Properties

Label 1638.2.a.o.1.1
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.0794958511\)
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 \(\chi\) \(=\) 1638.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -3.00000 q^{11} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{19} -1.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} -4.00000 q^{25} -1.00000 q^{26} -1.00000 q^{28} -5.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} -5.00000 q^{34} +1.00000 q^{35} -5.00000 q^{37} +1.00000 q^{38} -1.00000 q^{40} +8.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -3.00000 q^{46} -8.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -1.00000 q^{52} -6.00000 q^{53} +3.00000 q^{55} -1.00000 q^{56} -5.00000 q^{58} +13.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} -10.0000 q^{67} -5.00000 q^{68} +1.00000 q^{70} -8.00000 q^{71} -15.0000 q^{73} -5.00000 q^{74} +1.00000 q^{76} +3.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} +8.00000 q^{82} +2.00000 q^{83} +5.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +2.00000 q^{89} +1.00000 q^{91} -3.00000 q^{92} -8.00000 q^{94} -1.00000 q^{95} -2.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) −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
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −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
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −5.00000 −0.928477 −0.464238 0.885710i \(-0.653672\pi\)
−0.464238 + 0.885710i \(0.653672\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −5.00000 −0.857493
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) −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
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −1.00000 −0.138675
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −5.00000 −0.656532
\(59\) 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 \(0.187058\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −5.00000 −0.606339
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) −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 \(0.840995\pi\)
\(74\) −5.00000 −0.581238
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 8.00000 0.883452
\(83\) 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
\(86\) −1.00000 −0.107833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −4.00000 −0.400000
\(101\) 16.0000 1.59206 0.796030 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 0 0
\(103\) 1.00000 0.0985329 0.0492665 0.998786i \(-0.484312\pi\)
0.0492665 + 0.998786i \(0.484312\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 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 \(0.608817\pi\)
\(110\) 3.00000 0.286039
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) −5.00000 −0.464238
\(117\) 0 0
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 13.0000 1.17696
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −18.0000 −1.59724 −0.798621 0.601834i \(-0.794437\pi\)
−0.798621 + 0.601834i \(0.794437\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) −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
\(134\) −10.0000 −0.863868
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) 15.0000 1.28154 0.640768 0.767734i \(-0.278616\pi\)
0.640768 + 0.767734i \(0.278616\pi\)
\(138\) 0 0
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 5.00000 0.415227
\(146\) −15.0000 −1.24141
\(147\) 0 0
\(148\) −5.00000 −0.410997
\(149\) 16.0000 1.31077 0.655386 0.755295i \(-0.272506\pi\)
0.655386 + 0.755295i \(0.272506\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) 9.00000 0.718278 0.359139 0.933284i \(-0.383070\pi\)
0.359139 + 0.933284i \(0.383070\pi\)
\(158\) 6.00000 0.477334
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) 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
\(170\) 5.00000 0.383482
\(171\) 0 0
\(172\) −1.00000 −0.0762493
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 2.00000 0.149906
\(179\) 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 \(0.428419\pi\)
\(182\) 1.00000 0.0741249
\(183\) 0 0
\(184\) −3.00000 −0.221163
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 15.0000 1.09691
\(188\) −8.00000 −0.583460
\(189\) 0 0
\(190\) −1.00000 −0.0725476
\(191\) −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 \(0.755774\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 16.0000 1.12576
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 1.00000 0.0696733
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) −7.00000 −0.474100
\(219\) 0 0
\(220\) 3.00000 0.202260
\(221\) 5.00000 0.336336
\(222\) 0 0
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −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 \(0.563524\pi\)
\(230\) 3.00000 0.197814
\(231\) 0 0
\(232\) −5.00000 −0.328266
\(233\) 4.00000 0.262049 0.131024 0.991379i \(-0.458173\pi\)
0.131024 + 0.991379i \(0.458173\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 5.00000 0.324102
\(239\) 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 \(0.249340\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 13.0000 0.832240
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 4.00000 0.254000
\(249\) 0 0
\(250\) 9.00000 0.569210
\(251\) −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
\(254\) −18.0000 −1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 5.00000 0.310685
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) −17.0000 −1.05026
\(263\) −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
\(266\) −1.00000 −0.0613139
\(267\) 0 0
\(268\) −10.0000 −0.610847
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) 15.0000 0.906183
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) −18.0000 −1.08152 −0.540758 0.841178i \(-0.681862\pi\)
−0.540758 + 0.841178i \(0.681862\pi\)
\(278\) −16.0000 −0.959616
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −8.00000 −0.475551 −0.237775 0.971320i \(-0.576418\pi\)
−0.237775 + 0.971320i \(0.576418\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 5.00000 0.293610
\(291\) 0 0
\(292\) −15.0000 −0.877809
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.00000 −0.290619
\(297\) 0 0
\(298\) 16.0000 0.926855
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) −5.00000 −0.287718
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −13.0000 −0.744378
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −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 \(0.209391\pi\)
\(314\) 9.00000 0.507899
\(315\) 0 0
\(316\) 6.00000 0.337526
\(317\) 24.0000 1.34797 0.673987 0.738743i \(-0.264580\pi\)
0.673987 + 0.738743i \(0.264580\pi\)
\(318\) 0 0
\(319\) 15.0000 0.839839
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 3.00000 0.167183
\(323\) −5.00000 −0.278207
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 10.0000 0.553849
\(327\) 0 0
\(328\) 8.00000 0.441726
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) 34.0000 1.86881 0.934405 0.356214i \(-0.115932\pi\)
0.934405 + 0.356214i \(0.115932\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) 15.0000 0.820763
\(335\) 10.0000 0.546358
\(336\) 0 0
\(337\) −29.0000 −1.57973 −0.789865 0.613280i \(-0.789850\pi\)
−0.789865 + 0.613280i \(0.789850\pi\)
\(338\) 1.00000 0.0543928
\(339\) 0 0
\(340\) 5.00000 0.271163
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) −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 \(0.448662\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 16.0000 0.845626
\(359\) −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
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) 15.0000 0.785136
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) 5.00000 0.259938
\(371\) 6.00000 0.311504
\(372\) 0 0
\(373\) −32.0000 −1.65690 −0.828449 0.560065i \(-0.810776\pi\)
−0.828449 + 0.560065i \(0.810776\pi\)
\(374\) 15.0000 0.775632
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 5.00000 0.257513
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 0 0
\(382\) −11.0000 −0.562809
\(383\) −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
\(386\) −20.0000 −1.01797
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 15.0000 0.758583
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 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 \(0.228689\pi\)
\(398\) −5.00000 −0.250627
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 16.0000 0.796030
\(405\) 0 0
\(406\) 5.00000 0.248146
\(407\) 15.0000 0.743522
\(408\) 0 0
\(409\) 3.00000 0.148340 0.0741702 0.997246i \(-0.476369\pi\)
0.0741702 + 0.997246i \(0.476369\pi\)
\(410\) −8.00000 −0.395092
\(411\) 0 0
\(412\) 1.00000 0.0492665
\(413\) 0 0
\(414\) 0 0
\(415\) −2.00000 −0.0981761
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) −3.00000 −0.146735
\(419\) 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 \(0.680106\pi\)
\(422\) 5.00000 0.243396
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 20.0000 0.970143
\(426\) 0 0
\(427\) −13.0000 −0.629114
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 1.00000 0.0482243
\(431\) −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 \(0.340430\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −7.00000 −0.335239
\(437\) −3.00000 −0.143509
\(438\) 0 0
\(439\) −11.0000 −0.525001 −0.262501 0.964932i \(-0.584547\pi\)
−0.262501 + 0.964932i \(0.584547\pi\)
\(440\) 3.00000 0.143019
\(441\) 0 0
\(442\) 5.00000 0.237826
\(443\) 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
\(446\) 26.0000 1.23114
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) −2.00000 −0.0938647
\(455\) −1.00000 −0.0468807
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 3.00000 0.139876
\(461\) 13.0000 0.605470 0.302735 0.953075i \(-0.402100\pi\)
0.302735 + 0.953075i \(0.402100\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) −5.00000 −0.232119
\(465\) 0 0
\(466\) 4.00000 0.185296
\(467\) −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
\(470\) 8.00000 0.369012
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 5.00000 0.229175
\(477\) 0 0
\(478\) 12.0000 0.548867
\(479\) −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
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 13.0000 0.588482
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 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
\(494\) −1.00000 −0.0449921
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 8.00000 0.358129 0.179065 0.983837i \(-0.442693\pi\)
0.179065 + 0.983837i \(0.442693\pi\)
\(500\) 9.00000 0.402492
\(501\) 0 0
\(502\) −21.0000 −0.937276
\(503\) 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
\(506\) 9.00000 0.400099
\(507\) 0 0
\(508\) −18.0000 −0.798621
\(509\) −11.0000 −0.487566 −0.243783 0.969830i \(-0.578389\pi\)
−0.243783 + 0.969830i \(0.578389\pi\)
\(510\) 0 0
\(511\) 15.0000 0.663561
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) −1.00000 −0.0440653
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 5.00000 0.219687
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) −39.0000 −1.70862 −0.854311 0.519763i \(-0.826020\pi\)
−0.854311 + 0.519763i \(0.826020\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) −17.0000 −0.742648
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) −1.00000 −0.0433555
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) −6.00000 −0.259403
\(536\) −10.0000 −0.431934
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 41.0000 1.76273 0.881364 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 2.00000 0.0859074
\(543\) 0 0
\(544\) −5.00000 −0.214373
\(545\) 7.00000 0.299847
\(546\) 0 0
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 15.0000 0.640768
\(549\) 0 0
\(550\) 12.0000 0.511682
\(551\) −5.00000 −0.213007
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) −18.0000 −0.764747
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) 0 0
\(559\) 1.00000 0.0422955
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) −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
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 3.00000 0.125436
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 8.00000 0.332756
\(579\) 0 0
\(580\) 5.00000 0.207614
\(581\) −2.00000 −0.0829740
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) −15.0000 −0.620704
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 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
\(592\) −5.00000 −0.205499
\(593\) −38.0000 −1.56047 −0.780236 0.625485i \(-0.784901\pi\)
−0.780236 + 0.625485i \(0.784901\pi\)
\(594\) 0 0
\(595\) −5.00000 −0.204980
\(596\) 16.0000 0.655386
\(597\) 0 0
\(598\) 3.00000 0.122679
\(599\) −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 \(0.366258\pi\)
\(602\) 1.00000 0.0407570
\(603\) 0 0
\(604\) −5.00000 −0.203447
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −27.0000 −1.09590 −0.547948 0.836512i \(-0.684591\pi\)
−0.547948 + 0.836512i \(0.684591\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −13.0000 −0.526355
\(611\) 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
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −43.0000 −1.73111 −0.865557 0.500810i \(-0.833036\pi\)
−0.865557 + 0.500810i \(0.833036\pi\)
\(618\) 0 0
\(619\) −31.0000 −1.24600 −0.622998 0.782224i \(-0.714085\pi\)
−0.622998 + 0.782224i \(0.714085\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −26.0000 −1.04251
\(623\) −2.00000 −0.0801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) 9.00000 0.359139
\(629\) 25.0000 0.996815
\(630\) 0 0
\(631\) −9.00000 −0.358284 −0.179142 0.983823i \(-0.557332\pi\)
−0.179142 + 0.983823i \(0.557332\pi\)
\(632\) 6.00000 0.238667
\(633\) 0 0
\(634\) 24.0000 0.953162
\(635\) 18.0000 0.714308
\(636\) 0 0
\(637\) −1.00000 −0.0396214
\(638\) 15.0000 0.593856
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) −5.00000 −0.196722
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 10.0000 0.391630
\(653\) −3.00000 −0.117399 −0.0586995 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) 0 0
\(655\) 17.0000 0.664245
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) 8.00000 0.311872
\(659\) −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 \(0.437699\pi\)
\(662\) 34.0000 1.32145
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) 15.0000 0.580802
\(668\) 15.0000 0.580367
\(669\) 0 0
\(670\) 10.0000 0.386334
\(671\) −39.0000 −1.50558
\(672\) 0 0
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −29.0000 −1.11704
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 5.00000 0.191741
\(681\) 0 0
\(682\) −12.0000 −0.459504
\(683\) −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
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) −32.0000 −1.21470
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) −40.0000 −1.51511
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) −5.00000 −0.188579
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 2.00000 0.0749532
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 16.0000 0.597948
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −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
\(722\) −18.0000 −0.669891
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) −27.0000 −1.00137 −0.500687 0.865628i \(-0.666919\pi\)
−0.500687 + 0.865628i \(0.666919\pi\)
\(728\) 1.00000 0.0370625
\(729\) 0 0
\(730\) 15.0000 0.555175
\(731\) 5.00000 0.184932
\(732\) 0 0
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) 30.0000 1.10506
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 5.00000 0.183804
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) 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
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) 15.0000 0.548454
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) −8.00000 −0.291730
\(753\) 0 0
\(754\) 5.00000 0.182089
\(755\) 5.00000 0.181969
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) 7.00000 0.253417
\(764\) −11.0000 −0.397966
\(765\) 0 0
\(766\) −1.00000 −0.0361315
\(767\) 0 0
\(768\) 0 0
\(769\) 43.0000 1.55062 0.775310 0.631581i \(-0.217594\pi\)
0.775310 + 0.631581i \(0.217594\pi\)
\(770\) −3.00000 −0.108112
\(771\) 0 0
\(772\) −20.0000 −0.719816
\(773\) −11.0000 −0.395643 −0.197821 0.980238i \(-0.563387\pi\)
−0.197821 + 0.980238i \(0.563387\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 15.0000 0.536399
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −9.00000 −0.321224
\(786\) 0 0
\(787\) −47.0000 −1.67537 −0.837685 0.546154i \(-0.816091\pi\)
−0.837685 + 0.546154i \(0.816091\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) −6.00000 −0.213470
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −13.0000 −0.461644
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) −5.00000 −0.177220
\(797\) 28.0000 0.991811 0.495905 0.868377i \(-0.334836\pi\)
0.495905 + 0.868377i \(0.334836\pi\)
\(798\) 0 0
\(799\) 40.0000 1.41510
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) 30.0000 1.05934
\(803\) 45.0000 1.58802
\(804\) 0 0
\(805\) −3.00000 −0.105736
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 16.0000 0.562878
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) −19.0000 −0.667180 −0.333590 0.942718i \(-0.608260\pi\)
−0.333590 + 0.942718i \(0.608260\pi\)
\(812\) 5.00000 0.175466
\(813\) 0 0
\(814\) 15.0000 0.525750
\(815\) −10.0000 −0.350285
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 3.00000 0.104893
\(819\) 0 0
\(820\) −8.00000 −0.279372
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 1.00000 0.0348367
\(825\) 0 0
\(826\) 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 \(0.607031\pi\)
\(830\) −2.00000 −0.0694210
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) −5.00000 −0.173240
\(834\) 0 0
\(835\) −15.0000 −0.519096
\(836\) −3.00000 −0.103757
\(837\) 0 0
\(838\) 5.00000 0.172722
\(839\) 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
\(842\) −22.0000 −0.758170
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) −1.00000 −0.0344010
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 20.0000 0.685994
\(851\) 15.0000 0.514193
\(852\) 0 0
\(853\) −28.0000 −0.958702 −0.479351 0.877623i \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) −13.0000 −0.444851
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 1.00000 0.0340997
\(861\) 0 0
\(862\) −10.0000 −0.340601
\(863\) 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
\(866\) 20.0000 0.679628
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) −18.0000 −0.610608
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) −7.00000 −0.237050
\(873\) 0 0
\(874\) −3.00000 −0.101477
\(875\) −9.00000 −0.304256
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) −11.0000 −0.371232
\(879\) 0 0
\(880\) 3.00000 0.101130
\(881\) −37.0000 −1.24656 −0.623281 0.781998i \(-0.714201\pi\)
−0.623281 + 0.781998i \(0.714201\pi\)
\(882\) 0 0
\(883\) −41.0000 −1.37976 −0.689880 0.723924i \(-0.742337\pi\)
−0.689880 + 0.723924i \(0.742337\pi\)
\(884\) 5.00000 0.168168
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) 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
\(890\) −2.00000 −0.0670402
\(891\) 0 0
\(892\) 26.0000 0.870544
\(893\) −8.00000 −0.267710
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −13.0000 −0.433816
\(899\) −20.0000 −0.667037
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) −2.00000 −0.0663723
\(909\) 0 0
\(910\) −1.00000 −0.0331497
\(911\) −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
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 17.0000 0.561389
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 3.00000 0.0989071
\(921\) 0 0
\(922\) 13.0000 0.428132
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 20.0000 0.657596
\(926\) −13.0000 −0.427207
\(927\) 0 0
\(928\) −5.00000 −0.164133
\(929\) 4.00000 0.131236 0.0656179 0.997845i \(-0.479098\pi\)
0.0656179 + 0.997845i \(0.479098\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 4.00000 0.131024
\(933\) 0 0
\(934\) −21.0000 −0.687141
\(935\) −15.0000 −0.490552
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 10.0000 0.326512
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) −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
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 5.00000 0.162051
\(953\) 44.0000 1.42530 0.712650 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) 0 0
\(955\) 11.0000 0.355952
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) −27.0000 −0.872330
\(959\) −15.0000 −0.484375
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 5.00000 0.161206
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) 20.0000 0.643823
\(966\) 0 0
\(967\) −31.0000 −0.996893 −0.498446 0.866921i \(-0.666096\pi\)
−0.498446 + 0.866921i \(0.666096\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 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
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 13.0000 0.416120
\(977\) 25.0000 0.799821 0.399910 0.916554i \(-0.369041\pi\)
0.399910 + 0.916554i \(0.369041\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 38.0000 1.21263
\(983\) 5.00000 0.159475 0.0797376 0.996816i \(-0.474592\pi\)
0.0797376 + 0.996816i \(0.474592\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 25.0000 0.796162
\(987\) 0 0
\(988\) −1.00000 −0.0318142
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) −30.0000 −0.952981 −0.476491 0.879180i \(-0.658091\pi\)
−0.476491 + 0.879180i \(0.658091\pi\)
\(992\) 4.00000 0.127000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 8.00000 0.253236
\(999\) 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 1638.2.a.o.1.1 1
3.2 odd 2 546.2.a.c.1.1 1
12.11 even 2 4368.2.a.h.1.1 1
21.20 even 2 3822.2.a.e.1.1 1
39.38 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 3.2 odd 2
1638.2.a.o.1.1 1 1.1 even 1 trivial
3822.2.a.e.1.1 1 21.20 even 2
4368.2.a.h.1.1 1 12.11 even 2
7098.2.a.z.1.1 1 39.38 odd 2