Properties

Label 570.2.a.j.1.1
Level $570$
Weight $2$
Character 570.1
Self dual yes
Analytic conductor $4.551$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +6.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -10.0000 q^{29} -1.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +4.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +1.00000 q^{38} +6.00000 q^{39} -1.00000 q^{40} +8.00000 q^{41} +2.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +4.00000 q^{51} +6.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} +1.00000 q^{57} -10.0000 q^{58} -2.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -2.00000 q^{70} +1.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} -8.00000 q^{77} +6.00000 q^{78} -2.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +8.00000 q^{82} -10.0000 q^{83} +2.00000 q^{84} -4.00000 q^{85} -8.00000 q^{86} -10.0000 q^{87} -4.00000 q^{88} -12.0000 q^{89} -1.00000 q^{90} +12.0000 q^{91} -2.00000 q^{93} -1.00000 q^{95} +1.00000 q^{96} -2.00000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 2.00000 0.534522
\(15\) −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
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) −4.00000 −0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −1.00000 −0.182574
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 4.00000 0.685994
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 1.00000 0.162221
\(39\) 6.00000 0.960769
\(40\) −1.00000 −0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 2.00000 0.308607
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) 4.00000 0.560112
\(52\) 6.00000 0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 2.00000 0.267261
\(57\) 1.00000 0.132453
\(58\) −10.0000 −1.31306
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −2.00000 −0.254000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 1.00000 0.114708
\(77\) −8.00000 −0.911685
\(78\) 6.00000 0.679366
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 8.00000 0.883452
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 2.00000 0.218218
\(85\) −4.00000 −0.433861
\(86\) −8.00000 −0.862662
\(87\) −10.0000 −1.07211
\(88\) −4.00000 −0.426401
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) −1.00000 −0.105409
\(91\) 12.0000 1.25794
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −3.00000 −0.303046
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 4.00000 0.396059
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 6.00000 0.588348
\(105\) −2.00000 −0.195180
\(106\) −6.00000 −0.582772
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 4.00000 0.381385
\(111\) −2.00000 −0.189832
\(112\) 2.00000 0.188982
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −10.0000 −0.928477
\(117\) 6.00000 0.554700
\(118\) −2.00000 −0.184115
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 8.00000 0.721336
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) −6.00000 −0.526235
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 2.00000 0.173422
\(134\) 4.00000 0.345547
\(135\) −1.00000 −0.0860663
\(136\) 4.00000 0.342997
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) 10.0000 0.830455
\(146\) −10.0000 −0.827606
\(147\) −3.00000 −0.247436
\(148\) −2.00000 −0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) 18.0000 1.46482 0.732410 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 1.00000 0.0811107
\(153\) 4.00000 0.323381
\(154\) −8.00000 −0.644658
\(155\) 2.00000 0.160644
\(156\) 6.00000 0.480384
\(157\) 20.0000 1.59617 0.798087 0.602542i \(-0.205846\pi\)
0.798087 + 0.602542i \(0.205846\pi\)
\(158\) −2.00000 −0.159111
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 8.00000 0.624695
\(165\) 4.00000 0.311400
\(166\) −10.0000 −0.776151
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 2.00000 0.154303
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 1.00000 0.0764719
\(172\) −8.00000 −0.609994
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −10.0000 −0.758098
\(175\) 2.00000 0.151186
\(176\) −4.00000 −0.301511
\(177\) −2.00000 −0.150329
\(178\) −12.0000 −0.899438
\(179\) −22.0000 −1.64436 −0.822179 0.569230i \(-0.807242\pi\)
−0.822179 + 0.569230i \(0.807242\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 12.0000 0.889499
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) −2.00000 −0.146647
\(187\) −16.0000 −1.17004
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) −1.00000 −0.0725476
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −2.00000 −0.143592
\(195\) −6.00000 −0.429669
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) −4.00000 −0.284268
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) −20.0000 −1.40372
\(204\) 4.00000 0.280056
\(205\) −8.00000 −0.558744
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −4.00000 −0.276686
\(210\) −2.00000 −0.138013
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 8.00000 0.546869
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) 12.0000 0.812743
\(219\) −10.0000 −0.675737
\(220\) 4.00000 0.269680
\(221\) 24.0000 1.61441
\(222\) −2.00000 −0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) 14.0000 0.931266
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 1.00000 0.0662266
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) −10.0000 −0.656532
\(233\) −4.00000 −0.262049 −0.131024 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −2.00000 −0.130189
\(237\) −2.00000 −0.129914
\(238\) 8.00000 0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 5.00000 0.321412
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) 8.00000 0.510061
\(247\) 6.00000 0.381771
\(248\) −2.00000 −0.127000
\(249\) −10.0000 −0.633724
\(250\) −1.00000 −0.0632456
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) −20.0000 −1.25491
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −8.00000 −0.498058
\(259\) −4.00000 −0.248548
\(260\) −6.00000 −0.372104
\(261\) −10.0000 −0.618984
\(262\) −12.0000 −0.741362
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) −4.00000 −0.246183
\(265\) 6.00000 0.368577
\(266\) 2.00000 0.122628
\(267\) −12.0000 −0.734388
\(268\) 4.00000 0.244339
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) −1.00000 −0.0608581
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 4.00000 0.242536
\(273\) 12.0000 0.726273
\(274\) 16.0000 0.966595
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) 20.0000 1.20168 0.600842 0.799368i \(-0.294832\pi\)
0.600842 + 0.799368i \(0.294832\pi\)
\(278\) −12.0000 −0.719712
\(279\) −2.00000 −0.119737
\(280\) −2.00000 −0.119523
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) −24.0000 −1.41915
\(287\) 16.0000 0.944450
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 10.0000 0.587220
\(291\) −2.00000 −0.117242
\(292\) −10.0000 −0.585206
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) −3.00000 −0.174964
\(295\) 2.00000 0.116445
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −16.0000 −0.922225
\(302\) 18.0000 1.03578
\(303\) 6.00000 0.344691
\(304\) 1.00000 0.0573539
\(305\) −2.00000 −0.114520
\(306\) 4.00000 0.228665
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −8.00000 −0.455842
\(309\) 4.00000 0.227552
\(310\) 2.00000 0.113592
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 6.00000 0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 20.0000 1.12867
\(315\) −2.00000 −0.112687
\(316\) −2.00000 −0.112509
\(317\) 30.0000 1.68497 0.842484 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) −6.00000 −0.336463
\(319\) 40.0000 2.23957
\(320\) −1.00000 −0.0559017
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 6.00000 0.332820
\(326\) −20.0000 −1.10770
\(327\) 12.0000 0.663602
\(328\) 8.00000 0.441726
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) −10.0000 −0.548821
\(333\) −2.00000 −0.109599
\(334\) −16.0000 −0.875481
\(335\) −4.00000 −0.218543
\(336\) 2.00000 0.109109
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 23.0000 1.25104
\(339\) 14.0000 0.760376
\(340\) −4.00000 −0.216930
\(341\) 8.00000 0.433224
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) −6.00000 −0.322097 −0.161048 0.986947i \(-0.551488\pi\)
−0.161048 + 0.986947i \(0.551488\pi\)
\(348\) −10.0000 −0.536056
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 2.00000 0.106904
\(351\) 6.00000 0.320256
\(352\) −4.00000 −0.213201
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −2.00000 −0.106299
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) 8.00000 0.423405
\(358\) −22.0000 −1.16274
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 1.00000 0.0526316
\(362\) 12.0000 0.630706
\(363\) 5.00000 0.262432
\(364\) 12.0000 0.628971
\(365\) 10.0000 0.523424
\(366\) 2.00000 0.104542
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 0 0
\(369\) 8.00000 0.416463
\(370\) 2.00000 0.103975
\(371\) −12.0000 −0.623009
\(372\) −2.00000 −0.103695
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −16.0000 −0.827340
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −60.0000 −3.09016
\(378\) 2.00000 0.102869
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −1.00000 −0.0512989
\(381\) −20.0000 −1.02463
\(382\) 16.0000 0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 8.00000 0.407718
\(386\) 10.0000 0.508987
\(387\) −8.00000 −0.406663
\(388\) −2.00000 −0.101535
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) −6.00000 −0.303822
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) −12.0000 −0.605320
\(394\) 6.00000 0.302276
\(395\) 2.00000 0.100631
\(396\) −4.00000 −0.201008
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 20.0000 1.00251
\(399\) 2.00000 0.100125
\(400\) 1.00000 0.0500000
\(401\) −20.0000 −0.998752 −0.499376 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 4.00000 0.199502
\(403\) −12.0000 −0.597763
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) −20.0000 −0.992583
\(407\) 8.00000 0.396545
\(408\) 4.00000 0.198030
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) −8.00000 −0.395092
\(411\) 16.0000 0.789222
\(412\) 4.00000 0.197066
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 10.0000 0.490881
\(416\) 6.00000 0.294174
\(417\) −12.0000 −0.587643
\(418\) −4.00000 −0.195646
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 16.0000 0.778868
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 8.00000 0.386695
\(429\) −24.0000 −1.15873
\(430\) 8.00000 0.385794
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) −4.00000 −0.192006
\(435\) 10.0000 0.479463
\(436\) 12.0000 0.574696
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) −2.00000 −0.0954548 −0.0477274 0.998860i \(-0.515198\pi\)
−0.0477274 + 0.998860i \(0.515198\pi\)
\(440\) 4.00000 0.190693
\(441\) −3.00000 −0.142857
\(442\) 24.0000 1.14156
\(443\) −6.00000 −0.285069 −0.142534 0.989790i \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 12.0000 0.568855
\(446\) −16.0000 −0.757622
\(447\) −18.0000 −0.851371
\(448\) 2.00000 0.0944911
\(449\) −8.00000 −0.377543 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(450\) 1.00000 0.0471405
\(451\) −32.0000 −1.50682
\(452\) 14.0000 0.658505
\(453\) 18.0000 0.845714
\(454\) 12.0000 0.563188
\(455\) −12.0000 −0.562569
\(456\) 1.00000 0.0468293
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 22.0000 1.02799
\(459\) 4.00000 0.186704
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −8.00000 −0.372194
\(463\) 34.0000 1.58011 0.790057 0.613033i \(-0.210051\pi\)
0.790057 + 0.613033i \(0.210051\pi\)
\(464\) −10.0000 −0.464238
\(465\) 2.00000 0.0927478
\(466\) −4.00000 −0.185296
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 6.00000 0.277350
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) −2.00000 −0.0920575
\(473\) 32.0000 1.47136
\(474\) −2.00000 −0.0918630
\(475\) 1.00000 0.0458831
\(476\) 8.00000 0.366679
\(477\) −6.00000 −0.274721
\(478\) 24.0000 1.09773
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −12.0000 −0.547153
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) 4.00000 0.181257 0.0906287 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 2.00000 0.0905357
\(489\) −20.0000 −0.904431
\(490\) 3.00000 0.135526
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 8.00000 0.360668
\(493\) −40.0000 −1.80151
\(494\) 6.00000 0.269953
\(495\) 4.00000 0.179787
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −10.0000 −0.448111
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −16.0000 −0.714827
\(502\) 16.0000 0.714115
\(503\) −28.0000 −1.24846 −0.624229 0.781241i \(-0.714587\pi\)
−0.624229 + 0.781241i \(0.714587\pi\)
\(504\) 2.00000 0.0890871
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 23.0000 1.02147
\(508\) −20.0000 −0.887357
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) −4.00000 −0.177123
\(511\) −20.0000 −0.884748
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −6.00000 −0.264649
\(515\) −4.00000 −0.176261
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) −4.00000 −0.175750
\(519\) −14.0000 −0.614532
\(520\) −6.00000 −0.263117
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) −10.0000 −0.437688
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) −12.0000 −0.524222
\(525\) 2.00000 0.0872872
\(526\) 8.00000 0.348817
\(527\) −8.00000 −0.348485
\(528\) −4.00000 −0.174078
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −2.00000 −0.0867926
\(532\) 2.00000 0.0867110
\(533\) 48.0000 2.07911
\(534\) −12.0000 −0.519291
\(535\) −8.00000 −0.345870
\(536\) 4.00000 0.172774
\(537\) −22.0000 −0.949370
\(538\) −26.0000 −1.12094
\(539\) 12.0000 0.516877
\(540\) −1.00000 −0.0430331
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −20.0000 −0.859074
\(543\) 12.0000 0.514969
\(544\) 4.00000 0.171499
\(545\) −12.0000 −0.514024
\(546\) 12.0000 0.513553
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) 16.0000 0.683486
\(549\) 2.00000 0.0853579
\(550\) −4.00000 −0.170561
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 20.0000 0.849719
\(555\) 2.00000 0.0848953
\(556\) −12.0000 −0.508913
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −2.00000 −0.0846668
\(559\) −48.0000 −2.03018
\(560\) −2.00000 −0.0845154
\(561\) −16.0000 −0.675521
\(562\) 24.0000 1.01238
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 0 0
\(565\) −14.0000 −0.588984
\(566\) −16.0000 −0.672530
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) −24.0000 −1.00349
\(573\) 16.0000 0.668410
\(574\) 16.0000 0.667827
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 10.0000 0.415586
\(580\) 10.0000 0.415227
\(581\) −20.0000 −0.829740
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) −10.0000 −0.413803
\(585\) −6.00000 −0.248069
\(586\) 22.0000 0.908812
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) −3.00000 −0.123718
\(589\) −2.00000 −0.0824086
\(590\) 2.00000 0.0823387
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) 16.0000 0.657041 0.328521 0.944497i \(-0.393450\pi\)
0.328521 + 0.944497i \(0.393450\pi\)
\(594\) −4.00000 −0.164122
\(595\) −8.00000 −0.327968
\(596\) −18.0000 −0.737309
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 1.00000 0.0408248
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) −16.0000 −0.652111
\(603\) 4.00000 0.162893
\(604\) 18.0000 0.732410
\(605\) −5.00000 −0.203279
\(606\) 6.00000 0.243733
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 1.00000 0.0405554
\(609\) −20.0000 −0.810441
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 8.00000 0.323117 0.161558 0.986863i \(-0.448348\pi\)
0.161558 + 0.986863i \(0.448348\pi\)
\(614\) 4.00000 0.161427
\(615\) −8.00000 −0.322591
\(616\) −8.00000 −0.322329
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 4.00000 0.160904
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 32.0000 1.28308
\(623\) −24.0000 −0.961540
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) 6.00000 0.239808
\(627\) −4.00000 −0.159745
\(628\) 20.0000 0.798087
\(629\) −8.00000 −0.318981
\(630\) −2.00000 −0.0796819
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 16.0000 0.635943
\(634\) 30.0000 1.19145
\(635\) 20.0000 0.793676
\(636\) −6.00000 −0.237915
\(637\) −18.0000 −0.713186
\(638\) 40.0000 1.58362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −48.0000 −1.89589 −0.947943 0.318440i \(-0.896841\pi\)
−0.947943 + 0.318440i \(0.896841\pi\)
\(642\) 8.00000 0.315735
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 4.00000 0.157378
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 1.00000 0.0392837
\(649\) 8.00000 0.314027
\(650\) 6.00000 0.235339
\(651\) −4.00000 −0.156772
\(652\) −20.0000 −0.783260
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 12.0000 0.469237
\(655\) 12.0000 0.468879
\(656\) 8.00000 0.312348
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 4.00000 0.155700
\(661\) −20.0000 −0.777910 −0.388955 0.921257i \(-0.627164\pi\)
−0.388955 + 0.921257i \(0.627164\pi\)
\(662\) −28.0000 −1.08825
\(663\) 24.0000 0.932083
\(664\) −10.0000 −0.388075
\(665\) −2.00000 −0.0775567
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) −16.0000 −0.619059
\(669\) −16.0000 −0.618596
\(670\) −4.00000 −0.154533
\(671\) −8.00000 −0.308837
\(672\) 2.00000 0.0771517
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 14.0000 0.537667
\(679\) −4.00000 −0.153506
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 8.00000 0.306336
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 1.00000 0.0382360
\(685\) −16.0000 −0.611329
\(686\) −20.0000 −0.763604
\(687\) 22.0000 0.839352
\(688\) −8.00000 −0.304997
\(689\) −36.0000 −1.37149
\(690\) 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
\(693\) −8.00000 −0.303895
\(694\) −6.00000 −0.227757
\(695\) 12.0000 0.455186
\(696\) −10.0000 −0.379049
\(697\) 32.0000 1.21209
\(698\) 14.0000 0.529908
\(699\) −4.00000 −0.151294
\(700\) 2.00000 0.0755929
\(701\) 10.0000 0.377695 0.188847 0.982006i \(-0.439525\pi\)
0.188847 + 0.982006i \(0.439525\pi\)
\(702\) 6.00000 0.226455
\(703\) −2.00000 −0.0754314
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) −2.00000 −0.0751646
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) −12.0000 −0.449719
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 24.0000 0.897549
\(716\) −22.0000 −0.822179
\(717\) 24.0000 0.896296
\(718\) 16.0000 0.597115
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 1.00000 0.0372161
\(723\) 2.00000 0.0743808
\(724\) 12.0000 0.445976
\(725\) −10.0000 −0.371391
\(726\) 5.00000 0.185567
\(727\) 30.0000 1.11264 0.556319 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 12.0000 0.444750
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −32.0000 −1.18356
\(732\) 2.00000 0.0739221
\(733\) 8.00000 0.295487 0.147743 0.989026i \(-0.452799\pi\)
0.147743 + 0.989026i \(0.452799\pi\)
\(734\) −34.0000 −1.25496
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 8.00000 0.294484
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 2.00000 0.0735215
\(741\) 6.00000 0.220416
\(742\) −12.0000 −0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −2.00000 −0.0733236
\(745\) 18.0000 0.659469
\(746\) 26.0000 0.951928
\(747\) −10.0000 −0.365881
\(748\) −16.0000 −0.585018
\(749\) 16.0000 0.584627
\(750\) −1.00000 −0.0365148
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) 0 0
\(753\) 16.0000 0.583072
\(754\) −60.0000 −2.18507
\(755\) −18.0000 −0.655087
\(756\) 2.00000 0.0727393
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) −20.0000 −0.724524
\(763\) 24.0000 0.868858
\(764\) 16.0000 0.578860
\(765\) −4.00000 −0.144620
\(766\) 24.0000 0.867155
\(767\) −12.0000 −0.433295
\(768\) 1.00000 0.0360844
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 8.00000 0.288300
\(771\) −6.00000 −0.216085
\(772\) 10.0000 0.359908
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) −8.00000 −0.287554
\(775\) −2.00000 −0.0718421
\(776\) −2.00000 −0.0717958
\(777\) −4.00000 −0.143499
\(778\) 38.0000 1.36237
\(779\) 8.00000 0.286630
\(780\) −6.00000 −0.214834
\(781\) 0 0
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) −3.00000 −0.107143
\(785\) −20.0000 −0.713831
\(786\) −12.0000 −0.428026
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 6.00000 0.213741
\(789\) 8.00000 0.284808
\(790\) 2.00000 0.0711568
\(791\) 28.0000 0.995565
\(792\) −4.00000 −0.142134
\(793\) 12.0000 0.426132
\(794\) −16.0000 −0.567819
\(795\) 6.00000 0.212798
\(796\) 20.0000 0.708881
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 2.00000 0.0707992
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −12.0000 −0.423999
\(802\) −20.0000 −0.706225
\(803\) 40.0000 1.41157
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −12.0000 −0.422682
\(807\) −26.0000 −0.915243
\(808\) 6.00000 0.211079
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) −20.0000 −0.701862
\(813\) −20.0000 −0.701431
\(814\) 8.00000 0.280400
\(815\) 20.0000 0.700569
\(816\) 4.00000 0.140028
\(817\) −8.00000 −0.279885
\(818\) −6.00000 −0.209785
\(819\) 12.0000 0.419314
\(820\) −8.00000 −0.279372
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 16.0000 0.558064
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 4.00000 0.139347
\(825\) −4.00000 −0.139262
\(826\) −4.00000 −0.139178
\(827\) 32.0000 1.11275 0.556375 0.830932i \(-0.312192\pi\)
0.556375 + 0.830932i \(0.312192\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 10.0000 0.347105
\(831\) 20.0000 0.693792
\(832\) 6.00000 0.208013
\(833\) −12.0000 −0.415775
\(834\) −12.0000 −0.415526
\(835\) 16.0000 0.553703
\(836\) −4.00000 −0.138343
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 71.0000 2.44828
\(842\) −20.0000 −0.689246
\(843\) 24.0000 0.826604
\(844\) 16.0000 0.550743
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −6.00000 −0.206041
\(849\) −16.0000 −0.549119
\(850\) 4.00000 0.137199
\(851\) 0 0
\(852\) 0 0
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 4.00000 0.136877
\(855\) −1.00000 −0.0341993
\(856\) 8.00000 0.273434
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) −24.0000 −0.819346
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 8.00000 0.272798
\(861\) 16.0000 0.545279
\(862\) −8.00000 −0.272481
\(863\) −56.0000 −1.90626 −0.953131 0.302558i \(-0.902160\pi\)
−0.953131 + 0.302558i \(0.902160\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) 26.0000 0.883516
\(867\) −1.00000 −0.0339618
\(868\) −4.00000 −0.135769
\(869\) 8.00000 0.271381
\(870\) 10.0000 0.339032
\(871\) 24.0000 0.813209
\(872\) 12.0000 0.406371
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) −10.0000 −0.337869
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) −2.00000 −0.0674967
\(879\) 22.0000 0.742042
\(880\) 4.00000 0.134840
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) −3.00000 −0.101015
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 24.0000 0.807207
\(885\) 2.00000 0.0672293
\(886\) −6.00000 −0.201574
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −40.0000 −1.34156
\(890\) 12.0000 0.402241
\(891\) −4.00000 −0.134005
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 22.0000 0.735379
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 20.0000 0.667037
\(900\) 1.00000 0.0333333
\(901\) −24.0000 −0.799556
\(902\) −32.0000 −1.06548
\(903\) −16.0000 −0.532447
\(904\) 14.0000 0.465633
\(905\) −12.0000 −0.398893
\(906\) 18.0000 0.598010
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) −12.0000 −0.397796
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 1.00000 0.0331133
\(913\) 40.0000 1.32381
\(914\) −22.0000 −0.727695
\(915\) −2.00000 −0.0661180
\(916\) 22.0000 0.726900
\(917\) −24.0000 −0.792550
\(918\) 4.00000 0.132020
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 18.0000 0.592798
\(923\) 0 0
\(924\) −8.00000 −0.263181
\(925\) −2.00000 −0.0657596
\(926\) 34.0000 1.11731
\(927\) 4.00000 0.131377
\(928\) −10.0000 −0.328266
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 2.00000 0.0655826
\(931\) −3.00000 −0.0983210
\(932\) −4.00000 −0.131024
\(933\) 32.0000 1.04763
\(934\) 6.00000 0.196326
\(935\) 16.0000 0.523256
\(936\) 6.00000 0.196116
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 8.00000 0.261209
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 26.0000 0.847576 0.423788 0.905761i \(-0.360700\pi\)
0.423788 + 0.905761i \(0.360700\pi\)
\(942\) 20.0000 0.651635
\(943\) 0 0
\(944\) −2.00000 −0.0650945
\(945\) −2.00000 −0.0650600
\(946\) 32.0000 1.04041
\(947\) −10.0000 −0.324956 −0.162478 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) −2.00000 −0.0649570
\(949\) −60.0000 −1.94768
\(950\) 1.00000 0.0324443
\(951\) 30.0000 0.972817
\(952\) 8.00000 0.259281
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −6.00000 −0.194257
\(955\) −16.0000 −0.517748
\(956\) 24.0000 0.776215
\(957\) 40.0000 1.29302
\(958\) 0 0
\(959\) 32.0000 1.03333
\(960\) −1.00000 −0.0322749
\(961\) −27.0000 −0.870968
\(962\) −12.0000 −0.386896
\(963\) 8.00000 0.257796
\(964\) 2.00000 0.0644157
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) 26.0000 0.836104 0.418052 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 5.00000 0.160706
\(969\) 4.00000 0.128499
\(970\) 2.00000 0.0642161
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 1.00000 0.0320750
\(973\) −24.0000 −0.769405
\(974\) 4.00000 0.128168
\(975\) 6.00000 0.192154
\(976\) 2.00000 0.0640184
\(977\) 50.0000 1.59964 0.799821 0.600239i \(-0.204928\pi\)
0.799821 + 0.600239i \(0.204928\pi\)
\(978\) −20.0000 −0.639529
\(979\) 48.0000 1.53409
\(980\) 3.00000 0.0958315
\(981\) 12.0000 0.383131
\(982\) 4.00000 0.127645
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 8.00000 0.255031
\(985\) −6.00000 −0.191176
\(986\) −40.0000 −1.27386
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) 0 0
\(990\) 4.00000 0.127128
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) −2.00000 −0.0635001
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) −10.0000 −0.316862
\(997\) 20.0000 0.633406 0.316703 0.948525i \(-0.397424\pi\)
0.316703 + 0.948525i \(0.397424\pi\)
\(998\) −28.0000 −0.886325
\(999\) −2.00000 −0.0632772
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 570.2.a.j.1.1 1
3.2 odd 2 1710.2.a.k.1.1 1
4.3 odd 2 4560.2.a.d.1.1 1
5.2 odd 4 2850.2.d.l.799.2 2
5.3 odd 4 2850.2.d.l.799.1 2
5.4 even 2 2850.2.a.b.1.1 1
15.14 odd 2 8550.2.a.w.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.j.1.1 1 1.1 even 1 trivial
1710.2.a.k.1.1 1 3.2 odd 2
2850.2.a.b.1.1 1 5.4 even 2
2850.2.d.l.799.1 2 5.3 odd 4
2850.2.d.l.799.2 2 5.2 odd 4
4560.2.a.d.1.1 1 4.3 odd 2
8550.2.a.w.1.1 1 15.14 odd 2