Properties

Label 9702.2.a.cz.1.2
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9702.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(77.4708600410\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.64575 q^{5} +1.00000 q^{8} +1.64575 q^{10} -1.00000 q^{11} +5.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} -5.64575 q^{19} +1.64575 q^{20} -1.00000 q^{22} -1.64575 q^{23} -2.29150 q^{25} +5.00000 q^{26} -6.29150 q^{29} -4.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} +3.64575 q^{37} -5.64575 q^{38} +1.64575 q^{40} -10.9373 q^{41} -4.00000 q^{43} -1.00000 q^{44} -1.64575 q^{46} -2.70850 q^{47} -2.29150 q^{50} +5.00000 q^{52} -1.64575 q^{53} -1.64575 q^{55} -6.29150 q^{58} -4.64575 q^{59} +14.2915 q^{61} -4.00000 q^{62} +1.00000 q^{64} +8.22876 q^{65} -11.9373 q^{67} -6.00000 q^{68} -4.35425 q^{71} +0.354249 q^{73} +3.64575 q^{74} -5.64575 q^{76} -2.64575 q^{79} +1.64575 q^{80} -10.9373 q^{82} -2.70850 q^{83} -9.87451 q^{85} -4.00000 q^{86} -1.00000 q^{88} -6.58301 q^{89} -1.64575 q^{92} -2.70850 q^{94} -9.29150 q^{95} -16.2915 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 2 q^{8} - 2 q^{10} - 2 q^{11} + 10 q^{13} + 2 q^{16} - 12 q^{17} - 6 q^{19} - 2 q^{20} - 2 q^{22} + 2 q^{23} + 6 q^{25} + 10 q^{26} - 2 q^{29} - 8 q^{31} + 2 q^{32} - 12 q^{34} + 2 q^{37} - 6 q^{38} - 2 q^{40} - 6 q^{41} - 8 q^{43} - 2 q^{44} + 2 q^{46} - 16 q^{47} + 6 q^{50} + 10 q^{52} + 2 q^{53} + 2 q^{55} - 2 q^{58} - 4 q^{59} + 18 q^{61} - 8 q^{62} + 2 q^{64} - 10 q^{65} - 8 q^{67} - 12 q^{68} - 14 q^{71} + 6 q^{73} + 2 q^{74} - 6 q^{76} - 2 q^{80} - 6 q^{82} - 16 q^{83} + 12 q^{85} - 8 q^{86} - 2 q^{88} + 8 q^{89} + 2 q^{92} - 16 q^{94} - 8 q^{95} - 22 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.64575 0.736002 0.368001 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.64575 0.520432
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −5.64575 −1.29522 −0.647612 0.761970i \(-0.724232\pi\)
−0.647612 + 0.761970i \(0.724232\pi\)
\(20\) 1.64575 0.368001
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −1.64575 −0.343163 −0.171581 0.985170i \(-0.554888\pi\)
−0.171581 + 0.985170i \(0.554888\pi\)
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 5.00000 0.980581
\(27\) 0 0
\(28\) 0 0
\(29\) −6.29150 −1.16830 −0.584151 0.811645i \(-0.698573\pi\)
−0.584151 + 0.811645i \(0.698573\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 3.64575 0.599358 0.299679 0.954040i \(-0.403120\pi\)
0.299679 + 0.954040i \(0.403120\pi\)
\(38\) −5.64575 −0.915862
\(39\) 0 0
\(40\) 1.64575 0.260216
\(41\) −10.9373 −1.70811 −0.854056 0.520181i \(-0.825864\pi\)
−0.854056 + 0.520181i \(0.825864\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −1.64575 −0.242653
\(47\) −2.70850 −0.395075 −0.197537 0.980295i \(-0.563294\pi\)
−0.197537 + 0.980295i \(0.563294\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2.29150 −0.324067
\(51\) 0 0
\(52\) 5.00000 0.693375
\(53\) −1.64575 −0.226061 −0.113031 0.993592i \(-0.536056\pi\)
−0.113031 + 0.993592i \(0.536056\pi\)
\(54\) 0 0
\(55\) −1.64575 −0.221913
\(56\) 0 0
\(57\) 0 0
\(58\) −6.29150 −0.826115
\(59\) −4.64575 −0.604825 −0.302413 0.953177i \(-0.597792\pi\)
−0.302413 + 0.953177i \(0.597792\pi\)
\(60\) 0 0
\(61\) 14.2915 1.82984 0.914920 0.403636i \(-0.132254\pi\)
0.914920 + 0.403636i \(0.132254\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.22876 1.02065
\(66\) 0 0
\(67\) −11.9373 −1.45837 −0.729184 0.684318i \(-0.760100\pi\)
−0.729184 + 0.684318i \(0.760100\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −4.35425 −0.516754 −0.258377 0.966044i \(-0.583188\pi\)
−0.258377 + 0.966044i \(0.583188\pi\)
\(72\) 0 0
\(73\) 0.354249 0.0414617 0.0207308 0.999785i \(-0.493401\pi\)
0.0207308 + 0.999785i \(0.493401\pi\)
\(74\) 3.64575 0.423810
\(75\) 0 0
\(76\) −5.64575 −0.647612
\(77\) 0 0
\(78\) 0 0
\(79\) −2.64575 −0.297670 −0.148835 0.988862i \(-0.547552\pi\)
−0.148835 + 0.988862i \(0.547552\pi\)
\(80\) 1.64575 0.184001
\(81\) 0 0
\(82\) −10.9373 −1.20782
\(83\) −2.70850 −0.297296 −0.148648 0.988890i \(-0.547492\pi\)
−0.148648 + 0.988890i \(0.547492\pi\)
\(84\) 0 0
\(85\) −9.87451 −1.07104
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −6.58301 −0.697797 −0.348899 0.937160i \(-0.613444\pi\)
−0.348899 + 0.937160i \(0.613444\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.64575 −0.171581
\(93\) 0 0
\(94\) −2.70850 −0.279360
\(95\) −9.29150 −0.953288
\(96\) 0 0
\(97\) −16.2915 −1.65415 −0.827076 0.562090i \(-0.809997\pi\)
−0.827076 + 0.562090i \(0.809997\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.29150 −0.229150
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) −2.93725 −0.289416 −0.144708 0.989474i \(-0.546224\pi\)
−0.144708 + 0.989474i \(0.546224\pi\)
\(104\) 5.00000 0.490290
\(105\) 0 0
\(106\) −1.64575 −0.159849
\(107\) 10.9373 1.05734 0.528672 0.848826i \(-0.322690\pi\)
0.528672 + 0.848826i \(0.322690\pi\)
\(108\) 0 0
\(109\) −10.5830 −1.01367 −0.506834 0.862044i \(-0.669184\pi\)
−0.506834 + 0.862044i \(0.669184\pi\)
\(110\) −1.64575 −0.156916
\(111\) 0 0
\(112\) 0 0
\(113\) 18.2915 1.72072 0.860360 0.509687i \(-0.170239\pi\)
0.860360 + 0.509687i \(0.170239\pi\)
\(114\) 0 0
\(115\) −2.70850 −0.252569
\(116\) −6.29150 −0.584151
\(117\) 0 0
\(118\) −4.64575 −0.427676
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.2915 1.29389
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 15.9373 1.41420 0.707101 0.707112i \(-0.250002\pi\)
0.707101 + 0.707112i \(0.250002\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 8.22876 0.721710
\(131\) −10.3542 −0.904655 −0.452327 0.891852i \(-0.649406\pi\)
−0.452327 + 0.891852i \(0.649406\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −11.9373 −1.03122
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 12.8745 1.09994 0.549972 0.835183i \(-0.314638\pi\)
0.549972 + 0.835183i \(0.314638\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.35425 −0.365400
\(143\) −5.00000 −0.418121
\(144\) 0 0
\(145\) −10.3542 −0.859874
\(146\) 0.354249 0.0293178
\(147\) 0 0
\(148\) 3.64575 0.299679
\(149\) 15.2915 1.25273 0.626364 0.779530i \(-0.284542\pi\)
0.626364 + 0.779530i \(0.284542\pi\)
\(150\) 0 0
\(151\) −8.64575 −0.703581 −0.351791 0.936079i \(-0.614427\pi\)
−0.351791 + 0.936079i \(0.614427\pi\)
\(152\) −5.64575 −0.457931
\(153\) 0 0
\(154\) 0 0
\(155\) −6.58301 −0.528760
\(156\) 0 0
\(157\) 21.1660 1.68923 0.844616 0.535373i \(-0.179829\pi\)
0.844616 + 0.535373i \(0.179829\pi\)
\(158\) −2.64575 −0.210485
\(159\) 0 0
\(160\) 1.64575 0.130108
\(161\) 0 0
\(162\) 0 0
\(163\) 0.645751 0.0505791 0.0252896 0.999680i \(-0.491949\pi\)
0.0252896 + 0.999680i \(0.491949\pi\)
\(164\) −10.9373 −0.854056
\(165\) 0 0
\(166\) −2.70850 −0.210220
\(167\) 11.2288 0.868907 0.434454 0.900694i \(-0.356941\pi\)
0.434454 + 0.900694i \(0.356941\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) −9.87451 −0.757340
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 0.291503 0.0221625 0.0110813 0.999939i \(-0.496473\pi\)
0.0110813 + 0.999939i \(0.496473\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −6.58301 −0.493417
\(179\) 19.9373 1.49018 0.745090 0.666964i \(-0.232406\pi\)
0.745090 + 0.666964i \(0.232406\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.64575 −0.121326
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 6.00000 0.438763
\(188\) −2.70850 −0.197537
\(189\) 0 0
\(190\) −9.29150 −0.674076
\(191\) 2.70850 0.195980 0.0979900 0.995187i \(-0.468759\pi\)
0.0979900 + 0.995187i \(0.468759\pi\)
\(192\) 0 0
\(193\) 25.5203 1.83699 0.918494 0.395434i \(-0.129406\pi\)
0.918494 + 0.395434i \(0.129406\pi\)
\(194\) −16.2915 −1.16966
\(195\) 0 0
\(196\) 0 0
\(197\) 12.8745 0.917271 0.458635 0.888625i \(-0.348338\pi\)
0.458635 + 0.888625i \(0.348338\pi\)
\(198\) 0 0
\(199\) 4.22876 0.299769 0.149884 0.988704i \(-0.452110\pi\)
0.149884 + 0.988704i \(0.452110\pi\)
\(200\) −2.29150 −0.162034
\(201\) 0 0
\(202\) −3.00000 −0.211079
\(203\) 0 0
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) −2.93725 −0.204648
\(207\) 0 0
\(208\) 5.00000 0.346688
\(209\) 5.64575 0.390525
\(210\) 0 0
\(211\) 0.937254 0.0645232 0.0322616 0.999479i \(-0.489729\pi\)
0.0322616 + 0.999479i \(0.489729\pi\)
\(212\) −1.64575 −0.113031
\(213\) 0 0
\(214\) 10.9373 0.747655
\(215\) −6.58301 −0.448957
\(216\) 0 0
\(217\) 0 0
\(218\) −10.5830 −0.716772
\(219\) 0 0
\(220\) −1.64575 −0.110957
\(221\) −30.0000 −2.01802
\(222\) 0 0
\(223\) −17.6458 −1.18165 −0.590823 0.806801i \(-0.701197\pi\)
−0.590823 + 0.806801i \(0.701197\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.2915 1.21673
\(227\) 2.70850 0.179769 0.0898846 0.995952i \(-0.471350\pi\)
0.0898846 + 0.995952i \(0.471350\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −2.70850 −0.178593
\(231\) 0 0
\(232\) −6.29150 −0.413057
\(233\) −1.06275 −0.0696228 −0.0348114 0.999394i \(-0.511083\pi\)
−0.0348114 + 0.999394i \(0.511083\pi\)
\(234\) 0 0
\(235\) −4.45751 −0.290776
\(236\) −4.64575 −0.302413
\(237\) 0 0
\(238\) 0 0
\(239\) 17.2288 1.11444 0.557218 0.830366i \(-0.311869\pi\)
0.557218 + 0.830366i \(0.311869\pi\)
\(240\) 0 0
\(241\) −24.8118 −1.59827 −0.799133 0.601154i \(-0.794708\pi\)
−0.799133 + 0.601154i \(0.794708\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 14.2915 0.914920
\(245\) 0 0
\(246\) 0 0
\(247\) −28.2288 −1.79615
\(248\) −4.00000 −0.254000
\(249\) 0 0
\(250\) −12.0000 −0.758947
\(251\) 3.29150 0.207758 0.103879 0.994590i \(-0.466875\pi\)
0.103879 + 0.994590i \(0.466875\pi\)
\(252\) 0 0
\(253\) 1.64575 0.103467
\(254\) 15.9373 0.999992
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.5830 1.34631 0.673155 0.739501i \(-0.264938\pi\)
0.673155 + 0.739501i \(0.264938\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 8.22876 0.510326
\(261\) 0 0
\(262\) −10.3542 −0.639688
\(263\) −19.9373 −1.22938 −0.614692 0.788767i \(-0.710720\pi\)
−0.614692 + 0.788767i \(0.710720\pi\)
\(264\) 0 0
\(265\) −2.70850 −0.166382
\(266\) 0 0
\(267\) 0 0
\(268\) −11.9373 −0.729184
\(269\) −5.41699 −0.330280 −0.165140 0.986270i \(-0.552808\pi\)
−0.165140 + 0.986270i \(0.552808\pi\)
\(270\) 0 0
\(271\) −2.06275 −0.125303 −0.0626514 0.998035i \(-0.519956\pi\)
−0.0626514 + 0.998035i \(0.519956\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) 12.8745 0.777777
\(275\) 2.29150 0.138183
\(276\) 0 0
\(277\) −22.2915 −1.33937 −0.669683 0.742647i \(-0.733570\pi\)
−0.669683 + 0.742647i \(0.733570\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 22.9373 1.36832 0.684161 0.729331i \(-0.260169\pi\)
0.684161 + 0.729331i \(0.260169\pi\)
\(282\) 0 0
\(283\) −2.35425 −0.139946 −0.0699728 0.997549i \(-0.522291\pi\)
−0.0699728 + 0.997549i \(0.522291\pi\)
\(284\) −4.35425 −0.258377
\(285\) 0 0
\(286\) −5.00000 −0.295656
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −10.3542 −0.608022
\(291\) 0 0
\(292\) 0.354249 0.0207308
\(293\) −12.0000 −0.701047 −0.350524 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) −7.64575 −0.445153
\(296\) 3.64575 0.211905
\(297\) 0 0
\(298\) 15.2915 0.885813
\(299\) −8.22876 −0.475881
\(300\) 0 0
\(301\) 0 0
\(302\) −8.64575 −0.497507
\(303\) 0 0
\(304\) −5.64575 −0.323806
\(305\) 23.5203 1.34677
\(306\) 0 0
\(307\) 22.2288 1.26866 0.634331 0.773062i \(-0.281276\pi\)
0.634331 + 0.773062i \(0.281276\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −6.58301 −0.373890
\(311\) −1.06275 −0.0602628 −0.0301314 0.999546i \(-0.509593\pi\)
−0.0301314 + 0.999546i \(0.509593\pi\)
\(312\) 0 0
\(313\) 23.5830 1.33299 0.666495 0.745509i \(-0.267794\pi\)
0.666495 + 0.745509i \(0.267794\pi\)
\(314\) 21.1660 1.19447
\(315\) 0 0
\(316\) −2.64575 −0.148835
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 6.29150 0.352257
\(320\) 1.64575 0.0920003
\(321\) 0 0
\(322\) 0 0
\(323\) 33.8745 1.88483
\(324\) 0 0
\(325\) −11.4575 −0.635548
\(326\) 0.645751 0.0357649
\(327\) 0 0
\(328\) −10.9373 −0.603909
\(329\) 0 0
\(330\) 0 0
\(331\) −20.6458 −1.13479 −0.567397 0.823445i \(-0.692049\pi\)
−0.567397 + 0.823445i \(0.692049\pi\)
\(332\) −2.70850 −0.148648
\(333\) 0 0
\(334\) 11.2288 0.614410
\(335\) −19.6458 −1.07336
\(336\) 0 0
\(337\) 9.06275 0.493679 0.246840 0.969056i \(-0.420608\pi\)
0.246840 + 0.969056i \(0.420608\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) −9.87451 −0.535520
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 0.291503 0.0156713
\(347\) 20.8118 1.11723 0.558617 0.829426i \(-0.311332\pi\)
0.558617 + 0.829426i \(0.311332\pi\)
\(348\) 0 0
\(349\) −1.87451 −0.100340 −0.0501701 0.998741i \(-0.515976\pi\)
−0.0501701 + 0.998741i \(0.515976\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −7.16601 −0.381408 −0.190704 0.981648i \(-0.561077\pi\)
−0.190704 + 0.981648i \(0.561077\pi\)
\(354\) 0 0
\(355\) −7.16601 −0.380332
\(356\) −6.58301 −0.348899
\(357\) 0 0
\(358\) 19.9373 1.05372
\(359\) 25.9373 1.36892 0.684458 0.729052i \(-0.260039\pi\)
0.684458 + 0.729052i \(0.260039\pi\)
\(360\) 0 0
\(361\) 12.8745 0.677606
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 0 0
\(365\) 0.583005 0.0305159
\(366\) 0 0
\(367\) −36.2288 −1.89113 −0.945563 0.325440i \(-0.894488\pi\)
−0.945563 + 0.325440i \(0.894488\pi\)
\(368\) −1.64575 −0.0857907
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 0 0
\(373\) 20.8745 1.08084 0.540421 0.841395i \(-0.318265\pi\)
0.540421 + 0.841395i \(0.318265\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) −2.70850 −0.139680
\(377\) −31.4575 −1.62014
\(378\) 0 0
\(379\) 6.06275 0.311422 0.155711 0.987803i \(-0.450233\pi\)
0.155711 + 0.987803i \(0.450233\pi\)
\(380\) −9.29150 −0.476644
\(381\) 0 0
\(382\) 2.70850 0.138579
\(383\) −24.1033 −1.23162 −0.615810 0.787895i \(-0.711171\pi\)
−0.615810 + 0.787895i \(0.711171\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 25.5203 1.29895
\(387\) 0 0
\(388\) −16.2915 −0.827076
\(389\) −26.8118 −1.35941 −0.679705 0.733485i \(-0.737892\pi\)
−0.679705 + 0.733485i \(0.737892\pi\)
\(390\) 0 0
\(391\) 9.87451 0.499375
\(392\) 0 0
\(393\) 0 0
\(394\) 12.8745 0.648608
\(395\) −4.35425 −0.219086
\(396\) 0 0
\(397\) −11.1660 −0.560406 −0.280203 0.959941i \(-0.590402\pi\)
−0.280203 + 0.959941i \(0.590402\pi\)
\(398\) 4.22876 0.211968
\(399\) 0 0
\(400\) −2.29150 −0.114575
\(401\) −21.5830 −1.07780 −0.538902 0.842369i \(-0.681161\pi\)
−0.538902 + 0.842369i \(0.681161\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −3.00000 −0.149256
\(405\) 0 0
\(406\) 0 0
\(407\) −3.64575 −0.180713
\(408\) 0 0
\(409\) 3.06275 0.151443 0.0757215 0.997129i \(-0.475874\pi\)
0.0757215 + 0.997129i \(0.475874\pi\)
\(410\) −18.0000 −0.888957
\(411\) 0 0
\(412\) −2.93725 −0.144708
\(413\) 0 0
\(414\) 0 0
\(415\) −4.45751 −0.218811
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 5.64575 0.276143
\(419\) 9.87451 0.482401 0.241201 0.970475i \(-0.422459\pi\)
0.241201 + 0.970475i \(0.422459\pi\)
\(420\) 0 0
\(421\) 9.16601 0.446724 0.223362 0.974736i \(-0.428297\pi\)
0.223362 + 0.974736i \(0.428297\pi\)
\(422\) 0.937254 0.0456248
\(423\) 0 0
\(424\) −1.64575 −0.0799247
\(425\) 13.7490 0.666925
\(426\) 0 0
\(427\) 0 0
\(428\) 10.9373 0.528672
\(429\) 0 0
\(430\) −6.58301 −0.317461
\(431\) −29.2288 −1.40790 −0.703950 0.710250i \(-0.748582\pi\)
−0.703950 + 0.710250i \(0.748582\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.5830 −0.506834
\(437\) 9.29150 0.444473
\(438\) 0 0
\(439\) 3.93725 0.187915 0.0939574 0.995576i \(-0.470048\pi\)
0.0939574 + 0.995576i \(0.470048\pi\)
\(440\) −1.64575 −0.0784581
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) −34.4575 −1.63713 −0.818563 0.574417i \(-0.805229\pi\)
−0.818563 + 0.574417i \(0.805229\pi\)
\(444\) 0 0
\(445\) −10.8340 −0.513580
\(446\) −17.6458 −0.835551
\(447\) 0 0
\(448\) 0 0
\(449\) 21.8745 1.03232 0.516161 0.856492i \(-0.327361\pi\)
0.516161 + 0.856492i \(0.327361\pi\)
\(450\) 0 0
\(451\) 10.9373 0.515015
\(452\) 18.2915 0.860360
\(453\) 0 0
\(454\) 2.70850 0.127116
\(455\) 0 0
\(456\) 0 0
\(457\) 3.16601 0.148100 0.0740499 0.997255i \(-0.476408\pi\)
0.0740499 + 0.997255i \(0.476408\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) −2.70850 −0.126284
\(461\) −10.1660 −0.473478 −0.236739 0.971573i \(-0.576079\pi\)
−0.236739 + 0.971573i \(0.576079\pi\)
\(462\) 0 0
\(463\) 30.4575 1.41548 0.707740 0.706473i \(-0.249715\pi\)
0.707740 + 0.706473i \(0.249715\pi\)
\(464\) −6.29150 −0.292076
\(465\) 0 0
\(466\) −1.06275 −0.0492308
\(467\) −21.2915 −0.985253 −0.492627 0.870241i \(-0.663963\pi\)
−0.492627 + 0.870241i \(0.663963\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.45751 −0.205610
\(471\) 0 0
\(472\) −4.64575 −0.213838
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 12.9373 0.593602
\(476\) 0 0
\(477\) 0 0
\(478\) 17.2288 0.788025
\(479\) 10.0627 0.459779 0.229889 0.973217i \(-0.426164\pi\)
0.229889 + 0.973217i \(0.426164\pi\)
\(480\) 0 0
\(481\) 18.2288 0.831160
\(482\) −24.8118 −1.13014
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −26.8118 −1.21746
\(486\) 0 0
\(487\) −9.41699 −0.426725 −0.213362 0.976973i \(-0.568441\pi\)
−0.213362 + 0.976973i \(0.568441\pi\)
\(488\) 14.2915 0.646946
\(489\) 0 0
\(490\) 0 0
\(491\) −21.2915 −0.960872 −0.480436 0.877030i \(-0.659522\pi\)
−0.480436 + 0.877030i \(0.659522\pi\)
\(492\) 0 0
\(493\) 37.7490 1.70013
\(494\) −28.2288 −1.27007
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) 0 0
\(499\) −13.8745 −0.621108 −0.310554 0.950556i \(-0.600515\pi\)
−0.310554 + 0.950556i \(0.600515\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 3.29150 0.146907
\(503\) 4.06275 0.181149 0.0905744 0.995890i \(-0.471130\pi\)
0.0905744 + 0.995890i \(0.471130\pi\)
\(504\) 0 0
\(505\) −4.93725 −0.219705
\(506\) 1.64575 0.0731626
\(507\) 0 0
\(508\) 15.9373 0.707101
\(509\) 0.583005 0.0258413 0.0129206 0.999917i \(-0.495887\pi\)
0.0129206 + 0.999917i \(0.495887\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 21.5830 0.951986
\(515\) −4.83399 −0.213011
\(516\) 0 0
\(517\) 2.70850 0.119120
\(518\) 0 0
\(519\) 0 0
\(520\) 8.22876 0.360855
\(521\) 33.8745 1.48407 0.742035 0.670362i \(-0.233861\pi\)
0.742035 + 0.670362i \(0.233861\pi\)
\(522\) 0 0
\(523\) −21.5203 −0.941015 −0.470508 0.882396i \(-0.655929\pi\)
−0.470508 + 0.882396i \(0.655929\pi\)
\(524\) −10.3542 −0.452327
\(525\) 0 0
\(526\) −19.9373 −0.869306
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −20.2915 −0.882239
\(530\) −2.70850 −0.117650
\(531\) 0 0
\(532\) 0 0
\(533\) −54.6863 −2.36873
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) −11.9373 −0.515611
\(537\) 0 0
\(538\) −5.41699 −0.233543
\(539\) 0 0
\(540\) 0 0
\(541\) 2.29150 0.0985194 0.0492597 0.998786i \(-0.484314\pi\)
0.0492597 + 0.998786i \(0.484314\pi\)
\(542\) −2.06275 −0.0886025
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −17.4170 −0.746062
\(546\) 0 0
\(547\) −9.52026 −0.407057 −0.203528 0.979069i \(-0.565241\pi\)
−0.203528 + 0.979069i \(0.565241\pi\)
\(548\) 12.8745 0.549972
\(549\) 0 0
\(550\) 2.29150 0.0977100
\(551\) 35.5203 1.51321
\(552\) 0 0
\(553\) 0 0
\(554\) −22.2915 −0.947075
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 31.7490 1.34525 0.672624 0.739984i \(-0.265167\pi\)
0.672624 + 0.739984i \(0.265167\pi\)
\(558\) 0 0
\(559\) −20.0000 −0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 22.9373 0.967550
\(563\) −28.9373 −1.21956 −0.609780 0.792571i \(-0.708742\pi\)
−0.609780 + 0.792571i \(0.708742\pi\)
\(564\) 0 0
\(565\) 30.1033 1.26645
\(566\) −2.35425 −0.0989565
\(567\) 0 0
\(568\) −4.35425 −0.182700
\(569\) −1.16601 −0.0488817 −0.0244409 0.999701i \(-0.507781\pi\)
−0.0244409 + 0.999701i \(0.507781\pi\)
\(570\) 0 0
\(571\) −29.0627 −1.21624 −0.608119 0.793846i \(-0.708076\pi\)
−0.608119 + 0.793846i \(0.708076\pi\)
\(572\) −5.00000 −0.209061
\(573\) 0 0
\(574\) 0 0
\(575\) 3.77124 0.157272
\(576\) 0 0
\(577\) 27.4575 1.14307 0.571536 0.820577i \(-0.306348\pi\)
0.571536 + 0.820577i \(0.306348\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −10.3542 −0.429937
\(581\) 0 0
\(582\) 0 0
\(583\) 1.64575 0.0681601
\(584\) 0.354249 0.0146589
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) −7.93725 −0.327606 −0.163803 0.986493i \(-0.552376\pi\)
−0.163803 + 0.986493i \(0.552376\pi\)
\(588\) 0 0
\(589\) 22.5830 0.930517
\(590\) −7.64575 −0.314771
\(591\) 0 0
\(592\) 3.64575 0.149839
\(593\) −7.06275 −0.290032 −0.145016 0.989429i \(-0.546323\pi\)
−0.145016 + 0.989429i \(0.546323\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.2915 0.626364
\(597\) 0 0
\(598\) −8.22876 −0.336499
\(599\) −43.7490 −1.78754 −0.893768 0.448529i \(-0.851948\pi\)
−0.893768 + 0.448529i \(0.851948\pi\)
\(600\) 0 0
\(601\) −3.41699 −0.139382 −0.0696911 0.997569i \(-0.522201\pi\)
−0.0696911 + 0.997569i \(0.522201\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −8.64575 −0.351791
\(605\) 1.64575 0.0669093
\(606\) 0 0
\(607\) 10.7085 0.434645 0.217322 0.976100i \(-0.430268\pi\)
0.217322 + 0.976100i \(0.430268\pi\)
\(608\) −5.64575 −0.228965
\(609\) 0 0
\(610\) 23.5203 0.952307
\(611\) −13.5425 −0.547870
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 22.2288 0.897080
\(615\) 0 0
\(616\) 0 0
\(617\) 5.70850 0.229815 0.114908 0.993376i \(-0.463343\pi\)
0.114908 + 0.993376i \(0.463343\pi\)
\(618\) 0 0
\(619\) 13.4170 0.539275 0.269637 0.962962i \(-0.413096\pi\)
0.269637 + 0.962962i \(0.413096\pi\)
\(620\) −6.58301 −0.264380
\(621\) 0 0
\(622\) −1.06275 −0.0426122
\(623\) 0 0
\(624\) 0 0
\(625\) −8.29150 −0.331660
\(626\) 23.5830 0.942566
\(627\) 0 0
\(628\) 21.1660 0.844616
\(629\) −21.8745 −0.872194
\(630\) 0 0
\(631\) −12.8118 −0.510028 −0.255014 0.966937i \(-0.582080\pi\)
−0.255014 + 0.966937i \(0.582080\pi\)
\(632\) −2.64575 −0.105242
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 26.2288 1.04086
\(636\) 0 0
\(637\) 0 0
\(638\) 6.29150 0.249083
\(639\) 0 0
\(640\) 1.64575 0.0650540
\(641\) −12.8745 −0.508512 −0.254256 0.967137i \(-0.581831\pi\)
−0.254256 + 0.967137i \(0.581831\pi\)
\(642\) 0 0
\(643\) −30.5203 −1.20360 −0.601801 0.798646i \(-0.705550\pi\)
−0.601801 + 0.798646i \(0.705550\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 33.8745 1.33277
\(647\) 8.81176 0.346426 0.173213 0.984884i \(-0.444585\pi\)
0.173213 + 0.984884i \(0.444585\pi\)
\(648\) 0 0
\(649\) 4.64575 0.182362
\(650\) −11.4575 −0.449401
\(651\) 0 0
\(652\) 0.645751 0.0252896
\(653\) −7.64575 −0.299201 −0.149601 0.988746i \(-0.547799\pi\)
−0.149601 + 0.988746i \(0.547799\pi\)
\(654\) 0 0
\(655\) −17.0405 −0.665828
\(656\) −10.9373 −0.427028
\(657\) 0 0
\(658\) 0 0
\(659\) 6.58301 0.256437 0.128219 0.991746i \(-0.459074\pi\)
0.128219 + 0.991746i \(0.459074\pi\)
\(660\) 0 0
\(661\) 7.41699 0.288488 0.144244 0.989542i \(-0.453925\pi\)
0.144244 + 0.989542i \(0.453925\pi\)
\(662\) −20.6458 −0.802420
\(663\) 0 0
\(664\) −2.70850 −0.105110
\(665\) 0 0
\(666\) 0 0
\(667\) 10.3542 0.400918
\(668\) 11.2288 0.434454
\(669\) 0 0
\(670\) −19.6458 −0.758982
\(671\) −14.2915 −0.551717
\(672\) 0 0
\(673\) 0.937254 0.0361285 0.0180642 0.999837i \(-0.494250\pi\)
0.0180642 + 0.999837i \(0.494250\pi\)
\(674\) 9.06275 0.349084
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) 33.8745 1.30190 0.650952 0.759119i \(-0.274370\pi\)
0.650952 + 0.759119i \(0.274370\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −9.87451 −0.378670
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −1.93725 −0.0741270 −0.0370635 0.999313i \(-0.511800\pi\)
−0.0370635 + 0.999313i \(0.511800\pi\)
\(684\) 0 0
\(685\) 21.1882 0.809561
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −8.22876 −0.313491
\(690\) 0 0
\(691\) −45.2288 −1.72058 −0.860291 0.509802i \(-0.829718\pi\)
−0.860291 + 0.509802i \(0.829718\pi\)
\(692\) 0.291503 0.0110813
\(693\) 0 0
\(694\) 20.8118 0.790004
\(695\) −6.58301 −0.249708
\(696\) 0 0
\(697\) 65.6235 2.48567
\(698\) −1.87451 −0.0709512
\(699\) 0 0
\(700\) 0 0
\(701\) −24.8745 −0.939497 −0.469749 0.882800i \(-0.655655\pi\)
−0.469749 + 0.882800i \(0.655655\pi\)
\(702\) 0 0
\(703\) −20.5830 −0.776303
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −7.16601 −0.269696
\(707\) 0 0
\(708\) 0 0
\(709\) 40.8118 1.53272 0.766359 0.642413i \(-0.222066\pi\)
0.766359 + 0.642413i \(0.222066\pi\)
\(710\) −7.16601 −0.268936
\(711\) 0 0
\(712\) −6.58301 −0.246709
\(713\) 6.58301 0.246535
\(714\) 0 0
\(715\) −8.22876 −0.307738
\(716\) 19.9373 0.745090
\(717\) 0 0
\(718\) 25.9373 0.967970
\(719\) −27.8745 −1.03954 −0.519772 0.854305i \(-0.673983\pi\)
−0.519772 + 0.854305i \(0.673983\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12.8745 0.479140
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 14.4170 0.535434
\(726\) 0 0
\(727\) −6.70850 −0.248804 −0.124402 0.992232i \(-0.539701\pi\)
−0.124402 + 0.992232i \(0.539701\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0.583005 0.0215780
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −11.4575 −0.423193 −0.211596 0.977357i \(-0.567866\pi\)
−0.211596 + 0.977357i \(0.567866\pi\)
\(734\) −36.2288 −1.33723
\(735\) 0 0
\(736\) −1.64575 −0.0606632
\(737\) 11.9373 0.439714
\(738\) 0 0
\(739\) 23.8745 0.878238 0.439119 0.898429i \(-0.355291\pi\)
0.439119 + 0.898429i \(0.355291\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) 45.2915 1.66158 0.830792 0.556583i \(-0.187888\pi\)
0.830792 + 0.556583i \(0.187888\pi\)
\(744\) 0 0
\(745\) 25.1660 0.922011
\(746\) 20.8745 0.764270
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) −2.70850 −0.0987687
\(753\) 0 0
\(754\) −31.4575 −1.14562
\(755\) −14.2288 −0.517837
\(756\) 0 0
\(757\) −23.1660 −0.841983 −0.420991 0.907065i \(-0.638318\pi\)
−0.420991 + 0.907065i \(0.638318\pi\)
\(758\) 6.06275 0.220209
\(759\) 0 0
\(760\) −9.29150 −0.337038
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.70850 0.0979900
\(765\) 0 0
\(766\) −24.1033 −0.870886
\(767\) −23.2288 −0.838742
\(768\) 0 0
\(769\) 27.1660 0.979631 0.489816 0.871826i \(-0.337064\pi\)
0.489816 + 0.871826i \(0.337064\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 25.5203 0.918494
\(773\) 18.5830 0.668384 0.334192 0.942505i \(-0.391537\pi\)
0.334192 + 0.942505i \(0.391537\pi\)
\(774\) 0 0
\(775\) 9.16601 0.329253
\(776\) −16.2915 −0.584831
\(777\) 0 0
\(778\) −26.8118 −0.961248
\(779\) 61.7490 2.21239
\(780\) 0 0
\(781\) 4.35425 0.155807
\(782\) 9.87451 0.353112
\(783\) 0 0
\(784\) 0 0
\(785\) 34.8340 1.24328
\(786\) 0 0
\(787\) 46.8118 1.66866 0.834330 0.551266i \(-0.185855\pi\)
0.834330 + 0.551266i \(0.185855\pi\)
\(788\) 12.8745 0.458635
\(789\) 0 0
\(790\) −4.35425 −0.154917
\(791\) 0 0
\(792\) 0 0
\(793\) 71.4575 2.53753
\(794\) −11.1660 −0.396267
\(795\) 0 0
\(796\) 4.22876 0.149884
\(797\) −7.16601 −0.253833 −0.126917 0.991913i \(-0.540508\pi\)
−0.126917 + 0.991913i \(0.540508\pi\)
\(798\) 0 0
\(799\) 16.2510 0.574918
\(800\) −2.29150 −0.0810169
\(801\) 0 0
\(802\) −21.5830 −0.762122
\(803\) −0.354249 −0.0125012
\(804\) 0 0
\(805\) 0 0
\(806\) −20.0000 −0.704470
\(807\) 0 0
\(808\) −3.00000 −0.105540
\(809\) −29.4170 −1.03425 −0.517123 0.855911i \(-0.672997\pi\)
−0.517123 + 0.855911i \(0.672997\pi\)
\(810\) 0 0
\(811\) −35.7490 −1.25532 −0.627659 0.778489i \(-0.715987\pi\)
−0.627659 + 0.778489i \(0.715987\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3.64575 −0.127784
\(815\) 1.06275 0.0372264
\(816\) 0 0
\(817\) 22.5830 0.790079
\(818\) 3.06275 0.107086
\(819\) 0 0
\(820\) −18.0000 −0.628587
\(821\) 18.2915 0.638378 0.319189 0.947691i \(-0.396590\pi\)
0.319189 + 0.947691i \(0.396590\pi\)
\(822\) 0 0
\(823\) −12.1255 −0.422668 −0.211334 0.977414i \(-0.567781\pi\)
−0.211334 + 0.977414i \(0.567781\pi\)
\(824\) −2.93725 −0.102324
\(825\) 0 0
\(826\) 0 0
\(827\) −33.3948 −1.16125 −0.580625 0.814171i \(-0.697192\pi\)
−0.580625 + 0.814171i \(0.697192\pi\)
\(828\) 0 0
\(829\) −25.3948 −0.881997 −0.440998 0.897508i \(-0.645376\pi\)
−0.440998 + 0.897508i \(0.645376\pi\)
\(830\) −4.45751 −0.154723
\(831\) 0 0
\(832\) 5.00000 0.173344
\(833\) 0 0
\(834\) 0 0
\(835\) 18.4797 0.639518
\(836\) 5.64575 0.195262
\(837\) 0 0
\(838\) 9.87451 0.341109
\(839\) 3.87451 0.133763 0.0668814 0.997761i \(-0.478695\pi\)
0.0668814 + 0.997761i \(0.478695\pi\)
\(840\) 0 0
\(841\) 10.5830 0.364931
\(842\) 9.16601 0.315882
\(843\) 0 0
\(844\) 0.937254 0.0322616
\(845\) 19.7490 0.679387
\(846\) 0 0
\(847\) 0 0
\(848\) −1.64575 −0.0565153
\(849\) 0 0
\(850\) 13.7490 0.471587
\(851\) −6.00000 −0.205677
\(852\) 0 0
\(853\) 39.1660 1.34102 0.670509 0.741901i \(-0.266076\pi\)
0.670509 + 0.741901i \(0.266076\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 10.9373 0.373828
\(857\) −36.0000 −1.22974 −0.614868 0.788630i \(-0.710791\pi\)
−0.614868 + 0.788630i \(0.710791\pi\)
\(858\) 0 0
\(859\) −9.81176 −0.334773 −0.167386 0.985891i \(-0.553533\pi\)
−0.167386 + 0.985891i \(0.553533\pi\)
\(860\) −6.58301 −0.224479
\(861\) 0 0
\(862\) −29.2288 −0.995535
\(863\) 12.4797 0.424815 0.212408 0.977181i \(-0.431870\pi\)
0.212408 + 0.977181i \(0.431870\pi\)
\(864\) 0 0
\(865\) 0.479741 0.0163117
\(866\) −16.0000 −0.543702
\(867\) 0 0
\(868\) 0 0
\(869\) 2.64575 0.0897510
\(870\) 0 0
\(871\) −59.6863 −2.02239
\(872\) −10.5830 −0.358386
\(873\) 0 0
\(874\) 9.29150 0.314290
\(875\) 0 0
\(876\) 0 0
\(877\) −22.8745 −0.772417 −0.386209 0.922411i \(-0.626216\pi\)
−0.386209 + 0.922411i \(0.626216\pi\)
\(878\) 3.93725 0.132876
\(879\) 0 0
\(880\) −1.64575 −0.0554783
\(881\) 24.8745 0.838043 0.419022 0.907976i \(-0.362373\pi\)
0.419022 + 0.907976i \(0.362373\pi\)
\(882\) 0 0
\(883\) 21.9373 0.738247 0.369124 0.929380i \(-0.379658\pi\)
0.369124 + 0.929380i \(0.379658\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) −34.4575 −1.15762
\(887\) −45.1033 −1.51442 −0.757210 0.653172i \(-0.773438\pi\)
−0.757210 + 0.653172i \(0.773438\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −10.8340 −0.363156
\(891\) 0 0
\(892\) −17.6458 −0.590823
\(893\) 15.2915 0.511711
\(894\) 0 0
\(895\) 32.8118 1.09678
\(896\) 0 0
\(897\) 0 0
\(898\) 21.8745 0.729962
\(899\) 25.1660 0.839333
\(900\) 0 0
\(901\) 9.87451 0.328968
\(902\) 10.9373 0.364171
\(903\) 0 0
\(904\) 18.2915 0.608366
\(905\) −16.4575 −0.547066
\(906\) 0 0
\(907\) 42.4575 1.40978 0.704889 0.709317i \(-0.250997\pi\)
0.704889 + 0.709317i \(0.250997\pi\)
\(908\) 2.70850 0.0898846
\(909\) 0 0
\(910\) 0 0
\(911\) −9.29150 −0.307841 −0.153921 0.988083i \(-0.549190\pi\)
−0.153921 + 0.988083i \(0.549190\pi\)
\(912\) 0 0
\(913\) 2.70850 0.0896382
\(914\) 3.16601 0.104722
\(915\) 0 0
\(916\) −16.0000 −0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) −24.7085 −0.815058 −0.407529 0.913192i \(-0.633609\pi\)
−0.407529 + 0.913192i \(0.633609\pi\)
\(920\) −2.70850 −0.0892965
\(921\) 0 0
\(922\) −10.1660 −0.334800
\(923\) −21.7712 −0.716609
\(924\) 0 0
\(925\) −8.35425 −0.274686
\(926\) 30.4575 1.00090
\(927\) 0 0
\(928\) −6.29150 −0.206529
\(929\) −14.4170 −0.473006 −0.236503 0.971631i \(-0.576001\pi\)
−0.236503 + 0.971631i \(0.576001\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.06275 −0.0348114
\(933\) 0 0
\(934\) −21.2915 −0.696679
\(935\) 9.87451 0.322931
\(936\) 0 0
\(937\) 32.6863 1.06781 0.533907 0.845543i \(-0.320723\pi\)
0.533907 + 0.845543i \(0.320723\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.45751 −0.145388
\(941\) −50.6235 −1.65028 −0.825140 0.564929i \(-0.808904\pi\)
−0.825140 + 0.564929i \(0.808904\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) −4.64575 −0.151206
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 2.12549 0.0690692 0.0345346 0.999404i \(-0.489005\pi\)
0.0345346 + 0.999404i \(0.489005\pi\)
\(948\) 0 0
\(949\) 1.77124 0.0574970
\(950\) 12.9373 0.419740
\(951\) 0 0
\(952\) 0 0
\(953\) −11.5203 −0.373178 −0.186589 0.982438i \(-0.559743\pi\)
−0.186589 + 0.982438i \(0.559743\pi\)
\(954\) 0 0
\(955\) 4.45751 0.144242
\(956\) 17.2288 0.557218
\(957\) 0 0
\(958\) 10.0627 0.325113
\(959\) 0 0
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 18.2288 0.587719
\(963\) 0 0
\(964\) −24.8118 −0.799133
\(965\) 42.0000 1.35203
\(966\) 0 0
\(967\) 58.3320 1.87583 0.937916 0.346863i \(-0.112753\pi\)
0.937916 + 0.346863i \(0.112753\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −26.8118 −0.860874
\(971\) 10.0627 0.322929 0.161464 0.986879i \(-0.448378\pi\)
0.161464 + 0.986879i \(0.448378\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −9.41699 −0.301740
\(975\) 0 0
\(976\) 14.2915 0.457460
\(977\) 16.4575 0.526522 0.263261 0.964725i \(-0.415202\pi\)
0.263261 + 0.964725i \(0.415202\pi\)
\(978\) 0 0
\(979\) 6.58301 0.210394
\(980\) 0 0
\(981\) 0 0
\(982\) −21.2915 −0.679439
\(983\) 26.7085 0.851869 0.425934 0.904754i \(-0.359945\pi\)
0.425934 + 0.904754i \(0.359945\pi\)
\(984\) 0 0
\(985\) 21.1882 0.675113
\(986\) 37.7490 1.20217
\(987\) 0 0
\(988\) −28.2288 −0.898076
\(989\) 6.58301 0.209327
\(990\) 0 0
\(991\) −22.6863 −0.720653 −0.360327 0.932826i \(-0.617335\pi\)
−0.360327 + 0.932826i \(0.617335\pi\)
\(992\) −4.00000 −0.127000
\(993\) 0 0
\(994\) 0 0
\(995\) 6.95948 0.220630
\(996\) 0 0
\(997\) −21.4170 −0.678283 −0.339142 0.940735i \(-0.610137\pi\)
−0.339142 + 0.940735i \(0.610137\pi\)
\(998\) −13.8745 −0.439190
\(999\) 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 9702.2.a.cz.1.2 2
3.2 odd 2 1078.2.a.s.1.2 2
7.2 even 3 1386.2.k.s.991.1 4
7.4 even 3 1386.2.k.s.793.1 4
7.6 odd 2 9702.2.a.dr.1.1 2
12.11 even 2 8624.2.a.ca.1.1 2
21.2 odd 6 154.2.e.f.67.1 yes 4
21.5 even 6 1078.2.e.v.67.2 4
21.11 odd 6 154.2.e.f.23.1 4
21.17 even 6 1078.2.e.v.177.2 4
21.20 even 2 1078.2.a.n.1.1 2
84.11 even 6 1232.2.q.g.177.2 4
84.23 even 6 1232.2.q.g.529.2 4
84.83 odd 2 8624.2.a.bk.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.f.23.1 4 21.11 odd 6
154.2.e.f.67.1 yes 4 21.2 odd 6
1078.2.a.n.1.1 2 21.20 even 2
1078.2.a.s.1.2 2 3.2 odd 2
1078.2.e.v.67.2 4 21.5 even 6
1078.2.e.v.177.2 4 21.17 even 6
1232.2.q.g.177.2 4 84.11 even 6
1232.2.q.g.529.2 4 84.23 even 6
1386.2.k.s.793.1 4 7.4 even 3
1386.2.k.s.991.1 4 7.2 even 3
8624.2.a.bk.1.2 2 84.83 odd 2
8624.2.a.ca.1.1 2 12.11 even 2
9702.2.a.cz.1.2 2 1.1 even 1 trivial
9702.2.a.dr.1.1 2 7.6 odd 2