Properties

Label 9702.2.a.dz.1.3
Level $9702$
Weight $2$
Character 9702.1
Self dual yes
Analytic conductor $77.471$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4352.1
Defining polynomial: \(x^{4} - 6 x^{2} - 4 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3234)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.334904\) of defining polynomial
Character \(\chi\) \(=\) 9702.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.473626 q^{5} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +0.473626 q^{5} -1.00000 q^{8} -0.473626 q^{10} -1.00000 q^{11} +7.03127 q^{13} +1.00000 q^{16} +4.64167 q^{17} +6.55765 q^{19} +0.473626 q^{20} +1.00000 q^{22} -1.49824 q^{23} -4.77568 q^{25} -7.03127 q^{26} +2.11882 q^{29} +6.83509 q^{31} -1.00000 q^{32} -4.64167 q^{34} -4.32666 q^{37} -6.55765 q^{38} -0.473626 q^{40} +11.2458 q^{41} +0.158619 q^{43} -1.00000 q^{44} +1.49824 q^{46} -0.270780 q^{47} +4.77568 q^{50} +7.03127 q^{52} +9.66628 q^{53} -0.473626 q^{55} -2.11882 q^{58} -9.82490 q^{59} +15.1993 q^{61} -6.83509 q^{62} +1.00000 q^{64} +3.33019 q^{65} -1.21137 q^{67} +4.64167 q^{68} -15.0098 q^{71} -6.13048 q^{73} +4.32666 q^{74} +6.55765 q^{76} +11.4420 q^{79} +0.473626 q^{80} -11.2458 q^{82} +7.54469 q^{83} +2.19841 q^{85} -0.158619 q^{86} +1.00000 q^{88} -16.6881 q^{89} -1.49824 q^{92} +0.270780 q^{94} +3.10587 q^{95} +12.9656 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + O(q^{10}) \) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{8} + 4 q^{10} - 4 q^{11} + 8 q^{13} + 4 q^{16} - 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{22} + 8 q^{23} + 4 q^{25} - 8 q^{26} + 8 q^{29} + 4 q^{31} - 4 q^{32} + 4 q^{34} + 8 q^{37} - 12 q^{38} + 4 q^{40} - 12 q^{41} - 8 q^{43} - 4 q^{44} - 8 q^{46} - 4 q^{47} - 4 q^{50} + 8 q^{52} + 8 q^{53} + 4 q^{55} - 8 q^{58} + 24 q^{61} - 4 q^{62} + 4 q^{64} + 16 q^{65} - 8 q^{67} - 4 q^{68} - 8 q^{71} + 4 q^{73} - 8 q^{74} + 12 q^{76} - 8 q^{79} - 4 q^{80} + 12 q^{82} - 4 q^{83} - 8 q^{85} + 8 q^{86} + 4 q^{88} - 24 q^{89} + 8 q^{92} + 4 q^{94} - 8 q^{95} + O(q^{100}) \)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0.473626 0.211812 0.105906 0.994376i \(-0.466226\pi\)
0.105906 + 0.994376i \(0.466226\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −0.473626 −0.149774
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 7.03127 1.95012 0.975062 0.221932i \(-0.0712363\pi\)
0.975062 + 0.221932i \(0.0712363\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.64167 1.12577 0.562885 0.826535i \(-0.309691\pi\)
0.562885 + 0.826535i \(0.309691\pi\)
\(18\) 0 0
\(19\) 6.55765 1.50443 0.752214 0.658919i \(-0.228986\pi\)
0.752214 + 0.658919i \(0.228986\pi\)
\(20\) 0.473626 0.105906
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −1.49824 −0.312404 −0.156202 0.987725i \(-0.549925\pi\)
−0.156202 + 0.987725i \(0.549925\pi\)
\(24\) 0 0
\(25\) −4.77568 −0.955136
\(26\) −7.03127 −1.37895
\(27\) 0 0
\(28\) 0 0
\(29\) 2.11882 0.393456 0.196728 0.980458i \(-0.436968\pi\)
0.196728 + 0.980458i \(0.436968\pi\)
\(30\) 0 0
\(31\) 6.83509 1.22762 0.613809 0.789454i \(-0.289636\pi\)
0.613809 + 0.789454i \(0.289636\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.64167 −0.796040
\(35\) 0 0
\(36\) 0 0
\(37\) −4.32666 −0.711299 −0.355649 0.934619i \(-0.615740\pi\)
−0.355649 + 0.934619i \(0.615740\pi\)
\(38\) −6.55765 −1.06379
\(39\) 0 0
\(40\) −0.473626 −0.0748868
\(41\) 11.2458 1.75629 0.878147 0.478390i \(-0.158779\pi\)
0.878147 + 0.478390i \(0.158779\pi\)
\(42\) 0 0
\(43\) 0.158619 0.0241892 0.0120946 0.999927i \(-0.496150\pi\)
0.0120946 + 0.999927i \(0.496150\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 1.49824 0.220903
\(47\) −0.270780 −0.0394973 −0.0197486 0.999805i \(-0.506287\pi\)
−0.0197486 + 0.999805i \(0.506287\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 4.77568 0.675383
\(51\) 0 0
\(52\) 7.03127 0.975062
\(53\) 9.66628 1.32777 0.663883 0.747837i \(-0.268907\pi\)
0.663883 + 0.747837i \(0.268907\pi\)
\(54\) 0 0
\(55\) −0.473626 −0.0638637
\(56\) 0 0
\(57\) 0 0
\(58\) −2.11882 −0.278215
\(59\) −9.82490 −1.27909 −0.639546 0.768753i \(-0.720878\pi\)
−0.639546 + 0.768753i \(0.720878\pi\)
\(60\) 0 0
\(61\) 15.1993 1.94607 0.973037 0.230651i \(-0.0740856\pi\)
0.973037 + 0.230651i \(0.0740856\pi\)
\(62\) −6.83509 −0.868057
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.33019 0.413059
\(66\) 0 0
\(67\) −1.21137 −0.147992 −0.0739961 0.997259i \(-0.523575\pi\)
−0.0739961 + 0.997259i \(0.523575\pi\)
\(68\) 4.64167 0.562885
\(69\) 0 0
\(70\) 0 0
\(71\) −15.0098 −1.78134 −0.890668 0.454655i \(-0.849763\pi\)
−0.890668 + 0.454655i \(0.849763\pi\)
\(72\) 0 0
\(73\) −6.13048 −0.717518 −0.358759 0.933430i \(-0.616800\pi\)
−0.358759 + 0.933430i \(0.616800\pi\)
\(74\) 4.32666 0.502964
\(75\) 0 0
\(76\) 6.55765 0.752214
\(77\) 0 0
\(78\) 0 0
\(79\) 11.4420 1.28732 0.643660 0.765311i \(-0.277415\pi\)
0.643660 + 0.765311i \(0.277415\pi\)
\(80\) 0.473626 0.0529530
\(81\) 0 0
\(82\) −11.2458 −1.24189
\(83\) 7.54469 0.828138 0.414069 0.910246i \(-0.364107\pi\)
0.414069 + 0.910246i \(0.364107\pi\)
\(84\) 0 0
\(85\) 2.19841 0.238451
\(86\) −0.158619 −0.0171043
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −16.6881 −1.76894 −0.884469 0.466599i \(-0.845479\pi\)
−0.884469 + 0.466599i \(0.845479\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.49824 −0.156202
\(93\) 0 0
\(94\) 0.270780 0.0279288
\(95\) 3.10587 0.318656
\(96\) 0 0
\(97\) 12.9656 1.31645 0.658227 0.752819i \(-0.271307\pi\)
0.658227 + 0.752819i \(0.271307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −4.77568 −0.477568
\(101\) −0.298919 −0.0297436 −0.0148718 0.999889i \(-0.504734\pi\)
−0.0148718 + 0.999889i \(0.504734\pi\)
\(102\) 0 0
\(103\) −7.39550 −0.728700 −0.364350 0.931262i \(-0.618709\pi\)
−0.364350 + 0.931262i \(0.618709\pi\)
\(104\) −7.03127 −0.689473
\(105\) 0 0
\(106\) −9.66628 −0.938872
\(107\) 15.3231 1.48134 0.740672 0.671867i \(-0.234507\pi\)
0.740672 + 0.671867i \(0.234507\pi\)
\(108\) 0 0
\(109\) −0.422735 −0.0404907 −0.0202453 0.999795i \(-0.506445\pi\)
−0.0202453 + 0.999795i \(0.506445\pi\)
\(110\) 0.473626 0.0451584
\(111\) 0 0
\(112\) 0 0
\(113\) 8.40569 0.790741 0.395370 0.918522i \(-0.370616\pi\)
0.395370 + 0.918522i \(0.370616\pi\)
\(114\) 0 0
\(115\) −0.709603 −0.0661708
\(116\) 2.11882 0.196728
\(117\) 0 0
\(118\) 9.82490 0.904455
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −15.1993 −1.37608
\(123\) 0 0
\(124\) 6.83509 0.613809
\(125\) −4.63001 −0.414121
\(126\) 0 0
\(127\) 10.1023 0.896438 0.448219 0.893924i \(-0.352059\pi\)
0.448219 + 0.893924i \(0.352059\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.33019 −0.292077
\(131\) −20.8843 −1.82467 −0.912335 0.409444i \(-0.865723\pi\)
−0.912335 + 0.409444i \(0.865723\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.21137 0.104646
\(135\) 0 0
\(136\) −4.64167 −0.398020
\(137\) −7.35294 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(138\) 0 0
\(139\) −4.67647 −0.396653 −0.198327 0.980136i \(-0.563551\pi\)
−0.198327 + 0.980136i \(0.563551\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.0098 1.25959
\(143\) −7.03127 −0.587985
\(144\) 0 0
\(145\) 1.00353 0.0833386
\(146\) 6.13048 0.507362
\(147\) 0 0
\(148\) −4.32666 −0.355649
\(149\) −4.89097 −0.400684 −0.200342 0.979726i \(-0.564205\pi\)
−0.200342 + 0.979726i \(0.564205\pi\)
\(150\) 0 0
\(151\) −16.6533 −1.35523 −0.677614 0.735418i \(-0.736986\pi\)
−0.677614 + 0.735418i \(0.736986\pi\)
\(152\) −6.55765 −0.531895
\(153\) 0 0
\(154\) 0 0
\(155\) 3.23728 0.260024
\(156\) 0 0
\(157\) −14.5228 −1.15905 −0.579525 0.814955i \(-0.696762\pi\)
−0.579525 + 0.814955i \(0.696762\pi\)
\(158\) −11.4420 −0.910273
\(159\) 0 0
\(160\) −0.473626 −0.0374434
\(161\) 0 0
\(162\) 0 0
\(163\) −8.33609 −0.652933 −0.326466 0.945209i \(-0.605858\pi\)
−0.326466 + 0.945209i \(0.605858\pi\)
\(164\) 11.2458 0.878147
\(165\) 0 0
\(166\) −7.54469 −0.585582
\(167\) −2.84727 −0.220329 −0.110164 0.993913i \(-0.535138\pi\)
−0.110164 + 0.993913i \(0.535138\pi\)
\(168\) 0 0
\(169\) 36.4388 2.80298
\(170\) −2.19841 −0.168611
\(171\) 0 0
\(172\) 0.158619 0.0120946
\(173\) −9.17656 −0.697681 −0.348841 0.937182i \(-0.613425\pi\)
−0.348841 + 0.937182i \(0.613425\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 16.6881 1.25083
\(179\) 5.76625 0.430990 0.215495 0.976505i \(-0.430863\pi\)
0.215495 + 0.976505i \(0.430863\pi\)
\(180\) 0 0
\(181\) 3.38164 0.251356 0.125678 0.992071i \(-0.459889\pi\)
0.125678 + 0.992071i \(0.459889\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.49824 0.110451
\(185\) −2.04922 −0.150662
\(186\) 0 0
\(187\) −4.64167 −0.339432
\(188\) −0.270780 −0.0197486
\(189\) 0 0
\(190\) −3.10587 −0.225324
\(191\) 16.1586 1.16920 0.584598 0.811323i \(-0.301252\pi\)
0.584598 + 0.811323i \(0.301252\pi\)
\(192\) 0 0
\(193\) −24.8651 −1.78983 −0.894913 0.446240i \(-0.852763\pi\)
−0.894913 + 0.446240i \(0.852763\pi\)
\(194\) −12.9656 −0.930874
\(195\) 0 0
\(196\) 0 0
\(197\) −10.4290 −0.743036 −0.371518 0.928426i \(-0.621163\pi\)
−0.371518 + 0.928426i \(0.621163\pi\)
\(198\) 0 0
\(199\) −13.2712 −0.940767 −0.470384 0.882462i \(-0.655884\pi\)
−0.470384 + 0.882462i \(0.655884\pi\)
\(200\) 4.77568 0.337691
\(201\) 0 0
\(202\) 0.298919 0.0210319
\(203\) 0 0
\(204\) 0 0
\(205\) 5.32629 0.372004
\(206\) 7.39550 0.515269
\(207\) 0 0
\(208\) 7.03127 0.487531
\(209\) −6.55765 −0.453602
\(210\) 0 0
\(211\) −24.2837 −1.67176 −0.835880 0.548912i \(-0.815042\pi\)
−0.835880 + 0.548912i \(0.815042\pi\)
\(212\) 9.66628 0.663883
\(213\) 0 0
\(214\) −15.3231 −1.04747
\(215\) 0.0751261 0.00512356
\(216\) 0 0
\(217\) 0 0
\(218\) 0.422735 0.0286312
\(219\) 0 0
\(220\) −0.473626 −0.0319318
\(221\) 32.6368 2.19539
\(222\) 0 0
\(223\) −3.82529 −0.256161 −0.128080 0.991764i \(-0.540882\pi\)
−0.128080 + 0.991764i \(0.540882\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −8.40569 −0.559138
\(227\) 12.3106 0.817082 0.408541 0.912740i \(-0.366038\pi\)
0.408541 + 0.912740i \(0.366038\pi\)
\(228\) 0 0
\(229\) −20.2726 −1.33965 −0.669826 0.742518i \(-0.733631\pi\)
−0.669826 + 0.742518i \(0.733631\pi\)
\(230\) 0.709603 0.0467898
\(231\) 0 0
\(232\) −2.11882 −0.139108
\(233\) 10.8114 0.708277 0.354139 0.935193i \(-0.384774\pi\)
0.354139 + 0.935193i \(0.384774\pi\)
\(234\) 0 0
\(235\) −0.128248 −0.00836600
\(236\) −9.82490 −0.639546
\(237\) 0 0
\(238\) 0 0
\(239\) −4.07959 −0.263887 −0.131943 0.991257i \(-0.542122\pi\)
−0.131943 + 0.991257i \(0.542122\pi\)
\(240\) 0 0
\(241\) 21.9946 1.41680 0.708399 0.705812i \(-0.249418\pi\)
0.708399 + 0.705812i \(0.249418\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 15.1993 0.973037
\(245\) 0 0
\(246\) 0 0
\(247\) 46.1086 2.93382
\(248\) −6.83509 −0.434029
\(249\) 0 0
\(250\) 4.63001 0.292828
\(251\) 23.1082 1.45858 0.729289 0.684205i \(-0.239851\pi\)
0.729289 + 0.684205i \(0.239851\pi\)
\(252\) 0 0
\(253\) 1.49824 0.0941932
\(254\) −10.1023 −0.633877
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.75420 0.483694 0.241847 0.970314i \(-0.422247\pi\)
0.241847 + 0.970314i \(0.422247\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3.33019 0.206530
\(261\) 0 0
\(262\) 20.8843 1.29024
\(263\) 13.7851 0.850026 0.425013 0.905187i \(-0.360270\pi\)
0.425013 + 0.905187i \(0.360270\pi\)
\(264\) 0 0
\(265\) 4.57820 0.281236
\(266\) 0 0
\(267\) 0 0
\(268\) −1.21137 −0.0739961
\(269\) 6.13048 0.373782 0.186891 0.982381i \(-0.440159\pi\)
0.186891 + 0.982381i \(0.440159\pi\)
\(270\) 0 0
\(271\) −11.0661 −0.672216 −0.336108 0.941823i \(-0.609111\pi\)
−0.336108 + 0.941823i \(0.609111\pi\)
\(272\) 4.64167 0.281443
\(273\) 0 0
\(274\) 7.35294 0.444208
\(275\) 4.77568 0.287984
\(276\) 0 0
\(277\) −16.4986 −0.991305 −0.495653 0.868521i \(-0.665071\pi\)
−0.495653 + 0.868521i \(0.665071\pi\)
\(278\) 4.67647 0.280476
\(279\) 0 0
\(280\) 0 0
\(281\) 17.2341 1.02810 0.514051 0.857760i \(-0.328144\pi\)
0.514051 + 0.857760i \(0.328144\pi\)
\(282\) 0 0
\(283\) −18.3333 −1.08980 −0.544902 0.838500i \(-0.683433\pi\)
−0.544902 + 0.838500i \(0.683433\pi\)
\(284\) −15.0098 −0.890668
\(285\) 0 0
\(286\) 7.03127 0.415768
\(287\) 0 0
\(288\) 0 0
\(289\) 4.54509 0.267358
\(290\) −1.00353 −0.0589293
\(291\) 0 0
\(292\) −6.13048 −0.358759
\(293\) 26.1560 1.52805 0.764025 0.645187i \(-0.223221\pi\)
0.764025 + 0.645187i \(0.223221\pi\)
\(294\) 0 0
\(295\) −4.65332 −0.270927
\(296\) 4.32666 0.251482
\(297\) 0 0
\(298\) 4.89097 0.283326
\(299\) −10.5345 −0.609226
\(300\) 0 0
\(301\) 0 0
\(302\) 16.6533 0.958291
\(303\) 0 0
\(304\) 6.55765 0.376107
\(305\) 7.19879 0.412201
\(306\) 0 0
\(307\) 25.3109 1.44457 0.722286 0.691594i \(-0.243091\pi\)
0.722286 + 0.691594i \(0.243091\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.23728 −0.183865
\(311\) 24.6202 1.39608 0.698042 0.716057i \(-0.254055\pi\)
0.698042 + 0.716057i \(0.254055\pi\)
\(312\) 0 0
\(313\) 19.2713 1.08928 0.544639 0.838671i \(-0.316667\pi\)
0.544639 + 0.838671i \(0.316667\pi\)
\(314\) 14.5228 0.819572
\(315\) 0 0
\(316\) 11.4420 0.643660
\(317\) −4.34551 −0.244068 −0.122034 0.992526i \(-0.538942\pi\)
−0.122034 + 0.992526i \(0.538942\pi\)
\(318\) 0 0
\(319\) −2.11882 −0.118631
\(320\) 0.473626 0.0264765
\(321\) 0 0
\(322\) 0 0
\(323\) 30.4384 1.69364
\(324\) 0 0
\(325\) −33.5791 −1.86263
\(326\) 8.33609 0.461693
\(327\) 0 0
\(328\) −11.2458 −0.620944
\(329\) 0 0
\(330\) 0 0
\(331\) −5.46786 −0.300541 −0.150270 0.988645i \(-0.548014\pi\)
−0.150270 + 0.988645i \(0.548014\pi\)
\(332\) 7.54469 0.414069
\(333\) 0 0
\(334\) 2.84727 0.155796
\(335\) −0.573735 −0.0313465
\(336\) 0 0
\(337\) 7.43569 0.405048 0.202524 0.979277i \(-0.435086\pi\)
0.202524 + 0.979277i \(0.435086\pi\)
\(338\) −36.4388 −1.98201
\(339\) 0 0
\(340\) 2.19841 0.119226
\(341\) −6.83509 −0.370141
\(342\) 0 0
\(343\) 0 0
\(344\) −0.158619 −0.00855217
\(345\) 0 0
\(346\) 9.17656 0.493335
\(347\) −5.88508 −0.315928 −0.157964 0.987445i \(-0.550493\pi\)
−0.157964 + 0.987445i \(0.550493\pi\)
\(348\) 0 0
\(349\) 17.5658 0.940274 0.470137 0.882593i \(-0.344204\pi\)
0.470137 + 0.882593i \(0.344204\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 1.45401 0.0773891 0.0386945 0.999251i \(-0.487680\pi\)
0.0386945 + 0.999251i \(0.487680\pi\)
\(354\) 0 0
\(355\) −7.10903 −0.377308
\(356\) −16.6881 −0.884469
\(357\) 0 0
\(358\) −5.76625 −0.304756
\(359\) −19.6796 −1.03865 −0.519325 0.854577i \(-0.673817\pi\)
−0.519325 + 0.854577i \(0.673817\pi\)
\(360\) 0 0
\(361\) 24.0027 1.26330
\(362\) −3.38164 −0.177735
\(363\) 0 0
\(364\) 0 0
\(365\) −2.90355 −0.151979
\(366\) 0 0
\(367\) −18.0692 −0.943205 −0.471603 0.881811i \(-0.656324\pi\)
−0.471603 + 0.881811i \(0.656324\pi\)
\(368\) −1.49824 −0.0781009
\(369\) 0 0
\(370\) 2.04922 0.106534
\(371\) 0 0
\(372\) 0 0
\(373\) −21.3459 −1.10525 −0.552624 0.833431i \(-0.686373\pi\)
−0.552624 + 0.833431i \(0.686373\pi\)
\(374\) 4.64167 0.240015
\(375\) 0 0
\(376\) 0.270780 0.0139644
\(377\) 14.8980 0.767288
\(378\) 0 0
\(379\) −4.63448 −0.238057 −0.119029 0.992891i \(-0.537978\pi\)
−0.119029 + 0.992891i \(0.537978\pi\)
\(380\) 3.10587 0.159328
\(381\) 0 0
\(382\) −16.1586 −0.826747
\(383\) 7.37981 0.377091 0.188545 0.982065i \(-0.439623\pi\)
0.188545 + 0.982065i \(0.439623\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.8651 1.26560
\(387\) 0 0
\(388\) 12.9656 0.658227
\(389\) 1.34668 0.0682792 0.0341396 0.999417i \(-0.489131\pi\)
0.0341396 + 0.999417i \(0.489131\pi\)
\(390\) 0 0
\(391\) −6.95431 −0.351695
\(392\) 0 0
\(393\) 0 0
\(394\) 10.4290 0.525406
\(395\) 5.41921 0.272670
\(396\) 0 0
\(397\) −20.7346 −1.04064 −0.520320 0.853972i \(-0.674187\pi\)
−0.520320 + 0.853972i \(0.674187\pi\)
\(398\) 13.2712 0.665223
\(399\) 0 0
\(400\) −4.77568 −0.238784
\(401\) −7.96077 −0.397542 −0.198771 0.980046i \(-0.563695\pi\)
−0.198771 + 0.980046i \(0.563695\pi\)
\(402\) 0 0
\(403\) 48.0594 2.39401
\(404\) −0.298919 −0.0148718
\(405\) 0 0
\(406\) 0 0
\(407\) 4.32666 0.214465
\(408\) 0 0
\(409\) 30.1438 1.49052 0.745258 0.666777i \(-0.232326\pi\)
0.745258 + 0.666777i \(0.232326\pi\)
\(410\) −5.32629 −0.263047
\(411\) 0 0
\(412\) −7.39550 −0.364350
\(413\) 0 0
\(414\) 0 0
\(415\) 3.57336 0.175409
\(416\) −7.03127 −0.344737
\(417\) 0 0
\(418\) 6.55765 0.320745
\(419\) 26.7025 1.30450 0.652252 0.758002i \(-0.273824\pi\)
0.652252 + 0.758002i \(0.273824\pi\)
\(420\) 0 0
\(421\) 8.44158 0.411418 0.205709 0.978613i \(-0.434050\pi\)
0.205709 + 0.978613i \(0.434050\pi\)
\(422\) 24.2837 1.18211
\(423\) 0 0
\(424\) −9.66628 −0.469436
\(425\) −22.1671 −1.07526
\(426\) 0 0
\(427\) 0 0
\(428\) 15.3231 0.740672
\(429\) 0 0
\(430\) −0.0751261 −0.00362290
\(431\) −18.5510 −0.893569 −0.446785 0.894642i \(-0.647431\pi\)
−0.446785 + 0.894642i \(0.647431\pi\)
\(432\) 0 0
\(433\) −4.19969 −0.201824 −0.100912 0.994895i \(-0.532176\pi\)
−0.100912 + 0.994895i \(0.532176\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.422735 −0.0202453
\(437\) −9.82490 −0.469989
\(438\) 0 0
\(439\) −3.48175 −0.166175 −0.0830875 0.996542i \(-0.526478\pi\)
−0.0830875 + 0.996542i \(0.526478\pi\)
\(440\) 0.473626 0.0225792
\(441\) 0 0
\(442\) −32.6368 −1.55238
\(443\) −24.6690 −1.17206 −0.586030 0.810289i \(-0.699310\pi\)
−0.586030 + 0.810289i \(0.699310\pi\)
\(444\) 0 0
\(445\) −7.90393 −0.374682
\(446\) 3.82529 0.181133
\(447\) 0 0
\(448\) 0 0
\(449\) −6.39863 −0.301970 −0.150985 0.988536i \(-0.548245\pi\)
−0.150985 + 0.988536i \(0.548245\pi\)
\(450\) 0 0
\(451\) −11.2458 −0.529543
\(452\) 8.40569 0.395370
\(453\) 0 0
\(454\) −12.3106 −0.577764
\(455\) 0 0
\(456\) 0 0
\(457\) 14.5549 0.680849 0.340424 0.940272i \(-0.389429\pi\)
0.340424 + 0.940272i \(0.389429\pi\)
\(458\) 20.2726 0.947277
\(459\) 0 0
\(460\) −0.709603 −0.0330854
\(461\) 16.3115 0.759700 0.379850 0.925048i \(-0.375976\pi\)
0.379850 + 0.925048i \(0.375976\pi\)
\(462\) 0 0
\(463\) 8.81138 0.409500 0.204750 0.978814i \(-0.434362\pi\)
0.204750 + 0.978814i \(0.434362\pi\)
\(464\) 2.11882 0.0983640
\(465\) 0 0
\(466\) −10.8114 −0.500828
\(467\) −8.20468 −0.379667 −0.189834 0.981816i \(-0.560795\pi\)
−0.189834 + 0.981816i \(0.560795\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.128248 0.00591565
\(471\) 0 0
\(472\) 9.82490 0.452228
\(473\) −0.158619 −0.00729332
\(474\) 0 0
\(475\) −31.3172 −1.43693
\(476\) 0 0
\(477\) 0 0
\(478\) 4.07959 0.186596
\(479\) −6.72920 −0.307465 −0.153732 0.988113i \(-0.549129\pi\)
−0.153732 + 0.988113i \(0.549129\pi\)
\(480\) 0 0
\(481\) −30.4219 −1.38712
\(482\) −21.9946 −1.00183
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 6.14083 0.278841
\(486\) 0 0
\(487\) 18.9870 0.860385 0.430193 0.902737i \(-0.358446\pi\)
0.430193 + 0.902737i \(0.358446\pi\)
\(488\) −15.1993 −0.688041
\(489\) 0 0
\(490\) 0 0
\(491\) −19.1027 −0.862093 −0.431047 0.902330i \(-0.641856\pi\)
−0.431047 + 0.902330i \(0.641856\pi\)
\(492\) 0 0
\(493\) 9.83488 0.442941
\(494\) −46.1086 −2.07452
\(495\) 0 0
\(496\) 6.83509 0.306905
\(497\) 0 0
\(498\) 0 0
\(499\) −38.6243 −1.72906 −0.864530 0.502582i \(-0.832384\pi\)
−0.864530 + 0.502582i \(0.832384\pi\)
\(500\) −4.63001 −0.207060
\(501\) 0 0
\(502\) −23.1082 −1.03137
\(503\) −2.53784 −0.113157 −0.0565784 0.998398i \(-0.518019\pi\)
−0.0565784 + 0.998398i \(0.518019\pi\)
\(504\) 0 0
\(505\) −0.141576 −0.00630004
\(506\) −1.49824 −0.0666047
\(507\) 0 0
\(508\) 10.1023 0.448219
\(509\) 28.5987 1.26762 0.633808 0.773490i \(-0.281491\pi\)
0.633808 + 0.773490i \(0.281491\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −7.75420 −0.342023
\(515\) −3.50270 −0.154347
\(516\) 0 0
\(517\) 0.270780 0.0119089
\(518\) 0 0
\(519\) 0 0
\(520\) −3.33019 −0.146039
\(521\) −15.2789 −0.669381 −0.334691 0.942328i \(-0.608632\pi\)
−0.334691 + 0.942328i \(0.608632\pi\)
\(522\) 0 0
\(523\) −34.6031 −1.51309 −0.756545 0.653942i \(-0.773114\pi\)
−0.756545 + 0.653942i \(0.773114\pi\)
\(524\) −20.8843 −0.912335
\(525\) 0 0
\(526\) −13.7851 −0.601059
\(527\) 31.7262 1.38202
\(528\) 0 0
\(529\) −20.7553 −0.902404
\(530\) −4.57820 −0.198864
\(531\) 0 0
\(532\) 0 0
\(533\) 79.0721 3.42499
\(534\) 0 0
\(535\) 7.25743 0.313766
\(536\) 1.21137 0.0523231
\(537\) 0 0
\(538\) −6.13048 −0.264304
\(539\) 0 0
\(540\) 0 0
\(541\) 11.1153 0.477884 0.238942 0.971034i \(-0.423199\pi\)
0.238942 + 0.971034i \(0.423199\pi\)
\(542\) 11.0661 0.475329
\(543\) 0 0
\(544\) −4.64167 −0.199010
\(545\) −0.200218 −0.00857641
\(546\) 0 0
\(547\) 41.5576 1.77688 0.888438 0.458997i \(-0.151791\pi\)
0.888438 + 0.458997i \(0.151791\pi\)
\(548\) −7.35294 −0.314102
\(549\) 0 0
\(550\) −4.77568 −0.203636
\(551\) 13.8945 0.591926
\(552\) 0 0
\(553\) 0 0
\(554\) 16.4986 0.700959
\(555\) 0 0
\(556\) −4.67647 −0.198327
\(557\) −4.19841 −0.177893 −0.0889463 0.996036i \(-0.528350\pi\)
−0.0889463 + 0.996036i \(0.528350\pi\)
\(558\) 0 0
\(559\) 1.11529 0.0471719
\(560\) 0 0
\(561\) 0 0
\(562\) −17.2341 −0.726977
\(563\) −5.10311 −0.215070 −0.107535 0.994201i \(-0.534296\pi\)
−0.107535 + 0.994201i \(0.534296\pi\)
\(564\) 0 0
\(565\) 3.98115 0.167488
\(566\) 18.3333 0.770607
\(567\) 0 0
\(568\) 15.0098 0.629797
\(569\) 36.9635 1.54959 0.774795 0.632212i \(-0.217853\pi\)
0.774795 + 0.632212i \(0.217853\pi\)
\(570\) 0 0
\(571\) 16.7462 0.700808 0.350404 0.936599i \(-0.386044\pi\)
0.350404 + 0.936599i \(0.386044\pi\)
\(572\) −7.03127 −0.293992
\(573\) 0 0
\(574\) 0 0
\(575\) 7.15509 0.298388
\(576\) 0 0
\(577\) −20.9889 −0.873779 −0.436889 0.899515i \(-0.643920\pi\)
−0.436889 + 0.899515i \(0.643920\pi\)
\(578\) −4.54509 −0.189051
\(579\) 0 0
\(580\) 1.00353 0.0416693
\(581\) 0 0
\(582\) 0 0
\(583\) −9.66628 −0.400336
\(584\) 6.13048 0.253681
\(585\) 0 0
\(586\) −26.1560 −1.08049
\(587\) −2.56747 −0.105971 −0.0529854 0.998595i \(-0.516874\pi\)
−0.0529854 + 0.998595i \(0.516874\pi\)
\(588\) 0 0
\(589\) 44.8221 1.84686
\(590\) 4.65332 0.191574
\(591\) 0 0
\(592\) −4.32666 −0.177825
\(593\) −27.7177 −1.13823 −0.569115 0.822258i \(-0.692714\pi\)
−0.569115 + 0.822258i \(0.692714\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −4.89097 −0.200342
\(597\) 0 0
\(598\) 10.5345 0.430788
\(599\) −26.2173 −1.07121 −0.535604 0.844469i \(-0.679916\pi\)
−0.535604 + 0.844469i \(0.679916\pi\)
\(600\) 0 0
\(601\) 20.5621 0.838745 0.419372 0.907814i \(-0.362250\pi\)
0.419372 + 0.907814i \(0.362250\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −16.6533 −0.677614
\(605\) 0.473626 0.0192556
\(606\) 0 0
\(607\) 22.8839 0.928829 0.464415 0.885618i \(-0.346265\pi\)
0.464415 + 0.885618i \(0.346265\pi\)
\(608\) −6.55765 −0.265948
\(609\) 0 0
\(610\) −7.19879 −0.291470
\(611\) −1.90393 −0.0770246
\(612\) 0 0
\(613\) 6.82843 0.275798 0.137899 0.990446i \(-0.455965\pi\)
0.137899 + 0.990446i \(0.455965\pi\)
\(614\) −25.3109 −1.02147
\(615\) 0 0
\(616\) 0 0
\(617\) −22.4120 −0.902272 −0.451136 0.892455i \(-0.648981\pi\)
−0.451136 + 0.892455i \(0.648981\pi\)
\(618\) 0 0
\(619\) −15.6076 −0.627324 −0.313662 0.949535i \(-0.601556\pi\)
−0.313662 + 0.949535i \(0.601556\pi\)
\(620\) 3.23728 0.130012
\(621\) 0 0
\(622\) −24.6202 −0.987180
\(623\) 0 0
\(624\) 0 0
\(625\) 21.6855 0.867420
\(626\) −19.2713 −0.770236
\(627\) 0 0
\(628\) −14.5228 −0.579525
\(629\) −20.0829 −0.800759
\(630\) 0 0
\(631\) −18.4588 −0.734834 −0.367417 0.930056i \(-0.619758\pi\)
−0.367417 + 0.930056i \(0.619758\pi\)
\(632\) −11.4420 −0.455137
\(633\) 0 0
\(634\) 4.34551 0.172582
\(635\) 4.78473 0.189876
\(636\) 0 0
\(637\) 0 0
\(638\) 2.11882 0.0838851
\(639\) 0 0
\(640\) −0.473626 −0.0187217
\(641\) 29.6006 1.16915 0.584576 0.811339i \(-0.301261\pi\)
0.584576 + 0.811339i \(0.301261\pi\)
\(642\) 0 0
\(643\) −7.93039 −0.312744 −0.156372 0.987698i \(-0.549980\pi\)
−0.156372 + 0.987698i \(0.549980\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −30.4384 −1.19758
\(647\) 40.4810 1.59147 0.795736 0.605644i \(-0.207084\pi\)
0.795736 + 0.605644i \(0.207084\pi\)
\(648\) 0 0
\(649\) 9.82490 0.385661
\(650\) 33.5791 1.31708
\(651\) 0 0
\(652\) −8.33609 −0.326466
\(653\) 22.8643 0.894750 0.447375 0.894346i \(-0.352359\pi\)
0.447375 + 0.894346i \(0.352359\pi\)
\(654\) 0 0
\(655\) −9.89135 −0.386487
\(656\) 11.2458 0.439074
\(657\) 0 0
\(658\) 0 0
\(659\) −8.96114 −0.349076 −0.174538 0.984650i \(-0.555843\pi\)
−0.174538 + 0.984650i \(0.555843\pi\)
\(660\) 0 0
\(661\) 8.95891 0.348461 0.174231 0.984705i \(-0.444256\pi\)
0.174231 + 0.984705i \(0.444256\pi\)
\(662\) 5.46786 0.212515
\(663\) 0 0
\(664\) −7.54469 −0.292791
\(665\) 0 0
\(666\) 0 0
\(667\) −3.17450 −0.122917
\(668\) −2.84727 −0.110164
\(669\) 0 0
\(670\) 0.573735 0.0221653
\(671\) −15.1993 −0.586763
\(672\) 0 0
\(673\) −32.2141 −1.24176 −0.620881 0.783905i \(-0.713225\pi\)
−0.620881 + 0.783905i \(0.713225\pi\)
\(674\) −7.43569 −0.286412
\(675\) 0 0
\(676\) 36.4388 1.40149
\(677\) 2.96482 0.113947 0.0569737 0.998376i \(-0.481855\pi\)
0.0569737 + 0.998376i \(0.481855\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.19841 −0.0843053
\(681\) 0 0
\(682\) 6.83509 0.261729
\(683\) −25.4616 −0.974259 −0.487130 0.873330i \(-0.661956\pi\)
−0.487130 + 0.873330i \(0.661956\pi\)
\(684\) 0 0
\(685\) −3.48254 −0.133061
\(686\) 0 0
\(687\) 0 0
\(688\) 0.158619 0.00604730
\(689\) 67.9662 2.58931
\(690\) 0 0
\(691\) 8.44785 0.321371 0.160686 0.987006i \(-0.448629\pi\)
0.160686 + 0.987006i \(0.448629\pi\)
\(692\) −9.17656 −0.348841
\(693\) 0 0
\(694\) 5.88508 0.223395
\(695\) −2.21490 −0.0840158
\(696\) 0 0
\(697\) 52.1992 1.97718
\(698\) −17.5658 −0.664874
\(699\) 0 0
\(700\) 0 0
\(701\) −49.0886 −1.85405 −0.927025 0.374999i \(-0.877643\pi\)
−0.927025 + 0.374999i \(0.877643\pi\)
\(702\) 0 0
\(703\) −28.3727 −1.07010
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −1.45401 −0.0547223
\(707\) 0 0
\(708\) 0 0
\(709\) −0.167483 −0.00628996 −0.00314498 0.999995i \(-0.501001\pi\)
−0.00314498 + 0.999995i \(0.501001\pi\)
\(710\) 7.10903 0.266797
\(711\) 0 0
\(712\) 16.6881 0.625414
\(713\) −10.2406 −0.383512
\(714\) 0 0
\(715\) −3.33019 −0.124542
\(716\) 5.76625 0.215495
\(717\) 0 0
\(718\) 19.6796 0.734436
\(719\) 1.83472 0.0684234 0.0342117 0.999415i \(-0.489108\pi\)
0.0342117 + 0.999415i \(0.489108\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −24.0027 −0.893289
\(723\) 0 0
\(724\) 3.38164 0.125678
\(725\) −10.1188 −0.375804
\(726\) 0 0
\(727\) −27.7394 −1.02880 −0.514399 0.857551i \(-0.671985\pi\)
−0.514399 + 0.857551i \(0.671985\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2.90355 0.107465
\(731\) 0.736258 0.0272315
\(732\) 0 0
\(733\) 41.5283 1.53388 0.766942 0.641716i \(-0.221777\pi\)
0.766942 + 0.641716i \(0.221777\pi\)
\(734\) 18.0692 0.666947
\(735\) 0 0
\(736\) 1.49824 0.0552257
\(737\) 1.21137 0.0446213
\(738\) 0 0
\(739\) 1.19488 0.0439545 0.0219773 0.999758i \(-0.493004\pi\)
0.0219773 + 0.999758i \(0.493004\pi\)
\(740\) −2.04922 −0.0753308
\(741\) 0 0
\(742\) 0 0
\(743\) −4.25650 −0.156156 −0.0780779 0.996947i \(-0.524878\pi\)
−0.0780779 + 0.996947i \(0.524878\pi\)
\(744\) 0 0
\(745\) −2.31649 −0.0848697
\(746\) 21.3459 0.781528
\(747\) 0 0
\(748\) −4.64167 −0.169716
\(749\) 0 0
\(750\) 0 0
\(751\) −13.0925 −0.477754 −0.238877 0.971050i \(-0.576779\pi\)
−0.238877 + 0.971050i \(0.576779\pi\)
\(752\) −0.270780 −0.00987432
\(753\) 0 0
\(754\) −14.8980 −0.542554
\(755\) −7.88744 −0.287053
\(756\) 0 0
\(757\) −4.21411 −0.153164 −0.0765821 0.997063i \(-0.524401\pi\)
−0.0765821 + 0.997063i \(0.524401\pi\)
\(758\) 4.63448 0.168332
\(759\) 0 0
\(760\) −3.10587 −0.112662
\(761\) −45.8587 −1.66238 −0.831189 0.555990i \(-0.812339\pi\)
−0.831189 + 0.555990i \(0.812339\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 16.1586 0.584598
\(765\) 0 0
\(766\) −7.37981 −0.266643
\(767\) −69.0815 −2.49439
\(768\) 0 0
\(769\) 53.7185 1.93714 0.968569 0.248745i \(-0.0800183\pi\)
0.968569 + 0.248745i \(0.0800183\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −24.8651 −0.894913
\(773\) −6.90523 −0.248364 −0.124182 0.992259i \(-0.539631\pi\)
−0.124182 + 0.992259i \(0.539631\pi\)
\(774\) 0 0
\(775\) −32.6422 −1.17254
\(776\) −12.9656 −0.465437
\(777\) 0 0
\(778\) −1.34668 −0.0482807
\(779\) 73.7458 2.64222
\(780\) 0 0
\(781\) 15.0098 0.537093
\(782\) 6.95431 0.248686
\(783\) 0 0
\(784\) 0 0
\(785\) −6.87839 −0.245500
\(786\) 0 0
\(787\) −3.06703 −0.109328 −0.0546639 0.998505i \(-0.517409\pi\)
−0.0546639 + 0.998505i \(0.517409\pi\)
\(788\) −10.4290 −0.371518
\(789\) 0 0
\(790\) −5.41921 −0.192807
\(791\) 0 0
\(792\) 0 0
\(793\) 106.871 3.79508
\(794\) 20.7346 0.735843
\(795\) 0 0
\(796\) −13.2712 −0.470384
\(797\) 41.1028 1.45594 0.727969 0.685610i \(-0.240465\pi\)
0.727969 + 0.685610i \(0.240465\pi\)
\(798\) 0 0
\(799\) −1.25687 −0.0444649
\(800\) 4.77568 0.168846
\(801\) 0 0
\(802\) 7.96077 0.281104
\(803\) 6.13048 0.216340
\(804\) 0 0
\(805\) 0 0
\(806\) −48.0594 −1.69282
\(807\) 0 0
\(808\) 0.298919 0.0105159
\(809\) 36.2141 1.27322 0.636610 0.771186i \(-0.280336\pi\)
0.636610 + 0.771186i \(0.280336\pi\)
\(810\) 0 0
\(811\) 12.2752 0.431042 0.215521 0.976499i \(-0.430855\pi\)
0.215521 + 0.976499i \(0.430855\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −4.32666 −0.151649
\(815\) −3.94819 −0.138299
\(816\) 0 0
\(817\) 1.04017 0.0363909
\(818\) −30.1438 −1.05395
\(819\) 0 0
\(820\) 5.32629 0.186002
\(821\) 43.0697 1.50314 0.751572 0.659651i \(-0.229296\pi\)
0.751572 + 0.659651i \(0.229296\pi\)
\(822\) 0 0
\(823\) 26.9541 0.939560 0.469780 0.882783i \(-0.344333\pi\)
0.469780 + 0.882783i \(0.344333\pi\)
\(824\) 7.39550 0.257634
\(825\) 0 0
\(826\) 0 0
\(827\) 34.9902 1.21673 0.608364 0.793659i \(-0.291826\pi\)
0.608364 + 0.793659i \(0.291826\pi\)
\(828\) 0 0
\(829\) 20.7246 0.719795 0.359898 0.932992i \(-0.382812\pi\)
0.359898 + 0.932992i \(0.382812\pi\)
\(830\) −3.57336 −0.124033
\(831\) 0 0
\(832\) 7.03127 0.243766
\(833\) 0 0
\(834\) 0 0
\(835\) −1.34854 −0.0466682
\(836\) −6.55765 −0.226801
\(837\) 0 0
\(838\) −26.7025 −0.922424
\(839\) 31.3128 1.08104 0.540518 0.841332i \(-0.318228\pi\)
0.540518 + 0.841332i \(0.318228\pi\)
\(840\) 0 0
\(841\) −24.5106 −0.845193
\(842\) −8.44158 −0.290916
\(843\) 0 0
\(844\) −24.2837 −0.835880
\(845\) 17.2584 0.593705
\(846\) 0 0
\(847\) 0 0
\(848\) 9.66628 0.331941
\(849\) 0 0
\(850\) 22.1671 0.760326
\(851\) 6.48236 0.222212
\(852\) 0 0
\(853\) −29.5524 −1.01186 −0.505928 0.862576i \(-0.668850\pi\)
−0.505928 + 0.862576i \(0.668850\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −15.3231 −0.523734
\(857\) 36.7050 1.25382 0.626909 0.779093i \(-0.284320\pi\)
0.626909 + 0.779093i \(0.284320\pi\)
\(858\) 0 0
\(859\) 0.326102 0.0111265 0.00556323 0.999985i \(-0.498229\pi\)
0.00556323 + 0.999985i \(0.498229\pi\)
\(860\) 0.0751261 0.00256178
\(861\) 0 0
\(862\) 18.5510 0.631849
\(863\) −16.8727 −0.574353 −0.287176 0.957878i \(-0.592717\pi\)
−0.287176 + 0.957878i \(0.592717\pi\)
\(864\) 0 0
\(865\) −4.34626 −0.147777
\(866\) 4.19969 0.142711
\(867\) 0 0
\(868\) 0 0
\(869\) −11.4420 −0.388142
\(870\) 0 0
\(871\) −8.51746 −0.288603
\(872\) 0.422735 0.0143156
\(873\) 0 0
\(874\) 9.82490 0.332332
\(875\) 0 0
\(876\) 0 0
\(877\) −9.71054 −0.327902 −0.163951 0.986469i \(-0.552424\pi\)
−0.163951 + 0.986469i \(0.552424\pi\)
\(878\) 3.48175 0.117503
\(879\) 0 0
\(880\) −0.473626 −0.0159659
\(881\) −22.2976 −0.751224 −0.375612 0.926777i \(-0.622567\pi\)
−0.375612 + 0.926777i \(0.622567\pi\)
\(882\) 0 0
\(883\) 6.70589 0.225671 0.112836 0.993614i \(-0.464007\pi\)
0.112836 + 0.993614i \(0.464007\pi\)
\(884\) 32.6368 1.09770
\(885\) 0 0
\(886\) 24.6690 0.828772
\(887\) 58.3368 1.95876 0.979380 0.202029i \(-0.0647535\pi\)
0.979380 + 0.202029i \(0.0647535\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.90393 0.264940
\(891\) 0 0
\(892\) −3.82529 −0.128080
\(893\) −1.77568 −0.0594208
\(894\) 0 0
\(895\) 2.73105 0.0912888
\(896\) 0 0
\(897\) 0 0
\(898\) 6.39863 0.213525
\(899\) 14.4824 0.483014
\(900\) 0 0
\(901\) 44.8677 1.49476
\(902\) 11.2458 0.374443
\(903\) 0 0
\(904\) −8.40569 −0.279569
\(905\) 1.60163 0.0532401
\(906\) 0 0
\(907\) 46.2804 1.53671 0.768357 0.640021i \(-0.221074\pi\)
0.768357 + 0.640021i \(0.221074\pi\)
\(908\) 12.3106 0.408541
\(909\) 0 0
\(910\) 0 0
\(911\) 55.3864 1.83503 0.917517 0.397696i \(-0.130190\pi\)
0.917517 + 0.397696i \(0.130190\pi\)
\(912\) 0 0
\(913\) −7.54469 −0.249693
\(914\) −14.5549 −0.481433
\(915\) 0 0
\(916\) −20.2726 −0.669826
\(917\) 0 0
\(918\) 0 0
\(919\) 14.1851 0.467923 0.233961 0.972246i \(-0.424831\pi\)
0.233961 + 0.972246i \(0.424831\pi\)
\(920\) 0.709603 0.0233949
\(921\) 0 0
\(922\) −16.3115 −0.537189
\(923\) −105.538 −3.47383
\(924\) 0 0
\(925\) 20.6627 0.679387
\(926\) −8.81138 −0.289560
\(927\) 0 0
\(928\) −2.11882 −0.0695538
\(929\) 31.0150 1.01757 0.508784 0.860894i \(-0.330095\pi\)
0.508784 + 0.860894i \(0.330095\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 10.8114 0.354139
\(933\) 0 0
\(934\) 8.20468 0.268465
\(935\) −2.19841 −0.0718958
\(936\) 0 0
\(937\) 33.5238 1.09517 0.547587 0.836749i \(-0.315546\pi\)
0.547587 + 0.836749i \(0.315546\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.128248 −0.00418300
\(941\) −32.2097 −1.05001 −0.525003 0.851101i \(-0.675936\pi\)
−0.525003 + 0.851101i \(0.675936\pi\)
\(942\) 0 0
\(943\) −16.8488 −0.548673
\(944\) −9.82490 −0.319773
\(945\) 0 0
\(946\) 0.158619 0.00515715
\(947\) 45.4459 1.47679 0.738396 0.674367i \(-0.235584\pi\)
0.738396 + 0.674367i \(0.235584\pi\)
\(948\) 0 0
\(949\) −43.1051 −1.39925
\(950\) 31.3172 1.01606
\(951\) 0 0
\(952\) 0 0
\(953\) 38.5651 1.24924 0.624622 0.780927i \(-0.285253\pi\)
0.624622 + 0.780927i \(0.285253\pi\)
\(954\) 0 0
\(955\) 7.65314 0.247650
\(956\) −4.07959 −0.131943
\(957\) 0 0
\(958\) 6.72920 0.217411
\(959\) 0 0
\(960\) 0 0
\(961\) 15.7185 0.507047
\(962\) 30.4219 0.980843
\(963\) 0 0
\(964\) 21.9946 0.708399
\(965\) −11.7767 −0.379107
\(966\) 0 0
\(967\) 30.4086 0.977875 0.488938 0.872319i \(-0.337385\pi\)
0.488938 + 0.872319i \(0.337385\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −6.14083 −0.197170
\(971\) 18.1366 0.582032 0.291016 0.956718i \(-0.406007\pi\)
0.291016 + 0.956718i \(0.406007\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −18.9870 −0.608384
\(975\) 0 0
\(976\) 15.1993 0.486518
\(977\) −2.68370 −0.0858590 −0.0429295 0.999078i \(-0.513669\pi\)
−0.0429295 + 0.999078i \(0.513669\pi\)
\(978\) 0 0
\(979\) 16.6881 0.533355
\(980\) 0 0
\(981\) 0 0
\(982\) 19.1027 0.609592
\(983\) −39.2806 −1.25286 −0.626428 0.779479i \(-0.715484\pi\)
−0.626428 + 0.779479i \(0.715484\pi\)
\(984\) 0 0
\(985\) −4.93944 −0.157384
\(986\) −9.83488 −0.313206
\(987\) 0 0
\(988\) 46.1086 1.46691
\(989\) −0.237649 −0.00755679
\(990\) 0 0
\(991\) −48.8051 −1.55034 −0.775172 0.631750i \(-0.782337\pi\)
−0.775172 + 0.631750i \(0.782337\pi\)
\(992\) −6.83509 −0.217014
\(993\) 0 0
\(994\) 0 0
\(995\) −6.28556 −0.199266
\(996\) 0 0
\(997\) −0.552259 −0.0174902 −0.00874512 0.999962i \(-0.502784\pi\)
−0.00874512 + 0.999962i \(0.502784\pi\)
\(998\) 38.6243 1.22263
\(999\) 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.dz.1.3 4
3.2 odd 2 3234.2.a.bm.1.2 yes 4
7.6 odd 2 9702.2.a.ea.1.2 4
21.20 even 2 3234.2.a.bl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bl.1.3 4 21.20 even 2
3234.2.a.bm.1.2 yes 4 3.2 odd 2
9702.2.a.dz.1.3 4 1.1 even 1 trivial
9702.2.a.ea.1.2 4 7.6 odd 2