Properties

Label 6048.2.j.c.5615.5
Level 6048
Weight 2
Character 6048.5615
Analytic conductor 48.294
Analytic rank 0
Dimension 32
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.j (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(32\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 5615.5
Character \(\chi\) = 6048.5615
Dual form 6048.2.j.c.5615.6

$q$-expansion

\(f(q)\) \(=\) \(q-2.21305 q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-2.21305 q^{5} +1.00000i q^{7} -2.42373i q^{11} -1.11593i q^{13} +0.701326i q^{17} -0.938340 q^{19} +4.30502 q^{23} -0.102426 q^{25} +4.26130 q^{29} +7.02221i q^{31} -2.21305i q^{35} +3.12583i q^{37} +0.157922i q^{41} -7.08975 q^{43} -0.867970 q^{47} -1.00000 q^{49} +3.91444 q^{53} +5.36382i q^{55} -7.28820i q^{59} +4.57877i q^{61} +2.46961i q^{65} -15.1677 q^{67} -6.93451 q^{71} +7.02133 q^{73} +2.42373 q^{77} -6.65903i q^{79} -8.41084i q^{83} -1.55207i q^{85} -7.34335i q^{89} +1.11593 q^{91} +2.07659 q^{95} +7.99425 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q + O(q^{10}) \) \( 32q + 64q^{19} - 16q^{25} + 48q^{43} - 32q^{49} - 16q^{67} - 16q^{73} - 16q^{91} + 64q^{97} + O(q^{100}) \)

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
\(5\) −2.21305 −0.989704 −0.494852 0.868977i \(-0.664778\pi\)
−0.494852 + 0.868977i \(0.664778\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.42373i − 0.730781i −0.930854 0.365390i \(-0.880935\pi\)
0.930854 0.365390i \(-0.119065\pi\)
\(12\) 0 0
\(13\) − 1.11593i − 0.309504i −0.987953 0.154752i \(-0.950542\pi\)
0.987953 0.154752i \(-0.0494579\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.701326i 0.170097i 0.996377 + 0.0850483i \(0.0271044\pi\)
−0.996377 + 0.0850483i \(0.972896\pi\)
\(18\) 0 0
\(19\) −0.938340 −0.215270 −0.107635 0.994190i \(-0.534328\pi\)
−0.107635 + 0.994190i \(0.534328\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.30502 0.897658 0.448829 0.893618i \(-0.351841\pi\)
0.448829 + 0.893618i \(0.351841\pi\)
\(24\) 0 0
\(25\) −0.102426 −0.0204852
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.26130 0.791303 0.395652 0.918401i \(-0.370519\pi\)
0.395652 + 0.918401i \(0.370519\pi\)
\(30\) 0 0
\(31\) 7.02221i 1.26123i 0.776098 + 0.630613i \(0.217196\pi\)
−0.776098 + 0.630613i \(0.782804\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 2.21305i − 0.374073i
\(36\) 0 0
\(37\) 3.12583i 0.513883i 0.966427 + 0.256941i \(0.0827148\pi\)
−0.966427 + 0.256941i \(0.917285\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.157922i 0.0246632i 0.999924 + 0.0123316i \(0.00392537\pi\)
−0.999924 + 0.0123316i \(0.996075\pi\)
\(42\) 0 0
\(43\) −7.08975 −1.08118 −0.540588 0.841287i \(-0.681798\pi\)
−0.540588 + 0.841287i \(0.681798\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.867970 −0.126607 −0.0633033 0.997994i \(-0.520164\pi\)
−0.0633033 + 0.997994i \(0.520164\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.91444 0.537689 0.268845 0.963184i \(-0.413358\pi\)
0.268845 + 0.963184i \(0.413358\pi\)
\(54\) 0 0
\(55\) 5.36382i 0.723257i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 7.28820i − 0.948843i −0.880298 0.474421i \(-0.842657\pi\)
0.880298 0.474421i \(-0.157343\pi\)
\(60\) 0 0
\(61\) 4.57877i 0.586251i 0.956074 + 0.293126i \(0.0946955\pi\)
−0.956074 + 0.293126i \(0.905305\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.46961i 0.306318i
\(66\) 0 0
\(67\) −15.1677 −1.85303 −0.926513 0.376264i \(-0.877209\pi\)
−0.926513 + 0.376264i \(0.877209\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.93451 −0.822975 −0.411487 0.911416i \(-0.634991\pi\)
−0.411487 + 0.911416i \(0.634991\pi\)
\(72\) 0 0
\(73\) 7.02133 0.821785 0.410893 0.911684i \(-0.365217\pi\)
0.410893 + 0.911684i \(0.365217\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.42373 0.276209
\(78\) 0 0
\(79\) − 6.65903i − 0.749200i −0.927187 0.374600i \(-0.877780\pi\)
0.927187 0.374600i \(-0.122220\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 8.41084i − 0.923210i −0.887086 0.461605i \(-0.847274\pi\)
0.887086 0.461605i \(-0.152726\pi\)
\(84\) 0 0
\(85\) − 1.55207i − 0.168345i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 7.34335i − 0.778393i −0.921155 0.389197i \(-0.872753\pi\)
0.921155 0.389197i \(-0.127247\pi\)
\(90\) 0 0
\(91\) 1.11593 0.116982
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.07659 0.213054
\(96\) 0 0
\(97\) 7.99425 0.811693 0.405846 0.913941i \(-0.366977\pi\)
0.405846 + 0.913941i \(0.366977\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.6519 1.55742 0.778709 0.627385i \(-0.215875\pi\)
0.778709 + 0.627385i \(0.215875\pi\)
\(102\) 0 0
\(103\) 4.90314i 0.483121i 0.970386 + 0.241560i \(0.0776592\pi\)
−0.970386 + 0.241560i \(0.922341\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.1434i − 1.46397i −0.681320 0.731986i \(-0.738594\pi\)
0.681320 0.731986i \(-0.261406\pi\)
\(108\) 0 0
\(109\) − 2.72939i − 0.261428i −0.991420 0.130714i \(-0.958273\pi\)
0.991420 0.130714i \(-0.0417270\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 2.84331i − 0.267476i −0.991017 0.133738i \(-0.957302\pi\)
0.991017 0.133738i \(-0.0426980\pi\)
\(114\) 0 0
\(115\) −9.52720 −0.888416
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.701326 −0.0642905
\(120\) 0 0
\(121\) 5.12555 0.465959
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.2919 1.00998
\(126\) 0 0
\(127\) − 10.4243i − 0.925006i −0.886618 0.462503i \(-0.846951\pi\)
0.886618 0.462503i \(-0.153049\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.3573i 1.86600i 0.359881 + 0.932998i \(0.382817\pi\)
−0.359881 + 0.932998i \(0.617183\pi\)
\(132\) 0 0
\(133\) − 0.938340i − 0.0813644i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.2577i − 0.961812i −0.876772 0.480906i \(-0.840308\pi\)
0.876772 0.480906i \(-0.159692\pi\)
\(138\) 0 0
\(139\) −8.11586 −0.688379 −0.344189 0.938900i \(-0.611846\pi\)
−0.344189 + 0.938900i \(0.611846\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.70472 −0.226180
\(144\) 0 0
\(145\) −9.43045 −0.783156
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.90580 −0.401899 −0.200949 0.979602i \(-0.564403\pi\)
−0.200949 + 0.979602i \(0.564403\pi\)
\(150\) 0 0
\(151\) − 18.9521i − 1.54230i −0.636654 0.771150i \(-0.719682\pi\)
0.636654 0.771150i \(-0.280318\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 15.5405i − 1.24824i
\(156\) 0 0
\(157\) 10.0907i 0.805326i 0.915348 + 0.402663i \(0.131915\pi\)
−0.915348 + 0.402663i \(0.868085\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.30502i 0.339283i
\(162\) 0 0
\(163\) 19.9762 1.56466 0.782330 0.622864i \(-0.214031\pi\)
0.782330 + 0.622864i \(0.214031\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 13.8623 1.07270 0.536348 0.843997i \(-0.319803\pi\)
0.536348 + 0.843997i \(0.319803\pi\)
\(168\) 0 0
\(169\) 11.7547 0.904207
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.4796 1.63306 0.816531 0.577301i \(-0.195894\pi\)
0.816531 + 0.577301i \(0.195894\pi\)
\(174\) 0 0
\(175\) − 0.102426i − 0.00774268i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3.94291i − 0.294707i −0.989084 0.147354i \(-0.952925\pi\)
0.989084 0.147354i \(-0.0470755\pi\)
\(180\) 0 0
\(181\) − 11.7849i − 0.875962i −0.898984 0.437981i \(-0.855694\pi\)
0.898984 0.437981i \(-0.144306\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 6.91760i − 0.508592i
\(186\) 0 0
\(187\) 1.69982 0.124303
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.6064 1.20160 0.600798 0.799401i \(-0.294850\pi\)
0.600798 + 0.799401i \(0.294850\pi\)
\(192\) 0 0
\(193\) 6.90416 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.98273 −0.639993 −0.319997 0.947419i \(-0.603682\pi\)
−0.319997 + 0.947419i \(0.603682\pi\)
\(198\) 0 0
\(199\) 11.4427i 0.811153i 0.914061 + 0.405577i \(0.132929\pi\)
−0.914061 + 0.405577i \(0.867071\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.26130i 0.299085i
\(204\) 0 0
\(205\) − 0.349488i − 0.0244093i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.27428i 0.157315i
\(210\) 0 0
\(211\) −23.8035 −1.63870 −0.819351 0.573292i \(-0.805666\pi\)
−0.819351 + 0.573292i \(0.805666\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 15.6899 1.07004
\(216\) 0 0
\(217\) −7.02221 −0.476698
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.782633 0.0526456
\(222\) 0 0
\(223\) − 4.42841i − 0.296549i −0.988946 0.148274i \(-0.952628\pi\)
0.988946 0.148274i \(-0.0473718\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 6.46751i − 0.429264i −0.976695 0.214632i \(-0.931145\pi\)
0.976695 0.214632i \(-0.0688552\pi\)
\(228\) 0 0
\(229\) 11.4715i 0.758056i 0.925385 + 0.379028i \(0.123742\pi\)
−0.925385 + 0.379028i \(0.876258\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 19.8678i 1.30158i 0.759256 + 0.650792i \(0.225563\pi\)
−0.759256 + 0.650792i \(0.774437\pi\)
\(234\) 0 0
\(235\) 1.92086 0.125303
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.2210 0.661138 0.330569 0.943782i \(-0.392759\pi\)
0.330569 + 0.943782i \(0.392759\pi\)
\(240\) 0 0
\(241\) 28.1808 1.81529 0.907643 0.419744i \(-0.137880\pi\)
0.907643 + 0.419744i \(0.137880\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.21305 0.141386
\(246\) 0 0
\(247\) 1.04712i 0.0666269i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 4.62338i − 0.291825i −0.989297 0.145913i \(-0.953388\pi\)
0.989297 0.145913i \(-0.0466118\pi\)
\(252\) 0 0
\(253\) − 10.4342i − 0.655991i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 29.6682i − 1.85065i −0.379174 0.925326i \(-0.623792\pi\)
0.379174 0.925326i \(-0.376208\pi\)
\(258\) 0 0
\(259\) −3.12583 −0.194230
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.8635 0.978185 0.489093 0.872232i \(-0.337328\pi\)
0.489093 + 0.872232i \(0.337328\pi\)
\(264\) 0 0
\(265\) −8.66283 −0.532153
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.13803 −0.191329 −0.0956644 0.995414i \(-0.530498\pi\)
−0.0956644 + 0.995414i \(0.530498\pi\)
\(270\) 0 0
\(271\) − 22.7174i − 1.37998i −0.723817 0.689992i \(-0.757614\pi\)
0.723817 0.689992i \(-0.242386\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.248252i 0.0149702i
\(276\) 0 0
\(277\) 24.8267i 1.49169i 0.666119 + 0.745845i \(0.267954\pi\)
−0.666119 + 0.745845i \(0.732046\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 30.2207i − 1.80282i −0.432970 0.901408i \(-0.642535\pi\)
0.432970 0.901408i \(-0.357465\pi\)
\(282\) 0 0
\(283\) 14.9803 0.890488 0.445244 0.895409i \(-0.353117\pi\)
0.445244 + 0.895409i \(0.353117\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.157922 −0.00932182
\(288\) 0 0
\(289\) 16.5081 0.971067
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.58993 −0.443408 −0.221704 0.975114i \(-0.571162\pi\)
−0.221704 + 0.975114i \(0.571162\pi\)
\(294\) 0 0
\(295\) 16.1291i 0.939074i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 4.80411i − 0.277829i
\(300\) 0 0
\(301\) − 7.08975i − 0.408646i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 10.1330i − 0.580216i
\(306\) 0 0
\(307\) 18.1246 1.03443 0.517214 0.855856i \(-0.326969\pi\)
0.517214 + 0.855856i \(0.326969\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −17.0291 −0.965631 −0.482816 0.875722i \(-0.660386\pi\)
−0.482816 + 0.875722i \(0.660386\pi\)
\(312\) 0 0
\(313\) 7.44270 0.420686 0.210343 0.977628i \(-0.432542\pi\)
0.210343 + 0.977628i \(0.432542\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.40549 0.135106 0.0675528 0.997716i \(-0.478481\pi\)
0.0675528 + 0.997716i \(0.478481\pi\)
\(318\) 0 0
\(319\) − 10.3282i − 0.578269i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 0.658082i − 0.0366167i
\(324\) 0 0
\(325\) 0.114301i 0.00634025i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 0.867970i − 0.0478528i
\(330\) 0 0
\(331\) −14.3776 −0.790265 −0.395133 0.918624i \(-0.629301\pi\)
−0.395133 + 0.918624i \(0.629301\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 33.5667 1.83395
\(336\) 0 0
\(337\) 20.4430 1.11360 0.556800 0.830646i \(-0.312029\pi\)
0.556800 + 0.830646i \(0.312029\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 17.0199 0.921679
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 6.62453i − 0.355623i −0.984065 0.177812i \(-0.943098\pi\)
0.984065 0.177812i \(-0.0569018\pi\)
\(348\) 0 0
\(349\) − 33.7876i − 1.80861i −0.426889 0.904304i \(-0.640391\pi\)
0.426889 0.904304i \(-0.359609\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 26.5976i − 1.41565i −0.706390 0.707823i \(-0.749677\pi\)
0.706390 0.707823i \(-0.250323\pi\)
\(354\) 0 0
\(355\) 15.3464 0.814501
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.7098 1.56802 0.784012 0.620746i \(-0.213170\pi\)
0.784012 + 0.620746i \(0.213170\pi\)
\(360\) 0 0
\(361\) −18.1195 −0.953659
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.5385 −0.813324
\(366\) 0 0
\(367\) 15.6932i 0.819176i 0.912271 + 0.409588i \(0.134328\pi\)
−0.912271 + 0.409588i \(0.865672\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.91444i 0.203227i
\(372\) 0 0
\(373\) − 11.0269i − 0.570951i −0.958386 0.285475i \(-0.907849\pi\)
0.958386 0.285475i \(-0.0921515\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.75533i − 0.244912i
\(378\) 0 0
\(379\) 7.84736 0.403092 0.201546 0.979479i \(-0.435403\pi\)
0.201546 + 0.979479i \(0.435403\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.26238 −0.166700 −0.0833500 0.996520i \(-0.526562\pi\)
−0.0833500 + 0.996520i \(0.526562\pi\)
\(384\) 0 0
\(385\) −5.36382 −0.273365
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.15697 0.0586609 0.0293304 0.999570i \(-0.490662\pi\)
0.0293304 + 0.999570i \(0.490662\pi\)
\(390\) 0 0
\(391\) 3.01922i 0.152688i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 14.7367i 0.741486i
\(396\) 0 0
\(397\) 2.41295i 0.121103i 0.998165 + 0.0605513i \(0.0192859\pi\)
−0.998165 + 0.0605513i \(0.980714\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.48047i 0.373557i 0.982402 + 0.186778i \(0.0598047\pi\)
−0.982402 + 0.186778i \(0.940195\pi\)
\(402\) 0 0
\(403\) 7.83631 0.390355
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 7.57615 0.375536
\(408\) 0 0
\(409\) −7.49823 −0.370764 −0.185382 0.982667i \(-0.559352\pi\)
−0.185382 + 0.982667i \(0.559352\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.28820 0.358629
\(414\) 0 0
\(415\) 18.6136i 0.913705i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 26.7682i 1.30771i 0.756619 + 0.653856i \(0.226850\pi\)
−0.756619 + 0.653856i \(0.773150\pi\)
\(420\) 0 0
\(421\) − 28.1616i − 1.37251i −0.727359 0.686257i \(-0.759253\pi\)
0.727359 0.686257i \(-0.240747\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 0.0718340i − 0.00348446i
\(426\) 0 0
\(427\) −4.57877 −0.221582
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −29.9359 −1.44196 −0.720980 0.692955i \(-0.756308\pi\)
−0.720980 + 0.692955i \(0.756308\pi\)
\(432\) 0 0
\(433\) −39.1106 −1.87954 −0.939768 0.341814i \(-0.888959\pi\)
−0.939768 + 0.341814i \(0.888959\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4.03957 −0.193239
\(438\) 0 0
\(439\) 31.4351i 1.50031i 0.661260 + 0.750157i \(0.270022\pi\)
−0.661260 + 0.750157i \(0.729978\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 24.2344i − 1.15141i −0.817656 0.575707i \(-0.804727\pi\)
0.817656 0.575707i \(-0.195273\pi\)
\(444\) 0 0
\(445\) 16.2512i 0.770379i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 22.6066i − 1.06687i −0.845841 0.533435i \(-0.820901\pi\)
0.845841 0.533435i \(-0.179099\pi\)
\(450\) 0 0
\(451\) 0.382759 0.0180234
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.46961 −0.115777
\(456\) 0 0
\(457\) −9.35271 −0.437502 −0.218751 0.975781i \(-0.570198\pi\)
−0.218751 + 0.975781i \(0.570198\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.5898 0.539789 0.269895 0.962890i \(-0.413011\pi\)
0.269895 + 0.962890i \(0.413011\pi\)
\(462\) 0 0
\(463\) − 17.8299i − 0.828625i −0.910135 0.414312i \(-0.864022\pi\)
0.910135 0.414312i \(-0.135978\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.50180i 0.162044i 0.996712 + 0.0810221i \(0.0258184\pi\)
−0.996712 + 0.0810221i \(0.974182\pi\)
\(468\) 0 0
\(469\) − 15.1677i − 0.700378i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 17.1836i 0.790103i
\(474\) 0 0
\(475\) 0.0961103 0.00440985
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.2987 0.836087 0.418043 0.908427i \(-0.362716\pi\)
0.418043 + 0.908427i \(0.362716\pi\)
\(480\) 0 0
\(481\) 3.48822 0.159049
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −17.6916 −0.803336
\(486\) 0 0
\(487\) − 24.3559i − 1.10367i −0.833953 0.551836i \(-0.813928\pi\)
0.833953 0.551836i \(-0.186072\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 25.8479i − 1.16650i −0.812292 0.583250i \(-0.801781\pi\)
0.812292 0.583250i \(-0.198219\pi\)
\(492\) 0 0
\(493\) 2.98856i 0.134598i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6.93451i − 0.311055i
\(498\) 0 0
\(499\) 13.5430 0.606270 0.303135 0.952948i \(-0.401967\pi\)
0.303135 + 0.952948i \(0.401967\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −17.6930 −0.788890 −0.394445 0.918920i \(-0.629063\pi\)
−0.394445 + 0.918920i \(0.629063\pi\)
\(504\) 0 0
\(505\) −34.6383 −1.54138
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 12.0147 0.532544 0.266272 0.963898i \(-0.414208\pi\)
0.266272 + 0.963898i \(0.414208\pi\)
\(510\) 0 0
\(511\) 7.02133i 0.310606i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 10.8509i − 0.478147i
\(516\) 0 0
\(517\) 2.10372i 0.0925216i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 17.5217i − 0.767638i −0.923408 0.383819i \(-0.874609\pi\)
0.923408 0.383819i \(-0.125391\pi\)
\(522\) 0 0
\(523\) 5.61428 0.245495 0.122748 0.992438i \(-0.460829\pi\)
0.122748 + 0.992438i \(0.460829\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.92486 −0.214530
\(528\) 0 0
\(529\) −4.46684 −0.194210
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.176230 0.00763337
\(534\) 0 0
\(535\) 33.5131i 1.44890i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.42373i 0.104397i
\(540\) 0 0
\(541\) 34.1532i 1.46836i 0.678954 + 0.734181i \(0.262433\pi\)
−0.678954 + 0.734181i \(0.737567\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.04027i 0.258737i
\(546\) 0 0
\(547\) −21.6399 −0.925255 −0.462627 0.886553i \(-0.653093\pi\)
−0.462627 + 0.886553i \(0.653093\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.99855 −0.170344
\(552\) 0 0
\(553\) 6.65903 0.283171
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25.9554 1.09976 0.549882 0.835242i \(-0.314673\pi\)
0.549882 + 0.835242i \(0.314673\pi\)
\(558\) 0 0
\(559\) 7.91168i 0.334629i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 13.9262i − 0.586921i −0.955971 0.293460i \(-0.905193\pi\)
0.955971 0.293460i \(-0.0948069\pi\)
\(564\) 0 0
\(565\) 6.29237i 0.264722i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 18.6130i − 0.780296i −0.920752 0.390148i \(-0.872424\pi\)
0.920752 0.390148i \(-0.127576\pi\)
\(570\) 0 0
\(571\) −4.69112 −0.196317 −0.0981586 0.995171i \(-0.531295\pi\)
−0.0981586 + 0.995171i \(0.531295\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.440945 −0.0183887
\(576\) 0 0
\(577\) −9.27967 −0.386318 −0.193159 0.981167i \(-0.561873\pi\)
−0.193159 + 0.981167i \(0.561873\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.41084 0.348940
\(582\) 0 0
\(583\) − 9.48752i − 0.392933i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 34.3528i − 1.41789i −0.705264 0.708945i \(-0.749171\pi\)
0.705264 0.708945i \(-0.250829\pi\)
\(588\) 0 0
\(589\) − 6.58922i − 0.271504i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 44.4288i 1.82447i 0.409666 + 0.912236i \(0.365645\pi\)
−0.409666 + 0.912236i \(0.634355\pi\)
\(594\) 0 0
\(595\) 1.55207 0.0636285
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −13.8604 −0.566322 −0.283161 0.959072i \(-0.591383\pi\)
−0.283161 + 0.959072i \(0.591383\pi\)
\(600\) 0 0
\(601\) −6.51128 −0.265601 −0.132800 0.991143i \(-0.542397\pi\)
−0.132800 + 0.991143i \(0.542397\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.3431 −0.461162
\(606\) 0 0
\(607\) − 43.5994i − 1.76964i −0.465929 0.884822i \(-0.654280\pi\)
0.465929 0.884822i \(-0.345720\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0.968597i 0.0391852i
\(612\) 0 0
\(613\) 22.4072i 0.905017i 0.891760 + 0.452508i \(0.149471\pi\)
−0.891760 + 0.452508i \(0.850529\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.5128i 1.22840i 0.789151 + 0.614199i \(0.210521\pi\)
−0.789151 + 0.614199i \(0.789479\pi\)
\(618\) 0 0
\(619\) 13.5504 0.544637 0.272319 0.962207i \(-0.412210\pi\)
0.272319 + 0.962207i \(0.412210\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.34335 0.294205
\(624\) 0 0
\(625\) −24.4774 −0.979095
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.19222 −0.0874097
\(630\) 0 0
\(631\) 26.5393i 1.05651i 0.849085 + 0.528257i \(0.177154\pi\)
−0.849085 + 0.528257i \(0.822846\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 23.0694i 0.915482i
\(636\) 0 0
\(637\) 1.11593i 0.0442149i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 10.3318i − 0.408083i −0.978962 0.204041i \(-0.934592\pi\)
0.978962 0.204041i \(-0.0654077\pi\)
\(642\) 0 0
\(643\) −13.5130 −0.532901 −0.266450 0.963849i \(-0.585851\pi\)
−0.266450 + 0.963849i \(0.585851\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.31247 −0.326797 −0.163399 0.986560i \(-0.552246\pi\)
−0.163399 + 0.986560i \(0.552246\pi\)
\(648\) 0 0
\(649\) −17.6646 −0.693396
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.56899 −0.100533 −0.0502663 0.998736i \(-0.516007\pi\)
−0.0502663 + 0.998736i \(0.516007\pi\)
\(654\) 0 0
\(655\) − 47.2647i − 1.84678i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 12.7747i − 0.497632i −0.968551 0.248816i \(-0.919959\pi\)
0.968551 0.248816i \(-0.0800414\pi\)
\(660\) 0 0
\(661\) − 25.8624i − 1.00593i −0.864307 0.502965i \(-0.832242\pi\)
0.864307 0.502965i \(-0.167758\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.07659i 0.0805267i
\(666\) 0 0
\(667\) 18.3450 0.710320
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 11.0977 0.428421
\(672\) 0 0
\(673\) −33.9446 −1.30847 −0.654235 0.756292i \(-0.727009\pi\)
−0.654235 + 0.756292i \(0.727009\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34.1280 1.31165 0.655823 0.754915i \(-0.272322\pi\)
0.655823 + 0.754915i \(0.272322\pi\)
\(678\) 0 0
\(679\) 7.99425i 0.306791i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.80951i 0.260559i 0.991477 + 0.130279i \(0.0415874\pi\)
−0.991477 + 0.130279i \(0.958413\pi\)
\(684\) 0 0
\(685\) 24.9139i 0.951910i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 4.36825i − 0.166417i
\(690\) 0 0
\(691\) 5.06521 0.192690 0.0963448 0.995348i \(-0.469285\pi\)
0.0963448 + 0.995348i \(0.469285\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.9608 0.681291
\(696\) 0 0
\(697\) −0.110755 −0.00419513
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −44.7933 −1.69182 −0.845910 0.533326i \(-0.820942\pi\)
−0.845910 + 0.533326i \(0.820942\pi\)
\(702\) 0 0
\(703\) − 2.93309i − 0.110624i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 15.6519i 0.588649i
\(708\) 0 0
\(709\) 40.5597i 1.52325i 0.648018 + 0.761625i \(0.275598\pi\)
−0.648018 + 0.761625i \(0.724402\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 30.2307i 1.13215i
\(714\) 0 0
\(715\) 5.98566 0.223851
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −13.2926 −0.495729 −0.247864 0.968795i \(-0.579729\pi\)
−0.247864 + 0.968795i \(0.579729\pi\)
\(720\) 0 0
\(721\) −4.90314 −0.182603
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.436468 −0.0162100
\(726\) 0 0
\(727\) 28.6102i 1.06109i 0.847656 + 0.530546i \(0.178013\pi\)
−0.847656 + 0.530546i \(0.821987\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 4.97222i − 0.183904i
\(732\) 0 0
\(733\) 33.9007i 1.25215i 0.779763 + 0.626075i \(0.215340\pi\)
−0.779763 + 0.626075i \(0.784660\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.7623i 1.35416i
\(738\) 0 0
\(739\) 11.1200 0.409055 0.204528 0.978861i \(-0.434434\pi\)
0.204528 + 0.978861i \(0.434434\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −45.7857 −1.67971 −0.839857 0.542808i \(-0.817361\pi\)
−0.839857 + 0.542808i \(0.817361\pi\)
\(744\) 0 0
\(745\) 10.8568 0.397761
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 15.1434 0.553329
\(750\) 0 0
\(751\) − 13.8013i − 0.503616i −0.967777 0.251808i \(-0.918975\pi\)
0.967777 0.251808i \(-0.0810252\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 41.9418i 1.52642i
\(756\) 0 0
\(757\) 5.94985i 0.216251i 0.994137 + 0.108126i \(0.0344848\pi\)
−0.994137 + 0.108126i \(0.965515\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 38.1237i 1.38198i 0.722864 + 0.690991i \(0.242825\pi\)
−0.722864 + 0.690991i \(0.757175\pi\)
\(762\) 0 0
\(763\) 2.72939 0.0988106
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.13314 −0.293671
\(768\) 0 0
\(769\) 15.5207 0.559689 0.279845 0.960045i \(-0.409717\pi\)
0.279845 + 0.960045i \(0.409717\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.6525 1.42620 0.713101 0.701061i \(-0.247290\pi\)
0.713101 + 0.701061i \(0.247290\pi\)
\(774\) 0 0
\(775\) − 0.719256i − 0.0258365i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 0.148184i − 0.00530925i
\(780\) 0 0
\(781\) 16.8073i 0.601414i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 22.3312i − 0.797035i
\(786\) 0 0
\(787\) 50.2403 1.79087 0.895436 0.445189i \(-0.146864\pi\)
0.895436 + 0.445189i \(0.146864\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.84331 0.101096
\(792\) 0 0
\(793\) 5.10960 0.181447
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.78519 0.0632345 0.0316173 0.999500i \(-0.489934\pi\)
0.0316173 + 0.999500i \(0.489934\pi\)
\(798\) 0 0
\(799\) − 0.608730i − 0.0215353i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 17.0178i − 0.600545i
\(804\) 0 0
\(805\) − 9.52720i − 0.335790i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 31.9043i − 1.12169i −0.827919 0.560847i \(-0.810476\pi\)
0.827919 0.560847i \(-0.189524\pi\)
\(810\) 0 0
\(811\) 12.5528 0.440789 0.220394 0.975411i \(-0.429266\pi\)
0.220394 + 0.975411i \(0.429266\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −44.2084 −1.54855
\(816\) 0 0
\(817\) 6.65259 0.232745
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.1349 1.47052 0.735260 0.677786i \(-0.237060\pi\)
0.735260 + 0.677786i \(0.237060\pi\)
\(822\) 0 0
\(823\) − 1.50592i − 0.0524931i −0.999656 0.0262465i \(-0.991645\pi\)
0.999656 0.0262465i \(-0.00835549\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 40.6098i − 1.41214i −0.708141 0.706071i \(-0.750466\pi\)
0.708141 0.706071i \(-0.249534\pi\)
\(828\) 0 0
\(829\) − 35.4547i − 1.23139i −0.787984 0.615696i \(-0.788875\pi\)
0.787984 0.615696i \(-0.211125\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 0.701326i − 0.0242995i
\(834\) 0 0
\(835\) −30.6779 −1.06165
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.8372 1.09914 0.549571 0.835447i \(-0.314791\pi\)
0.549571 + 0.835447i \(0.314791\pi\)
\(840\) 0 0
\(841\) −10.8413 −0.373839
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −26.0137 −0.894898
\(846\) 0 0
\(847\) 5.12555i 0.176116i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 13.4567i 0.461291i
\(852\) 0 0
\(853\) − 30.0499i − 1.02889i −0.857523 0.514445i \(-0.827998\pi\)
0.857523 0.514445i \(-0.172002\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 26.7594i − 0.914085i −0.889445 0.457043i \(-0.848909\pi\)
0.889445 0.457043i \(-0.151091\pi\)
\(858\) 0 0
\(859\) 3.98544 0.135981 0.0679907 0.997686i \(-0.478341\pi\)
0.0679907 + 0.997686i \(0.478341\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 33.1318 1.12782 0.563910 0.825836i \(-0.309296\pi\)
0.563910 + 0.825836i \(0.309296\pi\)
\(864\) 0 0
\(865\) −47.5353 −1.61625
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −16.1397 −0.547501
\(870\) 0 0
\(871\) 16.9261i 0.573519i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.2919i 0.381736i
\(876\) 0 0
\(877\) − 4.35210i − 0.146960i −0.997297 0.0734800i \(-0.976589\pi\)
0.997297 0.0734800i \(-0.0234105\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.2945i 1.25648i 0.778018 + 0.628242i \(0.216225\pi\)
−0.778018 + 0.628242i \(0.783775\pi\)
\(882\) 0 0
\(883\) 44.7359 1.50548 0.752742 0.658315i \(-0.228731\pi\)
0.752742 + 0.658315i \(0.228731\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.69357 −0.124018 −0.0620090 0.998076i \(-0.519751\pi\)
−0.0620090 + 0.998076i \(0.519751\pi\)
\(888\) 0 0
\(889\) 10.4243 0.349619
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.814451 0.0272546
\(894\) 0 0
\(895\) 8.72584i 0.291673i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 29.9237i 0.998012i
\(900\) 0 0
\(901\) 2.74530i 0.0914591i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26.0804i 0.866943i
\(906\) 0 0
\(907\) 9.61629 0.319304 0.159652 0.987173i \(-0.448963\pi\)
0.159652 + 0.987173i \(0.448963\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −53.1598 −1.76126 −0.880632 0.473802i \(-0.842881\pi\)
−0.880632 + 0.473802i \(0.842881\pi\)
\(912\) 0 0
\(913\) −20.3856 −0.674664
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −21.3573 −0.705280
\(918\) 0 0
\(919\) 33.3183i 1.09907i 0.835471 + 0.549534i \(0.185195\pi\)
−0.835471 + 0.549534i \(0.814805\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.73845i 0.254714i
\(924\) 0 0
\(925\) − 0.320166i − 0.0105270i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.5434i 0.444343i 0.975008 + 0.222172i \(0.0713146\pi\)
−0.975008 + 0.222172i \(0.928685\pi\)
\(930\) 0 0
\(931\) 0.938340 0.0307528
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.76178 −0.123024
\(936\) 0 0
\(937\) 42.3432 1.38329 0.691647 0.722236i \(-0.256886\pi\)
0.691647 + 0.722236i \(0.256886\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −60.9773 −1.98780 −0.993902 0.110267i \(-0.964829\pi\)
−0.993902 + 0.110267i \(0.964829\pi\)
\(942\) 0 0
\(943\) 0.679855i 0.0221391i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.4879i 1.28318i 0.767047 + 0.641591i \(0.221725\pi\)
−0.767047 + 0.641591i \(0.778275\pi\)
\(948\) 0 0
\(949\) − 7.83534i − 0.254346i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 25.8210i − 0.836425i −0.908349 0.418213i \(-0.862657\pi\)
0.908349 0.418213i \(-0.137343\pi\)
\(954\) 0 0
\(955\) −36.7507 −1.18922
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 11.2577 0.363531
\(960\) 0 0
\(961\) −18.3114 −0.590690
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −15.2792 −0.491856
\(966\) 0 0
\(967\) 4.92813i 0.158478i 0.996856 + 0.0792390i \(0.0252490\pi\)
−0.996856 + 0.0792390i \(0.974751\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 20.9043i 0.670850i 0.942067 + 0.335425i \(0.108880\pi\)
−0.942067 + 0.335425i \(0.891120\pi\)
\(972\) 0 0
\(973\) − 8.11586i − 0.260183i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.6995i 0.534266i 0.963660 + 0.267133i \(0.0860763\pi\)
−0.963660 + 0.267133i \(0.913924\pi\)
\(978\) 0 0
\(979\) −17.7983 −0.568835
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −27.7932 −0.886465 −0.443232 0.896407i \(-0.646168\pi\)
−0.443232 + 0.896407i \(0.646168\pi\)
\(984\) 0 0
\(985\) 19.8792 0.633404
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −30.5215 −0.970526
\(990\) 0 0
\(991\) − 38.3482i − 1.21817i −0.793104 0.609086i \(-0.791536\pi\)
0.793104 0.609086i \(-0.208464\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 25.3233i − 0.802802i
\(996\) 0 0
\(997\) 27.7418i 0.878590i 0.898343 + 0.439295i \(0.144772\pi\)
−0.898343 + 0.439295i \(0.855228\pi\)
\(998\) 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.j.c.5615.5 32
3.2 odd 2 inner 6048.2.j.c.5615.28 32
4.3 odd 2 1512.2.j.c.323.12 yes 32
8.3 odd 2 inner 6048.2.j.c.5615.27 32
8.5 even 2 1512.2.j.c.323.22 yes 32
12.11 even 2 1512.2.j.c.323.21 yes 32
24.5 odd 2 1512.2.j.c.323.11 32
24.11 even 2 inner 6048.2.j.c.5615.6 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.11 32 24.5 odd 2
1512.2.j.c.323.12 yes 32 4.3 odd 2
1512.2.j.c.323.21 yes 32 12.11 even 2
1512.2.j.c.323.22 yes 32 8.5 even 2
6048.2.j.c.5615.5 32 1.1 even 1 trivial
6048.2.j.c.5615.6 32 24.11 even 2 inner
6048.2.j.c.5615.27 32 8.3 odd 2 inner
6048.2.j.c.5615.28 32 3.2 odd 2 inner