Properties

Label 8470.2.a.bf.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +1.00000 q^{12} -3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{18} -3.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -3.00000 q^{26} -5.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{35} -2.00000 q^{36} -6.00000 q^{37} -3.00000 q^{38} -3.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} +1.00000 q^{42} -6.00000 q^{43} -2.00000 q^{45} -7.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -3.00000 q^{52} -12.0000 q^{53} -5.00000 q^{54} +1.00000 q^{56} -3.00000 q^{57} -8.00000 q^{58} -1.00000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +4.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -3.00000 q^{65} -8.00000 q^{67} -7.00000 q^{69} +1.00000 q^{70} +8.00000 q^{71} -2.00000 q^{72} -8.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} -3.00000 q^{76} -3.00000 q^{78} +7.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -9.00000 q^{83} +1.00000 q^{84} -6.00000 q^{86} -8.00000 q^{87} +6.00000 q^{89} -2.00000 q^{90} -3.00000 q^{91} -7.00000 q^{92} +4.00000 q^{93} -12.0000 q^{94} -3.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) 1.00000 0.288675
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −2.00000 −0.471405
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −3.00000 −0.588348
\(27\) −5.00000 −0.962250
\(28\) 1.00000 0.188982
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) 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
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) −2.00000 −0.333333
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −3.00000 −0.486664
\(39\) −3.00000 −0.480384
\(40\) 1.00000 0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 1.00000 0.154303
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) −7.00000 −1.03209
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −3.00000 −0.416025
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −5.00000 −0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −3.00000 −0.397360
\(58\) −8.00000 −1.05045
\(59\) −1.00000 −0.130189 −0.0650945 0.997879i \(-0.520735\pi\)
−0.0650945 + 0.997879i \(0.520735\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −7.00000 −0.842701
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) −2.00000 −0.235702
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) −6.00000 −0.697486
\(75\) 1.00000 0.115470
\(76\) −3.00000 −0.344124
\(77\) 0 0
\(78\) −3.00000 −0.339683
\(79\) 7.00000 0.787562 0.393781 0.919204i \(-0.371167\pi\)
0.393781 + 0.919204i \(0.371167\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −9.00000 −0.987878 −0.493939 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −2.00000 −0.210819
\(91\) −3.00000 −0.314485
\(92\) −7.00000 −0.729800
\(93\) 4.00000 0.414781
\(94\) −12.0000 −1.23771
\(95\) −3.00000 −0.307794
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 3.00000 0.298511 0.149256 0.988799i \(-0.452312\pi\)
0.149256 + 0.988799i \(0.452312\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) −3.00000 −0.294174
\(105\) 1.00000 0.0975900
\(106\) −12.0000 −1.16554
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −5.00000 −0.481125
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) −3.00000 −0.280976
\(115\) −7.00000 −0.652753
\(116\) −8.00000 −0.742781
\(117\) 6.00000 0.554700
\(118\) −1.00000 −0.0920575
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 0 0
\(122\) 10.0000 0.905357
\(123\) 10.0000 0.901670
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) −17.0000 −1.50851 −0.754253 0.656584i \(-0.772001\pi\)
−0.754253 + 0.656584i \(0.772001\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.00000 −0.528271
\(130\) −3.00000 −0.263117
\(131\) −7.00000 −0.611593 −0.305796 0.952097i \(-0.598923\pi\)
−0.305796 + 0.952097i \(0.598923\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) −8.00000 −0.691095
\(135\) −5.00000 −0.430331
\(136\) 0 0
\(137\) −21.0000 −1.79415 −0.897076 0.441877i \(-0.854313\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(138\) −7.00000 −0.595880
\(139\) 1.00000 0.0848189 0.0424094 0.999100i \(-0.486497\pi\)
0.0424094 + 0.999100i \(0.486497\pi\)
\(140\) 1.00000 0.0845154
\(141\) −12.0000 −1.01058
\(142\) 8.00000 0.671345
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) −8.00000 −0.664364
\(146\) −8.00000 −0.662085
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 17.0000 1.38344 0.691720 0.722166i \(-0.256853\pi\)
0.691720 + 0.722166i \(0.256853\pi\)
\(152\) −3.00000 −0.243332
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −3.00000 −0.240192
\(157\) 15.0000 1.19713 0.598565 0.801074i \(-0.295738\pi\)
0.598565 + 0.801074i \(0.295738\pi\)
\(158\) 7.00000 0.556890
\(159\) −12.0000 −0.951662
\(160\) 1.00000 0.0790569
\(161\) −7.00000 −0.551677
\(162\) 1.00000 0.0785674
\(163\) 22.0000 1.72317 0.861586 0.507611i \(-0.169471\pi\)
0.861586 + 0.507611i \(0.169471\pi\)
\(164\) 10.0000 0.780869
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 1.00000 0.0771517
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −6.00000 −0.457496
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −8.00000 −0.606478
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −1.00000 −0.0751646
\(178\) 6.00000 0.449719
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) −2.00000 −0.149071
\(181\) −9.00000 −0.668965 −0.334482 0.942402i \(-0.608561\pi\)
−0.334482 + 0.942402i \(0.608561\pi\)
\(182\) −3.00000 −0.222375
\(183\) 10.0000 0.739221
\(184\) −7.00000 −0.516047
\(185\) −6.00000 −0.441129
\(186\) 4.00000 0.293294
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) −5.00000 −0.363696
\(190\) −3.00000 −0.217643
\(191\) 25.0000 1.80894 0.904468 0.426541i \(-0.140268\pi\)
0.904468 + 0.426541i \(0.140268\pi\)
\(192\) 1.00000 0.0721688
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 8.00000 0.574367
\(195\) −3.00000 −0.214834
\(196\) 1.00000 0.0714286
\(197\) 4.00000 0.284988 0.142494 0.989796i \(-0.454488\pi\)
0.142494 + 0.989796i \(0.454488\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 1.00000 0.0707107
\(201\) −8.00000 −0.564276
\(202\) 3.00000 0.211079
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) −2.00000 −0.139347
\(207\) 14.0000 0.973067
\(208\) −3.00000 −0.208013
\(209\) 0 0
\(210\) 1.00000 0.0690066
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) −12.0000 −0.824163
\(213\) 8.00000 0.548151
\(214\) 18.0000 1.23045
\(215\) −6.00000 −0.409197
\(216\) −5.00000 −0.340207
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 1.00000 0.0668153
\(225\) −2.00000 −0.133333
\(226\) −1.00000 −0.0665190
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −3.00000 −0.198680
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −7.00000 −0.461566
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 6.00000 0.392232
\(235\) −12.0000 −0.782794
\(236\) −1.00000 −0.0650945
\(237\) 7.00000 0.454699
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 1.00000 0.0645497
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 0 0
\(243\) 16.0000 1.02640
\(244\) 10.0000 0.640184
\(245\) 1.00000 0.0638877
\(246\) 10.0000 0.637577
\(247\) 9.00000 0.572656
\(248\) 4.00000 0.254000
\(249\) −9.00000 −0.570352
\(250\) 1.00000 0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0 0
\(254\) −17.0000 −1.06667
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) −6.00000 −0.373544
\(259\) −6.00000 −0.372822
\(260\) −3.00000 −0.186052
\(261\) 16.0000 0.990375
\(262\) −7.00000 −0.432461
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) −3.00000 −0.183942
\(267\) 6.00000 0.367194
\(268\) −8.00000 −0.488678
\(269\) 7.00000 0.426798 0.213399 0.976965i \(-0.431547\pi\)
0.213399 + 0.976965i \(0.431547\pi\)
\(270\) −5.00000 −0.304290
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) −3.00000 −0.181568
\(274\) −21.0000 −1.26866
\(275\) 0 0
\(276\) −7.00000 −0.421350
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 1.00000 0.0599760
\(279\) −8.00000 −0.478947
\(280\) 1.00000 0.0597614
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) −12.0000 −0.714590
\(283\) 11.0000 0.653882 0.326941 0.945045i \(-0.393982\pi\)
0.326941 + 0.945045i \(0.393982\pi\)
\(284\) 8.00000 0.474713
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) 10.0000 0.590281
\(288\) −2.00000 −0.117851
\(289\) −17.0000 −1.00000
\(290\) −8.00000 −0.469776
\(291\) 8.00000 0.468968
\(292\) −8.00000 −0.468165
\(293\) 31.0000 1.81104 0.905520 0.424304i \(-0.139481\pi\)
0.905520 + 0.424304i \(0.139481\pi\)
\(294\) 1.00000 0.0583212
\(295\) −1.00000 −0.0582223
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 21.0000 1.21446
\(300\) 1.00000 0.0577350
\(301\) −6.00000 −0.345834
\(302\) 17.0000 0.978240
\(303\) 3.00000 0.172345
\(304\) −3.00000 −0.172062
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) −2.00000 −0.113776
\(310\) 4.00000 0.227185
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −3.00000 −0.169842
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 15.0000 0.846499
\(315\) −2.00000 −0.112687
\(316\) 7.00000 0.393781
\(317\) 4.00000 0.224662 0.112331 0.993671i \(-0.464168\pi\)
0.112331 + 0.993671i \(0.464168\pi\)
\(318\) −12.0000 −0.672927
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) 18.0000 1.00466
\(322\) −7.00000 −0.390095
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −3.00000 −0.166410
\(326\) 22.0000 1.21847
\(327\) 0 0
\(328\) 10.0000 0.552158
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −9.00000 −0.493939
\(333\) 12.0000 0.657596
\(334\) −10.0000 −0.547176
\(335\) −8.00000 −0.437087
\(336\) 1.00000 0.0545545
\(337\) −17.0000 −0.926049 −0.463025 0.886345i \(-0.653236\pi\)
−0.463025 + 0.886345i \(0.653236\pi\)
\(338\) −4.00000 −0.217571
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 1.00000 0.0539949
\(344\) −6.00000 −0.323498
\(345\) −7.00000 −0.376867
\(346\) 18.0000 0.967686
\(347\) −22.0000 −1.18102 −0.590511 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) −8.00000 −0.428845
\(349\) 29.0000 1.55233 0.776167 0.630527i \(-0.217161\pi\)
0.776167 + 0.630527i \(0.217161\pi\)
\(350\) 1.00000 0.0534522
\(351\) 15.0000 0.800641
\(352\) 0 0
\(353\) −36.0000 −1.91609 −0.958043 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) −1.00000 −0.0531494
\(355\) 8.00000 0.424596
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) −18.0000 −0.951330
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) −2.00000 −0.105409
\(361\) −10.0000 −0.526316
\(362\) −9.00000 −0.473029
\(363\) 0 0
\(364\) −3.00000 −0.157243
\(365\) −8.00000 −0.418739
\(366\) 10.0000 0.522708
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) −7.00000 −0.364900
\(369\) −20.0000 −1.04116
\(370\) −6.00000 −0.311925
\(371\) −12.0000 −0.623009
\(372\) 4.00000 0.207390
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) 24.0000 1.23606
\(378\) −5.00000 −0.257172
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) −3.00000 −0.153897
\(381\) −17.0000 −0.870936
\(382\) 25.0000 1.27911
\(383\) −38.0000 −1.94171 −0.970855 0.239669i \(-0.922961\pi\)
−0.970855 + 0.239669i \(0.922961\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) 12.0000 0.609994
\(388\) 8.00000 0.406138
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) −3.00000 −0.151911
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) −7.00000 −0.353103
\(394\) 4.00000 0.201517
\(395\) 7.00000 0.352208
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −12.0000 −0.601506
\(399\) −3.00000 −0.150188
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) −8.00000 −0.399004
\(403\) −12.0000 −0.597763
\(404\) 3.00000 0.149256
\(405\) 1.00000 0.0496904
\(406\) −8.00000 −0.397033
\(407\) 0 0
\(408\) 0 0
\(409\) 20.0000 0.988936 0.494468 0.869196i \(-0.335363\pi\)
0.494468 + 0.869196i \(0.335363\pi\)
\(410\) 10.0000 0.493865
\(411\) −21.0000 −1.03585
\(412\) −2.00000 −0.0985329
\(413\) −1.00000 −0.0492068
\(414\) 14.0000 0.688062
\(415\) −9.00000 −0.441793
\(416\) −3.00000 −0.147087
\(417\) 1.00000 0.0489702
\(418\) 0 0
\(419\) 7.00000 0.341972 0.170986 0.985273i \(-0.445305\pi\)
0.170986 + 0.985273i \(0.445305\pi\)
\(420\) 1.00000 0.0487950
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 24.0000 1.16692
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 8.00000 0.387601
\(427\) 10.0000 0.483934
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 1.00000 0.0481683 0.0240842 0.999710i \(-0.492333\pi\)
0.0240842 + 0.999710i \(0.492333\pi\)
\(432\) −5.00000 −0.240563
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 4.00000 0.192006
\(435\) −8.00000 −0.383571
\(436\) 0 0
\(437\) 21.0000 1.00457
\(438\) −8.00000 −0.382255
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) −6.00000 −0.284747
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 1.00000 0.0472456
\(449\) 41.0000 1.93491 0.967455 0.253044i \(-0.0814317\pi\)
0.967455 + 0.253044i \(0.0814317\pi\)
\(450\) −2.00000 −0.0942809
\(451\) 0 0
\(452\) −1.00000 −0.0470360
\(453\) 17.0000 0.798730
\(454\) −4.00000 −0.187729
\(455\) −3.00000 −0.140642
\(456\) −3.00000 −0.140488
\(457\) 31.0000 1.45012 0.725059 0.688686i \(-0.241812\pi\)
0.725059 + 0.688686i \(0.241812\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) −7.00000 −0.326377
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) −23.0000 −1.06890 −0.534450 0.845200i \(-0.679481\pi\)
−0.534450 + 0.845200i \(0.679481\pi\)
\(464\) −8.00000 −0.371391
\(465\) 4.00000 0.185496
\(466\) 3.00000 0.138972
\(467\) 33.0000 1.52706 0.763529 0.645774i \(-0.223465\pi\)
0.763529 + 0.645774i \(0.223465\pi\)
\(468\) 6.00000 0.277350
\(469\) −8.00000 −0.369406
\(470\) −12.0000 −0.553519
\(471\) 15.0000 0.691164
\(472\) −1.00000 −0.0460287
\(473\) 0 0
\(474\) 7.00000 0.321521
\(475\) −3.00000 −0.137649
\(476\) 0 0
\(477\) 24.0000 1.09888
\(478\) 9.00000 0.411650
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 1.00000 0.0456435
\(481\) 18.0000 0.820729
\(482\) 4.00000 0.182195
\(483\) −7.00000 −0.318511
\(484\) 0 0
\(485\) 8.00000 0.363261
\(486\) 16.0000 0.725775
\(487\) −37.0000 −1.67663 −0.838315 0.545186i \(-0.816459\pi\)
−0.838315 + 0.545186i \(0.816459\pi\)
\(488\) 10.0000 0.452679
\(489\) 22.0000 0.994874
\(490\) 1.00000 0.0451754
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) 9.00000 0.404929
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 8.00000 0.358849
\(498\) −9.00000 −0.403300
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 1.00000 0.0447214
\(501\) −10.0000 −0.446767
\(502\) −20.0000 −0.892644
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) −4.00000 −0.177646
\(508\) −17.0000 −0.754253
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 1.00000 0.0441942
\(513\) 15.0000 0.662266
\(514\) −20.0000 −0.882162
\(515\) −2.00000 −0.0881305
\(516\) −6.00000 −0.264135
\(517\) 0 0
\(518\) −6.00000 −0.263625
\(519\) 18.0000 0.790112
\(520\) −3.00000 −0.131559
\(521\) −40.0000 −1.75243 −0.876216 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 16.0000 0.700301
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) −7.00000 −0.305796
\(525\) 1.00000 0.0436436
\(526\) 9.00000 0.392419
\(527\) 0 0
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) −12.0000 −0.521247
\(531\) 2.00000 0.0867926
\(532\) −3.00000 −0.130066
\(533\) −30.0000 −1.29944
\(534\) 6.00000 0.259645
\(535\) 18.0000 0.778208
\(536\) −8.00000 −0.345547
\(537\) −18.0000 −0.776757
\(538\) 7.00000 0.301791
\(539\) 0 0
\(540\) −5.00000 −0.215166
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 16.0000 0.687259
\(543\) −9.00000 −0.386227
\(544\) 0 0
\(545\) 0 0
\(546\) −3.00000 −0.128388
\(547\) 14.0000 0.598597 0.299298 0.954160i \(-0.403247\pi\)
0.299298 + 0.954160i \(0.403247\pi\)
\(548\) −21.0000 −0.897076
\(549\) −20.0000 −0.853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) −7.00000 −0.297940
\(553\) 7.00000 0.297670
\(554\) 26.0000 1.10463
\(555\) −6.00000 −0.254686
\(556\) 1.00000 0.0424094
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) −8.00000 −0.338667
\(559\) 18.0000 0.761319
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −13.0000 −0.548372
\(563\) 19.0000 0.800755 0.400377 0.916350i \(-0.368879\pi\)
0.400377 + 0.916350i \(0.368879\pi\)
\(564\) −12.0000 −0.505291
\(565\) −1.00000 −0.0420703
\(566\) 11.0000 0.462364
\(567\) 1.00000 0.0419961
\(568\) 8.00000 0.335673
\(569\) −39.0000 −1.63497 −0.817483 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) −3.00000 −0.125656
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 25.0000 1.04439
\(574\) 10.0000 0.417392
\(575\) −7.00000 −0.291920
\(576\) −2.00000 −0.0833333
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) −17.0000 −0.707107
\(579\) −5.00000 −0.207793
\(580\) −8.00000 −0.332182
\(581\) −9.00000 −0.373383
\(582\) 8.00000 0.331611
\(583\) 0 0
\(584\) −8.00000 −0.331042
\(585\) 6.00000 0.248069
\(586\) 31.0000 1.28060
\(587\) −1.00000 −0.0412744 −0.0206372 0.999787i \(-0.506569\pi\)
−0.0206372 + 0.999787i \(0.506569\pi\)
\(588\) 1.00000 0.0412393
\(589\) −12.0000 −0.494451
\(590\) −1.00000 −0.0411693
\(591\) 4.00000 0.164538
\(592\) −6.00000 −0.246598
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) −12.0000 −0.491127
\(598\) 21.0000 0.858754
\(599\) 17.0000 0.694601 0.347301 0.937754i \(-0.387098\pi\)
0.347301 + 0.937754i \(0.387098\pi\)
\(600\) 1.00000 0.0408248
\(601\) −16.0000 −0.652654 −0.326327 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) −6.00000 −0.244542
\(603\) 16.0000 0.651570
\(604\) 17.0000 0.691720
\(605\) 0 0
\(606\) 3.00000 0.121867
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) −3.00000 −0.121666
\(609\) −8.00000 −0.324176
\(610\) 10.0000 0.404888
\(611\) 36.0000 1.45640
\(612\) 0 0
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −4.00000 −0.161427
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −2.00000 −0.0804518
\(619\) 29.0000 1.16561 0.582804 0.812613i \(-0.301955\pi\)
0.582804 + 0.812613i \(0.301955\pi\)
\(620\) 4.00000 0.160644
\(621\) 35.0000 1.40450
\(622\) −18.0000 −0.721734
\(623\) 6.00000 0.240385
\(624\) −3.00000 −0.120096
\(625\) 1.00000 0.0400000
\(626\) −34.0000 −1.35891
\(627\) 0 0
\(628\) 15.0000 0.598565
\(629\) 0 0
\(630\) −2.00000 −0.0796819
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 7.00000 0.278445
\(633\) 0 0
\(634\) 4.00000 0.158860
\(635\) −17.0000 −0.674624
\(636\) −12.0000 −0.475831
\(637\) −3.00000 −0.118864
\(638\) 0 0
\(639\) −16.0000 −0.632950
\(640\) 1.00000 0.0395285
\(641\) −17.0000 −0.671460 −0.335730 0.941958i \(-0.608983\pi\)
−0.335730 + 0.941958i \(0.608983\pi\)
\(642\) 18.0000 0.710403
\(643\) −32.0000 −1.26196 −0.630978 0.775800i \(-0.717346\pi\)
−0.630978 + 0.775800i \(0.717346\pi\)
\(644\) −7.00000 −0.275839
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −3.00000 −0.117670
\(651\) 4.00000 0.156772
\(652\) 22.0000 0.861586
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) −7.00000 −0.273513
\(656\) 10.0000 0.390434
\(657\) 16.0000 0.624219
\(658\) −12.0000 −0.467809
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) −9.00000 −0.349268
\(665\) −3.00000 −0.116335
\(666\) 12.0000 0.464991
\(667\) 56.0000 2.16833
\(668\) −10.0000 −0.386912
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 0 0
\(672\) 1.00000 0.0385758
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) −17.0000 −0.654816
\(675\) −5.00000 −0.192450
\(676\) −4.00000 −0.153846
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) −1.00000 −0.0384048
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) 6.00000 0.229416
\(685\) −21.0000 −0.802369
\(686\) 1.00000 0.0381802
\(687\) 2.00000 0.0763048
\(688\) −6.00000 −0.228748
\(689\) 36.0000 1.37149
\(690\) −7.00000 −0.266485
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) 1.00000 0.0379322
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) 29.0000 1.09767
\(699\) 3.00000 0.113470
\(700\) 1.00000 0.0377964
\(701\) 20.0000 0.755390 0.377695 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 15.0000 0.566139
\(703\) 18.0000 0.678883
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) −36.0000 −1.35488
\(707\) 3.00000 0.112827
\(708\) −1.00000 −0.0375823
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 8.00000 0.300235
\(711\) −14.0000 −0.525041
\(712\) 6.00000 0.224860
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) −18.0000 −0.672692
\(717\) 9.00000 0.336111
\(718\) 8.00000 0.298557
\(719\) −28.0000 −1.04422 −0.522112 0.852877i \(-0.674856\pi\)
−0.522112 + 0.852877i \(0.674856\pi\)
\(720\) −2.00000 −0.0745356
\(721\) −2.00000 −0.0744839
\(722\) −10.0000 −0.372161
\(723\) 4.00000 0.148762
\(724\) −9.00000 −0.334482
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −3.00000 −0.111187
\(729\) 13.0000 0.481481
\(730\) −8.00000 −0.296093
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) 11.0000 0.406294 0.203147 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) 2.00000 0.0738213
\(735\) 1.00000 0.0368856
\(736\) −7.00000 −0.258023
\(737\) 0 0
\(738\) −20.0000 −0.736210
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −6.00000 −0.220564
\(741\) 9.00000 0.330623
\(742\) −12.0000 −0.440534
\(743\) −28.0000 −1.02722 −0.513610 0.858024i \(-0.671692\pi\)
−0.513610 + 0.858024i \(0.671692\pi\)
\(744\) 4.00000 0.146647
\(745\) −6.00000 −0.219823
\(746\) −14.0000 −0.512576
\(747\) 18.0000 0.658586
\(748\) 0 0
\(749\) 18.0000 0.657706
\(750\) 1.00000 0.0365148
\(751\) −47.0000 −1.71505 −0.857527 0.514439i \(-0.828000\pi\)
−0.857527 + 0.514439i \(0.828000\pi\)
\(752\) −12.0000 −0.437595
\(753\) −20.0000 −0.728841
\(754\) 24.0000 0.874028
\(755\) 17.0000 0.618693
\(756\) −5.00000 −0.181848
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 2.00000 0.0726433
\(759\) 0 0
\(760\) −3.00000 −0.108821
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) −17.0000 −0.615845
\(763\) 0 0
\(764\) 25.0000 0.904468
\(765\) 0 0
\(766\) −38.0000 −1.37300
\(767\) 3.00000 0.108324
\(768\) 1.00000 0.0360844
\(769\) −54.0000 −1.94729 −0.973645 0.228069i \(-0.926759\pi\)
−0.973645 + 0.228069i \(0.926759\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) −5.00000 −0.179954
\(773\) 21.0000 0.755318 0.377659 0.925945i \(-0.376729\pi\)
0.377659 + 0.925945i \(0.376729\pi\)
\(774\) 12.0000 0.431331
\(775\) 4.00000 0.143684
\(776\) 8.00000 0.287183
\(777\) −6.00000 −0.215249
\(778\) 4.00000 0.143407
\(779\) −30.0000 −1.07486
\(780\) −3.00000 −0.107417
\(781\) 0 0
\(782\) 0 0
\(783\) 40.0000 1.42948
\(784\) 1.00000 0.0357143
\(785\) 15.0000 0.535373
\(786\) −7.00000 −0.249682
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 4.00000 0.142494
\(789\) 9.00000 0.320408
\(790\) 7.00000 0.249049
\(791\) −1.00000 −0.0355559
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) −22.0000 −0.780751
\(795\) −12.0000 −0.425596
\(796\) −12.0000 −0.425329
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) −3.00000 −0.106199
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −12.0000 −0.423999
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) −7.00000 −0.246718
\(806\) −12.0000 −0.422682
\(807\) 7.00000 0.246412
\(808\) 3.00000 0.105540
\(809\) 42.0000 1.47664 0.738321 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 1.00000 0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −8.00000 −0.280745
\(813\) 16.0000 0.561144
\(814\) 0 0
\(815\) 22.0000 0.770626
\(816\) 0 0
\(817\) 18.0000 0.629740
\(818\) 20.0000 0.699284
\(819\) 6.00000 0.209657
\(820\) 10.0000 0.349215
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) −21.0000 −0.732459
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) −2.00000 −0.0696733
\(825\) 0 0
\(826\) −1.00000 −0.0347945
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 14.0000 0.486534
\(829\) −19.0000 −0.659897 −0.329949 0.943999i \(-0.607031\pi\)
−0.329949 + 0.943999i \(0.607031\pi\)
\(830\) −9.00000 −0.312395
\(831\) 26.0000 0.901930
\(832\) −3.00000 −0.104006
\(833\) 0 0
\(834\) 1.00000 0.0346272
\(835\) −10.0000 −0.346064
\(836\) 0 0
\(837\) −20.0000 −0.691301
\(838\) 7.00000 0.241811
\(839\) 14.0000 0.483334 0.241667 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 1.00000 0.0345033
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −13.0000 −0.447744
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) 24.0000 0.825137
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) 11.0000 0.377519
\(850\) 0 0
\(851\) 42.0000 1.43974
\(852\) 8.00000 0.274075
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 10.0000 0.342193
\(855\) 6.00000 0.205196
\(856\) 18.0000 0.615227
\(857\) 24.0000 0.819824 0.409912 0.912125i \(-0.365559\pi\)
0.409912 + 0.912125i \(0.365559\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −6.00000 −0.204598
\(861\) 10.0000 0.340799
\(862\) 1.00000 0.0340601
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) −5.00000 −0.170103
\(865\) 18.0000 0.612018
\(866\) 14.0000 0.475739
\(867\) −17.0000 −0.577350
\(868\) 4.00000 0.135769
\(869\) 0 0
\(870\) −8.00000 −0.271225
\(871\) 24.0000 0.813209
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) 21.0000 0.710336
\(875\) 1.00000 0.0338062
\(876\) −8.00000 −0.270295
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −22.0000 −0.742464
\(879\) 31.0000 1.04560
\(880\) 0 0
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) −2.00000 −0.0673435
\(883\) 46.0000 1.54802 0.774012 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 0 0
\(885\) −1.00000 −0.0336146
\(886\) 28.0000 0.940678
\(887\) −14.0000 −0.470074 −0.235037 0.971986i \(-0.575521\pi\)
−0.235037 + 0.971986i \(0.575521\pi\)
\(888\) −6.00000 −0.201347
\(889\) −17.0000 −0.570162
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) 0 0
\(893\) 36.0000 1.20469
\(894\) −6.00000 −0.200670
\(895\) −18.0000 −0.601674
\(896\) 1.00000 0.0334077
\(897\) 21.0000 0.701170
\(898\) 41.0000 1.36819
\(899\) −32.0000 −1.06726
\(900\) −2.00000 −0.0666667
\(901\) 0 0
\(902\) 0 0
\(903\) −6.00000 −0.199667
\(904\) −1.00000 −0.0332595
\(905\) −9.00000 −0.299170
\(906\) 17.0000 0.564787
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) −4.00000 −0.132745
\(909\) −6.00000 −0.199007
\(910\) −3.00000 −0.0994490
\(911\) 19.0000 0.629498 0.314749 0.949175i \(-0.398080\pi\)
0.314749 + 0.949175i \(0.398080\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 0 0
\(914\) 31.0000 1.02539
\(915\) 10.0000 0.330590
\(916\) 2.00000 0.0660819
\(917\) −7.00000 −0.231160
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −7.00000 −0.230783
\(921\) −4.00000 −0.131804
\(922\) −34.0000 −1.11973
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −23.0000 −0.755827
\(927\) 4.00000 0.131377
\(928\) −8.00000 −0.262613
\(929\) −36.0000 −1.18112 −0.590561 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) 4.00000 0.131165
\(931\) −3.00000 −0.0983210
\(932\) 3.00000 0.0982683
\(933\) −18.0000 −0.589294
\(934\) 33.0000 1.07979
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) 12.0000 0.392023 0.196011 0.980602i \(-0.437201\pi\)
0.196011 + 0.980602i \(0.437201\pi\)
\(938\) −8.00000 −0.261209
\(939\) −34.0000 −1.10955
\(940\) −12.0000 −0.391397
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 15.0000 0.488726
\(943\) −70.0000 −2.27951
\(944\) −1.00000 −0.0325472
\(945\) −5.00000 −0.162650
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 7.00000 0.227349
\(949\) 24.0000 0.779073
\(950\) −3.00000 −0.0973329
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 39.0000 1.26333 0.631667 0.775240i \(-0.282371\pi\)
0.631667 + 0.775240i \(0.282371\pi\)
\(954\) 24.0000 0.777029
\(955\) 25.0000 0.808981
\(956\) 9.00000 0.291081
\(957\) 0 0
\(958\) −26.0000 −0.840022
\(959\) −21.0000 −0.678125
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 18.0000 0.580343
\(963\) −36.0000 −1.16008
\(964\) 4.00000 0.128831
\(965\) −5.00000 −0.160956
\(966\) −7.00000 −0.225221
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 8.00000 0.256865
\(971\) −43.0000 −1.37994 −0.689968 0.723840i \(-0.742375\pi\)
−0.689968 + 0.723840i \(0.742375\pi\)
\(972\) 16.0000 0.513200
\(973\) 1.00000 0.0320585
\(974\) −37.0000 −1.18556
\(975\) −3.00000 −0.0960769
\(976\) 10.0000 0.320092
\(977\) −9.00000 −0.287936 −0.143968 0.989582i \(-0.545986\pi\)
−0.143968 + 0.989582i \(0.545986\pi\)
\(978\) 22.0000 0.703482
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) −58.0000 −1.84991 −0.924956 0.380073i \(-0.875899\pi\)
−0.924956 + 0.380073i \(0.875899\pi\)
\(984\) 10.0000 0.318788
\(985\) 4.00000 0.127451
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 9.00000 0.286328
\(989\) 42.0000 1.33552
\(990\) 0 0
\(991\) 41.0000 1.30241 0.651204 0.758903i \(-0.274264\pi\)
0.651204 + 0.758903i \(0.274264\pi\)
\(992\) 4.00000 0.127000
\(993\) 10.0000 0.317340
\(994\) 8.00000 0.253745
\(995\) −12.0000 −0.380426
\(996\) −9.00000 −0.285176
\(997\) −45.0000 −1.42516 −0.712582 0.701589i \(-0.752474\pi\)
−0.712582 + 0.701589i \(0.752474\pi\)
\(998\) 14.0000 0.443162
\(999\) 30.0000 0.949158
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.bf.1.1 yes 1
11.10 odd 2 8470.2.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.n.1.1 1 11.10 odd 2
8470.2.a.bf.1.1 yes 1 1.1 even 1 trivial