Properties

Label 512.2.a.c.1.1
Level $512$
Weight $2$
Character 512.1
Self dual yes
Analytic conductor $4.088$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.08834058349\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 512.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{3} +2.82843 q^{5} -4.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.41421 q^{3} +2.82843 q^{5} -4.00000 q^{7} -1.00000 q^{9} -1.41421 q^{11} -2.82843 q^{13} -4.00000 q^{15} -4.00000 q^{17} +7.07107 q^{19} +5.65685 q^{21} -4.00000 q^{23} +3.00000 q^{25} +5.65685 q^{27} -8.48528 q^{29} -8.00000 q^{31} +2.00000 q^{33} -11.3137 q^{35} -2.82843 q^{37} +4.00000 q^{39} +2.00000 q^{41} -4.24264 q^{43} -2.82843 q^{45} +9.00000 q^{49} +5.65685 q^{51} -2.82843 q^{53} -4.00000 q^{55} -10.0000 q^{57} -4.24264 q^{59} +8.48528 q^{61} +4.00000 q^{63} -8.00000 q^{65} -4.24264 q^{67} +5.65685 q^{69} +4.00000 q^{71} -4.00000 q^{73} -4.24264 q^{75} +5.65685 q^{77} +8.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} -11.3137 q^{85} +12.0000 q^{87} +12.0000 q^{89} +11.3137 q^{91} +11.3137 q^{93} +20.0000 q^{95} -4.00000 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q - 8 q^{7} - 2 q^{9} - 8 q^{15} - 8 q^{17} - 8 q^{23} + 6 q^{25} - 16 q^{31} + 4 q^{33} + 8 q^{39} + 4 q^{41} + 18 q^{49} - 8 q^{55} - 20 q^{57} + 8 q^{63} - 16 q^{65} + 8 q^{71} - 8 q^{73} + 16 q^{79} - 10 q^{81} + 24 q^{87} + 24 q^{89} + 40 q^{95} - 8 q^{97} + O(q^{100}) \)

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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) −2.82843 −0.784465 −0.392232 0.919866i \(-0.628297\pi\)
−0.392232 + 0.919866i \(0.628297\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 7.07107 1.62221 0.811107 0.584898i \(-0.198865\pi\)
0.811107 + 0.584898i \(0.198865\pi\)
\(20\) 0 0
\(21\) 5.65685 1.23443
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) −8.48528 −1.57568 −0.787839 0.615882i \(-0.788800\pi\)
−0.787839 + 0.615882i \(0.788800\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −11.3137 −1.91237
\(36\) 0 0
\(37\) −2.82843 −0.464991 −0.232495 0.972598i \(-0.574689\pi\)
−0.232495 + 0.972598i \(0.574689\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −4.24264 −0.646997 −0.323498 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 5.65685 0.792118
\(52\) 0 0
\(53\) −2.82843 −0.388514 −0.194257 0.980951i \(-0.562230\pi\)
−0.194257 + 0.980951i \(0.562230\pi\)
\(54\) 0 0
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) −10.0000 −1.32453
\(58\) 0 0
\(59\) −4.24264 −0.552345 −0.276172 0.961108i \(-0.589066\pi\)
−0.276172 + 0.961108i \(0.589066\pi\)
\(60\) 0 0
\(61\) 8.48528 1.08643 0.543214 0.839594i \(-0.317207\pi\)
0.543214 + 0.839594i \(0.317207\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −4.24264 −0.518321 −0.259161 0.965834i \(-0.583446\pi\)
−0.259161 + 0.965834i \(0.583446\pi\)
\(68\) 0 0
\(69\) 5.65685 0.681005
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −4.24264 −0.489898
\(76\) 0 0
\(77\) 5.65685 0.644658
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) −11.3137 −1.22714
\(86\) 0 0
\(87\) 12.0000 1.28654
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 11.3137 1.18600
\(92\) 0 0
\(93\) 11.3137 1.17318
\(94\) 0 0
\(95\) 20.0000 2.05196
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 1.41421 0.142134
\(100\) 0 0
\(101\) −2.82843 −0.281439 −0.140720 0.990050i \(-0.544942\pi\)
−0.140720 + 0.990050i \(0.544942\pi\)
\(102\) 0 0
\(103\) −12.0000 −1.18240 −0.591198 0.806527i \(-0.701345\pi\)
−0.591198 + 0.806527i \(0.701345\pi\)
\(104\) 0 0
\(105\) 16.0000 1.56144
\(106\) 0 0
\(107\) 9.89949 0.957020 0.478510 0.878082i \(-0.341177\pi\)
0.478510 + 0.878082i \(0.341177\pi\)
\(108\) 0 0
\(109\) 14.1421 1.35457 0.677285 0.735720i \(-0.263156\pi\)
0.677285 + 0.735720i \(0.263156\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) −11.3137 −1.05501
\(116\) 0 0
\(117\) 2.82843 0.261488
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −9.00000 −0.818182
\(122\) 0 0
\(123\) −2.82843 −0.255031
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 7.07107 0.617802 0.308901 0.951094i \(-0.400039\pi\)
0.308901 + 0.951094i \(0.400039\pi\)
\(132\) 0 0
\(133\) −28.2843 −2.45256
\(134\) 0 0
\(135\) 16.0000 1.37706
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −4.24264 −0.359856 −0.179928 0.983680i \(-0.557586\pi\)
−0.179928 + 0.983680i \(0.557586\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −24.0000 −1.99309
\(146\) 0 0
\(147\) −12.7279 −1.04978
\(148\) 0 0
\(149\) 2.82843 0.231714 0.115857 0.993266i \(-0.463039\pi\)
0.115857 + 0.993266i \(0.463039\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −22.6274 −1.81748
\(156\) 0 0
\(157\) 8.48528 0.677199 0.338600 0.940931i \(-0.390047\pi\)
0.338600 + 0.940931i \(0.390047\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 16.0000 1.26098
\(162\) 0 0
\(163\) 9.89949 0.775388 0.387694 0.921788i \(-0.373272\pi\)
0.387694 + 0.921788i \(0.373272\pi\)
\(164\) 0 0
\(165\) 5.65685 0.440386
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) −7.07107 −0.540738
\(172\) 0 0
\(173\) −14.1421 −1.07521 −0.537603 0.843198i \(-0.680670\pi\)
−0.537603 + 0.843198i \(0.680670\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 0 0
\(177\) 6.00000 0.450988
\(178\) 0 0
\(179\) −1.41421 −0.105703 −0.0528516 0.998602i \(-0.516831\pi\)
−0.0528516 + 0.998602i \(0.516831\pi\)
\(180\) 0 0
\(181\) −8.48528 −0.630706 −0.315353 0.948974i \(-0.602123\pi\)
−0.315353 + 0.948974i \(0.602123\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) −22.6274 −1.64590
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 11.3137 0.810191
\(196\) 0 0
\(197\) 14.1421 1.00759 0.503793 0.863825i \(-0.331938\pi\)
0.503793 + 0.863825i \(0.331938\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 0 0
\(203\) 33.9411 2.38220
\(204\) 0 0
\(205\) 5.65685 0.395092
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −10.0000 −0.691714
\(210\) 0 0
\(211\) −24.0416 −1.65509 −0.827547 0.561396i \(-0.810264\pi\)
−0.827547 + 0.561396i \(0.810264\pi\)
\(212\) 0 0
\(213\) −5.65685 −0.387601
\(214\) 0 0
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) 32.0000 2.17230
\(218\) 0 0
\(219\) 5.65685 0.382255
\(220\) 0 0
\(221\) 11.3137 0.761042
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 21.2132 1.40797 0.703985 0.710215i \(-0.251402\pi\)
0.703985 + 0.710215i \(0.251402\pi\)
\(228\) 0 0
\(229\) 25.4558 1.68217 0.841085 0.540903i \(-0.181918\pi\)
0.841085 + 0.540903i \(0.181918\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.3137 −0.734904
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) 25.4558 1.62631
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 0 0
\(249\) −14.0000 −0.887214
\(250\) 0 0
\(251\) −4.24264 −0.267793 −0.133897 0.990995i \(-0.542749\pi\)
−0.133897 + 0.990995i \(0.542749\pi\)
\(252\) 0 0
\(253\) 5.65685 0.355643
\(254\) 0 0
\(255\) 16.0000 1.00196
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 11.3137 0.703000
\(260\) 0 0
\(261\) 8.48528 0.525226
\(262\) 0 0
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) −16.9706 −1.03858
\(268\) 0 0
\(269\) −19.7990 −1.20717 −0.603583 0.797300i \(-0.706261\pi\)
−0.603583 + 0.797300i \(0.706261\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 0 0
\(273\) −16.0000 −0.968364
\(274\) 0 0
\(275\) −4.24264 −0.255841
\(276\) 0 0
\(277\) 8.48528 0.509831 0.254916 0.966963i \(-0.417952\pi\)
0.254916 + 0.966963i \(0.417952\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) −24.0416 −1.42913 −0.714563 0.699571i \(-0.753375\pi\)
−0.714563 + 0.699571i \(0.753375\pi\)
\(284\) 0 0
\(285\) −28.2843 −1.67542
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 5.65685 0.331611
\(292\) 0 0
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) −8.00000 −0.464207
\(298\) 0 0
\(299\) 11.3137 0.654289
\(300\) 0 0
\(301\) 16.9706 0.978167
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 7.07107 0.403567 0.201784 0.979430i \(-0.435326\pi\)
0.201784 + 0.979430i \(0.435326\pi\)
\(308\) 0 0
\(309\) 16.9706 0.965422
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 11.3137 0.637455
\(316\) 0 0
\(317\) 2.82843 0.158860 0.0794301 0.996840i \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) −14.0000 −0.781404
\(322\) 0 0
\(323\) −28.2843 −1.57378
\(324\) 0 0
\(325\) −8.48528 −0.470679
\(326\) 0 0
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.7279 −0.699590 −0.349795 0.936826i \(-0.613749\pi\)
−0.349795 + 0.936826i \(0.613749\pi\)
\(332\) 0 0
\(333\) 2.82843 0.154997
\(334\) 0 0
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) −19.7990 −1.07533
\(340\) 0 0
\(341\) 11.3137 0.612672
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −15.5563 −0.835109 −0.417554 0.908652i \(-0.637113\pi\)
−0.417554 + 0.908652i \(0.637113\pi\)
\(348\) 0 0
\(349\) −19.7990 −1.05982 −0.529908 0.848055i \(-0.677773\pi\)
−0.529908 + 0.848055i \(0.677773\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 11.3137 0.600469
\(356\) 0 0
\(357\) −22.6274 −1.19757
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) 31.0000 1.63158
\(362\) 0 0
\(363\) 12.7279 0.668043
\(364\) 0 0
\(365\) −11.3137 −0.592187
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 11.3137 0.587378
\(372\) 0 0
\(373\) −36.7696 −1.90386 −0.951928 0.306323i \(-0.900901\pi\)
−0.951928 + 0.306323i \(0.900901\pi\)
\(374\) 0 0
\(375\) 8.00000 0.413118
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 29.6985 1.52551 0.762754 0.646688i \(-0.223847\pi\)
0.762754 + 0.646688i \(0.223847\pi\)
\(380\) 0 0
\(381\) 11.3137 0.579619
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 4.24264 0.215666
\(388\) 0 0
\(389\) 19.7990 1.00385 0.501924 0.864912i \(-0.332626\pi\)
0.501924 + 0.864912i \(0.332626\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −10.0000 −0.504433
\(394\) 0 0
\(395\) 22.6274 1.13851
\(396\) 0 0
\(397\) 2.82843 0.141955 0.0709773 0.997478i \(-0.477388\pi\)
0.0709773 + 0.997478i \(0.477388\pi\)
\(398\) 0 0
\(399\) 40.0000 2.00250
\(400\) 0 0
\(401\) −36.0000 −1.79775 −0.898877 0.438201i \(-0.855616\pi\)
−0.898877 + 0.438201i \(0.855616\pi\)
\(402\) 0 0
\(403\) 22.6274 1.12715
\(404\) 0 0
\(405\) −14.1421 −0.702728
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 25.4558 1.25564
\(412\) 0 0
\(413\) 16.9706 0.835067
\(414\) 0 0
\(415\) 28.0000 1.37447
\(416\) 0 0
\(417\) 6.00000 0.293821
\(418\) 0 0
\(419\) −26.8701 −1.31269 −0.656344 0.754462i \(-0.727898\pi\)
−0.656344 + 0.754462i \(0.727898\pi\)
\(420\) 0 0
\(421\) −8.48528 −0.413547 −0.206774 0.978389i \(-0.566296\pi\)
−0.206774 + 0.978389i \(0.566296\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −33.9411 −1.64253
\(428\) 0 0
\(429\) −5.65685 −0.273115
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 33.9411 1.62735
\(436\) 0 0
\(437\) −28.2843 −1.35302
\(438\) 0 0
\(439\) 4.00000 0.190910 0.0954548 0.995434i \(-0.469569\pi\)
0.0954548 + 0.995434i \(0.469569\pi\)
\(440\) 0 0
\(441\) −9.00000 −0.428571
\(442\) 0 0
\(443\) −35.3553 −1.67978 −0.839891 0.542754i \(-0.817381\pi\)
−0.839891 + 0.542754i \(0.817381\pi\)
\(444\) 0 0
\(445\) 33.9411 1.60896
\(446\) 0 0
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −2.82843 −0.133185
\(452\) 0 0
\(453\) 5.65685 0.265782
\(454\) 0 0
\(455\) 32.0000 1.50018
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) −22.6274 −1.05616
\(460\) 0 0
\(461\) 14.1421 0.658665 0.329332 0.944214i \(-0.393176\pi\)
0.329332 + 0.944214i \(0.393176\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 32.0000 1.48396
\(466\) 0 0
\(467\) 18.3848 0.850746 0.425373 0.905018i \(-0.360143\pi\)
0.425373 + 0.905018i \(0.360143\pi\)
\(468\) 0 0
\(469\) 16.9706 0.783628
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 21.2132 0.973329
\(476\) 0 0
\(477\) 2.82843 0.129505
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) −22.6274 −1.02958
\(484\) 0 0
\(485\) −11.3137 −0.513729
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) −14.0000 −0.633102
\(490\) 0 0
\(491\) −15.5563 −0.702048 −0.351024 0.936366i \(-0.614166\pi\)
−0.351024 + 0.936366i \(0.614166\pi\)
\(492\) 0 0
\(493\) 33.9411 1.52863
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −12.7279 −0.569780 −0.284890 0.958560i \(-0.591957\pi\)
−0.284890 + 0.958560i \(0.591957\pi\)
\(500\) 0 0
\(501\) −16.9706 −0.758189
\(502\) 0 0
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 7.07107 0.314037
\(508\) 0 0
\(509\) −2.82843 −0.125368 −0.0626839 0.998033i \(-0.519966\pi\)
−0.0626839 + 0.998033i \(0.519966\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 40.0000 1.76604
\(514\) 0 0
\(515\) −33.9411 −1.49562
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 20.0000 0.877903
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 9.89949 0.432875 0.216437 0.976297i \(-0.430556\pi\)
0.216437 + 0.976297i \(0.430556\pi\)
\(524\) 0 0
\(525\) 16.9706 0.740656
\(526\) 0 0
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 4.24264 0.184115
\(532\) 0 0
\(533\) −5.65685 −0.245026
\(534\) 0 0
\(535\) 28.0000 1.21055
\(536\) 0 0
\(537\) 2.00000 0.0863064
\(538\) 0 0
\(539\) −12.7279 −0.548230
\(540\) 0 0
\(541\) 8.48528 0.364811 0.182405 0.983223i \(-0.441612\pi\)
0.182405 + 0.983223i \(0.441612\pi\)
\(542\) 0 0
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) 40.0000 1.71341
\(546\) 0 0
\(547\) −26.8701 −1.14888 −0.574440 0.818546i \(-0.694780\pi\)
−0.574440 + 0.818546i \(0.694780\pi\)
\(548\) 0 0
\(549\) −8.48528 −0.362143
\(550\) 0 0
\(551\) −60.0000 −2.55609
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 0 0
\(555\) 11.3137 0.480240
\(556\) 0 0
\(557\) 31.1127 1.31829 0.659144 0.752017i \(-0.270919\pi\)
0.659144 + 0.752017i \(0.270919\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 7.07107 0.298010 0.149005 0.988836i \(-0.452393\pi\)
0.149005 + 0.988836i \(0.452393\pi\)
\(564\) 0 0
\(565\) 39.5980 1.66590
\(566\) 0 0
\(567\) 20.0000 0.839921
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 32.5269 1.36121 0.680604 0.732651i \(-0.261717\pi\)
0.680604 + 0.732651i \(0.261717\pi\)
\(572\) 0 0
\(573\) 11.3137 0.472637
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 5.65685 0.235091
\(580\) 0 0
\(581\) −39.5980 −1.64280
\(582\) 0 0
\(583\) 4.00000 0.165663
\(584\) 0 0
\(585\) 8.00000 0.330759
\(586\) 0 0
\(587\) −1.41421 −0.0583708 −0.0291854 0.999574i \(-0.509291\pi\)
−0.0291854 + 0.999574i \(0.509291\pi\)
\(588\) 0 0
\(589\) −56.5685 −2.33087
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) 0 0
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 0 0
\(595\) 45.2548 1.85527
\(596\) 0 0
\(597\) 5.65685 0.231520
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 0 0
\(603\) 4.24264 0.172774
\(604\) 0 0
\(605\) −25.4558 −1.03493
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −48.0000 −1.94506
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −8.48528 −0.342717 −0.171359 0.985209i \(-0.554816\pi\)
−0.171359 + 0.985209i \(0.554816\pi\)
\(614\) 0 0
\(615\) −8.00000 −0.322591
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) −1.41421 −0.0568420 −0.0284210 0.999596i \(-0.509048\pi\)
−0.0284210 + 0.999596i \(0.509048\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) −48.0000 −1.92308
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 14.1421 0.564782
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 0 0
\(633\) 34.0000 1.35138
\(634\) 0 0
\(635\) −22.6274 −0.897942
\(636\) 0 0
\(637\) −25.4558 −1.00860
\(638\) 0 0
\(639\) −4.00000 −0.158238
\(640\) 0 0
\(641\) −4.00000 −0.157991 −0.0789953 0.996875i \(-0.525171\pi\)
−0.0789953 + 0.996875i \(0.525171\pi\)
\(642\) 0 0
\(643\) −4.24264 −0.167313 −0.0836567 0.996495i \(-0.526660\pi\)
−0.0836567 + 0.996495i \(0.526660\pi\)
\(644\) 0 0
\(645\) 16.9706 0.668215
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 6.00000 0.235521
\(650\) 0 0
\(651\) −45.2548 −1.77368
\(652\) 0 0
\(653\) 48.0833 1.88164 0.940822 0.338902i \(-0.110055\pi\)
0.940822 + 0.338902i \(0.110055\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 0 0
\(657\) 4.00000 0.156055
\(658\) 0 0
\(659\) 32.5269 1.26707 0.633534 0.773715i \(-0.281604\pi\)
0.633534 + 0.773715i \(0.281604\pi\)
\(660\) 0 0
\(661\) −31.1127 −1.21014 −0.605072 0.796171i \(-0.706856\pi\)
−0.605072 + 0.796171i \(0.706856\pi\)
\(662\) 0 0
\(663\) −16.0000 −0.621389
\(664\) 0 0
\(665\) −80.0000 −3.10227
\(666\) 0 0
\(667\) 33.9411 1.31421
\(668\) 0 0
\(669\) 33.9411 1.31224
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) 16.9706 0.653197
\(676\) 0 0
\(677\) 42.4264 1.63058 0.815290 0.579053i \(-0.196578\pi\)
0.815290 + 0.579053i \(0.196578\pi\)
\(678\) 0 0
\(679\) 16.0000 0.614024
\(680\) 0 0
\(681\) −30.0000 −1.14960
\(682\) 0 0
\(683\) 29.6985 1.13638 0.568190 0.822897i \(-0.307644\pi\)
0.568190 + 0.822897i \(0.307644\pi\)
\(684\) 0 0
\(685\) −50.9117 −1.94524
\(686\) 0 0
\(687\) −36.0000 −1.37349
\(688\) 0 0
\(689\) 8.00000 0.304776
\(690\) 0 0
\(691\) −46.6690 −1.77537 −0.887687 0.460447i \(-0.847689\pi\)
−0.887687 + 0.460447i \(0.847689\pi\)
\(692\) 0 0
\(693\) −5.65685 −0.214886
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) −16.9706 −0.641886
\(700\) 0 0
\(701\) −48.0833 −1.81608 −0.908040 0.418884i \(-0.862421\pi\)
−0.908040 + 0.418884i \(0.862421\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.3137 0.425496
\(708\) 0 0
\(709\) 8.48528 0.318671 0.159336 0.987224i \(-0.449065\pi\)
0.159336 + 0.987224i \(0.449065\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 11.3137 0.423109
\(716\) 0 0
\(717\) −11.3137 −0.422518
\(718\) 0 0
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) −16.9706 −0.631142
\(724\) 0 0
\(725\) −25.4558 −0.945406
\(726\) 0 0
\(727\) −28.0000 −1.03846 −0.519231 0.854634i \(-0.673782\pi\)
−0.519231 + 0.854634i \(0.673782\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 0 0
\(731\) 16.9706 0.627679
\(732\) 0 0
\(733\) −19.7990 −0.731292 −0.365646 0.930754i \(-0.619152\pi\)
−0.365646 + 0.930754i \(0.619152\pi\)
\(734\) 0 0
\(735\) −36.0000 −1.32788
\(736\) 0 0
\(737\) 6.00000 0.221013
\(738\) 0 0
\(739\) −24.0416 −0.884386 −0.442193 0.896920i \(-0.645799\pi\)
−0.442193 + 0.896920i \(0.645799\pi\)
\(740\) 0 0
\(741\) 28.2843 1.03905
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 0 0
\(747\) −9.89949 −0.362204
\(748\) 0 0
\(749\) −39.5980 −1.44688
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) −11.3137 −0.411748
\(756\) 0 0
\(757\) 25.4558 0.925208 0.462604 0.886565i \(-0.346915\pi\)
0.462604 + 0.886565i \(0.346915\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) −56.5685 −2.04792
\(764\) 0 0
\(765\) 11.3137 0.409048
\(766\) 0 0
\(767\) 12.0000 0.433295
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 0 0
\(771\) 2.82843 0.101863
\(772\) 0 0
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) 0 0
\(775\) −24.0000 −0.862105
\(776\) 0 0
\(777\) −16.0000 −0.573997
\(778\) 0 0
\(779\) 14.1421 0.506695
\(780\) 0 0
\(781\) −5.65685 −0.202418
\(782\) 0 0
\(783\) −48.0000 −1.71538
\(784\) 0 0
\(785\) 24.0000 0.856597
\(786\) 0 0
\(787\) −46.6690 −1.66357 −0.831786 0.555097i \(-0.812681\pi\)
−0.831786 + 0.555097i \(0.812681\pi\)
\(788\) 0 0
\(789\) 39.5980 1.40973
\(790\) 0 0
\(791\) −56.0000 −1.99113
\(792\) 0 0
\(793\) −24.0000 −0.852265
\(794\) 0 0
\(795\) 11.3137 0.401256
\(796\) 0 0
\(797\) 19.7990 0.701316 0.350658 0.936504i \(-0.385958\pi\)
0.350658 + 0.936504i \(0.385958\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 0 0
\(803\) 5.65685 0.199626
\(804\) 0 0
\(805\) 45.2548 1.59502
\(806\) 0 0
\(807\) 28.0000 0.985647
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −24.0416 −0.844216 −0.422108 0.906546i \(-0.638710\pi\)
−0.422108 + 0.906546i \(0.638710\pi\)
\(812\) 0 0
\(813\) 45.2548 1.58716
\(814\) 0 0
\(815\) 28.0000 0.980797
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) −11.3137 −0.395333
\(820\) 0 0
\(821\) 8.48528 0.296138 0.148069 0.988977i \(-0.452694\pi\)
0.148069 + 0.988977i \(0.452694\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) 21.2132 0.737655 0.368828 0.929498i \(-0.379759\pi\)
0.368828 + 0.929498i \(0.379759\pi\)
\(828\) 0 0
\(829\) −25.4558 −0.884118 −0.442059 0.896986i \(-0.645752\pi\)
−0.442059 + 0.896986i \(0.645752\pi\)
\(830\) 0 0
\(831\) −12.0000 −0.416275
\(832\) 0 0
\(833\) −36.0000 −1.24733
\(834\) 0 0
\(835\) 33.9411 1.17458
\(836\) 0 0
\(837\) −45.2548 −1.56424
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) 43.0000 1.48276
\(842\) 0 0
\(843\) 28.2843 0.974162
\(844\) 0 0
\(845\) −14.1421 −0.486504
\(846\) 0 0
\(847\) 36.0000 1.23697
\(848\) 0 0
\(849\) 34.0000 1.16688
\(850\) 0 0
\(851\) 11.3137 0.387829
\(852\) 0 0
\(853\) 42.4264 1.45265 0.726326 0.687350i \(-0.241226\pi\)
0.726326 + 0.687350i \(0.241226\pi\)
\(854\) 0 0
\(855\) −20.0000 −0.683986
\(856\) 0 0
\(857\) −34.0000 −1.16142 −0.580709 0.814111i \(-0.697225\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(858\) 0 0
\(859\) −38.1838 −1.30281 −0.651407 0.758729i \(-0.725821\pi\)
−0.651407 + 0.758729i \(0.725821\pi\)
\(860\) 0 0
\(861\) 11.3137 0.385570
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −40.0000 −1.36004
\(866\) 0 0
\(867\) 1.41421 0.0480292
\(868\) 0 0
\(869\) −11.3137 −0.383791
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 22.6274 0.764946
\(876\) 0 0
\(877\) −8.48528 −0.286528 −0.143264 0.989685i \(-0.545760\pi\)
−0.143264 + 0.989685i \(0.545760\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 0 0
\(883\) 41.0122 1.38017 0.690085 0.723728i \(-0.257573\pi\)
0.690085 + 0.723728i \(0.257573\pi\)
\(884\) 0 0
\(885\) 16.9706 0.570459
\(886\) 0 0
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 7.07107 0.236890
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) −16.0000 −0.534224
\(898\) 0 0
\(899\) 67.8823 2.26400
\(900\) 0 0
\(901\) 11.3137 0.376914
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) 0 0
\(905\) −24.0000 −0.797787
\(906\) 0 0
\(907\) 7.07107 0.234791 0.117395 0.993085i \(-0.462545\pi\)
0.117395 + 0.993085i \(0.462545\pi\)
\(908\) 0 0
\(909\) 2.82843 0.0938130
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) −14.0000 −0.463332
\(914\) 0 0
\(915\) −33.9411 −1.12206
\(916\) 0 0
\(917\) −28.2843 −0.934029
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) −11.3137 −0.372395
\(924\) 0 0
\(925\) −8.48528 −0.278994
\(926\) 0 0
\(927\) 12.0000 0.394132
\(928\) 0 0
\(929\) 60.0000 1.96854 0.984268 0.176682i \(-0.0565363\pi\)
0.984268 + 0.176682i \(0.0565363\pi\)
\(930\) 0 0
\(931\) 63.6396 2.08570
\(932\) 0 0
\(933\) −28.2843 −0.925985
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) −20.0000 −0.653372 −0.326686 0.945133i \(-0.605932\pi\)
−0.326686 + 0.945133i \(0.605932\pi\)
\(938\) 0 0
\(939\) 19.7990 0.646116
\(940\) 0 0
\(941\) 36.7696 1.19865 0.599327 0.800505i \(-0.295435\pi\)
0.599327 + 0.800505i \(0.295435\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) −64.0000 −2.08192
\(946\) 0 0
\(947\) −60.8112 −1.97610 −0.988049 0.154140i \(-0.950739\pi\)
−0.988049 + 0.154140i \(0.950739\pi\)
\(948\) 0 0
\(949\) 11.3137 0.367259
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) −22.6274 −0.732206
\(956\) 0 0
\(957\) −16.9706 −0.548580
\(958\) 0 0
\(959\) 72.0000 2.32500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −9.89949 −0.319007
\(964\) 0 0
\(965\) −11.3137 −0.364201
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) 40.0000 1.28499
\(970\) 0 0
\(971\) −15.5563 −0.499227 −0.249614 0.968346i \(-0.580304\pi\)
−0.249614 + 0.968346i \(0.580304\pi\)
\(972\) 0 0
\(973\) 16.9706 0.544051
\(974\) 0 0
\(975\) 12.0000 0.384308
\(976\) 0 0
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) −16.9706 −0.542382
\(980\) 0 0
\(981\) −14.1421 −0.451524
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 40.0000 1.27451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 18.0000 0.571213
\(994\) 0 0
\(995\) −11.3137 −0.358669
\(996\) 0 0
\(997\) −14.1421 −0.447886 −0.223943 0.974602i \(-0.571893\pi\)
−0.223943 + 0.974602i \(0.571893\pi\)
\(998\) 0 0
\(999\) −16.0000 −0.506218
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 512.2.a.c.1.1 2
3.2 odd 2 4608.2.a.f.1.1 2
4.3 odd 2 512.2.a.d.1.2 yes 2
8.3 odd 2 512.2.a.d.1.1 yes 2
8.5 even 2 inner 512.2.a.c.1.2 yes 2
12.11 even 2 4608.2.a.m.1.1 2
16.3 odd 4 512.2.b.a.257.2 2
16.5 even 4 512.2.b.b.257.2 2
16.11 odd 4 512.2.b.a.257.1 2
16.13 even 4 512.2.b.b.257.1 2
24.5 odd 2 4608.2.a.f.1.2 2
24.11 even 2 4608.2.a.m.1.2 2
32.3 odd 8 1024.2.e.d.257.1 2
32.5 even 8 1024.2.e.e.769.1 2
32.11 odd 8 1024.2.e.d.769.1 2
32.13 even 8 1024.2.e.e.257.1 2
32.19 odd 8 1024.2.e.c.257.1 2
32.21 even 8 1024.2.e.b.769.1 2
32.27 odd 8 1024.2.e.c.769.1 2
32.29 even 8 1024.2.e.b.257.1 2
48.5 odd 4 4608.2.d.b.2305.1 2
48.11 even 4 4608.2.d.a.2305.1 2
48.29 odd 4 4608.2.d.b.2305.2 2
48.35 even 4 4608.2.d.a.2305.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.c.1.1 2 1.1 even 1 trivial
512.2.a.c.1.2 yes 2 8.5 even 2 inner
512.2.a.d.1.1 yes 2 8.3 odd 2
512.2.a.d.1.2 yes 2 4.3 odd 2
512.2.b.a.257.1 2 16.11 odd 4
512.2.b.a.257.2 2 16.3 odd 4
512.2.b.b.257.1 2 16.13 even 4
512.2.b.b.257.2 2 16.5 even 4
1024.2.e.b.257.1 2 32.29 even 8
1024.2.e.b.769.1 2 32.21 even 8
1024.2.e.c.257.1 2 32.19 odd 8
1024.2.e.c.769.1 2 32.27 odd 8
1024.2.e.d.257.1 2 32.3 odd 8
1024.2.e.d.769.1 2 32.11 odd 8
1024.2.e.e.257.1 2 32.13 even 8
1024.2.e.e.769.1 2 32.5 even 8
4608.2.a.f.1.1 2 3.2 odd 2
4608.2.a.f.1.2 2 24.5 odd 2
4608.2.a.m.1.1 2 12.11 even 2
4608.2.a.m.1.2 2 24.11 even 2
4608.2.d.a.2305.1 2 48.11 even 4
4608.2.d.a.2305.2 2 48.35 even 4
4608.2.d.b.2305.1 2 48.5 odd 4
4608.2.d.b.2305.2 2 48.29 odd 4