Properties

Label 39.2.a.a.1.1
Level $39$
Weight $2$
Character 39.1
Self dual yes
Analytic conductor $0.311$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.311416567883\)
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 \(\chi\) \(=\) 39.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} -4.00000 q^{14} -2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.00000 q^{20} +4.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +4.00000 q^{28} -10.0000 q^{29} -2.00000 q^{30} +4.00000 q^{31} +5.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} -8.00000 q^{35} -1.00000 q^{36} -2.00000 q^{37} -1.00000 q^{39} -6.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} -4.00000 q^{44} +2.00000 q^{45} +1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} -1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +12.0000 q^{56} -10.0000 q^{58} +12.0000 q^{59} +2.00000 q^{60} -2.00000 q^{61} +4.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} -4.00000 q^{66} -8.00000 q^{67} -2.00000 q^{68} -8.00000 q^{70} -3.00000 q^{72} +2.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -16.0000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} +4.00000 q^{85} -12.0000 q^{86} +10.0000 q^{87} -12.0000 q^{88} -2.00000 q^{89} +2.00000 q^{90} -4.00000 q^{91} -4.00000 q^{93} -5.00000 q^{96} +10.0000 q^{97} +9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −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
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −4.00000 −1.06904
\(15\) −2.00000 −0.516398
\(16\) −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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −2.00000 −0.447214
\(21\) 4.00000 0.872872
\(22\) 4.00000 0.852803
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) −1.00000 −0.200000
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 4.00000 0.755929
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000 0.883883
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) −8.00000 −1.35225
\(36\) −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
\(39\) −1.00000 −0.160128
\(40\) −6.00000 −0.948683
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 4.00000 0.617213
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) −1.00000 −0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.00000 1.07872
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) −10.0000 −1.31306
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 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
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 2.00000 0.248069
\(66\) −4.00000 −0.492366
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) −8.00000 −0.956183
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −3.00000 −0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −16.0000 −1.82337
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −4.00000 −0.436436
\(85\) 4.00000 0.433861
\(86\) −12.0000 −1.29399
\(87\) 10.0000 1.07211
\(88\) −12.0000 −1.27920
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 2.00000 0.210819
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 9.00000 0.909137
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) −2.00000 −0.198030
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −3.00000 −0.294174
\(105\) 8.00000 0.780720
\(106\) 6.00000 0.582772
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 1.00000 0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 8.00000 0.762770
\(111\) 2.00000 0.189832
\(112\) 4.00000 0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 10.0000 0.928477
\(117\) 1.00000 0.0924500
\(118\) 12.0000 1.10469
\(119\) −8.00000 −0.733359
\(120\) 6.00000 0.547723
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) −4.00000 −0.359211
\(125\) −12.0000 −1.07331
\(126\) −4.00000 −0.356348
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −3.00000 −0.265165
\(129\) 12.0000 1.05654
\(130\) 2.00000 0.175412
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) −2.00000 −0.172133
\(136\) −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
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 8.00000 0.676123
\(141\) 0 0
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) −1.00000 −0.0833333
\(145\) −20.0000 −1.66091
\(146\) 2.00000 0.165521
\(147\) −9.00000 −0.742307
\(148\) 2.00000 0.164399
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 1.00000 0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) −16.0000 −1.28932
\(155\) 8.00000 0.642575
\(156\) 1.00000 0.0800641
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 8.00000 0.636446
\(159\) −6.00000 −0.475831
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) −6.00000 −0.468521
\(165\) −8.00000 −0.622799
\(166\) 4.00000 0.310460
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −12.0000 −0.925820
\(169\) 1.00000 0.0769231
\(170\) 4.00000 0.306786
\(171\) 0 0
\(172\) 12.0000 0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 10.0000 0.758098
\(175\) 4.00000 0.302372
\(176\) −4.00000 −0.301511
\(177\) −12.0000 −0.901975
\(178\) −2.00000 −0.149906
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) −2.00000 −0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −4.00000 −0.296500
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) −4.00000 −0.293294
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −7.00000 −0.505181
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 10.0000 0.717958
\(195\) −2.00000 −0.143223
\(196\) −9.00000 −0.642857
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 4.00000 0.284268
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 3.00000 0.212132
\(201\) 8.00000 0.564276
\(202\) −18.0000 −1.26648
\(203\) 40.0000 2.80745
\(204\) 2.00000 0.140028
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −0.0693375
\(209\) 0 0
\(210\) 8.00000 0.552052
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) −24.0000 −1.63679
\(216\) 3.00000 0.204124
\(217\) −16.0000 −1.08615
\(218\) −2.00000 −0.135457
\(219\) −2.00000 −0.135147
\(220\) −8.00000 −0.539360
\(221\) 2.00000 0.134535
\(222\) 2.00000 0.134231
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −20.0000 −1.33631
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) 30.0000 1.96960
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) −8.00000 −0.518563
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 2.00000 0.129099
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 18.0000 1.14998
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) −12.0000 −0.762001
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) −4.00000 −0.250490
\(256\) −17.0000 −1.06250
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 12.0000 0.747087
\(259\) 8.00000 0.497096
\(260\) −2.00000 −0.124035
\(261\) −10.0000 −0.618984
\(262\) 4.00000 0.247121
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 12.0000 0.738549
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) 8.00000 0.488678
\(269\) 22.0000 1.34136 0.670682 0.741745i \(-0.266002\pi\)
0.670682 + 0.741745i \(0.266002\pi\)
\(270\) −2.00000 −0.121716
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) −2.00000 −0.121268
\(273\) 4.00000 0.242091
\(274\) 6.00000 0.362473
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 12.0000 0.719712
\(279\) 4.00000 0.239474
\(280\) 24.0000 1.43427
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) 12.0000 0.713326 0.356663 0.934233i \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −24.0000 −1.41668
\(288\) 5.00000 0.294628
\(289\) −13.0000 −0.764706
\(290\) −20.0000 −1.17444
\(291\) −10.0000 −0.586210
\(292\) −2.00000 −0.117041
\(293\) −6.00000 −0.350524 −0.175262 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) −9.00000 −0.524891
\(295\) 24.0000 1.39733
\(296\) 6.00000 0.348743
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 48.0000 2.76667
\(302\) 4.00000 0.230174
\(303\) 18.0000 1.03407
\(304\) 0 0
\(305\) −4.00000 −0.229039
\(306\) 2.00000 0.114332
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 16.0000 0.911685
\(309\) 0 0
\(310\) 8.00000 0.454369
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 3.00000 0.169842
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −18.0000 −1.01580
\(315\) −8.00000 −0.450749
\(316\) −8.00000 −0.450035
\(317\) 26.0000 1.46031 0.730153 0.683284i \(-0.239449\pi\)
0.730153 + 0.683284i \(0.239449\pi\)
\(318\) −6.00000 −0.336463
\(319\) −40.0000 −2.23957
\(320\) 14.0000 0.782624
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 8.00000 0.443079
\(327\) 2.00000 0.110600
\(328\) −18.0000 −0.993884
\(329\) 0 0
\(330\) −8.00000 −0.440386
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) −4.00000 −0.219529
\(333\) −2.00000 −0.109599
\(334\) −8.00000 −0.437741
\(335\) −16.0000 −0.874173
\(336\) −4.00000 −0.218218
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 1.00000 0.0543928
\(339\) 6.00000 0.325875
\(340\) −4.00000 −0.216930
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −10.0000 −0.536056
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 4.00000 0.213809
\(351\) −1.00000 −0.0533761
\(352\) 20.0000 1.06600
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 2.00000 0.106000
\(357\) 8.00000 0.423405
\(358\) 4.00000 0.211407
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −6.00000 −0.316228
\(361\) −19.0000 −1.00000
\(362\) −10.0000 −0.525588
\(363\) −5.00000 −0.262432
\(364\) 4.00000 0.209657
\(365\) 4.00000 0.209370
\(366\) 2.00000 0.104542
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) −4.00000 −0.207950
\(371\) −24.0000 −1.24602
\(372\) 4.00000 0.207390
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 8.00000 0.413670
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) −10.0000 −0.515026
\(378\) 4.00000 0.205738
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 8.00000 0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 3.00000 0.153093
\(385\) −32.0000 −1.63087
\(386\) 18.0000 0.916176
\(387\) −12.0000 −0.609994
\(388\) −10.0000 −0.507673
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) −2.00000 −0.101274
\(391\) 0 0
\(392\) −27.0000 −1.36371
\(393\) −4.00000 −0.201773
\(394\) 18.0000 0.906827
\(395\) 16.0000 0.805047
\(396\) −4.00000 −0.201008
\(397\) 38.0000 1.90717 0.953583 0.301131i \(-0.0973643\pi\)
0.953583 + 0.301131i \(0.0973643\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 8.00000 0.399004
\(403\) 4.00000 0.199254
\(404\) 18.0000 0.895533
\(405\) 2.00000 0.0993808
\(406\) 40.0000 1.98517
\(407\) −8.00000 −0.396545
\(408\) 6.00000 0.297044
\(409\) 34.0000 1.68119 0.840596 0.541663i \(-0.182205\pi\)
0.840596 + 0.541663i \(0.182205\pi\)
\(410\) 12.0000 0.592638
\(411\) −6.00000 −0.295958
\(412\) 0 0
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 5.00000 0.245145
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) −8.00000 −0.390360
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) −12.0000 −0.580042
\(429\) −4.00000 −0.193122
\(430\) −24.0000 −1.15738
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) 34.0000 1.63394 0.816968 0.576683i \(-0.195653\pi\)
0.816968 + 0.576683i \(0.195653\pi\)
\(434\) −16.0000 −0.768025
\(435\) 20.0000 0.958927
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) −2.00000 −0.0955637
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −24.0000 −1.14416
\(441\) 9.00000 0.428571
\(442\) 2.00000 0.0951303
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) −2.00000 −0.0949158
\(445\) −4.00000 −0.189618
\(446\) 4.00000 0.189405
\(447\) 6.00000 0.283790
\(448\) −28.0000 −1.32288
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 24.0000 1.13012
\(452\) 6.00000 0.282216
\(453\) −4.00000 −0.187936
\(454\) −20.0000 −0.938647
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) −10.0000 −0.467269
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 16.0000 0.744387
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 10.0000 0.464238
\(465\) −8.00000 −0.370991
\(466\) −14.0000 −0.648537
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 32.0000 1.47762
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) −36.0000 −1.65703
\(473\) −48.0000 −2.20704
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) −24.0000 −1.09773
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) −10.0000 −0.456435
\(481\) −2.00000 −0.0911922
\(482\) 10.0000 0.455488
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 20.0000 0.908153
\(486\) −1.00000 −0.0453609
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 6.00000 0.271607
\(489\) −8.00000 −0.361773
\(490\) 18.0000 0.813157
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 6.00000 0.270501
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 8.00000 0.359573
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 12.0000 0.536656
\(501\) 8.00000 0.357414
\(502\) −12.0000 −0.535586
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 12.0000 0.534522
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 16.0000 0.709885
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) −4.00000 −0.177123
\(511\) −8.00000 −0.353899
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 26.0000 1.14681
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 8.00000 0.351500
\(519\) −6.00000 −0.263371
\(520\) −6.00000 −0.263117
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −10.0000 −0.437688
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) −4.00000 −0.174741
\(525\) −4.00000 −0.174574
\(526\) 24.0000 1.04645
\(527\) 8.00000 0.348485
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 12.0000 0.521247
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 2.00000 0.0865485
\(535\) 24.0000 1.03761
\(536\) 24.0000 1.03664
\(537\) −4.00000 −0.172613
\(538\) 22.0000 0.948487
\(539\) 36.0000 1.55063
\(540\) 2.00000 0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −12.0000 −0.515444
\(543\) 10.0000 0.429141
\(544\) 10.0000 0.428746
\(545\) −4.00000 −0.171341
\(546\) 4.00000 0.171184
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −6.00000 −0.256307
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) −10.0000 −0.424859
\(555\) 4.00000 0.169791
\(556\) −12.0000 −0.508913
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 4.00000 0.169334
\(559\) −12.0000 −0.507546
\(560\) 8.00000 0.338062
\(561\) −8.00000 −0.337760
\(562\) −10.0000 −0.421825
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 12.0000 0.504398
\(567\) −4.00000 −0.167984
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −4.00000 −0.167248
\(573\) −8.00000 −0.334205
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) −13.0000 −0.540729
\(579\) −18.0000 −0.748054
\(580\) 20.0000 0.830455
\(581\) −16.0000 −0.663792
\(582\) −10.0000 −0.414513
\(583\) 24.0000 0.993978
\(584\) −6.00000 −0.248282
\(585\) 2.00000 0.0826898
\(586\) −6.00000 −0.247858
\(587\) 28.0000 1.15568 0.577842 0.816149i \(-0.303895\pi\)
0.577842 + 0.816149i \(0.303895\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 24.0000 0.988064
\(591\) −18.0000 −0.740421
\(592\) 2.00000 0.0821995
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) −4.00000 −0.164122
\(595\) −16.0000 −0.655936
\(596\) 6.00000 0.245770
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) −3.00000 −0.122474
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 48.0000 1.95633
\(603\) −8.00000 −0.325785
\(604\) −4.00000 −0.162758
\(605\) 10.0000 0.406558
\(606\) 18.0000 0.731200
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) −40.0000 −1.62088
\(610\) −4.00000 −0.161955
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −16.0000 −0.645707
\(615\) −12.0000 −0.483887
\(616\) 48.0000 1.93398
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 0 0
\(619\) 24.0000 0.964641 0.482321 0.875995i \(-0.339794\pi\)
0.482321 + 0.875995i \(0.339794\pi\)
\(620\) −8.00000 −0.321288
\(621\) 0 0
\(622\) 0 0
\(623\) 8.00000 0.320513
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −4.00000 −0.159490
\(630\) −8.00000 −0.318728
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −24.0000 −0.954669
\(633\) 20.0000 0.794929
\(634\) 26.0000 1.03259
\(635\) −32.0000 −1.26988
\(636\) 6.00000 0.237915
\(637\) 9.00000 0.356593
\(638\) −40.0000 −1.58362
\(639\) 0 0
\(640\) −6.00000 −0.237171
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −12.0000 −0.473602
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) −8.00000 −0.314512 −0.157256 0.987558i \(-0.550265\pi\)
−0.157256 + 0.987558i \(0.550265\pi\)
\(648\) −3.00000 −0.117851
\(649\) 48.0000 1.88416
\(650\) −1.00000 −0.0392232
\(651\) 16.0000 0.627089
\(652\) −8.00000 −0.313304
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 2.00000 0.0782062
\(655\) 8.00000 0.312586
\(656\) −6.00000 −0.234261
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 8.00000 0.311400
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) −16.0000 −0.621858
\(663\) −2.00000 −0.0776736
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) −4.00000 −0.154649
\(670\) −16.0000 −0.618134
\(671\) −8.00000 −0.308837
\(672\) 20.0000 0.771517
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) −1.00000 −0.0384615
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 6.00000 0.230429
\(679\) −40.0000 −1.53506
\(680\) −12.0000 −0.460179
\(681\) 20.0000 0.766402
\(682\) 16.0000 0.612672
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) −8.00000 −0.305441
\(687\) 10.0000 0.381524
\(688\) 12.0000 0.457496
\(689\) 6.00000 0.228582
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) −6.00000 −0.228086
\(693\) −16.0000 −0.607790
\(694\) −12.0000 −0.455514
\(695\) 24.0000 0.910372
\(696\) −30.0000 −1.13715
\(697\) 12.0000 0.454532
\(698\) −26.0000 −0.984115
\(699\) 14.0000 0.529529
\(700\) −4.00000 −0.151186
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 0 0
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 72.0000 2.70784
\(708\) 12.0000 0.450988
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 8.00000 0.299392
\(715\) 8.00000 0.299183
\(716\) −4.00000 −0.149487
\(717\) 24.0000 0.896296
\(718\) 24.0000 0.895672
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) −19.0000 −0.707107
\(723\) −10.0000 −0.371904
\(724\) 10.0000 0.371647
\(725\) 10.0000 0.371391
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 12.0000 0.444750
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) −24.0000 −0.887672
\(732\) −2.00000 −0.0739221
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 16.0000 0.590571
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) −32.0000 −1.17874
\(738\) 6.00000 0.220863
\(739\) 32.0000 1.17714 0.588570 0.808447i \(-0.299691\pi\)
0.588570 + 0.808447i \(0.299691\pi\)
\(740\) 4.00000 0.147043
\(741\) 0 0
\(742\) −24.0000 −0.881068
\(743\) 48.0000 1.76095 0.880475 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 12.0000 0.439941
\(745\) −12.0000 −0.439646
\(746\) −26.0000 −0.951928
\(747\) 4.00000 0.146352
\(748\) −8.00000 −0.292509
\(749\) −48.0000 −1.75388
\(750\) 12.0000 0.438178
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) −10.0000 −0.364179
\(755\) 8.00000 0.291150
\(756\) −4.00000 −0.145479
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −24.0000 −0.871719
\(759\) 0 0
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 16.0000 0.579619
\(763\) 8.00000 0.289619
\(764\) −8.00000 −0.289430
\(765\) 4.00000 0.144620
\(766\) −16.0000 −0.578103
\(767\) 12.0000 0.433295
\(768\) 17.0000 0.613435
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) −32.0000 −1.15320
\(771\) −26.0000 −0.936367
\(772\) −18.0000 −0.647834
\(773\) 10.0000 0.359675 0.179838 0.983696i \(-0.442443\pi\)
0.179838 + 0.983696i \(0.442443\pi\)
\(774\) −12.0000 −0.431331
\(775\) −4.00000 −0.143684
\(776\) −30.0000 −1.07694
\(777\) −8.00000 −0.286998
\(778\) 22.0000 0.788738
\(779\) 0 0
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) −9.00000 −0.321429
\(785\) −36.0000 −1.28490
\(786\) −4.00000 −0.142675
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) −18.0000 −0.641223
\(789\) −24.0000 −0.854423
\(790\) 16.0000 0.569254
\(791\) 24.0000 0.853342
\(792\) −12.0000 −0.426401
\(793\) −2.00000 −0.0710221
\(794\) 38.0000 1.34857
\(795\) −12.0000 −0.425596
\(796\) −8.00000 −0.283552
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −5.00000 −0.176777
\(801\) −2.00000 −0.0706665
\(802\) 22.0000 0.776847
\(803\) 8.00000 0.282314
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −22.0000 −0.774437
\(808\) 54.0000 1.89971
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 2.00000 0.0702728
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) −40.0000 −1.40372
\(813\) 12.0000 0.420858
\(814\) −8.00000 −0.280400
\(815\) 16.0000 0.560456
\(816\) 2.00000 0.0700140
\(817\) 0 0
\(818\) 34.0000 1.18878
\(819\) −4.00000 −0.139771
\(820\) −12.0000 −0.419058
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) −6.00000 −0.209274
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 4.00000 0.139262
\(826\) −48.0000 −1.67013
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 8.00000 0.277684
\(831\) 10.0000 0.346896
\(832\) 7.00000 0.242681
\(833\) 18.0000 0.623663
\(834\) −12.0000 −0.415526
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 4.00000 0.138178
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) −24.0000 −0.828079
\(841\) 71.0000 2.44828
\(842\) −10.0000 −0.344623
\(843\) 10.0000 0.344418
\(844\) 20.0000 0.688428
\(845\) 2.00000 0.0688021
\(846\) 0 0
\(847\) −20.0000 −0.687208
\(848\) −6.00000 −0.206041
\(849\) −12.0000 −0.411839
\(850\) −2.00000 −0.0685994
\(851\) 0 0
\(852\) 0 0
\(853\) 30.0000 1.02718 0.513590 0.858036i \(-0.328315\pi\)
0.513590 + 0.858036i \(0.328315\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) −36.0000 −1.23045
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) −4.00000 −0.136558
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 24.0000 0.818393
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) 16.0000 0.544646 0.272323 0.962206i \(-0.412208\pi\)
0.272323 + 0.962206i \(0.412208\pi\)
\(864\) −5.00000 −0.170103
\(865\) 12.0000 0.408012
\(866\) 34.0000 1.15537
\(867\) 13.0000 0.441503
\(868\) 16.0000 0.543075
\(869\) 32.0000 1.08553
\(870\) 20.0000 0.678064
\(871\) −8.00000 −0.271070
\(872\) 6.00000 0.203186
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 48.0000 1.62270
\(876\) 2.00000 0.0675737
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 32.0000 1.07995
\(879\) 6.00000 0.202375
\(880\) −8.00000 −0.269680
\(881\) 58.0000 1.95407 0.977035 0.213080i \(-0.0683494\pi\)
0.977035 + 0.213080i \(0.0683494\pi\)
\(882\) 9.00000 0.303046
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) −2.00000 −0.0672673
\(885\) −24.0000 −0.806751
\(886\) −4.00000 −0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −6.00000 −0.201347
\(889\) 64.0000 2.14649
\(890\) −4.00000 −0.134080
\(891\) 4.00000 0.134005
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 6.00000 0.200670
\(895\) 8.00000 0.267411
\(896\) 12.0000 0.400892
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) −40.0000 −1.33407
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) −48.0000 −1.59734
\(904\) 18.0000 0.598671
\(905\) −20.0000 −0.664822
\(906\) −4.00000 −0.132891
\(907\) 12.0000 0.398453 0.199227 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 20.0000 0.663723
\(909\) −18.0000 −0.597022
\(910\) −8.00000 −0.265197
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) 16.0000 0.529523
\(914\) 2.00000 0.0661541
\(915\) 4.00000 0.132236
\(916\) 10.0000 0.330409
\(917\) −16.0000 −0.528367
\(918\) −2.00000 −0.0660098
\(919\) −48.0000 −1.58337 −0.791687 0.610927i \(-0.790797\pi\)
−0.791687 + 0.610927i \(0.790797\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) −38.0000 −1.25146
\(923\) 0 0
\(924\) −16.0000 −0.526361
\(925\) 2.00000 0.0657596
\(926\) 4.00000 0.131448
\(927\) 0 0
\(928\) −50.0000 −1.64133
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 0 0
\(934\) 12.0000 0.392652
\(935\) 16.0000 0.523256
\(936\) −3.00000 −0.0980581
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 32.0000 1.04484
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 18.0000 0.586472
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 8.00000 0.260240
\(946\) −48.0000 −1.56061
\(947\) −60.0000 −1.94974 −0.974869 0.222779i \(-0.928487\pi\)
−0.974869 + 0.222779i \(0.928487\pi\)
\(948\) 8.00000 0.259828
\(949\) 2.00000 0.0649227
\(950\) 0 0
\(951\) −26.0000 −0.843108
\(952\) 24.0000 0.777844
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 6.00000 0.194257
\(955\) 16.0000 0.517748
\(956\) 24.0000 0.776215
\(957\) 40.0000 1.29302
\(958\) −24.0000 −0.775405
\(959\) −24.0000 −0.775000
\(960\) −14.0000 −0.451848
\(961\) −15.0000 −0.483871
\(962\) −2.00000 −0.0644826
\(963\) 12.0000 0.386695
\(964\) −10.0000 −0.322078
\(965\) 36.0000 1.15888
\(966\) 0 0
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) −15.0000 −0.482118
\(969\) 0 0
\(970\) 20.0000 0.642161
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 1.00000 0.0320750
\(973\) −48.0000 −1.53881
\(974\) 12.0000 0.384505
\(975\) 1.00000 0.0320256
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −8.00000 −0.255812
\(979\) −8.00000 −0.255681
\(980\) −18.0000 −0.574989
\(981\) −2.00000 −0.0638551
\(982\) −12.0000 −0.382935
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 18.0000 0.573819
\(985\) 36.0000 1.14706
\(986\) −20.0000 −0.636930
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 8.00000 0.254257
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 20.0000 0.635001
\(993\) 16.0000 0.507745
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 4.00000 0.126745
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) −24.0000 −0.759707
\(999\) 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 39.2.a.a.1.1 1
3.2 odd 2 117.2.a.a.1.1 1
4.3 odd 2 624.2.a.i.1.1 1
5.2 odd 4 975.2.c.f.274.2 2
5.3 odd 4 975.2.c.f.274.1 2
5.4 even 2 975.2.a.f.1.1 1
7.6 odd 2 1911.2.a.f.1.1 1
8.3 odd 2 2496.2.a.e.1.1 1
8.5 even 2 2496.2.a.q.1.1 1
9.2 odd 6 1053.2.e.d.352.1 2
9.4 even 3 1053.2.e.b.703.1 2
9.5 odd 6 1053.2.e.d.703.1 2
9.7 even 3 1053.2.e.b.352.1 2
11.10 odd 2 4719.2.a.c.1.1 1
12.11 even 2 1872.2.a.h.1.1 1
13.2 odd 12 507.2.j.e.316.2 4
13.3 even 3 507.2.e.a.22.1 2
13.4 even 6 507.2.e.b.484.1 2
13.5 odd 4 507.2.b.a.337.1 2
13.6 odd 12 507.2.j.e.361.1 4
13.7 odd 12 507.2.j.e.361.2 4
13.8 odd 4 507.2.b.a.337.2 2
13.9 even 3 507.2.e.a.484.1 2
13.10 even 6 507.2.e.b.22.1 2
13.11 odd 12 507.2.j.e.316.1 4
13.12 even 2 507.2.a.a.1.1 1
15.2 even 4 2925.2.c.e.2224.1 2
15.8 even 4 2925.2.c.e.2224.2 2
15.14 odd 2 2925.2.a.p.1.1 1
21.20 even 2 5733.2.a.e.1.1 1
24.5 odd 2 7488.2.a.bl.1.1 1
24.11 even 2 7488.2.a.by.1.1 1
39.5 even 4 1521.2.b.b.1351.2 2
39.8 even 4 1521.2.b.b.1351.1 2
39.38 odd 2 1521.2.a.e.1.1 1
52.51 odd 2 8112.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.a.1.1 1 1.1 even 1 trivial
117.2.a.a.1.1 1 3.2 odd 2
507.2.a.a.1.1 1 13.12 even 2
507.2.b.a.337.1 2 13.5 odd 4
507.2.b.a.337.2 2 13.8 odd 4
507.2.e.a.22.1 2 13.3 even 3
507.2.e.a.484.1 2 13.9 even 3
507.2.e.b.22.1 2 13.10 even 6
507.2.e.b.484.1 2 13.4 even 6
507.2.j.e.316.1 4 13.11 odd 12
507.2.j.e.316.2 4 13.2 odd 12
507.2.j.e.361.1 4 13.6 odd 12
507.2.j.e.361.2 4 13.7 odd 12
624.2.a.i.1.1 1 4.3 odd 2
975.2.a.f.1.1 1 5.4 even 2
975.2.c.f.274.1 2 5.3 odd 4
975.2.c.f.274.2 2 5.2 odd 4
1053.2.e.b.352.1 2 9.7 even 3
1053.2.e.b.703.1 2 9.4 even 3
1053.2.e.d.352.1 2 9.2 odd 6
1053.2.e.d.703.1 2 9.5 odd 6
1521.2.a.e.1.1 1 39.38 odd 2
1521.2.b.b.1351.1 2 39.8 even 4
1521.2.b.b.1351.2 2 39.5 even 4
1872.2.a.h.1.1 1 12.11 even 2
1911.2.a.f.1.1 1 7.6 odd 2
2496.2.a.e.1.1 1 8.3 odd 2
2496.2.a.q.1.1 1 8.5 even 2
2925.2.a.p.1.1 1 15.14 odd 2
2925.2.c.e.2224.1 2 15.2 even 4
2925.2.c.e.2224.2 2 15.8 even 4
4719.2.a.c.1.1 1 11.10 odd 2
5733.2.a.e.1.1 1 21.20 even 2
7488.2.a.bl.1.1 1 24.5 odd 2
7488.2.a.by.1.1 1 24.11 even 2
8112.2.a.s.1.1 1 52.51 odd 2