Properties

Label 570.2.a.b.1.1
Level $570$
Weight $2$
Character 570.1
Self dual yes
Analytic conductor $4.551$
Analytic rank $1$
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: \(1\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -8.00000 q^{29} -1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} +1.00000 q^{40} -4.00000 q^{41} +2.00000 q^{42} -6.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} -4.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} -2.00000 q^{56} +1.00000 q^{57} +8.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} -4.00000 q^{69} +2.00000 q^{70} -1.00000 q^{72} +6.00000 q^{73} +4.00000 q^{74} -1.00000 q^{75} -1.00000 q^{76} -12.0000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +4.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} -2.00000 q^{85} +6.00000 q^{86} +8.00000 q^{87} +6.00000 q^{88} -4.00000 q^{89} +1.00000 q^{90} +4.00000 q^{92} +8.00000 q^{93} +12.0000 q^{94} +1.00000 q^{95} +1.00000 q^{96} +12.0000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) 6.00000 1.27920
\(23\) 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
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −1.00000 −0.182574
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 2.00000 0.308607
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −6.00000 −0.904534
\(45\) −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
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.00000 0.136083
\(55\) 6.00000 0.809040
\(56\) −2.00000 −0.267261
\(57\) 1.00000 0.132453
\(58\) 8.00000 1.05045
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 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
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) −4.00000 −0.481543
\(70\) 2.00000 0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 4.00000 0.464991
\(75\) −1.00000 −0.115470
\(76\) −1.00000 −0.114708
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 4.00000 0.441726
\(83\) 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
\(87\) 8.00000 0.857690
\(88\) 6.00000 0.639602
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 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 \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 3.00000 0.303046
\(99\) −6.00000 −0.603023
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 2.00000 0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 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
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −6.00000 −0.572078
\(111\) 4.00000 0.379663
\(112\) 2.00000 0.188982
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −4.00000 −0.373002
\(116\) −8.00000 −0.742781
\(117\) 0 0
\(118\) 4.00000 0.368230
\(119\) 4.00000 0.366679
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) −2.00000 −0.181071
\(123\) 4.00000 0.360668
\(124\) −8.00000 −0.718421
\(125\) −1.00000 −0.0894427
\(126\) −2.00000 −0.178174
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.00000 0.528271
\(130\) 0 0
\(131\) 22.0000 1.92215 0.961074 0.276289i \(-0.0891049\pi\)
0.961074 + 0.276289i \(0.0891049\pi\)
\(132\) 6.00000 0.522233
\(133\) −2.00000 −0.173422
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) −2.00000 −0.171499
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 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
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 8.00000 0.664364
\(146\) −6.00000 −0.496564
\(147\) 3.00000 0.247436
\(148\) −4.00000 −0.328798
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) 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
\(154\) 12.0000 0.966988
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) 6.00000 0.469956 0.234978 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −4.00000 −0.312348
\(165\) −6.00000 −0.467099
\(166\) −4.00000 −0.310460
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 2.00000 0.154303
\(169\) −13.0000 −1.00000
\(170\) 2.00000 0.153393
\(171\) −1.00000 −0.0764719
\(172\) −6.00000 −0.457496
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −8.00000 −0.606478
\(175\) 2.00000 0.151186
\(176\) −6.00000 −0.452267
\(177\) 4.00000 0.300658
\(178\) 4.00000 0.299813
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −4.00000 −0.294884
\(185\) 4.00000 0.294086
\(186\) −8.00000 −0.586588
\(187\) −12.0000 −0.877527
\(188\) −12.0000 −0.875190
\(189\) −2.00000 −0.145479
\(190\) −1.00000 −0.0725476
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −8.00000 −0.575853 −0.287926 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 6.00000 0.426401
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 8.00000 0.564276
\(202\) 10.0000 0.703598
\(203\) −16.0000 −1.12298
\(204\) −2.00000 −0.140028
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) 6.00000 0.415029
\(210\) −2.00000 −0.138013
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 6.00000 0.409197
\(216\) 1.00000 0.0680414
\(217\) −16.0000 −1.08615
\(218\) 14.0000 0.948200
\(219\) −6.00000 −0.405442
\(220\) 6.00000 0.404520
\(221\) 0 0
\(222\) −4.00000 −0.268462
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −2.00000 −0.133631
\(225\) 1.00000 0.0666667
\(226\) −14.0000 −0.931266
\(227\) −28.0000 −1.85843 −0.929213 0.369546i \(-0.879513\pi\)
−0.929213 + 0.369546i \(0.879513\pi\)
\(228\) 1.00000 0.0662266
\(229\) 18.0000 1.18947 0.594737 0.803921i \(-0.297256\pi\)
0.594737 + 0.803921i \(0.297256\pi\)
\(230\) 4.00000 0.263752
\(231\) 12.0000 0.789542
\(232\) 8.00000 0.525226
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) −4.00000 −0.260378
\(237\) −8.00000 −0.519656
\(238\) −4.00000 −0.259281
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) −4.00000 −0.255031
\(247\) 0 0
\(248\) 8.00000 0.508001
\(249\) −4.00000 −0.253490
\(250\) 1.00000 0.0632456
\(251\) −14.0000 −0.883672 −0.441836 0.897096i \(-0.645673\pi\)
−0.441836 + 0.897096i \(0.645673\pi\)
\(252\) 2.00000 0.125988
\(253\) −24.0000 −1.50887
\(254\) 12.0000 0.752947
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −6.00000 −0.373544
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) −22.0000 −1.35916
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −6.00000 −0.369274
\(265\) −6.00000 −0.368577
\(266\) 2.00000 0.122628
\(267\) 4.00000 0.244796
\(268\) −8.00000 −0.488678
\(269\) 8.00000 0.487769 0.243884 0.969804i \(-0.421578\pi\)
0.243884 + 0.969804i \(0.421578\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −6.00000 −0.361814
\(276\) −4.00000 −0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −8.00000 −0.478947
\(280\) 2.00000 0.119523
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) −12.0000 −0.714590
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −8.00000 −0.469776
\(291\) −12.0000 −0.703452
\(292\) 6.00000 0.351123
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) −3.00000 −0.174964
\(295\) 4.00000 0.232889
\(296\) 4.00000 0.232495
\(297\) 6.00000 0.348155
\(298\) −22.0000 −1.27443
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −12.0000 −0.691669
\(302\) 0 0
\(303\) 10.0000 0.574485
\(304\) −1.00000 −0.0573539
\(305\) −2.00000 −0.114520
\(306\) −2.00000 −0.114332
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) −12.0000 −0.683763
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −6.00000 −0.340229 −0.170114 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −6.00000 −0.338600
\(315\) −2.00000 −0.112687
\(316\) 8.00000 0.450035
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 6.00000 0.336463
\(319\) 48.0000 2.68748
\(320\) −1.00000 −0.0559017
\(321\) 12.0000 0.669775
\(322\) −8.00000 −0.445823
\(323\) −2.00000 −0.111283
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 14.0000 0.774202
\(328\) 4.00000 0.220863
\(329\) −24.0000 −1.32316
\(330\) 6.00000 0.330289
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 4.00000 0.219529
\(333\) −4.00000 −0.219199
\(334\) 16.0000 0.875481
\(335\) 8.00000 0.437087
\(336\) −2.00000 −0.109109
\(337\) −20.0000 −1.08947 −0.544735 0.838608i \(-0.683370\pi\)
−0.544735 + 0.838608i \(0.683370\pi\)
\(338\) 13.0000 0.707107
\(339\) −14.0000 −0.760376
\(340\) −2.00000 −0.108465
\(341\) 48.0000 2.59935
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) 6.00000 0.323498
\(345\) 4.00000 0.215353
\(346\) −18.0000 −0.967686
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) 8.00000 0.428845
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −2.00000 −0.106904
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −4.00000 −0.212000
\(357\) −4.00000 −0.211702
\(358\) −12.0000 −0.634220
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 1.00000 0.0527046
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −6.00000 −0.314054
\(366\) 2.00000 0.104542
\(367\) −10.0000 −0.521996 −0.260998 0.965339i \(-0.584052\pi\)
−0.260998 + 0.965339i \(0.584052\pi\)
\(368\) 4.00000 0.208514
\(369\) −4.00000 −0.208232
\(370\) −4.00000 −0.207950
\(371\) 12.0000 0.623009
\(372\) 8.00000 0.414781
\(373\) −24.0000 −1.24267 −0.621336 0.783544i \(-0.713410\pi\)
−0.621336 + 0.783544i \(0.713410\pi\)
\(374\) 12.0000 0.620505
\(375\) 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 \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 1.00000 0.0512989
\(381\) 12.0000 0.614779
\(382\) 22.0000 1.12562
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 12.0000 0.611577
\(386\) 8.00000 0.407189
\(387\) −6.00000 −0.304997
\(388\) 12.0000 0.609208
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 3.00000 0.151523
\(393\) −22.0000 −1.10975
\(394\) −14.0000 −0.705310
\(395\) −8.00000 −0.402524
\(396\) −6.00000 −0.301511
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) −20.0000 −1.00251
\(399\) 2.00000 0.100125
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) −1.00000 −0.0496904
\(406\) 16.0000 0.794067
\(407\) 24.0000 1.18964
\(408\) 2.00000 0.0990148
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −4.00000 −0.197546
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) −8.00000 −0.393654
\(414\) −4.00000 −0.196589
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 2.00000 0.0975900
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 28.0000 1.36302
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −6.00000 −0.289346
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 16.0000 0.768025
\(435\) −8.00000 −0.383571
\(436\) −14.0000 −0.670478
\(437\) −4.00000 −0.191346
\(438\) 6.00000 0.286691
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −6.00000 −0.286039
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 4.00000 0.189832
\(445\) 4.00000 0.189618
\(446\) 16.0000 0.757622
\(447\) −22.0000 −1.04056
\(448\) 2.00000 0.0944911
\(449\) −4.00000 −0.188772 −0.0943858 0.995536i \(-0.530089\pi\)
−0.0943858 + 0.995536i \(0.530089\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −14.0000 −0.654892 −0.327446 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) −18.0000 −0.841085
\(459\) −2.00000 −0.0933520
\(460\) −4.00000 −0.186501
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) −12.0000 −0.558291
\(463\) −26.0000 −1.20832 −0.604161 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −8.00000 −0.371391
\(465\) −8.00000 −0.370991
\(466\) −14.0000 −0.648537
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) −12.0000 −0.553519
\(471\) −6.00000 −0.276465
\(472\) 4.00000 0.184115
\(473\) 36.0000 1.65528
\(474\) 8.00000 0.367452
\(475\) −1.00000 −0.0458831
\(476\) 4.00000 0.183340
\(477\) 6.00000 0.274721
\(478\) 18.0000 0.823301
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 22.0000 1.00207
\(483\) −8.00000 −0.364013
\(484\) 25.0000 1.13636
\(485\) −12.0000 −0.544892
\(486\) 1.00000 0.0453609
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −6.00000 −0.271329
\(490\) −3.00000 −0.135526
\(491\) −34.0000 −1.53440 −0.767199 0.641409i \(-0.778350\pi\)
−0.767199 + 0.641409i \(0.778350\pi\)
\(492\) 4.00000 0.180334
\(493\) −16.0000 −0.720604
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 4.00000 0.179244
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 16.0000 0.714827
\(502\) 14.0000 0.624851
\(503\) 40.0000 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 10.0000 0.444994
\(506\) 24.0000 1.06693
\(507\) 13.0000 0.577350
\(508\) −12.0000 −0.532414
\(509\) 36.0000 1.59567 0.797836 0.602875i \(-0.205978\pi\)
0.797836 + 0.602875i \(0.205978\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 12.0000 0.530849
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) 6.00000 0.264135
\(517\) 72.0000 3.16656
\(518\) 8.00000 0.351500
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 24.0000 1.05146 0.525730 0.850652i \(-0.323792\pi\)
0.525730 + 0.850652i \(0.323792\pi\)
\(522\) 8.00000 0.350150
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 22.0000 0.961074
\(525\) −2.00000 −0.0872872
\(526\) −16.0000 −0.697633
\(527\) −16.0000 −0.696971
\(528\) 6.00000 0.261116
\(529\) −7.00000 −0.304348
\(530\) 6.00000 0.260623
\(531\) −4.00000 −0.173585
\(532\) −2.00000 −0.0867110
\(533\) 0 0
\(534\) −4.00000 −0.173097
\(535\) 12.0000 0.518805
\(536\) 8.00000 0.345547
\(537\) −12.0000 −0.517838
\(538\) −8.00000 −0.344904
\(539\) 18.0000 0.775315
\(540\) 1.00000 0.0430331
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −20.0000 −0.859074
\(543\) −2.00000 −0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 6.00000 0.255841
\(551\) 8.00000 0.340811
\(552\) 4.00000 0.170251
\(553\) 16.0000 0.680389
\(554\) 2.00000 0.0849719
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 8.00000 0.338667
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 12.0000 0.506640
\(562\) −16.0000 −0.674919
\(563\) 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
\(566\) −22.0000 −0.924729
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 1.00000 0.0418854
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 0 0
\(573\) 22.0000 0.919063
\(574\) 8.00000 0.333914
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 13.0000 0.540729
\(579\) 8.00000 0.332469
\(580\) 8.00000 0.332182
\(581\) 8.00000 0.331896
\(582\) 12.0000 0.497416
\(583\) −36.0000 −1.49097
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 24.0000 0.990586 0.495293 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 3.00000 0.123718
\(589\) 8.00000 0.329634
\(590\) −4.00000 −0.164677
\(591\) −14.0000 −0.575883
\(592\) −4.00000 −0.164399
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −6.00000 −0.246183
\(595\) −4.00000 −0.163984
\(596\) 22.0000 0.901155
\(597\) −20.0000 −0.818546
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 1.00000 0.0408248
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 12.0000 0.489083
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −25.0000 −1.01639
\(606\) −10.0000 −0.406222
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 1.00000 0.0405554
\(609\) 16.0000 0.648353
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 2.00000 0.0808452
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 8.00000 0.322854
\(615\) −4.00000 −0.161296
\(616\) 12.0000 0.483494
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 8.00000 0.321288
\(621\) −4.00000 −0.160514
\(622\) 6.00000 0.240578
\(623\) −8.00000 −0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 14.0000 0.559553
\(627\) −6.00000 −0.239617
\(628\) 6.00000 0.239426
\(629\) −8.00000 −0.318981
\(630\) 2.00000 0.0796819
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −8.00000 −0.318223
\(633\) 28.0000 1.11290
\(634\) −6.00000 −0.238290
\(635\) 12.0000 0.476205
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −48.0000 −1.90034
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) −12.0000 −0.473602
\(643\) −26.0000 −1.02534 −0.512670 0.858586i \(-0.671344\pi\)
−0.512670 + 0.858586i \(0.671344\pi\)
\(644\) 8.00000 0.315244
\(645\) −6.00000 −0.236250
\(646\) 2.00000 0.0786889
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 16.0000 0.627089
\(652\) 6.00000 0.234978
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −14.0000 −0.547443
\(655\) −22.0000 −0.859611
\(656\) −4.00000 −0.156174
\(657\) 6.00000 0.234082
\(658\) 24.0000 0.935617
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) −6.00000 −0.233550
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 2.00000 0.0775567
\(666\) 4.00000 0.154997
\(667\) −32.0000 −1.23904
\(668\) −16.0000 −0.619059
\(669\) 16.0000 0.618596
\(670\) −8.00000 −0.309067
\(671\) −12.0000 −0.463255
\(672\) 2.00000 0.0771517
\(673\) −28.0000 −1.07932 −0.539660 0.841883i \(-0.681447\pi\)
−0.539660 + 0.841883i \(0.681447\pi\)
\(674\) 20.0000 0.770371
\(675\) −1.00000 −0.0384900
\(676\) −13.0000 −0.500000
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 14.0000 0.537667
\(679\) 24.0000 0.921035
\(680\) 2.00000 0.0766965
\(681\) 28.0000 1.07296
\(682\) −48.0000 −1.83801
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −6.00000 −0.229248
\(686\) 20.0000 0.763604
\(687\) −18.0000 −0.686743
\(688\) −6.00000 −0.228748
\(689\) 0 0
\(690\) −4.00000 −0.152277
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 18.0000 0.684257
\(693\) −12.0000 −0.455842
\(694\) −36.0000 −1.36654
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) −8.00000 −0.303022
\(698\) 2.00000 0.0757011
\(699\) −14.0000 −0.529529
\(700\) 2.00000 0.0755929
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) −6.00000 −0.226134
\(705\) −12.0000 −0.451946
\(706\) 18.0000 0.677439
\(707\) −20.0000 −0.752177
\(708\) 4.00000 0.150329
\(709\) 50.0000 1.87779 0.938895 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 4.00000 0.149906
\(713\) −32.0000 −1.19841
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 18.0000 0.672222
\(718\) 6.00000 0.223918
\(719\) 14.0000 0.522112 0.261056 0.965324i \(-0.415929\pi\)
0.261056 + 0.965324i \(0.415929\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 22.0000 0.818189
\(724\) 2.00000 0.0743294
\(725\) −8.00000 −0.297113
\(726\) 25.0000 0.927837
\(727\) 14.0000 0.519231 0.259616 0.965712i \(-0.416404\pi\)
0.259616 + 0.965712i \(0.416404\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.00000 0.222070
\(731\) −12.0000 −0.443836
\(732\) −2.00000 −0.0739221
\(733\) 18.0000 0.664845 0.332423 0.943131i \(-0.392134\pi\)
0.332423 + 0.943131i \(0.392134\pi\)
\(734\) 10.0000 0.369107
\(735\) −3.00000 −0.110657
\(736\) −4.00000 −0.147442
\(737\) 48.0000 1.76810
\(738\) 4.00000 0.147242
\(739\) −28.0000 −1.03000 −0.514998 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −8.00000 −0.293294
\(745\) −22.0000 −0.806018
\(746\) 24.0000 0.878702
\(747\) 4.00000 0.146352
\(748\) −12.0000 −0.438763
\(749\) −24.0000 −0.876941
\(750\) −1.00000 −0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −12.0000 −0.437595
\(753\) 14.0000 0.510188
\(754\) 0 0
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) −20.0000 −0.726433
\(759\) 24.0000 0.871145
\(760\) −1.00000 −0.0362738
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −12.0000 −0.434714
\(763\) −28.0000 −1.01367
\(764\) −22.0000 −0.795932
\(765\) −2.00000 −0.0723102
\(766\) −24.0000 −0.867155
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −12.0000 −0.432450
\(771\) −6.00000 −0.216085
\(772\) −8.00000 −0.287926
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 6.00000 0.215666
\(775\) −8.00000 −0.287368
\(776\) −12.0000 −0.430775
\(777\) 8.00000 0.286998
\(778\) 30.0000 1.07555
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 0 0
\(782\) −8.00000 −0.286079
\(783\) 8.00000 0.285897
\(784\) −3.00000 −0.107143
\(785\) −6.00000 −0.214149
\(786\) 22.0000 0.784714
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) 14.0000 0.498729
\(789\) −16.0000 −0.569615
\(790\) 8.00000 0.284627
\(791\) 28.0000 0.995565
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −22.0000 −0.780751
\(795\) 6.00000 0.212798
\(796\) 20.0000 0.708881
\(797\) 2.00000 0.0708436 0.0354218 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) −2.00000 −0.0707992
\(799\) −24.0000 −0.849059
\(800\) −1.00000 −0.0353553
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) −36.0000 −1.27041
\(804\) 8.00000 0.282138
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) −8.00000 −0.281613
\(808\) 10.0000 0.351799
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 1.00000 0.0351364
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −16.0000 −0.561490
\(813\) −20.0000 −0.701431
\(814\) −24.0000 −0.841200
\(815\) −6.00000 −0.210171
\(816\) −2.00000 −0.0700140
\(817\) 6.00000 0.209913
\(818\) −14.0000 −0.489499
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −50.0000 −1.74501 −0.872506 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 6.00000 0.209274
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 8.00000 0.278356
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 4.00000 0.139010
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 4.00000 0.138842
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 6.00000 0.207514
\(837\) 8.00000 0.276520
\(838\) 14.0000 0.483622
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 35.0000 1.20690
\(842\) 2.00000 0.0689246
\(843\) −16.0000 −0.551069
\(844\) −28.0000 −0.963800
\(845\) 13.0000 0.447214
\(846\) 12.0000 0.412568
\(847\) 50.0000 1.71802
\(848\) 6.00000 0.206041
\(849\) −22.0000 −0.755038
\(850\) −2.00000 −0.0685994
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −4.00000 −0.136877
\(855\) 1.00000 0.0341993
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 6.00000 0.204598
\(861\) 8.00000 0.272639
\(862\) −36.0000 −1.22616
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) −16.0000 −0.543075
\(869\) −48.0000 −1.62829
\(870\) 8.00000 0.271225
\(871\) 0 0
\(872\) 14.0000 0.474100
\(873\) 12.0000 0.406138
\(874\) 4.00000 0.135302
\(875\) −2.00000 −0.0676123
\(876\) −6.00000 −0.202721
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 8.00000 0.269987
\(879\) −10.0000 −0.337292
\(880\) 6.00000 0.202260
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 3.00000 0.101015
\(883\) 14.0000 0.471138 0.235569 0.971858i \(-0.424305\pi\)
0.235569 + 0.971858i \(0.424305\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 24.0000 0.806296
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) −4.00000 −0.134231
\(889\) −24.0000 −0.804934
\(890\) −4.00000 −0.134080
\(891\) −6.00000 −0.201008
\(892\) −16.0000 −0.535720
\(893\) 12.0000 0.401565
\(894\) 22.0000 0.735790
\(895\) −12.0000 −0.401116
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 4.00000 0.133482
\(899\) 64.0000 2.13452
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) 12.0000 0.399335
\(904\) −14.0000 −0.465633
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −28.0000 −0.929213
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 1.00000 0.0331133
\(913\) −24.0000 −0.794284
\(914\) 14.0000 0.463079
\(915\) 2.00000 0.0661180
\(916\) 18.0000 0.594737
\(917\) 44.0000 1.45301
\(918\) 2.00000 0.0660098
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 4.00000 0.131876
\(921\) 8.00000 0.263609
\(922\) 34.0000 1.11973
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) −4.00000 −0.131519
\(926\) 26.0000 0.854413
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 8.00000 0.262330
\(931\) 3.00000 0.0983210
\(932\) 14.0000 0.458585
\(933\) 6.00000 0.196431
\(934\) 8.00000 0.261768
\(935\) 12.0000 0.392442
\(936\) 0 0
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 16.0000 0.522419
\(939\) 14.0000 0.456873
\(940\) 12.0000 0.391397
\(941\) −40.0000 −1.30396 −0.651981 0.758235i \(-0.726062\pi\)
−0.651981 + 0.758235i \(0.726062\pi\)
\(942\) 6.00000 0.195491
\(943\) −16.0000 −0.521032
\(944\) −4.00000 −0.130189
\(945\) 2.00000 0.0650600
\(946\) −36.0000 −1.17046
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) 1.00000 0.0324443
\(951\) −6.00000 −0.194563
\(952\) −4.00000 −0.129641
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) −6.00000 −0.194257
\(955\) 22.0000 0.711903
\(956\) −18.0000 −0.582162
\(957\) −48.0000 −1.55162
\(958\) −18.0000 −0.581554
\(959\) 12.0000 0.387500
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) −22.0000 −0.708572
\(965\) 8.00000 0.257529
\(966\) 8.00000 0.257396
\(967\) −18.0000 −0.578841 −0.289420 0.957202i \(-0.593463\pi\)
−0.289420 + 0.957202i \(0.593463\pi\)
\(968\) −25.0000 −0.803530
\(969\) 2.00000 0.0642493
\(970\) 12.0000 0.385297
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 6.00000 0.191859
\(979\) 24.0000 0.767043
\(980\) 3.00000 0.0958315
\(981\) −14.0000 −0.446986
\(982\) 34.0000 1.08498
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) −4.00000 −0.127515
\(985\) −14.0000 −0.446077
\(986\) 16.0000 0.509544
\(987\) 24.0000 0.763928
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) −6.00000 −0.190693
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 8.00000 0.254000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) −4.00000 −0.126745
\(997\) −58.0000 −1.83688 −0.918439 0.395562i \(-0.870550\pi\)
−0.918439 + 0.395562i \(0.870550\pi\)
\(998\) −24.0000 −0.759707
\(999\) 4.00000 0.126554
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.b.1.1 1
3.2 odd 2 1710.2.a.s.1.1 1
4.3 odd 2 4560.2.a.r.1.1 1
5.2 odd 4 2850.2.d.j.799.1 2
5.3 odd 4 2850.2.d.j.799.2 2
5.4 even 2 2850.2.a.y.1.1 1
15.14 odd 2 8550.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.b.1.1 1 1.1 even 1 trivial
1710.2.a.s.1.1 1 3.2 odd 2
2850.2.a.y.1.1 1 5.4 even 2
2850.2.d.j.799.1 2 5.2 odd 4
2850.2.d.j.799.2 2 5.3 odd 4
4560.2.a.r.1.1 1 4.3 odd 2
8550.2.a.g.1.1 1 15.14 odd 2