Properties

Label 8470.2.a.h.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -3.00000 q^{9} +1.00000 q^{10} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +3.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} +8.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{34} +1.00000 q^{35} -3.00000 q^{36} -4.00000 q^{37} -6.00000 q^{38} +1.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} +3.00000 q^{45} -8.00000 q^{46} -10.0000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} +1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -8.00000 q^{61} +8.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +6.00000 q^{65} -6.00000 q^{67} +6.00000 q^{68} -1.00000 q^{70} +8.00000 q^{71} +3.00000 q^{72} +10.0000 q^{73} +4.00000 q^{74} +6.00000 q^{76} +2.00000 q^{79} -1.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} -4.00000 q^{83} -6.00000 q^{85} -4.00000 q^{86} +6.00000 q^{89} -3.00000 q^{90} +6.00000 q^{91} +8.00000 q^{92} +10.0000 q^{94} -6.00000 q^{95} +16.0000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 3.00000 0.707107
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −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
\(35\) 1.00000 0.169031
\(36\) −3.00000 −0.500000
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −6.00000 −0.973329
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) −8.00000 −1.17954
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 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
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −1.00000 −0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 3.00000 0.353553
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) −1.00000 −0.111803
\(81\) 9.00000 1.00000
\(82\) −10.0000 −1.10432
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −3.00000 −0.316228
\(91\) 6.00000 0.628971
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) 10.0000 1.03142
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 10.0000 0.985329 0.492665 0.870219i \(-0.336023\pi\)
0.492665 + 0.870219i \(0.336023\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) −6.00000 −0.557086
\(117\) 18.0000 1.66410
\(118\) 4.00000 0.368230
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 0 0
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) −3.00000 −0.267261
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −6.00000 −0.526235
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 6.00000 0.518321
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −3.00000 −0.250000
\(145\) 6.00000 0.498273
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 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 \(0.633388\pi\)
\(152\) −6.00000 −0.486664
\(153\) −18.0000 −1.45521
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −8.00000 −0.630488
\(162\) −9.00000 −0.707107
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 6.00000 0.460179
\(171\) −18.0000 −1.37649
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 3.00000 0.223607
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) −6.00000 −0.444750
\(183\) 0 0
\(184\) −8.00000 −0.589768
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −4.00000 −0.281439
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) −10.0000 −0.696733
\(207\) −24.0000 −1.66812
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 0 0
\(221\) −36.0000 −2.42162
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 1.00000 0.0668153
\(225\) −3.00000 −0.200000
\(226\) 2.00000 0.133038
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) −18.0000 −1.17670
\(235\) 10.0000 0.652328
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −36.0000 −2.29063
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 3.00000 0.188982
\(253\) 0 0
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 6.00000 0.372104
\(261\) 18.0000 1.11417
\(262\) −6.00000 −0.370681
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) 0 0
\(268\) −6.00000 −0.366508
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 2.00000 0.119952
\(279\) 24.0000 1.43684
\(280\) −1.00000 −0.0597614
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 3.00000 0.176777
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) 4.00000 0.232495
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 8.00000 0.458079
\(306\) 18.0000 1.02899
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 10.0000 0.564333
\(315\) −3.00000 −0.169031
\(316\) 2.00000 0.112509
\(317\) −8.00000 −0.449325 −0.224662 0.974437i \(-0.572128\pi\)
−0.224662 + 0.974437i \(0.572128\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 8.00000 0.445823
\(323\) 36.0000 2.00309
\(324\) 9.00000 0.500000
\(325\) −6.00000 −0.332820
\(326\) 22.0000 1.21847
\(327\) 0 0
\(328\) −10.0000 −0.552158
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) −4.00000 −0.219529
\(333\) 12.0000 0.657596
\(334\) −8.00000 −0.437741
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 0 0
\(342\) 18.0000 0.973329
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −12.0000 −0.642345 −0.321173 0.947021i \(-0.604077\pi\)
−0.321173 + 0.947021i \(0.604077\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 0 0
\(353\) −4.00000 −0.212899 −0.106449 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 0 0
\(359\) −14.0000 −0.738892 −0.369446 0.929252i \(-0.620452\pi\)
−0.369446 + 0.929252i \(0.620452\pi\)
\(360\) −3.00000 −0.158114
\(361\) 17.0000 0.894737
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) 6.00000 0.314485
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 8.00000 0.417029
\(369\) −30.0000 −1.56174
\(370\) −4.00000 −0.207950
\(371\) 0 0
\(372\) 0 0
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 10.0000 0.515711
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −6.00000 −0.307794
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −38.0000 −1.94171 −0.970855 0.239669i \(-0.922961\pi\)
−0.970855 + 0.239669i \(0.922961\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −12.0000 −0.609994
\(388\) 16.0000 0.812277
\(389\) −38.0000 −1.92668 −0.963338 0.268290i \(-0.913542\pi\)
−0.963338 + 0.268290i \(0.913542\pi\)
\(390\) 0 0
\(391\) 48.0000 2.42746
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 48.0000 2.39105
\(404\) 4.00000 0.199007
\(405\) −9.00000 −0.447214
\(406\) −6.00000 −0.297775
\(407\) 0 0
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 10.0000 0.492665
\(413\) 4.00000 0.196827
\(414\) 24.0000 1.17954
\(415\) 4.00000 0.196352
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 20.0000 0.973585
\(423\) 30.0000 1.45865
\(424\) 0 0
\(425\) 6.00000 0.291043
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −2.00000 −0.0957826
\(437\) 48.0000 2.29615
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 36.0000 1.71235
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 10.0000 0.473514
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 3.00000 0.141421
\(451\) 0 0
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) −6.00000 −0.281284
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) −16.0000 −0.745194 −0.372597 0.927993i \(-0.621533\pi\)
−0.372597 + 0.927993i \(0.621533\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 18.0000 0.832050
\(469\) 6.00000 0.277054
\(470\) −10.0000 −0.461266
\(471\) 0 0
\(472\) 4.00000 0.184115
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 10.0000 0.457389
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 0 0
\(485\) −16.0000 −0.726523
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 36.0000 1.61972
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −8.00000 −0.358849
\(498\) 0 0
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −3.00000 −0.133631
\(505\) −4.00000 −0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) −10.0000 −0.440653
\(516\) 0 0
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) 0 0
\(520\) −6.00000 −0.263117
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) −18.0000 −0.787839
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) −48.0000 −2.09091
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) −6.00000 −0.260133
\(533\) −60.0000 −2.59889
\(534\) 0 0
\(535\) −4.00000 −0.172935
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) −6.00000 −0.257248
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −18.0000 −0.768922
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −24.0000 −1.01600
\(559\) −24.0000 −1.01509
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 0 0
\(563\) 36.0000 1.51722 0.758610 0.651546i \(-0.225879\pi\)
0.758610 + 0.651546i \(0.225879\pi\)
\(564\) 0 0
\(565\) 2.00000 0.0841406
\(566\) −24.0000 −1.00880
\(567\) −9.00000 −0.377964
\(568\) −8.00000 −0.335673
\(569\) −40.0000 −1.67689 −0.838444 0.544988i \(-0.816534\pi\)
−0.838444 + 0.544988i \(0.816534\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 10.0000 0.417392
\(575\) 8.00000 0.333623
\(576\) −3.00000 −0.125000
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 6.00000 0.249136
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 0 0
\(584\) −10.0000 −0.413803
\(585\) −18.0000 −0.744208
\(586\) 22.0000 0.908812
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −48.0000 −1.97781
\(590\) −4.00000 −0.164677
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −14.0000 −0.574911 −0.287456 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 48.0000 1.96287
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 4.00000 0.163028
\(603\) 18.0000 0.733017
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 0 0
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −6.00000 −0.243332
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) 60.0000 2.42734
\(612\) −18.0000 −0.727607
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −28.0000 −1.11911
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −24.0000 −0.956943
\(630\) 3.00000 0.119523
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) 8.00000 0.317721
\(635\) −20.0000 −0.793676
\(636\) 0 0
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) −24.0000 −0.949425
\(640\) 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
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −36.0000 −1.41640
\(647\) 46.0000 1.80845 0.904223 0.427060i \(-0.140451\pi\)
0.904223 + 0.427060i \(0.140451\pi\)
\(648\) −9.00000 −0.353553
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −24.0000 −0.939193 −0.469596 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) 0 0
\(655\) −6.00000 −0.234439
\(656\) 10.0000 0.390434
\(657\) −30.0000 −1.17041
\(658\) −10.0000 −0.389841
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 6.00000 0.232670
\(666\) −12.0000 −0.464991
\(667\) −48.0000 −1.85857
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −6.00000 −0.231800
\(671\) 0 0
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) 42.0000 1.60709 0.803543 0.595247i \(-0.202946\pi\)
0.803543 + 0.595247i \(0.202946\pi\)
\(684\) −18.0000 −0.688247
\(685\) 18.0000 0.687745
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 2.00000 0.0758643
\(696\) 0 0
\(697\) 60.0000 2.27266
\(698\) 12.0000 0.454207
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) 0 0
\(705\) 0 0
\(706\) 4.00000 0.150542
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 8.00000 0.300235
\(711\) −6.00000 −0.225018
\(712\) −6.00000 −0.224860
\(713\) −64.0000 −2.39682
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 14.0000 0.522475
\(719\) −40.0000 −1.49175 −0.745874 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 3.00000 0.111803
\(721\) −10.0000 −0.372419
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) −22.0000 −0.817624
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) −6.00000 −0.222528 −0.111264 0.993791i \(-0.535490\pi\)
−0.111264 + 0.993791i \(0.535490\pi\)
\(728\) −6.00000 −0.222375
\(729\) −27.0000 −1.00000
\(730\) 10.0000 0.370117
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) −2.00000 −0.0738213
\(735\) 0 0
\(736\) −8.00000 −0.294884
\(737\) 0 0
\(738\) 30.0000 1.10432
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) 0 0
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) −18.0000 −0.659027
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) −4.00000 −0.146157
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) −10.0000 −0.364662
\(753\) 0 0
\(754\) −36.0000 −1.31104
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) 44.0000 1.59921 0.799604 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) −8.00000 −0.289430
\(765\) 18.0000 0.650791
\(766\) 38.0000 1.37300
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 10.0000 0.359908
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) 12.0000 0.431331
\(775\) −8.00000 −0.287368
\(776\) −16.0000 −0.574367
\(777\) 0 0
\(778\) 38.0000 1.36237
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) 0 0
\(782\) −48.0000 −1.71648
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 2.00000 0.0711568
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 22.0000 0.779280 0.389640 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) 0 0
\(799\) −60.0000 −2.12265
\(800\) −1.00000 −0.0353553
\(801\) −18.0000 −0.635999
\(802\) 18.0000 0.635602
\(803\) 0 0
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) −48.0000 −1.69073
\(807\) 0 0
\(808\) −4.00000 −0.140720
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 9.00000 0.316228
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) 6.00000 0.210559
\(813\) 0 0
\(814\) 0 0
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) −30.0000 −1.04893
\(819\) −18.0000 −0.628971
\(820\) −10.0000 −0.349215
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) −10.0000 −0.348367
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) −24.0000 −0.834058
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 20.0000 0.690889
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 34.0000 1.17172
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −23.0000 −0.791224
\(846\) −30.0000 −1.03142
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −6.00000 −0.205798
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) −8.00000 −0.273754
\(855\) 18.0000 0.615587
\(856\) −4.00000 −0.136717
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −18.0000 −0.613082
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) −14.0000 −0.476014
\(866\) −4.00000 −0.135926
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 0 0
\(870\) 0 0
\(871\) 36.0000 1.21981
\(872\) 2.00000 0.0677285
\(873\) −48.0000 −1.62455
\(874\) −48.0000 −1.62362
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 28.0000 0.944954
\(879\) 0 0
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 3.00000 0.101015
\(883\) 54.0000 1.81724 0.908622 0.417619i \(-0.137135\pi\)
0.908622 + 0.417619i \(0.137135\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 32.0000 1.07445 0.537227 0.843437i \(-0.319472\pi\)
0.537227 + 0.843437i \(0.319472\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −10.0000 −0.334825
\(893\) −60.0000 −2.00782
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 34.0000 1.13459
\(899\) 48.0000 1.60089
\(900\) −3.00000 −0.100000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −8.00000 −0.265489
\(909\) −12.0000 −0.398015
\(910\) 6.00000 0.198898
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) 16.0000 0.526932
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −32.0000 −1.05159
\(927\) −30.0000 −0.985329
\(928\) 6.00000 0.196960
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −18.0000 −0.589610
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) −6.00000 −0.195907
\(939\) 0 0
\(940\) 10.0000 0.326164
\(941\) 12.0000 0.391189 0.195594 0.980685i \(-0.437336\pi\)
0.195594 + 0.980685i \(0.437336\pi\)
\(942\) 0 0
\(943\) 80.0000 2.60516
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 0 0
\(947\) −58.0000 −1.88475 −0.942373 0.334563i \(-0.891411\pi\)
−0.942373 + 0.334563i \(0.891411\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) 20.0000 0.646171
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −24.0000 −0.773791
\(963\) −12.0000 −0.386695
\(964\) 10.0000 0.322078
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 0 0
\(973\) 2.00000 0.0641171
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) 6.00000 0.191565
\(982\) −32.0000 −1.02116
\(983\) 42.0000 1.33959 0.669796 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) −18.0000 −0.573528
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) −36.0000 −1.14531
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −36.0000 −1.13956
\(999\) 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 8470.2.a.h.1.1 1
11.10 odd 2 8470.2.a.y.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.h.1.1 1 1.1 even 1 trivial
8470.2.a.y.1.1 yes 1 11.10 odd 2