Properties

Label 462.2.a.g.1.1
Level $462$
Weight $2$
Character 462.1
Self dual yes
Analytic conductor $3.689$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 462.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.68908857338\)
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 \(\chi\) \(=\) 462.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +2.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} +2.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +4.00000 q^{43} +1.00000 q^{44} +2.00000 q^{45} +4.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} +1.00000 q^{66} +12.0000 q^{67} -2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +6.00000 q^{73} -2.00000 q^{74} -1.00000 q^{75} -1.00000 q^{77} -2.00000 q^{78} -8.00000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -8.00000 q^{83} -1.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +1.00000 q^{88} -14.0000 q^{89} +2.00000 q^{90} +2.00000 q^{91} +4.00000 q^{93} +4.00000 q^{94} +1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 1.00000 0.301511
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 2.00000 0.316228
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.00000 −0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −4.00000 −0.496139
\(66\) 1.00000 0.123091
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) 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 \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −2.00000 −0.232495
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) −2.00000 −0.226455
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) −1.00000 −0.109109
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 1.00000 0.106600
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 2.00000 0.210819
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.00000 0.100504
\(100\) −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
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −2.00000 −0.196116
\(105\) −2.00000 −0.195180
\(106\) −2.00000 −0.194257
\(107\) −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 \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 2.00000 0.190693
\(111\) −2.00000 −0.189832
\(112\) −1.00000 −0.0944911
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 2.00000 0.183340
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) −2.00000 −0.181071
\(123\) −10.0000 −0.901670
\(124\) 4.00000 0.359211
\(125\) −12.0000 −1.07331
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.00000 0.352180
\(130\) −4.00000 −0.350823
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 2.00000 0.172133
\(136\) −2.00000 −0.171499
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) −2.00000 −0.169031
\(141\) 4.00000 0.336861
\(142\) 8.00000 0.671345
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) 6.00000 0.496564
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) −1.00000 −0.0805823
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −8.00000 −0.636446
\(159\) −2.00000 −0.158610
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −10.0000 −0.780869
\(165\) 2.00000 0.155700
\(166\) −8.00000 −0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −2.00000 −0.151620
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) −12.0000 −0.901975
\(178\) −14.0000 −1.04934
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 4.00000 0.293294
\(187\) −2.00000 −0.146254
\(188\) 4.00000 0.291730
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −14.0000 −1.00514
\(195\) −4.00000 −0.286446
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 1.00000 0.0710669
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000 0.846415
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) −2.00000 −0.140028
\(205\) −20.0000 −1.39686
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −2.00000 −0.137361
\(213\) 8.00000 0.548151
\(214\) −12.0000 −0.820303
\(215\) 8.00000 0.545595
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) 14.0000 0.948200
\(219\) 6.00000 0.405442
\(220\) 2.00000 0.134840
\(221\) 4.00000 0.269069
\(222\) −2.00000 −0.134231
\(223\) 20.0000 1.33930 0.669650 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −1.00000 −0.0666667
\(226\) 18.0000 1.19734
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 8.00000 0.521862
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) 2.00000 0.129641
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 2.00000 0.129099
\(241\) 30.0000 1.93247 0.966235 0.257663i \(-0.0829523\pi\)
0.966235 + 0.257663i \(0.0829523\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 2.00000 0.127775
\(246\) −10.0000 −0.637577
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) −8.00000 −0.506979
\(250\) −12.0000 −0.758947
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 4.00000 0.249029
\(259\) 2.00000 0.124274
\(260\) −4.00000 −0.248069
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 32.0000 1.97320 0.986602 0.163144i \(-0.0521635\pi\)
0.986602 + 0.163144i \(0.0521635\pi\)
\(264\) 1.00000 0.0615457
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) −14.0000 −0.856786
\(268\) 12.0000 0.733017
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 2.00000 0.121716
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −2.00000 −0.121268
\(273\) 2.00000 0.121046
\(274\) 10.0000 0.604122
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) −2.00000 −0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 4.00000 0.238197
\(283\) 16.0000 0.951101 0.475551 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 10.0000 0.590281
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) −14.0000 −0.820695
\(292\) 6.00000 0.351123
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) −2.00000 −0.116248
\(297\) 1.00000 0.0580259
\(298\) −10.0000 −0.579284
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −4.00000 −0.230556
\(302\) 8.00000 0.460348
\(303\) −10.0000 −0.574485
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) −2.00000 −0.114332
\(307\) 8.00000 0.456584 0.228292 0.973593i \(-0.426686\pi\)
0.228292 + 0.973593i \(0.426686\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 4.00000 0.227552
\(310\) 8.00000 0.454369
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) −2.00000 −0.113228
\(313\) −30.0000 −1.69570 −0.847850 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) 18.0000 1.01580
\(315\) −2.00000 −0.112687
\(316\) −8.00000 −0.450035
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −2.00000 −0.112154
\(319\) −2.00000 −0.111979
\(320\) 2.00000 0.111803
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) 4.00000 0.221540
\(327\) 14.0000 0.774202
\(328\) −10.0000 −0.552158
\(329\) −4.00000 −0.220527
\(330\) 2.00000 0.110096
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −8.00000 −0.439057
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 24.0000 1.31126
\(336\) −1.00000 −0.0545545
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −9.00000 −0.489535
\(339\) 18.0000 0.977626
\(340\) −4.00000 −0.216930
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −2.00000 −0.107211
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 1.00000 0.0534522
\(351\) −2.00000 −0.106752
\(352\) 1.00000 0.0533002
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −12.0000 −0.637793
\(355\) 16.0000 0.849192
\(356\) −14.0000 −0.741999
\(357\) 2.00000 0.105851
\(358\) 12.0000 0.634220
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 2.00000 0.105409
\(361\) −19.0000 −1.00000
\(362\) 2.00000 0.105118
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) 12.0000 0.628109
\(366\) −2.00000 −0.104542
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) −4.00000 −0.207950
\(371\) 2.00000 0.103835
\(372\) 4.00000 0.207390
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −2.00000 −0.103418
\(375\) −12.0000 −0.619677
\(376\) 4.00000 0.206284
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) −8.00000 −0.409316
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.00000 −0.101929
\(386\) 2.00000 0.101797
\(387\) 4.00000 0.203331
\(388\) −14.0000 −0.710742
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) −4.00000 −0.202548
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) 1.00000 0.0502519
\(397\) −30.0000 −1.50566 −0.752828 0.658217i \(-0.771311\pi\)
−0.752828 + 0.658217i \(0.771311\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) 12.0000 0.598506
\(403\) −8.00000 −0.398508
\(404\) −10.0000 −0.497519
\(405\) 2.00000 0.0993808
\(406\) 2.00000 0.0992583
\(407\) −2.00000 −0.0991363
\(408\) −2.00000 −0.0990148
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) −20.0000 −0.987730
\(411\) 10.0000 0.493264
\(412\) 4.00000 0.197066
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −16.0000 −0.785409
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −2.00000 −0.0975900
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 12.0000 0.584151
\(423\) 4.00000 0.194487
\(424\) −2.00000 −0.0971286
\(425\) 2.00000 0.0970143
\(426\) 8.00000 0.387601
\(427\) 2.00000 0.0967868
\(428\) −12.0000 −0.580042
\(429\) −2.00000 −0.0965609
\(430\) 8.00000 0.385794
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) −4.00000 −0.192006
\(435\) −4.00000 −0.191785
\(436\) 14.0000 0.670478
\(437\) 0 0
\(438\) 6.00000 0.286691
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 2.00000 0.0953463
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −28.0000 −1.32733
\(446\) 20.0000 0.947027
\(447\) −10.0000 −0.472984
\(448\) −1.00000 −0.0472456
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −10.0000 −0.470882
\(452\) 18.0000 0.846649
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −14.0000 −0.654177
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −1.00000 −0.0465242
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) −6.00000 −0.277945
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −12.0000 −0.554109
\(470\) 8.00000 0.369012
\(471\) 18.0000 0.829396
\(472\) −12.0000 −0.552345
\(473\) 4.00000 0.183920
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 2.00000 0.0916698
\(477\) −2.00000 −0.0915737
\(478\) 16.0000 0.731823
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 2.00000 0.0912871
\(481\) 4.00000 0.182384
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −28.0000 −1.27141
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 4.00000 0.180886
\(490\) 2.00000 0.0903508
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −10.0000 −0.450835
\(493\) 4.00000 0.180151
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 4.00000 0.179605
\(497\) −8.00000 −0.358849
\(498\) −8.00000 −0.358489
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −20.0000 −0.889988
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −4.00000 −0.177123
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) 8.00000 0.352522
\(516\) 4.00000 0.176090
\(517\) 4.00000 0.175920
\(518\) 2.00000 0.0878750
\(519\) 6.00000 0.263371
\(520\) −4.00000 −0.175412
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 32.0000 1.39527
\(527\) −8.00000 −0.348485
\(528\) 1.00000 0.0435194
\(529\) −23.0000 −1.00000
\(530\) −4.00000 −0.173749
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 20.0000 0.866296
\(534\) −14.0000 −0.605839
\(535\) −24.0000 −1.03761
\(536\) 12.0000 0.518321
\(537\) 12.0000 0.517838
\(538\) 10.0000 0.431131
\(539\) 1.00000 0.0430730
\(540\) 2.00000 0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −8.00000 −0.343629
\(543\) 2.00000 0.0858282
\(544\) −2.00000 −0.0857493
\(545\) 28.0000 1.19939
\(546\) 2.00000 0.0855921
\(547\) −44.0000 −1.88130 −0.940652 0.339372i \(-0.889785\pi\)
−0.940652 + 0.339372i \(0.889785\pi\)
\(548\) 10.0000 0.427179
\(549\) −2.00000 −0.0853579
\(550\) −1.00000 −0.0426401
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −10.0000 −0.424859
\(555\) −4.00000 −0.169791
\(556\) 0 0
\(557\) 14.0000 0.593199 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(558\) 4.00000 0.169334
\(559\) −8.00000 −0.338364
\(560\) −2.00000 −0.0845154
\(561\) −2.00000 −0.0844401
\(562\) 10.0000 0.421825
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 4.00000 0.168430
\(565\) 36.0000 1.51453
\(566\) 16.0000 0.672530
\(567\) −1.00000 −0.0419961
\(568\) 8.00000 0.335673
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −8.00000 −0.334205
\(574\) 10.0000 0.417392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) −13.0000 −0.540729
\(579\) 2.00000 0.0831172
\(580\) −4.00000 −0.166091
\(581\) 8.00000 0.331896
\(582\) −14.0000 −0.580319
\(583\) −2.00000 −0.0828315
\(584\) 6.00000 0.248282
\(585\) −4.00000 −0.165380
\(586\) 14.0000 0.578335
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −24.0000 −0.988064
\(591\) 6.00000 0.246807
\(592\) −2.00000 −0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 1.00000 0.0410305
\(595\) 4.00000 0.163984
\(596\) −10.0000 −0.409616
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) −4.00000 −0.163028
\(603\) 12.0000 0.488678
\(604\) 8.00000 0.325515
\(605\) 2.00000 0.0813116
\(606\) −10.0000 −0.406222
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) −4.00000 −0.161955
\(611\) −8.00000 −0.323645
\(612\) −2.00000 −0.0808452
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 8.00000 0.322854
\(615\) −20.0000 −0.806478
\(616\) −1.00000 −0.0402911
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 4.00000 0.160904
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 8.00000 0.321288
\(621\) 0 0
\(622\) 20.0000 0.801927
\(623\) 14.0000 0.560898
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) −30.0000 −1.19904
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 4.00000 0.159490
\(630\) −2.00000 −0.0796819
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −8.00000 −0.318223
\(633\) 12.0000 0.476957
\(634\) 22.0000 0.873732
\(635\) 16.0000 0.634941
\(636\) −2.00000 −0.0793052
\(637\) −2.00000 −0.0792429
\(638\) −2.00000 −0.0791808
\(639\) 8.00000 0.316475
\(640\) 2.00000 0.0790569
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) 44.0000 1.73519 0.867595 0.497271i \(-0.165665\pi\)
0.867595 + 0.497271i \(0.165665\pi\)
\(644\) 0 0
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 1.00000 0.0392837
\(649\) −12.0000 −0.471041
\(650\) 2.00000 0.0784465
\(651\) −4.00000 −0.156772
\(652\) 4.00000 0.156652
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 6.00000 0.234082
\(658\) −4.00000 −0.155936
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 2.00000 0.0778499
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 20.0000 0.777322
\(663\) 4.00000 0.155347
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) 20.0000 0.773245
\(670\) 24.0000 0.927201
\(671\) −2.00000 −0.0772091
\(672\) −1.00000 −0.0385758
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −22.0000 −0.847408
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) 18.0000 0.691286
\(679\) 14.0000 0.537271
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) 4.00000 0.153168
\(683\) −28.0000 −1.07139 −0.535695 0.844411i \(-0.679950\pi\)
−0.535695 + 0.844411i \(0.679950\pi\)
\(684\) 0 0
\(685\) 20.0000 0.764161
\(686\) −1.00000 −0.0381802
\(687\) −14.0000 −0.534133
\(688\) 4.00000 0.152499
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 6.00000 0.228086
\(693\) −1.00000 −0.0379869
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 20.0000 0.757554
\(698\) −26.0000 −0.984115
\(699\) −6.00000 −0.226941
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 0 0
\(704\) 1.00000 0.0376889
\(705\) 8.00000 0.301297
\(706\) −6.00000 −0.225813
\(707\) 10.0000 0.376089
\(708\) −12.0000 −0.450988
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 16.0000 0.600469
\(711\) −8.00000 −0.300023
\(712\) −14.0000 −0.524672
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) −4.00000 −0.149592
\(716\) 12.0000 0.448461
\(717\) 16.0000 0.597531
\(718\) −32.0000 −1.19423
\(719\) 28.0000 1.04422 0.522112 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 2.00000 0.0745356
\(721\) −4.00000 −0.148968
\(722\) −19.0000 −0.707107
\(723\) 30.0000 1.11571
\(724\) 2.00000 0.0743294
\(725\) 2.00000 0.0742781
\(726\) 1.00000 0.0371135
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 12.0000 0.444140
\(731\) −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 \(0.176889\pi\)
\(734\) −28.0000 −1.03350
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) −10.0000 −0.368105
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 2.00000 0.0734223
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 4.00000 0.146647
\(745\) −20.0000 −0.732743
\(746\) 22.0000 0.805477
\(747\) −8.00000 −0.292705
\(748\) −2.00000 −0.0731272
\(749\) 12.0000 0.438470
\(750\) −12.0000 −0.438178
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 4.00000 0.145865
\(753\) 4.00000 0.145768
\(754\) 4.00000 0.145671
\(755\) 16.0000 0.582300
\(756\) −1.00000 −0.0363696
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −20.0000 −0.726433
\(759\) 0 0
\(760\) 0 0
\(761\) −2.00000 −0.0724999 −0.0362500 0.999343i \(-0.511541\pi\)
−0.0362500 + 0.999343i \(0.511541\pi\)
\(762\) 8.00000 0.289809
\(763\) −14.0000 −0.506834
\(764\) −8.00000 −0.289430
\(765\) −4.00000 −0.144620
\(766\) 12.0000 0.433578
\(767\) 24.0000 0.866590
\(768\) 1.00000 0.0360844
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) −2.00000 −0.0720750
\(771\) −6.00000 −0.216085
\(772\) 2.00000 0.0719816
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) −14.0000 −0.502571
\(777\) 2.00000 0.0717496
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) −4.00000 −0.143223
\(781\) 8.00000 0.286263
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 6.00000 0.213741
\(789\) 32.0000 1.13923
\(790\) −16.0000 −0.569254
\(791\) −18.0000 −0.640006
\(792\) 1.00000 0.0355335
\(793\) 4.00000 0.142044
\(794\) −30.0000 −1.06466
\(795\) −4.00000 −0.141865
\(796\) 20.0000 0.708881
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) −1.00000 −0.0353553
\(801\) −14.0000 −0.494666
\(802\) 18.0000 0.635602
\(803\) 6.00000 0.211735
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 10.0000 0.352017
\(808\) −10.0000 −0.351799
\(809\) −14.0000 −0.492214 −0.246107 0.969243i \(-0.579151\pi\)
−0.246107 + 0.969243i \(0.579151\pi\)
\(810\) 2.00000 0.0702728
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 2.00000 0.0701862
\(813\) −8.00000 −0.280572
\(814\) −2.00000 −0.0701000
\(815\) 8.00000 0.280228
\(816\) −2.00000 −0.0700140
\(817\) 0 0
\(818\) 14.0000 0.489499
\(819\) 2.00000 0.0698857
\(820\) −20.0000 −0.698430
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 10.0000 0.348790
\(823\) −48.0000 −1.67317 −0.836587 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) 4.00000 0.139347
\(825\) −1.00000 −0.0348155
\(826\) 12.0000 0.417533
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) 0 0
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) −16.0000 −0.555368
\(831\) −10.0000 −0.346896
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4.00000 0.138260
\(838\) −12.0000 −0.414533
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 6.00000 0.206774
\(843\) 10.0000 0.344418
\(844\) 12.0000 0.413057
\(845\) −18.0000 −0.619219
\(846\) 4.00000 0.137523
\(847\) −1.00000 −0.0343604
\(848\) −2.00000 −0.0686803
\(849\) 16.0000 0.549119
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) 8.00000 0.274075
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −34.0000 −1.16142 −0.580709 0.814111i \(-0.697225\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 8.00000 0.272798
\(861\) 10.0000 0.340799
\(862\) −40.0000 −1.36241
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.0000 0.408012
\(866\) 18.0000 0.611665
\(867\) −13.0000 −0.441503
\(868\) −4.00000 −0.135769
\(869\) −8.00000 −0.271381
\(870\) −4.00000 −0.135613
\(871\) −24.0000 −0.813209
\(872\) 14.0000 0.474100
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 6.00000 0.202721
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 16.0000 0.539974
\(879\) 14.0000 0.472208
\(880\) 2.00000 0.0674200
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 1.00000 0.0336718
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 4.00000 0.134535
\(885\) −24.0000 −0.806751
\(886\) −12.0000 −0.403148
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −8.00000 −0.268311
\(890\) −28.0000 −0.938562
\(891\) 1.00000 0.0335013
\(892\) 20.0000 0.669650
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 24.0000 0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −30.0000 −1.00111
\(899\) −8.00000 −0.266815
\(900\) −1.00000 −0.0333333
\(901\) 4.00000 0.133259
\(902\) −10.0000 −0.332964
\(903\) −4.00000 −0.133112
\(904\) 18.0000 0.598671
\(905\) 4.00000 0.132964
\(906\) 8.00000 0.265782
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) 0 0
\(909\) −10.0000 −0.331679
\(910\) 4.00000 0.132599
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) −38.0000 −1.25693
\(915\) −4.00000 −0.132236
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 8.00000 0.263609
\(922\) −18.0000 −0.592798
\(923\) −16.0000 −0.526646
\(924\) −1.00000 −0.0328976
\(925\) 2.00000 0.0657596
\(926\) 16.0000 0.525793
\(927\) 4.00000 0.131377
\(928\) −2.00000 −0.0656532
\(929\) −54.0000 −1.77168 −0.885841 0.463988i \(-0.846418\pi\)
−0.885841 + 0.463988i \(0.846418\pi\)
\(930\) 8.00000 0.262330
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 20.0000 0.654771
\(934\) 4.00000 0.130884
\(935\) −4.00000 −0.130814
\(936\) −2.00000 −0.0653720
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) −12.0000 −0.391814
\(939\) −30.0000 −0.979013
\(940\) 8.00000 0.260931
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 18.0000 0.586472
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) −2.00000 −0.0650600
\(946\) 4.00000 0.130051
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −8.00000 −0.259828
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 2.00000 0.0648204
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) −2.00000 −0.0647524
\(955\) −16.0000 −0.517748
\(956\) 16.0000 0.517477
\(957\) −2.00000 −0.0646508
\(958\) −32.0000 −1.03387
\(959\) −10.0000 −0.322917
\(960\) 2.00000 0.0645497
\(961\) −15.0000 −0.483871
\(962\) 4.00000 0.128965
\(963\) −12.0000 −0.386695
\(964\) 30.0000 0.966235
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −28.0000 −0.899026
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −32.0000 −1.02535
\(975\) 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
\(979\) −14.0000 −0.447442
\(980\) 2.00000 0.0638877
\(981\) 14.0000 0.446986
\(982\) −36.0000 −1.14881
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −10.0000 −0.318788
\(985\) 12.0000 0.382352
\(986\) 4.00000 0.127386
\(987\) −4.00000 −0.127321
\(988\) 0 0
\(989\) 0 0
\(990\) 2.00000 0.0635642
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 4.00000 0.127000
\(993\) 20.0000 0.634681
\(994\) −8.00000 −0.253745
\(995\) 40.0000 1.26809
\(996\) −8.00000 −0.253490
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 28.0000 0.886325
\(999\) −2.00000 −0.0632772
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 462.2.a.g.1.1 1
3.2 odd 2 1386.2.a.a.1.1 1
4.3 odd 2 3696.2.a.m.1.1 1
7.6 odd 2 3234.2.a.p.1.1 1
11.10 odd 2 5082.2.a.n.1.1 1
21.20 even 2 9702.2.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.a.g.1.1 1 1.1 even 1 trivial
1386.2.a.a.1.1 1 3.2 odd 2
3234.2.a.p.1.1 1 7.6 odd 2
3696.2.a.m.1.1 1 4.3 odd 2
5082.2.a.n.1.1 1 11.10 odd 2
9702.2.a.r.1.1 1 21.20 even 2