Properties

Label 3072.2.a.n.1.1
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 48)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.27133\) of defining polynomial
Character \(\chi\) \(=\) 3072.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.79793 q^{5} +0.158942 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.79793 q^{5} +0.158942 q^{7} +1.00000 q^{9} +5.37109 q^{11} +5.95687 q^{13} +1.79793 q^{15} -3.05320 q^{17} -3.05320 q^{19} -0.158942 q^{21} +2.82843 q^{23} -1.76744 q^{25} -1.00000 q^{27} +2.96951 q^{29} -4.15894 q^{31} -5.37109 q^{33} -0.285766 q^{35} +8.46742 q^{37} -5.95687 q^{39} +2.60365 q^{41} -8.13853 q^{43} -1.79793 q^{45} -2.82843 q^{47} -6.97474 q^{49} +3.05320 q^{51} +5.03049 q^{53} -9.65685 q^{55} +3.05320 q^{57} +5.65685 q^{59} +5.18944 q^{61} +0.158942 q^{63} -10.7101 q^{65} -1.08532 q^{67} -2.82843 q^{69} -0.317883 q^{71} +1.33897 q^{73} +1.76744 q^{75} +0.853690 q^{77} -9.69382 q^{79} +1.00000 q^{81} +0.163788 q^{83} +5.48946 q^{85} -2.96951 q^{87} +14.3990 q^{89} +0.946795 q^{91} +4.15894 q^{93} +5.48946 q^{95} -0.571533 q^{97} +5.37109 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} + 4q^{5} - 4q^{7} + 4q^{9} + 8q^{13} - 4q^{15} + 4q^{21} + 4q^{25} - 4q^{27} + 12q^{29} - 12q^{31} + 16q^{37} - 8q^{39} + 4q^{45} + 4q^{49} + 20q^{53} - 16q^{55} + 16q^{61} - 4q^{63} - 8q^{65} + 16q^{67} + 8q^{71} - 8q^{73} - 4q^{75} + 24q^{77} - 12q^{79} + 4q^{81} + 24q^{85} - 12q^{87} - 8q^{89} + 16q^{91} + 12q^{93} + 24q^{95} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.79793 −0.804060 −0.402030 0.915627i \(-0.631695\pi\)
−0.402030 + 0.915627i \(0.631695\pi\)
\(6\) 0 0
\(7\) 0.158942 0.0600743 0.0300371 0.999549i \(-0.490437\pi\)
0.0300371 + 0.999549i \(0.490437\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.37109 1.61944 0.809722 0.586814i \(-0.199618\pi\)
0.809722 + 0.586814i \(0.199618\pi\)
\(12\) 0 0
\(13\) 5.95687 1.65214 0.826070 0.563568i \(-0.190572\pi\)
0.826070 + 0.563568i \(0.190572\pi\)
\(14\) 0 0
\(15\) 1.79793 0.464224
\(16\) 0 0
\(17\) −3.05320 −0.740511 −0.370255 0.928930i \(-0.620730\pi\)
−0.370255 + 0.928930i \(0.620730\pi\)
\(18\) 0 0
\(19\) −3.05320 −0.700453 −0.350227 0.936665i \(-0.613895\pi\)
−0.350227 + 0.936665i \(0.613895\pi\)
\(20\) 0 0
\(21\) −0.158942 −0.0346839
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −1.76744 −0.353488
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.96951 0.551423 0.275712 0.961240i \(-0.411087\pi\)
0.275712 + 0.961240i \(0.411087\pi\)
\(30\) 0 0
\(31\) −4.15894 −0.746968 −0.373484 0.927637i \(-0.621837\pi\)
−0.373484 + 0.927637i \(0.621837\pi\)
\(32\) 0 0
\(33\) −5.37109 −0.934986
\(34\) 0 0
\(35\) −0.285766 −0.0483033
\(36\) 0 0
\(37\) 8.46742 1.39203 0.696017 0.718025i \(-0.254954\pi\)
0.696017 + 0.718025i \(0.254954\pi\)
\(38\) 0 0
\(39\) −5.95687 −0.953863
\(40\) 0 0
\(41\) 2.60365 0.406622 0.203311 0.979114i \(-0.434830\pi\)
0.203311 + 0.979114i \(0.434830\pi\)
\(42\) 0 0
\(43\) −8.13853 −1.24111 −0.620557 0.784162i \(-0.713093\pi\)
−0.620557 + 0.784162i \(0.713093\pi\)
\(44\) 0 0
\(45\) −1.79793 −0.268020
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −6.97474 −0.996391
\(50\) 0 0
\(51\) 3.05320 0.427534
\(52\) 0 0
\(53\) 5.03049 0.690992 0.345496 0.938420i \(-0.387711\pi\)
0.345496 + 0.938420i \(0.387711\pi\)
\(54\) 0 0
\(55\) −9.65685 −1.30213
\(56\) 0 0
\(57\) 3.05320 0.404407
\(58\) 0 0
\(59\) 5.65685 0.736460 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(60\) 0 0
\(61\) 5.18944 0.664439 0.332220 0.943202i \(-0.392202\pi\)
0.332220 + 0.943202i \(0.392202\pi\)
\(62\) 0 0
\(63\) 0.158942 0.0200248
\(64\) 0 0
\(65\) −10.7101 −1.32842
\(66\) 0 0
\(67\) −1.08532 −0.132593 −0.0662966 0.997800i \(-0.521118\pi\)
−0.0662966 + 0.997800i \(0.521118\pi\)
\(68\) 0 0
\(69\) −2.82843 −0.340503
\(70\) 0 0
\(71\) −0.317883 −0.0377258 −0.0188629 0.999822i \(-0.506005\pi\)
−0.0188629 + 0.999822i \(0.506005\pi\)
\(72\) 0 0
\(73\) 1.33897 0.156715 0.0783573 0.996925i \(-0.475032\pi\)
0.0783573 + 0.996925i \(0.475032\pi\)
\(74\) 0 0
\(75\) 1.76744 0.204086
\(76\) 0 0
\(77\) 0.853690 0.0972870
\(78\) 0 0
\(79\) −9.69382 −1.09064 −0.545320 0.838228i \(-0.683592\pi\)
−0.545320 + 0.838228i \(0.683592\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.163788 0.0179781 0.00898906 0.999960i \(-0.497139\pi\)
0.00898906 + 0.999960i \(0.497139\pi\)
\(84\) 0 0
\(85\) 5.48946 0.595415
\(86\) 0 0
\(87\) −2.96951 −0.318364
\(88\) 0 0
\(89\) 14.3990 1.52629 0.763147 0.646225i \(-0.223653\pi\)
0.763147 + 0.646225i \(0.223653\pi\)
\(90\) 0 0
\(91\) 0.946795 0.0992511
\(92\) 0 0
\(93\) 4.15894 0.431262
\(94\) 0 0
\(95\) 5.48946 0.563206
\(96\) 0 0
\(97\) −0.571533 −0.0580304 −0.0290152 0.999579i \(-0.509237\pi\)
−0.0290152 + 0.999579i \(0.509237\pi\)
\(98\) 0 0
\(99\) 5.37109 0.539815
\(100\) 0 0
\(101\) 10.1158 1.00656 0.503281 0.864123i \(-0.332126\pi\)
0.503281 + 0.864123i \(0.332126\pi\)
\(102\) 0 0
\(103\) 11.3507 1.11841 0.559207 0.829028i \(-0.311106\pi\)
0.559207 + 0.829028i \(0.311106\pi\)
\(104\) 0 0
\(105\) 0.285766 0.0278879
\(106\) 0 0
\(107\) 1.02109 0.0987123 0.0493561 0.998781i \(-0.484283\pi\)
0.0493561 + 0.998781i \(0.484283\pi\)
\(108\) 0 0
\(109\) 2.04313 0.195696 0.0978480 0.995201i \(-0.468804\pi\)
0.0978480 + 0.995201i \(0.468804\pi\)
\(110\) 0 0
\(111\) −8.46742 −0.803692
\(112\) 0 0
\(113\) 3.53488 0.332533 0.166267 0.986081i \(-0.446829\pi\)
0.166267 + 0.986081i \(0.446829\pi\)
\(114\) 0 0
\(115\) −5.08532 −0.474209
\(116\) 0 0
\(117\) 5.95687 0.550713
\(118\) 0 0
\(119\) −0.485281 −0.0444857
\(120\) 0 0
\(121\) 17.8486 1.62260
\(122\) 0 0
\(123\) −2.60365 −0.234763
\(124\) 0 0
\(125\) 12.1674 1.08829
\(126\) 0 0
\(127\) 1.49791 0.132918 0.0664591 0.997789i \(-0.478830\pi\)
0.0664591 + 0.997789i \(0.478830\pi\)
\(128\) 0 0
\(129\) 8.13853 0.716557
\(130\) 0 0
\(131\) −14.7422 −1.28803 −0.644015 0.765013i \(-0.722733\pi\)
−0.644015 + 0.765013i \(0.722733\pi\)
\(132\) 0 0
\(133\) −0.485281 −0.0420792
\(134\) 0 0
\(135\) 1.79793 0.154741
\(136\) 0 0
\(137\) 13.7954 1.17862 0.589309 0.807907i \(-0.299400\pi\)
0.589309 + 0.807907i \(0.299400\pi\)
\(138\) 0 0
\(139\) 3.42847 0.290799 0.145399 0.989373i \(-0.453553\pi\)
0.145399 + 0.989373i \(0.453553\pi\)
\(140\) 0 0
\(141\) 2.82843 0.238197
\(142\) 0 0
\(143\) 31.9949 2.67555
\(144\) 0 0
\(145\) −5.33897 −0.443377
\(146\) 0 0
\(147\) 6.97474 0.575267
\(148\) 0 0
\(149\) 4.14108 0.339250 0.169625 0.985509i \(-0.445744\pi\)
0.169625 + 0.985509i \(0.445744\pi\)
\(150\) 0 0
\(151\) 22.6644 1.84440 0.922201 0.386712i \(-0.126389\pi\)
0.922201 + 0.386712i \(0.126389\pi\)
\(152\) 0 0
\(153\) −3.05320 −0.246837
\(154\) 0 0
\(155\) 7.47750 0.600607
\(156\) 0 0
\(157\) −3.93161 −0.313777 −0.156888 0.987616i \(-0.550146\pi\)
−0.156888 + 0.987616i \(0.550146\pi\)
\(158\) 0 0
\(159\) −5.03049 −0.398944
\(160\) 0 0
\(161\) 0.449555 0.0354299
\(162\) 0 0
\(163\) 7.68897 0.602247 0.301123 0.953585i \(-0.402638\pi\)
0.301123 + 0.953585i \(0.402638\pi\)
\(164\) 0 0
\(165\) 9.65685 0.751785
\(166\) 0 0
\(167\) 3.95458 0.306015 0.153007 0.988225i \(-0.451104\pi\)
0.153007 + 0.988225i \(0.451104\pi\)
\(168\) 0 0
\(169\) 22.4844 1.72957
\(170\) 0 0
\(171\) −3.05320 −0.233484
\(172\) 0 0
\(173\) 22.6011 1.71833 0.859165 0.511699i \(-0.170984\pi\)
0.859165 + 0.511699i \(0.170984\pi\)
\(174\) 0 0
\(175\) −0.280920 −0.0212355
\(176\) 0 0
\(177\) −5.65685 −0.425195
\(178\) 0 0
\(179\) −17.2981 −1.29292 −0.646462 0.762946i \(-0.723752\pi\)
−0.646462 + 0.762946i \(0.723752\pi\)
\(180\) 0 0
\(181\) 8.14953 0.605750 0.302875 0.953030i \(-0.402054\pi\)
0.302875 + 0.953030i \(0.402054\pi\)
\(182\) 0 0
\(183\) −5.18944 −0.383614
\(184\) 0 0
\(185\) −15.2238 −1.11928
\(186\) 0 0
\(187\) −16.3990 −1.19922
\(188\) 0 0
\(189\) −0.158942 −0.0115613
\(190\) 0 0
\(191\) 16.1674 1.16983 0.584916 0.811094i \(-0.301127\pi\)
0.584916 + 0.811094i \(0.301127\pi\)
\(192\) 0 0
\(193\) −22.1454 −1.59406 −0.797030 0.603940i \(-0.793597\pi\)
−0.797030 + 0.603940i \(0.793597\pi\)
\(194\) 0 0
\(195\) 10.7101 0.766963
\(196\) 0 0
\(197\) 20.2222 1.44077 0.720387 0.693572i \(-0.243964\pi\)
0.720387 + 0.693572i \(0.243964\pi\)
\(198\) 0 0
\(199\) −25.0075 −1.77274 −0.886368 0.462981i \(-0.846780\pi\)
−0.886368 + 0.462981i \(0.846780\pi\)
\(200\) 0 0
\(201\) 1.08532 0.0765527
\(202\) 0 0
\(203\) 0.471978 0.0331264
\(204\) 0 0
\(205\) −4.68119 −0.326948
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) −16.3990 −1.13434
\(210\) 0 0
\(211\) 26.0559 1.79376 0.896881 0.442273i \(-0.145828\pi\)
0.896881 + 0.442273i \(0.145828\pi\)
\(212\) 0 0
\(213\) 0.317883 0.0217810
\(214\) 0 0
\(215\) 14.6325 0.997930
\(216\) 0 0
\(217\) −0.661029 −0.0448736
\(218\) 0 0
\(219\) −1.33897 −0.0904793
\(220\) 0 0
\(221\) −18.1876 −1.22343
\(222\) 0 0
\(223\) −18.3465 −1.22857 −0.614286 0.789083i \(-0.710556\pi\)
−0.614286 + 0.789083i \(0.710556\pi\)
\(224\) 0 0
\(225\) −1.76744 −0.117829
\(226\) 0 0
\(227\) 0.163788 0.0108710 0.00543551 0.999985i \(-0.498270\pi\)
0.00543551 + 0.999985i \(0.498270\pi\)
\(228\) 0 0
\(229\) −4.02756 −0.266148 −0.133074 0.991106i \(-0.542485\pi\)
−0.133074 + 0.991106i \(0.542485\pi\)
\(230\) 0 0
\(231\) −0.853690 −0.0561687
\(232\) 0 0
\(233\) −11.7211 −0.767874 −0.383937 0.923359i \(-0.625432\pi\)
−0.383937 + 0.923359i \(0.625432\pi\)
\(234\) 0 0
\(235\) 5.08532 0.331730
\(236\) 0 0
\(237\) 9.69382 0.629681
\(238\) 0 0
\(239\) −13.6517 −0.883058 −0.441529 0.897247i \(-0.645564\pi\)
−0.441529 + 0.897247i \(0.645564\pi\)
\(240\) 0 0
\(241\) −2.13167 −0.137313 −0.0686565 0.997640i \(-0.521871\pi\)
−0.0686565 + 0.997640i \(0.521871\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 12.5401 0.801158
\(246\) 0 0
\(247\) −18.1876 −1.15725
\(248\) 0 0
\(249\) −0.163788 −0.0103797
\(250\) 0 0
\(251\) −6.27020 −0.395771 −0.197886 0.980225i \(-0.563407\pi\)
−0.197886 + 0.980225i \(0.563407\pi\)
\(252\) 0 0
\(253\) 15.1917 0.955096
\(254\) 0 0
\(255\) −5.48946 −0.343763
\(256\) 0 0
\(257\) 15.0853 0.940997 0.470498 0.882401i \(-0.344074\pi\)
0.470498 + 0.882401i \(0.344074\pi\)
\(258\) 0 0
\(259\) 1.34583 0.0836255
\(260\) 0 0
\(261\) 2.96951 0.183808
\(262\) 0 0
\(263\) 26.1706 1.61375 0.806875 0.590722i \(-0.201157\pi\)
0.806875 + 0.590722i \(0.201157\pi\)
\(264\) 0 0
\(265\) −9.04449 −0.555599
\(266\) 0 0
\(267\) −14.3990 −0.881206
\(268\) 0 0
\(269\) −12.1580 −0.741286 −0.370643 0.928775i \(-0.620863\pi\)
−0.370643 + 0.928775i \(0.620863\pi\)
\(270\) 0 0
\(271\) 10.6644 0.647815 0.323907 0.946089i \(-0.395003\pi\)
0.323907 + 0.946089i \(0.395003\pi\)
\(272\) 0 0
\(273\) −0.946795 −0.0573027
\(274\) 0 0
\(275\) −9.49307 −0.572453
\(276\) 0 0
\(277\) −3.76421 −0.226170 −0.113085 0.993585i \(-0.536073\pi\)
−0.113085 + 0.993585i \(0.536073\pi\)
\(278\) 0 0
\(279\) −4.15894 −0.248989
\(280\) 0 0
\(281\) 10.4496 0.623368 0.311684 0.950186i \(-0.399107\pi\)
0.311684 + 0.950186i \(0.399107\pi\)
\(282\) 0 0
\(283\) −17.6569 −1.04959 −0.524796 0.851228i \(-0.675858\pi\)
−0.524796 + 0.851228i \(0.675858\pi\)
\(284\) 0 0
\(285\) −5.48946 −0.325167
\(286\) 0 0
\(287\) 0.413828 0.0244275
\(288\) 0 0
\(289\) −7.67794 −0.451644
\(290\) 0 0
\(291\) 0.571533 0.0335038
\(292\) 0 0
\(293\) −30.7465 −1.79623 −0.898114 0.439762i \(-0.855063\pi\)
−0.898114 + 0.439762i \(0.855063\pi\)
\(294\) 0 0
\(295\) −10.1706 −0.592158
\(296\) 0 0
\(297\) −5.37109 −0.311662
\(298\) 0 0
\(299\) 16.8486 0.974379
\(300\) 0 0
\(301\) −1.29355 −0.0745590
\(302\) 0 0
\(303\) −10.1158 −0.581138
\(304\) 0 0
\(305\) −9.33026 −0.534249
\(306\) 0 0
\(307\) 21.2981 1.21555 0.607775 0.794110i \(-0.292062\pi\)
0.607775 + 0.794110i \(0.292062\pi\)
\(308\) 0 0
\(309\) −11.3507 −0.645717
\(310\) 0 0
\(311\) −1.77883 −0.100868 −0.0504342 0.998727i \(-0.516061\pi\)
−0.0504342 + 0.998727i \(0.516061\pi\)
\(312\) 0 0
\(313\) −2.70320 −0.152794 −0.0763971 0.997077i \(-0.524342\pi\)
−0.0763971 + 0.997077i \(0.524342\pi\)
\(314\) 0 0
\(315\) −0.285766 −0.0161011
\(316\) 0 0
\(317\) 22.0653 1.23931 0.619655 0.784874i \(-0.287272\pi\)
0.619655 + 0.784874i \(0.287272\pi\)
\(318\) 0 0
\(319\) 15.9495 0.892999
\(320\) 0 0
\(321\) −1.02109 −0.0569916
\(322\) 0 0
\(323\) 9.32206 0.518693
\(324\) 0 0
\(325\) −10.5284 −0.584011
\(326\) 0 0
\(327\) −2.04313 −0.112985
\(328\) 0 0
\(329\) −0.449555 −0.0247848
\(330\) 0 0
\(331\) −21.8431 −1.20060 −0.600302 0.799774i \(-0.704953\pi\)
−0.600302 + 0.799774i \(0.704953\pi\)
\(332\) 0 0
\(333\) 8.46742 0.464012
\(334\) 0 0
\(335\) 1.95133 0.106613
\(336\) 0 0
\(337\) 18.8738 1.02812 0.514062 0.857753i \(-0.328140\pi\)
0.514062 + 0.857753i \(0.328140\pi\)
\(338\) 0 0
\(339\) −3.53488 −0.191988
\(340\) 0 0
\(341\) −22.3380 −1.20967
\(342\) 0 0
\(343\) −2.22117 −0.119932
\(344\) 0 0
\(345\) 5.08532 0.273785
\(346\) 0 0
\(347\) 28.0490 1.50575 0.752875 0.658163i \(-0.228666\pi\)
0.752875 + 0.658163i \(0.228666\pi\)
\(348\) 0 0
\(349\) 16.9307 0.906279 0.453139 0.891440i \(-0.350304\pi\)
0.453139 + 0.891440i \(0.350304\pi\)
\(350\) 0 0
\(351\) −5.95687 −0.317954
\(352\) 0 0
\(353\) −12.6202 −0.671705 −0.335853 0.941915i \(-0.609024\pi\)
−0.335853 + 0.941915i \(0.609024\pi\)
\(354\) 0 0
\(355\) 0.571533 0.0303338
\(356\) 0 0
\(357\) 0.485281 0.0256838
\(358\) 0 0
\(359\) 27.0867 1.42958 0.714790 0.699339i \(-0.246522\pi\)
0.714790 + 0.699339i \(0.246522\pi\)
\(360\) 0 0
\(361\) −9.67794 −0.509365
\(362\) 0 0
\(363\) −17.8486 −0.936808
\(364\) 0 0
\(365\) −2.40738 −0.126008
\(366\) 0 0
\(367\) −20.4937 −1.06976 −0.534882 0.844927i \(-0.679644\pi\)
−0.534882 + 0.844927i \(0.679644\pi\)
\(368\) 0 0
\(369\) 2.60365 0.135541
\(370\) 0 0
\(371\) 0.799555 0.0415108
\(372\) 0 0
\(373\) 1.46190 0.0756943 0.0378471 0.999284i \(-0.487950\pi\)
0.0378471 + 0.999284i \(0.487950\pi\)
\(374\) 0 0
\(375\) −12.1674 −0.628322
\(376\) 0 0
\(377\) 17.6890 0.911028
\(378\) 0 0
\(379\) 24.9871 1.28350 0.641751 0.766913i \(-0.278208\pi\)
0.641751 + 0.766913i \(0.278208\pi\)
\(380\) 0 0
\(381\) −1.49791 −0.0767404
\(382\) 0 0
\(383\) 31.0958 1.58892 0.794460 0.607316i \(-0.207754\pi\)
0.794460 + 0.607316i \(0.207754\pi\)
\(384\) 0 0
\(385\) −1.53488 −0.0782245
\(386\) 0 0
\(387\) −8.13853 −0.413705
\(388\) 0 0
\(389\) 3.62218 0.183652 0.0918260 0.995775i \(-0.470730\pi\)
0.0918260 + 0.995775i \(0.470730\pi\)
\(390\) 0 0
\(391\) −8.63577 −0.436729
\(392\) 0 0
\(393\) 14.7422 0.743644
\(394\) 0 0
\(395\) 17.4288 0.876940
\(396\) 0 0
\(397\) 7.20959 0.361839 0.180920 0.983498i \(-0.442093\pi\)
0.180920 + 0.983498i \(0.442093\pi\)
\(398\) 0 0
\(399\) 0.485281 0.0242945
\(400\) 0 0
\(401\) 15.2660 0.762349 0.381174 0.924503i \(-0.375520\pi\)
0.381174 + 0.924503i \(0.375520\pi\)
\(402\) 0 0
\(403\) −24.7743 −1.23410
\(404\) 0 0
\(405\) −1.79793 −0.0893400
\(406\) 0 0
\(407\) 45.4792 2.25432
\(408\) 0 0
\(409\) −11.3779 −0.562603 −0.281302 0.959619i \(-0.590766\pi\)
−0.281302 + 0.959619i \(0.590766\pi\)
\(410\) 0 0
\(411\) −13.7954 −0.680476
\(412\) 0 0
\(413\) 0.899110 0.0442423
\(414\) 0 0
\(415\) −0.294481 −0.0144555
\(416\) 0 0
\(417\) −3.42847 −0.167893
\(418\) 0 0
\(419\) −32.9618 −1.61029 −0.805146 0.593077i \(-0.797913\pi\)
−0.805146 + 0.593077i \(0.797913\pi\)
\(420\) 0 0
\(421\) 24.9119 1.21413 0.607065 0.794652i \(-0.292347\pi\)
0.607065 + 0.794652i \(0.292347\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) 5.39635 0.261761
\(426\) 0 0
\(427\) 0.824818 0.0399157
\(428\) 0 0
\(429\) −31.9949 −1.54473
\(430\) 0 0
\(431\) 10.3211 0.497151 0.248576 0.968612i \(-0.420038\pi\)
0.248576 + 0.968612i \(0.420038\pi\)
\(432\) 0 0
\(433\) 15.3137 0.735930 0.367965 0.929840i \(-0.380055\pi\)
0.367965 + 0.929840i \(0.380055\pi\)
\(434\) 0 0
\(435\) 5.33897 0.255984
\(436\) 0 0
\(437\) −8.63577 −0.413105
\(438\) 0 0
\(439\) −22.5735 −1.07738 −0.538688 0.842505i \(-0.681080\pi\)
−0.538688 + 0.842505i \(0.681080\pi\)
\(440\) 0 0
\(441\) −6.97474 −0.332130
\(442\) 0 0
\(443\) 33.5334 1.59322 0.796610 0.604494i \(-0.206625\pi\)
0.796610 + 0.604494i \(0.206625\pi\)
\(444\) 0 0
\(445\) −25.8885 −1.22723
\(446\) 0 0
\(447\) −4.14108 −0.195866
\(448\) 0 0
\(449\) −1.75506 −0.0828266 −0.0414133 0.999142i \(-0.513186\pi\)
−0.0414133 + 0.999142i \(0.513186\pi\)
\(450\) 0 0
\(451\) 13.9844 0.658501
\(452\) 0 0
\(453\) −22.6644 −1.06487
\(454\) 0 0
\(455\) −1.70227 −0.0798039
\(456\) 0 0
\(457\) −26.7422 −1.25095 −0.625473 0.780246i \(-0.715094\pi\)
−0.625473 + 0.780246i \(0.715094\pi\)
\(458\) 0 0
\(459\) 3.05320 0.142511
\(460\) 0 0
\(461\) −13.0662 −0.608555 −0.304277 0.952584i \(-0.598415\pi\)
−0.304277 + 0.952584i \(0.598415\pi\)
\(462\) 0 0
\(463\) 29.4474 1.36854 0.684268 0.729231i \(-0.260122\pi\)
0.684268 + 0.729231i \(0.260122\pi\)
\(464\) 0 0
\(465\) −7.47750 −0.346761
\(466\) 0 0
\(467\) −27.7040 −1.28199 −0.640995 0.767545i \(-0.721478\pi\)
−0.640995 + 0.767545i \(0.721478\pi\)
\(468\) 0 0
\(469\) −0.172503 −0.00796544
\(470\) 0 0
\(471\) 3.93161 0.181159
\(472\) 0 0
\(473\) −43.7127 −2.00991
\(474\) 0 0
\(475\) 5.39635 0.247602
\(476\) 0 0
\(477\) 5.03049 0.230331
\(478\) 0 0
\(479\) −35.5499 −1.62432 −0.812159 0.583436i \(-0.801708\pi\)
−0.812159 + 0.583436i \(0.801708\pi\)
\(480\) 0 0
\(481\) 50.4393 2.29984
\(482\) 0 0
\(483\) −0.449555 −0.0204555
\(484\) 0 0
\(485\) 1.02758 0.0466599
\(486\) 0 0
\(487\) −9.86632 −0.447086 −0.223543 0.974694i \(-0.571762\pi\)
−0.223543 + 0.974694i \(0.571762\pi\)
\(488\) 0 0
\(489\) −7.68897 −0.347707
\(490\) 0 0
\(491\) 0.635767 0.0286917 0.0143459 0.999897i \(-0.495433\pi\)
0.0143459 + 0.999897i \(0.495433\pi\)
\(492\) 0 0
\(493\) −9.06651 −0.408335
\(494\) 0 0
\(495\) −9.65685 −0.434043
\(496\) 0 0
\(497\) −0.0505249 −0.00226635
\(498\) 0 0
\(499\) 3.82750 0.171342 0.0856712 0.996323i \(-0.472697\pi\)
0.0856712 + 0.996323i \(0.472697\pi\)
\(500\) 0 0
\(501\) −3.95458 −0.176678
\(502\) 0 0
\(503\) −23.6719 −1.05548 −0.527739 0.849407i \(-0.676960\pi\)
−0.527739 + 0.849407i \(0.676960\pi\)
\(504\) 0 0
\(505\) −18.1876 −0.809336
\(506\) 0 0
\(507\) −22.4844 −0.998565
\(508\) 0 0
\(509\) −34.7970 −1.54235 −0.771175 0.636623i \(-0.780331\pi\)
−0.771175 + 0.636623i \(0.780331\pi\)
\(510\) 0 0
\(511\) 0.212818 0.00941453
\(512\) 0 0
\(513\) 3.05320 0.134802
\(514\) 0 0
\(515\) −20.4077 −0.899273
\(516\) 0 0
\(517\) −15.1917 −0.668132
\(518\) 0 0
\(519\) −22.6011 −0.992078
\(520\) 0 0
\(521\) −14.4889 −0.634770 −0.317385 0.948297i \(-0.602805\pi\)
−0.317385 + 0.948297i \(0.602805\pi\)
\(522\) 0 0
\(523\) 27.5742 1.20574 0.602868 0.797841i \(-0.294025\pi\)
0.602868 + 0.797841i \(0.294025\pi\)
\(524\) 0 0
\(525\) 0.280920 0.0122603
\(526\) 0 0
\(527\) 12.6981 0.553138
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 5.65685 0.245487
\(532\) 0 0
\(533\) 15.5096 0.671796
\(534\) 0 0
\(535\) −1.83585 −0.0793706
\(536\) 0 0
\(537\) 17.2981 0.746470
\(538\) 0 0
\(539\) −37.4619 −1.61360
\(540\) 0 0
\(541\) −14.1982 −0.610428 −0.305214 0.952284i \(-0.598728\pi\)
−0.305214 + 0.952284i \(0.598728\pi\)
\(542\) 0 0
\(543\) −8.14953 −0.349730
\(544\) 0 0
\(545\) −3.67340 −0.157351
\(546\) 0 0
\(547\) 10.1807 0.435295 0.217648 0.976027i \(-0.430162\pi\)
0.217648 + 0.976027i \(0.430162\pi\)
\(548\) 0 0
\(549\) 5.18944 0.221480
\(550\) 0 0
\(551\) −9.06651 −0.386246
\(552\) 0 0
\(553\) −1.54075 −0.0655194
\(554\) 0 0
\(555\) 15.2238 0.646216
\(556\) 0 0
\(557\) −1.44432 −0.0611979 −0.0305990 0.999532i \(-0.509741\pi\)
−0.0305990 + 0.999532i \(0.509741\pi\)
\(558\) 0 0
\(559\) −48.4802 −2.05049
\(560\) 0 0
\(561\) 16.3990 0.692368
\(562\) 0 0
\(563\) 9.48585 0.399781 0.199890 0.979818i \(-0.435941\pi\)
0.199890 + 0.979818i \(0.435941\pi\)
\(564\) 0 0
\(565\) −6.35547 −0.267377
\(566\) 0 0
\(567\) 0.158942 0.00667492
\(568\) 0 0
\(569\) 8.98711 0.376759 0.188380 0.982096i \(-0.439676\pi\)
0.188380 + 0.982096i \(0.439676\pi\)
\(570\) 0 0
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 0 0
\(573\) −16.1674 −0.675403
\(574\) 0 0
\(575\) −4.99907 −0.208476
\(576\) 0 0
\(577\) 29.5013 1.22815 0.614077 0.789246i \(-0.289528\pi\)
0.614077 + 0.789246i \(0.289528\pi\)
\(578\) 0 0
\(579\) 22.1454 0.920331
\(580\) 0 0
\(581\) 0.0260328 0.00108002
\(582\) 0 0
\(583\) 27.0192 1.11902
\(584\) 0 0
\(585\) −10.7101 −0.442806
\(586\) 0 0
\(587\) −2.57988 −0.106483 −0.0532416 0.998582i \(-0.516955\pi\)
−0.0532416 + 0.998582i \(0.516955\pi\)
\(588\) 0 0
\(589\) 12.6981 0.523216
\(590\) 0 0
\(591\) −20.2222 −0.831831
\(592\) 0 0
\(593\) 35.4338 1.45509 0.727546 0.686058i \(-0.240661\pi\)
0.727546 + 0.686058i \(0.240661\pi\)
\(594\) 0 0
\(595\) 0.872503 0.0357691
\(596\) 0 0
\(597\) 25.0075 1.02349
\(598\) 0 0
\(599\) 27.1632 1.10986 0.554930 0.831897i \(-0.312745\pi\)
0.554930 + 0.831897i \(0.312745\pi\)
\(600\) 0 0
\(601\) −5.33897 −0.217781 −0.108891 0.994054i \(-0.534730\pi\)
−0.108891 + 0.994054i \(0.534730\pi\)
\(602\) 0 0
\(603\) −1.08532 −0.0441977
\(604\) 0 0
\(605\) −32.0906 −1.30467
\(606\) 0 0
\(607\) −16.1084 −0.653820 −0.326910 0.945055i \(-0.606007\pi\)
−0.326910 + 0.945055i \(0.606007\pi\)
\(608\) 0 0
\(609\) −0.471978 −0.0191255
\(610\) 0 0
\(611\) −16.8486 −0.681621
\(612\) 0 0
\(613\) −0.617903 −0.0249569 −0.0124784 0.999922i \(-0.503972\pi\)
−0.0124784 + 0.999922i \(0.503972\pi\)
\(614\) 0 0
\(615\) 4.68119 0.188764
\(616\) 0 0
\(617\) 8.80641 0.354533 0.177266 0.984163i \(-0.443275\pi\)
0.177266 + 0.984163i \(0.443275\pi\)
\(618\) 0 0
\(619\) −2.72847 −0.109666 −0.0548332 0.998496i \(-0.517463\pi\)
−0.0548332 + 0.998496i \(0.517463\pi\)
\(620\) 0 0
\(621\) −2.82843 −0.113501
\(622\) 0 0
\(623\) 2.28861 0.0916910
\(624\) 0 0
\(625\) −13.0390 −0.521559
\(626\) 0 0
\(627\) 16.3990 0.654914
\(628\) 0 0
\(629\) −25.8528 −1.03082
\(630\) 0 0
\(631\) −38.7864 −1.54406 −0.772030 0.635586i \(-0.780759\pi\)
−0.772030 + 0.635586i \(0.780759\pi\)
\(632\) 0 0
\(633\) −26.0559 −1.03563
\(634\) 0 0
\(635\) −2.69315 −0.106874
\(636\) 0 0
\(637\) −41.5476 −1.64618
\(638\) 0 0
\(639\) −0.317883 −0.0125753
\(640\) 0 0
\(641\) 33.1091 1.30773 0.653865 0.756611i \(-0.273146\pi\)
0.653865 + 0.756611i \(0.273146\pi\)
\(642\) 0 0
\(643\) 27.2797 1.07581 0.537904 0.843006i \(-0.319216\pi\)
0.537904 + 0.843006i \(0.319216\pi\)
\(644\) 0 0
\(645\) −14.6325 −0.576155
\(646\) 0 0
\(647\) −41.8477 −1.64520 −0.822601 0.568620i \(-0.807478\pi\)
−0.822601 + 0.568620i \(0.807478\pi\)
\(648\) 0 0
\(649\) 30.3835 1.19266
\(650\) 0 0
\(651\) 0.661029 0.0259078
\(652\) 0 0
\(653\) −20.8937 −0.817634 −0.408817 0.912616i \(-0.634059\pi\)
−0.408817 + 0.912616i \(0.634059\pi\)
\(654\) 0 0
\(655\) 26.5054 1.03565
\(656\) 0 0
\(657\) 1.33897 0.0522382
\(658\) 0 0
\(659\) −3.15142 −0.122762 −0.0613809 0.998114i \(-0.519550\pi\)
−0.0613809 + 0.998114i \(0.519550\pi\)
\(660\) 0 0
\(661\) −25.5527 −0.993886 −0.496943 0.867783i \(-0.665544\pi\)
−0.496943 + 0.867783i \(0.665544\pi\)
\(662\) 0 0
\(663\) 18.1876 0.706346
\(664\) 0 0
\(665\) 0.872503 0.0338342
\(666\) 0 0
\(667\) 8.39903 0.325212
\(668\) 0 0
\(669\) 18.3465 0.709317
\(670\) 0 0
\(671\) 27.8729 1.07602
\(672\) 0 0
\(673\) 20.7981 0.801706 0.400853 0.916142i \(-0.368714\pi\)
0.400853 + 0.916142i \(0.368714\pi\)
\(674\) 0 0
\(675\) 1.76744 0.0680287
\(676\) 0 0
\(677\) −41.0423 −1.57738 −0.788692 0.614789i \(-0.789241\pi\)
−0.788692 + 0.614789i \(0.789241\pi\)
\(678\) 0 0
\(679\) −0.0908404 −0.00348613
\(680\) 0 0
\(681\) −0.163788 −0.00627639
\(682\) 0 0
\(683\) 26.5784 1.01699 0.508497 0.861064i \(-0.330201\pi\)
0.508497 + 0.861064i \(0.330201\pi\)
\(684\) 0 0
\(685\) −24.8032 −0.947680
\(686\) 0 0
\(687\) 4.02756 0.153661
\(688\) 0 0
\(689\) 29.9660 1.14161
\(690\) 0 0
\(691\) 14.7899 0.562633 0.281316 0.959615i \(-0.409229\pi\)
0.281316 + 0.959615i \(0.409229\pi\)
\(692\) 0 0
\(693\) 0.853690 0.0324290
\(694\) 0 0
\(695\) −6.16415 −0.233820
\(696\) 0 0
\(697\) −7.94948 −0.301108
\(698\) 0 0
\(699\) 11.7211 0.443332
\(700\) 0 0
\(701\) 25.9245 0.979155 0.489577 0.871960i \(-0.337151\pi\)
0.489577 + 0.871960i \(0.337151\pi\)
\(702\) 0 0
\(703\) −25.8528 −0.975055
\(704\) 0 0
\(705\) −5.08532 −0.191524
\(706\) 0 0
\(707\) 1.60782 0.0604685
\(708\) 0 0
\(709\) −20.6082 −0.773958 −0.386979 0.922089i \(-0.626481\pi\)
−0.386979 + 0.922089i \(0.626481\pi\)
\(710\) 0 0
\(711\) −9.69382 −0.363547
\(712\) 0 0
\(713\) −11.7633 −0.440538
\(714\) 0 0
\(715\) −57.5247 −2.15130
\(716\) 0 0
\(717\) 13.6517 0.509834
\(718\) 0 0
\(719\) 44.0949 1.64446 0.822230 0.569155i \(-0.192730\pi\)
0.822230 + 0.569155i \(0.192730\pi\)
\(720\) 0 0
\(721\) 1.80409 0.0671880
\(722\) 0 0
\(723\) 2.13167 0.0792777
\(724\) 0 0
\(725\) −5.24842 −0.194921
\(726\) 0 0
\(727\) 9.23457 0.342491 0.171246 0.985228i \(-0.445221\pi\)
0.171246 + 0.985228i \(0.445221\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.8486 0.919058
\(732\) 0 0
\(733\) 25.8467 0.954670 0.477335 0.878721i \(-0.341603\pi\)
0.477335 + 0.878721i \(0.341603\pi\)
\(734\) 0 0
\(735\) −12.5401 −0.462549
\(736\) 0 0
\(737\) −5.82936 −0.214727
\(738\) 0 0
\(739\) −24.0403 −0.884337 −0.442169 0.896932i \(-0.645791\pi\)
−0.442169 + 0.896932i \(0.645791\pi\)
\(740\) 0 0
\(741\) 18.1876 0.668137
\(742\) 0 0
\(743\) 17.8748 0.655762 0.327881 0.944719i \(-0.393665\pi\)
0.327881 + 0.944719i \(0.393665\pi\)
\(744\) 0 0
\(745\) −7.44538 −0.272778
\(746\) 0 0
\(747\) 0.163788 0.00599271
\(748\) 0 0
\(749\) 0.162293 0.00593007
\(750\) 0 0
\(751\) −35.0731 −1.27984 −0.639918 0.768443i \(-0.721032\pi\)
−0.639918 + 0.768443i \(0.721032\pi\)
\(752\) 0 0
\(753\) 6.27020 0.228499
\(754\) 0 0
\(755\) −40.7490 −1.48301
\(756\) 0 0
\(757\) −46.3962 −1.68630 −0.843150 0.537679i \(-0.819301\pi\)
−0.843150 + 0.537679i \(0.819301\pi\)
\(758\) 0 0
\(759\) −15.1917 −0.551425
\(760\) 0 0
\(761\) −10.5531 −0.382550 −0.191275 0.981536i \(-0.561262\pi\)
−0.191275 + 0.981536i \(0.561262\pi\)
\(762\) 0 0
\(763\) 0.324738 0.0117563
\(764\) 0 0
\(765\) 5.48946 0.198472
\(766\) 0 0
\(767\) 33.6972 1.21673
\(768\) 0 0
\(769\) −35.2068 −1.26959 −0.634795 0.772681i \(-0.718915\pi\)
−0.634795 + 0.772681i \(0.718915\pi\)
\(770\) 0 0
\(771\) −15.0853 −0.543285
\(772\) 0 0
\(773\) 27.4212 0.986271 0.493136 0.869952i \(-0.335851\pi\)
0.493136 + 0.869952i \(0.335851\pi\)
\(774\) 0 0
\(775\) 7.35067 0.264044
\(776\) 0 0
\(777\) −1.34583 −0.0482812
\(778\) 0 0
\(779\) −7.94948 −0.284820
\(780\) 0 0
\(781\) −1.70738 −0.0610948
\(782\) 0 0
\(783\) −2.96951 −0.106121
\(784\) 0 0
\(785\) 7.06877 0.252295
\(786\) 0 0
\(787\) −9.46058 −0.337233 −0.168617 0.985682i \(-0.553930\pi\)
−0.168617 + 0.985682i \(0.553930\pi\)
\(788\) 0 0
\(789\) −26.1706 −0.931700
\(790\) 0 0
\(791\) 0.561839 0.0199767
\(792\) 0 0
\(793\) 30.9128 1.09775
\(794\) 0 0
\(795\) 9.04449 0.320775
\(796\) 0 0
\(797\) 19.1791 0.679359 0.339680 0.940541i \(-0.389681\pi\)
0.339680 + 0.940541i \(0.389681\pi\)
\(798\) 0 0
\(799\) 8.63577 0.305511
\(800\) 0 0
\(801\) 14.3990 0.508765
\(802\) 0 0
\(803\) 7.19173 0.253791
\(804\) 0 0
\(805\) −0.808269 −0.0284878
\(806\) 0 0
\(807\) 12.1580 0.427982
\(808\) 0 0
\(809\) −43.1578 −1.51735 −0.758673 0.651472i \(-0.774152\pi\)
−0.758673 + 0.651472i \(0.774152\pi\)
\(810\) 0 0
\(811\) −3.87518 −0.136076 −0.0680380 0.997683i \(-0.521674\pi\)
−0.0680380 + 0.997683i \(0.521674\pi\)
\(812\) 0 0
\(813\) −10.6644 −0.374016
\(814\) 0 0
\(815\) −13.8243 −0.484242
\(816\) 0 0
\(817\) 24.8486 0.869342
\(818\) 0 0
\(819\) 0.946795 0.0330837
\(820\) 0 0
\(821\) 5.62084 0.196169 0.0980843 0.995178i \(-0.468729\pi\)
0.0980843 + 0.995178i \(0.468729\pi\)
\(822\) 0 0
\(823\) 38.5255 1.34291 0.671457 0.741043i \(-0.265669\pi\)
0.671457 + 0.741043i \(0.265669\pi\)
\(824\) 0 0
\(825\) 9.49307 0.330506
\(826\) 0 0
\(827\) 4.23674 0.147326 0.0736629 0.997283i \(-0.476531\pi\)
0.0736629 + 0.997283i \(0.476531\pi\)
\(828\) 0 0
\(829\) −35.3128 −1.22646 −0.613231 0.789903i \(-0.710131\pi\)
−0.613231 + 0.789903i \(0.710131\pi\)
\(830\) 0 0
\(831\) 3.76421 0.130579
\(832\) 0 0
\(833\) 21.2953 0.737838
\(834\) 0 0
\(835\) −7.11007 −0.246054
\(836\) 0 0
\(837\) 4.15894 0.143754
\(838\) 0 0
\(839\) 39.6005 1.36716 0.683580 0.729876i \(-0.260422\pi\)
0.683580 + 0.729876i \(0.260422\pi\)
\(840\) 0 0
\(841\) −20.1820 −0.695932
\(842\) 0 0
\(843\) −10.4496 −0.359902
\(844\) 0 0
\(845\) −40.4253 −1.39067
\(846\) 0 0
\(847\) 2.83688 0.0974765
\(848\) 0 0
\(849\) 17.6569 0.605982
\(850\) 0 0
\(851\) 23.9495 0.820977
\(852\) 0 0
\(853\) 10.8683 0.372124 0.186062 0.982538i \(-0.440428\pi\)
0.186062 + 0.982538i \(0.440428\pi\)
\(854\) 0 0
\(855\) 5.48946 0.187735
\(856\) 0 0
\(857\) −34.0082 −1.16170 −0.580849 0.814011i \(-0.697279\pi\)
−0.580849 + 0.814011i \(0.697279\pi\)
\(858\) 0 0
\(859\) 12.9973 0.443463 0.221731 0.975108i \(-0.428829\pi\)
0.221731 + 0.975108i \(0.428829\pi\)
\(860\) 0 0
\(861\) −0.413828 −0.0141032
\(862\) 0 0
\(863\) 25.8307 0.879289 0.439644 0.898172i \(-0.355104\pi\)
0.439644 + 0.898172i \(0.355104\pi\)
\(864\) 0 0
\(865\) −40.6353 −1.38164
\(866\) 0 0
\(867\) 7.67794 0.260757
\(868\) 0 0
\(869\) −52.0663 −1.76623
\(870\) 0 0
\(871\) −6.46512 −0.219062
\(872\) 0 0
\(873\) −0.571533 −0.0193435
\(874\) 0 0
\(875\) 1.93391 0.0653780
\(876\) 0 0
\(877\) −45.2150 −1.52680 −0.763400 0.645926i \(-0.776472\pi\)
−0.763400 + 0.645926i \(0.776472\pi\)
\(878\) 0 0
\(879\) 30.7465 1.03705
\(880\) 0 0
\(881\) −34.7403 −1.17043 −0.585215 0.810878i \(-0.698990\pi\)
−0.585215 + 0.810878i \(0.698990\pi\)
\(882\) 0 0
\(883\) −48.9366 −1.64685 −0.823424 0.567427i \(-0.807939\pi\)
−0.823424 + 0.567427i \(0.807939\pi\)
\(884\) 0 0
\(885\) 10.1706 0.341882
\(886\) 0 0
\(887\) −22.9284 −0.769860 −0.384930 0.922946i \(-0.625774\pi\)
−0.384930 + 0.922946i \(0.625774\pi\)
\(888\) 0 0
\(889\) 0.238081 0.00798497
\(890\) 0 0
\(891\) 5.37109 0.179938
\(892\) 0 0
\(893\) 8.63577 0.288985
\(894\) 0 0
\(895\) 31.1009 1.03959
\(896\) 0 0
\(897\) −16.8486 −0.562558
\(898\) 0 0
\(899\) −12.3500 −0.411896
\(900\) 0 0
\(901\) −15.3591 −0.511687
\(902\) 0 0
\(903\) 1.29355 0.0430467
\(904\) 0 0
\(905\) −14.6523 −0.487059
\(906\) 0 0
\(907\) 23.1680 0.769280 0.384640 0.923067i \(-0.374326\pi\)
0.384640 + 0.923067i \(0.374326\pi\)
\(908\) 0 0
\(909\) 10.1158 0.335520
\(910\) 0 0
\(911\) −29.4078 −0.974324 −0.487162 0.873312i \(-0.661968\pi\)
−0.487162 + 0.873312i \(0.661968\pi\)
\(912\) 0 0
\(913\) 0.879722 0.0291146
\(914\) 0 0
\(915\) 9.33026 0.308449
\(916\) 0 0
\(917\) −2.34315 −0.0773775
\(918\) 0 0
\(919\) 6.86029 0.226300 0.113150 0.993578i \(-0.463906\pi\)
0.113150 + 0.993578i \(0.463906\pi\)
\(920\) 0 0
\(921\) −21.2981 −0.701798
\(922\) 0 0
\(923\) −1.89359 −0.0623283
\(924\) 0 0
\(925\) −14.9656 −0.492067
\(926\) 0 0
\(927\) 11.3507 0.372805
\(928\) 0 0
\(929\) 16.1385 0.529488 0.264744 0.964319i \(-0.414713\pi\)
0.264744 + 0.964319i \(0.414713\pi\)
\(930\) 0 0
\(931\) 21.2953 0.697925
\(932\) 0 0
\(933\) 1.77883 0.0582363
\(934\) 0 0
\(935\) 29.4844 0.964241
\(936\) 0 0
\(937\) −34.7669 −1.13579 −0.567893 0.823102i \(-0.692241\pi\)
−0.567893 + 0.823102i \(0.692241\pi\)
\(938\) 0 0
\(939\) 2.70320 0.0882157
\(940\) 0 0
\(941\) 52.7024 1.71805 0.859025 0.511934i \(-0.171071\pi\)
0.859025 + 0.511934i \(0.171071\pi\)
\(942\) 0 0
\(943\) 7.36423 0.239812
\(944\) 0 0
\(945\) 0.285766 0.00929598
\(946\) 0 0
\(947\) 26.1972 0.851296 0.425648 0.904889i \(-0.360046\pi\)
0.425648 + 0.904889i \(0.360046\pi\)
\(948\) 0 0
\(949\) 7.97608 0.258915
\(950\) 0 0
\(951\) −22.0653 −0.715516
\(952\) 0 0
\(953\) −11.1752 −0.362000 −0.181000 0.983483i \(-0.557933\pi\)
−0.181000 + 0.983483i \(0.557933\pi\)
\(954\) 0 0
\(955\) −29.0679 −0.940615
\(956\) 0 0
\(957\) −15.9495 −0.515573
\(958\) 0 0
\(959\) 2.19266 0.0708047
\(960\) 0 0
\(961\) −13.7032 −0.442039
\(962\) 0 0
\(963\) 1.02109 0.0329041
\(964\) 0 0
\(965\) 39.8159 1.28172
\(966\) 0 0
\(967\) 12.8452 0.413075 0.206537 0.978439i \(-0.433780\pi\)
0.206537 + 0.978439i \(0.433780\pi\)
\(968\) 0 0
\(969\) −9.32206 −0.299468
\(970\) 0 0
\(971\) −8.40774 −0.269817 −0.134909 0.990858i \(-0.543074\pi\)
−0.134909 + 0.990858i \(0.543074\pi\)
\(972\) 0 0
\(973\) 0.544926 0.0174695
\(974\) 0 0
\(975\) 10.5284 0.337179
\(976\) 0 0
\(977\) −40.4156 −1.29301 −0.646504 0.762910i \(-0.723770\pi\)
−0.646504 + 0.762910i \(0.723770\pi\)
\(978\) 0 0
\(979\) 77.3385 2.47175
\(980\) 0 0
\(981\) 2.04313 0.0652320
\(982\) 0 0
\(983\) 13.9202 0.443985 0.221993 0.975048i \(-0.428744\pi\)
0.221993 + 0.975048i \(0.428744\pi\)
\(984\) 0 0
\(985\) −36.3582 −1.15847
\(986\) 0 0
\(987\) 0.449555 0.0143095
\(988\) 0 0
\(989\) −23.0192 −0.731969
\(990\) 0 0
\(991\) −41.0309 −1.30339 −0.651695 0.758481i \(-0.725942\pi\)
−0.651695 + 0.758481i \(0.725942\pi\)
\(992\) 0 0
\(993\) 21.8431 0.693169
\(994\) 0 0
\(995\) 44.9618 1.42539
\(996\) 0 0
\(997\) −38.6427 −1.22383 −0.611913 0.790925i \(-0.709600\pi\)
−0.611913 + 0.790925i \(0.709600\pi\)
\(998\) 0 0
\(999\) −8.46742 −0.267897
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 3072.2.a.n.1.1 4
3.2 odd 2 9216.2.a.x.1.4 4
4.3 odd 2 3072.2.a.t.1.1 4
8.3 odd 2 3072.2.a.i.1.4 4
8.5 even 2 3072.2.a.o.1.4 4
12.11 even 2 9216.2.a.y.1.4 4
16.3 odd 4 3072.2.d.f.1537.8 8
16.5 even 4 3072.2.d.i.1537.5 8
16.11 odd 4 3072.2.d.f.1537.1 8
16.13 even 4 3072.2.d.i.1537.4 8
24.5 odd 2 9216.2.a.bn.1.1 4
24.11 even 2 9216.2.a.bo.1.1 4
32.3 odd 8 48.2.j.a.13.1 8
32.5 even 8 384.2.j.a.97.3 8
32.11 odd 8 48.2.j.a.37.1 yes 8
32.13 even 8 384.2.j.a.289.3 8
32.19 odd 8 384.2.j.b.289.1 8
32.21 even 8 192.2.j.a.49.2 8
32.27 odd 8 384.2.j.b.97.1 8
32.29 even 8 192.2.j.a.145.2 8
96.5 odd 8 1152.2.k.f.865.3 8
96.11 even 8 144.2.k.b.37.4 8
96.29 odd 8 576.2.k.b.145.2 8
96.35 even 8 144.2.k.b.109.4 8
96.53 odd 8 576.2.k.b.433.2 8
96.59 even 8 1152.2.k.c.865.3 8
96.77 odd 8 1152.2.k.f.289.3 8
96.83 even 8 1152.2.k.c.289.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.1 8 32.3 odd 8
48.2.j.a.37.1 yes 8 32.11 odd 8
144.2.k.b.37.4 8 96.11 even 8
144.2.k.b.109.4 8 96.35 even 8
192.2.j.a.49.2 8 32.21 even 8
192.2.j.a.145.2 8 32.29 even 8
384.2.j.a.97.3 8 32.5 even 8
384.2.j.a.289.3 8 32.13 even 8
384.2.j.b.97.1 8 32.27 odd 8
384.2.j.b.289.1 8 32.19 odd 8
576.2.k.b.145.2 8 96.29 odd 8
576.2.k.b.433.2 8 96.53 odd 8
1152.2.k.c.289.3 8 96.83 even 8
1152.2.k.c.865.3 8 96.59 even 8
1152.2.k.f.289.3 8 96.77 odd 8
1152.2.k.f.865.3 8 96.5 odd 8
3072.2.a.i.1.4 4 8.3 odd 2
3072.2.a.n.1.1 4 1.1 even 1 trivial
3072.2.a.o.1.4 4 8.5 even 2
3072.2.a.t.1.1 4 4.3 odd 2
3072.2.d.f.1537.1 8 16.11 odd 4
3072.2.d.f.1537.8 8 16.3 odd 4
3072.2.d.i.1537.4 8 16.13 even 4
3072.2.d.i.1537.5 8 16.5 even 4
9216.2.a.x.1.4 4 3.2 odd 2
9216.2.a.y.1.4 4 12.11 even 2
9216.2.a.bn.1.1 4 24.5 odd 2
9216.2.a.bo.1.1 4 24.11 even 2