Properties

Label 3072.2.a.d.1.2
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $0$
Dimension $2$
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.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 768)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3072.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +2.82843 q^{5} -4.24264 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +2.82843 q^{5} -4.24264 q^{7} +1.00000 q^{9} +4.00000 q^{11} +4.24264 q^{13} -2.82843 q^{15} +6.00000 q^{17} +2.00000 q^{19} +4.24264 q^{21} -2.82843 q^{23} +3.00000 q^{25} -1.00000 q^{27} -5.65685 q^{29} +4.24264 q^{31} -4.00000 q^{33} -12.0000 q^{35} +4.24264 q^{37} -4.24264 q^{39} -10.0000 q^{41} -6.00000 q^{43} +2.82843 q^{45} +2.82843 q^{47} +11.0000 q^{49} -6.00000 q^{51} -5.65685 q^{53} +11.3137 q^{55} -2.00000 q^{57} -4.24264 q^{61} -4.24264 q^{63} +12.0000 q^{65} +4.00000 q^{67} +2.82843 q^{69} -2.82843 q^{71} +16.0000 q^{73} -3.00000 q^{75} -16.9706 q^{77} +4.24264 q^{79} +1.00000 q^{81} +16.0000 q^{83} +16.9706 q^{85} +5.65685 q^{87} +14.0000 q^{89} -18.0000 q^{91} -4.24264 q^{93} +5.65685 q^{95} -4.00000 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 8q^{11} + 12q^{17} + 4q^{19} + 6q^{25} - 2q^{27} - 8q^{33} - 24q^{35} - 20q^{41} - 12q^{43} + 22q^{49} - 12q^{51} - 4q^{57} + 24q^{65} + 8q^{67} + 32q^{73} - 6q^{75} + 2q^{81} + 32q^{83} + 28q^{89} - 36q^{91} - 8q^{97} + 8q^{99} + O(q^{100}) \)

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
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) −2.82843 −0.730297
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 4.24264 0.925820
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.65685 −1.05045 −0.525226 0.850963i \(-0.676019\pi\)
−0.525226 + 0.850963i \(0.676019\pi\)
\(30\) 0 0
\(31\) 4.24264 0.762001 0.381000 0.924575i \(-0.375580\pi\)
0.381000 + 0.924575i \(0.375580\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) −12.0000 −2.02837
\(36\) 0 0
\(37\) 4.24264 0.697486 0.348743 0.937218i \(-0.386609\pi\)
0.348743 + 0.937218i \(0.386609\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) −5.65685 −0.777029 −0.388514 0.921443i \(-0.627012\pi\)
−0.388514 + 0.921443i \(0.627012\pi\)
\(54\) 0 0
\(55\) 11.3137 1.52554
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −4.24264 −0.543214 −0.271607 0.962408i \(-0.587555\pi\)
−0.271607 + 0.962408i \(0.587555\pi\)
\(62\) 0 0
\(63\) −4.24264 −0.534522
\(64\) 0 0
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) −2.82843 −0.335673 −0.167836 0.985815i \(-0.553678\pi\)
−0.167836 + 0.985815i \(0.553678\pi\)
\(72\) 0 0
\(73\) 16.0000 1.87266 0.936329 0.351123i \(-0.114200\pi\)
0.936329 + 0.351123i \(0.114200\pi\)
\(74\) 0 0
\(75\) −3.00000 −0.346410
\(76\) 0 0
\(77\) −16.9706 −1.93398
\(78\) 0 0
\(79\) 4.24264 0.477334 0.238667 0.971101i \(-0.423290\pi\)
0.238667 + 0.971101i \(0.423290\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.0000 1.75623 0.878114 0.478451i \(-0.158802\pi\)
0.878114 + 0.478451i \(0.158802\pi\)
\(84\) 0 0
\(85\) 16.9706 1.84072
\(86\) 0 0
\(87\) 5.65685 0.606478
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) 0 0
\(93\) −4.24264 −0.439941
\(94\) 0 0
\(95\) 5.65685 0.580381
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 11.3137 1.12576 0.562878 0.826540i \(-0.309694\pi\)
0.562878 + 0.826540i \(0.309694\pi\)
\(102\) 0 0
\(103\) 18.3848 1.81151 0.905753 0.423806i \(-0.139306\pi\)
0.905753 + 0.423806i \(0.139306\pi\)
\(104\) 0 0
\(105\) 12.0000 1.17108
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −9.89949 −0.948200 −0.474100 0.880471i \(-0.657226\pi\)
−0.474100 + 0.880471i \(0.657226\pi\)
\(110\) 0 0
\(111\) −4.24264 −0.402694
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 4.24264 0.392232
\(118\) 0 0
\(119\) −25.4558 −2.33353
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 10.0000 0.901670
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 4.24264 0.376473 0.188237 0.982124i \(-0.439723\pi\)
0.188237 + 0.982124i \(0.439723\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −8.48528 −0.735767
\(134\) 0 0
\(135\) −2.82843 −0.243432
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) −2.82843 −0.238197
\(142\) 0 0
\(143\) 16.9706 1.41915
\(144\) 0 0
\(145\) −16.0000 −1.32873
\(146\) 0 0
\(147\) −11.0000 −0.907265
\(148\) 0 0
\(149\) −8.48528 −0.695141 −0.347571 0.937654i \(-0.612993\pi\)
−0.347571 + 0.937654i \(0.612993\pi\)
\(150\) 0 0
\(151\) 1.41421 0.115087 0.0575435 0.998343i \(-0.481673\pi\)
0.0575435 + 0.998343i \(0.481673\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) −7.07107 −0.564333 −0.282166 0.959366i \(-0.591053\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(158\) 0 0
\(159\) 5.65685 0.448618
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) −11.3137 −0.880771
\(166\) 0 0
\(167\) 11.3137 0.875481 0.437741 0.899101i \(-0.355779\pi\)
0.437741 + 0.899101i \(0.355779\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) −12.7279 −0.962140
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 1.41421 0.105118 0.0525588 0.998618i \(-0.483262\pi\)
0.0525588 + 0.998618i \(0.483262\pi\)
\(182\) 0 0
\(183\) 4.24264 0.313625
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 24.0000 1.75505
\(188\) 0 0
\(189\) 4.24264 0.308607
\(190\) 0 0
\(191\) −22.6274 −1.63726 −0.818631 0.574320i \(-0.805267\pi\)
−0.818631 + 0.574320i \(0.805267\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) −12.0000 −0.859338
\(196\) 0 0
\(197\) 5.65685 0.403034 0.201517 0.979485i \(-0.435413\pi\)
0.201517 + 0.979485i \(0.435413\pi\)
\(198\) 0 0
\(199\) −7.07107 −0.501255 −0.250627 0.968084i \(-0.580637\pi\)
−0.250627 + 0.968084i \(0.580637\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 24.0000 1.68447
\(204\) 0 0
\(205\) −28.2843 −1.97546
\(206\) 0 0
\(207\) −2.82843 −0.196589
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 0 0
\(213\) 2.82843 0.193801
\(214\) 0 0
\(215\) −16.9706 −1.15738
\(216\) 0 0
\(217\) −18.0000 −1.22192
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) 25.4558 1.71235
\(222\) 0 0
\(223\) −21.2132 −1.42054 −0.710271 0.703929i \(-0.751427\pi\)
−0.710271 + 0.703929i \(0.751427\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 15.5563 1.02799 0.513996 0.857792i \(-0.328165\pi\)
0.513996 + 0.857792i \(0.328165\pi\)
\(230\) 0 0
\(231\) 16.9706 1.11658
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) −4.24264 −0.275589
\(238\) 0 0
\(239\) −11.3137 −0.731823 −0.365911 0.930650i \(-0.619243\pi\)
−0.365911 + 0.930650i \(0.619243\pi\)
\(240\) 0 0
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 31.1127 1.98772
\(246\) 0 0
\(247\) 8.48528 0.539906
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) −11.3137 −0.711287
\(254\) 0 0
\(255\) −16.9706 −1.06274
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −18.0000 −1.11847
\(260\) 0 0
\(261\) −5.65685 −0.350150
\(262\) 0 0
\(263\) −22.6274 −1.39527 −0.697633 0.716455i \(-0.745763\pi\)
−0.697633 + 0.716455i \(0.745763\pi\)
\(264\) 0 0
\(265\) −16.0000 −0.982872
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 1.41421 0.0859074 0.0429537 0.999077i \(-0.486323\pi\)
0.0429537 + 0.999077i \(0.486323\pi\)
\(272\) 0 0
\(273\) 18.0000 1.08941
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −1.41421 −0.0849719 −0.0424859 0.999097i \(-0.513528\pi\)
−0.0424859 + 0.999097i \(0.513528\pi\)
\(278\) 0 0
\(279\) 4.24264 0.254000
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) −5.65685 −0.335083
\(286\) 0 0
\(287\) 42.4264 2.50435
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) 5.65685 0.330477 0.165238 0.986254i \(-0.447161\pi\)
0.165238 + 0.986254i \(0.447161\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 25.4558 1.46725
\(302\) 0 0
\(303\) −11.3137 −0.649956
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −18.3848 −1.04587
\(310\) 0 0
\(311\) 28.2843 1.60385 0.801927 0.597422i \(-0.203808\pi\)
0.801927 + 0.597422i \(0.203808\pi\)
\(312\) 0 0
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) 0 0
\(315\) −12.0000 −0.676123
\(316\) 0 0
\(317\) −16.9706 −0.953162 −0.476581 0.879131i \(-0.658124\pi\)
−0.476581 + 0.879131i \(0.658124\pi\)
\(318\) 0 0
\(319\) −22.6274 −1.26689
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 12.7279 0.706018
\(326\) 0 0
\(327\) 9.89949 0.547443
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 4.24264 0.232495
\(334\) 0 0
\(335\) 11.3137 0.618134
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 16.9706 0.919007
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 25.4558 1.34727
\(358\) 0 0
\(359\) −2.82843 −0.149279 −0.0746393 0.997211i \(-0.523781\pi\)
−0.0746393 + 0.997211i \(0.523781\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 45.2548 2.36875
\(366\) 0 0
\(367\) −18.3848 −0.959678 −0.479839 0.877357i \(-0.659305\pi\)
−0.479839 + 0.877357i \(0.659305\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 0 0
\(373\) −18.3848 −0.951928 −0.475964 0.879465i \(-0.657901\pi\)
−0.475964 + 0.879465i \(0.657901\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −38.0000 −1.95193 −0.975964 0.217930i \(-0.930070\pi\)
−0.975964 + 0.217930i \(0.930070\pi\)
\(380\) 0 0
\(381\) −4.24264 −0.217357
\(382\) 0 0
\(383\) 28.2843 1.44526 0.722629 0.691236i \(-0.242933\pi\)
0.722629 + 0.691236i \(0.242933\pi\)
\(384\) 0 0
\(385\) −48.0000 −2.44631
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) −2.82843 −0.143407 −0.0717035 0.997426i \(-0.522844\pi\)
−0.0717035 + 0.997426i \(0.522844\pi\)
\(390\) 0 0
\(391\) −16.9706 −0.858238
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 32.5269 1.63248 0.816239 0.577714i \(-0.196055\pi\)
0.816239 + 0.577714i \(0.196055\pi\)
\(398\) 0 0
\(399\) 8.48528 0.424795
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 18.0000 0.896644
\(404\) 0 0
\(405\) 2.82843 0.140546
\(406\) 0 0
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) −28.0000 −1.38451 −0.692255 0.721653i \(-0.743383\pi\)
−0.692255 + 0.721653i \(0.743383\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 45.2548 2.22147
\(416\) 0 0
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −35.3553 −1.72311 −0.861557 0.507661i \(-0.830510\pi\)
−0.861557 + 0.507661i \(0.830510\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) 18.0000 0.873128
\(426\) 0 0
\(427\) 18.0000 0.871081
\(428\) 0 0
\(429\) −16.9706 −0.819346
\(430\) 0 0
\(431\) 36.7696 1.77113 0.885564 0.464518i \(-0.153773\pi\)
0.885564 + 0.464518i \(0.153773\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) 16.0000 0.767141
\(436\) 0 0
\(437\) −5.65685 −0.270604
\(438\) 0 0
\(439\) −1.41421 −0.0674967 −0.0337484 0.999430i \(-0.510744\pi\)
−0.0337484 + 0.999430i \(0.510744\pi\)
\(440\) 0 0
\(441\) 11.0000 0.523810
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) 39.5980 1.87712
\(446\) 0 0
\(447\) 8.48528 0.401340
\(448\) 0 0
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) 0 0
\(453\) −1.41421 −0.0664455
\(454\) 0 0
\(455\) −50.9117 −2.38678
\(456\) 0 0
\(457\) 12.0000 0.561336 0.280668 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 8.48528 0.395199 0.197599 0.980283i \(-0.436685\pi\)
0.197599 + 0.980283i \(0.436685\pi\)
\(462\) 0 0
\(463\) −12.7279 −0.591517 −0.295758 0.955263i \(-0.595572\pi\)
−0.295758 + 0.955263i \(0.595572\pi\)
\(464\) 0 0
\(465\) −12.0000 −0.556487
\(466\) 0 0
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) −16.9706 −0.783628
\(470\) 0 0
\(471\) 7.07107 0.325818
\(472\) 0 0
\(473\) −24.0000 −1.10352
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −5.65685 −0.259010
\(478\) 0 0
\(479\) 25.4558 1.16311 0.581554 0.813508i \(-0.302445\pi\)
0.581554 + 0.813508i \(0.302445\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) −11.3137 −0.513729
\(486\) 0 0
\(487\) 43.8406 1.98661 0.993304 0.115529i \(-0.0368564\pi\)
0.993304 + 0.115529i \(0.0368564\pi\)
\(488\) 0 0
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) −33.9411 −1.52863
\(494\) 0 0
\(495\) 11.3137 0.508513
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −11.3137 −0.505459
\(502\) 0 0
\(503\) 14.1421 0.630567 0.315283 0.948998i \(-0.397900\pi\)
0.315283 + 0.948998i \(0.397900\pi\)
\(504\) 0 0
\(505\) 32.0000 1.42398
\(506\) 0 0
\(507\) −5.00000 −0.222058
\(508\) 0 0
\(509\) −22.6274 −1.00294 −0.501471 0.865174i \(-0.667208\pi\)
−0.501471 + 0.865174i \(0.667208\pi\)
\(510\) 0 0
\(511\) −67.8823 −3.00293
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 52.0000 2.29139
\(516\) 0 0
\(517\) 11.3137 0.497576
\(518\) 0 0
\(519\) 2.82843 0.124154
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) 0 0
\(525\) 12.7279 0.555492
\(526\) 0 0
\(527\) 25.4558 1.10887
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −42.4264 −1.83769
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 44.0000 1.89521
\(540\) 0 0
\(541\) −41.0122 −1.76325 −0.881626 0.471949i \(-0.843551\pi\)
−0.881626 + 0.471949i \(0.843551\pi\)
\(542\) 0 0
\(543\) −1.41421 −0.0606897
\(544\) 0 0
\(545\) −28.0000 −1.19939
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 0 0
\(549\) −4.24264 −0.181071
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) −18.0000 −0.765438
\(554\) 0 0
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) −19.7990 −0.838910 −0.419455 0.907776i \(-0.637779\pi\)
−0.419455 + 0.907776i \(0.637779\pi\)
\(558\) 0 0
\(559\) −25.4558 −1.07667
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −16.9706 −0.713957
\(566\) 0 0
\(567\) −4.24264 −0.178174
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 22.6274 0.945274
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 0 0
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) −67.8823 −2.81623
\(582\) 0 0
\(583\) −22.6274 −0.937132
\(584\) 0 0
\(585\) 12.0000 0.496139
\(586\) 0 0
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) 8.48528 0.349630
\(590\) 0 0
\(591\) −5.65685 −0.232692
\(592\) 0 0
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −72.0000 −2.95171
\(596\) 0 0
\(597\) 7.07107 0.289400
\(598\) 0 0
\(599\) −48.0833 −1.96463 −0.982314 0.187239i \(-0.940046\pi\)
−0.982314 + 0.187239i \(0.940046\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 14.1421 0.574960
\(606\) 0 0
\(607\) 4.24264 0.172203 0.0861017 0.996286i \(-0.472559\pi\)
0.0861017 + 0.996286i \(0.472559\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −18.3848 −0.742554 −0.371277 0.928522i \(-0.621080\pi\)
−0.371277 + 0.928522i \(0.621080\pi\)
\(614\) 0 0
\(615\) 28.2843 1.14053
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) −36.0000 −1.44696 −0.723481 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 0 0
\(621\) 2.82843 0.113501
\(622\) 0 0
\(623\) −59.3970 −2.37969
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) 25.4558 1.01499
\(630\) 0 0
\(631\) −35.3553 −1.40747 −0.703737 0.710461i \(-0.748487\pi\)
−0.703737 + 0.710461i \(0.748487\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 46.6690 1.84909
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) 16.9706 0.668215
\(646\) 0 0
\(647\) −14.1421 −0.555985 −0.277992 0.960583i \(-0.589669\pi\)
−0.277992 + 0.960583i \(0.589669\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) 0 0
\(653\) −8.48528 −0.332055 −0.166027 0.986121i \(-0.553094\pi\)
−0.166027 + 0.986121i \(0.553094\pi\)
\(654\) 0 0
\(655\) 56.5685 2.21032
\(656\) 0 0
\(657\) 16.0000 0.624219
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 12.7279 0.495059 0.247529 0.968880i \(-0.420381\pi\)
0.247529 + 0.968880i \(0.420381\pi\)
\(662\) 0 0
\(663\) −25.4558 −0.988623
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 0 0
\(669\) 21.2132 0.820150
\(670\) 0 0
\(671\) −16.9706 −0.655141
\(672\) 0 0
\(673\) −40.0000 −1.54189 −0.770943 0.636904i \(-0.780215\pi\)
−0.770943 + 0.636904i \(0.780215\pi\)
\(674\) 0 0
\(675\) −3.00000 −0.115470
\(676\) 0 0
\(677\) −36.7696 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(678\) 0 0
\(679\) 16.9706 0.651270
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 28.2843 1.08069
\(686\) 0 0
\(687\) −15.5563 −0.593512
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −22.0000 −0.836919 −0.418460 0.908235i \(-0.637430\pi\)
−0.418460 + 0.908235i \(0.637430\pi\)
\(692\) 0 0
\(693\) −16.9706 −0.644658
\(694\) 0 0
\(695\) −33.9411 −1.28746
\(696\) 0 0
\(697\) −60.0000 −2.27266
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 5.65685 0.213656 0.106828 0.994277i \(-0.465931\pi\)
0.106828 + 0.994277i \(0.465931\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) −48.0000 −1.80523
\(708\) 0 0
\(709\) 18.3848 0.690455 0.345227 0.938519i \(-0.387802\pi\)
0.345227 + 0.938519i \(0.387802\pi\)
\(710\) 0 0
\(711\) 4.24264 0.159111
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 48.0000 1.79510
\(716\) 0 0
\(717\) 11.3137 0.422518
\(718\) 0 0
\(719\) −8.48528 −0.316448 −0.158224 0.987403i \(-0.550577\pi\)
−0.158224 + 0.987403i \(0.550577\pi\)
\(720\) 0 0
\(721\) −78.0000 −2.90487
\(722\) 0 0
\(723\) −18.0000 −0.669427
\(724\) 0 0
\(725\) −16.9706 −0.630271
\(726\) 0 0
\(727\) 26.8701 0.996555 0.498278 0.867018i \(-0.333966\pi\)
0.498278 + 0.867018i \(0.333966\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 0 0
\(733\) 35.3553 1.30588 0.652940 0.757410i \(-0.273536\pi\)
0.652940 + 0.757410i \(0.273536\pi\)
\(734\) 0 0
\(735\) −31.1127 −1.14761
\(736\) 0 0
\(737\) 16.0000 0.589368
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) −8.48528 −0.311715
\(742\) 0 0
\(743\) 22.6274 0.830119 0.415060 0.909794i \(-0.363761\pi\)
0.415060 + 0.909794i \(0.363761\pi\)
\(744\) 0 0
\(745\) −24.0000 −0.879292
\(746\) 0 0
\(747\) 16.0000 0.585409
\(748\) 0 0
\(749\) −16.9706 −0.620091
\(750\) 0 0
\(751\) 12.7279 0.464448 0.232224 0.972662i \(-0.425400\pi\)
0.232224 + 0.972662i \(0.425400\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) 0 0
\(755\) 4.00000 0.145575
\(756\) 0 0
\(757\) −38.1838 −1.38781 −0.693906 0.720065i \(-0.744112\pi\)
−0.693906 + 0.720065i \(0.744112\pi\)
\(758\) 0 0
\(759\) 11.3137 0.410662
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 0 0
\(763\) 42.0000 1.52050
\(764\) 0 0
\(765\) 16.9706 0.613572
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 0 0
\(773\) −5.65685 −0.203463 −0.101731 0.994812i \(-0.532438\pi\)
−0.101731 + 0.994812i \(0.532438\pi\)
\(774\) 0 0
\(775\) 12.7279 0.457200
\(776\) 0 0
\(777\) 18.0000 0.645746
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) 0 0
\(783\) 5.65685 0.202159
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 38.0000 1.35455 0.677277 0.735728i \(-0.263160\pi\)
0.677277 + 0.735728i \(0.263160\pi\)
\(788\) 0 0
\(789\) 22.6274 0.805557
\(790\) 0 0
\(791\) 25.4558 0.905106
\(792\) 0 0
\(793\) −18.0000 −0.639199
\(794\) 0 0
\(795\) 16.0000 0.567462
\(796\) 0 0
\(797\) 31.1127 1.10207 0.551034 0.834483i \(-0.314233\pi\)
0.551034 + 0.834483i \(0.314233\pi\)
\(798\) 0 0
\(799\) 16.9706 0.600375
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) 64.0000 2.25851
\(804\) 0 0
\(805\) 33.9411 1.19627
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −1.41421 −0.0495986
\(814\) 0 0
\(815\) 28.2843 0.990755
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) −18.0000 −0.628971
\(820\) 0 0
\(821\) 28.2843 0.987128 0.493564 0.869710i \(-0.335694\pi\)
0.493564 + 0.869710i \(0.335694\pi\)
\(822\) 0 0
\(823\) 4.24264 0.147889 0.0739446 0.997262i \(-0.476441\pi\)
0.0739446 + 0.997262i \(0.476441\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) −21.2132 −0.736765 −0.368383 0.929674i \(-0.620088\pi\)
−0.368383 + 0.929674i \(0.620088\pi\)
\(830\) 0 0
\(831\) 1.41421 0.0490585
\(832\) 0 0
\(833\) 66.0000 2.28676
\(834\) 0 0
\(835\) 32.0000 1.10741
\(836\) 0 0
\(837\) −4.24264 −0.146647
\(838\) 0 0
\(839\) −31.1127 −1.07413 −0.537065 0.843541i \(-0.680467\pi\)
−0.537065 + 0.843541i \(0.680467\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 0 0
\(845\) 14.1421 0.486504
\(846\) 0 0
\(847\) −21.2132 −0.728894
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 21.2132 0.726326 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(854\) 0 0
\(855\) 5.65685 0.193460
\(856\) 0 0
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) −10.0000 −0.341196 −0.170598 0.985341i \(-0.554570\pi\)
−0.170598 + 0.985341i \(0.554570\pi\)
\(860\) 0 0
\(861\) −42.4264 −1.44589
\(862\) 0 0
\(863\) 28.2843 0.962808 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) −19.0000 −0.645274
\(868\) 0 0
\(869\) 16.9706 0.575687
\(870\) 0 0
\(871\) 16.9706 0.575026
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) 0 0
\(877\) −15.5563 −0.525301 −0.262650 0.964891i \(-0.584597\pi\)
−0.262650 + 0.964891i \(0.584597\pi\)
\(878\) 0 0
\(879\) −5.65685 −0.190801
\(880\) 0 0
\(881\) 58.0000 1.95407 0.977035 0.213080i \(-0.0683494\pi\)
0.977035 + 0.213080i \(0.0683494\pi\)
\(882\) 0 0
\(883\) 34.0000 1.14419 0.572096 0.820187i \(-0.306131\pi\)
0.572096 + 0.820187i \(0.306131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.9706 −0.569816 −0.284908 0.958555i \(-0.591963\pi\)
−0.284908 + 0.958555i \(0.591963\pi\)
\(888\) 0 0
\(889\) −18.0000 −0.603701
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) 5.65685 0.189299
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −33.9411 −1.13074
\(902\) 0 0
\(903\) −25.4558 −0.847117
\(904\) 0 0
\(905\) 4.00000 0.132964
\(906\) 0 0
\(907\) −58.0000 −1.92586 −0.962929 0.269754i \(-0.913058\pi\)
−0.962929 + 0.269754i \(0.913058\pi\)
\(908\) 0 0
\(909\) 11.3137 0.375252
\(910\) 0 0
\(911\) 22.6274 0.749680 0.374840 0.927090i \(-0.377698\pi\)
0.374840 + 0.927090i \(0.377698\pi\)
\(912\) 0 0
\(913\) 64.0000 2.11809
\(914\) 0 0
\(915\) 12.0000 0.396708
\(916\) 0 0
\(917\) −84.8528 −2.80209
\(918\) 0 0
\(919\) −18.3848 −0.606458 −0.303229 0.952918i \(-0.598065\pi\)
−0.303229 + 0.952918i \(0.598065\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) 12.7279 0.418491
\(926\) 0 0
\(927\) 18.3848 0.603835
\(928\) 0 0
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) 22.0000 0.721021
\(932\) 0 0
\(933\) −28.2843 −0.925985
\(934\) 0 0
\(935\) 67.8823 2.21999
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 36.7696 1.19865 0.599327 0.800505i \(-0.295435\pi\)
0.599327 + 0.800505i \(0.295435\pi\)
\(942\) 0 0
\(943\) 28.2843 0.921063
\(944\) 0 0
\(945\) 12.0000 0.390360
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 67.8823 2.20355
\(950\) 0 0
\(951\) 16.9706 0.550308
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) −64.0000 −2.07099
\(956\) 0 0
\(957\) 22.6274 0.731441
\(958\) 0 0
\(959\) −42.4264 −1.37002
\(960\) 0 0
\(961\) −13.0000 −0.419355
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) 50.9117 1.63891
\(966\) 0 0
\(967\) 35.3553 1.13695 0.568476 0.822700i \(-0.307533\pi\)
0.568476 + 0.822700i \(0.307533\pi\)
\(968\) 0 0
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 32.0000 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) 0 0
\(973\) 50.9117 1.63215
\(974\) 0 0
\(975\) −12.7279 −0.407620
\(976\) 0 0
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 0 0
\(979\) 56.0000 1.78977
\(980\) 0 0
\(981\) −9.89949 −0.316067
\(982\) 0 0
\(983\) −22.6274 −0.721703 −0.360851 0.932623i \(-0.617514\pi\)
−0.360851 + 0.932623i \(0.617514\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) 46.6690 1.48249 0.741246 0.671234i \(-0.234235\pi\)
0.741246 + 0.671234i \(0.234235\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −46.6690 −1.47802 −0.739012 0.673693i \(-0.764707\pi\)
−0.739012 + 0.673693i \(0.764707\pi\)
\(998\) 0 0
\(999\) −4.24264 −0.134231
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.d.1.2 2
3.2 odd 2 9216.2.a.e.1.1 2
4.3 odd 2 3072.2.a.f.1.2 2
8.3 odd 2 inner 3072.2.a.d.1.1 2
8.5 even 2 3072.2.a.f.1.1 2
12.11 even 2 9216.2.a.q.1.1 2
16.3 odd 4 3072.2.d.d.1537.3 4
16.5 even 4 3072.2.d.d.1537.4 4
16.11 odd 4 3072.2.d.d.1537.2 4
16.13 even 4 3072.2.d.d.1537.1 4
24.5 odd 2 9216.2.a.q.1.2 2
24.11 even 2 9216.2.a.e.1.2 2
32.3 odd 8 768.2.j.a.577.2 yes 4
32.5 even 8 768.2.j.d.193.2 yes 4
32.11 odd 8 768.2.j.a.193.2 yes 4
32.13 even 8 768.2.j.d.577.2 yes 4
32.19 odd 8 768.2.j.d.577.1 yes 4
32.21 even 8 768.2.j.a.193.1 4
32.27 odd 8 768.2.j.d.193.1 yes 4
32.29 even 8 768.2.j.a.577.1 yes 4
96.5 odd 8 2304.2.k.a.1729.1 4
96.11 even 8 2304.2.k.d.1729.2 4
96.29 odd 8 2304.2.k.d.577.2 4
96.35 even 8 2304.2.k.d.577.1 4
96.53 odd 8 2304.2.k.d.1729.1 4
96.59 even 8 2304.2.k.a.1729.2 4
96.77 odd 8 2304.2.k.a.577.2 4
96.83 even 8 2304.2.k.a.577.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.a.193.1 4 32.21 even 8
768.2.j.a.193.2 yes 4 32.11 odd 8
768.2.j.a.577.1 yes 4 32.29 even 8
768.2.j.a.577.2 yes 4 32.3 odd 8
768.2.j.d.193.1 yes 4 32.27 odd 8
768.2.j.d.193.2 yes 4 32.5 even 8
768.2.j.d.577.1 yes 4 32.19 odd 8
768.2.j.d.577.2 yes 4 32.13 even 8
2304.2.k.a.577.1 4 96.83 even 8
2304.2.k.a.577.2 4 96.77 odd 8
2304.2.k.a.1729.1 4 96.5 odd 8
2304.2.k.a.1729.2 4 96.59 even 8
2304.2.k.d.577.1 4 96.35 even 8
2304.2.k.d.577.2 4 96.29 odd 8
2304.2.k.d.1729.1 4 96.53 odd 8
2304.2.k.d.1729.2 4 96.11 even 8
3072.2.a.d.1.1 2 8.3 odd 2 inner
3072.2.a.d.1.2 2 1.1 even 1 trivial
3072.2.a.f.1.1 2 8.5 even 2
3072.2.a.f.1.2 2 4.3 odd 2
3072.2.d.d.1537.1 4 16.13 even 4
3072.2.d.d.1537.2 4 16.11 odd 4
3072.2.d.d.1537.3 4 16.3 odd 4
3072.2.d.d.1537.4 4 16.5 even 4
9216.2.a.e.1.1 2 3.2 odd 2
9216.2.a.e.1.2 2 24.11 even 2
9216.2.a.q.1.1 2 12.11 even 2
9216.2.a.q.1.2 2 24.5 odd 2