Properties

Label 6048.2.h.i.2591.2
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 2591.2
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.i.2591.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.29730i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q+4.29730i q^{5} -1.00000i q^{7} +4.77104 q^{11} -0.0836472 q^{13} +2.60889i q^{17} -4.85636i q^{19} -4.78044 q^{23} -13.4668 q^{25} -0.263176i q^{29} -0.304131i q^{31} +4.29730 q^{35} +9.56782 q^{37} +10.5898i q^{41} +4.24414i q^{43} +9.85829 q^{47} -1.00000 q^{49} +11.9022i q^{53} +20.5026i q^{55} -5.50796 q^{59} -8.62260 q^{61} -0.359457i q^{65} -1.80727i q^{67} +10.8966 q^{71} -4.88339 q^{73} -4.77104i q^{77} +14.3421i q^{79} -10.4067 q^{83} -11.2112 q^{85} +7.85368i q^{89} +0.0836472i q^{91} +20.8693 q^{95} -1.29824 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24q + O(q^{10}) \) \( 24q - 8q^{13} - 24q^{25} + 48q^{37} - 24q^{49} - 48q^{61} - 32q^{73} - 80q^{85} + 64q^{97} + O(q^{100}) \)

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
\(5\) 4.29730i 1.92181i 0.276875 + 0.960906i \(0.410701\pi\)
−0.276875 + 0.960906i \(0.589299\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.77104 1.43852 0.719262 0.694739i \(-0.244480\pi\)
0.719262 + 0.694739i \(0.244480\pi\)
\(12\) 0 0
\(13\) −0.0836472 −0.0231996 −0.0115998 0.999933i \(-0.503692\pi\)
−0.0115998 + 0.999933i \(0.503692\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.60889i 0.632748i 0.948635 + 0.316374i \(0.102465\pi\)
−0.948635 + 0.316374i \(0.897535\pi\)
\(18\) 0 0
\(19\) − 4.85636i − 1.11413i −0.830470 0.557063i \(-0.811928\pi\)
0.830470 0.557063i \(-0.188072\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.78044 −0.996790 −0.498395 0.866950i \(-0.666077\pi\)
−0.498395 + 0.866950i \(0.666077\pi\)
\(24\) 0 0
\(25\) −13.4668 −2.69336
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.263176i − 0.0488706i −0.999701 0.0244353i \(-0.992221\pi\)
0.999701 0.0244353i \(-0.00777877\pi\)
\(30\) 0 0
\(31\) − 0.304131i − 0.0546236i −0.999627 0.0273118i \(-0.991305\pi\)
0.999627 0.0273118i \(-0.00869469\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.29730 0.726377
\(36\) 0 0
\(37\) 9.56782 1.57294 0.786470 0.617628i \(-0.211906\pi\)
0.786470 + 0.617628i \(0.211906\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.5898i 1.65385i 0.562310 + 0.826927i \(0.309913\pi\)
−0.562310 + 0.826927i \(0.690087\pi\)
\(42\) 0 0
\(43\) 4.24414i 0.647225i 0.946190 + 0.323613i \(0.104897\pi\)
−0.946190 + 0.323613i \(0.895103\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.85829 1.43798 0.718990 0.695021i \(-0.244605\pi\)
0.718990 + 0.695021i \(0.244605\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.9022i 1.63489i 0.576009 + 0.817443i \(0.304609\pi\)
−0.576009 + 0.817443i \(0.695391\pi\)
\(54\) 0 0
\(55\) 20.5026i 2.76457i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.50796 −0.717076 −0.358538 0.933515i \(-0.616725\pi\)
−0.358538 + 0.933515i \(0.616725\pi\)
\(60\) 0 0
\(61\) −8.62260 −1.10401 −0.552005 0.833841i \(-0.686137\pi\)
−0.552005 + 0.833841i \(0.686137\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 0.359457i − 0.0445852i
\(66\) 0 0
\(67\) − 1.80727i − 0.220793i −0.993888 0.110396i \(-0.964788\pi\)
0.993888 0.110396i \(-0.0352120\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.8966 1.29319 0.646597 0.762832i \(-0.276192\pi\)
0.646597 + 0.762832i \(0.276192\pi\)
\(72\) 0 0
\(73\) −4.88339 −0.571557 −0.285779 0.958296i \(-0.592252\pi\)
−0.285779 + 0.958296i \(0.592252\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.77104i − 0.543711i
\(78\) 0 0
\(79\) 14.3421i 1.61362i 0.590814 + 0.806808i \(0.298807\pi\)
−0.590814 + 0.806808i \(0.701193\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −10.4067 −1.14228 −0.571142 0.820852i \(-0.693499\pi\)
−0.571142 + 0.820852i \(0.693499\pi\)
\(84\) 0 0
\(85\) −11.2112 −1.21602
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.85368i 0.832488i 0.909253 + 0.416244i \(0.136654\pi\)
−0.909253 + 0.416244i \(0.863346\pi\)
\(90\) 0 0
\(91\) 0.0836472i 0.00876861i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 20.8693 2.14114
\(96\) 0 0
\(97\) −1.29824 −0.131816 −0.0659082 0.997826i \(-0.520994\pi\)
−0.0659082 + 0.997826i \(0.520994\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.5103i 1.14532i 0.819794 + 0.572659i \(0.194088\pi\)
−0.819794 + 0.572659i \(0.805912\pi\)
\(102\) 0 0
\(103\) 1.07843i 0.106261i 0.998588 + 0.0531306i \(0.0169200\pi\)
−0.998588 + 0.0531306i \(0.983080\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.95223 0.382077 0.191038 0.981583i \(-0.438814\pi\)
0.191038 + 0.981583i \(0.438814\pi\)
\(108\) 0 0
\(109\) 1.68780 0.161662 0.0808311 0.996728i \(-0.474243\pi\)
0.0808311 + 0.996728i \(0.474243\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 8.49874i − 0.799494i −0.916625 0.399747i \(-0.869098\pi\)
0.916625 0.399747i \(-0.130902\pi\)
\(114\) 0 0
\(115\) − 20.5430i − 1.91564i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.60889 0.239156
\(120\) 0 0
\(121\) 11.7629 1.06935
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 36.3844i − 3.25432i
\(126\) 0 0
\(127\) 12.2578i 1.08770i 0.839181 + 0.543852i \(0.183035\pi\)
−0.839181 + 0.543852i \(0.816965\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −10.2250 −0.893362 −0.446681 0.894693i \(-0.647394\pi\)
−0.446681 + 0.894693i \(0.647394\pi\)
\(132\) 0 0
\(133\) −4.85636 −0.421100
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 22.2500i − 1.90095i −0.310806 0.950473i \(-0.600599\pi\)
0.310806 0.950473i \(-0.399401\pi\)
\(138\) 0 0
\(139\) − 10.1276i − 0.859010i −0.903064 0.429505i \(-0.858688\pi\)
0.903064 0.429505i \(-0.141312\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.399084 −0.0333731
\(144\) 0 0
\(145\) 1.13095 0.0939201
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 8.83161i − 0.723514i −0.932272 0.361757i \(-0.882177\pi\)
0.932272 0.361757i \(-0.117823\pi\)
\(150\) 0 0
\(151\) − 4.18526i − 0.340592i −0.985393 0.170296i \(-0.945528\pi\)
0.985393 0.170296i \(-0.0544723\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.30694 0.104976
\(156\) 0 0
\(157\) −12.0045 −0.958062 −0.479031 0.877798i \(-0.659012\pi\)
−0.479031 + 0.877798i \(0.659012\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 4.78044i 0.376751i
\(162\) 0 0
\(163\) 12.7022i 0.994909i 0.867490 + 0.497455i \(0.165732\pi\)
−0.867490 + 0.497455i \(0.834268\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.2726 1.56874 0.784372 0.620290i \(-0.212985\pi\)
0.784372 + 0.620290i \(0.212985\pi\)
\(168\) 0 0
\(169\) −12.9930 −0.999462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.6754i 1.57192i 0.618278 + 0.785959i \(0.287830\pi\)
−0.618278 + 0.785959i \(0.712170\pi\)
\(174\) 0 0
\(175\) 13.4668i 1.01799i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.67796 0.349647 0.174824 0.984600i \(-0.444064\pi\)
0.174824 + 0.984600i \(0.444064\pi\)
\(180\) 0 0
\(181\) 23.9336 1.77897 0.889485 0.456965i \(-0.151063\pi\)
0.889485 + 0.456965i \(0.151063\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 41.1158i 3.02290i
\(186\) 0 0
\(187\) 12.4471i 0.910223i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −24.0841 −1.74267 −0.871334 0.490691i \(-0.836744\pi\)
−0.871334 + 0.490691i \(0.836744\pi\)
\(192\) 0 0
\(193\) −7.68081 −0.552876 −0.276438 0.961032i \(-0.589154\pi\)
−0.276438 + 0.961032i \(0.589154\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 22.5136i − 1.60403i −0.597305 0.802014i \(-0.703762\pi\)
0.597305 0.802014i \(-0.296238\pi\)
\(198\) 0 0
\(199\) 13.7258i 0.972996i 0.873682 + 0.486498i \(0.161726\pi\)
−0.873682 + 0.486498i \(0.838274\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.263176 −0.0184713
\(204\) 0 0
\(205\) −45.5077 −3.17840
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 23.1699i − 1.60270i
\(210\) 0 0
\(211\) − 2.67280i − 0.184003i −0.995759 0.0920017i \(-0.970673\pi\)
0.995759 0.0920017i \(-0.0293265\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −18.2384 −1.24385
\(216\) 0 0
\(217\) −0.304131 −0.0206458
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 0.218226i − 0.0146795i
\(222\) 0 0
\(223\) 13.8944i 0.930435i 0.885196 + 0.465217i \(0.154024\pi\)
−0.885196 + 0.465217i \(0.845976\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −16.7141 −1.10935 −0.554677 0.832065i \(-0.687158\pi\)
−0.554677 + 0.832065i \(0.687158\pi\)
\(228\) 0 0
\(229\) −9.78011 −0.646288 −0.323144 0.946350i \(-0.604740\pi\)
−0.323144 + 0.946350i \(0.604740\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 7.69348i − 0.504016i −0.967725 0.252008i \(-0.918909\pi\)
0.967725 0.252008i \(-0.0810910\pi\)
\(234\) 0 0
\(235\) 42.3641i 2.76353i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −22.5536 −1.45887 −0.729436 0.684049i \(-0.760217\pi\)
−0.729436 + 0.684049i \(0.760217\pi\)
\(240\) 0 0
\(241\) 4.08026 0.262833 0.131416 0.991327i \(-0.458047\pi\)
0.131416 + 0.991327i \(0.458047\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 4.29730i − 0.274545i
\(246\) 0 0
\(247\) 0.406221i 0.0258472i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.67785 −0.610861 −0.305430 0.952214i \(-0.598800\pi\)
−0.305430 + 0.952214i \(0.598800\pi\)
\(252\) 0 0
\(253\) −22.8077 −1.43391
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.83710i 0.239352i 0.992813 + 0.119676i \(0.0381856\pi\)
−0.992813 + 0.119676i \(0.961814\pi\)
\(258\) 0 0
\(259\) − 9.56782i − 0.594516i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.29994 0.573459 0.286729 0.958012i \(-0.407432\pi\)
0.286729 + 0.958012i \(0.407432\pi\)
\(264\) 0 0
\(265\) −51.1472 −3.14194
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.7782i 1.14493i 0.819931 + 0.572463i \(0.194012\pi\)
−0.819931 + 0.572463i \(0.805988\pi\)
\(270\) 0 0
\(271\) 30.7720i 1.86927i 0.355614 + 0.934633i \(0.384272\pi\)
−0.355614 + 0.934633i \(0.615728\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −64.2507 −3.87446
\(276\) 0 0
\(277\) 16.7278 1.00508 0.502538 0.864555i \(-0.332400\pi\)
0.502538 + 0.864555i \(0.332400\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 16.3993i − 0.978298i −0.872200 0.489149i \(-0.837307\pi\)
0.872200 0.489149i \(-0.162693\pi\)
\(282\) 0 0
\(283\) 11.0392i 0.656215i 0.944640 + 0.328108i \(0.106411\pi\)
−0.944640 + 0.328108i \(0.893589\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.5898 0.625098
\(288\) 0 0
\(289\) 10.1937 0.599630
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.1045i 0.707152i 0.935406 + 0.353576i \(0.115034\pi\)
−0.935406 + 0.353576i \(0.884966\pi\)
\(294\) 0 0
\(295\) − 23.6694i − 1.37808i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0.399870 0.0231251
\(300\) 0 0
\(301\) 4.24414 0.244628
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 37.0539i − 2.12170i
\(306\) 0 0
\(307\) − 6.71383i − 0.383179i −0.981475 0.191589i \(-0.938636\pi\)
0.981475 0.191589i \(-0.0613642\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −20.7974 −1.17931 −0.589656 0.807655i \(-0.700737\pi\)
−0.589656 + 0.807655i \(0.700737\pi\)
\(312\) 0 0
\(313\) 32.0502 1.81159 0.905793 0.423721i \(-0.139276\pi\)
0.905793 + 0.423721i \(0.139276\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.53433i 0.479336i 0.970855 + 0.239668i \(0.0770386\pi\)
−0.970855 + 0.239668i \(0.922961\pi\)
\(318\) 0 0
\(319\) − 1.25563i − 0.0703015i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.6697 0.704961
\(324\) 0 0
\(325\) 1.12646 0.0624848
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 9.85829i − 0.543505i
\(330\) 0 0
\(331\) − 4.90328i − 0.269509i −0.990879 0.134754i \(-0.956975\pi\)
0.990879 0.134754i \(-0.0430245\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.76637 0.424322
\(336\) 0 0
\(337\) 14.6839 0.799883 0.399942 0.916541i \(-0.369030\pi\)
0.399942 + 0.916541i \(0.369030\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 1.45102i − 0.0785773i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −23.1623 −1.24342 −0.621708 0.783249i \(-0.713561\pi\)
−0.621708 + 0.783249i \(0.713561\pi\)
\(348\) 0 0
\(349\) −0.685492 −0.0366936 −0.0183468 0.999832i \(-0.505840\pi\)
−0.0183468 + 0.999832i \(0.505840\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 4.88650i − 0.260082i −0.991509 0.130041i \(-0.958489\pi\)
0.991509 0.130041i \(-0.0415109\pi\)
\(354\) 0 0
\(355\) 46.8262i 2.48527i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.5643 0.874228 0.437114 0.899406i \(-0.356001\pi\)
0.437114 + 0.899406i \(0.356001\pi\)
\(360\) 0 0
\(361\) −4.58426 −0.241277
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 20.9854i − 1.09843i
\(366\) 0 0
\(367\) 3.56155i 0.185912i 0.995670 + 0.0929558i \(0.0296315\pi\)
−0.995670 + 0.0929558i \(0.970368\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.9022 0.617929
\(372\) 0 0
\(373\) 2.22788 0.115355 0.0576775 0.998335i \(-0.481630\pi\)
0.0576775 + 0.998335i \(0.481630\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.0220139i 0.00113378i
\(378\) 0 0
\(379\) − 31.3323i − 1.60943i −0.593660 0.804716i \(-0.702318\pi\)
0.593660 0.804716i \(-0.297682\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.47176 0.177399 0.0886994 0.996058i \(-0.471729\pi\)
0.0886994 + 0.996058i \(0.471729\pi\)
\(384\) 0 0
\(385\) 20.5026 1.04491
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 14.8062i − 0.750705i −0.926882 0.375353i \(-0.877522\pi\)
0.926882 0.375353i \(-0.122478\pi\)
\(390\) 0 0
\(391\) − 12.4716i − 0.630717i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −61.6324 −3.10106
\(396\) 0 0
\(397\) 2.64446 0.132722 0.0663609 0.997796i \(-0.478861\pi\)
0.0663609 + 0.997796i \(0.478861\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 32.2306i 1.60952i 0.593600 + 0.804760i \(0.297706\pi\)
−0.593600 + 0.804760i \(0.702294\pi\)
\(402\) 0 0
\(403\) 0.0254397i 0.00126724i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 45.6485 2.26271
\(408\) 0 0
\(409\) −29.1725 −1.44249 −0.721244 0.692681i \(-0.756430\pi\)
−0.721244 + 0.692681i \(0.756430\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.50796i 0.271029i
\(414\) 0 0
\(415\) − 44.7207i − 2.19525i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.33707 0.0653201 0.0326601 0.999467i \(-0.489602\pi\)
0.0326601 + 0.999467i \(0.489602\pi\)
\(420\) 0 0
\(421\) −28.9359 −1.41025 −0.705124 0.709084i \(-0.749109\pi\)
−0.705124 + 0.709084i \(0.749109\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 35.1334i − 1.70422i
\(426\) 0 0
\(427\) 8.62260i 0.417277i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 24.9335 1.20101 0.600503 0.799623i \(-0.294967\pi\)
0.600503 + 0.799623i \(0.294967\pi\)
\(432\) 0 0
\(433\) −15.2671 −0.733688 −0.366844 0.930282i \(-0.619562\pi\)
−0.366844 + 0.930282i \(0.619562\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 23.2155i 1.11055i
\(438\) 0 0
\(439\) − 21.0126i − 1.00288i −0.865193 0.501438i \(-0.832804\pi\)
0.865193 0.501438i \(-0.167196\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.73048 0.177240 0.0886202 0.996065i \(-0.471754\pi\)
0.0886202 + 0.996065i \(0.471754\pi\)
\(444\) 0 0
\(445\) −33.7496 −1.59989
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.1203i 1.04392i 0.852969 + 0.521961i \(0.174800\pi\)
−0.852969 + 0.521961i \(0.825200\pi\)
\(450\) 0 0
\(451\) 50.5246i 2.37911i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −0.359457 −0.0168516
\(456\) 0 0
\(457\) −28.5181 −1.33402 −0.667011 0.745048i \(-0.732427\pi\)
−0.667011 + 0.745048i \(0.732427\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.01774i 0.0474010i 0.999719 + 0.0237005i \(0.00754482\pi\)
−0.999719 + 0.0237005i \(0.992455\pi\)
\(462\) 0 0
\(463\) − 6.24844i − 0.290389i −0.989403 0.145195i \(-0.953619\pi\)
0.989403 0.145195i \(-0.0463809\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.0484 0.974002 0.487001 0.873401i \(-0.338091\pi\)
0.487001 + 0.873401i \(0.338091\pi\)
\(468\) 0 0
\(469\) −1.80727 −0.0834518
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20.2490i 0.931049i
\(474\) 0 0
\(475\) 65.3997i 3.00074i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.3215 0.837132 0.418566 0.908186i \(-0.362533\pi\)
0.418566 + 0.908186i \(0.362533\pi\)
\(480\) 0 0
\(481\) −0.800322 −0.0364915
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 5.57894i − 0.253326i
\(486\) 0 0
\(487\) − 36.0381i − 1.63304i −0.577317 0.816520i \(-0.695900\pi\)
0.577317 0.816520i \(-0.304100\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 19.2411 0.868341 0.434170 0.900831i \(-0.357042\pi\)
0.434170 + 0.900831i \(0.357042\pi\)
\(492\) 0 0
\(493\) 0.686597 0.0309228
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 10.8966i − 0.488781i
\(498\) 0 0
\(499\) 14.6319i 0.655015i 0.944849 + 0.327508i \(0.106209\pi\)
−0.944849 + 0.327508i \(0.893791\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 42.3558 1.88855 0.944275 0.329157i \(-0.106765\pi\)
0.944275 + 0.329157i \(0.106765\pi\)
\(504\) 0 0
\(505\) −49.4632 −2.20109
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 1.90879i − 0.0846056i −0.999105 0.0423028i \(-0.986531\pi\)
0.999105 0.0423028i \(-0.0134694\pi\)
\(510\) 0 0
\(511\) 4.88339i 0.216028i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.63435 −0.204214
\(516\) 0 0
\(517\) 47.0344 2.06857
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 0.934351i − 0.0409347i −0.999791 0.0204673i \(-0.993485\pi\)
0.999791 0.0204673i \(-0.00651541\pi\)
\(522\) 0 0
\(523\) − 13.3676i − 0.584525i −0.956338 0.292262i \(-0.905592\pi\)
0.956338 0.292262i \(-0.0944081\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.793444 0.0345630
\(528\) 0 0
\(529\) −0.147435 −0.00641022
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 0.885810i − 0.0383687i
\(534\) 0 0
\(535\) 16.9839i 0.734280i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.77104 −0.205503
\(540\) 0 0
\(541\) −45.2077 −1.94363 −0.971816 0.235740i \(-0.924249\pi\)
−0.971816 + 0.235740i \(0.924249\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.25300i 0.310684i
\(546\) 0 0
\(547\) − 6.59374i − 0.281928i −0.990015 0.140964i \(-0.954980\pi\)
0.990015 0.140964i \(-0.0450202\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.27808 −0.0544480
\(552\) 0 0
\(553\) 14.3421 0.609889
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 35.9511i − 1.52330i −0.647991 0.761648i \(-0.724391\pi\)
0.647991 0.761648i \(-0.275609\pi\)
\(558\) 0 0
\(559\) − 0.355011i − 0.0150153i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.93375 0.334368 0.167184 0.985926i \(-0.446533\pi\)
0.167184 + 0.985926i \(0.446533\pi\)
\(564\) 0 0
\(565\) 36.5217 1.53648
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 45.2629i 1.89752i 0.315999 + 0.948759i \(0.397660\pi\)
−0.315999 + 0.948759i \(0.602340\pi\)
\(570\) 0 0
\(571\) 14.7251i 0.616227i 0.951350 + 0.308114i \(0.0996977\pi\)
−0.951350 + 0.308114i \(0.900302\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 64.3772 2.68471
\(576\) 0 0
\(577\) −38.0893 −1.58568 −0.792840 0.609430i \(-0.791398\pi\)
−0.792840 + 0.609430i \(0.791398\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.4067i 0.431742i
\(582\) 0 0
\(583\) 56.7857i 2.35182i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.3877 0.717668 0.358834 0.933401i \(-0.383174\pi\)
0.358834 + 0.933401i \(0.383174\pi\)
\(588\) 0 0
\(589\) −1.47697 −0.0608575
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 0.551386i − 0.0226427i −0.999936 0.0113214i \(-0.996396\pi\)
0.999936 0.0113214i \(-0.00360378\pi\)
\(594\) 0 0
\(595\) 11.2112i 0.459613i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.43049 0.181025 0.0905124 0.995895i \(-0.471150\pi\)
0.0905124 + 0.995895i \(0.471150\pi\)
\(600\) 0 0
\(601\) −29.9890 −1.22328 −0.611638 0.791138i \(-0.709489\pi\)
−0.611638 + 0.791138i \(0.709489\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 50.5486i 2.05509i
\(606\) 0 0
\(607\) − 0.0521455i − 0.00211652i −0.999999 0.00105826i \(-0.999663\pi\)
0.999999 0.00105826i \(-0.000336854\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.824619 −0.0333605
\(612\) 0 0
\(613\) 21.4946 0.868157 0.434078 0.900875i \(-0.357074\pi\)
0.434078 + 0.900875i \(0.357074\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.81413i 0.153551i 0.997048 + 0.0767756i \(0.0244625\pi\)
−0.997048 + 0.0767756i \(0.975537\pi\)
\(618\) 0 0
\(619\) − 38.3553i − 1.54163i −0.637060 0.770814i \(-0.719850\pi\)
0.637060 0.770814i \(-0.280150\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.85368 0.314651
\(624\) 0 0
\(625\) 89.0208 3.56083
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.9614i 0.995275i
\(630\) 0 0
\(631\) − 13.7903i − 0.548982i −0.961590 0.274491i \(-0.911491\pi\)
0.961590 0.274491i \(-0.0885094\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −52.6755 −2.09036
\(636\) 0 0
\(637\) 0.0836472 0.00331422
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 39.7307i 1.56927i 0.619960 + 0.784634i \(0.287149\pi\)
−0.619960 + 0.784634i \(0.712851\pi\)
\(642\) 0 0
\(643\) − 35.6618i − 1.40637i −0.711009 0.703183i \(-0.751761\pi\)
0.711009 0.703183i \(-0.248239\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.5540 −1.00463 −0.502315 0.864685i \(-0.667518\pi\)
−0.502315 + 0.864685i \(0.667518\pi\)
\(648\) 0 0
\(649\) −26.2787 −1.03153
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.13082i 0.122518i 0.998122 + 0.0612591i \(0.0195116\pi\)
−0.998122 + 0.0612591i \(0.980488\pi\)
\(654\) 0 0
\(655\) − 43.9399i − 1.71687i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −17.9580 −0.699543 −0.349771 0.936835i \(-0.613741\pi\)
−0.349771 + 0.936835i \(0.613741\pi\)
\(660\) 0 0
\(661\) −18.3081 −0.712103 −0.356051 0.934466i \(-0.615877\pi\)
−0.356051 + 0.934466i \(0.615877\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 20.8693i − 0.809275i
\(666\) 0 0
\(667\) 1.25810i 0.0487137i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −41.1388 −1.58815
\(672\) 0 0
\(673\) 33.2016 1.27983 0.639914 0.768447i \(-0.278970\pi\)
0.639914 + 0.768447i \(0.278970\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22.3678i 0.859663i 0.902909 + 0.429832i \(0.141427\pi\)
−0.902909 + 0.429832i \(0.858573\pi\)
\(678\) 0 0
\(679\) 1.29824i 0.0498220i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.5984 1.70651 0.853256 0.521493i \(-0.174625\pi\)
0.853256 + 0.521493i \(0.174625\pi\)
\(684\) 0 0
\(685\) 95.6150 3.65326
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 0.995582i − 0.0379286i
\(690\) 0 0
\(691\) 25.9307i 0.986450i 0.869902 + 0.493225i \(0.164182\pi\)
−0.869902 + 0.493225i \(0.835818\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 43.5213 1.65086
\(696\) 0 0
\(697\) −27.6277 −1.04647
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 24.3122i − 0.918258i −0.888370 0.459129i \(-0.848162\pi\)
0.888370 0.459129i \(-0.151838\pi\)
\(702\) 0 0
\(703\) − 46.4648i − 1.75245i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.5103 0.432890
\(708\) 0 0
\(709\) −12.6409 −0.474740 −0.237370 0.971419i \(-0.576285\pi\)
−0.237370 + 0.971419i \(0.576285\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.45388i 0.0544482i
\(714\) 0 0
\(715\) − 1.71499i − 0.0641369i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 44.9044 1.67465 0.837325 0.546705i \(-0.184118\pi\)
0.837325 + 0.546705i \(0.184118\pi\)
\(720\) 0 0
\(721\) 1.07843 0.0401629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.54414i 0.131626i
\(726\) 0 0
\(727\) 19.8025i 0.734434i 0.930135 + 0.367217i \(0.119689\pi\)
−0.930135 + 0.367217i \(0.880311\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11.0725 −0.409531
\(732\) 0 0
\(733\) −28.3746 −1.04804 −0.524019 0.851707i \(-0.675568\pi\)
−0.524019 + 0.851707i \(0.675568\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8.62255i − 0.317616i
\(738\) 0 0
\(739\) − 13.8535i − 0.509608i −0.966993 0.254804i \(-0.917989\pi\)
0.966993 0.254804i \(-0.0820109\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.90143 0.363248 0.181624 0.983368i \(-0.441865\pi\)
0.181624 + 0.983368i \(0.441865\pi\)
\(744\) 0 0
\(745\) 37.9521 1.39046
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 3.95223i − 0.144411i
\(750\) 0 0
\(751\) − 5.39410i − 0.196833i −0.995145 0.0984167i \(-0.968622\pi\)
0.995145 0.0984167i \(-0.0313778\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 17.9853 0.654553
\(756\) 0 0
\(757\) −8.40323 −0.305421 −0.152710 0.988271i \(-0.548800\pi\)
−0.152710 + 0.988271i \(0.548800\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.7139i 1.25838i 0.777252 + 0.629190i \(0.216613\pi\)
−0.777252 + 0.629190i \(0.783387\pi\)
\(762\) 0 0
\(763\) − 1.68780i − 0.0611026i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.460726 0.0166358
\(768\) 0 0
\(769\) 45.7418 1.64949 0.824746 0.565504i \(-0.191318\pi\)
0.824746 + 0.565504i \(0.191318\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.53248i 0.306892i 0.988157 + 0.153446i \(0.0490371\pi\)
−0.988157 + 0.153446i \(0.950963\pi\)
\(774\) 0 0
\(775\) 4.09567i 0.147121i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 51.4281 1.84260
\(780\) 0 0
\(781\) 51.9884 1.86029
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 51.5869i − 1.84122i
\(786\) 0 0
\(787\) − 30.5916i − 1.09047i −0.838282 0.545237i \(-0.816440\pi\)
0.838282 0.545237i \(-0.183560\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.49874 −0.302180
\(792\) 0 0
\(793\) 0.721256 0.0256126
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 6.66452i − 0.236070i −0.993009 0.118035i \(-0.962341\pi\)
0.993009 0.118035i \(-0.0376594\pi\)
\(798\) 0 0
\(799\) 25.7192i 0.909879i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −23.2989 −0.822199
\(804\) 0 0
\(805\) −20.5430 −0.724045
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.5914i 1.14585i 0.819607 + 0.572926i \(0.194192\pi\)
−0.819607 + 0.572926i \(0.805808\pi\)
\(810\) 0 0
\(811\) − 1.99304i − 0.0699851i −0.999388 0.0349926i \(-0.988859\pi\)
0.999388 0.0349926i \(-0.0111408\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −54.5850 −1.91203
\(816\) 0 0
\(817\) 20.6111 0.721091
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.77861i 0.341276i 0.985334 + 0.170638i \(0.0545829\pi\)
−0.985334 + 0.170638i \(0.945417\pi\)
\(822\) 0 0
\(823\) 2.33642i 0.0814425i 0.999171 + 0.0407213i \(0.0129656\pi\)
−0.999171 + 0.0407213i \(0.987034\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15.1072 −0.525328 −0.262664 0.964887i \(-0.584601\pi\)
−0.262664 + 0.964887i \(0.584601\pi\)
\(828\) 0 0
\(829\) 6.56565 0.228034 0.114017 0.993479i \(-0.463628\pi\)
0.114017 + 0.993479i \(0.463628\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.60889i − 0.0903926i
\(834\) 0 0
\(835\) 87.1177i 3.01483i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.9747 1.10389 0.551944 0.833881i \(-0.313886\pi\)
0.551944 + 0.833881i \(0.313886\pi\)
\(840\) 0 0
\(841\) 28.9307 0.997612
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 55.8349i − 1.92078i
\(846\) 0 0
\(847\) − 11.7629i − 0.404177i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −45.7384 −1.56789
\(852\) 0 0
\(853\) 17.2085 0.589209 0.294604 0.955619i \(-0.404812\pi\)
0.294604 + 0.955619i \(0.404812\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 10.2007i 0.348450i 0.984706 + 0.174225i \(0.0557419\pi\)
−0.984706 + 0.174225i \(0.944258\pi\)
\(858\) 0 0
\(859\) 36.5760i 1.24796i 0.781441 + 0.623979i \(0.214485\pi\)
−0.781441 + 0.623979i \(0.785515\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −18.4332 −0.627472 −0.313736 0.949510i \(-0.601581\pi\)
−0.313736 + 0.949510i \(0.601581\pi\)
\(864\) 0 0
\(865\) −88.8482 −3.02093
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 68.4269i 2.32122i
\(870\) 0 0
\(871\) 0.151173i 0.00512229i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −36.3844 −1.23002
\(876\) 0 0
\(877\) 56.2219 1.89848 0.949240 0.314552i \(-0.101854\pi\)
0.949240 + 0.314552i \(0.101854\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.61624i 0.155525i 0.996972 + 0.0777625i \(0.0247776\pi\)
−0.996972 + 0.0777625i \(0.975222\pi\)
\(882\) 0 0
\(883\) − 52.5589i − 1.76875i −0.466778 0.884374i \(-0.654585\pi\)
0.466778 0.884374i \(-0.345415\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.4298 0.417351 0.208675 0.977985i \(-0.433085\pi\)
0.208675 + 0.977985i \(0.433085\pi\)
\(888\) 0 0
\(889\) 12.2578 0.411114
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 47.8754i − 1.60209i
\(894\) 0 0
\(895\) 20.1026i 0.671957i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.0800401 −0.00266949
\(900\) 0 0
\(901\) −31.0514 −1.03447
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 102.850i 3.41884i
\(906\) 0 0
\(907\) − 27.4245i − 0.910617i −0.890334 0.455308i \(-0.849529\pi\)
0.890334 0.455308i \(-0.150471\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −49.0785 −1.62604 −0.813022 0.582233i \(-0.802179\pi\)
−0.813022 + 0.582233i \(0.802179\pi\)
\(912\) 0 0
\(913\) −49.6508 −1.64320
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.2250i 0.337659i
\(918\) 0 0
\(919\) − 5.08992i − 0.167901i −0.996470 0.0839505i \(-0.973246\pi\)
0.996470 0.0839505i \(-0.0267538\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.911474 −0.0300015
\(924\) 0 0
\(925\) −128.848 −4.23650
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 28.5127i − 0.935473i −0.883868 0.467736i \(-0.845070\pi\)
0.883868 0.467736i \(-0.154930\pi\)
\(930\) 0 0
\(931\) 4.85636i 0.159161i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −53.4890 −1.74928
\(936\) 0 0
\(937\) −16.9605 −0.554075 −0.277037 0.960859i \(-0.589353\pi\)
−0.277037 + 0.960859i \(0.589353\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.636605i 0.0207527i 0.999946 + 0.0103764i \(0.00330296\pi\)
−0.999946 + 0.0103764i \(0.996697\pi\)
\(942\) 0 0
\(943\) − 50.6240i − 1.64854i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.81589 0.124000 0.0619999 0.998076i \(-0.480252\pi\)
0.0619999 + 0.998076i \(0.480252\pi\)
\(948\) 0 0
\(949\) 0.408482 0.0132599
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.41783i 0.143108i 0.997437 + 0.0715538i \(0.0227958\pi\)
−0.997437 + 0.0715538i \(0.977204\pi\)
\(954\) 0 0
\(955\) − 103.497i − 3.34908i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −22.2500 −0.718490
\(960\) 0 0
\(961\) 30.9075 0.997016
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 33.0067i − 1.06252i
\(966\) 0 0
\(967\) − 53.4421i − 1.71858i −0.511487 0.859291i \(-0.670905\pi\)
0.511487 0.859291i \(-0.329095\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −52.7188 −1.69183 −0.845913 0.533321i \(-0.820944\pi\)
−0.845913 + 0.533321i \(0.820944\pi\)
\(972\) 0 0
\(973\) −10.1276 −0.324675
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 42.0146i − 1.34417i −0.740476 0.672083i \(-0.765400\pi\)
0.740476 0.672083i \(-0.234600\pi\)
\(978\) 0 0
\(979\) 37.4702i 1.19755i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 34.9736 1.11549 0.557743 0.830014i \(-0.311668\pi\)
0.557743 + 0.830014i \(0.311668\pi\)
\(984\) 0 0
\(985\) 96.7478 3.08264
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 20.2888i − 0.645148i
\(990\) 0 0
\(991\) 42.1138i 1.33779i 0.743357 + 0.668894i \(0.233232\pi\)
−0.743357 + 0.668894i \(0.766768\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −58.9839 −1.86991
\(996\) 0 0
\(997\) 9.13760 0.289391 0.144695 0.989476i \(-0.453780\pi\)
0.144695 + 0.989476i \(0.453780\pi\)
\(998\) 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.2591.2 yes 24
3.2 odd 2 inner 6048.2.h.i.2591.23 yes 24
4.3 odd 2 inner 6048.2.h.i.2591.24 yes 24
12.11 even 2 inner 6048.2.h.i.2591.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.i.2591.1 24 12.11 even 2 inner
6048.2.h.i.2591.2 yes 24 1.1 even 1 trivial
6048.2.h.i.2591.23 yes 24 3.2 odd 2 inner
6048.2.h.i.2591.24 yes 24 4.3 odd 2 inner