Properties

Label 6048.2.h.i.2591.16
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 24
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.16
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.i.2591.15

$q$-expansion

\(f(q)\) \(=\) \(q+1.29686i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+1.29686i q^{5} -1.00000i q^{7} -2.21561 q^{11} -1.03730 q^{13} +2.24600i q^{17} +8.63898i q^{19} +8.96997 q^{23} +3.31815 q^{25} -3.26362i q^{29} +4.37958i q^{31} +1.29686 q^{35} -7.21713 q^{37} +7.58939i q^{41} -12.9813i q^{43} -5.24078 q^{47} -1.00000 q^{49} -11.1694i q^{53} -2.87334i q^{55} +2.46447 q^{59} -10.3300 q^{61} -1.34523i q^{65} -4.46833i q^{67} +13.5342 q^{71} -12.8567 q^{73} +2.21561i q^{77} +9.14521i q^{79} +9.96748 q^{83} -2.91275 q^{85} -3.48208i q^{89} +1.03730i q^{91} -11.2036 q^{95} -6.30706 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q + O(q^{10}) \) \( 24q - 8q^{13} - 24q^{25} + 48q^{37} - 24q^{49} - 48q^{61} - 32q^{73} - 80q^{85} + 64q^{97} + O(q^{100}) \)

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
\(5\) 1.29686i 0.579975i 0.957030 + 0.289987i \(0.0936511\pi\)
−0.957030 + 0.289987i \(0.906349\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.21561 −0.668031 −0.334016 0.942567i \(-0.608404\pi\)
−0.334016 + 0.942567i \(0.608404\pi\)
\(12\) 0 0
\(13\) −1.03730 −0.287695 −0.143847 0.989600i \(-0.545947\pi\)
−0.143847 + 0.989600i \(0.545947\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.24600i 0.544734i 0.962193 + 0.272367i \(0.0878065\pi\)
−0.962193 + 0.272367i \(0.912193\pi\)
\(18\) 0 0
\(19\) 8.63898i 1.98192i 0.134169 + 0.990959i \(0.457164\pi\)
−0.134169 + 0.990959i \(0.542836\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.96997 1.87037 0.935184 0.354161i \(-0.115234\pi\)
0.935184 + 0.354161i \(0.115234\pi\)
\(24\) 0 0
\(25\) 3.31815 0.663629
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 3.26362i − 0.606038i −0.952984 0.303019i \(-0.902005\pi\)
0.952984 0.303019i \(-0.0979946\pi\)
\(30\) 0 0
\(31\) 4.37958i 0.786595i 0.919411 + 0.393298i \(0.128666\pi\)
−0.919411 + 0.393298i \(0.871334\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.29686 0.219210
\(36\) 0 0
\(37\) −7.21713 −1.18649 −0.593244 0.805023i \(-0.702153\pi\)
−0.593244 + 0.805023i \(0.702153\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.58939i 1.18526i 0.805473 + 0.592632i \(0.201911\pi\)
−0.805473 + 0.592632i \(0.798089\pi\)
\(42\) 0 0
\(43\) − 12.9813i − 1.97962i −0.142384 0.989811i \(-0.545477\pi\)
0.142384 0.989811i \(-0.454523\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.24078 −0.764447 −0.382223 0.924070i \(-0.624842\pi\)
−0.382223 + 0.924070i \(0.624842\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 11.1694i − 1.53423i −0.641511 0.767114i \(-0.721692\pi\)
0.641511 0.767114i \(-0.278308\pi\)
\(54\) 0 0
\(55\) − 2.87334i − 0.387441i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.46447 0.320846 0.160423 0.987048i \(-0.448714\pi\)
0.160423 + 0.987048i \(0.448714\pi\)
\(60\) 0 0
\(61\) −10.3300 −1.32262 −0.661311 0.750112i \(-0.730000\pi\)
−0.661311 + 0.750112i \(0.730000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.34523i − 0.166856i
\(66\) 0 0
\(67\) − 4.46833i − 0.545894i −0.962029 0.272947i \(-0.912002\pi\)
0.962029 0.272947i \(-0.0879983\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 13.5342 1.60621 0.803107 0.595835i \(-0.203179\pi\)
0.803107 + 0.595835i \(0.203179\pi\)
\(72\) 0 0
\(73\) −12.8567 −1.50476 −0.752382 0.658727i \(-0.771095\pi\)
−0.752382 + 0.658727i \(0.771095\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.21561i 0.252492i
\(78\) 0 0
\(79\) 9.14521i 1.02892i 0.857515 + 0.514458i \(0.172007\pi\)
−0.857515 + 0.514458i \(0.827993\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.96748 1.09407 0.547037 0.837109i \(-0.315756\pi\)
0.547037 + 0.837109i \(0.315756\pi\)
\(84\) 0 0
\(85\) −2.91275 −0.315932
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 3.48208i − 0.369100i −0.982823 0.184550i \(-0.940917\pi\)
0.982823 0.184550i \(-0.0590828\pi\)
\(90\) 0 0
\(91\) 1.03730i 0.108738i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −11.2036 −1.14946
\(96\) 0 0
\(97\) −6.30706 −0.640385 −0.320192 0.947353i \(-0.603748\pi\)
−0.320192 + 0.947353i \(0.603748\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.7845i 1.07310i 0.843868 + 0.536550i \(0.180273\pi\)
−0.843868 + 0.536550i \(0.819727\pi\)
\(102\) 0 0
\(103\) 7.98276i 0.786565i 0.919418 + 0.393282i \(0.128661\pi\)
−0.919418 + 0.393282i \(0.871339\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −20.4082 −1.97293 −0.986465 0.163969i \(-0.947570\pi\)
−0.986465 + 0.163969i \(0.947570\pi\)
\(108\) 0 0
\(109\) 9.98623 0.956507 0.478254 0.878222i \(-0.341270\pi\)
0.478254 + 0.878222i \(0.341270\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.60077i 0.903165i 0.892229 + 0.451582i \(0.149140\pi\)
−0.892229 + 0.451582i \(0.850860\pi\)
\(114\) 0 0
\(115\) 11.6328i 1.08477i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.24600 0.205890
\(120\) 0 0
\(121\) −6.09107 −0.553734
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10.7875i 0.964863i
\(126\) 0 0
\(127\) 9.84909i 0.873965i 0.899470 + 0.436983i \(0.143953\pi\)
−0.899470 + 0.436983i \(0.856047\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −19.3232 −1.68827 −0.844137 0.536127i \(-0.819887\pi\)
−0.844137 + 0.536127i \(0.819887\pi\)
\(132\) 0 0
\(133\) 8.63898 0.749094
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 6.56515i − 0.560899i −0.959869 0.280449i \(-0.909517\pi\)
0.959869 0.280449i \(-0.0904834\pi\)
\(138\) 0 0
\(139\) 19.8276i 1.68176i 0.541225 + 0.840878i \(0.317961\pi\)
−0.541225 + 0.840878i \(0.682039\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.29825 0.192189
\(144\) 0 0
\(145\) 4.23246 0.351487
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.58523i 0.703330i 0.936126 + 0.351665i \(0.114384\pi\)
−0.936126 + 0.351665i \(0.885616\pi\)
\(150\) 0 0
\(151\) 14.8203i 1.20606i 0.797719 + 0.603030i \(0.206040\pi\)
−0.797719 + 0.603030i \(0.793960\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.67971 −0.456205
\(156\) 0 0
\(157\) −19.6744 −1.57019 −0.785093 0.619378i \(-0.787385\pi\)
−0.785093 + 0.619378i \(0.787385\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 8.96997i − 0.706933i
\(162\) 0 0
\(163\) 6.75148i 0.528816i 0.964411 + 0.264408i \(0.0851766\pi\)
−0.964411 + 0.264408i \(0.914823\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.10240 −0.472218 −0.236109 0.971727i \(-0.575872\pi\)
−0.236109 + 0.971727i \(0.575872\pi\)
\(168\) 0 0
\(169\) −11.9240 −0.917232
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 2.60237i − 0.197855i −0.995095 0.0989274i \(-0.968459\pi\)
0.995095 0.0989274i \(-0.0315412\pi\)
\(174\) 0 0
\(175\) − 3.31815i − 0.250828i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.0508 1.19969 0.599847 0.800114i \(-0.295228\pi\)
0.599847 + 0.800114i \(0.295228\pi\)
\(180\) 0 0
\(181\) 15.6876 1.16605 0.583026 0.812453i \(-0.301869\pi\)
0.583026 + 0.812453i \(0.301869\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 9.35962i − 0.688133i
\(186\) 0 0
\(187\) − 4.97625i − 0.363900i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.8131 1.50598 0.752991 0.658030i \(-0.228610\pi\)
0.752991 + 0.658030i \(0.228610\pi\)
\(192\) 0 0
\(193\) −14.9102 −1.07326 −0.536631 0.843817i \(-0.680303\pi\)
−0.536631 + 0.843817i \(0.680303\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.28412i 0.305231i 0.988286 + 0.152615i \(0.0487696\pi\)
−0.988286 + 0.152615i \(0.951230\pi\)
\(198\) 0 0
\(199\) 21.5110i 1.52488i 0.647061 + 0.762438i \(0.275998\pi\)
−0.647061 + 0.762438i \(0.724002\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.26362 −0.229061
\(204\) 0 0
\(205\) −9.84240 −0.687423
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 19.1406i − 1.32398i
\(210\) 0 0
\(211\) 1.01467i 0.0698529i 0.999390 + 0.0349264i \(0.0111197\pi\)
−0.999390 + 0.0349264i \(0.988880\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.8349 1.14813
\(216\) 0 0
\(217\) 4.37958 0.297305
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.32977i − 0.156717i
\(222\) 0 0
\(223\) − 0.114187i − 0.00764650i −0.999993 0.00382325i \(-0.998783\pi\)
0.999993 0.00382325i \(-0.00121698\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −25.0865 −1.66505 −0.832526 0.553986i \(-0.813106\pi\)
−0.832526 + 0.553986i \(0.813106\pi\)
\(228\) 0 0
\(229\) 4.54175 0.300127 0.150064 0.988676i \(-0.452052\pi\)
0.150064 + 0.988676i \(0.452052\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.4905i 1.53891i 0.638701 + 0.769455i \(0.279472\pi\)
−0.638701 + 0.769455i \(0.720528\pi\)
\(234\) 0 0
\(235\) − 6.79658i − 0.443360i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.821010 0.0531067 0.0265534 0.999647i \(-0.491547\pi\)
0.0265534 + 0.999647i \(0.491547\pi\)
\(240\) 0 0
\(241\) −2.69884 −0.173848 −0.0869239 0.996215i \(-0.527704\pi\)
−0.0869239 + 0.996215i \(0.527704\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.29686i − 0.0828535i
\(246\) 0 0
\(247\) − 8.96120i − 0.570187i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 14.7256 0.929475 0.464737 0.885449i \(-0.346149\pi\)
0.464737 + 0.885449i \(0.346149\pi\)
\(252\) 0 0
\(253\) −19.8740 −1.24947
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.11220i 0.568404i 0.958764 + 0.284202i \(0.0917286\pi\)
−0.958764 + 0.284202i \(0.908271\pi\)
\(258\) 0 0
\(259\) 7.21713i 0.448450i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.6206 0.963207 0.481604 0.876389i \(-0.340055\pi\)
0.481604 + 0.876389i \(0.340055\pi\)
\(264\) 0 0
\(265\) 14.4851 0.889814
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 28.3937i − 1.73120i −0.500739 0.865598i \(-0.666938\pi\)
0.500739 0.865598i \(-0.333062\pi\)
\(270\) 0 0
\(271\) 4.61964i 0.280623i 0.990107 + 0.140311i \(0.0448104\pi\)
−0.990107 + 0.140311i \(0.955190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.35172 −0.443325
\(276\) 0 0
\(277\) −23.6330 −1.41997 −0.709985 0.704217i \(-0.751298\pi\)
−0.709985 + 0.704217i \(0.751298\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 9.50949i − 0.567288i −0.958930 0.283644i \(-0.908457\pi\)
0.958930 0.283644i \(-0.0915435\pi\)
\(282\) 0 0
\(283\) − 8.52211i − 0.506587i −0.967390 0.253293i \(-0.918486\pi\)
0.967390 0.253293i \(-0.0815138\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.58939 0.447988
\(288\) 0 0
\(289\) 11.9555 0.703264
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 17.6710i − 1.03235i −0.856483 0.516175i \(-0.827355\pi\)
0.856483 0.516175i \(-0.172645\pi\)
\(294\) 0 0
\(295\) 3.19608i 0.186083i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.30454 −0.538095
\(300\) 0 0
\(301\) −12.9813 −0.748227
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 13.3966i − 0.767088i
\(306\) 0 0
\(307\) 9.51533i 0.543068i 0.962429 + 0.271534i \(0.0875310\pi\)
−0.962429 + 0.271534i \(0.912469\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.4716 −1.16084 −0.580419 0.814318i \(-0.697111\pi\)
−0.580419 + 0.814318i \(0.697111\pi\)
\(312\) 0 0
\(313\) −9.49300 −0.536576 −0.268288 0.963339i \(-0.586458\pi\)
−0.268288 + 0.963339i \(0.586458\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 16.5518i − 0.929644i −0.885404 0.464822i \(-0.846118\pi\)
0.885404 0.464822i \(-0.153882\pi\)
\(318\) 0 0
\(319\) 7.23090i 0.404853i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −19.4031 −1.07962
\(324\) 0 0
\(325\) −3.44191 −0.190923
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.24078i 0.288934i
\(330\) 0 0
\(331\) 30.8263i 1.69436i 0.531302 + 0.847182i \(0.321703\pi\)
−0.531302 + 0.847182i \(0.678297\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.79481 0.316605
\(336\) 0 0
\(337\) −15.4712 −0.842768 −0.421384 0.906882i \(-0.638456\pi\)
−0.421384 + 0.906882i \(0.638456\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 9.70344i − 0.525470i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.5979 −0.729971 −0.364986 0.931013i \(-0.618926\pi\)
−0.364986 + 0.931013i \(0.618926\pi\)
\(348\) 0 0
\(349\) −7.39184 −0.395676 −0.197838 0.980235i \(-0.563392\pi\)
−0.197838 + 0.980235i \(0.563392\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.5532i 0.721365i 0.932689 + 0.360683i \(0.117456\pi\)
−0.932689 + 0.360683i \(0.882544\pi\)
\(354\) 0 0
\(355\) 17.5520i 0.931563i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −6.96702 −0.367705 −0.183853 0.982954i \(-0.558857\pi\)
−0.183853 + 0.982954i \(0.558857\pi\)
\(360\) 0 0
\(361\) −55.6319 −2.92800
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 16.6734i − 0.872725i
\(366\) 0 0
\(367\) − 27.9129i − 1.45704i −0.685022 0.728522i \(-0.740208\pi\)
0.685022 0.728522i \(-0.259792\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −11.1694 −0.579884
\(372\) 0 0
\(373\) −3.28235 −0.169954 −0.0849769 0.996383i \(-0.527082\pi\)
−0.0849769 + 0.996383i \(0.527082\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.38534i 0.174354i
\(378\) 0 0
\(379\) 2.59351i 0.133220i 0.997779 + 0.0666098i \(0.0212183\pi\)
−0.997779 + 0.0666098i \(0.978782\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.5700 0.642295 0.321147 0.947029i \(-0.395931\pi\)
0.321147 + 0.947029i \(0.395931\pi\)
\(384\) 0 0
\(385\) −2.87334 −0.146439
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 33.0483i 1.67562i 0.545964 + 0.837809i \(0.316164\pi\)
−0.545964 + 0.837809i \(0.683836\pi\)
\(390\) 0 0
\(391\) 20.1465i 1.01885i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −11.8601 −0.596746
\(396\) 0 0
\(397\) 21.8925 1.09875 0.549377 0.835575i \(-0.314865\pi\)
0.549377 + 0.835575i \(0.314865\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.11600i 0.455231i 0.973751 + 0.227616i \(0.0730929\pi\)
−0.973751 + 0.227616i \(0.926907\pi\)
\(402\) 0 0
\(403\) − 4.54293i − 0.226299i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.9903 0.792612
\(408\) 0 0
\(409\) 15.6721 0.774935 0.387467 0.921883i \(-0.373350\pi\)
0.387467 + 0.921883i \(0.373350\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 2.46447i − 0.121269i
\(414\) 0 0
\(415\) 12.9265i 0.634535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −21.9407 −1.07187 −0.535936 0.844259i \(-0.680041\pi\)
−0.535936 + 0.844259i \(0.680041\pi\)
\(420\) 0 0
\(421\) −17.6611 −0.860752 −0.430376 0.902650i \(-0.641619\pi\)
−0.430376 + 0.902650i \(0.641619\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.45255i 0.361502i
\(426\) 0 0
\(427\) 10.3300i 0.499904i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.2362 1.07108 0.535540 0.844510i \(-0.320108\pi\)
0.535540 + 0.844510i \(0.320108\pi\)
\(432\) 0 0
\(433\) −36.2225 −1.74074 −0.870371 0.492396i \(-0.836121\pi\)
−0.870371 + 0.492396i \(0.836121\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 77.4914i 3.70692i
\(438\) 0 0
\(439\) − 4.36466i − 0.208314i −0.994561 0.104157i \(-0.966786\pi\)
0.994561 0.104157i \(-0.0332145\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.0758 1.09636 0.548181 0.836359i \(-0.315320\pi\)
0.548181 + 0.836359i \(0.315320\pi\)
\(444\) 0 0
\(445\) 4.51579 0.214069
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 25.6147i − 1.20883i −0.796669 0.604416i \(-0.793407\pi\)
0.796669 0.604416i \(-0.206593\pi\)
\(450\) 0 0
\(451\) − 16.8151i − 0.791794i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.34523 −0.0630655
\(456\) 0 0
\(457\) 19.0485 0.891052 0.445526 0.895269i \(-0.353017\pi\)
0.445526 + 0.895269i \(0.353017\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 18.6533i − 0.868773i −0.900727 0.434386i \(-0.856965\pi\)
0.900727 0.434386i \(-0.143035\pi\)
\(462\) 0 0
\(463\) − 26.5347i − 1.23317i −0.787288 0.616585i \(-0.788516\pi\)
0.787288 0.616585i \(-0.211484\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.04573 0.140939 0.0704697 0.997514i \(-0.477550\pi\)
0.0704697 + 0.997514i \(0.477550\pi\)
\(468\) 0 0
\(469\) −4.46833 −0.206328
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 28.7614i 1.32245i
\(474\) 0 0
\(475\) 28.6654i 1.31526i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.4984 0.662450 0.331225 0.943552i \(-0.392538\pi\)
0.331225 + 0.943552i \(0.392538\pi\)
\(480\) 0 0
\(481\) 7.48631 0.341347
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 8.17939i − 0.371407i
\(486\) 0 0
\(487\) 15.1551i 0.686743i 0.939200 + 0.343371i \(0.111569\pi\)
−0.939200 + 0.343371i \(0.888431\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −5.74814 −0.259410 −0.129705 0.991553i \(-0.541403\pi\)
−0.129705 + 0.991553i \(0.541403\pi\)
\(492\) 0 0
\(493\) 7.33007 0.330130
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 13.5342i − 0.607091i
\(498\) 0 0
\(499\) − 10.9764i − 0.491373i −0.969349 0.245686i \(-0.920987\pi\)
0.969349 0.245686i \(-0.0790134\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.71925 −0.210421 −0.105211 0.994450i \(-0.533552\pi\)
−0.105211 + 0.994450i \(0.533552\pi\)
\(504\) 0 0
\(505\) −13.9860 −0.622371
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.5925i 1.26734i 0.773604 + 0.633669i \(0.218452\pi\)
−0.773604 + 0.633669i \(0.781548\pi\)
\(510\) 0 0
\(511\) 12.8567i 0.568747i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.3525 −0.456188
\(516\) 0 0
\(517\) 11.6115 0.510674
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 29.2439i − 1.28120i −0.767876 0.640598i \(-0.778686\pi\)
0.767876 0.640598i \(-0.221314\pi\)
\(522\) 0 0
\(523\) 17.1008i 0.747765i 0.927476 + 0.373882i \(0.121974\pi\)
−0.927476 + 0.373882i \(0.878026\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.83652 −0.428486
\(528\) 0 0
\(529\) 57.4604 2.49828
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 7.87246i − 0.340994i
\(534\) 0 0
\(535\) − 26.4666i − 1.14425i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.21561 0.0954331
\(540\) 0 0
\(541\) 27.8421 1.19702 0.598512 0.801114i \(-0.295759\pi\)
0.598512 + 0.801114i \(0.295759\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.9508i 0.554750i
\(546\) 0 0
\(547\) − 0.515933i − 0.0220597i −0.999939 0.0110299i \(-0.996489\pi\)
0.999939 0.0110299i \(-0.00351099\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 28.1943 1.20112
\(552\) 0 0
\(553\) 9.14521 0.388894
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.9248i 0.759498i 0.925090 + 0.379749i \(0.123990\pi\)
−0.925090 + 0.379749i \(0.876010\pi\)
\(558\) 0 0
\(559\) 13.4654i 0.569527i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.4146 −1.36611 −0.683056 0.730366i \(-0.739350\pi\)
−0.683056 + 0.730366i \(0.739350\pi\)
\(564\) 0 0
\(565\) −12.4509 −0.523813
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.202528i 0.00849044i 0.999991 + 0.00424522i \(0.00135130\pi\)
−0.999991 + 0.00424522i \(0.998649\pi\)
\(570\) 0 0
\(571\) 26.2522i 1.09862i 0.835619 + 0.549310i \(0.185109\pi\)
−0.835619 + 0.549310i \(0.814891\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 29.7637 1.24123
\(576\) 0 0
\(577\) −6.79441 −0.282855 −0.141428 0.989949i \(-0.545169\pi\)
−0.141428 + 0.989949i \(0.545169\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 9.96748i − 0.413521i
\(582\) 0 0
\(583\) 24.7469i 1.02491i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −21.1087 −0.871248 −0.435624 0.900129i \(-0.643472\pi\)
−0.435624 + 0.900129i \(0.643472\pi\)
\(588\) 0 0
\(589\) −37.8351 −1.55897
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 36.1673i − 1.48521i −0.669727 0.742607i \(-0.733589\pi\)
0.669727 0.742607i \(-0.266411\pi\)
\(594\) 0 0
\(595\) 2.91275i 0.119411i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.5253 1.28809 0.644045 0.764988i \(-0.277255\pi\)
0.644045 + 0.764988i \(0.277255\pi\)
\(600\) 0 0
\(601\) −20.2761 −0.827081 −0.413540 0.910486i \(-0.635708\pi\)
−0.413540 + 0.910486i \(0.635708\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 7.89929i − 0.321152i
\(606\) 0 0
\(607\) 32.7857i 1.33073i 0.746519 + 0.665364i \(0.231724\pi\)
−0.746519 + 0.665364i \(0.768276\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.43626 0.219927
\(612\) 0 0
\(613\) 1.73331 0.0700077 0.0350039 0.999387i \(-0.488856\pi\)
0.0350039 + 0.999387i \(0.488856\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.2030i 1.05489i 0.849589 + 0.527446i \(0.176850\pi\)
−0.849589 + 0.527446i \(0.823150\pi\)
\(618\) 0 0
\(619\) − 17.2543i − 0.693509i −0.937956 0.346754i \(-0.887284\pi\)
0.937956 0.346754i \(-0.112716\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.48208 −0.139507
\(624\) 0 0
\(625\) 2.60083 0.104033
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 16.2096i − 0.646321i
\(630\) 0 0
\(631\) − 41.7871i − 1.66352i −0.555138 0.831758i \(-0.687334\pi\)
0.555138 0.831758i \(-0.312666\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −12.7729 −0.506878
\(636\) 0 0
\(637\) 1.03730 0.0410993
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 33.9794i − 1.34211i −0.741410 0.671053i \(-0.765842\pi\)
0.741410 0.671053i \(-0.234158\pi\)
\(642\) 0 0
\(643\) − 17.8079i − 0.702275i −0.936324 0.351138i \(-0.885795\pi\)
0.936324 0.351138i \(-0.114205\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.6184 0.967849 0.483924 0.875110i \(-0.339211\pi\)
0.483924 + 0.875110i \(0.339211\pi\)
\(648\) 0 0
\(649\) −5.46030 −0.214335
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.37615i 0.0538529i 0.999637 + 0.0269264i \(0.00857199\pi\)
−0.999637 + 0.0269264i \(0.991428\pi\)
\(654\) 0 0
\(655\) − 25.0595i − 0.979157i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 8.98020 0.349819 0.174909 0.984585i \(-0.444037\pi\)
0.174909 + 0.984585i \(0.444037\pi\)
\(660\) 0 0
\(661\) −1.39794 −0.0543734 −0.0271867 0.999630i \(-0.508655\pi\)
−0.0271867 + 0.999630i \(0.508655\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.2036i 0.434456i
\(666\) 0 0
\(667\) − 29.2745i − 1.13351i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 22.8873 0.883554
\(672\) 0 0
\(673\) −22.8752 −0.881773 −0.440886 0.897563i \(-0.645336\pi\)
−0.440886 + 0.897563i \(0.645336\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.3953i 0.553258i 0.960977 + 0.276629i \(0.0892173\pi\)
−0.960977 + 0.276629i \(0.910783\pi\)
\(678\) 0 0
\(679\) 6.30706i 0.242043i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.558805 −0.0213821 −0.0106910 0.999943i \(-0.503403\pi\)
−0.0106910 + 0.999943i \(0.503403\pi\)
\(684\) 0 0
\(685\) 8.51410 0.325307
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 11.5860i 0.441390i
\(690\) 0 0
\(691\) − 0.0964465i − 0.00366900i −0.999998 0.00183450i \(-0.999416\pi\)
0.999998 0.00183450i \(-0.000583940\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −25.7137 −0.975375
\(696\) 0 0
\(697\) −17.0458 −0.645654
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.5851i 0.664179i 0.943248 + 0.332090i \(0.107754\pi\)
−0.943248 + 0.332090i \(0.892246\pi\)
\(702\) 0 0
\(703\) − 62.3486i − 2.35152i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.7845 0.405594
\(708\) 0 0
\(709\) −22.4705 −0.843896 −0.421948 0.906620i \(-0.638654\pi\)
−0.421948 + 0.906620i \(0.638654\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 39.2847i 1.47122i
\(714\) 0 0
\(715\) 2.98051i 0.111465i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.41643 −0.127411 −0.0637057 0.997969i \(-0.520292\pi\)
−0.0637057 + 0.997969i \(0.520292\pi\)
\(720\) 0 0
\(721\) 7.98276 0.297294
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 10.8292i − 0.402185i
\(726\) 0 0
\(727\) 6.76927i 0.251059i 0.992090 + 0.125529i \(0.0400629\pi\)
−0.992090 + 0.125529i \(0.959937\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 29.1559 1.07837
\(732\) 0 0
\(733\) −8.85387 −0.327025 −0.163513 0.986541i \(-0.552282\pi\)
−0.163513 + 0.986541i \(0.552282\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.90008i 0.364674i
\(738\) 0 0
\(739\) − 7.90087i − 0.290638i −0.989385 0.145319i \(-0.953579\pi\)
0.989385 0.145319i \(-0.0464209\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10.8356 −0.397521 −0.198761 0.980048i \(-0.563692\pi\)
−0.198761 + 0.980048i \(0.563692\pi\)
\(744\) 0 0
\(745\) −11.1339 −0.407913
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 20.4082i 0.745698i
\(750\) 0 0
\(751\) 4.34951i 0.158716i 0.996846 + 0.0793580i \(0.0252870\pi\)
−0.996846 + 0.0793580i \(0.974713\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −19.2199 −0.699484
\(756\) 0 0
\(757\) 9.63880 0.350328 0.175164 0.984539i \(-0.443954\pi\)
0.175164 + 0.984539i \(0.443954\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.87867i 0.176852i 0.996083 + 0.0884259i \(0.0281836\pi\)
−0.996083 + 0.0884259i \(0.971816\pi\)
\(762\) 0 0
\(763\) − 9.98623i − 0.361526i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.55639 −0.0923058
\(768\) 0 0
\(769\) 19.2879 0.695540 0.347770 0.937580i \(-0.386939\pi\)
0.347770 + 0.937580i \(0.386939\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8.44127i − 0.303611i −0.988410 0.151806i \(-0.951491\pi\)
0.988410 0.151806i \(-0.0485088\pi\)
\(774\) 0 0
\(775\) 14.5321i 0.522008i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −65.5646 −2.34909
\(780\) 0 0
\(781\) −29.9865 −1.07300
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 25.5150i − 0.910668i
\(786\) 0 0
\(787\) 32.2439i 1.14937i 0.818374 + 0.574686i \(0.194876\pi\)
−0.818374 + 0.574686i \(0.805124\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.60077 0.341364
\(792\) 0 0
\(793\) 10.7153 0.380512
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.61145i 0.163346i 0.996659 + 0.0816730i \(0.0260263\pi\)
−0.996659 + 0.0816730i \(0.973974\pi\)
\(798\) 0 0
\(799\) − 11.7708i − 0.416420i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 28.4854 1.00523
\(804\) 0 0
\(805\) 11.6328 0.410003
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.69680i 0.200289i 0.994973 + 0.100144i \(0.0319305\pi\)
−0.994973 + 0.100144i \(0.968070\pi\)
\(810\) 0 0
\(811\) − 16.3693i − 0.574803i −0.957810 0.287401i \(-0.907209\pi\)
0.957810 0.287401i \(-0.0927914\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.75574 −0.306700
\(816\) 0 0
\(817\) 112.145 3.92345
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.5960i 0.439602i 0.975545 + 0.219801i \(0.0705408\pi\)
−0.975545 + 0.219801i \(0.929459\pi\)
\(822\) 0 0
\(823\) − 7.53241i − 0.262563i −0.991345 0.131282i \(-0.958091\pi\)
0.991345 0.131282i \(-0.0419092\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.9398 1.28452 0.642262 0.766485i \(-0.277996\pi\)
0.642262 + 0.766485i \(0.277996\pi\)
\(828\) 0 0
\(829\) −16.2971 −0.566022 −0.283011 0.959117i \(-0.591333\pi\)
−0.283011 + 0.959117i \(0.591333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.24600i − 0.0778192i
\(834\) 0 0
\(835\) − 7.91398i − 0.273875i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 29.0711 1.00365 0.501823 0.864970i \(-0.332663\pi\)
0.501823 + 0.864970i \(0.332663\pi\)
\(840\) 0 0
\(841\) 18.3488 0.632718
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 15.4638i − 0.531971i
\(846\) 0 0
\(847\) 6.09107i 0.209292i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −64.7374 −2.21917
\(852\) 0 0
\(853\) 45.3207 1.55175 0.775875 0.630887i \(-0.217309\pi\)
0.775875 + 0.630887i \(0.217309\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 2.68393i − 0.0916813i −0.998949 0.0458407i \(-0.985403\pi\)
0.998949 0.0458407i \(-0.0145967\pi\)
\(858\) 0 0
\(859\) − 18.2862i − 0.623917i −0.950096 0.311959i \(-0.899015\pi\)
0.950096 0.311959i \(-0.100985\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 13.0706 0.444927 0.222463 0.974941i \(-0.428590\pi\)
0.222463 + 0.974941i \(0.428590\pi\)
\(864\) 0 0
\(865\) 3.37492 0.114751
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 20.2622i − 0.687349i
\(870\) 0 0
\(871\) 4.63499i 0.157051i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.7875 0.364684
\(876\) 0 0
\(877\) 34.1346 1.15264 0.576322 0.817223i \(-0.304487\pi\)
0.576322 + 0.817223i \(0.304487\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 6.07996i − 0.204839i −0.994741 0.102420i \(-0.967342\pi\)
0.994741 0.102420i \(-0.0326584\pi\)
\(882\) 0 0
\(883\) 8.63019i 0.290429i 0.989400 + 0.145214i \(0.0463872\pi\)
−0.989400 + 0.145214i \(0.953613\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43.4177 −1.45782 −0.728911 0.684608i \(-0.759973\pi\)
−0.728911 + 0.684608i \(0.759973\pi\)
\(888\) 0 0
\(889\) 9.84909 0.330328
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 45.2750i − 1.51507i
\(894\) 0 0
\(895\) 20.8157i 0.695793i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.2933 0.476707
\(900\) 0 0
\(901\) 25.0863 0.835747
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.3447i 0.676281i
\(906\) 0 0
\(907\) − 16.9055i − 0.561337i −0.959805 0.280669i \(-0.909444\pi\)
0.959805 0.280669i \(-0.0905562\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −14.7515 −0.488737 −0.244369 0.969682i \(-0.578581\pi\)
−0.244369 + 0.969682i \(0.578581\pi\)
\(912\) 0 0
\(913\) −22.0840 −0.730875
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 19.3232i 0.638108i
\(918\) 0 0
\(919\) − 0.0621024i − 0.00204857i −0.999999 0.00102428i \(-0.999674\pi\)
0.999999 0.00102428i \(-0.000326040\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −14.0390 −0.462099
\(924\) 0 0
\(925\) −23.9475 −0.787388
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.38318i 0.242234i 0.992638 + 0.121117i \(0.0386476\pi\)
−0.992638 + 0.121117i \(0.961352\pi\)
\(930\) 0 0
\(931\) − 8.63898i − 0.283131i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.45352 0.211053
\(936\) 0 0
\(937\) 33.9194 1.10810 0.554049 0.832484i \(-0.313082\pi\)
0.554049 + 0.832484i \(0.313082\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.63805i − 0.0533990i −0.999644 0.0266995i \(-0.991500\pi\)
0.999644 0.0266995i \(-0.00849973\pi\)
\(942\) 0 0
\(943\) 68.0766i 2.21688i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −7.85963 −0.255404 −0.127702 0.991813i \(-0.540760\pi\)
−0.127702 + 0.991813i \(0.540760\pi\)
\(948\) 0 0
\(949\) 13.3362 0.432913
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.0249i 1.16696i 0.812127 + 0.583480i \(0.198309\pi\)
−0.812127 + 0.583480i \(0.801691\pi\)
\(954\) 0 0
\(955\) 26.9917i 0.873432i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.56515 −0.212000
\(960\) 0 0
\(961\) 11.8193 0.381268
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 19.3365i − 0.622465i
\(966\) 0 0
\(967\) − 60.0449i − 1.93091i −0.260562 0.965457i \(-0.583908\pi\)
0.260562 0.965457i \(-0.416092\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.72865 0.0875665 0.0437832 0.999041i \(-0.486059\pi\)
0.0437832 + 0.999041i \(0.486059\pi\)
\(972\) 0 0
\(973\) 19.8276 0.635644
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 53.0413i − 1.69694i −0.529243 0.848470i \(-0.677524\pi\)
0.529243 0.848470i \(-0.322476\pi\)
\(978\) 0 0
\(979\) 7.71494i 0.246571i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 39.3167 1.25401 0.627004 0.779016i \(-0.284281\pi\)
0.627004 + 0.779016i \(0.284281\pi\)
\(984\) 0 0
\(985\) −5.55591 −0.177026
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 116.442i − 3.70262i
\(990\) 0 0
\(991\) − 13.6496i − 0.433595i −0.976217 0.216798i \(-0.930439\pi\)
0.976217 0.216798i \(-0.0695612\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −27.8968 −0.884389
\(996\) 0 0
\(997\) 30.6397 0.970370 0.485185 0.874412i \(-0.338752\pi\)
0.485185 + 0.874412i \(0.338752\pi\)
\(998\) 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.h.i.2591.16 yes 24
3.2 odd 2 inner 6048.2.h.i.2591.9 24
4.3 odd 2 inner 6048.2.h.i.2591.10 yes 24
12.11 even 2 inner 6048.2.h.i.2591.15 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.i.2591.9 24 3.2 odd 2 inner
6048.2.h.i.2591.10 yes 24 4.3 odd 2 inner
6048.2.h.i.2591.15 yes 24 12.11 even 2 inner
6048.2.h.i.2591.16 yes 24 1.1 even 1 trivial