Properties

Label 6400.2.a.cu.1.1
Level $6400$
Weight $2$
Character 6400.1
Self dual yes
Analytic conductor $51.104$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 6400 = 2^{8} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.1042572936\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 320)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.517638\) of defining polynomial
Character \(\chi\) \(=\) 6400.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.44949 q^{3} -1.41421 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-2.44949 q^{3} -1.41421 q^{7} +3.00000 q^{9} +2.00000 q^{11} -5.65685 q^{13} -4.89898 q^{17} +6.00000 q^{19} +3.46410 q^{21} -7.07107 q^{23} -6.92820 q^{29} -6.92820 q^{31} -4.89898 q^{33} +2.82843 q^{37} +13.8564 q^{39} +4.00000 q^{41} +2.44949 q^{43} +4.24264 q^{47} -5.00000 q^{49} +12.0000 q^{51} -14.6969 q^{57} +2.00000 q^{59} -3.46410 q^{61} -4.24264 q^{63} -2.44949 q^{67} +17.3205 q^{69} -6.92820 q^{71} -4.89898 q^{73} -2.82843 q^{77} -6.92820 q^{79} -9.00000 q^{81} +12.2474 q^{83} +16.9706 q^{87} -2.00000 q^{89} +8.00000 q^{91} +16.9706 q^{93} -14.6969 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + O(q^{10}) \) \( 4 q + 12 q^{9} + 8 q^{11} + 24 q^{19} + 16 q^{41} - 20 q^{49} + 48 q^{51} + 8 q^{59} - 36 q^{81} - 8 q^{89} + 32 q^{91} + 24 q^{99} + O(q^{100}) \)

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
\(3\) −2.44949 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.89898 −1.18818 −0.594089 0.804400i \(-0.702487\pi\)
−0.594089 + 0.804400i \(0.702487\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 0 0
\(23\) −7.07107 −1.47442 −0.737210 0.675664i \(-0.763857\pi\)
−0.737210 + 0.675664i \(0.763857\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) −4.89898 −0.852803
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 0 0
\(39\) 13.8564 2.21880
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 2.44949 0.373544 0.186772 0.982403i \(-0.440197\pi\)
0.186772 + 0.982403i \(0.440197\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.24264 0.618853 0.309426 0.950923i \(-0.399863\pi\)
0.309426 + 0.950923i \(0.399863\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −14.6969 −1.94666
\(58\) 0 0
\(59\) 2.00000 0.260378 0.130189 0.991489i \(-0.458442\pi\)
0.130189 + 0.991489i \(0.458442\pi\)
\(60\) 0 0
\(61\) −3.46410 −0.443533 −0.221766 0.975100i \(-0.571182\pi\)
−0.221766 + 0.975100i \(0.571182\pi\)
\(62\) 0 0
\(63\) −4.24264 −0.534522
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.44949 −0.299253 −0.149626 0.988743i \(-0.547807\pi\)
−0.149626 + 0.988743i \(0.547807\pi\)
\(68\) 0 0
\(69\) 17.3205 2.08514
\(70\) 0 0
\(71\) −6.92820 −0.822226 −0.411113 0.911584i \(-0.634860\pi\)
−0.411113 + 0.911584i \(0.634860\pi\)
\(72\) 0 0
\(73\) −4.89898 −0.573382 −0.286691 0.958023i \(-0.592555\pi\)
−0.286691 + 0.958023i \(0.592555\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.82843 −0.322329
\(78\) 0 0
\(79\) −6.92820 −0.779484 −0.389742 0.920924i \(-0.627436\pi\)
−0.389742 + 0.920924i \(0.627436\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 12.2474 1.34433 0.672166 0.740400i \(-0.265364\pi\)
0.672166 + 0.740400i \(0.265364\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 16.9706 1.81944
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 16.9706 1.75977
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.6969 −1.49225 −0.746124 0.665807i \(-0.768087\pi\)
−0.746124 + 0.665807i \(0.768087\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) −6.92820 −0.689382 −0.344691 0.938716i \(-0.612016\pi\)
−0.344691 + 0.938716i \(0.612016\pi\)
\(102\) 0 0
\(103\) 9.89949 0.975426 0.487713 0.873004i \(-0.337831\pi\)
0.487713 + 0.873004i \(0.337831\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.44949 0.236801 0.118401 0.992966i \(-0.462223\pi\)
0.118401 + 0.992966i \(0.462223\pi\)
\(108\) 0 0
\(109\) −3.46410 −0.331801 −0.165900 0.986143i \(-0.553053\pi\)
−0.165900 + 0.986143i \(0.553053\pi\)
\(110\) 0 0
\(111\) −6.92820 −0.657596
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −16.9706 −1.56893
\(118\) 0 0
\(119\) 6.92820 0.635107
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −9.79796 −0.883452
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.07107 0.627456 0.313728 0.949513i \(-0.398422\pi\)
0.313728 + 0.949513i \(0.398422\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 0 0
\(133\) −8.48528 −0.735767
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.79796 −0.837096 −0.418548 0.908195i \(-0.637461\pi\)
−0.418548 + 0.908195i \(0.637461\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) −10.3923 −0.875190
\(142\) 0 0
\(143\) −11.3137 −0.946100
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 12.2474 1.01015
\(148\) 0 0
\(149\) −10.3923 −0.851371 −0.425685 0.904871i \(-0.639967\pi\)
−0.425685 + 0.904871i \(0.639967\pi\)
\(150\) 0 0
\(151\) 13.8564 1.12762 0.563809 0.825905i \(-0.309335\pi\)
0.563809 + 0.825905i \(0.309335\pi\)
\(152\) 0 0
\(153\) −14.6969 −1.18818
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.82843 0.225733 0.112867 0.993610i \(-0.463997\pi\)
0.112867 + 0.993610i \(0.463997\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.0000 0.788110
\(162\) 0 0
\(163\) −22.0454 −1.72673 −0.863365 0.504580i \(-0.831647\pi\)
−0.863365 + 0.504580i \(0.831647\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.89949 −0.766046 −0.383023 0.923739i \(-0.625117\pi\)
−0.383023 + 0.923739i \(0.625117\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 18.0000 1.37649
\(172\) 0 0
\(173\) −8.48528 −0.645124 −0.322562 0.946548i \(-0.604544\pi\)
−0.322562 + 0.946548i \(0.604544\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.89898 −0.368230
\(178\) 0 0
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 20.7846 1.54491 0.772454 0.635071i \(-0.219029\pi\)
0.772454 + 0.635071i \(0.219029\pi\)
\(182\) 0 0
\(183\) 8.48528 0.627250
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.79796 −0.716498
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 4.89898 0.352636 0.176318 0.984333i \(-0.443581\pi\)
0.176318 + 0.984333i \(0.443581\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 22.6274 1.61214 0.806068 0.591822i \(-0.201591\pi\)
0.806068 + 0.591822i \(0.201591\pi\)
\(198\) 0 0
\(199\) −20.7846 −1.47338 −0.736691 0.676230i \(-0.763613\pi\)
−0.736691 + 0.676230i \(0.763613\pi\)
\(200\) 0 0
\(201\) 6.00000 0.423207
\(202\) 0 0
\(203\) 9.79796 0.687682
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −21.2132 −1.47442
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 16.9706 1.16280
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 9.79796 0.665129
\(218\) 0 0
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 27.7128 1.86417
\(222\) 0 0
\(223\) 26.8701 1.79935 0.899676 0.436558i \(-0.143803\pi\)
0.899676 + 0.436558i \(0.143803\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.1464 −1.13805 −0.569024 0.822321i \(-0.692679\pi\)
−0.569024 + 0.822321i \(0.692679\pi\)
\(228\) 0 0
\(229\) −6.92820 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(230\) 0 0
\(231\) 6.92820 0.455842
\(232\) 0 0
\(233\) 14.6969 0.962828 0.481414 0.876493i \(-0.340123\pi\)
0.481414 + 0.876493i \(0.340123\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 16.9706 1.10236
\(238\) 0 0
\(239\) 6.92820 0.448148 0.224074 0.974572i \(-0.428064\pi\)
0.224074 + 0.974572i \(0.428064\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 22.0454 1.41421
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −33.9411 −2.15962
\(248\) 0 0
\(249\) −30.0000 −1.90117
\(250\) 0 0
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 0 0
\(253\) −14.1421 −0.889108
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.79796 0.611180 0.305590 0.952163i \(-0.401146\pi\)
0.305590 + 0.952163i \(0.401146\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −20.7846 −1.28654
\(262\) 0 0
\(263\) −26.8701 −1.65688 −0.828439 0.560079i \(-0.810771\pi\)
−0.828439 + 0.560079i \(0.810771\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.89898 0.299813
\(268\) 0 0
\(269\) 17.3205 1.05605 0.528025 0.849229i \(-0.322933\pi\)
0.528025 + 0.849229i \(0.322933\pi\)
\(270\) 0 0
\(271\) 20.7846 1.26258 0.631288 0.775549i \(-0.282527\pi\)
0.631288 + 0.775549i \(0.282527\pi\)
\(272\) 0 0
\(273\) −19.5959 −1.18600
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 14.1421 0.849719 0.424859 0.905259i \(-0.360324\pi\)
0.424859 + 0.905259i \(0.360324\pi\)
\(278\) 0 0
\(279\) −20.7846 −1.24434
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −2.44949 −0.145607 −0.0728035 0.997346i \(-0.523195\pi\)
−0.0728035 + 0.997346i \(0.523195\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5.65685 −0.333914
\(288\) 0 0
\(289\) 7.00000 0.411765
\(290\) 0 0
\(291\) 36.0000 2.11036
\(292\) 0 0
\(293\) −2.82843 −0.165238 −0.0826192 0.996581i \(-0.526329\pi\)
−0.0826192 + 0.996581i \(0.526329\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 40.0000 2.31326
\(300\) 0 0
\(301\) −3.46410 −0.199667
\(302\) 0 0
\(303\) 16.9706 0.974933
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.44949 −0.139800 −0.0698999 0.997554i \(-0.522268\pi\)
−0.0698999 + 0.997554i \(0.522268\pi\)
\(308\) 0 0
\(309\) −24.2487 −1.37946
\(310\) 0 0
\(311\) −27.7128 −1.57145 −0.785725 0.618576i \(-0.787710\pi\)
−0.785725 + 0.618576i \(0.787710\pi\)
\(312\) 0 0
\(313\) 29.3939 1.66144 0.830720 0.556690i \(-0.187929\pi\)
0.830720 + 0.556690i \(0.187929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.6274 −1.27088 −0.635441 0.772149i \(-0.719182\pi\)
−0.635441 + 0.772149i \(0.719182\pi\)
\(318\) 0 0
\(319\) −13.8564 −0.775810
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −29.3939 −1.63552
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.48528 0.469237
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) 0 0
\(333\) 8.48528 0.464991
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.5959 1.06746 0.533729 0.845656i \(-0.320790\pi\)
0.533729 + 0.845656i \(0.320790\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −13.8564 −0.750366
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 17.1464 0.920468 0.460234 0.887798i \(-0.347765\pi\)
0.460234 + 0.887798i \(0.347765\pi\)
\(348\) 0 0
\(349\) 6.92820 0.370858 0.185429 0.982658i \(-0.440632\pi\)
0.185429 + 0.982658i \(0.440632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 9.79796 0.521493 0.260746 0.965407i \(-0.416031\pi\)
0.260746 + 0.965407i \(0.416031\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −16.9706 −0.898177
\(358\) 0 0
\(359\) −27.7128 −1.46263 −0.731313 0.682042i \(-0.761092\pi\)
−0.731313 + 0.682042i \(0.761092\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 17.1464 0.899954
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.89949 −0.516749 −0.258375 0.966045i \(-0.583187\pi\)
−0.258375 + 0.966045i \(0.583187\pi\)
\(368\) 0 0
\(369\) 12.0000 0.624695
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 19.7990 1.02515 0.512576 0.858642i \(-0.328691\pi\)
0.512576 + 0.858642i \(0.328691\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 39.1918 2.01848
\(378\) 0 0
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) 0 0
\(381\) −17.3205 −0.887357
\(382\) 0 0
\(383\) −12.7279 −0.650366 −0.325183 0.945651i \(-0.605426\pi\)
−0.325183 + 0.945651i \(0.605426\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.34847 0.373544
\(388\) 0 0
\(389\) 3.46410 0.175637 0.0878185 0.996136i \(-0.472010\pi\)
0.0878185 + 0.996136i \(0.472010\pi\)
\(390\) 0 0
\(391\) 34.6410 1.75187
\(392\) 0 0
\(393\) −24.4949 −1.23560
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −11.3137 −0.567819 −0.283909 0.958851i \(-0.591631\pi\)
−0.283909 + 0.958851i \(0.591631\pi\)
\(398\) 0 0
\(399\) 20.7846 1.04053
\(400\) 0 0
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 39.1918 1.95228
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.65685 0.280400
\(408\) 0 0
\(409\) −20.0000 −0.988936 −0.494468 0.869196i \(-0.664637\pi\)
−0.494468 + 0.869196i \(0.664637\pi\)
\(410\) 0 0
\(411\) 24.0000 1.18383
\(412\) 0 0
\(413\) −2.82843 −0.139178
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 24.4949 1.19952
\(418\) 0 0
\(419\) 26.0000 1.27018 0.635092 0.772437i \(-0.280962\pi\)
0.635092 + 0.772437i \(0.280962\pi\)
\(420\) 0 0
\(421\) −3.46410 −0.168830 −0.0844150 0.996431i \(-0.526902\pi\)
−0.0844150 + 0.996431i \(0.526902\pi\)
\(422\) 0 0
\(423\) 12.7279 0.618853
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 4.89898 0.237078
\(428\) 0 0
\(429\) 27.7128 1.33799
\(430\) 0 0
\(431\) 6.92820 0.333720 0.166860 0.985981i \(-0.446637\pi\)
0.166860 + 0.985981i \(0.446637\pi\)
\(432\) 0 0
\(433\) 34.2929 1.64801 0.824005 0.566583i \(-0.191735\pi\)
0.824005 + 0.566583i \(0.191735\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −42.4264 −2.02953
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) −22.0454 −1.04741 −0.523704 0.851900i \(-0.675450\pi\)
−0.523704 + 0.851900i \(0.675450\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 25.4558 1.20402
\(448\) 0 0
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) 8.00000 0.376705
\(452\) 0 0
\(453\) −33.9411 −1.59469
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.5959 0.916658 0.458329 0.888783i \(-0.348448\pi\)
0.458329 + 0.888783i \(0.348448\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 20.7846 0.968036 0.484018 0.875058i \(-0.339177\pi\)
0.484018 + 0.875058i \(0.339177\pi\)
\(462\) 0 0
\(463\) 1.41421 0.0657241 0.0328620 0.999460i \(-0.489538\pi\)
0.0328620 + 0.999460i \(0.489538\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 31.8434 1.47354 0.736768 0.676146i \(-0.236351\pi\)
0.736768 + 0.676146i \(0.236351\pi\)
\(468\) 0 0
\(469\) 3.46410 0.159957
\(470\) 0 0
\(471\) −6.92820 −0.319235
\(472\) 0 0
\(473\) 4.89898 0.225255
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 41.5692 1.89935 0.949673 0.313243i \(-0.101415\pi\)
0.949673 + 0.313243i \(0.101415\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) −24.4949 −1.11456
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 26.8701 1.21760 0.608799 0.793324i \(-0.291651\pi\)
0.608799 + 0.793324i \(0.291651\pi\)
\(488\) 0 0
\(489\) 54.0000 2.44196
\(490\) 0 0
\(491\) 38.0000 1.71492 0.857458 0.514554i \(-0.172042\pi\)
0.857458 + 0.514554i \(0.172042\pi\)
\(492\) 0 0
\(493\) 33.9411 1.52863
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.79796 0.439499
\(498\) 0 0
\(499\) −6.00000 −0.268597 −0.134298 0.990941i \(-0.542878\pi\)
−0.134298 + 0.990941i \(0.542878\pi\)
\(500\) 0 0
\(501\) 24.2487 1.08335
\(502\) 0 0
\(503\) 21.2132 0.945850 0.472925 0.881103i \(-0.343198\pi\)
0.472925 + 0.881103i \(0.343198\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −46.5403 −2.06693
\(508\) 0 0
\(509\) 20.7846 0.921262 0.460631 0.887592i \(-0.347623\pi\)
0.460631 + 0.887592i \(0.347623\pi\)
\(510\) 0 0
\(511\) 6.92820 0.306486
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.48528 0.373182
\(518\) 0 0
\(519\) 20.7846 0.912343
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −2.44949 −0.107109 −0.0535544 0.998565i \(-0.517055\pi\)
−0.0535544 + 0.998565i \(0.517055\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 33.9411 1.47850
\(528\) 0 0
\(529\) 27.0000 1.17391
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −22.6274 −0.980102
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.89898 −0.211407
\(538\) 0 0
\(539\) −10.0000 −0.430730
\(540\) 0 0
\(541\) −6.92820 −0.297867 −0.148933 0.988847i \(-0.547584\pi\)
−0.148933 + 0.988847i \(0.547584\pi\)
\(542\) 0 0
\(543\) −50.9117 −2.18483
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −26.9444 −1.15206 −0.576029 0.817429i \(-0.695399\pi\)
−0.576029 + 0.817429i \(0.695399\pi\)
\(548\) 0 0
\(549\) −10.3923 −0.443533
\(550\) 0 0
\(551\) −41.5692 −1.77091
\(552\) 0 0
\(553\) 9.79796 0.416652
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 25.4558 1.07860 0.539299 0.842114i \(-0.318689\pi\)
0.539299 + 0.842114i \(0.318689\pi\)
\(558\) 0 0
\(559\) −13.8564 −0.586064
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) 7.34847 0.309701 0.154851 0.987938i \(-0.450510\pi\)
0.154851 + 0.987938i \(0.450510\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 12.7279 0.534522
\(568\) 0 0
\(569\) −28.0000 −1.17382 −0.586911 0.809652i \(-0.699656\pi\)
−0.586911 + 0.809652i \(0.699656\pi\)
\(570\) 0 0
\(571\) −30.0000 −1.25546 −0.627730 0.778431i \(-0.716016\pi\)
−0.627730 + 0.778431i \(0.716016\pi\)
\(572\) 0 0
\(573\) −33.9411 −1.41791
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 29.3939 1.22368 0.611842 0.790980i \(-0.290429\pi\)
0.611842 + 0.790980i \(0.290429\pi\)
\(578\) 0 0
\(579\) −12.0000 −0.498703
\(580\) 0 0
\(581\) −17.3205 −0.718576
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.2474 −0.505506 −0.252753 0.967531i \(-0.581336\pi\)
−0.252753 + 0.967531i \(0.581336\pi\)
\(588\) 0 0
\(589\) −41.5692 −1.71283
\(590\) 0 0
\(591\) −55.4256 −2.27991
\(592\) 0 0
\(593\) 29.3939 1.20706 0.603531 0.797340i \(-0.293760\pi\)
0.603531 + 0.797340i \(0.293760\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 50.9117 2.08368
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 0 0
\(603\) −7.34847 −0.299253
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −41.0122 −1.66463 −0.832317 0.554300i \(-0.812986\pi\)
−0.832317 + 0.554300i \(0.812986\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 5.65685 0.228478 0.114239 0.993453i \(-0.463557\pi\)
0.114239 + 0.993453i \(0.463557\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4949 0.986127 0.493064 0.869993i \(-0.335877\pi\)
0.493064 + 0.869993i \(0.335877\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.82843 0.113319
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −29.3939 −1.17388
\(628\) 0 0
\(629\) −13.8564 −0.552491
\(630\) 0 0
\(631\) 13.8564 0.551615 0.275807 0.961213i \(-0.411055\pi\)
0.275807 + 0.961213i \(0.411055\pi\)
\(632\) 0 0
\(633\) −24.4949 −0.973585
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 28.2843 1.12066
\(638\) 0 0
\(639\) −20.7846 −0.822226
\(640\) 0 0
\(641\) −40.0000 −1.57991 −0.789953 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 2.44949 0.0965984 0.0482992 0.998833i \(-0.484620\pi\)
0.0482992 + 0.998833i \(0.484620\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.24264 0.166795 0.0833977 0.996516i \(-0.473423\pi\)
0.0833977 + 0.996516i \(0.473423\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 0 0
\(653\) −5.65685 −0.221370 −0.110685 0.993856i \(-0.535304\pi\)
−0.110685 + 0.993856i \(0.535304\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −14.6969 −0.573382
\(658\) 0 0
\(659\) 2.00000 0.0779089 0.0389545 0.999241i \(-0.487597\pi\)
0.0389545 + 0.999241i \(0.487597\pi\)
\(660\) 0 0
\(661\) −10.3923 −0.404214 −0.202107 0.979363i \(-0.564779\pi\)
−0.202107 + 0.979363i \(0.564779\pi\)
\(662\) 0 0
\(663\) −67.8823 −2.63633
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 48.9898 1.89689
\(668\) 0 0
\(669\) −65.8179 −2.54467
\(670\) 0 0
\(671\) −6.92820 −0.267460
\(672\) 0 0
\(673\) −24.4949 −0.944209 −0.472104 0.881543i \(-0.656505\pi\)
−0.472104 + 0.881543i \(0.656505\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.65685 −0.217411 −0.108705 0.994074i \(-0.534670\pi\)
−0.108705 + 0.994074i \(0.534670\pi\)
\(678\) 0 0
\(679\) 20.7846 0.797640
\(680\) 0 0
\(681\) 42.0000 1.60944
\(682\) 0 0
\(683\) 2.44949 0.0937271 0.0468636 0.998901i \(-0.485077\pi\)
0.0468636 + 0.998901i \(0.485077\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 16.9706 0.647467
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) 0 0
\(693\) −8.48528 −0.322329
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −19.5959 −0.742248
\(698\) 0 0
\(699\) −36.0000 −1.36165
\(700\) 0 0
\(701\) 3.46410 0.130837 0.0654187 0.997858i \(-0.479162\pi\)
0.0654187 + 0.997858i \(0.479162\pi\)
\(702\) 0 0
\(703\) 16.9706 0.640057
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.79796 0.368490
\(708\) 0 0
\(709\) 34.6410 1.30097 0.650485 0.759519i \(-0.274566\pi\)
0.650485 + 0.759519i \(0.274566\pi\)
\(710\) 0 0
\(711\) −20.7846 −0.779484
\(712\) 0 0
\(713\) 48.9898 1.83468
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −16.9706 −0.633777
\(718\) 0 0
\(719\) −6.92820 −0.258378 −0.129189 0.991620i \(-0.541237\pi\)
−0.129189 + 0.991620i \(0.541237\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −18.3848 −0.681854 −0.340927 0.940090i \(-0.610741\pi\)
−0.340927 + 0.940090i \(0.610741\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 19.7990 0.731292 0.365646 0.930754i \(-0.380848\pi\)
0.365646 + 0.930754i \(0.380848\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.89898 −0.180456
\(738\) 0 0
\(739\) −30.0000 −1.10357 −0.551784 0.833987i \(-0.686053\pi\)
−0.551784 + 0.833987i \(0.686053\pi\)
\(740\) 0 0
\(741\) 83.1384 3.05417
\(742\) 0 0
\(743\) −4.24264 −0.155647 −0.0778237 0.996967i \(-0.524797\pi\)
−0.0778237 + 0.996967i \(0.524797\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 36.7423 1.34433
\(748\) 0 0
\(749\) −3.46410 −0.126576
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) −63.6867 −2.32087
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.82843 −0.102801 −0.0514005 0.998678i \(-0.516368\pi\)
−0.0514005 + 0.998678i \(0.516368\pi\)
\(758\) 0 0
\(759\) 34.6410 1.25739
\(760\) 0 0
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) 4.89898 0.177355
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.3137 −0.408514
\(768\) 0 0
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 0 0
\(773\) 45.2548 1.62770 0.813852 0.581073i \(-0.197367\pi\)
0.813852 + 0.581073i \(0.197367\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 9.79796 0.351500
\(778\) 0 0
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.5403 1.65898 0.829491 0.558520i \(-0.188630\pi\)
0.829491 + 0.558520i \(0.188630\pi\)
\(788\) 0 0
\(789\) 65.8179 2.34318
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 19.5959 0.695871
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −33.9411 −1.20226 −0.601128 0.799153i \(-0.705282\pi\)
−0.601128 + 0.799153i \(0.705282\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) −9.79796 −0.345762
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −42.4264 −1.49348
\(808\) 0 0
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) −22.0000 −0.772524 −0.386262 0.922389i \(-0.626234\pi\)
−0.386262 + 0.922389i \(0.626234\pi\)
\(812\) 0 0
\(813\) −50.9117 −1.78555
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 14.6969 0.514181
\(818\) 0 0
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) −51.9615 −1.81347 −0.906735 0.421701i \(-0.861433\pi\)
−0.906735 + 0.421701i \(0.861433\pi\)
\(822\) 0 0
\(823\) −35.3553 −1.23241 −0.616205 0.787586i \(-0.711331\pi\)
−0.616205 + 0.787586i \(0.711331\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −31.8434 −1.10730 −0.553651 0.832749i \(-0.686766\pi\)
−0.553651 + 0.832749i \(0.686766\pi\)
\(828\) 0 0
\(829\) 45.0333 1.56407 0.782036 0.623233i \(-0.214181\pi\)
0.782036 + 0.623233i \(0.214181\pi\)
\(830\) 0 0
\(831\) −34.6410 −1.20168
\(832\) 0 0
\(833\) 24.4949 0.848698
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 41.5692 1.43513 0.717564 0.696492i \(-0.245257\pi\)
0.717564 + 0.696492i \(0.245257\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 0 0
\(843\) 19.5959 0.674919
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 9.89949 0.340151
\(848\) 0 0
\(849\) 6.00000 0.205919
\(850\) 0 0
\(851\) −20.0000 −0.685591
\(852\) 0 0
\(853\) 5.65685 0.193687 0.0968435 0.995300i \(-0.469125\pi\)
0.0968435 + 0.995300i \(0.469125\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.1918 −1.33877 −0.669384 0.742917i \(-0.733442\pi\)
−0.669384 + 0.742917i \(0.733442\pi\)
\(858\) 0 0
\(859\) −22.0000 −0.750630 −0.375315 0.926897i \(-0.622466\pi\)
−0.375315 + 0.926897i \(0.622466\pi\)
\(860\) 0 0
\(861\) 13.8564 0.472225
\(862\) 0 0
\(863\) 24.0416 0.818387 0.409193 0.912448i \(-0.365810\pi\)
0.409193 + 0.912448i \(0.365810\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −17.1464 −0.582323
\(868\) 0 0
\(869\) −13.8564 −0.470046
\(870\) 0 0
\(871\) 13.8564 0.469506
\(872\) 0 0
\(873\) −44.0908 −1.49225
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36.7696 −1.24162 −0.620810 0.783961i \(-0.713196\pi\)
−0.620810 + 0.783961i \(0.713196\pi\)
\(878\) 0 0
\(879\) 6.92820 0.233682
\(880\) 0 0
\(881\) 52.0000 1.75192 0.875962 0.482380i \(-0.160227\pi\)
0.875962 + 0.482380i \(0.160227\pi\)
\(882\) 0 0
\(883\) −12.2474 −0.412159 −0.206080 0.978535i \(-0.566071\pi\)
−0.206080 + 0.978535i \(0.566071\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.4975 1.66196 0.830981 0.556300i \(-0.187780\pi\)
0.830981 + 0.556300i \(0.187780\pi\)
\(888\) 0 0
\(889\) −10.0000 −0.335389
\(890\) 0 0
\(891\) −18.0000 −0.603023
\(892\) 0 0
\(893\) 25.4558 0.851847
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −97.9796 −3.27144
\(898\) 0 0
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 8.48528 0.282372
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −26.9444 −0.894674 −0.447337 0.894366i \(-0.647627\pi\)
−0.447337 + 0.894366i \(0.647627\pi\)
\(908\) 0 0
\(909\) −20.7846 −0.689382
\(910\) 0 0
\(911\) 48.4974 1.60679 0.803396 0.595446i \(-0.203024\pi\)
0.803396 + 0.595446i \(0.203024\pi\)
\(912\) 0 0
\(913\) 24.4949 0.810663
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −14.1421 −0.467014
\(918\) 0 0
\(919\) −6.92820 −0.228540 −0.114270 0.993450i \(-0.536453\pi\)
−0.114270 + 0.993450i \(0.536453\pi\)
\(920\) 0 0
\(921\) 6.00000 0.197707
\(922\) 0 0
\(923\) 39.1918 1.29001
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 29.6985 0.975426
\(928\) 0 0
\(929\) 8.00000 0.262471 0.131236 0.991351i \(-0.458106\pi\)
0.131236 + 0.991351i \(0.458106\pi\)
\(930\) 0 0
\(931\) −30.0000 −0.983210
\(932\) 0 0
\(933\) 67.8823 2.22237
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24.4949 −0.800213 −0.400107 0.916469i \(-0.631027\pi\)
−0.400107 + 0.916469i \(0.631027\pi\)
\(938\) 0 0
\(939\) −72.0000 −2.34963
\(940\) 0 0
\(941\) −34.6410 −1.12926 −0.564632 0.825342i \(-0.690982\pi\)
−0.564632 + 0.825342i \(0.690982\pi\)
\(942\) 0 0
\(943\) −28.2843 −0.921063
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −41.6413 −1.35316 −0.676581 0.736369i \(-0.736539\pi\)
−0.676581 + 0.736369i \(0.736539\pi\)
\(948\) 0 0
\(949\) 27.7128 0.899596
\(950\) 0 0
\(951\) 55.4256 1.79730
\(952\) 0 0
\(953\) −39.1918 −1.26955 −0.634774 0.772698i \(-0.718907\pi\)
−0.634774 + 0.772698i \(0.718907\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 33.9411 1.09716
\(958\) 0 0
\(959\) 13.8564 0.447447
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) 7.34847 0.236801
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −15.5563 −0.500258 −0.250129 0.968212i \(-0.580473\pi\)
−0.250129 + 0.968212i \(0.580473\pi\)
\(968\) 0 0
\(969\) 72.0000 2.31297
\(970\) 0 0
\(971\) 22.0000 0.706014 0.353007 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 0 0
\(973\) 14.1421 0.453376
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.89898 0.156732 0.0783661 0.996925i \(-0.475030\pi\)
0.0783661 + 0.996925i \(0.475030\pi\)
\(978\) 0 0
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) −10.3923 −0.331801
\(982\) 0 0
\(983\) 46.6690 1.48851 0.744256 0.667895i \(-0.232804\pi\)
0.744256 + 0.667895i \(0.232804\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 14.6969 0.467809
\(988\) 0 0
\(989\) −17.3205 −0.550760
\(990\) 0 0
\(991\) −27.7128 −0.880327 −0.440163 0.897918i \(-0.645079\pi\)
−0.440163 + 0.897918i \(0.645079\pi\)
\(992\) 0 0
\(993\) −14.6969 −0.466393
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −56.5685 −1.79154 −0.895772 0.444514i \(-0.853376\pi\)
−0.895772 + 0.444514i \(0.853376\pi\)
\(998\) 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 6400.2.a.cu.1.1 4
4.3 odd 2 6400.2.a.ct.1.4 4
5.2 odd 4 1280.2.c.h.769.4 4
5.3 odd 4 1280.2.c.h.769.2 4
5.4 even 2 inner 6400.2.a.cu.1.4 4
8.3 odd 2 inner 6400.2.a.cu.1.2 4
8.5 even 2 6400.2.a.ct.1.3 4
16.3 odd 4 1600.2.d.i.801.6 8
16.5 even 4 1600.2.d.i.801.8 8
16.11 odd 4 1600.2.d.i.801.1 8
16.13 even 4 1600.2.d.i.801.3 8
20.3 even 4 1280.2.c.g.769.4 4
20.7 even 4 1280.2.c.g.769.2 4
20.19 odd 2 6400.2.a.ct.1.1 4
40.3 even 4 1280.2.c.h.769.1 4
40.13 odd 4 1280.2.c.g.769.3 4
40.19 odd 2 inner 6400.2.a.cu.1.3 4
40.27 even 4 1280.2.c.h.769.3 4
40.29 even 2 6400.2.a.ct.1.2 4
40.37 odd 4 1280.2.c.g.769.1 4
80.3 even 4 320.2.f.b.289.1 8
80.13 odd 4 320.2.f.b.289.5 yes 8
80.19 odd 4 1600.2.d.i.801.4 8
80.27 even 4 320.2.f.b.289.2 yes 8
80.29 even 4 1600.2.d.i.801.5 8
80.37 odd 4 320.2.f.b.289.6 yes 8
80.43 even 4 320.2.f.b.289.8 yes 8
80.53 odd 4 320.2.f.b.289.4 yes 8
80.59 odd 4 1600.2.d.i.801.7 8
80.67 even 4 320.2.f.b.289.7 yes 8
80.69 even 4 1600.2.d.i.801.2 8
80.77 odd 4 320.2.f.b.289.3 yes 8
240.53 even 4 2880.2.d.g.289.1 8
240.77 even 4 2880.2.d.g.289.4 8
240.83 odd 4 2880.2.d.g.289.8 8
240.107 odd 4 2880.2.d.g.289.5 8
240.173 even 4 2880.2.d.g.289.7 8
240.197 even 4 2880.2.d.g.289.6 8
240.203 odd 4 2880.2.d.g.289.2 8
240.227 odd 4 2880.2.d.g.289.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
320.2.f.b.289.1 8 80.3 even 4
320.2.f.b.289.2 yes 8 80.27 even 4
320.2.f.b.289.3 yes 8 80.77 odd 4
320.2.f.b.289.4 yes 8 80.53 odd 4
320.2.f.b.289.5 yes 8 80.13 odd 4
320.2.f.b.289.6 yes 8 80.37 odd 4
320.2.f.b.289.7 yes 8 80.67 even 4
320.2.f.b.289.8 yes 8 80.43 even 4
1280.2.c.g.769.1 4 40.37 odd 4
1280.2.c.g.769.2 4 20.7 even 4
1280.2.c.g.769.3 4 40.13 odd 4
1280.2.c.g.769.4 4 20.3 even 4
1280.2.c.h.769.1 4 40.3 even 4
1280.2.c.h.769.2 4 5.3 odd 4
1280.2.c.h.769.3 4 40.27 even 4
1280.2.c.h.769.4 4 5.2 odd 4
1600.2.d.i.801.1 8 16.11 odd 4
1600.2.d.i.801.2 8 80.69 even 4
1600.2.d.i.801.3 8 16.13 even 4
1600.2.d.i.801.4 8 80.19 odd 4
1600.2.d.i.801.5 8 80.29 even 4
1600.2.d.i.801.6 8 16.3 odd 4
1600.2.d.i.801.7 8 80.59 odd 4
1600.2.d.i.801.8 8 16.5 even 4
2880.2.d.g.289.1 8 240.53 even 4
2880.2.d.g.289.2 8 240.203 odd 4
2880.2.d.g.289.3 8 240.227 odd 4
2880.2.d.g.289.4 8 240.77 even 4
2880.2.d.g.289.5 8 240.107 odd 4
2880.2.d.g.289.6 8 240.197 even 4
2880.2.d.g.289.7 8 240.173 even 4
2880.2.d.g.289.8 8 240.83 odd 4
6400.2.a.ct.1.1 4 20.19 odd 2
6400.2.a.ct.1.2 4 40.29 even 2
6400.2.a.ct.1.3 4 8.5 even 2
6400.2.a.ct.1.4 4 4.3 odd 2
6400.2.a.cu.1.1 4 1.1 even 1 trivial
6400.2.a.cu.1.2 4 8.3 odd 2 inner
6400.2.a.cu.1.3 4 40.19 odd 2 inner
6400.2.a.cu.1.4 4 5.4 even 2 inner