Properties

Label 570.2.a.g.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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} +6.00000 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -1.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} +10.0000 q^{41} -4.00000 q^{43} +4.00000 q^{44} -1.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +1.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -6.00000 q^{74} -1.00000 q^{75} -1.00000 q^{76} -2.00000 q^{78} -4.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -12.0000 q^{83} -2.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} +4.00000 q^{92} -4.00000 q^{93} -12.0000 q^{94} +1.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 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
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 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
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −1.00000 −0.162221
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 4.00000 0.603023
\(45\) −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
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 6.00000 0.787839
\(59\) −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 \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) −4.00000 −0.431331
\(87\) −6.00000 −0.643268
\(88\) 4.00000 0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −4.00000 −0.414781
\(94\) −12.0000 −1.23771
\(95\) 1.00000 0.102598
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −7.00000 −0.707107
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −2.00000 −0.198030
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) −4.00000 −0.381385
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 1.00000 0.0936586
\(115\) −4.00000 −0.373002
\(116\) 6.00000 0.557086
\(117\) 2.00000 0.184900
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) −2.00000 −0.175412
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 1.00000 0.0860663
\(136\) 2.00000 0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) −4.00000 −0.340503
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 8.00000 0.671345
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −6.00000 −0.498273
\(146\) −6.00000 −0.496564
\(147\) 7.00000 0.577350
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) −2.00000 −0.160128
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −4.00000 −0.318223
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 10.0000 0.780869
\(165\) 4.00000 0.311400
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) −1.00000 −0.0764719
\(172\) −4.00000 −0.304997
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 12.0000 0.901975
\(178\) 10.0000 0.749532
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 18.0000 1.33793 0.668965 0.743294i \(-0.266738\pi\)
0.668965 + 0.743294i \(0.266738\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 4.00000 0.294884
\(185\) 6.00000 0.441129
\(186\) −4.00000 −0.293294
\(187\) 8.00000 0.585018
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) 20.0000 1.44715 0.723575 0.690246i \(-0.242498\pi\)
0.723575 + 0.690246i \(0.242498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 2.00000 0.143592
\(195\) 2.00000 0.143223
\(196\) −7.00000 −0.500000
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 4.00000 0.284268
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 1.00000 0.0707107
\(201\) −4.00000 −0.282138
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) −10.0000 −0.698430
\(206\) −4.00000 −0.278693
\(207\) 4.00000 0.278019
\(208\) 2.00000 0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) −4.00000 −0.273434
\(215\) 4.00000 0.272798
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −6.00000 −0.406371
\(219\) 6.00000 0.405442
\(220\) −4.00000 −0.269680
\(221\) 4.00000 0.269069
\(222\) 6.00000 0.402694
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 1.00000 0.0662266
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −4.00000 −0.263752
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 2.00000 0.130744
\(235\) 12.0000 0.782794
\(236\) −12.0000 −0.781133
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 1.00000 0.0645497
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 7.00000 0.447214
\(246\) −10.0000 −0.637577
\(247\) −2.00000 −0.127257
\(248\) 4.00000 0.254000
\(249\) 12.0000 0.760469
\(250\) −1.00000 −0.0632456
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) −4.00000 −0.250982
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 6.00000 0.371391
\(262\) −12.0000 −0.741362
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) −4.00000 −0.246183
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 4.00000 0.244339
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) 4.00000 0.241209
\(276\) −4.00000 −0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −20.0000 −1.19952
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 12.0000 0.714590
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 8.00000 0.474713
\(285\) −1.00000 −0.0592349
\(286\) 8.00000 0.473050
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) −2.00000 −0.117242
\(292\) −6.00000 −0.351123
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 7.00000 0.408248
\(295\) 12.0000 0.698667
\(296\) −6.00000 −0.348743
\(297\) −4.00000 −0.232104
\(298\) 18.0000 1.04271
\(299\) 8.00000 0.462652
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) −10.0000 −0.574485
\(304\) −1.00000 −0.0573539
\(305\) 2.00000 0.114520
\(306\) 2.00000 0.114332
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −4.00000 −0.227185
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) −2.00000 −0.113228
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) −6.00000 −0.336463
\(319\) 24.0000 1.34374
\(320\) −1.00000 −0.0559017
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 6.00000 0.331801
\(328\) 10.0000 0.552158
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −12.0000 −0.658586
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −9.00000 −0.489535
\(339\) 14.0000 0.760376
\(340\) −2.00000 −0.108465
\(341\) 16.0000 0.866449
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 4.00000 0.215353
\(346\) −10.0000 −0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −6.00000 −0.321634
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 12.0000 0.637793
\(355\) −8.00000 −0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 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
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 6.00000 0.314054
\(366\) 2.00000 0.104542
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 4.00000 0.208514
\(369\) 10.0000 0.520579
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) −4.00000 −0.207390
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 8.00000 0.413670
\(375\) 1.00000 0.0516398
\(376\) −12.0000 −0.618853
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 1.00000 0.0512989
\(381\) 4.00000 0.204926
\(382\) 20.0000 1.02329
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −4.00000 −0.203331
\(388\) 2.00000 0.101535
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 2.00000 0.101274
\(391\) 8.00000 0.404577
\(392\) −7.00000 −0.353553
\(393\) 12.0000 0.605320
\(394\) 2.00000 0.100759
\(395\) 4.00000 0.201262
\(396\) 4.00000 0.201008
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) −4.00000 −0.199502
\(403\) 8.00000 0.398508
\(404\) 10.0000 0.497519
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) −2.00000 −0.0990148
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) −10.0000 −0.493865
\(411\) −10.0000 −0.493264
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) 20.0000 0.979404
\(418\) −4.00000 −0.195646
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 26.0000 1.26716 0.633581 0.773676i \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −4.00000 −0.194717
\(423\) −12.0000 −0.583460
\(424\) 6.00000 0.291386
\(425\) 2.00000 0.0970143
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −8.00000 −0.386244
\(430\) 4.00000 0.192897
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) −6.00000 −0.287348
\(437\) −4.00000 −0.191346
\(438\) 6.00000 0.286691
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) 4.00000 0.190261
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 6.00000 0.284747
\(445\) −10.0000 −0.474045
\(446\) 12.0000 0.568216
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) −14.0000 −0.658505
\(453\) 12.0000 0.563809
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −6.00000 −0.280668 −0.140334 0.990104i \(-0.544818\pi\)
−0.140334 + 0.990104i \(0.544818\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) −4.00000 −0.186501
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 6.00000 0.278543
\(465\) 4.00000 0.185496
\(466\) 18.0000 0.833834
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 12.0000 0.553519
\(471\) 2.00000 0.0921551
\(472\) −12.0000 −0.552345
\(473\) −16.0000 −0.735681
\(474\) 4.00000 0.183726
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 12.0000 0.548867
\(479\) 20.0000 0.913823 0.456912 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 1.00000 0.0456435
\(481\) −12.0000 −0.547153
\(482\) 26.0000 1.18427
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) −2.00000 −0.0908153
\(486\) −1.00000 −0.0453609
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 4.00000 0.180886
\(490\) 7.00000 0.316228
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −10.0000 −0.450835
\(493\) 12.0000 0.540453
\(494\) −2.00000 −0.0899843
\(495\) −4.00000 −0.179787
\(496\) 4.00000 0.179605
\(497\) 0 0
\(498\) 12.0000 0.537733
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −20.0000 −0.892644
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 16.0000 0.711287
\(507\) 9.00000 0.399704
\(508\) −4.00000 −0.177471
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 2.00000 0.0885615
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) 2.00000 0.0882162
\(515\) 4.00000 0.176261
\(516\) 4.00000 0.176090
\(517\) −48.0000 −2.11104
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) −2.00000 −0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 6.00000 0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 8.00000 0.348485
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) −6.00000 −0.260623
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) −10.0000 −0.432742
\(535\) 4.00000 0.172935
\(536\) 4.00000 0.172774
\(537\) −20.0000 −0.863064
\(538\) −2.00000 −0.0862261
\(539\) −28.0000 −1.20605
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 8.00000 0.343629
\(543\) −18.0000 −0.772454
\(544\) 2.00000 0.0857493
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) −6.00000 −0.255609
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) −6.00000 −0.254686
\(556\) −20.0000 −0.848189
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) −6.00000 −0.253095
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 12.0000 0.505291
\(565\) 14.0000 0.588984
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −38.0000 −1.59304 −0.796521 0.604610i \(-0.793329\pi\)
−0.796521 + 0.604610i \(0.793329\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 8.00000 0.334497
\(573\) −20.0000 −0.835512
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −13.0000 −0.540729
\(579\) −2.00000 −0.0831172
\(580\) −6.00000 −0.249136
\(581\) 0 0
\(582\) −2.00000 −0.0829027
\(583\) 24.0000 0.993978
\(584\) −6.00000 −0.248282
\(585\) −2.00000 −0.0826898
\(586\) −2.00000 −0.0826192
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 7.00000 0.288675
\(589\) −4.00000 −0.164817
\(590\) 12.0000 0.494032
\(591\) −2.00000 −0.0822690
\(592\) −6.00000 −0.246598
\(593\) 10.0000 0.410651 0.205325 0.978694i \(-0.434175\pi\)
0.205325 + 0.978694i \(0.434175\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) −12.0000 −0.488273
\(605\) −5.00000 −0.203279
\(606\) −10.0000 −0.406222
\(607\) 4.00000 0.162355 0.0811775 0.996700i \(-0.474132\pi\)
0.0811775 + 0.996700i \(0.474132\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) −24.0000 −0.970936
\(612\) 2.00000 0.0808452
\(613\) 22.0000 0.888572 0.444286 0.895885i \(-0.353457\pi\)
0.444286 + 0.895885i \(0.353457\pi\)
\(614\) 4.00000 0.161427
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 4.00000 0.160904
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −4.00000 −0.160644
\(621\) −4.00000 −0.160514
\(622\) −28.0000 −1.12270
\(623\) 0 0
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) 4.00000 0.159745
\(628\) −2.00000 −0.0798087
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 4.00000 0.158986
\(634\) −26.0000 −1.03259
\(635\) 4.00000 0.158735
\(636\) −6.00000 −0.237915
\(637\) −14.0000 −0.554700
\(638\) 24.0000 0.950169
\(639\) 8.00000 0.316475
\(640\) −1.00000 −0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −2.00000 −0.0786889
\(647\) 36.0000 1.41531 0.707653 0.706560i \(-0.249754\pi\)
0.707653 + 0.706560i \(0.249754\pi\)
\(648\) 1.00000 0.0392837
\(649\) −48.0000 −1.88416
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 6.00000 0.234619
\(655\) 12.0000 0.468879
\(656\) 10.0000 0.390434
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 4.00000 0.155700
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 28.0000 1.08825
\(663\) −4.00000 −0.155347
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) −12.0000 −0.463947
\(670\) −4.00000 −0.154533
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 26.0000 1.00148
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) −2.00000 −0.0766965
\(681\) 20.0000 0.766402
\(682\) 16.0000 0.612672
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) −4.00000 −0.152499
\(689\) 12.0000 0.457164
\(690\) 4.00000 0.152277
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 20.0000 0.758643
\(696\) −6.00000 −0.227429
\(697\) 20.0000 0.757554
\(698\) −10.0000 −0.378506
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 6.00000 0.226294
\(704\) 4.00000 0.150756
\(705\) −12.0000 −0.451946
\(706\) 26.0000 0.978523
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −8.00000 −0.300235
\(711\) −4.00000 −0.150012
\(712\) 10.0000 0.374766
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 20.0000 0.747435
\(717\) −12.0000 −0.448148
\(718\) 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
\(721\) 0 0
\(722\) 1.00000 0.0372161
\(723\) −26.0000 −0.966950
\(724\) 18.0000 0.668965
\(725\) 6.00000 0.222834
\(726\) −5.00000 −0.185567
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −8.00000 −0.295891
\(732\) 2.00000 0.0739221
\(733\) 46.0000 1.69905 0.849524 0.527549i \(-0.176889\pi\)
0.849524 + 0.527549i \(0.176889\pi\)
\(734\) 24.0000 0.885856
\(735\) −7.00000 −0.258199
\(736\) 4.00000 0.147442
\(737\) 16.0000 0.589368
\(738\) 10.0000 0.368105
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 6.00000 0.220564
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) −4.00000 −0.146647
\(745\) −18.0000 −0.659469
\(746\) −22.0000 −0.805477
\(747\) −12.0000 −0.439057
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) 1.00000 0.0365148
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −12.0000 −0.437595
\(753\) 20.0000 0.728841
\(754\) 12.0000 0.437014
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −20.0000 −0.726433
\(759\) −16.0000 −0.580763
\(760\) 1.00000 0.0362738
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 4.00000 0.144905
\(763\) 0 0
\(764\) 20.0000 0.723575
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) −46.0000 −1.65880 −0.829401 0.558653i \(-0.811318\pi\)
−0.829401 + 0.558653i \(0.811318\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 2.00000 0.0719816
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) −4.00000 −0.143777
\(775\) 4.00000 0.143684
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −22.0000 −0.788738
\(779\) −10.0000 −0.358287
\(780\) 2.00000 0.0716115
\(781\) 32.0000 1.14505
\(782\) 8.00000 0.286079
\(783\) −6.00000 −0.214423
\(784\) −7.00000 −0.250000
\(785\) 2.00000 0.0713831
\(786\) 12.0000 0.428026
\(787\) −52.0000 −1.85360 −0.926800 0.375555i \(-0.877452\pi\)
−0.926800 + 0.375555i \(0.877452\pi\)
\(788\) 2.00000 0.0712470
\(789\) 4.00000 0.142404
\(790\) 4.00000 0.142314
\(791\) 0 0
\(792\) 4.00000 0.142134
\(793\) −4.00000 −0.142044
\(794\) −18.0000 −0.638796
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) 54.0000 1.91278 0.956389 0.292096i \(-0.0943526\pi\)
0.956389 + 0.292096i \(0.0943526\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) −14.0000 −0.494357
\(803\) −24.0000 −0.846942
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 2.00000 0.0704033
\(808\) 10.0000 0.351799
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 0 0
\(813\) −8.00000 −0.280572
\(814\) −24.0000 −0.841200
\(815\) 4.00000 0.140114
\(816\) −2.00000 −0.0700140
\(817\) 4.00000 0.139942
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) −10.0000 −0.349215
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) −10.0000 −0.348790
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −4.00000 −0.139347
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 4.00000 0.139010
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 12.0000 0.416526
\(831\) 2.00000 0.0693792
\(832\) 2.00000 0.0693375
\(833\) −14.0000 −0.485071
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) −4.00000 −0.138260
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 6.00000 0.206651
\(844\) −4.00000 −0.137686
\(845\) 9.00000 0.309609
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 28.0000 0.960958
\(850\) 2.00000 0.0685994
\(851\) −24.0000 −0.822709
\(852\) −8.00000 −0.274075
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) −4.00000 −0.136717
\(857\) 26.0000 0.888143 0.444072 0.895991i \(-0.353534\pi\)
0.444072 + 0.895991i \(0.353534\pi\)
\(858\) −8.00000 −0.273115
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 8.00000 0.272323 0.136162 0.990687i \(-0.456523\pi\)
0.136162 + 0.990687i \(0.456523\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 10.0000 0.340010
\(866\) 34.0000 1.15537
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −16.0000 −0.542763
\(870\) 6.00000 0.203419
\(871\) 8.00000 0.271070
\(872\) −6.00000 −0.203186
\(873\) 2.00000 0.0676897
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) −46.0000 −1.55331 −0.776655 0.629926i \(-0.783085\pi\)
−0.776655 + 0.629926i \(0.783085\pi\)
\(878\) 12.0000 0.404980
\(879\) 2.00000 0.0674583
\(880\) −4.00000 −0.134840
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −7.00000 −0.235702
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 4.00000 0.134535
\(885\) −12.0000 −0.403376
\(886\) 4.00000 0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) −10.0000 −0.335201
\(891\) 4.00000 0.134005
\(892\) 12.0000 0.401790
\(893\) 12.0000 0.401565
\(894\) −18.0000 −0.602010
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) −30.0000 −1.00111
\(899\) 24.0000 0.800445
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 40.0000 1.33185
\(903\) 0 0
\(904\) −14.0000 −0.465633
\(905\) −18.0000 −0.598340
\(906\) 12.0000 0.398673
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −20.0000 −0.663723
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 1.00000 0.0331133
\(913\) −48.0000 −1.58857
\(914\) −6.00000 −0.198462
\(915\) −2.00000 −0.0661180
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) −4.00000 −0.131876
\(921\) −4.00000 −0.131804
\(922\) −38.0000 −1.25146
\(923\) 16.0000 0.526646
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 8.00000 0.262896
\(927\) −4.00000 −0.131377
\(928\) 6.00000 0.196960
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 4.00000 0.131165
\(931\) 7.00000 0.229416
\(932\) 18.0000 0.589610
\(933\) 28.0000 0.916679
\(934\) −28.0000 −0.916188
\(935\) −8.00000 −0.261628
\(936\) 2.00000 0.0653720
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 12.0000 0.391397
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 2.00000 0.0651635
\(943\) 40.0000 1.30258
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 4.00000 0.129914
\(949\) −12.0000 −0.389536
\(950\) −1.00000 −0.0324443
\(951\) 26.0000 0.843108
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 6.00000 0.194257
\(955\) −20.0000 −0.647185
\(956\) 12.0000 0.388108
\(957\) −24.0000 −0.775810
\(958\) 20.0000 0.646171
\(959\) 0 0
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) −12.0000 −0.386896
\(963\) −4.00000 −0.128898
\(964\) 26.0000 0.837404
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 5.00000 0.160706
\(969\) 2.00000 0.0642493
\(970\) −2.00000 −0.0642161
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −20.0000 −0.640841
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 4.00000 0.127906
\(979\) 40.0000 1.27841
\(980\) 7.00000 0.223607
\(981\) −6.00000 −0.191565
\(982\) −36.0000 −1.14881
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) −10.0000 −0.318788
\(985\) −2.00000 −0.0637253
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) −16.0000 −0.508770
\(990\) −4.00000 −0.127128
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000 0.127000
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −36.0000 −1.13956
\(999\) 6.00000 0.189832
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.g.1.1 1
3.2 odd 2 1710.2.a.i.1.1 1
4.3 odd 2 4560.2.a.s.1.1 1
5.2 odd 4 2850.2.d.i.799.2 2
5.3 odd 4 2850.2.d.i.799.1 2
5.4 even 2 2850.2.a.m.1.1 1
15.14 odd 2 8550.2.a.x.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.g.1.1 1 1.1 even 1 trivial
1710.2.a.i.1.1 1 3.2 odd 2
2850.2.a.m.1.1 1 5.4 even 2
2850.2.d.i.799.1 2 5.3 odd 4
2850.2.d.i.799.2 2 5.2 odd 4
4560.2.a.s.1.1 1 4.3 odd 2
8550.2.a.x.1.1 1 15.14 odd 2