\pi\)
\(912\) 0 0
\(913\) 2.01532 0.0666974
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.5084i 0.611202i
\(918\) 0 0
\(919\) 1.64588 0.0542924 0.0271462 0.999631i \(-0.491358\pi\)
0.0271462 + 0.999631i \(0.491358\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 1.05587i − 0.0347543i
\(924\) 0 0
\(925\) 21.8973i 0.719979i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 52.1072 1.70958 0.854791 0.518973i \(-0.173685\pi\)
0.854791 + 0.518973i \(0.173685\pi\)
\(930\) 0 0
\(931\) − 6.85261i − 0.224585i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.118664 0.00388072
\(936\) 0 0
\(937\) 10.2023 0.333294 0.166647 0.986017i \(-0.446706\pi\)
0.166647 + 0.986017i \(0.446706\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.500685i 0.0163219i 0.999967 + 0.00816093i \(0.00259773\pi\)
−0.999967 + 0.00816093i \(0.997402\pi\)
\(942\) 0 0
\(943\) 10.6093 0.345487
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 49.4247i − 1.60609i −0.595921 0.803043i \(-0.703213\pi\)
0.595921 0.803043i \(-0.296787\pi\)
\(948\) 0 0
\(949\) 24.5707i 0.797599i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −32.0038 −1.03670 −0.518352 0.855167i \(-0.673454\pi\)
−0.518352 + 0.855167i \(0.673454\pi\)
\(954\) 0 0
\(955\) 2.82727i 0.0914882i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.42852 −0.272171
\(960\) 0 0
\(961\) 0.401788 0.0129609
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 0.0384818i − 0.00123877i
\(966\) 0 0
\(967\) 45.6934 1.46940 0.734700 0.678392i \(-0.237323\pi\)
0.734700 + 0.678392i \(0.237323\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 9.06582i 0.290936i 0.989363 + 0.145468i \(0.0464688\pi\)
−0.989363 + 0.145468i \(0.953531\pi\)
\(972\) 0 0
\(973\) − 13.8639i − 0.444456i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −20.6361 −0.660207 −0.330103 0.943945i \(-0.607084\pi\)
−0.330103 + 0.943945i \(0.607084\pi\)
\(978\) 0 0
\(979\) 4.30369i 0.137547i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 15.4308 0.492167 0.246083 0.969249i \(-0.420856\pi\)
0.246083 + 0.969249i \(0.420856\pi\)
\(984\) 0 0
\(985\) −1.52509 −0.0485933
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19.2640i 0.612560i
\(990\) 0 0
\(991\) 25.7039 0.816510 0.408255 0.912868i \(-0.366137\pi\)
0.408255 + 0.912868i \(0.366137\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.05961i 0.0335918i
\(996\) 0 0
\(997\) 18.0217i 0.570753i 0.958416 + 0.285376i \(0.0921186\pi\)
−0.958416 + 0.285376i \(0.907881\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6048.2.c.e.3025.11 20
3.2 odd 2 inner 6048.2.c.e.3025.9 20
4.3 odd 2 1512.2.c.e.757.20 yes 20
8.3 odd 2 1512.2.c.e.757.19 yes 20
8.5 even 2 inner 6048.2.c.e.3025.10 20
12.11 even 2 1512.2.c.e.757.1 20
24.5 odd 2 inner 6048.2.c.e.3025.12 20
24.11 even 2 1512.2.c.e.757.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.e.757.1 20 12.11 even 2
1512.2.c.e.757.2 yes 20 24.11 even 2
1512.2.c.e.757.19 yes 20 8.3 odd 2
1512.2.c.e.757.20 yes 20 4.3 odd 2
6048.2.c.e.3025.9 20 3.2 odd 2 inner
6048.2.c.e.3025.10 20 8.5 even 2 inner
6048.2.c.e.3025.11 20 1.1 even 1 trivial
6048.2.c.e.3025.12 20 24.5 odd 2 inner