Properties

Label 570.2.a.l.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -1.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -8.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -2.00000 q^{56} -1.00000 q^{57} +1.00000 q^{60} +2.00000 q^{61} -8.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{66} +8.00000 q^{67} -2.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +14.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} -1.00000 q^{76} -4.00000 q^{77} +4.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -2.00000 q^{85} -6.00000 q^{86} +2.00000 q^{88} +1.00000 q^{90} -8.00000 q^{91} +4.00000 q^{92} -8.00000 q^{93} -12.0000 q^{94} -1.00000 q^{95} +1.00000 q^{96} -12.0000 q^{97} -3.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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 \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 1.00000 0.223607
\(21\) −2.00000 −0.436436
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −1.00000 −0.162221
\(39\) 4.00000 0.640513
\(40\) 1.00000 0.158114
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) −2.00000 −0.308607
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 4.00000 0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 4.00000 0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) −2.00000 −0.267261
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −8.00000 −1.01600
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 2.00000 0.246183
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.00000 −0.242536
\(69\) 4.00000 0.481543
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) −1.00000 −0.114708
\(77\) −4.00000 −0.455842
\(78\) 4.00000 0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −2.00000 −0.218218
\(85\) −2.00000 −0.216930
\(86\) −6.00000 −0.646997
\(87\) 0 0
\(88\) 2.00000 0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 0.105409
\(91\) −8.00000 −0.838628
\(92\) 4.00000 0.417029
\(93\) −8.00000 −0.829561
\(94\) −12.0000 −1.23771
\(95\) −1.00000 −0.102598
\(96\) 1.00000 0.102062
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) −3.00000 −0.303046
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) −2.00000 −0.198030
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 4.00000 0.392232
\(105\) −2.00000 −0.195180
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 2.00000 0.190693
\(111\) 8.00000 0.759326
\(112\) −2.00000 −0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 4.00000 0.373002
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −8.00000 −0.721336
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) −2.00000 −0.178174
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.00000 −0.528271
\(130\) 4.00000 0.350823
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 2.00000 0.174078
\(133\) 2.00000 0.173422
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 4.00000 0.340503
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −2.00000 −0.169031
\(141\) −12.0000 −1.01058
\(142\) −8.00000 −0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) −3.00000 −0.247436
\(148\) 8.00000 0.657596
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 1.00000 0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −2.00000 −0.161690
\(154\) −4.00000 −0.322329
\(155\) −8.00000 −0.642575
\(156\) 4.00000 0.320256
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) −8.00000 −0.630488
\(162\) 1.00000 0.0785674
\(163\) 14.0000 1.09656 0.548282 0.836293i \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −8.00000 −0.624695
\(165\) 2.00000 0.155700
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) −2.00000 −0.153393
\(171\) −1.00000 −0.0764719
\(172\) −6.00000 −0.457496
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −8.00000 −0.592999
\(183\) 2.00000 0.147844
\(184\) 4.00000 0.294884
\(185\) 8.00000 0.588172
\(186\) −8.00000 −0.586588
\(187\) −4.00000 −0.292509
\(188\) −12.0000 −0.875190
\(189\) −2.00000 −0.145479
\(190\) −1.00000 −0.0725476
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 1.00000 0.0721688
\(193\) 24.0000 1.72756 0.863779 0.503871i \(-0.168091\pi\)
0.863779 + 0.503871i \(0.168091\pi\)
\(194\) −12.0000 −0.861550
\(195\) 4.00000 0.286446
\(196\) −3.00000 −0.214286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 2.00000 0.142134
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −8.00000 −0.558744
\(206\) 4.00000 0.278693
\(207\) 4.00000 0.278019
\(208\) 4.00000 0.277350
\(209\) −2.00000 −0.138343
\(210\) −2.00000 −0.138013
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −6.00000 −0.412082
\(213\) −8.00000 −0.548151
\(214\) −12.0000 −0.820303
\(215\) −6.00000 −0.409197
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) 10.0000 0.677285
\(219\) 14.0000 0.946032
\(220\) 2.00000 0.134840
\(221\) −8.00000 −0.538138
\(222\) 8.00000 0.536925
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 4.00000 0.263752
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 4.00000 0.261488
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 4.00000 0.259281
\(239\) 10.0000 0.646846 0.323423 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 1.00000 0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) −3.00000 −0.191663
\(246\) −8.00000 −0.510061
\(247\) −4.00000 −0.254514
\(248\) −8.00000 −0.508001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) −2.00000 −0.125988
\(253\) 8.00000 0.502956
\(254\) 8.00000 0.501965
\(255\) −2.00000 −0.125245
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) −6.00000 −0.373544
\(259\) −16.0000 −0.994192
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 2.00000 0.123091
\(265\) −6.00000 −0.368577
\(266\) 2.00000 0.122628
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 1.00000 0.0608581
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −2.00000 −0.121268
\(273\) −8.00000 −0.484182
\(274\) −22.0000 −1.32907
\(275\) 2.00000 0.120605
\(276\) 4.00000 0.240772
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) −2.00000 −0.119523
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) −12.0000 −0.714590
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) −8.00000 −0.474713
\(285\) −1.00000 −0.0592349
\(286\) 8.00000 0.473050
\(287\) 16.0000 0.944450
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 14.0000 0.819288
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 2.00000 0.116052
\(298\) 10.0000 0.579284
\(299\) 16.0000 0.925304
\(300\) 1.00000 0.0577350
\(301\) 12.0000 0.691669
\(302\) −8.00000 −0.460348
\(303\) 2.00000 0.114897
\(304\) −1.00000 −0.0573539
\(305\) 2.00000 0.114520
\(306\) −2.00000 −0.114332
\(307\) −32.0000 −1.82634 −0.913168 0.407583i \(-0.866372\pi\)
−0.913168 + 0.407583i \(0.866372\pi\)
\(308\) −4.00000 −0.227921
\(309\) 4.00000 0.227552
\(310\) −8.00000 −0.454369
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 4.00000 0.226455
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 18.0000 1.01580
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) −8.00000 −0.445823
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 14.0000 0.775388
\(327\) 10.0000 0.553001
\(328\) −8.00000 −0.441726
\(329\) 24.0000 1.32316
\(330\) 2.00000 0.110096
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 4.00000 0.219529
\(333\) 8.00000 0.438397
\(334\) 8.00000 0.437741
\(335\) 8.00000 0.437087
\(336\) −2.00000 −0.109109
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 3.00000 0.163178
\(339\) −6.00000 −0.325875
\(340\) −2.00000 −0.108465
\(341\) −16.0000 −0.866449
\(342\) −1.00000 −0.0540738
\(343\) 20.0000 1.07990
\(344\) −6.00000 −0.323498
\(345\) 4.00000 0.215353
\(346\) 14.0000 0.752645
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −2.00000 −0.106904
\(351\) 4.00000 0.213504
\(352\) 2.00000 0.106600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −10.0000 −0.527780 −0.263890 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 1.00000 0.0527046
\(361\) 1.00000 0.0526316
\(362\) 2.00000 0.105118
\(363\) −7.00000 −0.367405
\(364\) −8.00000 −0.419314
\(365\) 14.0000 0.732793
\(366\) 2.00000 0.104542
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 4.00000 0.208514
\(369\) −8.00000 −0.416463
\(370\) 8.00000 0.415900
\(371\) 12.0000 0.623009
\(372\) −8.00000 −0.414781
\(373\) −36.0000 −1.86401 −0.932005 0.362446i \(-0.881942\pi\)
−0.932005 + 0.362446i \(0.881942\pi\)
\(374\) −4.00000 −0.206835
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 8.00000 0.409852
\(382\) −18.0000 −0.920960
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) −4.00000 −0.203859
\(386\) 24.0000 1.22157
\(387\) −6.00000 −0.304997
\(388\) −12.0000 −0.609208
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 4.00000 0.202548
\(391\) −8.00000 −0.404577
\(392\) −3.00000 −0.151523
\(393\) −18.0000 −0.907980
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −20.0000 −1.00251
\(399\) 2.00000 0.100125
\(400\) 1.00000 0.0500000
\(401\) 12.0000 0.599251 0.299626 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) 8.00000 0.399004
\(403\) −32.0000 −1.59403
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 16.0000 0.793091
\(408\) −2.00000 −0.0990148
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −8.00000 −0.395092
\(411\) −22.0000 −1.08518
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 4.00000 0.196352
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) −2.00000 −0.0978232
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 12.0000 0.584151
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) −8.00000 −0.387601
\(427\) −4.00000 −0.193574
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) −6.00000 −0.289346
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −4.00000 −0.191346
\(438\) 14.0000 0.668946
\(439\) −40.0000 −1.90910 −0.954548 0.298057i \(-0.903661\pi\)
−0.954548 + 0.298057i \(0.903661\pi\)
\(440\) 2.00000 0.0953463
\(441\) −3.00000 −0.142857
\(442\) −8.00000 −0.380521
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) 10.0000 0.472984
\(448\) −2.00000 −0.0944911
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 1.00000 0.0471405
\(451\) −16.0000 −0.753411
\(452\) −6.00000 −0.282216
\(453\) −8.00000 −0.375873
\(454\) −12.0000 −0.563188
\(455\) −8.00000 −0.375046
\(456\) −1.00000 −0.0468293
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 10.0000 0.467269
\(459\) −2.00000 −0.0933520
\(460\) 4.00000 0.186501
\(461\) 2.00000 0.0931493 0.0465746 0.998915i \(-0.485169\pi\)
0.0465746 + 0.998915i \(0.485169\pi\)
\(462\) −4.00000 −0.186097
\(463\) −6.00000 −0.278844 −0.139422 0.990233i \(-0.544524\pi\)
−0.139422 + 0.990233i \(0.544524\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) −6.00000 −0.277945
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 4.00000 0.184900
\(469\) −16.0000 −0.738811
\(470\) −12.0000 −0.553519
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 4.00000 0.183340
\(477\) −6.00000 −0.274721
\(478\) 10.0000 0.457389
\(479\) −10.0000 −0.456912 −0.228456 0.973554i \(-0.573368\pi\)
−0.228456 + 0.973554i \(0.573368\pi\)
\(480\) 1.00000 0.0456435
\(481\) 32.0000 1.45907
\(482\) 2.00000 0.0910975
\(483\) −8.00000 −0.364013
\(484\) −7.00000 −0.318182
\(485\) −12.0000 −0.544892
\(486\) 1.00000 0.0453609
\(487\) 28.0000 1.26880 0.634401 0.773004i \(-0.281247\pi\)
0.634401 + 0.773004i \(0.281247\pi\)
\(488\) 2.00000 0.0905357
\(489\) 14.0000 0.633102
\(490\) −3.00000 −0.135526
\(491\) 22.0000 0.992846 0.496423 0.868081i \(-0.334646\pi\)
0.496423 + 0.868081i \(0.334646\pi\)
\(492\) −8.00000 −0.360668
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) 2.00000 0.0898933
\(496\) −8.00000 −0.359211
\(497\) 16.0000 0.717698
\(498\) 4.00000 0.179244
\(499\) −40.0000 −1.79065 −0.895323 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.00000 0.357414
\(502\) 2.00000 0.0892644
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 2.00000 0.0889988
\(506\) 8.00000 0.355643
\(507\) 3.00000 0.133235
\(508\) 8.00000 0.354943
\(509\) 20.0000 0.886484 0.443242 0.896402i \(-0.353828\pi\)
0.443242 + 0.896402i \(0.353828\pi\)
\(510\) −2.00000 −0.0885615
\(511\) −28.0000 −1.23865
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 18.0000 0.793946
\(515\) 4.00000 0.176261
\(516\) −6.00000 −0.264135
\(517\) −24.0000 −1.05552
\(518\) −16.0000 −0.703000
\(519\) 14.0000 0.614532
\(520\) 4.00000 0.175412
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −18.0000 −0.786334
\(525\) −2.00000 −0.0872872
\(526\) 24.0000 1.04645
\(527\) 16.0000 0.696971
\(528\) 2.00000 0.0870388
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) −32.0000 −1.38607
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 1.00000 0.0430331
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 12.0000 0.515444
\(543\) 2.00000 0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 10.0000 0.428353
\(546\) −8.00000 −0.342368
\(547\) −32.0000 −1.36822 −0.684111 0.729378i \(-0.739809\pi\)
−0.684111 + 0.729378i \(0.739809\pi\)
\(548\) −22.0000 −0.939793
\(549\) 2.00000 0.0853579
\(550\) 2.00000 0.0852803
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 0 0
\(554\) 18.0000 0.764747
\(555\) 8.00000 0.339581
\(556\) 0 0
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) −8.00000 −0.338667
\(559\) −24.0000 −1.01509
\(560\) −2.00000 −0.0845154
\(561\) −4.00000 −0.168880
\(562\) 12.0000 0.506189
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) −12.0000 −0.505291
\(565\) −6.00000 −0.252422
\(566\) 14.0000 0.588464
\(567\) −2.00000 −0.0839921
\(568\) −8.00000 −0.335673
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) −1.00000 −0.0418854
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) 8.00000 0.334497
\(573\) −18.0000 −0.751961
\(574\) 16.0000 0.667827
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −13.0000 −0.540729
\(579\) 24.0000 0.997406
\(580\) 0 0
\(581\) −8.00000 −0.331896
\(582\) −12.0000 −0.497416
\(583\) −12.0000 −0.496989
\(584\) 14.0000 0.579324
\(585\) 4.00000 0.165380
\(586\) 14.0000 0.578335
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) 8.00000 0.328798
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 2.00000 0.0820610
\(595\) 4.00000 0.163984
\(596\) 10.0000 0.409616
\(597\) −20.0000 −0.818546
\(598\) 16.0000 0.654289
\(599\) 20.0000 0.817178 0.408589 0.912719i \(-0.366021\pi\)
0.408589 + 0.912719i \(0.366021\pi\)
\(600\) 1.00000 0.0408248
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 12.0000 0.489083
\(603\) 8.00000 0.325785
\(604\) −8.00000 −0.325515
\(605\) −7.00000 −0.284590
\(606\) 2.00000 0.0812444
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −48.0000 −1.94187
\(612\) −2.00000 −0.0808452
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) −32.0000 −1.29141
\(615\) −8.00000 −0.322591
\(616\) −4.00000 −0.161165
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 4.00000 0.160904
\(619\) −40.0000 −1.60774 −0.803868 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) −8.00000 −0.321288
\(621\) 4.00000 0.160514
\(622\) −18.0000 −0.721734
\(623\) 0 0
\(624\) 4.00000 0.160128
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) −2.00000 −0.0798723
\(628\) 18.0000 0.718278
\(629\) −16.0000 −0.637962
\(630\) −2.00000 −0.0796819
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 18.0000 0.714871
\(635\) 8.00000 0.317470
\(636\) −6.00000 −0.237915
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) 32.0000 1.26392 0.631962 0.774999i \(-0.282250\pi\)
0.631962 + 0.774999i \(0.282250\pi\)
\(642\) −12.0000 −0.473602
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) −8.00000 −0.315244
\(645\) −6.00000 −0.236250
\(646\) 2.00000 0.0786889
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 16.0000 0.627089
\(652\) 14.0000 0.548282
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 10.0000 0.391031
\(655\) −18.0000 −0.703318
\(656\) −8.00000 −0.312348
\(657\) 14.0000 0.546192
\(658\) 24.0000 0.935617
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 2.00000 0.0778499
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 12.0000 0.466393
\(663\) −8.00000 −0.310694
\(664\) 4.00000 0.155230
\(665\) 2.00000 0.0775567
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 4.00000 0.154649
\(670\) 8.00000 0.309067
\(671\) 4.00000 0.154418
\(672\) −2.00000 −0.0771517
\(673\) 4.00000 0.154189 0.0770943 0.997024i \(-0.475436\pi\)
0.0770943 + 0.997024i \(0.475436\pi\)
\(674\) 28.0000 1.07852
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −6.00000 −0.230429
\(679\) 24.0000 0.921035
\(680\) −2.00000 −0.0766965
\(681\) −12.0000 −0.459841
\(682\) −16.0000 −0.612672
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −22.0000 −0.840577
\(686\) 20.0000 0.763604
\(687\) 10.0000 0.381524
\(688\) −6.00000 −0.228748
\(689\) −24.0000 −0.914327
\(690\) 4.00000 0.152277
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 14.0000 0.532200
\(693\) −4.00000 −0.151947
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) −10.0000 −0.378506
\(699\) −6.00000 −0.226941
\(700\) −2.00000 −0.0755929
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 4.00000 0.150970
\(703\) −8.00000 −0.301726
\(704\) 2.00000 0.0753778
\(705\) −12.0000 −0.451946
\(706\) −6.00000 −0.225813
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 4.00000 0.149696
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 10.0000 0.373457
\(718\) −10.0000 −0.373197
\(719\) 50.0000 1.86469 0.932343 0.361576i \(-0.117761\pi\)
0.932343 + 0.361576i \(0.117761\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.00000 −0.297936
\(722\) 1.00000 0.0372161
\(723\) 2.00000 0.0743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 14.0000 0.518163
\(731\) 12.0000 0.443836
\(732\) 2.00000 0.0739221
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 18.0000 0.664392
\(735\) −3.00000 −0.110657
\(736\) 4.00000 0.147442
\(737\) 16.0000 0.589368
\(738\) −8.00000 −0.294484
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 8.00000 0.294086
\(741\) −4.00000 −0.146944
\(742\) 12.0000 0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) −8.00000 −0.293294
\(745\) 10.0000 0.366372
\(746\) −36.0000 −1.31805
\(747\) 4.00000 0.146352
\(748\) −4.00000 −0.146254
\(749\) 24.0000 0.876941
\(750\) 1.00000 0.0365148
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −12.0000 −0.437595
\(753\) 2.00000 0.0728841
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) −2.00000 −0.0727393
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −20.0000 −0.726433
\(759\) 8.00000 0.290382
\(760\) −1.00000 −0.0362738
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 8.00000 0.289809
\(763\) −20.0000 −0.724049
\(764\) −18.0000 −0.651217
\(765\) −2.00000 −0.0723102
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) −4.00000 −0.144150
\(771\) 18.0000 0.648254
\(772\) 24.0000 0.863779
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) −6.00000 −0.215666
\(775\) −8.00000 −0.287368
\(776\) −12.0000 −0.430775
\(777\) −16.0000 −0.573997
\(778\) 30.0000 1.07555
\(779\) 8.00000 0.286630
\(780\) 4.00000 0.143223
\(781\) −16.0000 −0.572525
\(782\) −8.00000 −0.286079
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 18.0000 0.642448
\(786\) −18.0000 −0.642039
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 18.0000 0.641223
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 2.00000 0.0710669
\(793\) 8.00000 0.284088
\(794\) −22.0000 −0.780751
\(795\) −6.00000 −0.212798
\(796\) −20.0000 −0.708881
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 2.00000 0.0707992
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 12.0000 0.423735
\(803\) 28.0000 0.988099
\(804\) 8.00000 0.282138
\(805\) −8.00000 −0.281963
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) 2.00000 0.0703598
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 16.0000 0.560800
\(815\) 14.0000 0.490399
\(816\) −2.00000 −0.0700140
\(817\) 6.00000 0.209913
\(818\) 30.0000 1.04893
\(819\) −8.00000 −0.279543
\(820\) −8.00000 −0.279372
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −22.0000 −0.767338
\(823\) 34.0000 1.18517 0.592583 0.805510i \(-0.298108\pi\)
0.592583 + 0.805510i \(0.298108\pi\)
\(824\) 4.00000 0.139347
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 4.00000 0.139010
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 4.00000 0.138842
\(831\) 18.0000 0.624413
\(832\) 4.00000 0.138675
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) −2.00000 −0.0691714
\(837\) −8.00000 −0.276520
\(838\) −30.0000 −1.03633
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) 22.0000 0.758170
\(843\) 12.0000 0.413302
\(844\) 12.0000 0.413057
\(845\) 3.00000 0.103203
\(846\) −12.0000 −0.412568
\(847\) 14.0000 0.481046
\(848\) −6.00000 −0.206041
\(849\) 14.0000 0.480479
\(850\) −2.00000 −0.0685994
\(851\) 32.0000 1.09695
\(852\) −8.00000 −0.274075
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) −4.00000 −0.136877
\(855\) −1.00000 −0.0341993
\(856\) −12.0000 −0.410152
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 8.00000 0.273115
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) −6.00000 −0.204598
\(861\) 16.0000 0.545279
\(862\) 12.0000 0.408722
\(863\) −16.0000 −0.544646 −0.272323 0.962206i \(-0.587792\pi\)
−0.272323 + 0.962206i \(0.587792\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.0000 0.476014
\(866\) −16.0000 −0.543702
\(867\) −13.0000 −0.441503
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 10.0000 0.338643
\(873\) −12.0000 −0.406138
\(874\) −4.00000 −0.135302
\(875\) −2.00000 −0.0676123
\(876\) 14.0000 0.473016
\(877\) −12.0000 −0.405211 −0.202606 0.979260i \(-0.564941\pi\)
−0.202606 + 0.979260i \(0.564941\pi\)
\(878\) −40.0000 −1.34993
\(879\) 14.0000 0.472208
\(880\) 2.00000 0.0674200
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −3.00000 −0.101015
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 8.00000 0.268462
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 4.00000 0.133930
\(893\) 12.0000 0.401565
\(894\) 10.0000 0.334450
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 16.0000 0.534224
\(898\) 0 0
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) −16.0000 −0.532742
\(903\) 12.0000 0.399335
\(904\) −6.00000 −0.199557
\(905\) 2.00000 0.0664822
\(906\) −8.00000 −0.265782
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −12.0000 −0.398234
\(909\) 2.00000 0.0663358
\(910\) −8.00000 −0.265197
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 8.00000 0.264761
\(914\) 18.0000 0.595387
\(915\) 2.00000 0.0661180
\(916\) 10.0000 0.330409
\(917\) 36.0000 1.18882
\(918\) −2.00000 −0.0660098
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 4.00000 0.131876
\(921\) −32.0000 −1.05444
\(922\) 2.00000 0.0658665
\(923\) −32.0000 −1.05329
\(924\) −4.00000 −0.131590
\(925\) 8.00000 0.263038
\(926\) −6.00000 −0.197172
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) −8.00000 −0.262330
\(931\) 3.00000 0.0983210
\(932\) −6.00000 −0.196537
\(933\) −18.0000 −0.589294
\(934\) 8.00000 0.261768
\(935\) −4.00000 −0.130814
\(936\) 4.00000 0.130744
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −16.0000 −0.522419
\(939\) −6.00000 −0.195803
\(940\) −12.0000 −0.391397
\(941\) −8.00000 −0.260793 −0.130396 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) 18.0000 0.586472
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) −12.0000 −0.390154
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) −1.00000 −0.0324443
\(951\) 18.0000 0.583690
\(952\) 4.00000 0.129641
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −6.00000 −0.194257
\(955\) −18.0000 −0.582466
\(956\) 10.0000 0.323423
\(957\) 0 0
\(958\) −10.0000 −0.323085
\(959\) 44.0000 1.42083
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 32.0000 1.03172
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) 24.0000 0.772587
\(966\) −8.00000 −0.257396
\(967\) 58.0000 1.86515 0.932577 0.360971i \(-0.117555\pi\)
0.932577 + 0.360971i \(0.117555\pi\)
\(968\) −7.00000 −0.224989
\(969\) 2.00000 0.0642493
\(970\) −12.0000 −0.385297
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 4.00000 0.128103
\(976\) 2.00000 0.0640184
\(977\) 38.0000 1.21573 0.607864 0.794041i \(-0.292027\pi\)
0.607864 + 0.794041i \(0.292027\pi\)
\(978\) 14.0000 0.447671
\(979\) 0 0
\(980\) −3.00000 −0.0958315
\(981\) 10.0000 0.319275
\(982\) 22.0000 0.702048
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −8.00000 −0.255031
\(985\) 18.0000 0.573528
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) −4.00000 −0.127257
\(989\) −24.0000 −0.763156
\(990\) 2.00000 0.0635642
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −8.00000 −0.254000
\(993\) 12.0000 0.380808
\(994\) 16.0000 0.507489
\(995\) −20.0000 −0.634043
\(996\) 4.00000 0.126745
\(997\) −62.0000 −1.96356 −0.981780 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) −40.0000 −1.26618
\(999\) 8.00000 0.253109
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.l.1.1 1
3.2 odd 2 1710.2.a.b.1.1 1
4.3 odd 2 4560.2.a.o.1.1 1
5.2 odd 4 2850.2.d.q.799.2 2
5.3 odd 4 2850.2.d.q.799.1 2
5.4 even 2 2850.2.a.e.1.1 1
15.14 odd 2 8550.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.l.1.1 1 1.1 even 1 trivial
1710.2.a.b.1.1 1 3.2 odd 2
2850.2.a.e.1.1 1 5.4 even 2
2850.2.d.q.799.1 2 5.3 odd 4
2850.2.d.q.799.2 2 5.2 odd 4
4560.2.a.o.1.1 1 4.3 odd 2
8550.2.a.bf.1.1 1 15.14 odd 2