Properties

Label 3072.2.d.i.1537.5
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.5
Root \(0.500000 - 2.10607i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.i.1537.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} -1.79793i q^{5} -0.158942 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} -1.79793i q^{5} -0.158942 q^{7} -1.00000 q^{9} +5.37109i q^{11} -5.95687i q^{13} +1.79793 q^{15} -3.05320 q^{17} +3.05320i q^{19} -0.158942i q^{21} -2.82843 q^{23} +1.76744 q^{25} -1.00000i q^{27} -2.96951i q^{29} -4.15894 q^{31} -5.37109 q^{33} +0.285766i q^{35} +8.46742i q^{37} +5.95687 q^{39} -2.60365 q^{41} -8.13853i q^{43} +1.79793i q^{45} -2.82843 q^{47} -6.97474 q^{49} -3.05320i q^{51} +5.03049i q^{53} +9.65685 q^{55} -3.05320 q^{57} +5.65685i q^{59} -5.18944i q^{61} +0.158942 q^{63} -10.7101 q^{65} +1.08532i q^{67} -2.82843i q^{69} +0.317883 q^{71} -1.33897 q^{73} +1.76744i q^{75} -0.853690i q^{77} -9.69382 q^{79} +1.00000 q^{81} -0.163788i q^{83} +5.48946i q^{85} +2.96951 q^{87} -14.3990 q^{89} +0.946795i q^{91} -4.15894i q^{93} +5.48946 q^{95} -0.571533 q^{97} -5.37109i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{9} - 8q^{15} - 8q^{25} - 24q^{31} + 16q^{39} + 8q^{49} + 32q^{55} - 8q^{63} - 16q^{65} - 16q^{71} + 16q^{73} - 24q^{79} + 8q^{81} + 24q^{87} + 16q^{89} + 48q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) − 1.79793i − 0.804060i −0.915627 0.402030i \(-0.868305\pi\)
0.915627 0.402030i \(-0.131695\pi\)
\(6\) 0 0
\(7\) −0.158942 −0.0600743 −0.0300371 0.999549i \(-0.509563\pi\)
−0.0300371 + 0.999549i \(0.509563\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.37109i 1.61944i 0.586814 + 0.809722i \(0.300382\pi\)
−0.586814 + 0.809722i \(0.699618\pi\)
\(12\) 0 0
\(13\) − 5.95687i − 1.65214i −0.563568 0.826070i \(-0.690572\pi\)
0.563568 0.826070i \(-0.309428\pi\)
\(14\) 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
\(19\) 3.05320i 0.700453i 0.936665 + 0.350227i \(0.113895\pi\)
−0.936665 + 0.350227i \(0.886105\pi\)
\(20\) 0 0
\(21\) − 0.158942i − 0.0346839i
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 1.76744 0.353488
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 2.96951i − 0.551423i −0.961240 0.275712i \(-0.911087\pi\)
0.961240 0.275712i \(-0.0889135\pi\)
\(30\) 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
\(35\) 0.285766i 0.0483033i
\(36\) 0 0
\(37\) 8.46742i 1.39203i 0.718025 + 0.696017i \(0.245046\pi\)
−0.718025 + 0.696017i \(0.754954\pi\)
\(38\) 0 0
\(39\) 5.95687 0.953863
\(40\) 0 0
\(41\) −2.60365 −0.406622 −0.203311 0.979114i \(-0.565170\pi\)
−0.203311 + 0.979114i \(0.565170\pi\)
\(42\) 0 0
\(43\) − 8.13853i − 1.24111i −0.784162 0.620557i \(-0.786907\pi\)
0.784162 0.620557i \(-0.213093\pi\)
\(44\) 0 0
\(45\) 1.79793i 0.268020i
\(46\) 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
\(51\) − 3.05320i − 0.427534i
\(52\) 0 0
\(53\) 5.03049i 0.690992i 0.938420 + 0.345496i \(0.112289\pi\)
−0.938420 + 0.345496i \(0.887711\pi\)
\(54\) 0 0
\(55\) 9.65685 1.30213
\(56\) 0 0
\(57\) −3.05320 −0.404407
\(58\) 0 0
\(59\) 5.65685i 0.736460i 0.929735 + 0.368230i \(0.120036\pi\)
−0.929735 + 0.368230i \(0.879964\pi\)
\(60\) 0 0
\(61\) − 5.18944i − 0.664439i −0.943202 0.332220i \(-0.892202\pi\)
0.943202 0.332220i \(-0.107798\pi\)
\(62\) 0 0
\(63\) 0.158942 0.0200248
\(64\) 0 0
\(65\) −10.7101 −1.32842
\(66\) 0 0
\(67\) 1.08532i 0.132593i 0.997800 + 0.0662966i \(0.0211183\pi\)
−0.997800 + 0.0662966i \(0.978882\pi\)
\(68\) 0 0
\(69\) − 2.82843i − 0.340503i
\(70\) 0 0
\(71\) 0.317883 0.0377258 0.0188629 0.999822i \(-0.493995\pi\)
0.0188629 + 0.999822i \(0.493995\pi\)
\(72\) 0 0
\(73\) −1.33897 −0.156715 −0.0783573 0.996925i \(-0.524968\pi\)
−0.0783573 + 0.996925i \(0.524968\pi\)
\(74\) 0 0
\(75\) 1.76744i 0.204086i
\(76\) 0 0
\(77\) − 0.853690i − 0.0972870i
\(78\) 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
\(83\) − 0.163788i − 0.0179781i −0.999960 0.00898906i \(-0.997139\pi\)
0.999960 0.00898906i \(-0.00286135\pi\)
\(84\) 0 0
\(85\) 5.48946i 0.595415i
\(86\) 0 0
\(87\) 2.96951 0.318364
\(88\) 0 0
\(89\) −14.3990 −1.52629 −0.763147 0.646225i \(-0.776347\pi\)
−0.763147 + 0.646225i \(0.776347\pi\)
\(90\) 0 0
\(91\) 0.946795i 0.0992511i
\(92\) 0 0
\(93\) − 4.15894i − 0.431262i
\(94\) 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
\(99\) − 5.37109i − 0.539815i
\(100\) 0 0
\(101\) 10.1158i 1.00656i 0.864123 + 0.503281i \(0.167874\pi\)
−0.864123 + 0.503281i \(0.832126\pi\)
\(102\) 0 0
\(103\) −11.3507 −1.11841 −0.559207 0.829028i \(-0.688894\pi\)
−0.559207 + 0.829028i \(0.688894\pi\)
\(104\) 0 0
\(105\) −0.285766 −0.0278879
\(106\) 0 0
\(107\) 1.02109i 0.0987123i 0.998781 + 0.0493561i \(0.0157169\pi\)
−0.998781 + 0.0493561i \(0.984283\pi\)
\(108\) 0 0
\(109\) − 2.04313i − 0.195696i −0.995201 0.0978480i \(-0.968804\pi\)
0.995201 0.0978480i \(-0.0311959\pi\)
\(110\) 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
\(115\) 5.08532i 0.474209i
\(116\) 0 0
\(117\) 5.95687i 0.550713i
\(118\) 0 0
\(119\) 0.485281 0.0444857
\(120\) 0 0
\(121\) −17.8486 −1.62260
\(122\) 0 0
\(123\) − 2.60365i − 0.234763i
\(124\) 0 0
\(125\) − 12.1674i − 1.08829i
\(126\) 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
\(131\) 14.7422i 1.28803i 0.765013 + 0.644015i \(0.222733\pi\)
−0.765013 + 0.644015i \(0.777267\pi\)
\(132\) 0 0
\(133\) − 0.485281i − 0.0420792i
\(134\) 0 0
\(135\) −1.79793 −0.154741
\(136\) 0 0
\(137\) −13.7954 −1.17862 −0.589309 0.807907i \(-0.700600\pi\)
−0.589309 + 0.807907i \(0.700600\pi\)
\(138\) 0 0
\(139\) 3.42847i 0.290799i 0.989373 + 0.145399i \(0.0464467\pi\)
−0.989373 + 0.145399i \(0.953553\pi\)
\(140\) 0 0
\(141\) − 2.82843i − 0.238197i
\(142\) 0 0
\(143\) 31.9949 2.67555
\(144\) 0 0
\(145\) −5.33897 −0.443377
\(146\) 0 0
\(147\) − 6.97474i − 0.575267i
\(148\) 0 0
\(149\) 4.14108i 0.339250i 0.985509 + 0.169625i \(0.0542557\pi\)
−0.985509 + 0.169625i \(0.945744\pi\)
\(150\) 0 0
\(151\) −22.6644 −1.84440 −0.922201 0.386712i \(-0.873611\pi\)
−0.922201 + 0.386712i \(0.873611\pi\)
\(152\) 0 0
\(153\) 3.05320 0.246837
\(154\) 0 0
\(155\) 7.47750i 0.600607i
\(156\) 0 0
\(157\) 3.93161i 0.313777i 0.987616 + 0.156888i \(0.0501463\pi\)
−0.987616 + 0.156888i \(0.949854\pi\)
\(158\) 0 0
\(159\) −5.03049 −0.398944
\(160\) 0 0
\(161\) 0.449555 0.0354299
\(162\) 0 0
\(163\) − 7.68897i − 0.602247i −0.953585 0.301123i \(-0.902638\pi\)
0.953585 0.301123i \(-0.0973616\pi\)
\(164\) 0 0
\(165\) 9.65685i 0.751785i
\(166\) 0 0
\(167\) −3.95458 −0.306015 −0.153007 0.988225i \(-0.548896\pi\)
−0.153007 + 0.988225i \(0.548896\pi\)
\(168\) 0 0
\(169\) −22.4844 −1.72957
\(170\) 0 0
\(171\) − 3.05320i − 0.233484i
\(172\) 0 0
\(173\) − 22.6011i − 1.71833i −0.511699 0.859165i \(-0.670984\pi\)
0.511699 0.859165i \(-0.329016\pi\)
\(174\) 0 0
\(175\) −0.280920 −0.0212355
\(176\) 0 0
\(177\) −5.65685 −0.425195
\(178\) 0 0
\(179\) 17.2981i 1.29292i 0.762946 + 0.646462i \(0.223752\pi\)
−0.762946 + 0.646462i \(0.776248\pi\)
\(180\) 0 0
\(181\) 8.14953i 0.605750i 0.953030 + 0.302875i \(0.0979465\pi\)
−0.953030 + 0.302875i \(0.902054\pi\)
\(182\) 0 0
\(183\) 5.18944 0.383614
\(184\) 0 0
\(185\) 15.2238 1.11928
\(186\) 0 0
\(187\) − 16.3990i − 1.19922i
\(188\) 0 0
\(189\) 0.158942i 0.0115613i
\(190\) 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
\(195\) − 10.7101i − 0.766963i
\(196\) 0 0
\(197\) 20.2222i 1.44077i 0.693572 + 0.720387i \(0.256036\pi\)
−0.693572 + 0.720387i \(0.743964\pi\)
\(198\) 0 0
\(199\) 25.0075 1.77274 0.886368 0.462981i \(-0.153220\pi\)
0.886368 + 0.462981i \(0.153220\pi\)
\(200\) 0 0
\(201\) −1.08532 −0.0765527
\(202\) 0 0
\(203\) 0.471978i 0.0331264i
\(204\) 0 0
\(205\) 4.68119i 0.326948i
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) −16.3990 −1.13434
\(210\) 0 0
\(211\) − 26.0559i − 1.79376i −0.442273 0.896881i \(-0.645828\pi\)
0.442273 0.896881i \(-0.354172\pi\)
\(212\) 0 0
\(213\) 0.317883i 0.0217810i
\(214\) 0 0
\(215\) −14.6325 −0.997930
\(216\) 0 0
\(217\) 0.661029 0.0448736
\(218\) 0 0
\(219\) − 1.33897i − 0.0904793i
\(220\) 0 0
\(221\) 18.1876i 1.22343i
\(222\) 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
\(227\) − 0.163788i − 0.0108710i −0.999985 0.00543551i \(-0.998270\pi\)
0.999985 0.00543551i \(-0.00173019\pi\)
\(228\) 0 0
\(229\) − 4.02756i − 0.266148i −0.991106 0.133074i \(-0.957515\pi\)
0.991106 0.133074i \(-0.0424849\pi\)
\(230\) 0 0
\(231\) 0.853690 0.0561687
\(232\) 0 0
\(233\) 11.7211 0.767874 0.383937 0.923359i \(-0.374568\pi\)
0.383937 + 0.923359i \(0.374568\pi\)
\(234\) 0 0
\(235\) 5.08532i 0.331730i
\(236\) 0 0
\(237\) − 9.69382i − 0.629681i
\(238\) 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
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 12.5401i 0.801158i
\(246\) 0 0
\(247\) 18.1876 1.15725
\(248\) 0 0
\(249\) 0.163788 0.0103797
\(250\) 0 0
\(251\) − 6.27020i − 0.395771i −0.980225 0.197886i \(-0.936593\pi\)
0.980225 0.197886i \(-0.0634075\pi\)
\(252\) 0 0
\(253\) − 15.1917i − 0.955096i
\(254\) 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
\(259\) − 1.34583i − 0.0836255i
\(260\) 0 0
\(261\) 2.96951i 0.183808i
\(262\) 0 0
\(263\) −26.1706 −1.61375 −0.806875 0.590722i \(-0.798843\pi\)
−0.806875 + 0.590722i \(0.798843\pi\)
\(264\) 0 0
\(265\) 9.04449 0.555599
\(266\) 0 0
\(267\) − 14.3990i − 0.881206i
\(268\) 0 0
\(269\) 12.1580i 0.741286i 0.928775 + 0.370643i \(0.120863\pi\)
−0.928775 + 0.370643i \(0.879137\pi\)
\(270\) 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
\(275\) 9.49307i 0.572453i
\(276\) 0 0
\(277\) − 3.76421i − 0.226170i −0.993585 0.113085i \(-0.963927\pi\)
0.993585 0.113085i \(-0.0360732\pi\)
\(278\) 0 0
\(279\) 4.15894 0.248989
\(280\) 0 0
\(281\) −10.4496 −0.623368 −0.311684 0.950186i \(-0.600893\pi\)
−0.311684 + 0.950186i \(0.600893\pi\)
\(282\) 0 0
\(283\) − 17.6569i − 1.04959i −0.851228 0.524796i \(-0.824142\pi\)
0.851228 0.524796i \(-0.175858\pi\)
\(284\) 0 0
\(285\) 5.48946i 0.325167i
\(286\) 0 0
\(287\) 0.413828 0.0244275
\(288\) 0 0
\(289\) −7.67794 −0.451644
\(290\) 0 0
\(291\) − 0.571533i − 0.0335038i
\(292\) 0 0
\(293\) − 30.7465i − 1.79623i −0.439762 0.898114i \(-0.644937\pi\)
0.439762 0.898114i \(-0.355063\pi\)
\(294\) 0 0
\(295\) 10.1706 0.592158
\(296\) 0 0
\(297\) 5.37109 0.311662
\(298\) 0 0
\(299\) 16.8486i 0.974379i
\(300\) 0 0
\(301\) 1.29355i 0.0745590i
\(302\) 0 0
\(303\) −10.1158 −0.581138
\(304\) 0 0
\(305\) −9.33026 −0.534249
\(306\) 0 0
\(307\) − 21.2981i − 1.21555i −0.794110 0.607775i \(-0.792062\pi\)
0.794110 0.607775i \(-0.207938\pi\)
\(308\) 0 0
\(309\) − 11.3507i − 0.645717i
\(310\) 0 0
\(311\) 1.77883 0.100868 0.0504342 0.998727i \(-0.483939\pi\)
0.0504342 + 0.998727i \(0.483939\pi\)
\(312\) 0 0
\(313\) 2.70320 0.152794 0.0763971 0.997077i \(-0.475658\pi\)
0.0763971 + 0.997077i \(0.475658\pi\)
\(314\) 0 0
\(315\) − 0.285766i − 0.0161011i
\(316\) 0 0
\(317\) − 22.0653i − 1.23931i −0.784874 0.619655i \(-0.787272\pi\)
0.784874 0.619655i \(-0.212728\pi\)
\(318\) 0 0
\(319\) 15.9495 0.892999
\(320\) 0 0
\(321\) −1.02109 −0.0569916
\(322\) 0 0
\(323\) − 9.32206i − 0.518693i
\(324\) 0 0
\(325\) − 10.5284i − 0.584011i
\(326\) 0 0
\(327\) 2.04313 0.112985
\(328\) 0 0
\(329\) 0.449555 0.0247848
\(330\) 0 0
\(331\) − 21.8431i − 1.20060i −0.799774 0.600302i \(-0.795047\pi\)
0.799774 0.600302i \(-0.204953\pi\)
\(332\) 0 0
\(333\) − 8.46742i − 0.464012i
\(334\) 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
\(339\) 3.53488i 0.191988i
\(340\) 0 0
\(341\) − 22.3380i − 1.20967i
\(342\) 0 0
\(343\) 2.22117 0.119932
\(344\) 0 0
\(345\) −5.08532 −0.273785
\(346\) 0 0
\(347\) 28.0490i 1.50575i 0.658163 + 0.752875i \(0.271334\pi\)
−0.658163 + 0.752875i \(0.728666\pi\)
\(348\) 0 0
\(349\) − 16.9307i − 0.906279i −0.891440 0.453139i \(-0.850304\pi\)
0.891440 0.453139i \(-0.149696\pi\)
\(350\) 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
\(355\) − 0.571533i − 0.0303338i
\(356\) 0 0
\(357\) 0.485281i 0.0256838i
\(358\) 0 0
\(359\) −27.0867 −1.42958 −0.714790 0.699339i \(-0.753478\pi\)
−0.714790 + 0.699339i \(0.753478\pi\)
\(360\) 0 0
\(361\) 9.67794 0.509365
\(362\) 0 0
\(363\) − 17.8486i − 0.936808i
\(364\) 0 0
\(365\) 2.40738i 0.126008i
\(366\) 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
\(371\) − 0.799555i − 0.0415108i
\(372\) 0 0
\(373\) 1.46190i 0.0756943i 0.999284 + 0.0378471i \(0.0120500\pi\)
−0.999284 + 0.0378471i \(0.987950\pi\)
\(374\) 0 0
\(375\) 12.1674 0.628322
\(376\) 0 0
\(377\) −17.6890 −0.911028
\(378\) 0 0
\(379\) 24.9871i 1.28350i 0.766913 + 0.641751i \(0.221792\pi\)
−0.766913 + 0.641751i \(0.778208\pi\)
\(380\) 0 0
\(381\) 1.49791i 0.0767404i
\(382\) 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
\(387\) 8.13853i 0.413705i
\(388\) 0 0
\(389\) 3.62218i 0.183652i 0.995775 + 0.0918260i \(0.0292704\pi\)
−0.995775 + 0.0918260i \(0.970730\pi\)
\(390\) 0 0
\(391\) 8.63577 0.436729
\(392\) 0 0
\(393\) −14.7422 −0.743644
\(394\) 0 0
\(395\) 17.4288i 0.876940i
\(396\) 0 0
\(397\) − 7.20959i − 0.361839i −0.983498 0.180920i \(-0.942093\pi\)
0.983498 0.180920i \(-0.0579074\pi\)
\(398\) 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
\(403\) 24.7743i 1.23410i
\(404\) 0 0
\(405\) − 1.79793i − 0.0893400i
\(406\) 0 0
\(407\) −45.4792 −2.25432
\(408\) 0 0
\(409\) 11.3779 0.562603 0.281302 0.959619i \(-0.409234\pi\)
0.281302 + 0.959619i \(0.409234\pi\)
\(410\) 0 0
\(411\) − 13.7954i − 0.680476i
\(412\) 0 0
\(413\) − 0.899110i − 0.0442423i
\(414\) 0 0
\(415\) −0.294481 −0.0144555
\(416\) 0 0
\(417\) −3.42847 −0.167893
\(418\) 0 0
\(419\) 32.9618i 1.61029i 0.593077 + 0.805146i \(0.297913\pi\)
−0.593077 + 0.805146i \(0.702087\pi\)
\(420\) 0 0
\(421\) 24.9119i 1.21413i 0.794652 + 0.607065i \(0.207653\pi\)
−0.794652 + 0.607065i \(0.792347\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) −5.39635 −0.261761
\(426\) 0 0
\(427\) 0.824818i 0.0399157i
\(428\) 0 0
\(429\) 31.9949i 1.54473i
\(430\) 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
\(435\) − 5.33897i − 0.255984i
\(436\) 0 0
\(437\) − 8.63577i − 0.413105i
\(438\) 0 0
\(439\) 22.5735 1.07738 0.538688 0.842505i \(-0.318920\pi\)
0.538688 + 0.842505i \(0.318920\pi\)
\(440\) 0 0
\(441\) 6.97474 0.332130
\(442\) 0 0
\(443\) 33.5334i 1.59322i 0.604494 + 0.796610i \(0.293375\pi\)
−0.604494 + 0.796610i \(0.706625\pi\)
\(444\) 0 0
\(445\) 25.8885i 1.22723i
\(446\) 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
\(451\) − 13.9844i − 0.658501i
\(452\) 0 0
\(453\) − 22.6644i − 1.06487i
\(454\) 0 0
\(455\) 1.70227 0.0798039
\(456\) 0 0
\(457\) 26.7422 1.25095 0.625473 0.780246i \(-0.284906\pi\)
0.625473 + 0.780246i \(0.284906\pi\)
\(458\) 0 0
\(459\) 3.05320i 0.142511i
\(460\) 0 0
\(461\) 13.0662i 0.608555i 0.952584 + 0.304277i \(0.0984149\pi\)
−0.952584 + 0.304277i \(0.901585\pi\)
\(462\) 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
\(467\) 27.7040i 1.28199i 0.767545 + 0.640995i \(0.221478\pi\)
−0.767545 + 0.640995i \(0.778522\pi\)
\(468\) 0 0
\(469\) − 0.172503i − 0.00796544i
\(470\) 0 0
\(471\) −3.93161 −0.181159
\(472\) 0 0
\(473\) 43.7127 2.00991
\(474\) 0 0
\(475\) 5.39635i 0.247602i
\(476\) 0 0
\(477\) − 5.03049i − 0.230331i
\(478\) 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
\(483\) 0.449555i 0.0204555i
\(484\) 0 0
\(485\) 1.02758i 0.0466599i
\(486\) 0 0
\(487\) 9.86632 0.447086 0.223543 0.974694i \(-0.428238\pi\)
0.223543 + 0.974694i \(0.428238\pi\)
\(488\) 0 0
\(489\) 7.68897 0.347707
\(490\) 0 0
\(491\) 0.635767i 0.0286917i 0.999897 + 0.0143459i \(0.00456659\pi\)
−0.999897 + 0.0143459i \(0.995433\pi\)
\(492\) 0 0
\(493\) 9.06651i 0.408335i
\(494\) 0 0
\(495\) −9.65685 −0.434043
\(496\) 0 0
\(497\) −0.0505249 −0.00226635
\(498\) 0 0
\(499\) − 3.82750i − 0.171342i −0.996323 0.0856712i \(-0.972697\pi\)
0.996323 0.0856712i \(-0.0273034\pi\)
\(500\) 0 0
\(501\) − 3.95458i − 0.176678i
\(502\) 0 0
\(503\) 23.6719 1.05548 0.527739 0.849407i \(-0.323040\pi\)
0.527739 + 0.849407i \(0.323040\pi\)
\(504\) 0 0
\(505\) 18.1876 0.809336
\(506\) 0 0
\(507\) − 22.4844i − 0.998565i
\(508\) 0 0
\(509\) 34.7970i 1.54235i 0.636623 + 0.771175i \(0.280331\pi\)
−0.636623 + 0.771175i \(0.719669\pi\)
\(510\) 0 0
\(511\) 0.212818 0.00941453
\(512\) 0 0
\(513\) 3.05320 0.134802
\(514\) 0 0
\(515\) 20.4077i 0.899273i
\(516\) 0 0
\(517\) − 15.1917i − 0.668132i
\(518\) 0 0
\(519\) 22.6011 0.992078
\(520\) 0 0
\(521\) 14.4889 0.634770 0.317385 0.948297i \(-0.397195\pi\)
0.317385 + 0.948297i \(0.397195\pi\)
\(522\) 0 0
\(523\) 27.5742i 1.20574i 0.797841 + 0.602868i \(0.205975\pi\)
−0.797841 + 0.602868i \(0.794025\pi\)
\(524\) 0 0
\(525\) − 0.280920i − 0.0122603i
\(526\) 0 0
\(527\) 12.6981 0.553138
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) − 5.65685i − 0.245487i
\(532\) 0 0
\(533\) 15.5096i 0.671796i
\(534\) 0 0
\(535\) 1.83585 0.0793706
\(536\) 0 0
\(537\) −17.2981 −0.746470
\(538\) 0 0
\(539\) − 37.4619i − 1.61360i
\(540\) 0 0
\(541\) 14.1982i 0.610428i 0.952284 + 0.305214i \(0.0987280\pi\)
−0.952284 + 0.305214i \(0.901272\pi\)
\(542\) 0 0
\(543\) −8.14953 −0.349730
\(544\) 0 0
\(545\) −3.67340 −0.157351
\(546\) 0 0
\(547\) − 10.1807i − 0.435295i −0.976027 0.217648i \(-0.930162\pi\)
0.976027 0.217648i \(-0.0698384\pi\)
\(548\) 0 0
\(549\) 5.18944i 0.221480i
\(550\) 0 0
\(551\) 9.06651 0.386246
\(552\) 0 0
\(553\) 1.54075 0.0655194
\(554\) 0 0
\(555\) 15.2238i 0.646216i
\(556\) 0 0
\(557\) 1.44432i 0.0611979i 0.999532 + 0.0305990i \(0.00974147\pi\)
−0.999532 + 0.0305990i \(0.990259\pi\)
\(558\) 0 0
\(559\) −48.4802 −2.05049
\(560\) 0 0
\(561\) 16.3990 0.692368
\(562\) 0 0
\(563\) − 9.48585i − 0.399781i −0.979818 0.199890i \(-0.935941\pi\)
0.979818 0.199890i \(-0.0640586\pi\)
\(564\) 0 0
\(565\) − 6.35547i − 0.267377i
\(566\) 0 0
\(567\) −0.158942 −0.00667492
\(568\) 0 0
\(569\) −8.98711 −0.376759 −0.188380 0.982096i \(-0.560324\pi\)
−0.188380 + 0.982096i \(0.560324\pi\)
\(570\) 0 0
\(571\) − 12.9706i − 0.542801i −0.962466 0.271401i \(-0.912513\pi\)
0.962466 0.271401i \(-0.0874868\pi\)
\(572\) 0 0
\(573\) 16.1674i 0.675403i
\(574\) 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
\(579\) − 22.1454i − 0.920331i
\(580\) 0 0
\(581\) 0.0260328i 0.00108002i
\(582\) 0 0
\(583\) −27.0192 −1.11902
\(584\) 0 0
\(585\) 10.7101 0.442806
\(586\) 0 0
\(587\) − 2.57988i − 0.106483i −0.998582 0.0532416i \(-0.983045\pi\)
0.998582 0.0532416i \(-0.0169553\pi\)
\(588\) 0 0
\(589\) − 12.6981i − 0.523216i
\(590\) 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
\(595\) − 0.872503i − 0.0357691i
\(596\) 0 0
\(597\) 25.0075i 1.02349i
\(598\) 0 0
\(599\) −27.1632 −1.10986 −0.554930 0.831897i \(-0.687255\pi\)
−0.554930 + 0.831897i \(0.687255\pi\)
\(600\) 0 0
\(601\) 5.33897 0.217781 0.108891 0.994054i \(-0.465270\pi\)
0.108891 + 0.994054i \(0.465270\pi\)
\(602\) 0 0
\(603\) − 1.08532i − 0.0441977i
\(604\) 0 0
\(605\) 32.0906i 1.30467i
\(606\) 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
\(611\) 16.8486i 0.681621i
\(612\) 0 0
\(613\) − 0.617903i − 0.0249569i −0.999922 0.0124784i \(-0.996028\pi\)
0.999922 0.0124784i \(-0.00397211\pi\)
\(614\) 0 0
\(615\) −4.68119 −0.188764
\(616\) 0 0
\(617\) −8.80641 −0.354533 −0.177266 0.984163i \(-0.556725\pi\)
−0.177266 + 0.984163i \(0.556725\pi\)
\(618\) 0 0
\(619\) − 2.72847i − 0.109666i −0.998496 0.0548332i \(-0.982537\pi\)
0.998496 0.0548332i \(-0.0174627\pi\)
\(620\) 0 0
\(621\) 2.82843i 0.113501i
\(622\) 0 0
\(623\) 2.28861 0.0916910
\(624\) 0 0
\(625\) −13.0390 −0.521559
\(626\) 0 0
\(627\) − 16.3990i − 0.654914i
\(628\) 0 0
\(629\) − 25.8528i − 1.03082i
\(630\) 0 0
\(631\) 38.7864 1.54406 0.772030 0.635586i \(-0.219241\pi\)
0.772030 + 0.635586i \(0.219241\pi\)
\(632\) 0 0
\(633\) 26.0559 1.03563
\(634\) 0 0
\(635\) − 2.69315i − 0.106874i
\(636\) 0 0
\(637\) 41.5476i 1.64618i
\(638\) 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
\(643\) − 27.2797i − 1.07581i −0.843006 0.537904i \(-0.819216\pi\)
0.843006 0.537904i \(-0.180784\pi\)
\(644\) 0 0
\(645\) − 14.6325i − 0.576155i
\(646\) 0 0
\(647\) 41.8477 1.64520 0.822601 0.568620i \(-0.192522\pi\)
0.822601 + 0.568620i \(0.192522\pi\)
\(648\) 0 0
\(649\) −30.3835 −1.19266
\(650\) 0 0
\(651\) 0.661029i 0.0259078i
\(652\) 0 0
\(653\) 20.8937i 0.817634i 0.912616 + 0.408817i \(0.134059\pi\)
−0.912616 + 0.408817i \(0.865941\pi\)
\(654\) 0 0
\(655\) 26.5054 1.03565
\(656\) 0 0
\(657\) 1.33897 0.0522382
\(658\) 0 0
\(659\) 3.15142i 0.122762i 0.998114 + 0.0613809i \(0.0195504\pi\)
−0.998114 + 0.0613809i \(0.980450\pi\)
\(660\) 0 0
\(661\) − 25.5527i − 0.993886i −0.867783 0.496943i \(-0.834456\pi\)
0.867783 0.496943i \(-0.165544\pi\)
\(662\) 0 0
\(663\) −18.1876 −0.706346
\(664\) 0 0
\(665\) −0.872503 −0.0338342
\(666\) 0 0
\(667\) 8.39903i 0.325212i
\(668\) 0 0
\(669\) − 18.3465i − 0.709317i
\(670\) 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
\(675\) − 1.76744i − 0.0680287i
\(676\) 0 0
\(677\) − 41.0423i − 1.57738i −0.614789 0.788692i \(-0.710759\pi\)
0.614789 0.788692i \(-0.289241\pi\)
\(678\) 0 0
\(679\) 0.0908404 0.00348613
\(680\) 0 0
\(681\) 0.163788 0.00627639
\(682\) 0 0
\(683\) 26.5784i 1.01699i 0.861064 + 0.508497i \(0.169799\pi\)
−0.861064 + 0.508497i \(0.830201\pi\)
\(684\) 0 0
\(685\) 24.8032i 0.947680i
\(686\) 0 0
\(687\) 4.02756 0.153661
\(688\) 0 0
\(689\) 29.9660 1.14161
\(690\) 0 0
\(691\) − 14.7899i − 0.562633i −0.959615 0.281316i \(-0.909229\pi\)
0.959615 0.281316i \(-0.0907710\pi\)
\(692\) 0 0
\(693\) 0.853690i 0.0324290i
\(694\) 0 0
\(695\) 6.16415 0.233820
\(696\) 0 0
\(697\) 7.94948 0.301108
\(698\) 0 0
\(699\) 11.7211i 0.443332i
\(700\) 0 0
\(701\) − 25.9245i − 0.979155i −0.871960 0.489577i \(-0.837151\pi\)
0.871960 0.489577i \(-0.162849\pi\)
\(702\) 0 0
\(703\) −25.8528 −0.975055
\(704\) 0 0
\(705\) −5.08532 −0.191524
\(706\) 0 0
\(707\) − 1.60782i − 0.0604685i
\(708\) 0 0
\(709\) − 20.6082i − 0.773958i −0.922089 0.386979i \(-0.873519\pi\)
0.922089 0.386979i \(-0.126481\pi\)
\(710\) 0 0
\(711\) 9.69382 0.363547
\(712\) 0 0
\(713\) 11.7633 0.440538
\(714\) 0 0
\(715\) − 57.5247i − 2.15130i
\(716\) 0 0
\(717\) − 13.6517i − 0.509834i
\(718\) 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
\(723\) − 2.13167i − 0.0792777i
\(724\) 0 0
\(725\) − 5.24842i − 0.194921i
\(726\) 0 0
\(727\) −9.23457 −0.342491 −0.171246 0.985228i \(-0.554779\pi\)
−0.171246 + 0.985228i \(0.554779\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.8486i 0.919058i
\(732\) 0 0
\(733\) − 25.8467i − 0.954670i −0.878721 0.477335i \(-0.841603\pi\)
0.878721 0.477335i \(-0.158397\pi\)
\(734\) 0 0
\(735\) −12.5401 −0.462549
\(736\) 0 0
\(737\) −5.82936 −0.214727
\(738\) 0 0
\(739\) 24.0403i 0.884337i 0.896932 + 0.442169i \(0.145791\pi\)
−0.896932 + 0.442169i \(0.854209\pi\)
\(740\) 0 0
\(741\) 18.1876i 0.668137i
\(742\) 0 0
\(743\) −17.8748 −0.655762 −0.327881 0.944719i \(-0.606335\pi\)
−0.327881 + 0.944719i \(0.606335\pi\)
\(744\) 0 0
\(745\) 7.44538 0.272778
\(746\) 0 0
\(747\) 0.163788i 0.00599271i
\(748\) 0 0
\(749\) − 0.162293i − 0.00593007i
\(750\) 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
\(755\) 40.7490i 1.48301i
\(756\) 0 0
\(757\) − 46.3962i − 1.68630i −0.537679 0.843150i \(-0.680699\pi\)
0.537679 0.843150i \(-0.319301\pi\)
\(758\) 0 0
\(759\) 15.1917 0.551425
\(760\) 0 0
\(761\) 10.5531 0.382550 0.191275 0.981536i \(-0.438738\pi\)
0.191275 + 0.981536i \(0.438738\pi\)
\(762\) 0 0
\(763\) 0.324738i 0.0117563i
\(764\) 0 0
\(765\) − 5.48946i − 0.198472i
\(766\) 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
\(771\) 15.0853i 0.543285i
\(772\) 0 0
\(773\) 27.4212i 0.986271i 0.869952 + 0.493136i \(0.164149\pi\)
−0.869952 + 0.493136i \(0.835851\pi\)
\(774\) 0 0
\(775\) −7.35067 −0.264044
\(776\) 0 0
\(777\) 1.34583 0.0482812
\(778\) 0 0
\(779\) − 7.94948i − 0.284820i
\(780\) 0 0
\(781\) 1.70738i 0.0610948i
\(782\) 0 0
\(783\) −2.96951 −0.106121
\(784\) 0 0
\(785\) 7.06877 0.252295
\(786\) 0 0
\(787\) 9.46058i 0.337233i 0.985682 + 0.168617i \(0.0539300\pi\)
−0.985682 + 0.168617i \(0.946070\pi\)
\(788\) 0 0
\(789\) − 26.1706i − 0.931700i
\(790\) 0 0
\(791\) −0.561839 −0.0199767
\(792\) 0 0
\(793\) −30.9128 −1.09775
\(794\) 0 0
\(795\) 9.04449i 0.320775i
\(796\) 0 0
\(797\) − 19.1791i − 0.679359i −0.940541 0.339680i \(-0.889681\pi\)
0.940541 0.339680i \(-0.110319\pi\)
\(798\) 0 0
\(799\) 8.63577 0.305511
\(800\) 0 0
\(801\) 14.3990 0.508765
\(802\) 0 0
\(803\) − 7.19173i − 0.253791i
\(804\) 0 0
\(805\) − 0.808269i − 0.0284878i
\(806\) 0 0
\(807\) −12.1580 −0.427982
\(808\) 0 0
\(809\) 43.1578 1.51735 0.758673 0.651472i \(-0.225848\pi\)
0.758673 + 0.651472i \(0.225848\pi\)
\(810\) 0 0
\(811\) − 3.87518i − 0.136076i −0.997683 0.0680380i \(-0.978326\pi\)
0.997683 0.0680380i \(-0.0216739\pi\)
\(812\) 0 0
\(813\) 10.6644i 0.374016i
\(814\) 0 0
\(815\) −13.8243 −0.484242
\(816\) 0 0
\(817\) 24.8486 0.869342
\(818\) 0 0
\(819\) − 0.946795i − 0.0330837i
\(820\) 0 0
\(821\) 5.62084i 0.196169i 0.995178 + 0.0980843i \(0.0312715\pi\)
−0.995178 + 0.0980843i \(0.968729\pi\)
\(822\) 0 0
\(823\) −38.5255 −1.34291 −0.671457 0.741043i \(-0.734331\pi\)
−0.671457 + 0.741043i \(0.734331\pi\)
\(824\) 0 0
\(825\) −9.49307 −0.330506
\(826\) 0 0
\(827\) 4.23674i 0.147326i 0.997283 + 0.0736629i \(0.0234689\pi\)
−0.997283 + 0.0736629i \(0.976531\pi\)
\(828\) 0 0
\(829\) 35.3128i 1.22646i 0.789903 + 0.613231i \(0.210131\pi\)
−0.789903 + 0.613231i \(0.789869\pi\)
\(830\) 0 0
\(831\) 3.76421 0.130579
\(832\) 0 0
\(833\) 21.2953 0.737838
\(834\) 0 0
\(835\) 7.11007i 0.246054i
\(836\) 0 0
\(837\) 4.15894i 0.143754i
\(838\) 0 0
\(839\) −39.6005 −1.36716 −0.683580 0.729876i \(-0.739578\pi\)
−0.683580 + 0.729876i \(0.739578\pi\)
\(840\) 0 0
\(841\) 20.1820 0.695932
\(842\) 0 0
\(843\) − 10.4496i − 0.359902i
\(844\) 0 0
\(845\) 40.4253i 1.39067i
\(846\) 0 0
\(847\) 2.83688 0.0974765
\(848\) 0 0
\(849\) 17.6569 0.605982
\(850\) 0 0
\(851\) − 23.9495i − 0.820977i
\(852\) 0 0
\(853\) 10.8683i 0.372124i 0.982538 + 0.186062i \(0.0595725\pi\)
−0.982538 + 0.186062i \(0.940428\pi\)
\(854\) 0 0
\(855\) −5.48946 −0.187735
\(856\) 0 0
\(857\) 34.0082 1.16170 0.580849 0.814011i \(-0.302721\pi\)
0.580849 + 0.814011i \(0.302721\pi\)
\(858\) 0 0
\(859\) 12.9973i 0.443463i 0.975108 + 0.221731i \(0.0711708\pi\)
−0.975108 + 0.221731i \(0.928829\pi\)
\(860\) 0 0
\(861\) 0.413828i 0.0141032i
\(862\) 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
\(867\) − 7.67794i − 0.260757i
\(868\) 0 0
\(869\) − 52.0663i − 1.76623i
\(870\) 0 0
\(871\) 6.46512 0.219062
\(872\) 0 0
\(873\) 0.571533 0.0193435
\(874\) 0 0
\(875\) 1.93391i 0.0653780i
\(876\) 0 0
\(877\) 45.2150i 1.52680i 0.645926 + 0.763400i \(0.276472\pi\)
−0.645926 + 0.763400i \(0.723528\pi\)
\(878\) 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
\(883\) 48.9366i 1.64685i 0.567427 + 0.823424i \(0.307939\pi\)
−0.567427 + 0.823424i \(0.692061\pi\)
\(884\) 0 0
\(885\) 10.1706i 0.341882i
\(886\) 0 0
\(887\) 22.9284 0.769860 0.384930 0.922946i \(-0.374226\pi\)
0.384930 + 0.922946i \(0.374226\pi\)
\(888\) 0 0
\(889\) −0.238081 −0.00798497
\(890\) 0 0
\(891\) 5.37109i 0.179938i
\(892\) 0 0
\(893\) − 8.63577i − 0.288985i
\(894\) 0 0
\(895\) 31.1009 1.03959
\(896\) 0 0
\(897\) −16.8486 −0.562558
\(898\) 0 0
\(899\) 12.3500i 0.411896i
\(900\) 0 0
\(901\) − 15.3591i − 0.511687i
\(902\) 0 0
\(903\) −1.29355 −0.0430467
\(904\) 0 0
\(905\) 14.6523 0.487059
\(906\) 0 0
\(907\) 23.1680i 0.769280i 0.923067 + 0.384640i \(0.125674\pi\)
−0.923067 + 0.384640i \(0.874326\pi\)
\(908\) 0 0
\(909\) − 10.1158i − 0.335520i
\(910\) 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
\(915\) − 9.33026i − 0.308449i
\(916\) 0 0
\(917\) − 2.34315i − 0.0773775i
\(918\) 0 0
\(919\) −6.86029 −0.226300 −0.113150 0.993578i \(-0.536094\pi\)
−0.113150 + 0.993578i \(0.536094\pi\)
\(920\) 0 0
\(921\) 21.2981 0.701798
\(922\) 0 0
\(923\) − 1.89359i − 0.0623283i
\(924\) 0 0
\(925\) 14.9656i 0.492067i
\(926\) 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
\(931\) − 21.2953i − 0.697925i
\(932\) 0 0
\(933\) 1.77883i 0.0582363i
\(934\) 0 0
\(935\) −29.4844 −0.964241
\(936\) 0 0
\(937\) 34.7669 1.13579 0.567893 0.823102i \(-0.307759\pi\)
0.567893 + 0.823102i \(0.307759\pi\)
\(938\) 0 0
\(939\) 2.70320i 0.0882157i
\(940\) 0 0
\(941\) − 52.7024i − 1.71805i −0.511934 0.859025i \(-0.671071\pi\)
0.511934 0.859025i \(-0.328929\pi\)
\(942\) 0 0
\(943\) 7.36423 0.239812
\(944\) 0 0
\(945\) 0.285766 0.00929598
\(946\) 0 0
\(947\) − 26.1972i − 0.851296i −0.904889 0.425648i \(-0.860046\pi\)
0.904889 0.425648i \(-0.139954\pi\)
\(948\) 0 0
\(949\) 7.97608i 0.258915i
\(950\) 0 0
\(951\) 22.0653 0.715516
\(952\) 0 0
\(953\) 11.1752 0.362000 0.181000 0.983483i \(-0.442067\pi\)
0.181000 + 0.983483i \(0.442067\pi\)
\(954\) 0 0
\(955\) − 29.0679i − 0.940615i
\(956\) 0 0
\(957\) 15.9495i 0.515573i
\(958\) 0 0
\(959\) 2.19266 0.0708047
\(960\) 0 0
\(961\) −13.7032 −0.442039
\(962\) 0 0
\(963\) − 1.02109i − 0.0329041i
\(964\) 0 0
\(965\) 39.8159i 1.28172i
\(966\) 0 0
\(967\) −12.8452 −0.413075 −0.206537 0.978439i \(-0.566220\pi\)
−0.206537 + 0.978439i \(0.566220\pi\)
\(968\) 0 0
\(969\) 9.32206 0.299468
\(970\) 0 0
\(971\) − 8.40774i − 0.269817i −0.990858 0.134909i \(-0.956926\pi\)
0.990858 0.134909i \(-0.0430741\pi\)
\(972\) 0 0
\(973\) − 0.544926i − 0.0174695i
\(974\) 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
\(979\) − 77.3385i − 2.47175i
\(980\) 0 0
\(981\) 2.04313i 0.0652320i
\(982\) 0 0
\(983\) −13.9202 −0.443985 −0.221993 0.975048i \(-0.571256\pi\)
−0.221993 + 0.975048i \(0.571256\pi\)
\(984\) 0 0
\(985\) 36.3582 1.15847
\(986\) 0 0
\(987\) 0.449555i 0.0143095i
\(988\) 0 0
\(989\) 23.0192i 0.731969i
\(990\) 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
\(995\) − 44.9618i − 1.42539i
\(996\) 0 0
\(997\) − 38.6427i − 1.22383i −0.790925 0.611913i \(-0.790400\pi\)
0.790925 0.611913i \(-0.209600\pi\)
\(998\) 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.d.i.1537.5 8
4.3 odd 2 3072.2.d.f.1537.1 8
8.3 odd 2 3072.2.d.f.1537.8 8
8.5 even 2 inner 3072.2.d.i.1537.4 8
16.3 odd 4 3072.2.a.t.1.1 4
16.5 even 4 3072.2.a.o.1.4 4
16.11 odd 4 3072.2.a.i.1.4 4
16.13 even 4 3072.2.a.n.1.1 4
32.3 odd 8 48.2.j.a.37.1 yes 8
32.5 even 8 192.2.j.a.145.2 8
32.11 odd 8 384.2.j.b.289.1 8
32.13 even 8 384.2.j.a.97.3 8
32.19 odd 8 384.2.j.b.97.1 8
32.21 even 8 384.2.j.a.289.3 8
32.27 odd 8 48.2.j.a.13.1 8
32.29 even 8 192.2.j.a.49.2 8
48.5 odd 4 9216.2.a.bn.1.1 4
48.11 even 4 9216.2.a.bo.1.1 4
48.29 odd 4 9216.2.a.x.1.4 4
48.35 even 4 9216.2.a.y.1.4 4
96.5 odd 8 576.2.k.b.145.2 8
96.11 even 8 1152.2.k.c.289.3 8
96.29 odd 8 576.2.k.b.433.2 8
96.35 even 8 144.2.k.b.37.4 8
96.53 odd 8 1152.2.k.f.289.3 8
96.59 even 8 144.2.k.b.109.4 8
96.77 odd 8 1152.2.k.f.865.3 8
96.83 even 8 1152.2.k.c.865.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.1 8 32.27 odd 8
48.2.j.a.37.1 yes 8 32.3 odd 8
144.2.k.b.37.4 8 96.35 even 8
144.2.k.b.109.4 8 96.59 even 8
192.2.j.a.49.2 8 32.29 even 8
192.2.j.a.145.2 8 32.5 even 8
384.2.j.a.97.3 8 32.13 even 8
384.2.j.a.289.3 8 32.21 even 8
384.2.j.b.97.1 8 32.19 odd 8
384.2.j.b.289.1 8 32.11 odd 8
576.2.k.b.145.2 8 96.5 odd 8
576.2.k.b.433.2 8 96.29 odd 8
1152.2.k.c.289.3 8 96.11 even 8
1152.2.k.c.865.3 8 96.83 even 8
1152.2.k.f.289.3 8 96.53 odd 8
1152.2.k.f.865.3 8 96.77 odd 8
3072.2.a.i.1.4 4 16.11 odd 4
3072.2.a.n.1.1 4 16.13 even 4
3072.2.a.o.1.4 4 16.5 even 4
3072.2.a.t.1.1 4 16.3 odd 4
3072.2.d.f.1537.1 8 4.3 odd 2
3072.2.d.f.1537.8 8 8.3 odd 2
3072.2.d.i.1537.4 8 8.5 even 2 inner
3072.2.d.i.1537.5 8 1.1 even 1 trivial
9216.2.a.x.1.4 4 48.29 odd 4
9216.2.a.y.1.4 4 48.35 even 4
9216.2.a.bn.1.1 4 48.5 odd 4
9216.2.a.bo.1.1 4 48.11 even 4