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Properties

Label 4368.2.a.h.1.1
Level $4368$
Weight $2$
Character 4368.1
Self dual yes
Analytic conductor $34.879$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.8786556029\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4368.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} +5.00000 q^{17} -1.00000 q^{19} -1.00000 q^{21} -3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} -4.00000 q^{31} +3.00000 q^{33} +1.00000 q^{35} -5.00000 q^{37} +1.00000 q^{39} -8.00000 q^{41} +1.00000 q^{43} +1.00000 q^{45} -8.00000 q^{47} +1.00000 q^{49} -5.00000 q^{51} +6.00000 q^{53} -3.00000 q^{55} +1.00000 q^{57} +13.0000 q^{61} +1.00000 q^{63} -1.00000 q^{65} +10.0000 q^{67} +3.00000 q^{69} -8.00000 q^{71} -15.0000 q^{73} +4.00000 q^{75} -3.00000 q^{77} -6.00000 q^{79} +1.00000 q^{81} +2.00000 q^{83} +5.00000 q^{85} -5.00000 q^{87} -2.00000 q^{89} -1.00000 q^{91} +4.00000 q^{93} -1.00000 q^{95} -2.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 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
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 − 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 3.00000 0.522233
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 − 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 − 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −15.0000 −1.75562 −0.877809 − 0.479012i \(-0.840995\pi\)
−0.877809 + 0.479012i \(0.840995\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −3.00000 −0.341882
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 − 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) −5.00000 −0.536056
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −16.0000 −1.59206 −0.796030 − 0.605257i \(-0.793070\pi\)
−0.796030 + 0.605257i \(0.793070\pi\)
\(102\) 0 0
\(103\) −1.00000 −0.0985329 −0.0492665 − 0.998786i \(-0.515688\pi\)
−0.0492665 + 0.998786i \(0.515688\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 − 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −3.00000 −0.279751
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 18.0000 1.59724 0.798621 − 0.601834i \(-0.205563\pi\)
0.798621 + 0.601834i \(0.205563\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) −17.0000 −1.48530 −0.742648 − 0.669681i \(-0.766431\pi\)
−0.742648 + 0.669681i \(0.766431\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −15.0000 −1.28154 −0.640768 − 0.767734i \(-0.721384\pi\)
−0.640768 + 0.767734i \(0.721384\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 − 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −16.0000 −1.31077 −0.655386 − 0.755295i \(-0.727494\pi\)
−0.655386 + 0.755295i \(0.727494\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 − 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) 5.00000 0.404226
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 9.00000 0.718278 0.359139 − 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −10.0000 −0.783260 −0.391630 − 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) 0 0
\(165\) 3.00000 0.233550
\(166\) 0 0
\(167\) 15.0000 1.16073 0.580367 − 0.814355i \(-0.302909\pi\)
0.580367 + 0.814355i \(0.302909\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) 6.00000 0.456172 0.228086 − 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) 0 0
\(185\) −5.00000 −0.367607
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −11.0000 −0.795932 −0.397966 − 0.917400i \(-0.630284\pi\)
−0.397966 + 0.917400i \(0.630284\pi\)
\(192\) 0 0
\(193\) −20.0000 −1.43963 −0.719816 − 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 − 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 8.00000 0.548151
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 15.0000 1.01361
\(220\) 0 0
\(221\) −5.00000 −0.336336
\(222\) 0 0
\(223\) −26.0000 −1.74109 −0.870544 − 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) −2.00000 −0.132745 −0.0663723 − 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 − 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) 0 0
\(233\) −4.00000 −0.262049 −0.131024 − 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 6.00000 0.389742
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 − 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 − 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) 0 0
\(249\) −2.00000 −0.126745
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 − 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 9.00000 0.565825
\(254\) 0 0
\(255\) −5.00000 −0.313112
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 5.00000 0.309492
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 − 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 − 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 − 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 − 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 0 0
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) 0 0
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 16.0000 0.919176
\(304\) 0 0
\(305\) 13.0000 0.744378
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) −26.0000 −1.47432 −0.737162 − 0.675716i \(-0.763835\pi\)
−0.737162 + 0.675716i \(0.763835\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 − 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −24.0000 −1.34797 −0.673987 − 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 7.00000 0.387101
\(328\) 0 0
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) −34.0000 −1.86881 −0.934405 − 0.356214i \(-0.884068\pi\)
−0.934405 + 0.356214i \(0.884068\pi\)
\(332\) 0 0
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 − 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) −32.0000 −1.71785 −0.858925 − 0.512101i \(-0.828867\pi\)
−0.858925 + 0.512101i \(0.828867\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 − 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 − 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) −5.00000 −0.264628
\(358\) 0 0
\(359\) −24.0000 −1.26667 −0.633336 − 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −15.0000 −0.785136
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 − 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 − 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 − 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) −1.00000 −0.0510976 −0.0255488 − 0.999674i \(-0.508133\pi\)
−0.0255488 + 0.999674i \(0.508133\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) −18.0000 −0.912636 −0.456318 − 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) −15.0000 −0.758583
\(392\) 0 0
\(393\) 17.0000 0.857537
\(394\) 0 0
\(395\) −6.00000 −0.301893
\(396\) 0 0
\(397\) 30.0000 1.50566 0.752828 − 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 − 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 15.0000 0.743522
\(408\) 0 0
\(409\) 3.00000 0.148340 0.0741702 − 0.997246i \(-0.476369\pi\)
0.0741702 + 0.997246i \(0.476369\pi\)
\(410\) 0 0
\(411\) 15.0000 0.739895
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) 5.00000 0.244266 0.122133 − 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 − 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) −20.0000 −0.970143
\(426\) 0 0
\(427\) 13.0000 0.629114
\(428\) 0 0
\(429\) −3.00000 −0.144841
\(430\) 0 0
\(431\) −10.0000 −0.481683 −0.240842 − 0.970564i \(-0.577423\pi\)
−0.240842 + 0.970564i \(0.577423\pi\)
\(432\) 0 0
\(433\) 20.0000 0.961139 0.480569 − 0.876957i \(-0.340430\pi\)
0.480569 + 0.876957i \(0.340430\pi\)
\(434\) 0 0
\(435\) −5.00000 −0.239732
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 11.0000 0.525001 0.262501 − 0.964932i \(-0.415453\pi\)
0.262501 + 0.964932i \(0.415453\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 − 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) 0 0
\(445\) −2.00000 −0.0948091
\(446\) 0 0
\(447\) 16.0000 0.756774
\(448\) 0 0
\(449\) 13.0000 0.613508 0.306754 − 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 0 0
\(453\) −5.00000 −0.234920
\(454\) 0 0
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 − 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) −13.0000 −0.605470 −0.302735 − 0.953075i \(-0.597900\pi\)
−0.302735 + 0.953075i \(0.597900\pi\)
\(462\) 0 0
\(463\) 13.0000 0.604161 0.302081 − 0.953282i \(-0.402319\pi\)
0.302081 + 0.953282i \(0.402319\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) −21.0000 −0.971764 −0.485882 − 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 0 0
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) −9.00000 −0.414698
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −27.0000 −1.23366 −0.616831 − 0.787096i \(-0.711584\pi\)
−0.616831 + 0.787096i \(0.711584\pi\)
\(480\) 0 0
\(481\) 5.00000 0.227980
\(482\) 0 0
\(483\) 3.00000 0.136505
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 − 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) 38.0000 1.71492 0.857458 − 0.514554i \(-0.172042\pi\)
0.857458 + 0.514554i \(0.172042\pi\)
\(492\) 0 0
\(493\) 25.0000 1.12594
\(494\) 0 0
\(495\) −3.00000 −0.134840
\(496\) 0 0
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 − 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 0 0
\(501\) −15.0000 −0.670151
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −16.0000 −0.711991
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 11.0000 0.487566 0.243783 − 0.969830i \(-0.421611\pi\)
0.243783 + 0.969830i \(0.421611\pi\)
\(510\) 0 0
\(511\) −15.0000 −0.663561
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −1.00000 −0.0440653
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 − 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 − 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 0 0
\(525\) 4.00000 0.174574
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 41.0000 1.76273 0.881364 − 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) −7.00000 −0.299847
\(546\) 0 0
\(547\) 44.0000 1.88130 0.940652 − 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) 0 0
\(549\) 13.0000 0.554826
\(550\) 0 0
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 5.00000 0.212238
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 − 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 15.0000 0.633300
\(562\) 0 0
\(563\) −9.00000 −0.379305 −0.189652 − 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 − 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 11.0000 0.459532
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 − 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 0 0
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) 26.0000 1.07313 0.536567 − 0.843857i \(-0.319721\pi\)
0.536567 + 0.843857i \(0.319721\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 38.0000 1.56047 0.780236 − 0.625485i \(-0.215099\pi\)
0.780236 + 0.625485i \(0.215099\pi\)
\(594\) 0 0
\(595\) 5.00000 0.204980
\(596\) 0 0
\(597\) −5.00000 −0.204636
\(598\) 0 0
\(599\) −37.0000 −1.51178 −0.755890 − 0.654699i \(-0.772795\pi\)
−0.755890 + 0.654699i \(0.772795\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 − 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 27.0000 1.09590 0.547948 − 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) 41.0000 1.65597 0.827987 − 0.560747i \(-0.189486\pi\)
0.827987 + 0.560747i \(0.189486\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 43.0000 1.73111 0.865557 − 0.500810i \(-0.166964\pi\)
0.865557 + 0.500810i \(0.166964\pi\)
\(618\) 0 0
\(619\) 31.0000 1.24600 0.622998 − 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 0 0
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) −25.0000 −0.996815
\(630\) 0 0
\(631\) 9.00000 0.358284 0.179142 − 0.983823i \(-0.442668\pi\)
0.179142 + 0.983823i \(0.442668\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 − 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 − 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 − 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) 3.00000 0.117399 0.0586995 − 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 0 0
\(655\) −17.0000 −0.664245
\(656\) 0 0
\(657\) −15.0000 −0.585206
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 5.00000 0.194184
\(664\) 0 0
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) −15.0000 −0.580802
\(668\) 0 0
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) −39.0000 −1.50558
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 − 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 − 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 0 0
\(683\) −33.0000 −1.26271 −0.631355 − 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) 0 0
\(685\) −15.0000 −0.573121
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 − 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 0 0
\(693\) −3.00000 −0.113961
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) −40.0000 −1.51511
\(698\) 0 0
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) −14.0000 −0.528773 −0.264386 − 0.964417i \(-0.585169\pi\)
−0.264386 + 0.964417i \(0.585169\pi\)
\(702\) 0 0
\(703\) 5.00000 0.188579
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 − 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 0 0
\(713\) 12.0000 0.449404
\(714\) 0 0
\(715\) 3.00000 0.112194
\(716\) 0 0
\(717\) −12.0000 −0.448148
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −1.00000 −0.0372419
\(722\) 0 0
\(723\) −22.0000 −0.818189
\(724\) 0 0
\(725\) −20.0000 −0.742781
\(726\) 0 0
\(727\) 27.0000 1.00137 0.500687 − 0.865628i \(-0.333081\pi\)
0.500687 + 0.865628i \(0.333081\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 5.00000 0.184932
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 − 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) 26.0000 0.956425 0.478213 − 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(740\) 0 0
\(741\) −1.00000 −0.0367359
\(742\) 0 0
\(743\) 42.0000 1.54083 0.770415 − 0.637542i \(-0.220049\pi\)
0.770415 + 0.637542i \(0.220049\pi\)
\(744\) 0 0
\(745\) −16.0000 −0.586195
\(746\) 0 0
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 21.0000 0.765283
\(754\) 0 0
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 − 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) −9.00000 −0.326679
\(760\) 0 0
\(761\) −24.0000 −0.869999 −0.435000 − 0.900431i \(-0.643252\pi\)
−0.435000 + 0.900431i \(0.643252\pi\)
\(762\) 0 0
\(763\) −7.00000 −0.253417
\(764\) 0 0
\(765\) 5.00000 0.180775
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 43.0000 1.55062 0.775310 − 0.631581i \(-0.217594\pi\)
0.775310 + 0.631581i \(0.217594\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 11.0000 0.395643 0.197821 − 0.980238i \(-0.436613\pi\)
0.197821 + 0.980238i \(0.436613\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) 5.00000 0.179374
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 9.00000 0.321224
\(786\) 0 0
\(787\) 47.0000 1.67537 0.837685 − 0.546154i \(-0.183909\pi\)
0.837685 + 0.546154i \(0.183909\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −13.0000 −0.461644
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) −28.0000 −0.991811 −0.495905 − 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 0 0
\(803\) 45.0000 1.58802
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 19.0000 0.667180 0.333590 − 0.942718i \(-0.391740\pi\)
0.333590 + 0.942718i \(0.391740\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 24.0000 0.837606 0.418803 − 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −43.0000 −1.49526 −0.747628 − 0.664117i \(-0.768807\pi\)
−0.747628 + 0.664117i \(0.768807\pi\)
\(828\) 0 0
\(829\) −19.0000 −0.659897 −0.329949 − 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) 0 0
\(831\) 18.0000 0.624413
\(832\) 0 0
\(833\) 5.00000 0.173240
\(834\) 0 0
\(835\) 15.0000 0.519096
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) 0 0
\(839\) 12.0000 0.414286 0.207143 − 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 0 0
\(843\) −10.0000 −0.344418
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 15.0000 0.514193
\(852\) 0 0
\(853\) −28.0000 −0.958702 −0.479351 − 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 − 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) 2.00000 0.0680808 0.0340404 − 0.999420i \(-0.489163\pi\)
0.0340404 + 0.999420i \(0.489163\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 18.0000 0.610608
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 − 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 0 0
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) 37.0000 1.24656 0.623281 − 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) 0 0
\(883\) 41.0000 1.37976 0.689880 − 0.723924i \(-0.257663\pi\)
0.689880 + 0.723924i \(0.257663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 20.0000 0.671534 0.335767 − 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) 0 0
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 0 0
\(893\) 8.00000 0.267710
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) −3.00000 −0.100167
\(898\) 0 0
\(899\) −20.0000 −0.667037
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) 0 0
\(903\) −1.00000 −0.0332779
\(904\) 0 0
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) −16.0000 −0.530687
\(910\) 0 0
\(911\) −29.0000 −0.960813 −0.480406 − 0.877046i \(-0.659511\pi\)
−0.480406 + 0.877046i \(0.659511\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 0 0
\(915\) −13.0000 −0.429767
\(916\) 0 0
\(917\) −17.0000 −0.561389
\(918\) 0 0
\(919\) 50.0000 1.64935 0.824674 − 0.565608i \(-0.191359\pi\)
0.824674 + 0.565608i \(0.191359\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) 0 0
\(927\) −1.00000 −0.0328443
\(928\) 0 0
\(929\) −4.00000 −0.131236 −0.0656179 − 0.997845i \(-0.520902\pi\)
−0.0656179 + 0.997845i \(0.520902\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) 26.0000 0.851202
\(934\) 0 0
\(935\) −15.0000 −0.490552
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 − 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) −28.0000 −0.913745
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 − 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 24.0000 0.781548
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −47.0000 −1.52729 −0.763647 − 0.645634i \(-0.776593\pi\)
−0.763647 + 0.645634i \(0.776593\pi\)
\(948\) 0 0
\(949\) 15.0000 0.486921
\(950\) 0 0
\(951\) 24.0000 0.778253
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 − 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) −11.0000 −0.355952
\(956\) 0 0
\(957\) 15.0000 0.484881
\(958\) 0 0
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 0 0
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) 31.0000 0.996893 0.498446 − 0.866921i \(-0.333904\pi\)
0.498446 + 0.866921i \(0.333904\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) 32.0000 1.02693 0.513464 − 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −25.0000 −0.799821 −0.399910 − 0.916554i \(-0.630959\pi\)
−0.399910 + 0.916554i \(0.630959\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 0 0
\(983\) 5.00000 0.159475 0.0797376 − 0.996816i \(-0.474592\pi\)
0.0797376 + 0.996816i \(0.474592\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) 30.0000 0.952981 0.476491 − 0.879180i \(-0.341909\pi\)
0.476491 + 0.879180i \(0.341909\pi\)
\(992\) 0 0
\(993\) 34.0000 1.07896
\(994\) 0 0
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 − 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 4368.2.a.h.1.1 1
4.3 odd 2 546.2.a.c.1.1 ✓ 1
12.11 even 2 1638.2.a.o.1.1 1
28.27 even 2 3822.2.a.e.1.1 1
52.51 odd 2 7098.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.c.1.1 ✓ 1 4.3 odd 2
1638.2.a.o.1.1 1 12.11 even 2
3822.2.a.e.1.1 1 28.27 even 2
4368.2.a.h.1.1 1 1.1 even 1 trivial
7098.2.a.z.1.1 1 52.51 odd 2