Properties

Label 6762.2.a.cp.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.42048.1
Defining polynomial: \(x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.94433\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.82843 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -3.82843 q^{10} +5.27981 q^{11} +1.00000 q^{12} +4.35854 q^{13} -3.82843 q^{15} +1.00000 q^{16} -2.46988 q^{17} +1.00000 q^{18} -1.92127 q^{19} -3.82843 q^{20} +5.27981 q^{22} +1.00000 q^{23} +1.00000 q^{24} +9.65685 q^{25} +4.35854 q^{26} +1.00000 q^{27} -1.35854 q^{29} -3.82843 q^{30} -1.43727 q^{31} +1.00000 q^{32} +5.27981 q^{33} -2.46988 q^{34} +1.00000 q^{36} +6.63836 q^{37} -1.92127 q^{38} +4.35854 q^{39} -3.82843 q^{40} +1.92127 q^{41} -8.21648 q^{43} +5.27981 q^{44} -3.82843 q^{45} +1.00000 q^{46} -6.99690 q^{47} +1.00000 q^{48} +9.65685 q^{50} -2.46988 q^{51} +4.35854 q^{52} -2.88866 q^{53} +1.00000 q^{54} -20.2134 q^{55} -1.92127 q^{57} -1.35854 q^{58} +4.29831 q^{59} -3.82843 q^{60} +11.4342 q^{61} -1.43727 q^{62} +1.00000 q^{64} -16.6864 q^{65} +5.27981 q^{66} +5.33092 q^{67} -2.46988 q^{68} +1.00000 q^{69} +7.64146 q^{71} +1.00000 q^{72} -0.437273 q^{73} +6.63836 q^{74} +9.65685 q^{75} -1.92127 q^{76} +4.35854 q^{78} +1.35854 q^{79} -3.82843 q^{80} +1.00000 q^{81} +1.92127 q^{82} -4.37704 q^{83} +9.45577 q^{85} -8.21648 q^{86} -1.35854 q^{87} +5.27981 q^{88} -12.2952 q^{89} -3.82843 q^{90} +1.00000 q^{92} -1.43727 q^{93} -6.99690 q^{94} +7.35544 q^{95} +1.00000 q^{96} +0.298311 q^{97} +5.27981 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{8} + 4q^{9} - 4q^{10} + 6q^{11} + 4q^{12} + 6q^{13} - 4q^{15} + 4q^{16} - 10q^{17} + 4q^{18} - 4q^{19} - 4q^{20} + 6q^{22} + 4q^{23} + 4q^{24} + 16q^{25} + 6q^{26} + 4q^{27} + 6q^{29} - 4q^{30} + 2q^{31} + 4q^{32} + 6q^{33} - 10q^{34} + 4q^{36} - 4q^{38} + 6q^{39} - 4q^{40} + 4q^{41} + 20q^{43} + 6q^{44} - 4q^{45} + 4q^{46} + 10q^{47} + 4q^{48} + 16q^{50} - 10q^{51} + 6q^{52} + 4q^{54} + 10q^{55} - 4q^{57} + 6q^{58} + 6q^{59} - 4q^{60} + 2q^{62} + 4q^{64} - 14q^{65} + 6q^{66} + 18q^{67} - 10q^{68} + 4q^{69} + 42q^{71} + 4q^{72} + 6q^{73} + 16q^{75} - 4q^{76} + 6q^{78} - 6q^{79} - 4q^{80} + 4q^{81} + 4q^{82} - 10q^{83} + 34q^{85} + 20q^{86} + 6q^{87} + 6q^{88} - 4q^{90} + 4q^{92} + 2q^{93} + 10q^{94} - 20q^{95} + 4q^{96} - 10q^{97} + 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.82843 −1.71212 −0.856062 0.516873i \(-0.827096\pi\)
−0.856062 + 0.516873i \(0.827096\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −3.82843 −1.21065
\(11\) 5.27981 1.59192 0.795962 0.605347i \(-0.206966\pi\)
0.795962 + 0.605347i \(0.206966\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.35854 1.20884 0.604421 0.796665i \(-0.293404\pi\)
0.604421 + 0.796665i \(0.293404\pi\)
\(14\) 0 0
\(15\) −3.82843 −0.988496
\(16\) 1.00000 0.250000
\(17\) −2.46988 −0.599035 −0.299517 0.954091i \(-0.596826\pi\)
−0.299517 + 0.954091i \(0.596826\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.92127 −0.440770 −0.220385 0.975413i \(-0.570731\pi\)
−0.220385 + 0.975413i \(0.570731\pi\)
\(20\) −3.82843 −0.856062
\(21\) 0 0
\(22\) 5.27981 1.12566
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 9.65685 1.93137
\(26\) 4.35854 0.854781
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.35854 −0.252275 −0.126138 0.992013i \(-0.540258\pi\)
−0.126138 + 0.992013i \(0.540258\pi\)
\(30\) −3.82843 −0.698972
\(31\) −1.43727 −0.258142 −0.129071 0.991635i \(-0.541199\pi\)
−0.129071 + 0.991635i \(0.541199\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.27981 0.919098
\(34\) −2.46988 −0.423582
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.63836 1.09134 0.545670 0.838000i \(-0.316275\pi\)
0.545670 + 0.838000i \(0.316275\pi\)
\(38\) −1.92127 −0.311671
\(39\) 4.35854 0.697926
\(40\) −3.82843 −0.605327
\(41\) 1.92127 0.300052 0.150026 0.988682i \(-0.452064\pi\)
0.150026 + 0.988682i \(0.452064\pi\)
\(42\) 0 0
\(43\) −8.21648 −1.25300 −0.626501 0.779421i \(-0.715514\pi\)
−0.626501 + 0.779421i \(0.715514\pi\)
\(44\) 5.27981 0.795962
\(45\) −3.82843 −0.570708
\(46\) 1.00000 0.147442
\(47\) −6.99690 −1.02060 −0.510301 0.859996i \(-0.670466\pi\)
−0.510301 + 0.859996i \(0.670466\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 9.65685 1.36569
\(51\) −2.46988 −0.345853
\(52\) 4.35854 0.604421
\(53\) −2.88866 −0.396788 −0.198394 0.980122i \(-0.563573\pi\)
−0.198394 + 0.980122i \(0.563573\pi\)
\(54\) 1.00000 0.136083
\(55\) −20.2134 −2.72557
\(56\) 0 0
\(57\) −1.92127 −0.254479
\(58\) −1.35854 −0.178385
\(59\) 4.29831 0.559592 0.279796 0.960059i \(-0.409733\pi\)
0.279796 + 0.960059i \(0.409733\pi\)
\(60\) −3.82843 −0.494248
\(61\) 11.4342 1.46400 0.731998 0.681307i \(-0.238588\pi\)
0.731998 + 0.681307i \(0.238588\pi\)
\(62\) −1.43727 −0.182534
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.6864 −2.06969
\(66\) 5.27981 0.649900
\(67\) 5.33092 0.651276 0.325638 0.945495i \(-0.394421\pi\)
0.325638 + 0.945495i \(0.394421\pi\)
\(68\) −2.46988 −0.299517
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 7.64146 0.906874 0.453437 0.891288i \(-0.350198\pi\)
0.453437 + 0.891288i \(0.350198\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.437273 −0.0511789 −0.0255895 0.999673i \(-0.508146\pi\)
−0.0255895 + 0.999673i \(0.508146\pi\)
\(74\) 6.63836 0.771693
\(75\) 9.65685 1.11508
\(76\) −1.92127 −0.220385
\(77\) 0 0
\(78\) 4.35854 0.493508
\(79\) 1.35854 0.152848 0.0764240 0.997075i \(-0.475650\pi\)
0.0764240 + 0.997075i \(0.475650\pi\)
\(80\) −3.82843 −0.428031
\(81\) 1.00000 0.111111
\(82\) 1.92127 0.212169
\(83\) −4.37704 −0.480443 −0.240221 0.970718i \(-0.577220\pi\)
−0.240221 + 0.970718i \(0.577220\pi\)
\(84\) 0 0
\(85\) 9.45577 1.02562
\(86\) −8.21648 −0.886006
\(87\) −1.35854 −0.145651
\(88\) 5.27981 0.562830
\(89\) −12.2952 −1.30329 −0.651645 0.758524i \(-0.725921\pi\)
−0.651645 + 0.758524i \(0.725921\pi\)
\(90\) −3.82843 −0.403552
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −1.43727 −0.149038
\(94\) −6.99690 −0.721675
\(95\) 7.35544 0.754653
\(96\) 1.00000 0.102062
\(97\) 0.298311 0.0302889 0.0151444 0.999885i \(-0.495179\pi\)
0.0151444 + 0.999885i \(0.495179\pi\)
\(98\) 0 0
\(99\) 5.27981 0.530641
\(100\) 9.65685 0.965685
\(101\) 18.0590 1.79694 0.898470 0.439035i \(-0.144680\pi\)
0.898470 + 0.439035i \(0.144680\pi\)
\(102\) −2.46988 −0.244555
\(103\) −0.669078 −0.0659263 −0.0329631 0.999457i \(-0.510494\pi\)
−0.0329631 + 0.999457i \(0.510494\pi\)
\(104\) 4.35854 0.427390
\(105\) 0 0
\(106\) −2.88866 −0.280571
\(107\) −18.4361 −1.78228 −0.891141 0.453727i \(-0.850094\pi\)
−0.891141 + 0.453727i \(0.850094\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.5314 1.58342 0.791710 0.610896i \(-0.209191\pi\)
0.791710 + 0.610896i \(0.209191\pi\)
\(110\) −20.2134 −1.92727
\(111\) 6.63836 0.630085
\(112\) 0 0
\(113\) 19.7466 1.85760 0.928802 0.370577i \(-0.120840\pi\)
0.928802 + 0.370577i \(0.120840\pi\)
\(114\) −1.92127 −0.179943
\(115\) −3.82843 −0.357003
\(116\) −1.35854 −0.126138
\(117\) 4.35854 0.402947
\(118\) 4.29831 0.395692
\(119\) 0 0
\(120\) −3.82843 −0.349486
\(121\) 16.8764 1.53422
\(122\) 11.4342 1.03520
\(123\) 1.92127 0.173235
\(124\) −1.43727 −0.129071
\(125\) −17.8284 −1.59462
\(126\) 0 0
\(127\) 18.9367 1.68036 0.840179 0.542310i \(-0.182450\pi\)
0.840179 + 0.542310i \(0.182450\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.21648 −0.723421
\(130\) −16.6864 −1.46349
\(131\) 16.3554 1.42898 0.714491 0.699645i \(-0.246658\pi\)
0.714491 + 0.699645i \(0.246658\pi\)
\(132\) 5.27981 0.459549
\(133\) 0 0
\(134\) 5.33092 0.460521
\(135\) −3.82843 −0.329499
\(136\) −2.46988 −0.211791
\(137\) −5.10824 −0.436426 −0.218213 0.975901i \(-0.570023\pi\)
−0.218213 + 0.975901i \(0.570023\pi\)
\(138\) 1.00000 0.0851257
\(139\) 9.31371 0.789978 0.394989 0.918686i \(-0.370748\pi\)
0.394989 + 0.918686i \(0.370748\pi\)
\(140\) 0 0
\(141\) −6.99690 −0.589245
\(142\) 7.64146 0.641257
\(143\) 23.0123 1.92439
\(144\) 1.00000 0.0833333
\(145\) 5.20108 0.431927
\(146\) −0.437273 −0.0361890
\(147\) 0 0
\(148\) 6.63836 0.545670
\(149\) 2.23370 0.182991 0.0914957 0.995805i \(-0.470835\pi\)
0.0914957 + 0.995805i \(0.470835\pi\)
\(150\) 9.65685 0.788479
\(151\) −22.2134 −1.80770 −0.903850 0.427850i \(-0.859271\pi\)
−0.903850 + 0.427850i \(0.859271\pi\)
\(152\) −1.92127 −0.155836
\(153\) −2.46988 −0.199678
\(154\) 0 0
\(155\) 5.50249 0.441971
\(156\) 4.35854 0.348963
\(157\) 10.5731 0.843828 0.421914 0.906636i \(-0.361358\pi\)
0.421914 + 0.906636i \(0.361358\pi\)
\(158\) 1.35854 0.108080
\(159\) −2.88866 −0.229086
\(160\) −3.82843 −0.302664
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 19.7294 1.54532 0.772662 0.634817i \(-0.218925\pi\)
0.772662 + 0.634817i \(0.218925\pi\)
\(164\) 1.92127 0.150026
\(165\) −20.2134 −1.57361
\(166\) −4.37704 −0.339724
\(167\) 10.6907 0.827275 0.413637 0.910442i \(-0.364258\pi\)
0.413637 + 0.910442i \(0.364258\pi\)
\(168\) 0 0
\(169\) 5.99690 0.461300
\(170\) 9.45577 0.725224
\(171\) −1.92127 −0.146923
\(172\) −8.21648 −0.626501
\(173\) 22.0308 1.67497 0.837485 0.546460i \(-0.184025\pi\)
0.837485 + 0.546460i \(0.184025\pi\)
\(174\) −1.35854 −0.102991
\(175\) 0 0
\(176\) 5.27981 0.397981
\(177\) 4.29831 0.323081
\(178\) −12.2952 −0.921565
\(179\) −16.9969 −1.27041 −0.635204 0.772344i \(-0.719084\pi\)
−0.635204 + 0.772344i \(0.719084\pi\)
\(180\) −3.82843 −0.285354
\(181\) −12.4022 −0.921846 −0.460923 0.887440i \(-0.652482\pi\)
−0.460923 + 0.887440i \(0.652482\pi\)
\(182\) 0 0
\(183\) 11.4342 0.845238
\(184\) 1.00000 0.0737210
\(185\) −25.4145 −1.86851
\(186\) −1.43727 −0.105386
\(187\) −13.0405 −0.953618
\(188\) −6.99690 −0.510301
\(189\) 0 0
\(190\) 7.35544 0.533620
\(191\) −16.6384 −1.20391 −0.601955 0.798530i \(-0.705611\pi\)
−0.601955 + 0.798530i \(0.705611\pi\)
\(192\) 1.00000 0.0721688
\(193\) 25.2165 1.81512 0.907561 0.419920i \(-0.137942\pi\)
0.907561 + 0.419920i \(0.137942\pi\)
\(194\) 0.298311 0.0214175
\(195\) −16.6864 −1.19494
\(196\) 0 0
\(197\) 13.6199 0.970375 0.485187 0.874410i \(-0.338751\pi\)
0.485187 + 0.874410i \(0.338751\pi\)
\(198\) 5.27981 0.375220
\(199\) −0.902774 −0.0639959 −0.0319980 0.999488i \(-0.510187\pi\)
−0.0319980 + 0.999488i \(0.510187\pi\)
\(200\) 9.65685 0.682843
\(201\) 5.33092 0.376014
\(202\) 18.0590 1.27063
\(203\) 0 0
\(204\) −2.46988 −0.172926
\(205\) −7.35544 −0.513727
\(206\) −0.669078 −0.0466169
\(207\) 1.00000 0.0695048
\(208\) 4.35854 0.302211
\(209\) −10.1440 −0.701672
\(210\) 0 0
\(211\) −22.8131 −1.57052 −0.785259 0.619167i \(-0.787470\pi\)
−0.785259 + 0.619167i \(0.787470\pi\)
\(212\) −2.88866 −0.198394
\(213\) 7.64146 0.523584
\(214\) −18.4361 −1.26026
\(215\) 31.4562 2.14530
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 16.5314 1.11965
\(219\) −0.437273 −0.0295482
\(220\) −20.2134 −1.36279
\(221\) −10.7651 −0.724139
\(222\) 6.63836 0.445537
\(223\) −0.942867 −0.0631390 −0.0315695 0.999502i \(-0.510051\pi\)
−0.0315695 + 0.999502i \(0.510051\pi\)
\(224\) 0 0
\(225\) 9.65685 0.643790
\(226\) 19.7466 1.31352
\(227\) 15.3106 1.01620 0.508100 0.861298i \(-0.330348\pi\)
0.508100 + 0.861298i \(0.330348\pi\)
\(228\) −1.92127 −0.127239
\(229\) −6.07873 −0.401694 −0.200847 0.979623i \(-0.564369\pi\)
−0.200847 + 0.979623i \(0.564369\pi\)
\(230\) −3.82843 −0.252439
\(231\) 0 0
\(232\) −1.35854 −0.0891927
\(233\) −7.57812 −0.496459 −0.248230 0.968701i \(-0.579849\pi\)
−0.248230 + 0.968701i \(0.579849\pi\)
\(234\) 4.35854 0.284927
\(235\) 26.7871 1.74740
\(236\) 4.29831 0.279796
\(237\) 1.35854 0.0882469
\(238\) 0 0
\(239\) −16.4330 −1.06296 −0.531480 0.847071i \(-0.678364\pi\)
−0.531480 + 0.847071i \(0.678364\pi\)
\(240\) −3.82843 −0.247124
\(241\) −13.2319 −0.852340 −0.426170 0.904643i \(-0.640138\pi\)
−0.426170 + 0.904643i \(0.640138\pi\)
\(242\) 16.8764 1.08486
\(243\) 1.00000 0.0641500
\(244\) 11.4342 0.731998
\(245\) 0 0
\(246\) 1.92127 0.122496
\(247\) −8.37394 −0.532821
\(248\) −1.43727 −0.0912669
\(249\) −4.37704 −0.277384
\(250\) −17.8284 −1.12757
\(251\) 9.87644 0.623395 0.311698 0.950181i \(-0.399102\pi\)
0.311698 + 0.950181i \(0.399102\pi\)
\(252\) 0 0
\(253\) 5.27981 0.331939
\(254\) 18.9367 1.18819
\(255\) 9.45577 0.592143
\(256\) 1.00000 0.0625000
\(257\) −21.3554 −1.33212 −0.666058 0.745900i \(-0.732020\pi\)
−0.666058 + 0.745900i \(0.732020\pi\)
\(258\) −8.21648 −0.511536
\(259\) 0 0
\(260\) −16.6864 −1.03484
\(261\) −1.35854 −0.0840917
\(262\) 16.3554 1.01044
\(263\) 8.68009 0.535237 0.267619 0.963525i \(-0.413763\pi\)
0.267619 + 0.963525i \(0.413763\pi\)
\(264\) 5.27981 0.324950
\(265\) 11.0590 0.679350
\(266\) 0 0
\(267\) −12.2952 −0.752455
\(268\) 5.33092 0.325638
\(269\) −26.0142 −1.58611 −0.793057 0.609148i \(-0.791512\pi\)
−0.793057 + 0.609148i \(0.791512\pi\)
\(270\) −3.82843 −0.232991
\(271\) 1.43727 0.0873081 0.0436541 0.999047i \(-0.486100\pi\)
0.0436541 + 0.999047i \(0.486100\pi\)
\(272\) −2.46988 −0.149759
\(273\) 0 0
\(274\) −5.10824 −0.308600
\(275\) 50.9864 3.07460
\(276\) 1.00000 0.0601929
\(277\) 18.0289 1.08325 0.541626 0.840620i \(-0.317809\pi\)
0.541626 + 0.840620i \(0.317809\pi\)
\(278\) 9.31371 0.558599
\(279\) −1.43727 −0.0860473
\(280\) 0 0
\(281\) 24.9643 1.48924 0.744622 0.667486i \(-0.232630\pi\)
0.744622 + 0.667486i \(0.232630\pi\)
\(282\) −6.99690 −0.416659
\(283\) 2.73430 0.162537 0.0812686 0.996692i \(-0.474103\pi\)
0.0812686 + 0.996692i \(0.474103\pi\)
\(284\) 7.64146 0.453437
\(285\) 7.35544 0.435699
\(286\) 23.0123 1.36075
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −10.8997 −0.641157
\(290\) 5.20108 0.305418
\(291\) 0.298311 0.0174873
\(292\) −0.437273 −0.0255895
\(293\) −14.5147 −0.847959 −0.423979 0.905672i \(-0.639367\pi\)
−0.423979 + 0.905672i \(0.639367\pi\)
\(294\) 0 0
\(295\) −16.4558 −0.958092
\(296\) 6.63836 0.385847
\(297\) 5.27981 0.306366
\(298\) 2.23370 0.129394
\(299\) 4.35854 0.252061
\(300\) 9.65685 0.557539
\(301\) 0 0
\(302\) −22.2134 −1.27824
\(303\) 18.0590 1.03746
\(304\) −1.92127 −0.110192
\(305\) −43.7749 −2.50654
\(306\) −2.46988 −0.141194
\(307\) 31.9521 1.82360 0.911800 0.410634i \(-0.134693\pi\)
0.911800 + 0.410634i \(0.134693\pi\)
\(308\) 0 0
\(309\) −0.669078 −0.0380625
\(310\) 5.50249 0.312521
\(311\) 31.0142 1.75865 0.879327 0.476219i \(-0.157993\pi\)
0.879327 + 0.476219i \(0.157993\pi\)
\(312\) 4.35854 0.246754
\(313\) −6.57623 −0.371711 −0.185855 0.982577i \(-0.559506\pi\)
−0.185855 + 0.982577i \(0.559506\pi\)
\(314\) 10.5731 0.596677
\(315\) 0 0
\(316\) 1.35854 0.0764240
\(317\) −28.0142 −1.57343 −0.786717 0.617314i \(-0.788221\pi\)
−0.786717 + 0.617314i \(0.788221\pi\)
\(318\) −2.88866 −0.161988
\(319\) −7.17286 −0.401603
\(320\) −3.82843 −0.214016
\(321\) −18.4361 −1.02900
\(322\) 0 0
\(323\) 4.74531 0.264036
\(324\) 1.00000 0.0555556
\(325\) 42.0898 2.33472
\(326\) 19.7294 1.09271
\(327\) 16.5314 0.914189
\(328\) 1.92127 0.106084
\(329\) 0 0
\(330\) −20.2134 −1.11271
\(331\) 24.8266 1.36459 0.682297 0.731075i \(-0.260981\pi\)
0.682297 + 0.731075i \(0.260981\pi\)
\(332\) −4.37704 −0.240221
\(333\) 6.63836 0.363780
\(334\) 10.6907 0.584972
\(335\) −20.4090 −1.11507
\(336\) 0 0
\(337\) 17.4176 0.948795 0.474398 0.880311i \(-0.342666\pi\)
0.474398 + 0.880311i \(0.342666\pi\)
\(338\) 5.99690 0.326188
\(339\) 19.7466 1.07249
\(340\) 9.45577 0.512811
\(341\) −7.58853 −0.410942
\(342\) −1.92127 −0.103890
\(343\) 0 0
\(344\) −8.21648 −0.443003
\(345\) −3.82843 −0.206116
\(346\) 22.0308 1.18438
\(347\) −9.87145 −0.529927 −0.264964 0.964258i \(-0.585360\pi\)
−0.264964 + 0.964258i \(0.585360\pi\)
\(348\) −1.35854 −0.0728256
\(349\) 33.9170 1.81553 0.907767 0.419475i \(-0.137786\pi\)
0.907767 + 0.419475i \(0.137786\pi\)
\(350\) 0 0
\(351\) 4.35854 0.232642
\(352\) 5.27981 0.281415
\(353\) −7.35544 −0.391491 −0.195745 0.980655i \(-0.562713\pi\)
−0.195745 + 0.980655i \(0.562713\pi\)
\(354\) 4.29831 0.228453
\(355\) −29.2548 −1.55268
\(356\) −12.2952 −0.651645
\(357\) 0 0
\(358\) −16.9969 −0.898314
\(359\) 15.1932 0.801869 0.400934 0.916107i \(-0.368686\pi\)
0.400934 + 0.916107i \(0.368686\pi\)
\(360\) −3.82843 −0.201776
\(361\) −15.3087 −0.805722
\(362\) −12.4022 −0.651844
\(363\) 16.8764 0.885783
\(364\) 0 0
\(365\) 1.67407 0.0876247
\(366\) 11.4342 0.597674
\(367\) 8.34632 0.435674 0.217837 0.975985i \(-0.430100\pi\)
0.217837 + 0.975985i \(0.430100\pi\)
\(368\) 1.00000 0.0521286
\(369\) 1.92127 0.100017
\(370\) −25.4145 −1.32124
\(371\) 0 0
\(372\) −1.43727 −0.0745191
\(373\) 20.5596 1.06454 0.532269 0.846576i \(-0.321340\pi\)
0.532269 + 0.846576i \(0.321340\pi\)
\(374\) −13.0405 −0.674310
\(375\) −17.8284 −0.920656
\(376\) −6.99690 −0.360838
\(377\) −5.92127 −0.304961
\(378\) 0 0
\(379\) −1.45139 −0.0745527 −0.0372764 0.999305i \(-0.511868\pi\)
−0.0372764 + 0.999305i \(0.511868\pi\)
\(380\) 7.35544 0.377326
\(381\) 18.9367 0.970155
\(382\) −16.6384 −0.851292
\(383\) −29.4145 −1.50301 −0.751505 0.659728i \(-0.770672\pi\)
−0.751505 + 0.659728i \(0.770672\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 25.2165 1.28349
\(387\) −8.21648 −0.417667
\(388\) 0.298311 0.0151444
\(389\) 22.1513 1.12311 0.561557 0.827438i \(-0.310203\pi\)
0.561557 + 0.827438i \(0.310203\pi\)
\(390\) −16.6864 −0.844947
\(391\) −2.46988 −0.124907
\(392\) 0 0
\(393\) 16.3554 0.825023
\(394\) 13.6199 0.686159
\(395\) −5.20108 −0.261695
\(396\) 5.27981 0.265321
\(397\) −20.8567 −1.04677 −0.523385 0.852097i \(-0.675331\pi\)
−0.523385 + 0.852097i \(0.675331\pi\)
\(398\) −0.902774 −0.0452520
\(399\) 0 0
\(400\) 9.65685 0.482843
\(401\) −15.0430 −0.751213 −0.375606 0.926779i \(-0.622565\pi\)
−0.375606 + 0.926779i \(0.622565\pi\)
\(402\) 5.33092 0.265882
\(403\) −6.26442 −0.312053
\(404\) 18.0590 0.898470
\(405\) −3.82843 −0.190236
\(406\) 0 0
\(407\) 35.0493 1.73733
\(408\) −2.46988 −0.122277
\(409\) −19.0960 −0.944237 −0.472119 0.881535i \(-0.656511\pi\)
−0.472119 + 0.881535i \(0.656511\pi\)
\(410\) −7.35544 −0.363259
\(411\) −5.10824 −0.251971
\(412\) −0.669078 −0.0329631
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 16.7572 0.822578
\(416\) 4.35854 0.213695
\(417\) 9.31371 0.456094
\(418\) −10.1440 −0.496157
\(419\) −20.2165 −0.987640 −0.493820 0.869564i \(-0.664400\pi\)
−0.493820 + 0.869564i \(0.664400\pi\)
\(420\) 0 0
\(421\) 8.59662 0.418974 0.209487 0.977811i \(-0.432821\pi\)
0.209487 + 0.977811i \(0.432821\pi\)
\(422\) −22.8131 −1.11052
\(423\) −6.99690 −0.340201
\(424\) −2.88866 −0.140286
\(425\) −23.8513 −1.15696
\(426\) 7.64146 0.370230
\(427\) 0 0
\(428\) −18.4361 −0.891141
\(429\) 23.0123 1.11104
\(430\) 31.4562 1.51695
\(431\) 36.0616 1.73703 0.868513 0.495667i \(-0.165076\pi\)
0.868513 + 0.495667i \(0.165076\pi\)
\(432\) 1.00000 0.0481125
\(433\) 7.56462 0.363532 0.181766 0.983342i \(-0.441819\pi\)
0.181766 + 0.983342i \(0.441819\pi\)
\(434\) 0 0
\(435\) 5.20108 0.249373
\(436\) 16.5314 0.791710
\(437\) −1.92127 −0.0919068
\(438\) −0.437273 −0.0208937
\(439\) 35.1470 1.67747 0.838736 0.544538i \(-0.183295\pi\)
0.838736 + 0.544538i \(0.183295\pi\)
\(440\) −20.2134 −0.963635
\(441\) 0 0
\(442\) −10.7651 −0.512043
\(443\) −6.35544 −0.301956 −0.150978 0.988537i \(-0.548242\pi\)
−0.150978 + 0.988537i \(0.548242\pi\)
\(444\) 6.63836 0.315043
\(445\) 47.0713 2.23139
\(446\) −0.942867 −0.0446460
\(447\) 2.23370 0.105650
\(448\) 0 0
\(449\) −4.36043 −0.205782 −0.102891 0.994693i \(-0.532809\pi\)
−0.102891 + 0.994693i \(0.532809\pi\)
\(450\) 9.65685 0.455228
\(451\) 10.1440 0.477660
\(452\) 19.7466 0.928802
\(453\) −22.2134 −1.04368
\(454\) 15.3106 0.718563
\(455\) 0 0
\(456\) −1.92127 −0.0899717
\(457\) −33.8567 −1.58375 −0.791876 0.610683i \(-0.790895\pi\)
−0.791876 + 0.610683i \(0.790895\pi\)
\(458\) −6.07873 −0.284040
\(459\) −2.46988 −0.115284
\(460\) −3.82843 −0.178501
\(461\) −17.6851 −0.823676 −0.411838 0.911257i \(-0.635113\pi\)
−0.411838 + 0.911257i \(0.635113\pi\)
\(462\) 0 0
\(463\) 35.5954 1.65426 0.827130 0.562011i \(-0.189972\pi\)
0.827130 + 0.562011i \(0.189972\pi\)
\(464\) −1.35854 −0.0630688
\(465\) 5.50249 0.255172
\(466\) −7.57812 −0.351050
\(467\) −14.4022 −0.666453 −0.333226 0.942847i \(-0.608137\pi\)
−0.333226 + 0.942847i \(0.608137\pi\)
\(468\) 4.35854 0.201474
\(469\) 0 0
\(470\) 26.7871 1.23560
\(471\) 10.5731 0.487184
\(472\) 4.29831 0.197846
\(473\) −43.3815 −1.99468
\(474\) 1.35854 0.0624000
\(475\) −18.5534 −0.851290
\(476\) 0 0
\(477\) −2.88866 −0.132263
\(478\) −16.4330 −0.751626
\(479\) −38.3542 −1.75245 −0.876225 0.481902i \(-0.839946\pi\)
−0.876225 + 0.481902i \(0.839946\pi\)
\(480\) −3.82843 −0.174743
\(481\) 28.9336 1.31926
\(482\) −13.2319 −0.602696
\(483\) 0 0
\(484\) 16.8764 0.767111
\(485\) −1.14206 −0.0518583
\(486\) 1.00000 0.0453609
\(487\) 20.2504 0.917632 0.458816 0.888531i \(-0.348274\pi\)
0.458816 + 0.888531i \(0.348274\pi\)
\(488\) 11.4342 0.517601
\(489\) 19.7294 0.892194
\(490\) 0 0
\(491\) −26.6459 −1.20251 −0.601257 0.799056i \(-0.705333\pi\)
−0.601257 + 0.799056i \(0.705333\pi\)
\(492\) 1.92127 0.0866176
\(493\) 3.35544 0.151122
\(494\) −8.37394 −0.376761
\(495\) −20.2134 −0.908524
\(496\) −1.43727 −0.0645355
\(497\) 0 0
\(498\) −4.37704 −0.196140
\(499\) −13.0602 −0.584656 −0.292328 0.956318i \(-0.594430\pi\)
−0.292328 + 0.956318i \(0.594430\pi\)
\(500\) −17.8284 −0.797311
\(501\) 10.6907 0.477627
\(502\) 9.87644 0.440807
\(503\) −11.1525 −0.497264 −0.248632 0.968598i \(-0.579981\pi\)
−0.248632 + 0.968598i \(0.579981\pi\)
\(504\) 0 0
\(505\) −69.1377 −3.07659
\(506\) 5.27981 0.234716
\(507\) 5.99690 0.266332
\(508\) 18.9367 0.840179
\(509\) −11.8874 −0.526899 −0.263449 0.964673i \(-0.584860\pi\)
−0.263449 + 0.964673i \(0.584860\pi\)
\(510\) 9.45577 0.418709
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −1.92127 −0.0848262
\(514\) −21.3554 −0.941948
\(515\) 2.56152 0.112874
\(516\) −8.21648 −0.361710
\(517\) −36.9423 −1.62472
\(518\) 0 0
\(519\) 22.0308 0.967045
\(520\) −16.6864 −0.731746
\(521\) 25.4452 1.11477 0.557387 0.830253i \(-0.311804\pi\)
0.557387 + 0.830253i \(0.311804\pi\)
\(522\) −1.35854 −0.0594618
\(523\) −24.5424 −1.07316 −0.536582 0.843848i \(-0.680285\pi\)
−0.536582 + 0.843848i \(0.680285\pi\)
\(524\) 16.3554 0.714491
\(525\) 0 0
\(526\) 8.68009 0.378470
\(527\) 3.54990 0.154636
\(528\) 5.27981 0.229774
\(529\) 1.00000 0.0434783
\(530\) 11.0590 0.480373
\(531\) 4.29831 0.186531
\(532\) 0 0
\(533\) 8.37394 0.362716
\(534\) −12.2952 −0.532066
\(535\) 70.5811 3.05149
\(536\) 5.33092 0.230261
\(537\) −16.9969 −0.733470
\(538\) −26.0142 −1.12155
\(539\) 0 0
\(540\) −3.82843 −0.164749
\(541\) −29.9909 −1.28941 −0.644706 0.764431i \(-0.723020\pi\)
−0.644706 + 0.764431i \(0.723020\pi\)
\(542\) 1.43727 0.0617362
\(543\) −12.4022 −0.532228
\(544\) −2.46988 −0.105895
\(545\) −63.2893 −2.71101
\(546\) 0 0
\(547\) −35.6986 −1.52636 −0.763181 0.646185i \(-0.776363\pi\)
−0.763181 + 0.646185i \(0.776363\pi\)
\(548\) −5.10824 −0.218213
\(549\) 11.4342 0.487999
\(550\) 50.9864 2.17407
\(551\) 2.61013 0.111195
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 18.0289 0.765975
\(555\) −25.4145 −1.07878
\(556\) 9.31371 0.394989
\(557\) −34.3426 −1.45514 −0.727572 0.686032i \(-0.759351\pi\)
−0.727572 + 0.686032i \(0.759351\pi\)
\(558\) −1.43727 −0.0608446
\(559\) −35.8119 −1.51468
\(560\) 0 0
\(561\) −13.0405 −0.550571
\(562\) 24.9643 1.05306
\(563\) −32.4078 −1.36583 −0.682914 0.730499i \(-0.739288\pi\)
−0.682914 + 0.730499i \(0.739288\pi\)
\(564\) −6.99690 −0.294623
\(565\) −75.5984 −3.18045
\(566\) 2.73430 0.114931
\(567\) 0 0
\(568\) 7.64146 0.320628
\(569\) −19.4884 −0.816995 −0.408498 0.912759i \(-0.633947\pi\)
−0.408498 + 0.912759i \(0.633947\pi\)
\(570\) 7.35544 0.308086
\(571\) −9.13173 −0.382151 −0.191075 0.981575i \(-0.561198\pi\)
−0.191075 + 0.981575i \(0.561198\pi\)
\(572\) 23.0123 0.962193
\(573\) −16.6384 −0.695077
\(574\) 0 0
\(575\) 9.65685 0.402719
\(576\) 1.00000 0.0416667
\(577\) 34.9957 1.45689 0.728445 0.685104i \(-0.240243\pi\)
0.728445 + 0.685104i \(0.240243\pi\)
\(578\) −10.8997 −0.453367
\(579\) 25.2165 1.04796
\(580\) 5.20108 0.215963
\(581\) 0 0
\(582\) 0.298311 0.0123654
\(583\) −15.2516 −0.631656
\(584\) −0.437273 −0.0180945
\(585\) −16.6864 −0.689896
\(586\) −14.5147 −0.599597
\(587\) −3.60136 −0.148644 −0.0743221 0.997234i \(-0.523679\pi\)
−0.0743221 + 0.997234i \(0.523679\pi\)
\(588\) 0 0
\(589\) 2.76139 0.113781
\(590\) −16.4558 −0.677473
\(591\) 13.6199 0.560246
\(592\) 6.63836 0.272835
\(593\) 7.65066 0.314175 0.157087 0.987585i \(-0.449790\pi\)
0.157087 + 0.987585i \(0.449790\pi\)
\(594\) 5.27981 0.216633
\(595\) 0 0
\(596\) 2.23370 0.0914957
\(597\) −0.902774 −0.0369481
\(598\) 4.35854 0.178234
\(599\) −19.7030 −0.805045 −0.402522 0.915410i \(-0.631866\pi\)
−0.402522 + 0.915410i \(0.631866\pi\)
\(600\) 9.65685 0.394239
\(601\) −22.3094 −0.910019 −0.455010 0.890486i \(-0.650364\pi\)
−0.455010 + 0.890486i \(0.650364\pi\)
\(602\) 0 0
\(603\) 5.33092 0.217092
\(604\) −22.2134 −0.903850
\(605\) −64.6102 −2.62678
\(606\) 18.0590 0.733598
\(607\) 4.21958 0.171268 0.0856338 0.996327i \(-0.472709\pi\)
0.0856338 + 0.996327i \(0.472709\pi\)
\(608\) −1.92127 −0.0779178
\(609\) 0 0
\(610\) −43.7749 −1.77239
\(611\) −30.4963 −1.23375
\(612\) −2.46988 −0.0998391
\(613\) −9.33816 −0.377165 −0.188582 0.982057i \(-0.560389\pi\)
−0.188582 + 0.982057i \(0.560389\pi\)
\(614\) 31.9521 1.28948
\(615\) −7.35544 −0.296600
\(616\) 0 0
\(617\) 1.89055 0.0761107 0.0380553 0.999276i \(-0.487884\pi\)
0.0380553 + 0.999276i \(0.487884\pi\)
\(618\) −0.669078 −0.0269143
\(619\) −15.3665 −0.617630 −0.308815 0.951122i \(-0.599932\pi\)
−0.308815 + 0.951122i \(0.599932\pi\)
\(620\) 5.50249 0.220985
\(621\) 1.00000 0.0401286
\(622\) 31.0142 1.24356
\(623\) 0 0
\(624\) 4.35854 0.174481
\(625\) 19.9706 0.798823
\(626\) −6.57623 −0.262839
\(627\) −10.1440 −0.405110
\(628\) 10.5731 0.421914
\(629\) −16.3960 −0.653750
\(630\) 0 0
\(631\) −1.49622 −0.0595636 −0.0297818 0.999556i \(-0.509481\pi\)
−0.0297818 + 0.999556i \(0.509481\pi\)
\(632\) 1.35854 0.0540400
\(633\) −22.8131 −0.906739
\(634\) −28.0142 −1.11259
\(635\) −72.4977 −2.87698
\(636\) −2.88866 −0.114543
\(637\) 0 0
\(638\) −7.17286 −0.283976
\(639\) 7.64146 0.302291
\(640\) −3.82843 −0.151332
\(641\) −1.33566 −0.0527555 −0.0263778 0.999652i \(-0.508397\pi\)
−0.0263778 + 0.999652i \(0.508397\pi\)
\(642\) −18.4361 −0.727613
\(643\) −17.5167 −0.690791 −0.345395 0.938457i \(-0.612255\pi\)
−0.345395 + 0.938457i \(0.612255\pi\)
\(644\) 0 0
\(645\) 31.4562 1.23859
\(646\) 4.74531 0.186702
\(647\) 22.1612 0.871248 0.435624 0.900129i \(-0.356528\pi\)
0.435624 + 0.900129i \(0.356528\pi\)
\(648\) 1.00000 0.0392837
\(649\) 22.6943 0.890828
\(650\) 42.0898 1.65090
\(651\) 0 0
\(652\) 19.7294 0.772662
\(653\) −9.17286 −0.358962 −0.179481 0.983761i \(-0.557442\pi\)
−0.179481 + 0.983761i \(0.557442\pi\)
\(654\) 16.5314 0.646429
\(655\) −62.6156 −2.44659
\(656\) 1.92127 0.0750130
\(657\) −0.437273 −0.0170596
\(658\) 0 0
\(659\) −23.3419 −0.909273 −0.454636 0.890677i \(-0.650231\pi\)
−0.454636 + 0.890677i \(0.650231\pi\)
\(660\) −20.2134 −0.786805
\(661\) −35.9632 −1.39880 −0.699402 0.714728i \(-0.746550\pi\)
−0.699402 + 0.714728i \(0.746550\pi\)
\(662\) 24.8266 0.964914
\(663\) −10.7651 −0.418082
\(664\) −4.37704 −0.169862
\(665\) 0 0
\(666\) 6.63836 0.257231
\(667\) −1.35854 −0.0526030
\(668\) 10.6907 0.413637
\(669\) −0.942867 −0.0364533
\(670\) −20.4090 −0.788470
\(671\) 60.3703 2.33057
\(672\) 0 0
\(673\) 22.9957 0.886419 0.443209 0.896418i \(-0.353840\pi\)
0.443209 + 0.896418i \(0.353840\pi\)
\(674\) 17.4176 0.670900
\(675\) 9.65685 0.371692
\(676\) 5.99690 0.230650
\(677\) 13.3648 0.513652 0.256826 0.966458i \(-0.417323\pi\)
0.256826 + 0.966458i \(0.417323\pi\)
\(678\) 19.7466 0.758364
\(679\) 0 0
\(680\) 9.45577 0.362612
\(681\) 15.3106 0.586704
\(682\) −7.58853 −0.290580
\(683\) −11.1660 −0.427254 −0.213627 0.976915i \(-0.568528\pi\)
−0.213627 + 0.976915i \(0.568528\pi\)
\(684\) −1.92127 −0.0734616
\(685\) 19.5565 0.747217
\(686\) 0 0
\(687\) −6.07873 −0.231918
\(688\) −8.21648 −0.313250
\(689\) −12.5903 −0.479654
\(690\) −3.82843 −0.145746
\(691\) 43.9829 1.67319 0.836593 0.547824i \(-0.184544\pi\)
0.836593 + 0.547824i \(0.184544\pi\)
\(692\) 22.0308 0.837485
\(693\) 0 0
\(694\) −9.87145 −0.374715
\(695\) −35.6569 −1.35254
\(696\) −1.35854 −0.0514955
\(697\) −4.74531 −0.179742
\(698\) 33.9170 1.28378
\(699\) −7.57812 −0.286631
\(700\) 0 0
\(701\) −2.86043 −0.108037 −0.0540185 0.998540i \(-0.517203\pi\)
−0.0540185 + 0.998540i \(0.517203\pi\)
\(702\) 4.35854 0.164503
\(703\) −12.7541 −0.481029
\(704\) 5.27981 0.198990
\(705\) 26.7871 1.00886
\(706\) −7.35544 −0.276826
\(707\) 0 0
\(708\) 4.29831 0.161540
\(709\) −34.5795 −1.29866 −0.649330 0.760507i \(-0.724951\pi\)
−0.649330 + 0.760507i \(0.724951\pi\)
\(710\) −29.2548 −1.09791
\(711\) 1.35854 0.0509494
\(712\) −12.2952 −0.460783
\(713\) −1.43727 −0.0538263
\(714\) 0 0
\(715\) −88.1009 −3.29479
\(716\) −16.9969 −0.635204
\(717\) −16.4330 −0.613700
\(718\) 15.1932 0.567007
\(719\) 23.3991 0.872638 0.436319 0.899792i \(-0.356282\pi\)
0.436319 + 0.899792i \(0.356282\pi\)
\(720\) −3.82843 −0.142677
\(721\) 0 0
\(722\) −15.3087 −0.569732
\(723\) −13.2319 −0.492099
\(724\) −12.4022 −0.460923
\(725\) −13.1193 −0.487237
\(726\) 16.8764 0.626343
\(727\) 0.604463 0.0224183 0.0112091 0.999937i \(-0.496432\pi\)
0.0112091 + 0.999937i \(0.496432\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.67407 0.0619600
\(731\) 20.2938 0.750592
\(732\) 11.4342 0.422619
\(733\) 20.2670 0.748578 0.374289 0.927312i \(-0.377887\pi\)
0.374289 + 0.927312i \(0.377887\pi\)
\(734\) 8.34632 0.308068
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 28.1463 1.03678
\(738\) 1.92127 0.0707229
\(739\) −20.8376 −0.766522 −0.383261 0.923640i \(-0.625199\pi\)
−0.383261 + 0.923640i \(0.625199\pi\)
\(740\) −25.4145 −0.934254
\(741\) −8.37394 −0.307624
\(742\) 0 0
\(743\) −29.3222 −1.07573 −0.537864 0.843032i \(-0.680769\pi\)
−0.537864 + 0.843032i \(0.680769\pi\)
\(744\) −1.43727 −0.0526930
\(745\) −8.55154 −0.313304
\(746\) 20.5596 0.752741
\(747\) −4.37704 −0.160148
\(748\) −13.0405 −0.476809
\(749\) 0 0
\(750\) −17.8284 −0.651002
\(751\) 3.60325 0.131485 0.0657423 0.997837i \(-0.479058\pi\)
0.0657423 + 0.997837i \(0.479058\pi\)
\(752\) −6.99690 −0.255151
\(753\) 9.87644 0.359917
\(754\) −5.92127 −0.215640
\(755\) 85.0423 3.09501
\(756\) 0 0
\(757\) 51.0161 1.85421 0.927105 0.374801i \(-0.122289\pi\)
0.927105 + 0.374801i \(0.122289\pi\)
\(758\) −1.45139 −0.0527167
\(759\) 5.27981 0.191645
\(760\) 7.35544 0.266810
\(761\) 38.8524 1.40840 0.704200 0.710002i \(-0.251306\pi\)
0.704200 + 0.710002i \(0.251306\pi\)
\(762\) 18.9367 0.686003
\(763\) 0 0
\(764\) −16.6384 −0.601955
\(765\) 9.45577 0.341874
\(766\) −29.4145 −1.06279
\(767\) 18.7344 0.676459
\(768\) 1.00000 0.0360844
\(769\) −23.4176 −0.844459 −0.422230 0.906489i \(-0.638752\pi\)
−0.422230 + 0.906489i \(0.638752\pi\)
\(770\) 0 0
\(771\) −21.3554 −0.769098
\(772\) 25.2165 0.907561
\(773\) 26.1685 0.941215 0.470607 0.882343i \(-0.344035\pi\)
0.470607 + 0.882343i \(0.344035\pi\)
\(774\) −8.21648 −0.295335
\(775\) −13.8795 −0.498568
\(776\) 0.298311 0.0107087
\(777\) 0 0
\(778\) 22.1513 0.794161
\(779\) −3.69128 −0.132254
\(780\) −16.6864 −0.597468
\(781\) 40.3455 1.44367
\(782\) −2.46988 −0.0883229
\(783\) −1.35854 −0.0485504
\(784\) 0 0
\(785\) −40.4785 −1.44474
\(786\) 16.3554 0.583379
\(787\) −12.4872 −0.445120 −0.222560 0.974919i \(-0.571441\pi\)
−0.222560 + 0.974919i \(0.571441\pi\)
\(788\) 13.6199 0.485187
\(789\) 8.68009 0.309019
\(790\) −5.20108 −0.185046
\(791\) 0 0
\(792\) 5.27981 0.187610
\(793\) 49.8363 1.76974
\(794\) −20.8567 −0.740178
\(795\) 11.0590 0.392223
\(796\) −0.902774 −0.0319980
\(797\) −15.2626 −0.540629 −0.270315 0.962772i \(-0.587128\pi\)
−0.270315 + 0.962772i \(0.587128\pi\)
\(798\) 0 0
\(799\) 17.2815 0.611377
\(800\) 9.65685 0.341421
\(801\) −12.2952 −0.434430
\(802\) −15.0430 −0.531187
\(803\) −2.30872 −0.0814729
\(804\) 5.33092 0.188007
\(805\) 0 0
\(806\) −6.26442 −0.220655
\(807\) −26.0142 −0.915743
\(808\) 18.0590 0.635314
\(809\) −21.3492 −0.750600 −0.375300 0.926903i \(-0.622460\pi\)
−0.375300 + 0.926903i \(0.622460\pi\)
\(810\) −3.82843 −0.134517
\(811\) 2.14869 0.0754508 0.0377254 0.999288i \(-0.487989\pi\)
0.0377254 + 0.999288i \(0.487989\pi\)
\(812\) 0 0
\(813\) 1.43727 0.0504074
\(814\) 35.0493 1.22848
\(815\) −75.5325 −2.64579
\(816\) −2.46988 −0.0864632
\(817\) 15.7861 0.552285
\(818\) −19.0960 −0.667677
\(819\) 0 0
\(820\) −7.35544 −0.256863
\(821\) 23.5688 0.822558 0.411279 0.911510i \(-0.365082\pi\)
0.411279 + 0.911510i \(0.365082\pi\)
\(822\) −5.10824 −0.178170
\(823\) −19.0948 −0.665603 −0.332802 0.942997i \(-0.607994\pi\)
−0.332802 + 0.942997i \(0.607994\pi\)
\(824\) −0.669078 −0.0233084
\(825\) 50.9864 1.77512
\(826\) 0 0
\(827\) −13.2428 −0.460498 −0.230249 0.973132i \(-0.573954\pi\)
−0.230249 + 0.973132i \(0.573954\pi\)
\(828\) 1.00000 0.0347524
\(829\) 17.6218 0.612029 0.306014 0.952027i \(-0.401004\pi\)
0.306014 + 0.952027i \(0.401004\pi\)
\(830\) 16.7572 0.581650
\(831\) 18.0289 0.625416
\(832\) 4.35854 0.151105
\(833\) 0 0
\(834\) 9.31371 0.322507
\(835\) −40.9288 −1.41640
\(836\) −10.1440 −0.350836
\(837\) −1.43727 −0.0496794
\(838\) −20.2165 −0.698367
\(839\) −5.89682 −0.203581 −0.101790 0.994806i \(-0.532457\pi\)
−0.101790 + 0.994806i \(0.532457\pi\)
\(840\) 0 0
\(841\) −27.1544 −0.936357
\(842\) 8.59662 0.296259
\(843\) 24.9643 0.859816
\(844\) −22.8131 −0.785259
\(845\) −22.9587 −0.789803
\(846\) −6.99690 −0.240558
\(847\) 0 0
\(848\) −2.88866 −0.0991970
\(849\) 2.73430 0.0938409
\(850\) −23.8513 −0.818093
\(851\) 6.63836 0.227560
\(852\) 7.64146 0.261792
\(853\) 26.1575 0.895614 0.447807 0.894130i \(-0.352205\pi\)
0.447807 + 0.894130i \(0.352205\pi\)
\(854\) 0 0
\(855\) 7.35544 0.251551
\(856\) −18.4361 −0.630132
\(857\) −26.3825 −0.901208 −0.450604 0.892724i \(-0.648791\pi\)
−0.450604 + 0.892724i \(0.648791\pi\)
\(858\) 23.0123 0.785627
\(859\) 40.4565 1.38036 0.690178 0.723640i \(-0.257532\pi\)
0.690178 + 0.723640i \(0.257532\pi\)
\(860\) 31.4562 1.07265
\(861\) 0 0
\(862\) 36.0616 1.22826
\(863\) −20.2501 −0.689322 −0.344661 0.938727i \(-0.612006\pi\)
−0.344661 + 0.938727i \(0.612006\pi\)
\(864\) 1.00000 0.0340207
\(865\) −84.3433 −2.86776
\(866\) 7.56462 0.257056
\(867\) −10.8997 −0.370172
\(868\) 0 0
\(869\) 7.17286 0.243322
\(870\) 5.20108 0.176333
\(871\) 23.2351 0.787290
\(872\) 16.5314 0.559824
\(873\) 0.298311 0.0100963
\(874\) −1.92127 −0.0649879
\(875\) 0 0
\(876\) −0.437273 −0.0147741
\(877\) −9.88873 −0.333919 −0.166959 0.985964i \(-0.553395\pi\)
−0.166959 + 0.985964i \(0.553395\pi\)
\(878\) 35.1470 1.18615
\(879\) −14.5147 −0.489569
\(880\) −20.2134 −0.681393
\(881\) 4.39115 0.147942 0.0739709 0.997260i \(-0.476433\pi\)
0.0739709 + 0.997260i \(0.476433\pi\)
\(882\) 0 0
\(883\) −6.05428 −0.203743 −0.101871 0.994798i \(-0.532483\pi\)
−0.101871 + 0.994798i \(0.532483\pi\)
\(884\) −10.7651 −0.362069
\(885\) −16.4558 −0.553155
\(886\) −6.35544 −0.213515
\(887\) −44.4946 −1.49398 −0.746990 0.664835i \(-0.768502\pi\)
−0.746990 + 0.664835i \(0.768502\pi\)
\(888\) 6.63836 0.222769
\(889\) 0 0
\(890\) 47.0713 1.57783
\(891\) 5.27981 0.176880
\(892\) −0.942867 −0.0315695
\(893\) 13.4429 0.449851
\(894\) 2.23370 0.0747059
\(895\) 65.0714 2.17510
\(896\) 0 0
\(897\) 4.35854 0.145528
\(898\) −4.36043 −0.145510
\(899\) 1.95260 0.0651228
\(900\) 9.65685 0.321895
\(901\) 7.13465 0.237690
\(902\) 10.1440 0.337757
\(903\) 0 0
\(904\) 19.7466 0.656762
\(905\) 47.4808 1.57832
\(906\) −22.2134 −0.737990
\(907\) −20.0063 −0.664297 −0.332149 0.943227i \(-0.607774\pi\)
−0.332149 + 0.943227i \(0.607774\pi\)
\(908\) 15.3106 0.508100
\(909\) 18.0590 0.598980
\(910\) 0 0
\(911\) 36.4085 1.20627 0.603134 0.797640i \(-0.293918\pi\)
0.603134 + 0.797640i \(0.293918\pi\)
\(912\) −1.92127 −0.0636196
\(913\) −23.1100 −0.764828
\(914\) −33.8567 −1.11988
\(915\) −43.7749 −1.44715
\(916\) −6.07873 −0.200847
\(917\) 0 0
\(918\) −2.46988 −0.0815183
\(919\) −33.7883 −1.11457 −0.557287 0.830320i \(-0.688158\pi\)
−0.557287 + 0.830320i \(0.688158\pi\)
\(920\) −3.82843 −0.126220
\(921\) 31.9521 1.05286
\(922\) −17.6851 −0.582427
\(923\) 33.3056 1.09627
\(924\) 0 0
\(925\) 64.1056 2.10778
\(926\) 35.5954 1.16974
\(927\) −0.669078 −0.0219754
\(928\) −1.35854 −0.0445964
\(929\) −28.8610 −0.946900 −0.473450 0.880821i \(-0.656992\pi\)
−0.473450 + 0.880821i \(0.656992\pi\)
\(930\) 5.50249 0.180434
\(931\) 0 0
\(932\) −7.57812 −0.248230
\(933\) 31.0142 1.01536
\(934\) −14.4022 −0.471253
\(935\) 49.9247 1.63271
\(936\) 4.35854 0.142463
\(937\) 15.6710 0.511951 0.255975 0.966683i \(-0.417603\pi\)
0.255975 + 0.966683i \(0.417603\pi\)
\(938\) 0 0
\(939\) −6.57623 −0.214607
\(940\) 26.7871 0.873700
\(941\) 21.7460 0.708899 0.354450 0.935075i \(-0.384668\pi\)
0.354450 + 0.935075i \(0.384668\pi\)
\(942\) 10.5731 0.344491
\(943\) 1.92127 0.0625652
\(944\) 4.29831 0.139898
\(945\) 0 0
\(946\) −43.3815 −1.41045
\(947\) 24.3784 0.792192 0.396096 0.918209i \(-0.370365\pi\)
0.396096 + 0.918209i \(0.370365\pi\)
\(948\) 1.35854 0.0441234
\(949\) −1.90587 −0.0618673
\(950\) −18.5534 −0.601953
\(951\) −28.0142 −0.908422
\(952\) 0 0
\(953\) 41.2178 1.33518 0.667588 0.744531i \(-0.267327\pi\)
0.667588 + 0.744531i \(0.267327\pi\)
\(954\) −2.88866 −0.0935238
\(955\) 63.6987 2.06124
\(956\) −16.4330 −0.531480
\(957\) −7.17286 −0.231866
\(958\) −38.3542 −1.23917
\(959\) 0 0
\(960\) −3.82843 −0.123562
\(961\) −28.9342 −0.933363
\(962\) 28.9336 0.932856
\(963\) −18.4361 −0.594094
\(964\) −13.2319 −0.426170
\(965\) −96.5395 −3.10772
\(966\) 0 0
\(967\) 36.7730 1.18254 0.591270 0.806474i \(-0.298627\pi\)
0.591270 + 0.806474i \(0.298627\pi\)
\(968\) 16.8764 0.542429
\(969\) 4.74531 0.152441
\(970\) −1.14206 −0.0366694
\(971\) 15.9661 0.512377 0.256188 0.966627i \(-0.417533\pi\)
0.256188 + 0.966627i \(0.417533\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 20.2504 0.648864
\(975\) 42.0898 1.34795
\(976\) 11.4342 0.365999
\(977\) 18.0392 0.577127 0.288563 0.957461i \(-0.406822\pi\)
0.288563 + 0.957461i \(0.406822\pi\)
\(978\) 19.7294 0.630876
\(979\) −64.9164 −2.07474
\(980\) 0 0
\(981\) 16.5314 0.527807
\(982\) −26.6459 −0.850305
\(983\) −14.0417 −0.447862 −0.223931 0.974605i \(-0.571889\pi\)
−0.223931 + 0.974605i \(0.571889\pi\)
\(984\) 1.92127 0.0612479
\(985\) −52.1426 −1.66140
\(986\) 3.35544 0.106859
\(987\) 0 0
\(988\) −8.37394 −0.266411
\(989\) −8.21648 −0.261269
\(990\) −20.2134 −0.642423
\(991\) 14.1581 0.449748 0.224874 0.974388i \(-0.427803\pi\)
0.224874 + 0.974388i \(0.427803\pi\)
\(992\) −1.43727 −0.0456335
\(993\) 24.8266 0.787849
\(994\) 0 0
\(995\) 3.45620 0.109569
\(996\) −4.37704 −0.138692
\(997\) −25.1735 −0.797254 −0.398627 0.917113i \(-0.630513\pi\)
−0.398627 + 0.917113i \(0.630513\pi\)
\(998\) −13.0602 −0.413414
\(999\) 6.63836 0.210028
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 6762.2.a.cp.1.2 4
7.2 even 3 966.2.i.l.277.3 8
7.4 even 3 966.2.i.l.415.3 yes 8
7.6 odd 2 6762.2.a.cm.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.l.277.3 8 7.2 even 3
966.2.i.l.415.3 yes 8 7.4 even 3
6762.2.a.cm.1.4 4 7.6 odd 2
6762.2.a.cp.1.2 4 1.1 even 1 trivial