Properties

Label 6048.2.c.g.3025.10
Level 6048
Weight 2
Character 6048.3025
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(24\)
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 3025.10
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.g.3025.15

$q$-expansion

\(f(q)\) \(=\) \(q-1.53368i q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-1.53368i q^{5} -1.00000 q^{7} -2.28335i q^{11} +7.10447i q^{13} -6.81307 q^{17} +1.60676i q^{19} +1.16757 q^{23} +2.64784 q^{25} -4.07059i q^{29} +7.90568 q^{31} +1.53368i q^{35} -7.04836i q^{37} -10.1710 q^{41} -0.344719i q^{43} +3.10859 q^{47} +1.00000 q^{49} +5.66835i q^{53} -3.50191 q^{55} -1.38165i q^{59} +5.50516i q^{61} +10.8960 q^{65} -8.35907i q^{67} +12.2819 q^{71} -5.95132 q^{73} +2.28335i q^{77} +15.0027 q^{79} -14.7419i q^{83} +10.4490i q^{85} +11.8923 q^{89} -7.10447i q^{91} +2.46425 q^{95} -2.60256 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q - 24q^{7} + O(q^{10}) \) \( 24q - 24q^{7} - 24q^{25} + 16q^{31} + 24q^{49} + 8q^{55} - 32q^{79} + O(q^{100}) \)

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
\(5\) − 1.53368i − 0.685881i −0.939357 0.342940i \(-0.888577\pi\)
0.939357 0.342940i \(-0.111423\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.28335i − 0.688455i −0.938886 0.344228i \(-0.888141\pi\)
0.938886 0.344228i \(-0.111859\pi\)
\(12\) 0 0
\(13\) 7.10447i 1.97043i 0.171333 + 0.985213i \(0.445193\pi\)
−0.171333 + 0.985213i \(0.554807\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.81307 −1.65241 −0.826206 0.563368i \(-0.809505\pi\)
−0.826206 + 0.563368i \(0.809505\pi\)
\(18\) 0 0
\(19\) 1.60676i 0.368616i 0.982869 + 0.184308i \(0.0590044\pi\)
−0.982869 + 0.184308i \(0.940996\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.16757 0.243455 0.121727 0.992564i \(-0.461157\pi\)
0.121727 + 0.992564i \(0.461157\pi\)
\(24\) 0 0
\(25\) 2.64784 0.529568
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.07059i − 0.755889i −0.925828 0.377944i \(-0.876631\pi\)
0.925828 0.377944i \(-0.123369\pi\)
\(30\) 0 0
\(31\) 7.90568 1.41990 0.709951 0.704251i \(-0.248717\pi\)
0.709951 + 0.704251i \(0.248717\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.53368i 0.259238i
\(36\) 0 0
\(37\) − 7.04836i − 1.15874i −0.815063 0.579372i \(-0.803298\pi\)
0.815063 0.579372i \(-0.196702\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.1710 −1.58844 −0.794218 0.607633i \(-0.792119\pi\)
−0.794218 + 0.607633i \(0.792119\pi\)
\(42\) 0 0
\(43\) − 0.344719i − 0.0525691i −0.999655 0.0262846i \(-0.991632\pi\)
0.999655 0.0262846i \(-0.00836760\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.10859 0.453434 0.226717 0.973961i \(-0.427201\pi\)
0.226717 + 0.973961i \(0.427201\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.66835i 0.778608i 0.921109 + 0.389304i \(0.127284\pi\)
−0.921109 + 0.389304i \(0.872716\pi\)
\(54\) 0 0
\(55\) −3.50191 −0.472198
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1.38165i − 0.179875i −0.995947 0.0899374i \(-0.971333\pi\)
0.995947 0.0899374i \(-0.0286667\pi\)
\(60\) 0 0
\(61\) 5.50516i 0.704863i 0.935838 + 0.352431i \(0.114645\pi\)
−0.935838 + 0.352431i \(0.885355\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.8960 1.35148
\(66\) 0 0
\(67\) − 8.35907i − 1.02122i −0.859811 0.510612i \(-0.829419\pi\)
0.859811 0.510612i \(-0.170581\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.2819 1.45759 0.728795 0.684732i \(-0.240081\pi\)
0.728795 + 0.684732i \(0.240081\pi\)
\(72\) 0 0
\(73\) −5.95132 −0.696549 −0.348275 0.937393i \(-0.613232\pi\)
−0.348275 + 0.937393i \(0.613232\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.28335i 0.260212i
\(78\) 0 0
\(79\) 15.0027 1.68794 0.843968 0.536393i \(-0.180213\pi\)
0.843968 + 0.536393i \(0.180213\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 14.7419i − 1.61814i −0.587715 0.809068i \(-0.699973\pi\)
0.587715 0.809068i \(-0.300027\pi\)
\(84\) 0 0
\(85\) 10.4490i 1.13336i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.8923 1.26058 0.630290 0.776360i \(-0.282936\pi\)
0.630290 + 0.776360i \(0.282936\pi\)
\(90\) 0 0
\(91\) − 7.10447i − 0.744751i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.46425 0.252827
\(96\) 0 0
\(97\) −2.60256 −0.264250 −0.132125 0.991233i \(-0.542180\pi\)
−0.132125 + 0.991233i \(0.542180\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 9.58182i − 0.953427i −0.879059 0.476713i \(-0.841828\pi\)
0.879059 0.476713i \(-0.158172\pi\)
\(102\) 0 0
\(103\) −9.44175 −0.930323 −0.465162 0.885226i \(-0.654004\pi\)
−0.465162 + 0.885226i \(0.654004\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.04240i 0.487467i 0.969842 + 0.243733i \(0.0783722\pi\)
−0.969842 + 0.243733i \(0.921628\pi\)
\(108\) 0 0
\(109\) − 2.35923i − 0.225973i −0.993597 0.112987i \(-0.963958\pi\)
0.993597 0.112987i \(-0.0360417\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.0317 1.31999 0.659994 0.751271i \(-0.270559\pi\)
0.659994 + 0.751271i \(0.270559\pi\)
\(114\) 0 0
\(115\) − 1.79067i − 0.166981i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.81307 0.624553
\(120\) 0 0
\(121\) 5.78632 0.526029
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.7293i − 1.04910i
\(126\) 0 0
\(127\) 17.0218 1.51044 0.755220 0.655472i \(-0.227530\pi\)
0.755220 + 0.655472i \(0.227530\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 17.8861i − 1.56272i −0.624082 0.781359i \(-0.714527\pi\)
0.624082 0.781359i \(-0.285473\pi\)
\(132\) 0 0
\(133\) − 1.60676i − 0.139324i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.18009 −0.784308 −0.392154 0.919900i \(-0.628270\pi\)
−0.392154 + 0.919900i \(0.628270\pi\)
\(138\) 0 0
\(139\) − 12.0518i − 1.02222i −0.859516 0.511108i \(-0.829235\pi\)
0.859516 0.511108i \(-0.170765\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.2220 1.35655
\(144\) 0 0
\(145\) −6.24296 −0.518449
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 7.71091i − 0.631702i −0.948809 0.315851i \(-0.897710\pi\)
0.948809 0.315851i \(-0.102290\pi\)
\(150\) 0 0
\(151\) −5.91312 −0.481203 −0.240601 0.970624i \(-0.577345\pi\)
−0.240601 + 0.970624i \(0.577345\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 12.1247i − 0.973883i
\(156\) 0 0
\(157\) 0.906198i 0.0723225i 0.999346 + 0.0361612i \(0.0115130\pi\)
−0.999346 + 0.0361612i \(0.988487\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.16757 −0.0920172
\(162\) 0 0
\(163\) 4.46083i 0.349400i 0.984622 + 0.174700i \(0.0558955\pi\)
−0.984622 + 0.174700i \(0.944105\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −22.6885 −1.75569 −0.877845 0.478944i \(-0.841020\pi\)
−0.877845 + 0.478944i \(0.841020\pi\)
\(168\) 0 0
\(169\) −37.4735 −2.88258
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 24.2316i 1.84229i 0.389218 + 0.921145i \(0.372745\pi\)
−0.389218 + 0.921145i \(0.627255\pi\)
\(174\) 0 0
\(175\) −2.64784 −0.200158
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 8.82249i − 0.659424i −0.944082 0.329712i \(-0.893048\pi\)
0.944082 0.329712i \(-0.106952\pi\)
\(180\) 0 0
\(181\) 4.29583i 0.319306i 0.987173 + 0.159653i \(0.0510376\pi\)
−0.987173 + 0.159653i \(0.948962\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −10.8099 −0.794760
\(186\) 0 0
\(187\) 15.5566i 1.13761i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.224849 −0.0162695 −0.00813476 0.999967i \(-0.502589\pi\)
−0.00813476 + 0.999967i \(0.502589\pi\)
\(192\) 0 0
\(193\) −11.7447 −0.845404 −0.422702 0.906269i \(-0.638918\pi\)
−0.422702 + 0.906269i \(0.638918\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.7998i − 1.05444i −0.849729 0.527220i \(-0.823234\pi\)
0.849729 0.527220i \(-0.176766\pi\)
\(198\) 0 0
\(199\) 19.6242 1.39112 0.695561 0.718467i \(-0.255156\pi\)
0.695561 + 0.718467i \(0.255156\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.07059i 0.285699i
\(204\) 0 0
\(205\) 15.5989i 1.08948i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.66879 0.253776
\(210\) 0 0
\(211\) − 0.913482i − 0.0628867i −0.999506 0.0314434i \(-0.989990\pi\)
0.999506 0.0314434i \(-0.0100104\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −0.528687 −0.0360561
\(216\) 0 0
\(217\) −7.90568 −0.536673
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 48.4033i − 3.25596i
\(222\) 0 0
\(223\) 11.8646 0.794512 0.397256 0.917708i \(-0.369963\pi\)
0.397256 + 0.917708i \(0.369963\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 19.2403i − 1.27702i −0.769613 0.638510i \(-0.779551\pi\)
0.769613 0.638510i \(-0.220449\pi\)
\(228\) 0 0
\(229\) 7.11176i 0.469958i 0.972000 + 0.234979i \(0.0755022\pi\)
−0.972000 + 0.234979i \(0.924498\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.5527 1.21542 0.607712 0.794157i \(-0.292087\pi\)
0.607712 + 0.794157i \(0.292087\pi\)
\(234\) 0 0
\(235\) − 4.76757i − 0.311002i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.86610 0.250077 0.125039 0.992152i \(-0.460095\pi\)
0.125039 + 0.992152i \(0.460095\pi\)
\(240\) 0 0
\(241\) −11.5687 −0.745205 −0.372603 0.927991i \(-0.621535\pi\)
−0.372603 + 0.927991i \(0.621535\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.53368i − 0.0979829i
\(246\) 0 0
\(247\) −11.4152 −0.726331
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.5653i 0.856235i 0.903723 + 0.428117i \(0.140823\pi\)
−0.903723 + 0.428117i \(0.859177\pi\)
\(252\) 0 0
\(253\) − 2.66596i − 0.167608i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.1567 −0.695938 −0.347969 0.937506i \(-0.613129\pi\)
−0.347969 + 0.937506i \(0.613129\pi\)
\(258\) 0 0
\(259\) 7.04836i 0.437964i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.9609 −0.860864 −0.430432 0.902623i \(-0.641639\pi\)
−0.430432 + 0.902623i \(0.641639\pi\)
\(264\) 0 0
\(265\) 8.69341 0.534032
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 16.1185i − 0.982761i −0.870945 0.491380i \(-0.836493\pi\)
0.870945 0.491380i \(-0.163507\pi\)
\(270\) 0 0
\(271\) −8.91716 −0.541679 −0.270840 0.962624i \(-0.587301\pi\)
−0.270840 + 0.962624i \(0.587301\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 6.04594i − 0.364584i
\(276\) 0 0
\(277\) − 26.8189i − 1.61139i −0.592328 0.805697i \(-0.701791\pi\)
0.592328 0.805697i \(-0.298209\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.0688 0.660306 0.330153 0.943927i \(-0.392900\pi\)
0.330153 + 0.943927i \(0.392900\pi\)
\(282\) 0 0
\(283\) − 21.7405i − 1.29234i −0.763194 0.646170i \(-0.776370\pi\)
0.763194 0.646170i \(-0.223630\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.1710 0.600373
\(288\) 0 0
\(289\) 29.4179 1.73046
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 8.80042i − 0.514126i −0.966395 0.257063i \(-0.917245\pi\)
0.966395 0.257063i \(-0.0827548\pi\)
\(294\) 0 0
\(295\) −2.11900 −0.123373
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.29496i 0.479710i
\(300\) 0 0
\(301\) 0.344719i 0.0198693i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.44312 0.483452
\(306\) 0 0
\(307\) − 3.11170i − 0.177594i −0.996050 0.0887971i \(-0.971698\pi\)
0.996050 0.0887971i \(-0.0283023\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.443815 0.0251664 0.0125832 0.999921i \(-0.495995\pi\)
0.0125832 + 0.999921i \(0.495995\pi\)
\(312\) 0 0
\(313\) 19.1601 1.08299 0.541497 0.840703i \(-0.317858\pi\)
0.541497 + 0.840703i \(0.317858\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 11.2142i − 0.629854i −0.949116 0.314927i \(-0.898020\pi\)
0.949116 0.314927i \(-0.101980\pi\)
\(318\) 0 0
\(319\) −9.29456 −0.520396
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 10.9470i − 0.609105i
\(324\) 0 0
\(325\) 18.8115i 1.04347i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.10859 −0.171382
\(330\) 0 0
\(331\) − 17.8007i − 0.978414i −0.872168 0.489207i \(-0.837286\pi\)
0.872168 0.489207i \(-0.162714\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.8201 −0.700437
\(336\) 0 0
\(337\) −3.46393 −0.188692 −0.0943461 0.995539i \(-0.530076\pi\)
−0.0943461 + 0.995539i \(0.530076\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 18.0514i − 0.977539i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 27.3962i − 1.47070i −0.677686 0.735352i \(-0.737017\pi\)
0.677686 0.735352i \(-0.262983\pi\)
\(348\) 0 0
\(349\) − 1.62149i − 0.0867961i −0.999058 0.0433981i \(-0.986182\pi\)
0.999058 0.0433981i \(-0.0138184\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.5882 0.882903 0.441452 0.897285i \(-0.354464\pi\)
0.441452 + 0.897285i \(0.354464\pi\)
\(354\) 0 0
\(355\) − 18.8364i − 0.999732i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.84388 −0.413984 −0.206992 0.978343i \(-0.566367\pi\)
−0.206992 + 0.978343i \(0.566367\pi\)
\(360\) 0 0
\(361\) 16.4183 0.864122
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.12740i 0.477750i
\(366\) 0 0
\(367\) −0.761448 −0.0397472 −0.0198736 0.999803i \(-0.506326\pi\)
−0.0198736 + 0.999803i \(0.506326\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 5.66835i − 0.294286i
\(372\) 0 0
\(373\) 13.3240i 0.689888i 0.938623 + 0.344944i \(0.112102\pi\)
−0.938623 + 0.344944i \(0.887898\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 28.9194 1.48942
\(378\) 0 0
\(379\) − 13.6595i − 0.701640i −0.936443 0.350820i \(-0.885903\pi\)
0.936443 0.350820i \(-0.114097\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 9.70332 0.495816 0.247908 0.968784i \(-0.420257\pi\)
0.247908 + 0.968784i \(0.420257\pi\)
\(384\) 0 0
\(385\) 3.50191 0.178474
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 24.7223i 1.25347i 0.779233 + 0.626734i \(0.215609\pi\)
−0.779233 + 0.626734i \(0.784391\pi\)
\(390\) 0 0
\(391\) −7.95472 −0.402287
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 23.0093i − 1.15772i
\(396\) 0 0
\(397\) − 29.1070i − 1.46084i −0.682999 0.730420i \(-0.739325\pi\)
0.682999 0.730420i \(-0.260675\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.7247 0.685380 0.342690 0.939449i \(-0.388662\pi\)
0.342690 + 0.939449i \(0.388662\pi\)
\(402\) 0 0
\(403\) 56.1657i 2.79781i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.0939 −0.797743
\(408\) 0 0
\(409\) 18.1707 0.898486 0.449243 0.893410i \(-0.351694\pi\)
0.449243 + 0.893410i \(0.351694\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.38165i 0.0679863i
\(414\) 0 0
\(415\) −22.6093 −1.10985
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 32.9489i 1.60966i 0.593507 + 0.804829i \(0.297743\pi\)
−0.593507 + 0.804829i \(0.702257\pi\)
\(420\) 0 0
\(421\) − 27.5222i − 1.34135i −0.741751 0.670675i \(-0.766004\pi\)
0.741751 0.670675i \(-0.233996\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −18.0399 −0.875064
\(426\) 0 0
\(427\) − 5.50516i − 0.266413i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 40.3524 1.94371 0.971855 0.235582i \(-0.0756996\pi\)
0.971855 + 0.235582i \(0.0756996\pi\)
\(432\) 0 0
\(433\) −38.5928 −1.85465 −0.927327 0.374253i \(-0.877899\pi\)
−0.927327 + 0.374253i \(0.877899\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.87600i 0.0897413i
\(438\) 0 0
\(439\) 1.68908 0.0806152 0.0403076 0.999187i \(-0.487166\pi\)
0.0403076 + 0.999187i \(0.487166\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.3190i 0.870361i 0.900343 + 0.435180i \(0.143315\pi\)
−0.900343 + 0.435180i \(0.856685\pi\)
\(444\) 0 0
\(445\) − 18.2389i − 0.864607i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 34.2644 1.61704 0.808518 0.588471i \(-0.200270\pi\)
0.808518 + 0.588471i \(0.200270\pi\)
\(450\) 0 0
\(451\) 23.2238i 1.09357i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −10.8960 −0.510810
\(456\) 0 0
\(457\) 3.08541 0.144330 0.0721648 0.997393i \(-0.477009\pi\)
0.0721648 + 0.997393i \(0.477009\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.62369i 0.401645i 0.979628 + 0.200823i \(0.0643615\pi\)
−0.979628 + 0.200823i \(0.935639\pi\)
\(462\) 0 0
\(463\) −3.29178 −0.152982 −0.0764910 0.997070i \(-0.524372\pi\)
−0.0764910 + 0.997070i \(0.524372\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.3473i 1.45058i 0.688443 + 0.725290i \(0.258294\pi\)
−0.688443 + 0.725290i \(0.741706\pi\)
\(468\) 0 0
\(469\) 8.35907i 0.385986i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.787113 −0.0361915
\(474\) 0 0
\(475\) 4.25444i 0.195207i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 23.7139 1.08351 0.541757 0.840535i \(-0.317759\pi\)
0.541757 + 0.840535i \(0.317759\pi\)
\(480\) 0 0
\(481\) 50.0749 2.28322
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.99148i 0.181244i
\(486\) 0 0
\(487\) −35.7317 −1.61916 −0.809580 0.587010i \(-0.800305\pi\)
−0.809580 + 0.587010i \(0.800305\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 40.3543i − 1.82116i −0.413330 0.910581i \(-0.635634\pi\)
0.413330 0.910581i \(-0.364366\pi\)
\(492\) 0 0
\(493\) 27.7332i 1.24904i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.2819 −0.550917
\(498\) 0 0
\(499\) 41.0646i 1.83831i 0.393902 + 0.919153i \(0.371125\pi\)
−0.393902 + 0.919153i \(0.628875\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 13.5744 0.605252 0.302626 0.953109i \(-0.402137\pi\)
0.302626 + 0.953109i \(0.402137\pi\)
\(504\) 0 0
\(505\) −14.6954 −0.653937
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 13.3591i 0.592132i 0.955167 + 0.296066i \(0.0956749\pi\)
−0.955167 + 0.296066i \(0.904325\pi\)
\(510\) 0 0
\(511\) 5.95132 0.263271
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.4806i 0.638091i
\(516\) 0 0
\(517\) − 7.09799i − 0.312169i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.5338 0.899600 0.449800 0.893129i \(-0.351495\pi\)
0.449800 + 0.893129i \(0.351495\pi\)
\(522\) 0 0
\(523\) 4.37298i 0.191217i 0.995419 + 0.0956086i \(0.0304797\pi\)
−0.995419 + 0.0956086i \(0.969520\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −53.8619 −2.34626
\(528\) 0 0
\(529\) −21.6368 −0.940730
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 72.2593i − 3.12990i
\(534\) 0 0
\(535\) 7.73340 0.334344
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.28335i − 0.0983507i
\(540\) 0 0
\(541\) − 11.8762i − 0.510598i −0.966862 0.255299i \(-0.917826\pi\)
0.966862 0.255299i \(-0.0821740\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.61829 −0.154991
\(546\) 0 0
\(547\) − 5.50044i − 0.235182i −0.993062 0.117591i \(-0.962483\pi\)
0.993062 0.117591i \(-0.0375171\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 6.54045 0.278633
\(552\) 0 0
\(553\) −15.0027 −0.637980
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.19596i 0.347274i 0.984810 + 0.173637i \(0.0555519\pi\)
−0.984810 + 0.173637i \(0.944448\pi\)
\(558\) 0 0
\(559\) 2.44905 0.103584
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 0.145215i − 0.00612008i −0.999995 0.00306004i \(-0.999026\pi\)
0.999995 0.00306004i \(-0.000974043\pi\)
\(564\) 0 0
\(565\) − 21.5200i − 0.905354i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.09548 −0.213613 −0.106807 0.994280i \(-0.534063\pi\)
−0.106807 + 0.994280i \(0.534063\pi\)
\(570\) 0 0
\(571\) 25.3061i 1.05903i 0.848301 + 0.529514i \(0.177626\pi\)
−0.848301 + 0.529514i \(0.822374\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.09153 0.128926
\(576\) 0 0
\(577\) 19.5004 0.811814 0.405907 0.913914i \(-0.366956\pi\)
0.405907 + 0.913914i \(0.366956\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14.7419i 0.611598i
\(582\) 0 0
\(583\) 12.9428 0.536037
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 36.0783i − 1.48911i −0.667561 0.744555i \(-0.732662\pi\)
0.667561 0.744555i \(-0.267338\pi\)
\(588\) 0 0
\(589\) 12.7025i 0.523399i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −16.7669 −0.688535 −0.344268 0.938872i \(-0.611873\pi\)
−0.344268 + 0.938872i \(0.611873\pi\)
\(594\) 0 0
\(595\) − 10.4490i − 0.428369i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.1008 1.27074 0.635372 0.772206i \(-0.280847\pi\)
0.635372 + 0.772206i \(0.280847\pi\)
\(600\) 0 0
\(601\) 17.1858 0.701022 0.350511 0.936559i \(-0.386008\pi\)
0.350511 + 0.936559i \(0.386008\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 8.87434i − 0.360793i
\(606\) 0 0
\(607\) −12.9692 −0.526404 −0.263202 0.964741i \(-0.584779\pi\)
−0.263202 + 0.964741i \(0.584779\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.0849i 0.893459i
\(612\) 0 0
\(613\) 26.5265i 1.07139i 0.844410 + 0.535697i \(0.179951\pi\)
−0.844410 + 0.535697i \(0.820049\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −37.3263 −1.50270 −0.751350 0.659904i \(-0.770597\pi\)
−0.751350 + 0.659904i \(0.770597\pi\)
\(618\) 0 0
\(619\) − 10.8774i − 0.437200i −0.975815 0.218600i \(-0.929851\pi\)
0.975815 0.218600i \(-0.0701489\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −11.8923 −0.476454
\(624\) 0 0
\(625\) −4.74975 −0.189990
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.0210i 1.91472i
\(630\) 0 0
\(631\) 24.2943 0.967140 0.483570 0.875306i \(-0.339340\pi\)
0.483570 + 0.875306i \(0.339340\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 26.1059i − 1.03598i
\(636\) 0 0
\(637\) 7.10447i 0.281489i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −16.5765 −0.654731 −0.327365 0.944898i \(-0.606161\pi\)
−0.327365 + 0.944898i \(0.606161\pi\)
\(642\) 0 0
\(643\) − 7.15319i − 0.282094i −0.990003 0.141047i \(-0.954953\pi\)
0.990003 0.141047i \(-0.0450469\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.94968 0.312534 0.156267 0.987715i \(-0.450054\pi\)
0.156267 + 0.987715i \(0.450054\pi\)
\(648\) 0 0
\(649\) −3.15478 −0.123836
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 7.45217i − 0.291626i −0.989312 0.145813i \(-0.953420\pi\)
0.989312 0.145813i \(-0.0465798\pi\)
\(654\) 0 0
\(655\) −27.4315 −1.07184
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 26.2741i − 1.02349i −0.859136 0.511747i \(-0.828999\pi\)
0.859136 0.511747i \(-0.171001\pi\)
\(660\) 0 0
\(661\) − 3.98821i − 0.155123i −0.996988 0.0775617i \(-0.975287\pi\)
0.996988 0.0775617i \(-0.0247135\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.46425 −0.0955594
\(666\) 0 0
\(667\) − 4.75268i − 0.184025i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.5702 0.485267
\(672\) 0 0
\(673\) 30.5709 1.17842 0.589211 0.807979i \(-0.299439\pi\)
0.589211 + 0.807979i \(0.299439\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 21.9032i − 0.841808i −0.907105 0.420904i \(-0.861713\pi\)
0.907105 0.420904i \(-0.138287\pi\)
\(678\) 0 0
\(679\) 2.60256 0.0998770
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 13.0423i 0.499051i 0.968368 + 0.249525i \(0.0802746\pi\)
−0.968368 + 0.249525i \(0.919725\pi\)
\(684\) 0 0
\(685\) 14.0793i 0.537942i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −40.2707 −1.53419
\(690\) 0 0
\(691\) − 3.25399i − 0.123788i −0.998083 0.0618938i \(-0.980286\pi\)
0.998083 0.0618938i \(-0.0197140\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −18.4835 −0.701118
\(696\) 0 0
\(697\) 69.2954 2.62475
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 2.70404i − 0.102130i −0.998695 0.0510650i \(-0.983738\pi\)
0.998695 0.0510650i \(-0.0162616\pi\)
\(702\) 0 0
\(703\) 11.3250 0.427132
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.58182i 0.360361i
\(708\) 0 0
\(709\) − 49.0447i − 1.84191i −0.389668 0.920955i \(-0.627410\pi\)
0.389668 0.920955i \(-0.372590\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.23042 0.345682
\(714\) 0 0
\(715\) − 24.8793i − 0.930431i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 21.3291 0.795442 0.397721 0.917506i \(-0.369801\pi\)
0.397721 + 0.917506i \(0.369801\pi\)
\(720\) 0 0
\(721\) 9.44175 0.351629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 10.7783i − 0.400294i
\(726\) 0 0
\(727\) 23.5713 0.874210 0.437105 0.899411i \(-0.356004\pi\)
0.437105 + 0.899411i \(0.356004\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.34859i 0.0868658i
\(732\) 0 0
\(733\) − 2.28803i − 0.0845105i −0.999107 0.0422552i \(-0.986546\pi\)
0.999107 0.0422552i \(-0.0134543\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −19.0867 −0.703067
\(738\) 0 0
\(739\) 33.6953i 1.23950i 0.784798 + 0.619751i \(0.212766\pi\)
−0.784798 + 0.619751i \(0.787234\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −35.7898 −1.31300 −0.656501 0.754326i \(-0.727964\pi\)
−0.656501 + 0.754326i \(0.727964\pi\)
\(744\) 0 0
\(745\) −11.8260 −0.433272
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 5.04240i − 0.184245i
\(750\) 0 0
\(751\) −7.35858 −0.268518 −0.134259 0.990946i \(-0.542865\pi\)
−0.134259 + 0.990946i \(0.542865\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 9.06881i 0.330048i
\(756\) 0 0
\(757\) 23.2086i 0.843530i 0.906705 + 0.421765i \(0.138589\pi\)
−0.906705 + 0.421765i \(0.861411\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.1491 −0.802902 −0.401451 0.915880i \(-0.631494\pi\)
−0.401451 + 0.915880i \(0.631494\pi\)
\(762\) 0 0
\(763\) 2.35923i 0.0854099i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.81586 0.354430
\(768\) 0 0
\(769\) −27.4818 −0.991020 −0.495510 0.868602i \(-0.665019\pi\)
−0.495510 + 0.868602i \(0.665019\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 39.9693i 1.43760i 0.695219 + 0.718798i \(0.255308\pi\)
−0.695219 + 0.718798i \(0.744692\pi\)
\(774\) 0 0
\(775\) 20.9330 0.751935
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 16.3423i − 0.585523i
\(780\) 0 0
\(781\) − 28.0438i − 1.00348i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.38981 0.0496046
\(786\) 0 0
\(787\) 34.5651i 1.23211i 0.787702 + 0.616057i \(0.211271\pi\)
−0.787702 + 0.616057i \(0.788729\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0317 −0.498908
\(792\) 0 0
\(793\) −39.1112 −1.38888
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 16.9968i 0.602059i 0.953615 + 0.301029i \(0.0973302\pi\)
−0.953615 + 0.301029i \(0.902670\pi\)
\(798\) 0 0
\(799\) −21.1790 −0.749260
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 13.5889i 0.479543i
\(804\) 0 0
\(805\) 1.79067i 0.0631128i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.33018 0.0819249 0.0409624 0.999161i \(-0.486958\pi\)
0.0409624 + 0.999161i \(0.486958\pi\)
\(810\) 0 0
\(811\) − 8.79524i − 0.308843i −0.988005 0.154421i \(-0.950649\pi\)
0.988005 0.154421i \(-0.0493513\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.84147 0.239646
\(816\) 0 0
\(817\) 0.553880 0.0193778
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 14.1689i 0.494497i 0.968952 + 0.247249i \(0.0795265\pi\)
−0.968952 + 0.247249i \(0.920474\pi\)
\(822\) 0 0
\(823\) −5.53482 −0.192932 −0.0964659 0.995336i \(-0.530754\pi\)
−0.0964659 + 0.995336i \(0.530754\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 7.32030i − 0.254552i −0.991867 0.127276i \(-0.959377\pi\)
0.991867 0.127276i \(-0.0406234\pi\)
\(828\) 0 0
\(829\) − 32.7046i − 1.13588i −0.823071 0.567939i \(-0.807741\pi\)
0.823071 0.567939i \(-0.192259\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.81307 −0.236059
\(834\) 0 0
\(835\) 34.7968i 1.20419i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 19.2836 0.665744 0.332872 0.942972i \(-0.391982\pi\)
0.332872 + 0.942972i \(0.391982\pi\)
\(840\) 0 0
\(841\) 12.4303 0.428632
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 57.4723i 1.97711i
\(846\) 0 0
\(847\) −5.78632 −0.198820
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 8.22944i − 0.282102i
\(852\) 0 0
\(853\) − 24.5003i − 0.838874i −0.907784 0.419437i \(-0.862228\pi\)
0.907784 0.419437i \(-0.137772\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 24.0113 0.820211 0.410105 0.912038i \(-0.365492\pi\)
0.410105 + 0.912038i \(0.365492\pi\)
\(858\) 0 0
\(859\) − 2.15328i − 0.0734689i −0.999325 0.0367344i \(-0.988304\pi\)
0.999325 0.0367344i \(-0.0116956\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.94727 −0.236488 −0.118244 0.992985i \(-0.537726\pi\)
−0.118244 + 0.992985i \(0.537726\pi\)
\(864\) 0 0
\(865\) 37.1633 1.26359
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 34.2564i − 1.16207i
\(870\) 0 0
\(871\) 59.3868 2.01225
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.7293i 0.396523i
\(876\) 0 0
\(877\) 24.2307i 0.818211i 0.912487 + 0.409106i \(0.134159\pi\)
−0.912487 + 0.409106i \(0.865841\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.4462 1.43005 0.715024 0.699100i \(-0.246416\pi\)
0.715024 + 0.699100i \(0.246416\pi\)
\(882\) 0 0
\(883\) 0.397509i 0.0133773i 0.999978 + 0.00668863i \(0.00212907\pi\)
−0.999978 + 0.00668863i \(0.997871\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −36.3698 −1.22118 −0.610590 0.791947i \(-0.709068\pi\)
−0.610590 + 0.791947i \(0.709068\pi\)
\(888\) 0 0
\(889\) −17.0218 −0.570892
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.99476i 0.167143i
\(894\) 0 0
\(895\) −13.5308 −0.452286
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 32.1807i − 1.07329i
\(900\) 0 0
\(901\) − 38.6189i − 1.28658i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.58840 0.219006
\(906\) 0 0
\(907\) − 52.1212i − 1.73066i −0.501205 0.865329i \(-0.667110\pi\)
0.501205 0.865329i \(-0.332890\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25.8160 0.855322 0.427661 0.903939i \(-0.359338\pi\)
0.427661 + 0.903939i \(0.359338\pi\)
\(912\) 0 0
\(913\) −33.6609 −1.11401
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.8861i 0.590652i
\(918\) 0 0
\(919\) 4.71609 0.155569 0.0777847 0.996970i \(-0.475215\pi\)
0.0777847 + 0.996970i \(0.475215\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 87.2562i 2.87207i
\(924\) 0 0
\(925\) − 18.6629i − 0.613634i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −48.4774 −1.59049 −0.795246 0.606287i \(-0.792658\pi\)
−0.795246 + 0.606287i \(0.792658\pi\)
\(930\) 0 0
\(931\) 1.60676i 0.0526594i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 23.8588 0.780266
\(936\) 0 0
\(937\) 11.5193 0.376319 0.188159 0.982138i \(-0.439748\pi\)
0.188159 + 0.982138i \(0.439748\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 32.3390i − 1.05422i −0.849796 0.527111i \(-0.823275\pi\)
0.849796 0.527111i \(-0.176725\pi\)
\(942\) 0 0
\(943\) −11.8753 −0.386712
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 37.4382i 1.21658i 0.793715 + 0.608289i \(0.208144\pi\)
−0.793715 + 0.608289i \(0.791856\pi\)
\(948\) 0 0
\(949\) − 42.2810i − 1.37250i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.4974 0.599190 0.299595 0.954066i \(-0.403148\pi\)
0.299595 + 0.954066i \(0.403148\pi\)
\(954\) 0 0
\(955\) 0.344846i 0.0111590i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.18009 0.296441
\(960\) 0 0
\(961\) 31.4998 1.01612
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 18.0126i 0.579846i
\(966\) 0 0
\(967\) −44.4221 −1.42852 −0.714259 0.699882i \(-0.753236\pi\)
−0.714259 + 0.699882i \(0.753236\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 37.5434i 1.20482i 0.798185 + 0.602412i \(0.205794\pi\)
−0.798185 + 0.602412i \(0.794206\pi\)
\(972\) 0 0
\(973\) 12.0518i 0.386361i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.89949 0.220734 0.110367 0.993891i \(-0.464797\pi\)
0.110367 + 0.993891i \(0.464797\pi\)
\(978\) 0 0
\(979\) − 27.1542i − 0.867853i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 28.9252 0.922571 0.461285 0.887252i \(-0.347388\pi\)
0.461285 + 0.887252i \(0.347388\pi\)
\(984\) 0 0
\(985\) −22.6980 −0.723220
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 0.402483i − 0.0127982i
\(990\) 0 0
\(991\) −37.7278 −1.19846 −0.599232 0.800575i \(-0.704527\pi\)
−0.599232 + 0.800575i \(0.704527\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 30.0971i − 0.954144i
\(996\) 0 0
\(997\) 53.0914i 1.68142i 0.541484 + 0.840711i \(0.317863\pi\)
−0.541484 + 0.840711i \(0.682137\pi\)
\(998\) 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.g.3025.10 24
3.2 odd 2 inner 6048.2.c.g.3025.16 24
4.3 odd 2 1512.2.c.f.757.4 yes 24
8.3 odd 2 1512.2.c.f.757.3 24
8.5 even 2 inner 6048.2.c.g.3025.15 24
12.11 even 2 1512.2.c.f.757.21 yes 24
24.5 odd 2 inner 6048.2.c.g.3025.9 24
24.11 even 2 1512.2.c.f.757.22 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.f.757.3 24 8.3 odd 2
1512.2.c.f.757.4 yes 24 4.3 odd 2
1512.2.c.f.757.21 yes 24 12.11 even 2
1512.2.c.f.757.22 yes 24 24.11 even 2
6048.2.c.g.3025.9 24 24.5 odd 2 inner
6048.2.c.g.3025.10 24 1.1 even 1 trivial
6048.2.c.g.3025.15 24 8.5 even 2 inner
6048.2.c.g.3025.16 24 3.2 odd 2 inner