Properties

Label 1470.2.a.k.1.1
Level 1470
Weight 2
Character 1470.1
Self dual yes
Analytic conductor 11.738
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7380090971\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} -7.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -7.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} +1.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +1.00000 q^{36} -3.00000 q^{37} -1.00000 q^{38} +7.00000 q^{39} -1.00000 q^{40} -9.00000 q^{41} -4.00000 q^{43} -1.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +3.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} -4.00000 q^{51} -7.00000 q^{52} -1.00000 q^{53} -1.00000 q^{54} +1.00000 q^{55} +1.00000 q^{57} -8.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} +4.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +7.00000 q^{65} +1.00000 q^{66} +12.0000 q^{67} +4.00000 q^{68} -1.00000 q^{69} -14.0000 q^{71} +1.00000 q^{72} +14.0000 q^{73} -3.00000 q^{74} -1.00000 q^{75} -1.00000 q^{76} +7.00000 q^{78} +4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.00000 q^{82} -12.0000 q^{83} -4.00000 q^{85} -4.00000 q^{86} +8.00000 q^{87} -1.00000 q^{88} +2.00000 q^{89} -1.00000 q^{90} +1.00000 q^{92} +6.00000 q^{93} +3.00000 q^{94} +1.00000 q^{95} -1.00000 q^{96} +16.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) −1.00000 −0.288675
\(13\) −7.00000 −1.94145 −0.970725 0.240192i \(-0.922790\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 1.00000 0.208514 0.104257 0.994550i \(-0.466753\pi\)
0.104257 + 0.994550i \(0.466753\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −7.00000 −1.37281
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −3.00000 −0.493197 −0.246598 0.969118i \(-0.579313\pi\)
−0.246598 + 0.969118i \(0.579313\pi\)
\(38\) −1.00000 −0.162221
\(39\) 7.00000 1.12090
\(40\) −1.00000 −0.158114
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) −7.00000 −0.970725
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) −8.00000 −1.05045
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 1.00000 0.129099
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 7.00000 0.868243
\(66\) 1.00000 0.123091
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 4.00000 0.485071
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −14.0000 −1.66149 −0.830747 0.556650i \(-0.812086\pi\)
−0.830747 + 0.556650i \(0.812086\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −3.00000 −0.348743
\(75\) −1.00000 −0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 7.00000 0.792594
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −9.00000 −0.993884
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −4.00000 −0.431331
\(87\) 8.00000 0.857690
\(88\) −1.00000 −0.106600
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 6.00000 0.622171
\(94\) 3.00000 0.309426
\(95\) 1.00000 0.102598
\(96\) −1.00000 −0.102062
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 1.00000 0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −4.00000 −0.396059
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −7.00000 −0.686406
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) −18.0000 −1.74013 −0.870063 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 1.00000 0.0953463
\(111\) 3.00000 0.284747
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 1.00000 0.0936586
\(115\) −1.00000 −0.0932505
\(116\) −8.00000 −0.742781
\(117\) −7.00000 −0.647150
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −10.0000 −0.909091
\(122\) 4.00000 0.362143
\(123\) 9.00000 0.811503
\(124\) −6.00000 −0.538816
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) 7.00000 0.613941
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 1.00000 0.0860663
\(136\) 4.00000 0.342997
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) −14.0000 −1.17485
\(143\) 7.00000 0.585369
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −3.00000 −0.246598
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 7.00000 0.560449
\(157\) −15.0000 −1.19713 −0.598565 0.801074i \(-0.704262\pi\)
−0.598565 + 0.801074i \(0.704262\pi\)
\(158\) 4.00000 0.318223
\(159\) 1.00000 0.0793052
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −9.00000 −0.702782
\(165\) −1.00000 −0.0778499
\(166\) −12.0000 −0.931381
\(167\) 5.00000 0.386912 0.193456 0.981109i \(-0.438030\pi\)
0.193456 + 0.981109i \(0.438030\pi\)
\(168\) 0 0
\(169\) 36.0000 2.76923
\(170\) −4.00000 −0.306786
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) 21.0000 1.59660 0.798300 0.602260i \(-0.205733\pi\)
0.798300 + 0.602260i \(0.205733\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 12.0000 0.901975
\(178\) 2.00000 0.149906
\(179\) 13.0000 0.971666 0.485833 0.874052i \(-0.338516\pi\)
0.485833 + 0.874052i \(0.338516\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) −4.00000 −0.295689
\(184\) 1.00000 0.0737210
\(185\) 3.00000 0.220564
\(186\) 6.00000 0.439941
\(187\) −4.00000 −0.292509
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 26.0000 1.87152 0.935760 0.352636i \(-0.114715\pi\)
0.935760 + 0.352636i \(0.114715\pi\)
\(194\) 16.0000 1.14873
\(195\) −7.00000 −0.501280
\(196\) 0 0
\(197\) −3.00000 −0.213741 −0.106871 0.994273i \(-0.534083\pi\)
−0.106871 + 0.994273i \(0.534083\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 9.00000 0.628587
\(206\) −16.0000 −1.11477
\(207\) 1.00000 0.0695048
\(208\) −7.00000 −0.485363
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −15.0000 −1.03264 −0.516321 0.856395i \(-0.672699\pi\)
−0.516321 + 0.856395i \(0.672699\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 14.0000 0.959264
\(214\) −18.0000 −1.23045
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) −14.0000 −0.946032
\(220\) 1.00000 0.0674200
\(221\) −28.0000 −1.88348
\(222\) 3.00000 0.201347
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 1.00000 0.0662266
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −7.00000 −0.457604
\(235\) −3.00000 −0.195698
\(236\) −12.0000 −0.781133
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 1.00000 0.0645497
\(241\) −7.00000 −0.450910 −0.225455 0.974254i \(-0.572387\pi\)
−0.225455 + 0.974254i \(0.572387\pi\)
\(242\) −10.0000 −0.642824
\(243\) −1.00000 −0.0641500
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) 9.00000 0.573819
\(247\) 7.00000 0.445399
\(248\) −6.00000 −0.381000
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) 3.00000 0.189358 0.0946792 0.995508i \(-0.469817\pi\)
0.0946792 + 0.995508i \(0.469817\pi\)
\(252\) 0 0
\(253\) −1.00000 −0.0628695
\(254\) 5.00000 0.313728
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) 7.00000 0.434122
\(261\) −8.00000 −0.495188
\(262\) 13.0000 0.803143
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 1.00000 0.0615457
\(265\) 1.00000 0.0614295
\(266\) 0 0
\(267\) −2.00000 −0.122398
\(268\) 12.0000 0.733017
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 1.00000 0.0608581
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −1.00000 −0.0603023
\(276\) −1.00000 −0.0601929
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 4.00000 0.239904
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) −3.00000 −0.178647
\(283\) 2.00000 0.118888 0.0594438 0.998232i \(-0.481067\pi\)
0.0594438 + 0.998232i \(0.481067\pi\)
\(284\) −14.0000 −0.830747
\(285\) −1.00000 −0.0592349
\(286\) 7.00000 0.413919
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 8.00000 0.469776
\(291\) −16.0000 −0.937937
\(292\) 14.0000 0.819288
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −3.00000 −0.174371
\(297\) 1.00000 0.0580259
\(298\) −4.00000 −0.231714
\(299\) −7.00000 −0.404820
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −2.00000 −0.115087
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) −4.00000 −0.229039
\(306\) 4.00000 0.228665
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 6.00000 0.340777
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 7.00000 0.396297
\(313\) 24.0000 1.35656 0.678280 0.734803i \(-0.262726\pi\)
0.678280 + 0.734803i \(0.262726\pi\)
\(314\) −15.0000 −0.846499
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 10.0000 0.561656 0.280828 0.959758i \(-0.409391\pi\)
0.280828 + 0.959758i \(0.409391\pi\)
\(318\) 1.00000 0.0560772
\(319\) 8.00000 0.447914
\(320\) −1.00000 −0.0559017
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 1.00000 0.0555556
\(325\) −7.00000 −0.388290
\(326\) 8.00000 0.443079
\(327\) 10.0000 0.553001
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) −1.00000 −0.0550482
\(331\) −9.00000 −0.494685 −0.247342 0.968928i \(-0.579557\pi\)
−0.247342 + 0.968928i \(0.579557\pi\)
\(332\) −12.0000 −0.658586
\(333\) −3.00000 −0.164399
\(334\) 5.00000 0.273588
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 36.0000 1.95814
\(339\) 6.00000 0.325875
\(340\) −4.00000 −0.216930
\(341\) 6.00000 0.324918
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 1.00000 0.0538382
\(346\) 21.0000 1.12897
\(347\) −34.0000 −1.82522 −0.912608 0.408836i \(-0.865935\pi\)
−0.912608 + 0.408836i \(0.865935\pi\)
\(348\) 8.00000 0.428845
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 0 0
\(351\) 7.00000 0.373632
\(352\) −1.00000 −0.0533002
\(353\) 8.00000 0.425797 0.212899 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 12.0000 0.637793
\(355\) 14.0000 0.743043
\(356\) 2.00000 0.106000
\(357\) 0 0
\(358\) 13.0000 0.687071
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) 12.0000 0.630706
\(363\) 10.0000 0.524864
\(364\) 0 0
\(365\) −14.0000 −0.732793
\(366\) −4.00000 −0.209083
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 1.00000 0.0521286
\(369\) −9.00000 −0.468521
\(370\) 3.00000 0.155963
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −4.00000 −0.206835
\(375\) 1.00000 0.0516398
\(376\) 3.00000 0.154713
\(377\) 56.0000 2.88415
\(378\) 0 0
\(379\) 1.00000 0.0513665 0.0256833 0.999670i \(-0.491824\pi\)
0.0256833 + 0.999670i \(0.491824\pi\)
\(380\) 1.00000 0.0512989
\(381\) −5.00000 −0.256158
\(382\) −10.0000 −0.511645
\(383\) −13.0000 −0.664269 −0.332134 0.943232i \(-0.607769\pi\)
−0.332134 + 0.943232i \(0.607769\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 26.0000 1.32337
\(387\) −4.00000 −0.203331
\(388\) 16.0000 0.812277
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) −7.00000 −0.354459
\(391\) 4.00000 0.202289
\(392\) 0 0
\(393\) −13.0000 −0.655763
\(394\) −3.00000 −0.151138
\(395\) −4.00000 −0.201262
\(396\) −1.00000 −0.0502519
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) −12.0000 −0.598506
\(403\) 42.0000 2.09217
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 3.00000 0.148704
\(408\) −4.00000 −0.198030
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 9.00000 0.444478
\(411\) 2.00000 0.0986527
\(412\) −16.0000 −0.788263
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 12.0000 0.589057
\(416\) −7.00000 −0.343203
\(417\) −4.00000 −0.195881
\(418\) 1.00000 0.0489116
\(419\) −11.0000 −0.537385 −0.268693 0.963226i \(-0.586592\pi\)
−0.268693 + 0.963226i \(0.586592\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −15.0000 −0.730189
\(423\) 3.00000 0.145865
\(424\) −1.00000 −0.0485643
\(425\) 4.00000 0.194029
\(426\) 14.0000 0.678302
\(427\) 0 0
\(428\) −18.0000 −0.870063
\(429\) −7.00000 −0.337963
\(430\) 4.00000 0.192897
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −40.0000 −1.92228 −0.961139 0.276066i \(-0.910969\pi\)
−0.961139 + 0.276066i \(0.910969\pi\)
\(434\) 0 0
\(435\) −8.00000 −0.383571
\(436\) −10.0000 −0.478913
\(437\) −1.00000 −0.0478365
\(438\) −14.0000 −0.668946
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −28.0000 −1.33182
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 3.00000 0.142374
\(445\) −2.00000 −0.0948091
\(446\) 4.00000 0.189405
\(447\) 4.00000 0.189194
\(448\) 0 0
\(449\) −25.0000 −1.17982 −0.589911 0.807468i \(-0.700837\pi\)
−0.589911 + 0.807468i \(0.700837\pi\)
\(450\) 1.00000 0.0471405
\(451\) 9.00000 0.423793
\(452\) −6.00000 −0.282216
\(453\) 2.00000 0.0939682
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 22.0000 1.02799
\(459\) −4.00000 −0.186704
\(460\) −1.00000 −0.0466252
\(461\) 28.0000 1.30409 0.652045 0.758180i \(-0.273911\pi\)
0.652045 + 0.758180i \(0.273911\pi\)
\(462\) 0 0
\(463\) 33.0000 1.53364 0.766820 0.641862i \(-0.221838\pi\)
0.766820 + 0.641862i \(0.221838\pi\)
\(464\) −8.00000 −0.371391
\(465\) −6.00000 −0.278243
\(466\) 26.0000 1.20443
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −7.00000 −0.323575
\(469\) 0 0
\(470\) −3.00000 −0.138380
\(471\) 15.0000 0.691164
\(472\) −12.0000 −0.552345
\(473\) 4.00000 0.183920
\(474\) −4.00000 −0.183726
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −1.00000 −0.0457869
\(478\) −6.00000 −0.274434
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 1.00000 0.0456435
\(481\) 21.0000 0.957518
\(482\) −7.00000 −0.318841
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −16.0000 −0.726523
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 4.00000 0.181071
\(489\) −8.00000 −0.361773
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 9.00000 0.405751
\(493\) −32.0000 −1.44121
\(494\) 7.00000 0.314945
\(495\) 1.00000 0.0449467
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −5.00000 −0.223384
\(502\) 3.00000 0.133897
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.00000 −0.0444554
\(507\) −36.0000 −1.59882
\(508\) 5.00000 0.221839
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 8.00000 0.352865
\(515\) 16.0000 0.705044
\(516\) 4.00000 0.176090
\(517\) −3.00000 −0.131940
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 7.00000 0.306970
\(521\) 21.0000 0.920027 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) −8.00000 −0.350150
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 13.0000 0.567908
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −24.0000 −1.04546
\(528\) 1.00000 0.0435194
\(529\) −22.0000 −0.956522
\(530\) 1.00000 0.0434372
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 63.0000 2.72883
\(534\) −2.00000 −0.0865485
\(535\) 18.0000 0.778208
\(536\) 12.0000 0.518321
\(537\) −13.0000 −0.560991
\(538\) 0 0
\(539\) 0 0
\(540\) 1.00000 0.0430331
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −16.0000 −0.687259
\(543\) −12.0000 −0.514969
\(544\) 4.00000 0.171499
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 4.00000 0.170716
\(550\) −1.00000 −0.0426401
\(551\) 8.00000 0.340811
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −3.00000 −0.127343
\(556\) 4.00000 0.169638
\(557\) 45.0000 1.90671 0.953356 0.301849i \(-0.0976040\pi\)
0.953356 + 0.301849i \(0.0976040\pi\)
\(558\) −6.00000 −0.254000
\(559\) 28.0000 1.18427
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 3.00000 0.126547
\(563\) −14.0000 −0.590030 −0.295015 0.955493i \(-0.595325\pi\)
−0.295015 + 0.955493i \(0.595325\pi\)
\(564\) −3.00000 −0.126323
\(565\) 6.00000 0.252422
\(566\) 2.00000 0.0840663
\(567\) 0 0
\(568\) −14.0000 −0.587427
\(569\) −37.0000 −1.55112 −0.775560 0.631273i \(-0.782533\pi\)
−0.775560 + 0.631273i \(0.782533\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −8.00000 −0.334790 −0.167395 0.985890i \(-0.553535\pi\)
−0.167395 + 0.985890i \(0.553535\pi\)
\(572\) 7.00000 0.292685
\(573\) 10.0000 0.417756
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −26.0000 −1.08052
\(580\) 8.00000 0.332182
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 1.00000 0.0414158
\(584\) 14.0000 0.579324
\(585\) 7.00000 0.289414
\(586\) −9.00000 −0.371787
\(587\) −42.0000 −1.73353 −0.866763 0.498721i \(-0.833803\pi\)
−0.866763 + 0.498721i \(0.833803\pi\)
\(588\) 0 0
\(589\) 6.00000 0.247226
\(590\) 12.0000 0.494032
\(591\) 3.00000 0.123404
\(592\) −3.00000 −0.123299
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −4.00000 −0.163846
\(597\) −12.0000 −0.491127
\(598\) −7.00000 −0.286251
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −2.00000 −0.0813788
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) −25.0000 −1.01472 −0.507359 0.861735i \(-0.669378\pi\)
−0.507359 + 0.861735i \(0.669378\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −21.0000 −0.849569
\(612\) 4.00000 0.161690
\(613\) −15.0000 −0.605844 −0.302922 0.953015i \(-0.597962\pi\)
−0.302922 + 0.953015i \(0.597962\pi\)
\(614\) −8.00000 −0.322854
\(615\) −9.00000 −0.362915
\(616\) 0 0
\(617\) 8.00000 0.322068 0.161034 0.986949i \(-0.448517\pi\)
0.161034 + 0.986949i \(0.448517\pi\)
\(618\) 16.0000 0.643614
\(619\) 7.00000 0.281354 0.140677 0.990056i \(-0.455072\pi\)
0.140677 + 0.990056i \(0.455072\pi\)
\(620\) 6.00000 0.240966
\(621\) −1.00000 −0.0401286
\(622\) −16.0000 −0.641542
\(623\) 0 0
\(624\) 7.00000 0.280224
\(625\) 1.00000 0.0400000
\(626\) 24.0000 0.959233
\(627\) −1.00000 −0.0399362
\(628\) −15.0000 −0.598565
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 14.0000 0.557331 0.278666 0.960388i \(-0.410108\pi\)
0.278666 + 0.960388i \(0.410108\pi\)
\(632\) 4.00000 0.159111
\(633\) 15.0000 0.596196
\(634\) 10.0000 0.397151
\(635\) −5.00000 −0.198419
\(636\) 1.00000 0.0396526
\(637\) 0 0
\(638\) 8.00000 0.316723
\(639\) −14.0000 −0.553831
\(640\) −1.00000 −0.0395285
\(641\) −23.0000 −0.908445 −0.454223 0.890888i \(-0.650083\pi\)
−0.454223 + 0.890888i \(0.650083\pi\)
\(642\) 18.0000 0.710403
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −4.00000 −0.157378
\(647\) 15.0000 0.589711 0.294855 0.955542i \(-0.404729\pi\)
0.294855 + 0.955542i \(0.404729\pi\)
\(648\) 1.00000 0.0392837
\(649\) 12.0000 0.471041
\(650\) −7.00000 −0.274563
\(651\) 0 0
\(652\) 8.00000 0.313304
\(653\) 29.0000 1.13486 0.567429 0.823422i \(-0.307938\pi\)
0.567429 + 0.823422i \(0.307938\pi\)
\(654\) 10.0000 0.391031
\(655\) −13.0000 −0.507952
\(656\) −9.00000 −0.351391
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −1.00000 −0.0389249
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) −9.00000 −0.349795
\(663\) 28.0000 1.08743
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −3.00000 −0.116248
\(667\) −8.00000 −0.309761
\(668\) 5.00000 0.193456
\(669\) −4.00000 −0.154649
\(670\) −12.0000 −0.463600
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −12.0000 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 36.0000 1.38462
\(677\) −1.00000 −0.0384331 −0.0192166 0.999815i \(-0.506117\pi\)
−0.0192166 + 0.999815i \(0.506117\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) −20.0000 −0.766402
\(682\) 6.00000 0.229752
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) 7.00000 0.266679
\(690\) 1.00000 0.0380693
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) 21.0000 0.798300
\(693\) 0 0
\(694\) −34.0000 −1.29062
\(695\) −4.00000 −0.151729
\(696\) 8.00000 0.303239
\(697\) −36.0000 −1.36360
\(698\) 28.0000 1.05982
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 7.00000 0.264198
\(703\) 3.00000 0.113147
\(704\) −1.00000 −0.0376889
\(705\) 3.00000 0.112987
\(706\) 8.00000 0.301084
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 14.0000 0.525411
\(711\) 4.00000 0.150012
\(712\) 2.00000 0.0749532
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −7.00000 −0.261785
\(716\) 13.0000 0.485833
\(717\) 6.00000 0.224074
\(718\) 36.0000 1.34351
\(719\) −26.0000 −0.969636 −0.484818 0.874615i \(-0.661114\pi\)
−0.484818 + 0.874615i \(0.661114\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −18.0000 −0.669891
\(723\) 7.00000 0.260333
\(724\) 12.0000 0.445976
\(725\) −8.00000 −0.297113
\(726\) 10.0000 0.371135
\(727\) −17.0000 −0.630495 −0.315248 0.949009i \(-0.602088\pi\)
−0.315248 + 0.949009i \(0.602088\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −14.0000 −0.518163
\(731\) −16.0000 −0.591781
\(732\) −4.00000 −0.147844
\(733\) −37.0000 −1.36663 −0.683313 0.730125i \(-0.739462\pi\)
−0.683313 + 0.730125i \(0.739462\pi\)
\(734\) −19.0000 −0.701303
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −12.0000 −0.442026
\(738\) −9.00000 −0.331295
\(739\) 41.0000 1.50821 0.754105 0.656754i \(-0.228071\pi\)
0.754105 + 0.656754i \(0.228071\pi\)
\(740\) 3.00000 0.110282
\(741\) −7.00000 −0.257151
\(742\) 0 0
\(743\) −9.00000 −0.330178 −0.165089 0.986279i \(-0.552791\pi\)
−0.165089 + 0.986279i \(0.552791\pi\)
\(744\) 6.00000 0.219971
\(745\) 4.00000 0.146549
\(746\) 26.0000 0.951928
\(747\) −12.0000 −0.439057
\(748\) −4.00000 −0.146254
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 3.00000 0.109399
\(753\) −3.00000 −0.109326
\(754\) 56.0000 2.03940
\(755\) 2.00000 0.0727875
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 1.00000 0.0363216
\(759\) 1.00000 0.0362977
\(760\) 1.00000 0.0362738
\(761\) 17.0000 0.616250 0.308125 0.951346i \(-0.400299\pi\)
0.308125 + 0.951346i \(0.400299\pi\)
\(762\) −5.00000 −0.181131
\(763\) 0 0
\(764\) −10.0000 −0.361787
\(765\) −4.00000 −0.144620
\(766\) −13.0000 −0.469709
\(767\) 84.0000 3.03306
\(768\) −1.00000 −0.0360844
\(769\) 29.0000 1.04577 0.522883 0.852404i \(-0.324856\pi\)
0.522883 + 0.852404i \(0.324856\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 26.0000 0.935760
\(773\) −43.0000 −1.54660 −0.773301 0.634039i \(-0.781396\pi\)
−0.773301 + 0.634039i \(0.781396\pi\)
\(774\) −4.00000 −0.143777
\(775\) −6.00000 −0.215526
\(776\) 16.0000 0.574367
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) 9.00000 0.322458
\(780\) −7.00000 −0.250640
\(781\) 14.0000 0.500959
\(782\) 4.00000 0.143040
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 15.0000 0.535373
\(786\) −13.0000 −0.463695
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −3.00000 −0.106871
\(789\) 16.0000 0.569615
\(790\) −4.00000 −0.142314
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −28.0000 −0.994309
\(794\) −18.0000 −0.638796
\(795\) −1.00000 −0.0354663
\(796\) 12.0000 0.425329
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 1.00000 0.0353553
\(801\) 2.00000 0.0706665
\(802\) −17.0000 −0.600291
\(803\) −14.0000 −0.494049
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 42.0000 1.47939
\(807\) 0 0
\(808\) 0 0
\(809\) 53.0000 1.86338 0.931690 0.363253i \(-0.118334\pi\)
0.931690 + 0.363253i \(0.118334\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −37.0000 −1.29925 −0.649623 0.760257i \(-0.725073\pi\)
−0.649623 + 0.760257i \(0.725073\pi\)
\(812\) 0 0
\(813\) 16.0000 0.561144
\(814\) 3.00000 0.105150
\(815\) −8.00000 −0.280228
\(816\) −4.00000 −0.140028
\(817\) 4.00000 0.139942
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) 9.00000 0.314294
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 2.00000 0.0697580
\(823\) 48.0000 1.67317 0.836587 0.547833i \(-0.184547\pi\)
0.836587 + 0.547833i \(0.184547\pi\)
\(824\) −16.0000 −0.557386
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 10.0000 0.347734 0.173867 0.984769i \(-0.444374\pi\)
0.173867 + 0.984769i \(0.444374\pi\)
\(828\) 1.00000 0.0347524
\(829\) −12.0000 −0.416777 −0.208389 0.978046i \(-0.566822\pi\)
−0.208389 + 0.978046i \(0.566822\pi\)
\(830\) 12.0000 0.416526
\(831\) 2.00000 0.0693792
\(832\) −7.00000 −0.242681
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) −5.00000 −0.173032
\(836\) 1.00000 0.0345857
\(837\) 6.00000 0.207390
\(838\) −11.0000 −0.379989
\(839\) −44.0000 −1.51905 −0.759524 0.650479i \(-0.774568\pi\)
−0.759524 + 0.650479i \(0.774568\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) −14.0000 −0.482472
\(843\) −3.00000 −0.103325
\(844\) −15.0000 −0.516321
\(845\) −36.0000 −1.23844
\(846\) 3.00000 0.103142
\(847\) 0 0
\(848\) −1.00000 −0.0343401
\(849\) −2.00000 −0.0686398
\(850\) 4.00000 0.137199
\(851\) −3.00000 −0.102839
\(852\) 14.0000 0.479632
\(853\) −1.00000 −0.0342393 −0.0171197 0.999853i \(-0.505450\pi\)
−0.0171197 + 0.999853i \(0.505450\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) −18.0000 −0.615227
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) −7.00000 −0.238976
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 29.0000 0.987171 0.493586 0.869697i \(-0.335686\pi\)
0.493586 + 0.869697i \(0.335686\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −21.0000 −0.714021
\(866\) −40.0000 −1.35926
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) −8.00000 −0.271225
\(871\) −84.0000 −2.84623
\(872\) −10.0000 −0.338643
\(873\) 16.0000 0.541518
\(874\) −1.00000 −0.0338255
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) −16.0000 −0.539974
\(879\) 9.00000 0.303562
\(880\) 1.00000 0.0337100
\(881\) 25.0000 0.842271 0.421136 0.906998i \(-0.361632\pi\)
0.421136 + 0.906998i \(0.361632\pi\)
\(882\) 0 0
\(883\) −58.0000 −1.95186 −0.975928 0.218094i \(-0.930016\pi\)
−0.975928 + 0.218094i \(0.930016\pi\)
\(884\) −28.0000 −0.941742
\(885\) −12.0000 −0.403376
\(886\) −36.0000 −1.20944
\(887\) −20.0000 −0.671534 −0.335767 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 3.00000 0.100673
\(889\) 0 0
\(890\) −2.00000 −0.0670402
\(891\) −1.00000 −0.0335013
\(892\) 4.00000 0.133930
\(893\) −3.00000 −0.100391
\(894\) 4.00000 0.133780
\(895\) −13.0000 −0.434542
\(896\) 0 0
\(897\) 7.00000 0.233723
\(898\) −25.0000 −0.834261
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) 9.00000 0.299667
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) −12.0000 −0.398893
\(906\) 2.00000 0.0664455
\(907\) 46.0000 1.52740 0.763702 0.645568i \(-0.223379\pi\)
0.763702 + 0.645568i \(0.223379\pi\)
\(908\) 20.0000 0.663723
\(909\) 0 0
\(910\) 0 0
\(911\) −58.0000 −1.92163 −0.960813 0.277198i \(-0.910594\pi\)
−0.960813 + 0.277198i \(0.910594\pi\)
\(912\) 1.00000 0.0331133
\(913\) 12.0000 0.397142
\(914\) 10.0000 0.330771
\(915\) 4.00000 0.132236
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −4.00000 −0.132020
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 8.00000 0.263609
\(922\) 28.0000 0.922131
\(923\) 98.0000 3.22571
\(924\) 0 0
\(925\) −3.00000 −0.0986394
\(926\) 33.0000 1.08445
\(927\) −16.0000 −0.525509
\(928\) −8.00000 −0.262613
\(929\) −31.0000 −1.01708 −0.508539 0.861039i \(-0.669814\pi\)
−0.508539 + 0.861039i \(0.669814\pi\)
\(930\) −6.00000 −0.196748
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) 16.0000 0.523816
\(934\) −12.0000 −0.392652
\(935\) 4.00000 0.130814
\(936\) −7.00000 −0.228802
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) −3.00000 −0.0978492
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 15.0000 0.488726
\(943\) −9.00000 −0.293080
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 46.0000 1.49480 0.747400 0.664375i \(-0.231302\pi\)
0.747400 + 0.664375i \(0.231302\pi\)
\(948\) −4.00000 −0.129914
\(949\) −98.0000 −3.18121
\(950\) −1.00000 −0.0324443
\(951\) −10.0000 −0.324272
\(952\) 0 0
\(953\) 44.0000 1.42530 0.712650 0.701520i \(-0.247495\pi\)
0.712650 + 0.701520i \(0.247495\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 10.0000 0.323592
\(956\) −6.00000 −0.194054
\(957\) −8.00000 −0.258603
\(958\) −26.0000 −0.840022
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) 5.00000 0.161290
\(962\) 21.0000 0.677067
\(963\) −18.0000 −0.580042
\(964\) −7.00000 −0.225455
\(965\) −26.0000 −0.836970
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) −10.0000 −0.321412
\(969\) 4.00000 0.128499
\(970\) −16.0000 −0.513729
\(971\) −43.0000 −1.37994 −0.689968 0.723840i \(-0.742375\pi\)
−0.689968 + 0.723840i \(0.742375\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 7.00000 0.224179
\(976\) 4.00000 0.128037
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) −8.00000 −0.255812
\(979\) −2.00000 −0.0639203
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −12.0000 −0.382935
\(983\) 33.0000 1.05254 0.526268 0.850319i \(-0.323591\pi\)
0.526268 + 0.850319i \(0.323591\pi\)
\(984\) 9.00000 0.286910
\(985\) 3.00000 0.0955879
\(986\) −32.0000 −1.01909
\(987\) 0 0
\(988\) 7.00000 0.222700
\(989\) −4.00000 −0.127193
\(990\) 1.00000 0.0317821
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) −6.00000 −0.190500
\(993\) 9.00000 0.285606
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 12.0000 0.380235
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 24.0000 0.759707
\(999\) 3.00000 0.0949158
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 1470.2.a.k.1.1 1
3.2 odd 2 4410.2.a.q.1.1 1
5.4 even 2 7350.2.a.ba.1.1 1
7.2 even 3 1470.2.i.i.361.1 2
7.3 odd 6 210.2.i.a.121.1 2
7.4 even 3 1470.2.i.i.961.1 2
7.5 odd 6 210.2.i.a.151.1 yes 2
7.6 odd 2 1470.2.a.r.1.1 1
21.5 even 6 630.2.k.h.361.1 2
21.17 even 6 630.2.k.h.541.1 2
21.20 even 2 4410.2.a.g.1.1 1
28.3 even 6 1680.2.bg.k.961.1 2
28.19 even 6 1680.2.bg.k.1201.1 2
35.3 even 12 1050.2.o.j.499.2 4
35.12 even 12 1050.2.o.j.949.2 4
35.17 even 12 1050.2.o.j.499.1 4
35.19 odd 6 1050.2.i.s.151.1 2
35.24 odd 6 1050.2.i.s.751.1 2
35.33 even 12 1050.2.o.j.949.1 4
35.34 odd 2 7350.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.2.i.a.121.1 2 7.3 odd 6
210.2.i.a.151.1 yes 2 7.5 odd 6
630.2.k.h.361.1 2 21.5 even 6
630.2.k.h.541.1 2 21.17 even 6
1050.2.i.s.151.1 2 35.19 odd 6
1050.2.i.s.751.1 2 35.24 odd 6
1050.2.o.j.499.1 4 35.17 even 12
1050.2.o.j.499.2 4 35.3 even 12
1050.2.o.j.949.1 4 35.33 even 12
1050.2.o.j.949.2 4 35.12 even 12
1470.2.a.k.1.1 1 1.1 even 1 trivial
1470.2.a.r.1.1 1 7.6 odd 2
1470.2.i.i.361.1 2 7.2 even 3
1470.2.i.i.961.1 2 7.4 even 3
1680.2.bg.k.961.1 2 28.3 even 6
1680.2.bg.k.1201.1 2 28.19 even 6
4410.2.a.g.1.1 1 21.20 even 2
4410.2.a.q.1.1 1 3.2 odd 2
7350.2.a.j.1.1 1 35.34 odd 2
7350.2.a.ba.1.1 1 5.4 even 2