Properties

Label 570.2.a.i.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.00000 q^{12} -6.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} +1.00000 q^{20} -4.00000 q^{21} +8.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +4.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +1.00000 q^{38} +6.00000 q^{39} +1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} -8.00000 q^{43} +1.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} +9.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} -6.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{56} -1.00000 q^{57} +2.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +4.00000 q^{63} +1.00000 q^{64} -6.00000 q^{65} -12.0000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} -1.00000 q^{75} +1.00000 q^{76} +6.00000 q^{78} -16.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -4.00000 q^{83} -4.00000 q^{84} +2.00000 q^{85} -8.00000 q^{86} -2.00000 q^{87} +6.00000 q^{89} +1.00000 q^{90} -24.0000 q^{91} +8.00000 q^{92} +8.00000 q^{93} +1.00000 q^{95} -1.00000 q^{96} -10.0000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 4.00000 1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) 1.00000 0.223607
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −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
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 4.00000 0.676123
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 1.00000 0.162221
\(39\) 6.00000 0.960769
\(40\) 1.00000 0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −4.00000 −0.617213
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 8.00000 1.17954
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) −6.00000 −0.832050
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) −1.00000 −0.132453
\(58\) 2.00000 0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 4.00000 0.478091
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 10.0000 1.16248
\(75\) −1.00000 −0.115470
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −4.00000 −0.436436
\(85\) 2.00000 0.216930
\(86\) −8.00000 −0.862662
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 1.00000 0.105409
\(91\) −24.0000 −2.51588
\(92\) 8.00000 0.834058
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −2.00000 −0.198030
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) −4.00000 −0.390360
\(106\) −2.00000 −0.194257
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 4.00000 0.377964
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 8.00000 0.746004
\(116\) 2.00000 0.185695
\(117\) −6.00000 −0.554700
\(118\) −12.0000 −1.10469
\(119\) 8.00000 0.733359
\(120\) −1.00000 −0.0912871
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) −8.00000 −0.718421
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) −6.00000 −0.526235
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) −1.00000 −0.0860663
\(136\) 2.00000 0.171499
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −8.00000 −0.681005
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 4.00000 0.338062
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.00000 0.166091
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) 10.0000 0.821995
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 1.00000 0.0811107
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 6.00000 0.480384
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −16.0000 −1.27289
\(159\) 2.00000 0.158610
\(160\) 1.00000 0.0790569
\(161\) 32.0000 2.52195
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −4.00000 −0.308607
\(169\) 23.0000 1.76923
\(170\) 2.00000 0.153393
\(171\) 1.00000 0.0764719
\(172\) −8.00000 −0.609994
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −2.00000 −0.151620
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 1.00000 0.0745356
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) −24.0000 −1.77900
\(183\) 2.00000 0.147844
\(184\) 8.00000 0.589768
\(185\) 10.0000 0.735215
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 1.00000 0.0725476
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −10.0000 −0.717958
\(195\) 6.00000 0.429669
\(196\) 9.00000 0.642857
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 12.0000 0.846415
\(202\) −18.0000 −1.26648
\(203\) 8.00000 0.561490
\(204\) −2.00000 −0.140028
\(205\) 6.00000 0.419058
\(206\) 8.00000 0.557386
\(207\) 8.00000 0.556038
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −2.00000 −0.137361
\(213\) 8.00000 0.548151
\(214\) 20.0000 1.36717
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) −32.0000 −2.17230
\(218\) 6.00000 0.406371
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −10.0000 −0.671156
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 4.00000 0.267261
\(225\) 1.00000 0.0666667
\(226\) 18.0000 1.19734
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 16.0000 1.03931
\(238\) 8.00000 0.518563
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −11.0000 −0.707107
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 9.00000 0.574989
\(246\) −6.00000 −0.382546
\(247\) −6.00000 −0.381771
\(248\) −8.00000 −0.508001
\(249\) 4.00000 0.253490
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) −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 \(0.310261\pi\)
\(258\) 8.00000 0.498058
\(259\) 40.0000 2.48548
\(260\) −6.00000 −0.372104
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 4.00000 0.245256
\(267\) −6.00000 −0.367194
\(268\) −12.0000 −0.733017
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 24.0000 1.45255
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −12.0000 −0.719712
\(279\) −8.00000 −0.478947
\(280\) 4.00000 0.239046
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −8.00000 −0.474713
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 10.0000 0.586210
\(292\) 2.00000 0.117041
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −9.00000 −0.524891
\(295\) −12.0000 −0.698667
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) −2.00000 −0.115857
\(299\) −48.0000 −2.77591
\(300\) −1.00000 −0.0577350
\(301\) −32.0000 −1.84445
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) 1.00000 0.0573539
\(305\) −2.00000 −0.114520
\(306\) 2.00000 0.114332
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −8.00000 −0.454369
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 6.00000 0.339683
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −10.0000 −0.564333
\(315\) 4.00000 0.225374
\(316\) −16.0000 −0.900070
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) 2.00000 0.112154
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −20.0000 −1.11629
\(322\) 32.0000 1.78329
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) 8.00000 0.443079
\(327\) −6.00000 −0.331801
\(328\) 6.00000 0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −4.00000 −0.219529
\(333\) 10.0000 0.547997
\(334\) 8.00000 0.437741
\(335\) −12.0000 −0.655630
\(336\) −4.00000 −0.218218
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 23.0000 1.25104
\(339\) −18.0000 −0.977626
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) 1.00000 0.0540738
\(343\) 8.00000 0.431959
\(344\) −8.00000 −0.431331
\(345\) −8.00000 −0.430706
\(346\) 14.0000 0.752645
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) −2.00000 −0.107211
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 4.00000 0.213809
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) 12.0000 0.637793
\(355\) −8.00000 −0.424596
\(356\) 6.00000 0.317999
\(357\) −8.00000 −0.423405
\(358\) −20.0000 −1.05703
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 1.00000 0.0527046
\(361\) 1.00000 0.0526316
\(362\) −18.0000 −0.946059
\(363\) 11.0000 0.577350
\(364\) −24.0000 −1.25794
\(365\) 2.00000 0.104685
\(366\) 2.00000 0.104542
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 8.00000 0.417029
\(369\) 6.00000 0.312348
\(370\) 10.0000 0.519875
\(371\) −8.00000 −0.415339
\(372\) 8.00000 0.414781
\(373\) 34.0000 1.76045 0.880227 0.474554i \(-0.157390\pi\)
0.880227 + 0.474554i \(0.157390\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) −4.00000 −0.205738
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 1.00000 0.0512989
\(381\) 8.00000 0.409852
\(382\) −12.0000 −0.613973
\(383\) −24.0000 −1.22634 −0.613171 0.789950i \(-0.710106\pi\)
−0.613171 + 0.789950i \(0.710106\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −8.00000 −0.406663
\(388\) −10.0000 −0.507673
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 6.00000 0.303822
\(391\) 16.0000 0.809155
\(392\) 9.00000 0.454569
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 16.0000 0.802008
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 12.0000 0.598506
\(403\) 48.0000 2.39105
\(404\) −18.0000 −0.895533
\(405\) 1.00000 0.0496904
\(406\) 8.00000 0.397033
\(407\) 0 0
\(408\) −2.00000 −0.0990148
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 6.00000 0.296319
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) −48.0000 −2.36193
\(414\) 8.00000 0.393179
\(415\) −4.00000 −0.196352
\(416\) −6.00000 −0.294174
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) −40.0000 −1.95413 −0.977064 0.212946i \(-0.931694\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) −4.00000 −0.195180
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 8.00000 0.387601
\(427\) −8.00000 −0.387147
\(428\) 20.0000 0.966736
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −32.0000 −1.53605
\(435\) −2.00000 −0.0958927
\(436\) 6.00000 0.287348
\(437\) 8.00000 0.382692
\(438\) −2.00000 −0.0955637
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −12.0000 −0.570782
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −10.0000 −0.474579
\(445\) 6.00000 0.284427
\(446\) −16.0000 −0.757622
\(447\) 2.00000 0.0945968
\(448\) 4.00000 0.188982
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) −24.0000 −1.12514
\(456\) −1.00000 −0.0468293
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 6.00000 0.280362
\(459\) −2.00000 −0.0933520
\(460\) 8.00000 0.373002
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 2.00000 0.0928477
\(465\) 8.00000 0.370991
\(466\) −6.00000 −0.277945
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) −6.00000 −0.277350
\(469\) −48.0000 −2.21643
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 16.0000 0.734904
\(475\) 1.00000 0.0458831
\(476\) 8.00000 0.366679
\(477\) −2.00000 −0.0915737
\(478\) −20.0000 −0.914779
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −60.0000 −2.73576
\(482\) −14.0000 −0.637683
\(483\) −32.0000 −1.45605
\(484\) −11.0000 −0.500000
\(485\) −10.0000 −0.454077
\(486\) −1.00000 −0.0453609
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −8.00000 −0.361773
\(490\) 9.00000 0.406579
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) −6.00000 −0.270501
\(493\) 4.00000 0.180151
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −32.0000 −1.43540
\(498\) 4.00000 0.179244
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 4.00000 0.178174
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) −8.00000 −0.354943
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 8.00000 0.353899
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 18.0000 0.793946
\(515\) 8.00000 0.352522
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 40.0000 1.75750
\(519\) −14.0000 −0.614532
\(520\) −6.00000 −0.263117
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 2.00000 0.0875376
\(523\) −44.0000 −1.92399 −0.961993 0.273075i \(-0.911959\pi\)
−0.961993 + 0.273075i \(0.911959\pi\)
\(524\) 0 0
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −2.00000 −0.0868744
\(531\) −12.0000 −0.520756
\(532\) 4.00000 0.173422
\(533\) −36.0000 −1.55933
\(534\) −6.00000 −0.259645
\(535\) 20.0000 0.864675
\(536\) −12.0000 −0.518321
\(537\) 20.0000 0.863064
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 8.00000 0.343629
\(543\) 18.0000 0.772454
\(544\) 2.00000 0.0857493
\(545\) 6.00000 0.257012
\(546\) 24.0000 1.02711
\(547\) 36.0000 1.53925 0.769624 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) −8.00000 −0.340503
\(553\) −64.0000 −2.72156
\(554\) −2.00000 −0.0849719
\(555\) −10.0000 −0.424476
\(556\) −12.0000 −0.508913
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) 48.0000 2.03018
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −2.00000 −0.0843649
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) −16.0000 −0.672530
\(567\) 4.00000 0.167984
\(568\) −8.00000 −0.335673
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −12.0000 −0.502184 −0.251092 0.967963i \(-0.580790\pi\)
−0.251092 + 0.967963i \(0.580790\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 24.0000 1.00174
\(575\) 8.00000 0.333623
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) −13.0000 −0.540729
\(579\) 2.00000 0.0831172
\(580\) 2.00000 0.0830455
\(581\) −16.0000 −0.663792
\(582\) 10.0000 0.414513
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) −6.00000 −0.248069
\(586\) 6.00000 0.247858
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) −9.00000 −0.371154
\(589\) −8.00000 −0.329634
\(590\) −12.0000 −0.494032
\(591\) −6.00000 −0.246807
\(592\) 10.0000 0.410997
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) 8.00000 0.327968
\(596\) −2.00000 −0.0819232
\(597\) −16.0000 −0.654836
\(598\) −48.0000 −1.96287
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) −32.0000 −1.30422
\(603\) −12.0000 −0.488678
\(604\) 0 0
\(605\) −11.0000 −0.447214
\(606\) 18.0000 0.731200
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 1.00000 0.0405554
\(609\) −8.00000 −0.324176
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 20.0000 0.807134
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) −8.00000 −0.321807
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −8.00000 −0.321288
\(621\) −8.00000 −0.321029
\(622\) 12.0000 0.481156
\(623\) 24.0000 0.961540
\(624\) 6.00000 0.240192
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 20.0000 0.797452
\(630\) 4.00000 0.159364
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −16.0000 −0.636446
\(633\) −20.0000 −0.794929
\(634\) −26.0000 −1.03259
\(635\) −8.00000 −0.317470
\(636\) 2.00000 0.0793052
\(637\) −54.0000 −2.13956
\(638\) 0 0
\(639\) −8.00000 −0.316475
\(640\) 1.00000 0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −20.0000 −0.789337
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 32.0000 1.26098
\(645\) 8.00000 0.315000
\(646\) 2.00000 0.0786889
\(647\) 32.0000 1.25805 0.629025 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) −6.00000 −0.235339
\(651\) 32.0000 1.25418
\(652\) 8.00000 0.313304
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 20.0000 0.777322
\(663\) 12.0000 0.466041
\(664\) −4.00000 −0.155230
\(665\) 4.00000 0.155113
\(666\) 10.0000 0.387492
\(667\) 16.0000 0.619522
\(668\) 8.00000 0.309529
\(669\) 16.0000 0.618596
\(670\) −12.0000 −0.463600
\(671\) 0 0
\(672\) −4.00000 −0.154303
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 22.0000 0.847408
\(675\) −1.00000 −0.0384900
\(676\) 23.0000 0.884615
\(677\) 30.0000 1.15299 0.576497 0.817099i \(-0.304419\pi\)
0.576497 + 0.817099i \(0.304419\pi\)
\(678\) −18.0000 −0.691286
\(679\) −40.0000 −1.53506
\(680\) 2.00000 0.0766965
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 1.00000 0.0382360
\(685\) −6.00000 −0.229248
\(686\) 8.00000 0.305441
\(687\) −6.00000 −0.228914
\(688\) −8.00000 −0.304997
\(689\) 12.0000 0.457164
\(690\) −8.00000 −0.304555
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 20.0000 0.759190
\(695\) −12.0000 −0.455186
\(696\) −2.00000 −0.0758098
\(697\) 12.0000 0.454532
\(698\) 14.0000 0.529908
\(699\) 6.00000 0.226941
\(700\) 4.00000 0.151186
\(701\) 38.0000 1.43524 0.717620 0.696435i \(-0.245231\pi\)
0.717620 + 0.696435i \(0.245231\pi\)
\(702\) 6.00000 0.226455
\(703\) 10.0000 0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) −72.0000 −2.70784
\(708\) 12.0000 0.450988
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −8.00000 −0.300235
\(711\) −16.0000 −0.600047
\(712\) 6.00000 0.224860
\(713\) −64.0000 −2.39682
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 20.0000 0.746914
\(718\) 12.0000 0.447836
\(719\) 28.0000 1.04422 0.522112 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 1.00000 0.0372678
\(721\) 32.0000 1.19174
\(722\) 1.00000 0.0372161
\(723\) 14.0000 0.520666
\(724\) −18.0000 −0.668965
\(725\) 2.00000 0.0742781
\(726\) 11.0000 0.408248
\(727\) 12.0000 0.445055 0.222528 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) −24.0000 −0.889499
\(729\) 1.00000 0.0370370
\(730\) 2.00000 0.0740233
\(731\) −16.0000 −0.591781
\(732\) 2.00000 0.0739221
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 4.00000 0.147643
\(735\) −9.00000 −0.331970
\(736\) 8.00000 0.294884
\(737\) 0 0
\(738\) 6.00000 0.220863
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 10.0000 0.367607
\(741\) 6.00000 0.220416
\(742\) −8.00000 −0.293689
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 8.00000 0.293294
\(745\) −2.00000 −0.0732743
\(746\) 34.0000 1.24483
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 80.0000 2.92314
\(750\) −1.00000 −0.0365148
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 1.00000 0.0362738
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 8.00000 0.289809
\(763\) 24.0000 0.868858
\(764\) −12.0000 −0.434145
\(765\) 2.00000 0.0723102
\(766\) −24.0000 −0.867155
\(767\) 72.0000 2.59977
\(768\) −1.00000 −0.0360844
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −2.00000 −0.0719816
\(773\) 46.0000 1.65451 0.827253 0.561830i \(-0.189903\pi\)
0.827253 + 0.561830i \(0.189903\pi\)
\(774\) −8.00000 −0.287554
\(775\) −8.00000 −0.287368
\(776\) −10.0000 −0.358979
\(777\) −40.0000 −1.43499
\(778\) 6.00000 0.215110
\(779\) 6.00000 0.214972
\(780\) 6.00000 0.214834
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) −2.00000 −0.0714742
\(784\) 9.00000 0.321429
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −16.0000 −0.569254
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 38.0000 1.34857
\(795\) 2.00000 0.0709327
\(796\) 16.0000 0.567105
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 6.00000 0.212000
\(802\) 6.00000 0.211867
\(803\) 0 0
\(804\) 12.0000 0.423207
\(805\) 32.0000 1.12785
\(806\) 48.0000 1.69073
\(807\) −18.0000 −0.633630
\(808\) −18.0000 −0.633238
\(809\) −22.0000 −0.773479 −0.386739 0.922189i \(-0.626399\pi\)
−0.386739 + 0.922189i \(0.626399\pi\)
\(810\) 1.00000 0.0351364
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) 8.00000 0.280745
\(813\) −8.00000 −0.280572
\(814\) 0 0
\(815\) 8.00000 0.280228
\(816\) −2.00000 −0.0700140
\(817\) −8.00000 −0.279885
\(818\) 18.0000 0.629355
\(819\) −24.0000 −0.838628
\(820\) 6.00000 0.209529
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 6.00000 0.209274
\(823\) 44.0000 1.53374 0.766872 0.641800i \(-0.221812\pi\)
0.766872 + 0.641800i \(0.221812\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 8.00000 0.278019
\(829\) 22.0000 0.764092 0.382046 0.924143i \(-0.375220\pi\)
0.382046 + 0.924143i \(0.375220\pi\)
\(830\) −4.00000 −0.138842
\(831\) 2.00000 0.0693792
\(832\) −6.00000 −0.208013
\(833\) 18.0000 0.623663
\(834\) 12.0000 0.415526
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −40.0000 −1.38178
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) −4.00000 −0.138013
\(841\) −25.0000 −0.862069
\(842\) 22.0000 0.758170
\(843\) 2.00000 0.0688837
\(844\) 20.0000 0.688428
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) −44.0000 −1.51186
\(848\) −2.00000 −0.0686803
\(849\) 16.0000 0.549119
\(850\) 2.00000 0.0685994
\(851\) 80.0000 2.74236
\(852\) 8.00000 0.274075
\(853\) −10.0000 −0.342393 −0.171197 0.985237i \(-0.554763\pi\)
−0.171197 + 0.985237i \(0.554763\pi\)
\(854\) −8.00000 −0.273754
\(855\) 1.00000 0.0341993
\(856\) 20.0000 0.683586
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) −8.00000 −0.272798
\(861\) −24.0000 −0.817918
\(862\) 24.0000 0.817443
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 14.0000 0.476014
\(866\) −34.0000 −1.15537
\(867\) 13.0000 0.441503
\(868\) −32.0000 −1.08615
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 72.0000 2.43963
\(872\) 6.00000 0.203186
\(873\) −10.0000 −0.338449
\(874\) 8.00000 0.270604
\(875\) 4.00000 0.135225
\(876\) −2.00000 −0.0675737
\(877\) −6.00000 −0.202606 −0.101303 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 8.00000 0.269987
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 9.00000 0.303046
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −12.0000 −0.403604
\(885\) 12.0000 0.403376
\(886\) −4.00000 −0.134383
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) −10.0000 −0.335578
\(889\) −32.0000 −1.07325
\(890\) 6.00000 0.201120
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 2.00000 0.0668900
\(895\) −20.0000 −0.668526
\(896\) 4.00000 0.133631
\(897\) 48.0000 1.60267
\(898\) 30.0000 1.00111
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) 32.0000 1.06489
\(904\) 18.0000 0.598671
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) −60.0000 −1.99227 −0.996134 0.0878507i \(-0.972000\pi\)
−0.996134 + 0.0878507i \(0.972000\pi\)
\(908\) 20.0000 0.663723
\(909\) −18.0000 −0.597022
\(910\) −24.0000 −0.795592
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 2.00000 0.0661180
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 8.00000 0.263752
\(921\) −20.0000 −0.659022
\(922\) 6.00000 0.197599
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 4.00000 0.131448
\(927\) 8.00000 0.262754
\(928\) 2.00000 0.0656532
\(929\) −14.0000 −0.459325 −0.229663 0.973270i \(-0.573762\pi\)
−0.229663 + 0.973270i \(0.573762\pi\)
\(930\) 8.00000 0.262330
\(931\) 9.00000 0.294963
\(932\) −6.00000 −0.196537
\(933\) −12.0000 −0.392862
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −48.0000 −1.56726
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 10.0000 0.325818
\(943\) 48.0000 1.56310
\(944\) −12.0000 −0.390567
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 16.0000 0.519656
\(949\) −12.0000 −0.389536
\(950\) 1.00000 0.0324443
\(951\) 26.0000 0.843108
\(952\) 8.00000 0.259281
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −12.0000 −0.388311
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) −24.0000 −0.775000
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −60.0000 −1.93448
\(963\) 20.0000 0.644491
\(964\) −14.0000 −0.450910
\(965\) −2.00000 −0.0643823
\(966\) −32.0000 −1.02958
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) −11.0000 −0.353553
\(969\) −2.00000 −0.0642493
\(970\) −10.0000 −0.321081
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −48.0000 −1.53881
\(974\) −8.00000 −0.256337
\(975\) 6.00000 0.192154
\(976\) −2.00000 −0.0640184
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) −8.00000 −0.255812
\(979\) 0 0
\(980\) 9.00000 0.287494
\(981\) 6.00000 0.191565
\(982\) −40.0000 −1.27645
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 6.00000 0.191176
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) −6.00000 −0.190885
\(989\) −64.0000 −2.03508
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) −8.00000 −0.254000
\(993\) −20.0000 −0.634681
\(994\) −32.0000 −1.01498
\(995\) 16.0000 0.507234
\(996\) 4.00000 0.126745
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) −20.0000 −0.633089
\(999\) −10.0000 −0.316386
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.i.1.1 1
3.2 odd 2 1710.2.a.e.1.1 1
4.3 odd 2 4560.2.a.x.1.1 1
5.2 odd 4 2850.2.d.c.799.2 2
5.3 odd 4 2850.2.d.c.799.1 2
5.4 even 2 2850.2.a.h.1.1 1
15.14 odd 2 8550.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.i.1.1 1 1.1 even 1 trivial
1710.2.a.e.1.1 1 3.2 odd 2
2850.2.a.h.1.1 1 5.4 even 2
2850.2.d.c.799.1 2 5.3 odd 4
2850.2.d.c.799.2 2 5.2 odd 4
4560.2.a.x.1.1 1 4.3 odd 2
8550.2.a.s.1.1 1 15.14 odd 2