Properties

Label 1856.2.a.f.1.1
Level $1856$
Weight $2$
Character 1856.1
Self dual yes
Analytic conductor $14.820$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.8202346151\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} +2.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{15} +8.00000 q^{17} -2.00000 q^{21} -4.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +1.00000 q^{29} +3.00000 q^{31} +3.00000 q^{33} -2.00000 q^{35} -8.00000 q^{37} -1.00000 q^{39} +2.00000 q^{41} -11.0000 q^{43} +2.00000 q^{45} -13.0000 q^{47} -3.00000 q^{49} -8.00000 q^{51} +11.0000 q^{53} +3.00000 q^{55} +8.00000 q^{61} -4.00000 q^{63} -1.00000 q^{65} -12.0000 q^{67} +4.00000 q^{69} -2.00000 q^{71} +4.00000 q^{73} +4.00000 q^{75} -6.00000 q^{77} -15.0000 q^{79} +1.00000 q^{81} +4.00000 q^{83} -8.00000 q^{85} -1.00000 q^{87} -10.0000 q^{89} +2.00000 q^{91} -3.00000 q^{93} -2.00000 q^{97} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −13.0000 −1.89624 −0.948122 0.317905i \(-0.897021\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 11.0000 1.51097 0.755483 0.655168i \(-0.227402\pi\)
0.755483 + 0.655168i \(0.227402\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −15.0000 −1.68763 −0.843816 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 13.0000 1.09480
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −16.0000 −1.29352
\(154\) 0 0
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) −11.0000 −0.872357
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) 9.00000 0.704934 0.352467 0.935824i \(-0.385343\pi\)
0.352467 + 0.935824i \(0.385343\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −8.00000 −0.604743
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 0 0
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 8.00000 0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 0 0
\(213\) 2.00000 0.137038
\(214\) 0 0
\(215\) 11.0000 0.750194
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 0 0
\(225\) 8.00000 0.533333
\(226\) 0 0
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 0 0
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 0 0
\(235\) 13.0000 0.848026
\(236\) 0 0
\(237\) 15.0000 0.974355
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 0 0
\(255\) 8.00000 0.500979
\(256\) 0 0
\(257\) 13.0000 0.810918 0.405459 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −15.0000 −0.870388
\(298\) 0 0
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −22.0000 −1.26806
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 5.00000 0.276501
\(328\) 0 0
\(329\) −26.0000 −1.43343
\(330\) 0 0
\(331\) −23.0000 −1.26419 −0.632097 0.774889i \(-0.717806\pi\)
−0.632097 + 0.774889i \(0.717806\pi\)
\(332\) 0 0
\(333\) 16.0000 0.876795
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −9.00000 −0.487377
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 0 0
\(357\) −16.0000 −0.846810
\(358\) 0 0
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 32.0000 1.67039 0.835193 0.549957i \(-0.185356\pi\)
0.835193 + 0.549957i \(0.185356\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 22.0000 1.14218
\(372\) 0 0
\(373\) 21.0000 1.08734 0.543669 0.839299i \(-0.317035\pi\)
0.543669 + 0.839299i \(0.317035\pi\)
\(374\) 0 0
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 0 0
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) 6.00000 0.305788
\(386\) 0 0
\(387\) 22.0000 1.11832
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −32.0000 −1.61831
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 15.0000 0.754732
\(396\) 0 0
\(397\) 17.0000 0.853206 0.426603 0.904439i \(-0.359710\pi\)
0.426603 + 0.904439i \(0.359710\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) 3.00000 0.149441
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 0 0
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 0 0
\(423\) 26.0000 1.26416
\(424\) 0 0
\(425\) −32.0000 −1.55223
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 0 0
\(429\) 3.00000 0.144841
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) 15.0000 0.709476
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 2.00000 0.0939682
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 0 0
\(459\) 40.0000 1.86704
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 3.00000 0.139122
\(466\) 0 0
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 33.0000 1.51734
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −22.0000 −1.00731
\(478\) 0 0
\(479\) 5.00000 0.228456 0.114228 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 8.00000 0.364013
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 0 0
\(489\) −9.00000 −0.406994
\(490\) 0 0
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 0 0
\(503\) −19.0000 −0.847168 −0.423584 0.905857i \(-0.639228\pi\)
−0.423584 + 0.905857i \(0.639228\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 12.0000 0.532939
\(508\) 0 0
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.0000 0.616914
\(516\) 0 0
\(517\) 39.0000 1.71522
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 0 0
\(525\) 8.00000 0.349149
\(526\) 0 0
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) 0 0
\(549\) −16.0000 −0.682863
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −30.0000 −1.27573
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 0 0
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 0 0
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) −8.00000 −0.334205
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) 8.00000 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 0 0
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −33.0000 −1.36672
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 0 0
\(593\) 39.0000 1.60154 0.800769 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 5.00000 0.204294 0.102147 0.994769i \(-0.467429\pi\)
0.102147 + 0.994769i \(0.467429\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 24.0000 0.977356
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −3.00000 −0.121766 −0.0608831 0.998145i \(-0.519392\pi\)
−0.0608831 + 0.998145i \(0.519392\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) −13.0000 −0.525924
\(612\) 0 0
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 0 0
\(619\) −35.0000 −1.40677 −0.703384 0.710810i \(-0.748329\pi\)
−0.703384 + 0.710810i \(0.748329\pi\)
\(620\) 0 0
\(621\) −20.0000 −0.802572
\(622\) 0 0
\(623\) −20.0000 −0.801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −64.0000 −2.55185
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) 3.00000 0.119239
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) −3.00000 −0.118864
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) 0 0
\(645\) −11.0000 −0.433125
\(646\) 0 0
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4.00000 −0.154881
\(668\) 0 0
\(669\) −26.0000 −1.00522
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) 0 0
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) 0 0
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) −18.0000 −0.689761
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 11.0000 0.419067
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 0 0
\(693\) 12.0000 0.455842
\(694\) 0 0
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) 1.00000 0.0378235
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −13.0000 −0.489608
\(706\) 0 0
\(707\) 16.0000 0.601742
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) 0 0
\(711\) 30.0000 1.12509
\(712\) 0 0
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 0 0
\(721\) −28.0000 −1.04277
\(722\) 0 0
\(723\) −17.0000 −0.632237
\(724\) 0 0
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −88.0000 −3.25480
\(732\) 0 0
\(733\) −24.0000 −0.886460 −0.443230 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 0 0
\(735\) −3.00000 −0.110657
\(736\) 0 0
\(737\) 36.0000 1.32608
\(738\) 0 0
\(739\) −5.00000 −0.183928 −0.0919640 0.995762i \(-0.529314\pi\)
−0.0919640 + 0.995762i \(0.529314\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) 0 0
\(747\) −8.00000 −0.292705
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 0 0
\(753\) −27.0000 −0.983935
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) 0 0
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) 0 0
\(765\) 16.0000 0.578481
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −20.0000 −0.721218 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(770\) 0 0
\(771\) −13.0000 −0.468184
\(772\) 0 0
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) −12.0000 −0.431053
\(776\) 0 0
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 5.00000 0.178685
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 0 0
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 11.0000 0.390130
\(796\) 0 0
\(797\) 32.0000 1.13350 0.566749 0.823890i \(-0.308201\pi\)
0.566749 + 0.823890i \(0.308201\pi\)
\(798\) 0 0
\(799\) −104.000 −3.67926
\(800\) 0 0
\(801\) 20.0000 0.706665
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −18.0000 −0.632065 −0.316033 0.948748i \(-0.602351\pi\)
−0.316033 + 0.948748i \(0.602351\pi\)
\(812\) 0 0
\(813\) −13.0000 −0.455930
\(814\) 0 0
\(815\) −9.00000 −0.315256
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 13.0000 0.452054 0.226027 0.974121i \(-0.427426\pi\)
0.226027 + 0.974121i \(0.427426\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 0 0
\(833\) −24.0000 −0.831551
\(834\) 0 0
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) 0 0
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −27.0000 −0.929929
\(844\) 0 0
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) −4.00000 −0.137442
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −47.0000 −1.59620
\(868\) 0 0
\(869\) 45.0000 1.52652
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 18.0000 0.608511
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) 0 0
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.0000 −1.10803 −0.554016 0.832506i \(-0.686905\pi\)
−0.554016 + 0.832506i \(0.686905\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) 88.0000 2.93171
\(902\) 0 0
\(903\) 22.0000 0.732114
\(904\) 0 0
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) 13.0000 0.430709 0.215355 0.976536i \(-0.430909\pi\)
0.215355 + 0.976536i \(0.430909\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) 8.00000 0.264472
\(916\) 0 0
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) −30.0000 −0.989609 −0.494804 0.869004i \(-0.664760\pi\)
−0.494804 + 0.869004i \(0.664760\pi\)
\(920\) 0 0
\(921\) 7.00000 0.230658
\(922\) 0 0
\(923\) −2.00000 −0.0658308
\(924\) 0 0
\(925\) 32.0000 1.05215
\(926\) 0 0
\(927\) 28.0000 0.919641
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −8.00000 −0.261908
\(934\) 0 0
\(935\) 24.0000 0.784884
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) −9.00000 −0.293704
\(940\) 0 0
\(941\) −37.0000 −1.20617 −0.603083 0.797679i \(-0.706061\pi\)
−0.603083 + 0.797679i \(0.706061\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) −10.0000 −0.325300
\(946\) 0 0
\(947\) 33.0000 1.07236 0.536178 0.844105i \(-0.319868\pi\)
0.536178 + 0.844105i \(0.319868\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) −1.00000 −0.0323932 −0.0161966 0.999869i \(-0.505156\pi\)
−0.0161966 + 0.999869i \(0.505156\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) 0 0
\(957\) 3.00000 0.0969762
\(958\) 0 0
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 0 0
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −40.0000 −1.28234
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) 13.0000 0.415907 0.207953 0.978139i \(-0.433320\pi\)
0.207953 + 0.978139i \(0.433320\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) 0 0
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 26.0000 0.827589
\(988\) 0 0
\(989\) 44.0000 1.39912
\(990\) 0 0
\(991\) −22.0000 −0.698853 −0.349427 0.936964i \(-0.613624\pi\)
−0.349427 + 0.936964i \(0.613624\pi\)
\(992\) 0 0
\(993\) 23.0000 0.729883
\(994\) 0 0
\(995\) −10.0000 −0.317021
\(996\) 0 0
\(997\) −8.00000 −0.253363 −0.126681 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) 0 0
\(999\) −40.0000 −1.26554
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 1856.2.a.f.1.1 1
4.3 odd 2 1856.2.a.k.1.1 1
8.3 odd 2 58.2.a.b.1.1 1
8.5 even 2 464.2.a.e.1.1 1
24.5 odd 2 4176.2.a.n.1.1 1
24.11 even 2 522.2.a.b.1.1 1
40.3 even 4 1450.2.b.b.349.1 2
40.19 odd 2 1450.2.a.c.1.1 1
40.27 even 4 1450.2.b.b.349.2 2
56.27 even 2 2842.2.a.e.1.1 1
88.43 even 2 7018.2.a.a.1.1 1
104.51 odd 2 9802.2.a.a.1.1 1
232.75 even 4 1682.2.b.a.1681.1 2
232.99 even 4 1682.2.b.a.1681.2 2
232.115 odd 2 1682.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 8.3 odd 2
464.2.a.e.1.1 1 8.5 even 2
522.2.a.b.1.1 1 24.11 even 2
1450.2.a.c.1.1 1 40.19 odd 2
1450.2.b.b.349.1 2 40.3 even 4
1450.2.b.b.349.2 2 40.27 even 4
1682.2.a.d.1.1 1 232.115 odd 2
1682.2.b.a.1681.1 2 232.75 even 4
1682.2.b.a.1681.2 2 232.99 even 4
1856.2.a.f.1.1 1 1.1 even 1 trivial
1856.2.a.k.1.1 1 4.3 odd 2
2842.2.a.e.1.1 1 56.27 even 2
4176.2.a.n.1.1 1 24.5 odd 2
7018.2.a.a.1.1 1 88.43 even 2
9802.2.a.a.1.1 1 104.51 odd 2