Properties

Label 1728.2.c.d.1727.4
Level $1728$
Weight $2$
Character 1728.1727
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1727.4
Root \(1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1727
Dual form 1728.2.c.d.1727.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{5} +1.73205i q^{7} +O(q^{10})\) \(q+2.82843i q^{5} +1.73205i q^{7} +4.89898 q^{11} +1.00000 q^{13} -2.82843i q^{17} +5.19615i q^{19} +4.89898 q^{23} -3.00000 q^{25} -5.65685i q^{29} +3.46410i q^{31} -4.89898 q^{35} +1.00000 q^{37} +5.65685i q^{41} +3.46410i q^{43} -4.89898 q^{47} +4.00000 q^{49} -5.65685i q^{53} +13.8564i q^{55} -4.89898 q^{59} -11.0000 q^{61} +2.82843i q^{65} +12.1244i q^{67} -1.00000 q^{73} +8.48528i q^{77} +1.73205i q^{79} -9.79796 q^{83} +8.00000 q^{85} -2.82843i q^{89} +1.73205i q^{91} -14.6969 q^{95} -13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 4q^{13} - 12q^{25} + 4q^{37} + 16q^{49} - 44q^{61} - 4q^{73} + 32q^{85} - 52q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.82843i 1.26491i 0.774597 + 0.632456i \(0.217953\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 1.73205i 0.654654i 0.944911 + 0.327327i \(0.106148\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.89898 1.47710 0.738549 0.674200i \(-0.235511\pi\)
0.738549 + 0.674200i \(0.235511\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.82843i − 0.685994i −0.939336 0.342997i \(-0.888558\pi\)
0.939336 0.342997i \(-0.111442\pi\)
\(18\) 0 0
\(19\) 5.19615i 1.19208i 0.802955 + 0.596040i \(0.203260\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.89898 1.02151 0.510754 0.859727i \(-0.329366\pi\)
0.510754 + 0.859727i \(0.329366\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 5.65685i − 1.05045i −0.850963 0.525226i \(-0.823981\pi\)
0.850963 0.525226i \(-0.176019\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.89898 −0.828079
\(36\) 0 0
\(37\) 1.00000 0.164399 0.0821995 0.996616i \(-0.473806\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.65685i 0.883452i 0.897150 + 0.441726i \(0.145634\pi\)
−0.897150 + 0.441726i \(0.854366\pi\)
\(42\) 0 0
\(43\) 3.46410i 0.528271i 0.964486 + 0.264135i \(0.0850865\pi\)
−0.964486 + 0.264135i \(0.914913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.89898 −0.714590 −0.357295 0.933992i \(-0.616301\pi\)
−0.357295 + 0.933992i \(0.616301\pi\)
\(48\) 0 0
\(49\) 4.00000 0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.65685i − 0.777029i −0.921443 0.388514i \(-0.872988\pi\)
0.921443 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) 13.8564i 1.86840i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.89898 −0.637793 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.82843i 0.350823i
\(66\) 0 0
\(67\) 12.1244i 1.48123i 0.671932 + 0.740613i \(0.265465\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.48528i 0.966988i
\(78\) 0 0
\(79\) 1.73205i 0.194871i 0.995242 + 0.0974355i \(0.0310640\pi\)
−0.995242 + 0.0974355i \(0.968936\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.79796 −1.07547 −0.537733 0.843115i \(-0.680719\pi\)
−0.537733 + 0.843115i \(0.680719\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.82843i − 0.299813i −0.988700 0.149906i \(-0.952103\pi\)
0.988700 0.149906i \(-0.0478972\pi\)
\(90\) 0 0
\(91\) 1.73205i 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −14.6969 −1.50787
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.3137i 1.12576i 0.826540 + 0.562878i \(0.190306\pi\)
−0.826540 + 0.562878i \(0.809694\pi\)
\(102\) 0 0
\(103\) 8.66025i 0.853320i 0.904412 + 0.426660i \(0.140310\pi\)
−0.904412 + 0.426660i \(0.859690\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.6969 1.42081 0.710403 0.703795i \(-0.248513\pi\)
0.710403 + 0.703795i \(0.248513\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 2.82843i − 0.266076i −0.991111 0.133038i \(-0.957527\pi\)
0.991111 0.133038i \(-0.0424732\pi\)
\(114\) 0 0
\(115\) 13.8564i 1.29212i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.89898 0.449089
\(120\) 0 0
\(121\) 13.0000 1.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) − 10.3923i − 0.922168i −0.887357 0.461084i \(-0.847461\pi\)
0.887357 0.461084i \(-0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.79796 0.856052 0.428026 0.903767i \(-0.359209\pi\)
0.428026 + 0.903767i \(0.359209\pi\)
\(132\) 0 0
\(133\) −9.00000 −0.780399
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 19.7990i − 1.69154i −0.533546 0.845771i \(-0.679141\pi\)
0.533546 0.845771i \(-0.320859\pi\)
\(138\) 0 0
\(139\) − 8.66025i − 0.734553i −0.930112 0.367277i \(-0.880290\pi\)
0.930112 0.367277i \(-0.119710\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.89898 0.409673
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 22.6274i − 1.85371i −0.375419 0.926855i \(-0.622501\pi\)
0.375419 0.926855i \(-0.377499\pi\)
\(150\) 0 0
\(151\) 22.5167i 1.83238i 0.400744 + 0.916190i \(0.368752\pi\)
−0.400744 + 0.916190i \(0.631248\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.79796 −0.786991
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.48528i 0.668734i
\(162\) 0 0
\(163\) 5.19615i 0.406994i 0.979076 + 0.203497i \(0.0652307\pi\)
−0.979076 + 0.203497i \(0.934769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4.89898 0.379094 0.189547 0.981872i \(-0.439298\pi\)
0.189547 + 0.981872i \(0.439298\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 5.65685i − 0.430083i −0.976605 0.215041i \(-0.931011\pi\)
0.976605 0.215041i \(-0.0689886\pi\)
\(174\) 0 0
\(175\) − 5.19615i − 0.392792i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −11.0000 −0.817624 −0.408812 0.912619i \(-0.634057\pi\)
−0.408812 + 0.912619i \(0.634057\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.82843i 0.207950i
\(186\) 0 0
\(187\) − 13.8564i − 1.01328i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.4949 1.77239 0.886194 0.463314i \(-0.153340\pi\)
0.886194 + 0.463314i \(0.153340\pi\)
\(192\) 0 0
\(193\) −1.00000 −0.0719816 −0.0359908 0.999352i \(-0.511459\pi\)
−0.0359908 + 0.999352i \(0.511459\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.82843i 0.201517i 0.994911 + 0.100759i \(0.0321270\pi\)
−0.994911 + 0.100759i \(0.967873\pi\)
\(198\) 0 0
\(199\) − 5.19615i − 0.368345i −0.982894 0.184173i \(-0.941039\pi\)
0.982894 0.184173i \(-0.0589606\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.79796 0.687682
\(204\) 0 0
\(205\) −16.0000 −1.11749
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 25.4558i 1.76082i
\(210\) 0 0
\(211\) 12.1244i 0.834675i 0.908752 + 0.417338i \(0.137037\pi\)
−0.908752 + 0.417338i \(0.862963\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.79796 −0.668215
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.82843i − 0.190261i
\(222\) 0 0
\(223\) − 24.2487i − 1.62381i −0.583787 0.811907i \(-0.698430\pi\)
0.583787 0.811907i \(-0.301570\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.5959 1.30063 0.650313 0.759666i \(-0.274638\pi\)
0.650313 + 0.759666i \(0.274638\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.6274i 1.48237i 0.671300 + 0.741186i \(0.265736\pi\)
−0.671300 + 0.741186i \(0.734264\pi\)
\(234\) 0 0
\(235\) − 13.8564i − 0.903892i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.5959 1.26755 0.633777 0.773516i \(-0.281504\pi\)
0.633777 + 0.773516i \(0.281504\pi\)
\(240\) 0 0
\(241\) −25.0000 −1.61039 −0.805196 0.593009i \(-0.797940\pi\)
−0.805196 + 0.593009i \(0.797940\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.3137i 0.722806i
\(246\) 0 0
\(247\) 5.19615i 0.330623i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −29.3939 −1.85533 −0.927663 0.373420i \(-0.878185\pi\)
−0.927663 + 0.373420i \(0.878185\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 28.2843i − 1.76432i −0.470946 0.882162i \(-0.656087\pi\)
0.470946 0.882162i \(-0.343913\pi\)
\(258\) 0 0
\(259\) 1.73205i 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −19.5959 −1.20834 −0.604168 0.796857i \(-0.706494\pi\)
−0.604168 + 0.796857i \(0.706494\pi\)
\(264\) 0 0
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.82843i 0.172452i 0.996276 + 0.0862261i \(0.0274808\pi\)
−0.996276 + 0.0862261i \(0.972519\pi\)
\(270\) 0 0
\(271\) − 5.19615i − 0.315644i −0.987468 0.157822i \(-0.949553\pi\)
0.987468 0.157822i \(-0.0504472\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −14.6969 −0.886259
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.65685i 0.337460i 0.985662 + 0.168730i \(0.0539665\pi\)
−0.985662 + 0.168730i \(0.946033\pi\)
\(282\) 0 0
\(283\) − 24.2487i − 1.44144i −0.693228 0.720718i \(-0.743812\pi\)
0.693228 0.720718i \(-0.256188\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.79796 −0.578355
\(288\) 0 0
\(289\) 9.00000 0.529412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.82843i 0.165238i 0.996581 + 0.0826192i \(0.0263285\pi\)
−0.996581 + 0.0826192i \(0.973671\pi\)
\(294\) 0 0
\(295\) − 13.8564i − 0.806751i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.89898 0.283315
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 31.1127i − 1.78151i
\(306\) 0 0
\(307\) − 10.3923i − 0.593120i −0.955014 0.296560i \(-0.904160\pi\)
0.955014 0.296560i \(-0.0958395\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.4949 −1.38898 −0.694489 0.719503i \(-0.744370\pi\)
−0.694489 + 0.719503i \(0.744370\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 5.65685i − 0.317721i −0.987301 0.158860i \(-0.949218\pi\)
0.987301 0.158860i \(-0.0507819\pi\)
\(318\) 0 0
\(319\) − 27.7128i − 1.55162i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 14.6969 0.817760
\(324\) 0 0
\(325\) −3.00000 −0.166410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 8.48528i − 0.467809i
\(330\) 0 0
\(331\) − 22.5167i − 1.23763i −0.785538 0.618814i \(-0.787614\pi\)
0.785538 0.618814i \(-0.212386\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −34.2929 −1.87362
\(336\) 0 0
\(337\) 11.0000 0.599208 0.299604 0.954064i \(-0.403145\pi\)
0.299604 + 0.954064i \(0.403145\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 16.9706i 0.919007i
\(342\) 0 0
\(343\) 19.0526i 1.02874i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.5959 −1.05196 −0.525982 0.850496i \(-0.676302\pi\)
−0.525982 + 0.850496i \(0.676302\pi\)
\(348\) 0 0
\(349\) −23.0000 −1.23116 −0.615581 0.788074i \(-0.711079\pi\)
−0.615581 + 0.788074i \(0.711079\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 11.3137i − 0.602168i −0.953598 0.301084i \(-0.902652\pi\)
0.953598 0.301084i \(-0.0973484\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.6969 0.775675 0.387837 0.921728i \(-0.373222\pi\)
0.387837 + 0.921728i \(0.373222\pi\)
\(360\) 0 0
\(361\) −8.00000 −0.421053
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 2.82843i − 0.148047i
\(366\) 0 0
\(367\) − 19.0526i − 0.994535i −0.867597 0.497268i \(-0.834337\pi\)
0.867597 0.497268i \(-0.165663\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 9.79796 0.508685
\(372\) 0 0
\(373\) 25.0000 1.29445 0.647225 0.762299i \(-0.275929\pi\)
0.647225 + 0.762299i \(0.275929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 5.65685i − 0.291343i
\(378\) 0 0
\(379\) − 15.5885i − 0.800725i −0.916357 0.400363i \(-0.868884\pi\)
0.916357 0.400363i \(-0.131116\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.5959 1.00130 0.500652 0.865648i \(-0.333094\pi\)
0.500652 + 0.865648i \(0.333094\pi\)
\(384\) 0 0
\(385\) −24.0000 −1.22315
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 14.1421i − 0.717035i −0.933523 0.358517i \(-0.883282\pi\)
0.933523 0.358517i \(-0.116718\pi\)
\(390\) 0 0
\(391\) − 13.8564i − 0.700749i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.89898 −0.246494
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.6274i 1.12996i 0.825105 + 0.564980i \(0.191116\pi\)
−0.825105 + 0.564980i \(0.808884\pi\)
\(402\) 0 0
\(403\) 3.46410i 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.89898 0.242833
\(408\) 0 0
\(409\) −1.00000 −0.0494468 −0.0247234 0.999694i \(-0.507871\pi\)
−0.0247234 + 0.999694i \(0.507871\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 8.48528i − 0.417533i
\(414\) 0 0
\(415\) − 27.7128i − 1.36037i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.89898 −0.239331 −0.119665 0.992814i \(-0.538182\pi\)
−0.119665 + 0.992814i \(0.538182\pi\)
\(420\) 0 0
\(421\) 25.0000 1.21843 0.609213 0.793007i \(-0.291486\pi\)
0.609213 + 0.793007i \(0.291486\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.48528i 0.411597i
\(426\) 0 0
\(427\) − 19.0526i − 0.922018i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.6969 0.707927 0.353963 0.935259i \(-0.384834\pi\)
0.353963 + 0.935259i \(0.384834\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 25.4558i 1.21772i
\(438\) 0 0
\(439\) 17.3205i 0.826663i 0.910581 + 0.413331i \(0.135635\pi\)
−0.910581 + 0.413331i \(0.864365\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.79796 −0.465515 −0.232758 0.972535i \(-0.574775\pi\)
−0.232758 + 0.972535i \(0.574775\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.1127i 1.46830i 0.678988 + 0.734150i \(0.262419\pi\)
−0.678988 + 0.734150i \(0.737581\pi\)
\(450\) 0 0
\(451\) 27.7128i 1.30495i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.89898 −0.229668
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 14.1421i − 0.658665i −0.944214 0.329332i \(-0.893176\pi\)
0.944214 0.329332i \(-0.106824\pi\)
\(462\) 0 0
\(463\) − 12.1244i − 0.563467i −0.959493 0.281733i \(-0.909091\pi\)
0.959493 0.281733i \(-0.0909093\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.6969 0.680093 0.340047 0.940409i \(-0.389557\pi\)
0.340047 + 0.940409i \(0.389557\pi\)
\(468\) 0 0
\(469\) −21.0000 −0.969690
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 16.9706i 0.780307i
\(474\) 0 0
\(475\) − 15.5885i − 0.715247i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.79796 0.447680 0.223840 0.974626i \(-0.428141\pi\)
0.223840 + 0.974626i \(0.428141\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 36.7696i − 1.66962i
\(486\) 0 0
\(487\) − 25.9808i − 1.17730i −0.808388 0.588650i \(-0.799659\pi\)
0.808388 0.588650i \(-0.200341\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.4949 1.10544 0.552720 0.833367i \(-0.313590\pi\)
0.552720 + 0.833367i \(0.313590\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 24.2487i − 1.08552i −0.839887 0.542761i \(-0.817379\pi\)
0.839887 0.542761i \(-0.182621\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.6969 0.655304 0.327652 0.944798i \(-0.393743\pi\)
0.327652 + 0.944798i \(0.393743\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.7990i 0.877575i 0.898591 + 0.438787i \(0.144592\pi\)
−0.898591 + 0.438787i \(0.855408\pi\)
\(510\) 0 0
\(511\) − 1.73205i − 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.4949 −1.07937
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 19.7990i − 0.867409i −0.901055 0.433705i \(-0.857206\pi\)
0.901055 0.433705i \(-0.142794\pi\)
\(522\) 0 0
\(523\) − 15.5885i − 0.681636i −0.940129 0.340818i \(-0.889296\pi\)
0.940129 0.340818i \(-0.110704\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.79796 0.426806
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.65685i 0.245026i
\(534\) 0 0
\(535\) 41.5692i 1.79719i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 19.5959 0.844056
\(540\) 0 0
\(541\) 1.00000 0.0429934 0.0214967 0.999769i \(-0.493157\pi\)
0.0214967 + 0.999769i \(0.493157\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 28.2843i 1.21157i
\(546\) 0 0
\(547\) − 22.5167i − 0.962743i −0.876517 0.481371i \(-0.840139\pi\)
0.876517 0.481371i \(-0.159861\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 29.3939 1.25222
\(552\) 0 0
\(553\) −3.00000 −0.127573
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.82843i 0.119844i 0.998203 + 0.0599222i \(0.0190852\pi\)
−0.998203 + 0.0599222i \(0.980915\pi\)
\(558\) 0 0
\(559\) 3.46410i 0.146516i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.79796 0.412935 0.206467 0.978453i \(-0.433803\pi\)
0.206467 + 0.978453i \(0.433803\pi\)
\(564\) 0 0
\(565\) 8.00000 0.336563
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 19.7990i − 0.830017i −0.909818 0.415008i \(-0.863779\pi\)
0.909818 0.415008i \(-0.136221\pi\)
\(570\) 0 0
\(571\) 32.9090i 1.37720i 0.725143 + 0.688599i \(0.241774\pi\)
−0.725143 + 0.688599i \(0.758226\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −14.6969 −0.612905
\(576\) 0 0
\(577\) 35.0000 1.45707 0.728535 0.685009i \(-0.240202\pi\)
0.728535 + 0.685009i \(0.240202\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 16.9706i − 0.704058i
\(582\) 0 0
\(583\) − 27.7128i − 1.14775i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.89898 0.202203 0.101101 0.994876i \(-0.467763\pi\)
0.101101 + 0.994876i \(0.467763\pi\)
\(588\) 0 0
\(589\) −18.0000 −0.741677
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 28.2843i − 1.16150i −0.814083 0.580748i \(-0.802760\pi\)
0.814083 0.580748i \(-0.197240\pi\)
\(594\) 0 0
\(595\) 13.8564i 0.568057i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.79796 −0.400334 −0.200167 0.979762i \(-0.564148\pi\)
−0.200167 + 0.979762i \(0.564148\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 36.7696i 1.49489i
\(606\) 0 0
\(607\) − 12.1244i − 0.492112i −0.969256 0.246056i \(-0.920865\pi\)
0.969256 0.246056i \(-0.0791348\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −4.89898 −0.198191
\(612\) 0 0
\(613\) −11.0000 −0.444286 −0.222143 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.82843i − 0.113868i −0.998378 0.0569341i \(-0.981868\pi\)
0.998378 0.0569341i \(-0.0181325\pi\)
\(618\) 0 0
\(619\) 19.0526i 0.765787i 0.923792 + 0.382893i \(0.125072\pi\)
−0.923792 + 0.382893i \(0.874928\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.89898 0.196273
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 2.82843i − 0.112777i
\(630\) 0 0
\(631\) 36.3731i 1.44799i 0.689806 + 0.723994i \(0.257696\pi\)
−0.689806 + 0.723994i \(0.742304\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 29.3939 1.16646
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 45.2548i − 1.78746i −0.448607 0.893729i \(-0.648080\pi\)
0.448607 0.893729i \(-0.351920\pi\)
\(642\) 0 0
\(643\) 17.3205i 0.683054i 0.939872 + 0.341527i \(0.110944\pi\)
−0.939872 + 0.341527i \(0.889056\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.3939 −1.15559 −0.577796 0.816181i \(-0.696087\pi\)
−0.577796 + 0.816181i \(0.696087\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 5.65685i − 0.221370i −0.993856 0.110685i \(-0.964696\pi\)
0.993856 0.110685i \(-0.0353044\pi\)
\(654\) 0 0
\(655\) 27.7128i 1.08283i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.79796 −0.381674 −0.190837 0.981622i \(-0.561120\pi\)
−0.190837 + 0.981622i \(0.561120\pi\)
\(660\) 0 0
\(661\) 37.0000 1.43913 0.719567 0.694423i \(-0.244340\pi\)
0.719567 + 0.694423i \(0.244340\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 25.4558i − 0.987135i
\(666\) 0 0
\(667\) − 27.7128i − 1.07304i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −53.8888 −2.08035
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 14.1421i − 0.543526i −0.962364 0.271763i \(-0.912393\pi\)
0.962364 0.271763i \(-0.0876068\pi\)
\(678\) 0 0
\(679\) − 22.5167i − 0.864110i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 56.0000 2.13965
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 5.65685i − 0.215509i
\(690\) 0 0
\(691\) 45.0333i 1.71315i 0.516024 + 0.856574i \(0.327412\pi\)
−0.516024 + 0.856574i \(0.672588\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.4949 0.929144
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.82843i 0.106828i 0.998572 + 0.0534141i \(0.0170103\pi\)
−0.998572 + 0.0534141i \(0.982990\pi\)
\(702\) 0 0
\(703\) 5.19615i 0.195977i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −19.5959 −0.736980
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 16.9706i 0.635553i
\(714\) 0 0
\(715\) 13.8564i 0.518200i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −29.3939 −1.09621 −0.548103 0.836411i \(-0.684650\pi\)
−0.548103 + 0.836411i \(0.684650\pi\)
\(720\) 0 0
\(721\) −15.0000 −0.558629
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16.9706i 0.630271i
\(726\) 0 0
\(727\) 17.3205i 0.642382i 0.947014 + 0.321191i \(0.104083\pi\)
−0.947014 + 0.321191i \(0.895917\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.79796 0.362391
\(732\) 0 0
\(733\) −38.0000 −1.40356 −0.701781 0.712393i \(-0.747612\pi\)
−0.701781 + 0.712393i \(0.747612\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 59.3970i 2.18792i
\(738\) 0 0
\(739\) − 10.3923i − 0.382287i −0.981562 0.191144i \(-0.938780\pi\)
0.981562 0.191144i \(-0.0612196\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 34.2929 1.25808 0.629041 0.777372i \(-0.283448\pi\)
0.629041 + 0.777372i \(0.283448\pi\)
\(744\) 0 0
\(745\) 64.0000 2.34478
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 25.4558i 0.930136i
\(750\) 0 0
\(751\) 29.4449i 1.07446i 0.843436 + 0.537229i \(0.180529\pi\)
−0.843436 + 0.537229i \(0.819471\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −63.6867 −2.31780
\(756\) 0 0
\(757\) −47.0000 −1.70824 −0.854122 0.520073i \(-0.825905\pi\)
−0.854122 + 0.520073i \(0.825905\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 2.82843i − 0.102530i −0.998685 0.0512652i \(-0.983675\pi\)
0.998685 0.0512652i \(-0.0163254\pi\)
\(762\) 0 0
\(763\) 17.3205i 0.627044i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −4.89898 −0.176892
\(768\) 0 0
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 28.2843i 1.01731i 0.860969 + 0.508657i \(0.169858\pi\)
−0.860969 + 0.508657i \(0.830142\pi\)
\(774\) 0 0
\(775\) − 10.3923i − 0.373303i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −29.3939 −1.05314
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 28.2843i 1.00951i
\(786\) 0 0
\(787\) − 8.66025i − 0.308705i −0.988016 0.154352i \(-0.950671\pi\)
0.988016 0.154352i \(-0.0493291\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.89898 0.174188
\(792\) 0 0
\(793\) −11.0000 −0.390621
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 22.6274i − 0.801504i −0.916187 0.400752i \(-0.868749\pi\)
0.916187 0.400752i \(-0.131251\pi\)
\(798\) 0 0
\(799\) 13.8564i 0.490204i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.89898 −0.172881
\(804\) 0 0
\(805\) −24.0000 −0.845889
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 22.6274i 0.795538i 0.917486 + 0.397769i \(0.130215\pi\)
−0.917486 + 0.397769i \(0.869785\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i 0.836881 + 0.547385i \(0.184377\pi\)
−0.836881 + 0.547385i \(0.815623\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −14.6969 −0.514811
\(816\) 0 0
\(817\) −18.0000 −0.629740
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 31.1127i − 1.08584i −0.839784 0.542920i \(-0.817319\pi\)
0.839784 0.542920i \(-0.182681\pi\)
\(822\) 0 0
\(823\) 50.2295i 1.75089i 0.483318 + 0.875445i \(0.339431\pi\)
−0.483318 + 0.875445i \(0.660569\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14.6969 0.511063 0.255531 0.966801i \(-0.417750\pi\)
0.255531 + 0.966801i \(0.417750\pi\)
\(828\) 0 0
\(829\) −11.0000 −0.382046 −0.191023 0.981586i \(-0.561180\pi\)
−0.191023 + 0.981586i \(0.561180\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 11.3137i − 0.391997i
\(834\) 0 0
\(835\) 13.8564i 0.479521i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −48.9898 −1.69132 −0.845658 0.533726i \(-0.820792\pi\)
−0.845658 + 0.533726i \(0.820792\pi\)
\(840\) 0 0
\(841\) −3.00000 −0.103448
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 33.9411i − 1.16761i
\(846\) 0 0
\(847\) 22.5167i 0.773682i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.89898 0.167935
\(852\) 0 0
\(853\) 49.0000 1.67773 0.838864 0.544341i \(-0.183220\pi\)
0.838864 + 0.544341i \(0.183220\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 56.5685i 1.93234i 0.257897 + 0.966172i \(0.416970\pi\)
−0.257897 + 0.966172i \(0.583030\pi\)
\(858\) 0 0
\(859\) 12.1244i 0.413678i 0.978375 + 0.206839i \(0.0663176\pi\)
−0.978375 + 0.206839i \(0.933682\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.6969 0.500290 0.250145 0.968208i \(-0.419522\pi\)
0.250145 + 0.968208i \(0.419522\pi\)
\(864\) 0 0
\(865\) 16.0000 0.544016
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.48528i 0.287843i
\(870\) 0 0
\(871\) 12.1244i 0.410818i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.79796 −0.331231
\(876\) 0 0
\(877\) 1.00000 0.0337676 0.0168838 0.999857i \(-0.494625\pi\)
0.0168838 + 0.999857i \(0.494625\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 19.7990i − 0.667045i −0.942742 0.333522i \(-0.891763\pi\)
0.942742 0.333522i \(-0.108237\pi\)
\(882\) 0 0
\(883\) 5.19615i 0.174864i 0.996170 + 0.0874322i \(0.0278661\pi\)
−0.996170 + 0.0874322i \(0.972134\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.9898 1.64492 0.822458 0.568826i \(-0.192602\pi\)
0.822458 + 0.568826i \(0.192602\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 25.4558i − 0.851847i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 19.5959 0.653560
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 31.1127i − 1.03422i
\(906\) 0 0
\(907\) − 1.73205i − 0.0575118i −0.999586 0.0287559i \(-0.990845\pi\)
0.999586 0.0287559i \(-0.00915455\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 9.79796 0.324621 0.162310 0.986740i \(-0.448105\pi\)
0.162310 + 0.986740i \(0.448105\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.9706i 0.560417i
\(918\) 0 0
\(919\) − 10.3923i − 0.342811i −0.985201 0.171405i \(-0.945169\pi\)
0.985201 0.171405i \(-0.0548307\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 45.2548i − 1.48476i −0.669977 0.742381i \(-0.733696\pi\)
0.669977 0.742381i \(-0.266304\pi\)
\(930\) 0 0
\(931\) 20.7846i 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 39.1918 1.28171
\(936\) 0 0
\(937\) −37.0000 −1.20874 −0.604369 0.796705i \(-0.706575\pi\)
−0.604369 + 0.796705i \(0.706575\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 48.0833i − 1.56747i −0.621096 0.783735i \(-0.713312\pi\)
0.621096 0.783735i \(-0.286688\pi\)
\(942\) 0 0
\(943\) 27.7128i 0.902453i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 34.2929 1.11437 0.557184 0.830389i \(-0.311882\pi\)
0.557184 + 0.830389i \(0.311882\pi\)
\(948\) 0 0
\(949\) −1.00000 −0.0324614
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 2.82843i − 0.0916217i −0.998950 0.0458109i \(-0.985413\pi\)
0.998950 0.0458109i \(-0.0145872\pi\)
\(954\) 0 0
\(955\) 69.2820i 2.24191i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 34.2929 1.10737
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 2.82843i − 0.0910503i
\(966\) 0 0
\(967\) − 53.6936i − 1.72667i −0.504632 0.863334i \(-0.668372\pi\)
0.504632 0.863334i \(-0.331628\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.6969 −0.471647 −0.235824 0.971796i \(-0.575779\pi\)
−0.235824 + 0.971796i \(0.575779\pi\)
\(972\) 0 0
\(973\) 15.0000 0.480878
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.65685i 0.180979i 0.995897 + 0.0904894i \(0.0288431\pi\)
−0.995897 + 0.0904894i \(0.971157\pi\)
\(978\) 0 0
\(979\) − 13.8564i − 0.442853i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −34.2929 −1.09377 −0.546886 0.837207i \(-0.684187\pi\)
−0.546886 + 0.837207i \(0.684187\pi\)
\(984\) 0 0
\(985\) −8.00000 −0.254901
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706i 0.539633i
\(990\) 0 0
\(991\) 36.3731i 1.15543i 0.816239 + 0.577714i \(0.196055\pi\)
−0.816239 + 0.577714i \(0.803945\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.6969 0.465924
\(996\) 0 0
\(997\) 34.0000 1.07679 0.538395 0.842692i \(-0.319031\pi\)
0.538395 + 0.842692i \(0.319031\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.2.c.d.1727.4 4
3.2 odd 2 inner 1728.2.c.d.1727.2 4
4.3 odd 2 inner 1728.2.c.d.1727.3 4
8.3 odd 2 108.2.b.b.107.2 yes 4
8.5 even 2 108.2.b.b.107.4 yes 4
12.11 even 2 inner 1728.2.c.d.1727.1 4
24.5 odd 2 108.2.b.b.107.1 4
24.11 even 2 108.2.b.b.107.3 yes 4
72.5 odd 6 324.2.h.b.107.2 4
72.11 even 6 324.2.h.b.215.1 4
72.13 even 6 324.2.h.b.107.1 4
72.29 odd 6 324.2.h.a.215.2 4
72.43 odd 6 324.2.h.b.215.2 4
72.59 even 6 324.2.h.a.107.2 4
72.61 even 6 324.2.h.a.215.1 4
72.67 odd 6 324.2.h.a.107.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.b.b.107.1 4 24.5 odd 2
108.2.b.b.107.2 yes 4 8.3 odd 2
108.2.b.b.107.3 yes 4 24.11 even 2
108.2.b.b.107.4 yes 4 8.5 even 2
324.2.h.a.107.1 4 72.67 odd 6
324.2.h.a.107.2 4 72.59 even 6
324.2.h.a.215.1 4 72.61 even 6
324.2.h.a.215.2 4 72.29 odd 6
324.2.h.b.107.1 4 72.13 even 6
324.2.h.b.107.2 4 72.5 odd 6
324.2.h.b.215.1 4 72.11 even 6
324.2.h.b.215.2 4 72.43 odd 6
1728.2.c.d.1727.1 4 12.11 even 2 inner
1728.2.c.d.1727.2 4 3.2 odd 2 inner
1728.2.c.d.1727.3 4 4.3 odd 2 inner
1728.2.c.d.1727.4 4 1.1 even 1 trivial