Properties

Label 4410.2.a.bg.1.1
Level $4410$
Weight $2$
Character 4410.1
Self dual yes
Analytic conductor $35.214$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4410.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} -9.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -7.00000 q^{37} +1.00000 q^{38} +1.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} -3.00000 q^{44} -9.00000 q^{46} +9.00000 q^{47} +1.00000 q^{50} +1.00000 q^{52} -9.00000 q^{53} -3.00000 q^{55} -6.00000 q^{58} -8.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} +4.00000 q^{73} -7.00000 q^{74} +1.00000 q^{76} -10.0000 q^{79} +1.00000 q^{80} +3.00000 q^{82} -6.00000 q^{85} +2.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} -9.00000 q^{92} +9.00000 q^{94} +1.00000 q^{95} +10.0000 q^{97} +O(q^{100})\)

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
\(6\) 0 0
\(7\) 0 0
\(8\) 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 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 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
\(23\) −9.00000 −1.87663 −0.938315 0.345782i \(-0.887614\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −7.00000 −0.813733
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 3.00000 0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.00000 −0.938315
\(93\) 0 0
\(94\) 9.00000 0.928279
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −3.00000 −0.286039
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 1.00000 0.0877058
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −23.0000 −1.83560 −0.917800 0.397043i \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 3.00000 0.234261
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −6.00000 −0.460179
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 3.00000 0.224231 0.112115 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −9.00000 −0.663489
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) 18.0000 1.31629
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) 0 0
\(204\) 0 0
\(205\) 3.00000 0.209529
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) −16.0000 −1.08366
\(219\) 0 0
\(220\) −3.00000 −0.202260
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157 0.0322078 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 0.0636285
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 27.0000 1.69748
\(254\) −1.00000 −0.0627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 3.00000 0.185341
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −3.00000 −0.180907
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −3.00000 −0.177394
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.00000 −0.406867
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) −9.00000 −0.520483
\(300\) 0 0
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −14.0000 −0.799022 −0.399511 0.916728i \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) −23.0000 −1.29797
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 20.0000 1.10770
\(327\) 0 0
\(328\) 3.00000 0.165647
\(329\) 0 0
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 3.00000 0.164153
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −12.0000 −0.652714
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 0 0
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 9.00000 0.483843
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 3.00000 0.158555
\(359\) −18.0000 −0.950004 −0.475002 0.879985i \(-0.657553\pi\)
−0.475002 + 0.879985i \(0.657553\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 19.0000 0.991792 0.495896 0.868382i \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) −9.00000 −0.469157
\(369\) 0 0
\(370\) −7.00000 −0.363913
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 1.00000 0.0512989
\(381\) 0 0
\(382\) −12.0000 −0.613973
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −16.0000 −0.814379
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 54.0000 2.73090
\(392\) 0 0
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) −10.0000 −0.503155
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 27.0000 1.34832 0.674158 0.738587i \(-0.264507\pi\)
0.674158 + 0.738587i \(0.264507\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) 21.0000 1.04093
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 3.00000 0.148159
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 0 0
\(418\) −3.00000 −0.146735
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 23.0000 1.11962
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 40.0000 1.92228 0.961139 0.276066i \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) −9.00000 −0.430528
\(438\) 0 0
\(439\) −26.0000 −1.24091 −0.620456 0.784241i \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) −8.00000 −0.378811
\(447\) 0 0
\(448\) 0 0
\(449\) −21.0000 −0.991051 −0.495526 0.868593i \(-0.665025\pi\)
−0.495526 + 0.868593i \(0.665025\pi\)
\(450\) 0 0
\(451\) −9.00000 −0.423793
\(452\) 0 0
\(453\) 0 0
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) −9.00000 −0.419627
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −1.00000 −0.0464739 −0.0232370 0.999730i \(-0.507397\pi\)
−0.0232370 + 0.999730i \(0.507397\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 9.00000 0.415139
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 1.00000 0.0455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) 10.0000 0.454077
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −8.00000 −0.362143
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 1.00000 0.0449921
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −15.0000 −0.669483
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 27.0000 1.20030
\(507\) 0 0
\(508\) −1.00000 −0.0443678
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 0 0
\(515\) 4.00000 0.176261
\(516\) 0 0
\(517\) −27.0000 −1.18746
\(518\) 0 0
\(519\) 0 0
\(520\) 1.00000 0.0438529
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000 2.09091
\(528\) 0 0
\(529\) 58.0000 2.52174
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) 0 0
\(533\) 3.00000 0.129944
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −12.0000 −0.512615
\(549\) 0 0
\(550\) −3.00000 −0.127920
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 9.00000 0.381342 0.190671 0.981654i \(-0.438934\pi\)
0.190671 + 0.981654i \(0.438934\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 27.0000 1.13893
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −3.00000 −0.125436
\(573\) 0 0
\(574\) 0 0
\(575\) −9.00000 −0.375326
\(576\) 0 0
\(577\) −44.0000 −1.83174 −0.915872 0.401470i \(-0.868499\pi\)
−0.915872 + 0.401470i \(0.868499\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) 0 0
\(583\) 27.0000 1.11823
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) −9.00000 −0.371787
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −9.00000 −0.368037
\(599\) 42.0000 1.71607 0.858037 0.513588i \(-0.171684\pi\)
0.858037 + 0.513588i \(0.171684\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) 1.00000 0.0405887 0.0202944 0.999794i \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) −14.0000 −0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) −23.0000 −0.924448 −0.462224 0.886763i \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) −23.0000 −0.917800
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −10.0000 −0.397779
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) −1.00000 −0.0396838
\(636\) 0 0
\(637\) 0 0
\(638\) 18.0000 0.712627
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −27.0000 −1.06644 −0.533218 0.845978i \(-0.679017\pi\)
−0.533218 + 0.845978i \(0.679017\pi\)
\(642\) 0 0
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) 9.00000 0.352197 0.176099 0.984373i \(-0.443652\pi\)
0.176099 + 0.984373i \(0.443652\pi\)
\(654\) 0 0
\(655\) 3.00000 0.117220
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 28.0000 1.08907 0.544537 0.838737i \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) −7.00000 −0.272063
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 54.0000 2.09089
\(668\) 3.00000 0.116073
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) −22.0000 −0.847408
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 24.0000 0.919007
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 2.00000 0.0762493
\(689\) −9.00000 −0.342873
\(690\) 0 0
\(691\) −32.0000 −1.21734 −0.608669 0.793424i \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −18.0000 −0.681799
\(698\) −26.0000 −0.984115
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −7.00000 −0.264010
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) 72.0000 2.69642
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 3.00000 0.112115
\(717\) 0 0
\(718\) −18.0000 −0.671754
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −18.0000 −0.669891
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 43.0000 1.58824 0.794121 0.607760i \(-0.207932\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) −7.00000 −0.257325
\(741\) 0 0
\(742\) 0 0
\(743\) 45.0000 1.65089 0.825445 0.564483i \(-0.190924\pi\)
0.825445 + 0.564483i \(0.190924\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 2.00000 0.0732252
\(747\) 0 0
\(748\) 18.0000 0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 9.00000 0.328196
\(753\) 0 0
\(754\) −6.00000 −0.218507
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 23.0000 0.835398
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 21.0000 0.758761
\(767\) 0 0
\(768\) 0 0
\(769\) −23.0000 −0.829401 −0.414701 0.909958i \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) −51.0000 −1.83434 −0.917171 0.398493i \(-0.869533\pi\)
−0.917171 + 0.398493i \(0.869533\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) 3.00000 0.107486
\(780\) 0 0
\(781\) 0 0
\(782\) 54.0000 1.93104
\(783\) 0 0
\(784\) 0 0
\(785\) −23.0000 −0.820905
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −15.0000 −0.534353
\(789\) 0 0
\(790\) −10.0000 −0.355784
\(791\) 0 0
\(792\) 0 0
\(793\) −8.00000 −0.284088
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) −54.0000 −1.91038
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 27.0000 0.953403
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) 25.0000 0.877869 0.438934 0.898519i \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 21.0000 0.736050
\(815\) 20.0000 0.700569
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 3.00000 0.104765
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) −6.00000 −0.208640 −0.104320 0.994544i \(-0.533267\pi\)
−0.104320 + 0.994544i \(0.533267\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 0 0
\(835\) 3.00000 0.103819
\(836\) −3.00000 −0.103757
\(837\) 0 0
\(838\) −9.00000 −0.310900
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 2.00000 0.0689246
\(843\) 0 0
\(844\) 23.0000 0.791693
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) 63.0000 2.15961
\(852\) 0 0
\(853\) 19.0000 0.650548 0.325274 0.945620i \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 3.00000 0.102121 0.0510606 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) 0 0
\(865\) 9.00000 0.306009
\(866\) 40.0000 1.35926
\(867\) 0 0
\(868\) 0 0
\(869\) 30.0000 1.01768
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −16.0000 −0.541828
\(873\) 0 0
\(874\) −9.00000 −0.304430
\(875\) 0 0
\(876\) 0 0
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −26.0000 −0.877457
\(879\) 0 0
\(880\) −3.00000 −0.101130
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −8.00000 −0.267860
\(893\) 9.00000 0.301174
\(894\) 0 0
\(895\) 3.00000 0.100279
\(896\) 0 0
\(897\) 0 0
\(898\) −21.0000 −0.700779
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 54.0000 1.79900
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 4.00000 0.132164
\(917\) 0 0
\(918\) 0 0
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) −9.00000 −0.296721
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) −1.00000 −0.0328620
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 57.0000 1.87011 0.935055 0.354504i \(-0.115350\pi\)
0.935055 + 0.354504i \(0.115350\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −6.00000 −0.196326
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 9.00000 0.293548
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 0 0
\(943\) −27.0000 −0.879241
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −6.00000 −0.194974 −0.0974869 0.995237i \(-0.531080\pi\)
−0.0974869 + 0.995237i \(0.531080\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 1.00000 0.0324443
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 6.00000 0.194054
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −7.00000 −0.225689
\(963\) 0 0
\(964\) 1.00000 0.0322078
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −2.00000 −0.0642824
\(969\) 0 0
\(970\) 10.0000 0.321081
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) −36.0000 −1.14881
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 0 0
\(985\) −15.0000 −0.477940
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) −18.0000 −0.572367
\(990\) 0 0
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) −8.00000 −0.254000
\(993\) 0 0
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −4.00000 −0.126618
\(999\) 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 4410.2.a.bg.1.1 1
3.2 odd 2 490.2.a.d.1.1 1
7.3 odd 6 630.2.k.d.541.1 2
7.5 odd 6 630.2.k.d.361.1 2
7.6 odd 2 4410.2.a.x.1.1 1
12.11 even 2 3920.2.a.e.1.1 1
15.2 even 4 2450.2.c.e.99.1 2
15.8 even 4 2450.2.c.e.99.2 2
15.14 odd 2 2450.2.a.v.1.1 1
21.2 odd 6 490.2.e.g.361.1 2
21.5 even 6 70.2.e.d.11.1 2
21.11 odd 6 490.2.e.g.471.1 2
21.17 even 6 70.2.e.d.51.1 yes 2
21.20 even 2 490.2.a.a.1.1 1
84.47 odd 6 560.2.q.b.81.1 2
84.59 odd 6 560.2.q.b.401.1 2
84.83 odd 2 3920.2.a.bh.1.1 1
105.17 odd 12 350.2.j.d.149.2 4
105.38 odd 12 350.2.j.d.149.1 4
105.47 odd 12 350.2.j.d.249.1 4
105.59 even 6 350.2.e.b.51.1 2
105.62 odd 4 2450.2.c.q.99.1 2
105.68 odd 12 350.2.j.d.249.2 4
105.83 odd 4 2450.2.c.q.99.2 2
105.89 even 6 350.2.e.b.151.1 2
105.104 even 2 2450.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 21.5 even 6
70.2.e.d.51.1 yes 2 21.17 even 6
350.2.e.b.51.1 2 105.59 even 6
350.2.e.b.151.1 2 105.89 even 6
350.2.j.d.149.1 4 105.38 odd 12
350.2.j.d.149.2 4 105.17 odd 12
350.2.j.d.249.1 4 105.47 odd 12
350.2.j.d.249.2 4 105.68 odd 12
490.2.a.a.1.1 1 21.20 even 2
490.2.a.d.1.1 1 3.2 odd 2
490.2.e.g.361.1 2 21.2 odd 6
490.2.e.g.471.1 2 21.11 odd 6
560.2.q.b.81.1 2 84.47 odd 6
560.2.q.b.401.1 2 84.59 odd 6
630.2.k.d.361.1 2 7.5 odd 6
630.2.k.d.541.1 2 7.3 odd 6
2450.2.a.v.1.1 1 15.14 odd 2
2450.2.a.bf.1.1 1 105.104 even 2
2450.2.c.e.99.1 2 15.2 even 4
2450.2.c.e.99.2 2 15.8 even 4
2450.2.c.q.99.1 2 105.62 odd 4
2450.2.c.q.99.2 2 105.83 odd 4
3920.2.a.e.1.1 1 12.11 even 2
3920.2.a.bh.1.1 1 84.83 odd 2
4410.2.a.x.1.1 1 7.6 odd 2
4410.2.a.bg.1.1 1 1.1 even 1 trivial