Properties

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

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +6.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +6.00000 q^{22} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} +8.00000 q^{31} +1.00000 q^{32} +6.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -12.0000 q^{41} +2.00000 q^{42} +2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{45} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -6.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} -6.00000 q^{55} +2.00000 q^{56} +1.00000 q^{57} -12.0000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +8.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +6.00000 q^{66} -16.0000 q^{67} -6.00000 q^{68} -2.00000 q^{70} +1.00000 q^{72} -10.0000 q^{73} +8.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +12.0000 q^{77} -4.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +2.00000 q^{84} +6.00000 q^{85} +2.00000 q^{86} +6.00000 q^{88} -12.0000 q^{89} -1.00000 q^{90} -8.00000 q^{91} +8.00000 q^{93} -1.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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 \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 2.00000 0.534522
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 1.00000 0.235702
\(19\) 1.00000 0.229416
\(20\) −1.00000 −0.223607
\(21\) 2.00000 0.436436
\(22\) 6.00000 1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 6.00000 1.04447
\(34\) −6.00000 −1.02899
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 1.00000 0.162221
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) −12.0000 −1.87409 −0.937043 0.349215i \(-0.886448\pi\)
−0.937043 + 0.349215i \(0.886448\pi\)
\(42\) 2.00000 0.308607
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 6.00000 0.904534
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −6.00000 −0.840168
\(52\) −4.00000 −0.554700
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000 0.136083
\(55\) −6.00000 −0.809040
\(56\) 2.00000 0.267261
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 8.00000 1.01600
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) 6.00000 0.738549
\(67\) −16.0000 −1.95471 −0.977356 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 8.00000 0.929981
\(75\) 1.00000 0.115470
\(76\) 1.00000 0.114708
\(77\) 12.0000 1.36753
\(78\) −4.00000 −0.452911
\(79\) 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
\(82\) −12.0000 −1.32518
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 2.00000 0.218218
\(85\) 6.00000 0.650791
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −3.00000 −0.303046
\(99\) 6.00000 0.603023
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −6.00000 −0.594089
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −4.00000 −0.392232
\(105\) −2.00000 −0.195180
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −6.00000 −0.572078
\(111\) 8.00000 0.759326
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) −12.0000 −1.10469
\(119\) −12.0000 −1.10004
\(120\) −1.00000 −0.0912871
\(121\) 25.0000 2.27273
\(122\) 2.00000 0.181071
\(123\) −12.0000 −1.08200
\(124\) 8.00000 0.718421
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) 2.00000 0.176090
\(130\) 4.00000 0.350823
\(131\) 18.0000 1.57267 0.786334 0.617802i \(-0.211977\pi\)
0.786334 + 0.617802i \(0.211977\pi\)
\(132\) 6.00000 0.522233
\(133\) 2.00000 0.173422
\(134\) −16.0000 −1.38219
\(135\) −1.00000 −0.0860663
\(136\) −6.00000 −0.514496
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −10.0000 −0.827606
\(147\) −3.00000 −0.247436
\(148\) 8.00000 0.657596
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 1.00000 0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 1.00000 0.0811107
\(153\) −6.00000 −0.485071
\(154\) 12.0000 0.966988
\(155\) −8.00000 −0.642575
\(156\) −4.00000 −0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −10.0000 −0.783260 −0.391630 0.920123i \(-0.628089\pi\)
−0.391630 + 0.920123i \(0.628089\pi\)
\(164\) −12.0000 −0.937043
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 2.00000 0.154303
\(169\) 3.00000 0.230769
\(170\) 6.00000 0.460179
\(171\) 1.00000 0.0764719
\(172\) 2.00000 0.152499
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 6.00000 0.452267
\(177\) −12.0000 −0.901975
\(178\) −12.0000 −0.899438
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −8.00000 −0.592999
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 8.00000 0.586588
\(187\) −36.0000 −2.63258
\(188\) 0 0
\(189\) 2.00000 0.145479
\(190\) −1.00000 −0.0725476
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 1.00000 0.0721688
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) 8.00000 0.574367
\(195\) 4.00000 0.286446
\(196\) −3.00000 −0.214286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 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
\(201\) −16.0000 −1.12855
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 12.0000 0.838116
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 6.00000 0.415029
\(210\) −2.00000 −0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −2.00000 −0.136399
\(216\) 1.00000 0.0680414
\(217\) 16.0000 1.08615
\(218\) 2.00000 0.135457
\(219\) −10.0000 −0.675737
\(220\) −6.00000 −0.404520
\(221\) 24.0000 1.61441
\(222\) 8.00000 0.536925
\(223\) −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
\(226\) −6.00000 −0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 1.00000 0.0662266
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 12.0000 0.789542
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) 8.00000 0.519656
\(238\) −12.0000 −0.777844
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 25.0000 1.60706
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 3.00000 0.191663
\(246\) −12.0000 −0.765092
\(247\) −4.00000 −0.254514
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000 0.125988
\(253\) 0 0
\(254\) 20.0000 1.25491
\(255\) 6.00000 0.375735
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 2.00000 0.124515
\(259\) 16.0000 0.994192
\(260\) 4.00000 0.248069
\(261\) 0 0
\(262\) 18.0000 1.11204
\(263\) −12.0000 −0.739952 −0.369976 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 6.00000 0.369274
\(265\) 6.00000 0.368577
\(266\) 2.00000 0.122628
\(267\) −12.0000 −0.734388
\(268\) −16.0000 −0.977356
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) −6.00000 −0.363803
\(273\) −8.00000 −0.484182
\(274\) 6.00000 0.362473
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 8.00000 0.479808
\(279\) 8.00000 0.478947
\(280\) −2.00000 −0.119523
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) −24.0000 −1.41915
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −10.0000 −0.585206
\(293\) −18.0000 −1.05157 −0.525786 0.850617i \(-0.676229\pi\)
−0.525786 + 0.850617i \(0.676229\pi\)
\(294\) −3.00000 −0.174964
\(295\) 12.0000 0.698667
\(296\) 8.00000 0.464991
\(297\) 6.00000 0.348155
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 4.00000 0.230556
\(302\) 8.00000 0.460348
\(303\) 6.00000 0.344691
\(304\) 1.00000 0.0573539
\(305\) −2.00000 −0.114520
\(306\) −6.00000 −0.342997
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 12.0000 0.683763
\(309\) −16.0000 −0.910208
\(310\) −8.00000 −0.454369
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) −4.00000 −0.226455
\(313\) 26.0000 1.46961 0.734803 0.678280i \(-0.237274\pi\)
0.734803 + 0.678280i \(0.237274\pi\)
\(314\) −10.0000 −0.564333
\(315\) −2.00000 −0.112687
\(316\) 8.00000 0.450035
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) −10.0000 −0.553849
\(327\) 2.00000 0.110600
\(328\) −12.0000 −0.662589
\(329\) 0 0
\(330\) −6.00000 −0.330289
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 24.0000 1.31322
\(335\) 16.0000 0.874173
\(336\) 2.00000 0.109109
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 3.00000 0.163178
\(339\) −6.00000 −0.325875
\(340\) 6.00000 0.325396
\(341\) 48.0000 2.59935
\(342\) 1.00000 0.0540738
\(343\) −20.0000 −1.07990
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 2.00000 0.106904
\(351\) −4.00000 −0.213504
\(352\) 6.00000 0.319801
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) −12.0000 −0.635107
\(358\) −12.0000 −0.634220
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 1.00000 0.0526316
\(362\) 2.00000 0.105118
\(363\) 25.0000 1.31216
\(364\) −8.00000 −0.419314
\(365\) 10.0000 0.523424
\(366\) 2.00000 0.104542
\(367\) −34.0000 −1.77479 −0.887393 0.461014i \(-0.847486\pi\)
−0.887393 + 0.461014i \(0.847486\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) −8.00000 −0.415900
\(371\) −12.0000 −0.623009
\(372\) 8.00000 0.414781
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) −36.0000 −1.86152
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 2.00000 0.102869
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 20.0000 1.02463
\(382\) 6.00000 0.306987
\(383\) 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
\(386\) 20.0000 1.01797
\(387\) 2.00000 0.101666
\(388\) 8.00000 0.406138
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 18.0000 0.907980
\(394\) 6.00000 0.302276
\(395\) −8.00000 −0.402524
\(396\) 6.00000 0.301511
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 20.0000 1.00251
\(399\) 2.00000 0.100125
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −16.0000 −0.798007
\(403\) −32.0000 −1.59403
\(404\) 6.00000 0.298511
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 48.0000 2.37927
\(408\) −6.00000 −0.297044
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 12.0000 0.592638
\(411\) 6.00000 0.295958
\(412\) −16.0000 −0.788263
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 8.00000 0.391762
\(418\) 6.00000 0.293470
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) −2.00000 −0.0975900
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) −12.0000 −0.580042
\(429\) −24.0000 −1.15873
\(430\) −2.00000 −0.0964486
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 16.0000 0.768025
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −10.0000 −0.477818
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) −6.00000 −0.286039
\(441\) −3.00000 −0.142857
\(442\) 24.0000 1.14156
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 8.00000 0.379663
\(445\) 12.0000 0.568855
\(446\) −16.0000 −0.757622
\(447\) −18.0000 −0.851371
\(448\) 2.00000 0.0944911
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) 1.00000 0.0471405
\(451\) −72.0000 −3.39035
\(452\) −6.00000 −0.282216
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) 8.00000 0.375046
\(456\) 1.00000 0.0468293
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 2.00000 0.0934539
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −42.0000 −1.95614 −0.978068 0.208288i \(-0.933211\pi\)
−0.978068 + 0.208288i \(0.933211\pi\)
\(462\) 12.0000 0.558291
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) −8.00000 −0.370991
\(466\) 6.00000 0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −4.00000 −0.184900
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) −12.0000 −0.552345
\(473\) 12.0000 0.551761
\(474\) 8.00000 0.367452
\(475\) 1.00000 0.0458831
\(476\) −12.0000 −0.550019
\(477\) −6.00000 −0.274721
\(478\) −6.00000 −0.274434
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −32.0000 −1.45907
\(482\) 2.00000 0.0910975
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −8.00000 −0.363261
\(486\) 1.00000 0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 2.00000 0.0905357
\(489\) −10.0000 −0.452216
\(490\) 3.00000 0.135526
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) −12.0000 −0.541002
\(493\) 0 0
\(494\) −4.00000 −0.179969
\(495\) −6.00000 −0.269680
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 24.0000 1.07224
\(502\) 6.00000 0.267793
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 2.00000 0.0890871
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 20.0000 0.887357
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 6.00000 0.265684
\(511\) −20.0000 −0.884748
\(512\) 1.00000 0.0441942
\(513\) 1.00000 0.0441511
\(514\) −6.00000 −0.264649
\(515\) 16.0000 0.705044
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) 16.0000 0.703000
\(519\) 6.00000 0.263371
\(520\) 4.00000 0.175412
\(521\) −24.0000 −1.05146 −0.525730 0.850652i \(-0.676208\pi\)
−0.525730 + 0.850652i \(0.676208\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 18.0000 0.786334
\(525\) 2.00000 0.0872872
\(526\) −12.0000 −0.523225
\(527\) −48.0000 −2.09091
\(528\) 6.00000 0.261116
\(529\) −23.0000 −1.00000
\(530\) 6.00000 0.260623
\(531\) −12.0000 −0.520756
\(532\) 2.00000 0.0867110
\(533\) 48.0000 2.07911
\(534\) −12.0000 −0.519291
\(535\) 12.0000 0.518805
\(536\) −16.0000 −0.691095
\(537\) −12.0000 −0.517838
\(538\) 24.0000 1.03471
\(539\) −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
\(544\) −6.00000 −0.257248
\(545\) −2.00000 −0.0856706
\(546\) −8.00000 −0.342368
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 6.00000 0.255841
\(551\) 0 0
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) −10.0000 −0.424859
\(555\) −8.00000 −0.339581
\(556\) 8.00000 0.339276
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 8.00000 0.338667
\(559\) −8.00000 −0.338364
\(560\) −2.00000 −0.0845154
\(561\) −36.0000 −1.51992
\(562\) 24.0000 1.01238
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 14.0000 0.588464
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) −1.00000 −0.0418854
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) −24.0000 −1.00349
\(573\) 6.00000 0.250654
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 19.0000 0.790296
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) −36.0000 −1.49097
\(584\) −10.0000 −0.413803
\(585\) 4.00000 0.165380
\(586\) −18.0000 −0.743573
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 12.0000 0.494032
\(591\) 6.00000 0.246807
\(592\) 8.00000 0.328798
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 6.00000 0.246183
\(595\) 12.0000 0.491952
\(596\) −18.0000 −0.737309
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 1.00000 0.0408248
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 4.00000 0.163028
\(603\) −16.0000 −0.651570
\(604\) 8.00000 0.325515
\(605\) −25.0000 −1.01639
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) −16.0000 −0.645707
\(615\) 12.0000 0.483887
\(616\) 12.0000 0.483494
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) −16.0000 −0.643614
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) −18.0000 −0.721734
\(623\) −24.0000 −0.961540
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) 26.0000 1.03917
\(627\) 6.00000 0.239617
\(628\) −10.0000 −0.399043
\(629\) −48.0000 −1.91389
\(630\) −2.00000 −0.0796819
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 8.00000 0.318223
\(633\) −4.00000 −0.158986
\(634\) −30.0000 −1.19145
\(635\) −20.0000 −0.793676
\(636\) −6.00000 −0.237915
\(637\) 12.0000 0.475457
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) −12.0000 −0.473602
\(643\) 14.0000 0.552106 0.276053 0.961142i \(-0.410973\pi\)
0.276053 + 0.961142i \(0.410973\pi\)
\(644\) 0 0
\(645\) −2.00000 −0.0787499
\(646\) −6.00000 −0.236067
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 1.00000 0.0392837
\(649\) −72.0000 −2.82625
\(650\) −4.00000 −0.156893
\(651\) 16.0000 0.627089
\(652\) −10.0000 −0.391630
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 2.00000 0.0782062
\(655\) −18.0000 −0.703318
\(656\) −12.0000 −0.468521
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) −6.00000 −0.233550
\(661\) 50.0000 1.94477 0.972387 0.233373i \(-0.0749763\pi\)
0.972387 + 0.233373i \(0.0749763\pi\)
\(662\) −28.0000 −1.08825
\(663\) 24.0000 0.932083
\(664\) 0 0
\(665\) −2.00000 −0.0775567
\(666\) 8.00000 0.309994
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) −16.0000 −0.618596
\(670\) 16.0000 0.618134
\(671\) 12.0000 0.463255
\(672\) 2.00000 0.0771517
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 8.00000 0.308148
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) −6.00000 −0.230429
\(679\) 16.0000 0.614024
\(680\) 6.00000 0.230089
\(681\) 12.0000 0.459841
\(682\) 48.0000 1.83801
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 1.00000 0.0382360
\(685\) −6.00000 −0.229248
\(686\) −20.0000 −0.763604
\(687\) 2.00000 0.0763048
\(688\) 2.00000 0.0762493
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 6.00000 0.228086
\(693\) 12.0000 0.455842
\(694\) 24.0000 0.911028
\(695\) −8.00000 −0.303457
\(696\) 0 0
\(697\) 72.0000 2.72719
\(698\) 14.0000 0.529908
\(699\) 6.00000 0.226941
\(700\) 2.00000 0.0755929
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −4.00000 −0.150970
\(703\) 8.00000 0.301726
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 12.0000 0.451306
\(708\) −12.0000 −0.450988
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −12.0000 −0.449719
\(713\) 0 0
\(714\) −12.0000 −0.449089
\(715\) 24.0000 0.897549
\(716\) −12.0000 −0.448461
\(717\) −6.00000 −0.224074
\(718\) 6.00000 0.223918
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −32.0000 −1.19174
\(722\) 1.00000 0.0372161
\(723\) 2.00000 0.0743808
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 25.0000 0.927837
\(727\) −10.0000 −0.370879 −0.185440 0.982656i \(-0.559371\pi\)
−0.185440 + 0.982656i \(0.559371\pi\)
\(728\) −8.00000 −0.296500
\(729\) 1.00000 0.0370370
\(730\) 10.0000 0.370117
\(731\) −12.0000 −0.443836
\(732\) 2.00000 0.0739221
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −34.0000 −1.25496
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) −96.0000 −3.53621
\(738\) −12.0000 −0.441726
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) −8.00000 −0.294086
\(741\) −4.00000 −0.146944
\(742\) −12.0000 −0.440534
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 8.00000 0.293294
\(745\) 18.0000 0.659469
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) −36.0000 −1.31629
\(749\) −24.0000 −0.876941
\(750\) −1.00000 −0.0365148
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) −8.00000 −0.291150
\(756\) 2.00000 0.0727393
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) −1.00000 −0.0362738
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 20.0000 0.724524
\(763\) 4.00000 0.144810
\(764\) 6.00000 0.217072
\(765\) 6.00000 0.216930
\(766\) 24.0000 0.867155
\(767\) 48.0000 1.73318
\(768\) 1.00000 0.0360844
\(769\) 50.0000 1.80305 0.901523 0.432731i \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) −12.0000 −0.432450
\(771\) −6.00000 −0.216085
\(772\) 20.0000 0.719816
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 2.00000 0.0718885
\(775\) 8.00000 0.287368
\(776\) 8.00000 0.287183
\(777\) 16.0000 0.573997
\(778\) 18.0000 0.645331
\(779\) −12.0000 −0.429945
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 10.0000 0.356915
\(786\) 18.0000 0.642039
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 6.00000 0.213741
\(789\) −12.0000 −0.427211
\(790\) −8.00000 −0.284627
\(791\) −12.0000 −0.426671
\(792\) 6.00000 0.213201
\(793\) −8.00000 −0.284088
\(794\) 14.0000 0.496841
\(795\) 6.00000 0.212798
\(796\) 20.0000 0.708881
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 2.00000 0.0707992
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) −12.0000 −0.423999
\(802\) 0 0
\(803\) −60.0000 −2.11735
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 24.0000 0.844840
\(808\) 6.00000 0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 20.0000 0.701431
\(814\) 48.0000 1.68240
\(815\) 10.0000 0.350285
\(816\) −6.00000 −0.210042
\(817\) 2.00000 0.0699711
\(818\) 14.0000 0.489499
\(819\) −8.00000 −0.279543
\(820\) 12.0000 0.419058
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 6.00000 0.209274
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −16.0000 −0.557386
\(825\) 6.00000 0.208893
\(826\) −24.0000 −0.835067
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) −4.00000 −0.138675
\(833\) 18.0000 0.623663
\(834\) 8.00000 0.277017
\(835\) −24.0000 −0.830554
\(836\) 6.00000 0.207514
\(837\) 8.00000 0.276520
\(838\) 30.0000 1.03633
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −29.0000 −1.00000
\(842\) −10.0000 −0.344623
\(843\) 24.0000 0.826604
\(844\) −4.00000 −0.137686
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 50.0000 1.71802
\(848\) −6.00000 −0.206041
\(849\) 14.0000 0.480479
\(850\) −6.00000 −0.205798
\(851\) 0 0
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 4.00000 0.136877
\(855\) −1.00000 −0.0341993
\(856\) −12.0000 −0.410152
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) −24.0000 −0.819346
\(859\) −52.0000 −1.77422 −0.887109 0.461561i \(-0.847290\pi\)
−0.887109 + 0.461561i \(0.847290\pi\)
\(860\) −2.00000 −0.0681994
\(861\) −24.0000 −0.817918
\(862\) 12.0000 0.408722
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) −6.00000 −0.204006
\(866\) −4.00000 −0.135926
\(867\) 19.0000 0.645274
\(868\) 16.0000 0.543075
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) 64.0000 2.16856
\(872\) 2.00000 0.0677285
\(873\) 8.00000 0.270759
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) −10.0000 −0.337869
\(877\) 20.0000 0.675352 0.337676 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 8.00000 0.269987
\(879\) −18.0000 −0.607125
\(880\) −6.00000 −0.202260
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −3.00000 −0.101015
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 24.0000 0.807207
\(885\) 12.0000 0.403376
\(886\) −36.0000 −1.20944
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 8.00000 0.268462
\(889\) 40.0000 1.34156
\(890\) 12.0000 0.402241
\(891\) 6.00000 0.201008
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) −18.0000 −0.602010
\(895\) 12.0000 0.401116
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 36.0000 1.19933
\(902\) −72.0000 −2.39734
\(903\) 4.00000 0.133112
\(904\) −6.00000 −0.199557
\(905\) −2.00000 −0.0664822
\(906\) 8.00000 0.265782
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) 8.00000 0.265197
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 1.00000 0.0331133
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) −2.00000 −0.0661180
\(916\) 2.00000 0.0660819
\(917\) 36.0000 1.18882
\(918\) −6.00000 −0.198030
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) −42.0000 −1.38320
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) 8.00000 0.263038
\(926\) 14.0000 0.460069
\(927\) −16.0000 −0.525509
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −8.00000 −0.262330
\(931\) −3.00000 −0.0983210
\(932\) 6.00000 0.196537
\(933\) −18.0000 −0.589294
\(934\) 36.0000 1.17796
\(935\) 36.0000 1.17733
\(936\) −4.00000 −0.130744
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) −32.0000 −1.04484
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) −24.0000 −0.782378 −0.391189 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) −10.0000 −0.325818
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) −2.00000 −0.0650600
\(946\) 12.0000 0.390154
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 8.00000 0.259828
\(949\) 40.0000 1.29845
\(950\) 1.00000 0.0324443
\(951\) −30.0000 −0.972817
\(952\) −12.0000 −0.388922
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −6.00000 −0.194257
\(955\) −6.00000 −0.194155
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) 12.0000 0.387500
\(960\) −1.00000 −0.0322749
\(961\) 33.0000 1.06452
\(962\) −32.0000 −1.03172
\(963\) −12.0000 −0.386695
\(964\) 2.00000 0.0644157
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −34.0000 −1.09337 −0.546683 0.837340i \(-0.684110\pi\)
−0.546683 + 0.837340i \(0.684110\pi\)
\(968\) 25.0000 0.803530
\(969\) −6.00000 −0.192748
\(970\) −8.00000 −0.256865
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) −16.0000 −0.512673
\(975\) −4.00000 −0.128103
\(976\) 2.00000 0.0640184
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) −10.0000 −0.319765
\(979\) −72.0000 −2.30113
\(980\) 3.00000 0.0958315
\(981\) 2.00000 0.0638551
\(982\) −6.00000 −0.191468
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) −12.0000 −0.382546
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) 0 0
\(990\) −6.00000 −0.190693
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 8.00000 0.254000
\(993\) −28.0000 −0.888553
\(994\) 0 0
\(995\) −20.0000 −0.634043
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 32.0000 1.01294
\(999\) 8.00000 0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 570.2.a.k.1.1 1
3.2 odd 2 1710.2.a.j.1.1 1
4.3 odd 2 4560.2.a.b.1.1 1
5.2 odd 4 2850.2.d.t.799.2 2
5.3 odd 4 2850.2.d.t.799.1 2
5.4 even 2 2850.2.a.c.1.1 1
15.14 odd 2 8550.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.k.1.1 1 1.1 even 1 trivial
1710.2.a.j.1.1 1 3.2 odd 2
2850.2.a.c.1.1 1 5.4 even 2
2850.2.d.t.799.1 2 5.3 odd 4
2850.2.d.t.799.2 2 5.2 odd 4
4560.2.a.b.1.1 1 4.3 odd 2
8550.2.a.v.1.1 1 15.14 odd 2