Properties

Label 38.2.a.a.1.1
Level 38
Weight 2
Character 38.1
Self dual yes
Analytic conductor 0.303
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 38 = 2 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.303431527681\)
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
Root \(0\)
Character \(\chi\) = 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} -6.00000 q^{11} +1.00000 q^{12} +5.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +2.00000 q^{18} +1.00000 q^{19} -1.00000 q^{21} +6.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -5.00000 q^{26} -5.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -3.00000 q^{34} -2.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} +5.00000 q^{39} +1.00000 q^{42} +8.00000 q^{43} -6.00000 q^{44} -3.00000 q^{46} +1.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +3.00000 q^{51} +5.00000 q^{52} -3.00000 q^{53} +5.00000 q^{54} +1.00000 q^{56} +1.00000 q^{57} -9.00000 q^{58} +9.00000 q^{59} -10.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +6.00000 q^{66} +5.00000 q^{67} +3.00000 q^{68} +3.00000 q^{69} -6.00000 q^{71} +2.00000 q^{72} -7.00000 q^{73} -2.00000 q^{74} -5.00000 q^{75} +1.00000 q^{76} +6.00000 q^{77} -5.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -1.00000 q^{84} -8.00000 q^{86} +9.00000 q^{87} +6.00000 q^{88} -12.0000 q^{89} -5.00000 q^{91} +3.00000 q^{92} -4.00000 q^{93} -1.00000 q^{96} -10.0000 q^{97} +6.00000 q^{98} +12.0000 q^{99} +O(q^{100})\)

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 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 1.00000 0.288675
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 2.00000 0.471405
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 6.00000 1.27920
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −5.00000 −0.980581
\(27\) −5.00000 −0.962250
\(28\) −1.00000 −0.188982
\(29\) 9.00000 1.67126 0.835629 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 1.00000 0.154303
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 5.00000 0.707107
\(51\) 3.00000 0.420084
\(52\) 5.00000 0.693375
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.00000 0.132453
\(58\) −9.00000 −1.18176
\(59\) 9.00000 1.17170 0.585850 0.810419i \(-0.300761\pi\)
0.585850 + 0.810419i \(0.300761\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 4.00000 0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 3.00000 0.363803
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 2.00000 0.235702
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) −2.00000 −0.232495
\(75\) −5.00000 −0.577350
\(76\) 1.00000 0.114708
\(77\) 6.00000 0.683763
\(78\) −5.00000 −0.566139
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.00000 −0.658586 −0.329293 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 9.00000 0.964901
\(88\) 6.00000 0.639602
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) −5.00000 −0.524142
\(92\) 3.00000 0.312772
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 6.00000 0.606092
\(99\) 12.0000 1.20605
\(100\) −5.00000 −0.500000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) −3.00000 −0.297044
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) −5.00000 −0.481125
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −10.0000 −0.924500
\(118\) −9.00000 −0.828517
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) −6.00000 −0.522233
\(133\) −1.00000 −0.0867110
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) −3.00000 −0.255377
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) −30.0000 −2.50873
\(144\) −2.00000 −0.166667
\(145\) 0 0
\(146\) 7.00000 0.579324
\(147\) −6.00000 −0.494872
\(148\) 2.00000 0.164399
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 5.00000 0.408248
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −6.00000 −0.485071
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 5.00000 0.400320
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) 10.0000 0.795557
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) −1.00000 −0.0785674
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 8.00000 0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) −9.00000 −0.682288
\(175\) 5.00000 0.377964
\(176\) −6.00000 −0.452267
\(177\) 9.00000 0.676481
\(178\) 12.0000 0.899438
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 5.00000 0.370625
\(183\) −10.0000 −0.739221
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −18.0000 −1.31629
\(188\) 0 0
\(189\) 5.00000 0.363696
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) −12.0000 −0.852803
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 5.00000 0.353553
\(201\) 5.00000 0.352673
\(202\) −18.0000 −1.26648
\(203\) −9.00000 −0.631676
\(204\) 3.00000 0.210042
\(205\) 0 0
\(206\) −14.0000 −0.975426
\(207\) −6.00000 −0.417029
\(208\) 5.00000 0.346688
\(209\) −6.00000 −0.415029
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −3.00000 −0.206041
\(213\) −6.00000 −0.411113
\(214\) 9.00000 0.615227
\(215\) 0 0
\(216\) 5.00000 0.340207
\(217\) 4.00000 0.271538
\(218\) −11.0000 −0.745014
\(219\) −7.00000 −0.473016
\(220\) 0 0
\(221\) 15.0000 1.00901
\(222\) −2.00000 −0.134231
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) 1.00000 0.0668153
\(225\) 10.0000 0.666667
\(226\) −6.00000 −0.399114
\(227\) −15.0000 −0.995585 −0.497792 0.867296i \(-0.665856\pi\)
−0.497792 + 0.867296i \(0.665856\pi\)
\(228\) 1.00000 0.0662266
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) −9.00000 −0.590879
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 10.0000 0.653720
\(235\) 0 0
\(236\) 9.00000 0.585850
\(237\) −10.0000 −0.649570
\(238\) 3.00000 0.194461
\(239\) −21.0000 −1.35838 −0.679189 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) −25.0000 −1.60706
\(243\) 16.0000 1.02640
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 5.00000 0.318142
\(248\) 4.00000 0.254000
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 2.00000 0.125988
\(253\) −18.0000 −1.13165
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) −8.00000 −0.498058
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −18.0000 −1.11417
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 1.00000 0.0613139
\(267\) −12.0000 −0.734388
\(268\) 5.00000 0.305424
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 11.0000 0.668202 0.334101 0.942537i \(-0.391567\pi\)
0.334101 + 0.942537i \(0.391567\pi\)
\(272\) 3.00000 0.181902
\(273\) −5.00000 −0.302614
\(274\) 9.00000 0.543710
\(275\) 30.0000 1.80907
\(276\) 3.00000 0.180579
\(277\) 8.00000 0.480673 0.240337 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) 4.00000 0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −22.0000 −1.30776 −0.653882 0.756596i \(-0.726861\pi\)
−0.653882 + 0.756596i \(0.726861\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 30.0000 1.77394
\(287\) 0 0
\(288\) 2.00000 0.117851
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) −7.00000 −0.409644
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 30.0000 1.74078
\(298\) 0 0
\(299\) 15.0000 0.867472
\(300\) −5.00000 −0.288675
\(301\) −8.00000 −0.461112
\(302\) 10.0000 0.575435
\(303\) 18.0000 1.03407
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 6.00000 0.341882
\(309\) 14.0000 0.796432
\(310\) 0 0
\(311\) −21.0000 −1.19080 −0.595400 0.803429i \(-0.703007\pi\)
−0.595400 + 0.803429i \(0.703007\pi\)
\(312\) −5.00000 −0.283069
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) 3.00000 0.168232
\(319\) −54.0000 −3.02342
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 3.00000 0.167183
\(323\) 3.00000 0.166924
\(324\) 1.00000 0.0555556
\(325\) −25.0000 −1.38675
\(326\) −20.0000 −1.10770
\(327\) 11.0000 0.608301
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) −6.00000 −0.329293
\(333\) −4.00000 −0.219199
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) −12.0000 −0.652714
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 2.00000 0.108148
\(343\) 13.0000 0.701934
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 18.0000 0.966291 0.483145 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 9.00000 0.482451
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −5.00000 −0.267261
\(351\) −25.0000 −1.33440
\(352\) 6.00000 0.319801
\(353\) −15.0000 −0.798369 −0.399185 0.916871i \(-0.630707\pi\)
−0.399185 + 0.916871i \(0.630707\pi\)
\(354\) −9.00000 −0.478345
\(355\) 0 0
\(356\) −12.0000 −0.635999
\(357\) −3.00000 −0.158777
\(358\) 0 0
\(359\) 21.0000 1.10834 0.554169 0.832404i \(-0.313036\pi\)
0.554169 + 0.832404i \(0.313036\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −2.00000 −0.105118
\(363\) 25.0000 1.31216
\(364\) −5.00000 −0.262071
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 3.00000 0.156386
\(369\) 0 0
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) −4.00000 −0.207390
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) 45.0000 2.31762
\(378\) −5.00000 −0.257172
\(379\) −7.00000 −0.359566 −0.179783 0.983706i \(-0.557540\pi\)
−0.179783 + 0.983706i \(0.557540\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −3.00000 −0.153493
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) −16.0000 −0.813326
\(388\) −10.0000 −0.507673
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 9.00000 0.455150
\(392\) 6.00000 0.303046
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 12.0000 0.603023
\(397\) 20.0000 1.00377 0.501886 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) −11.0000 −0.551380
\(399\) −1.00000 −0.0500626
\(400\) −5.00000 −0.250000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) −5.00000 −0.249377
\(403\) −20.0000 −0.996271
\(404\) 18.0000 0.895533
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) −12.0000 −0.594818
\(408\) −3.00000 −0.148522
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) −9.00000 −0.443937
\(412\) 14.0000 0.689730
\(413\) −9.00000 −0.442861
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) −4.00000 −0.195881
\(418\) 6.00000 0.293470
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) −5.00000 −0.243396
\(423\) 0 0
\(424\) 3.00000 0.145693
\(425\) −15.0000 −0.727607
\(426\) 6.00000 0.290701
\(427\) 10.0000 0.483934
\(428\) −9.00000 −0.435031
\(429\) −30.0000 −1.44841
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) −5.00000 −0.240563
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 3.00000 0.143509
\(438\) 7.00000 0.334473
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 12.0000 0.571429
\(442\) −15.0000 −0.713477
\(443\) −18.0000 −0.855206 −0.427603 0.903967i \(-0.640642\pi\)
−0.427603 + 0.903967i \(0.640642\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −10.0000 −0.471405
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) −10.0000 −0.469841
\(454\) 15.0000 0.703985
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) 22.0000 1.02799
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) −6.00000 −0.279145
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) −10.0000 −0.462250
\(469\) −5.00000 −0.230879
\(470\) 0 0
\(471\) −22.0000 −1.01371
\(472\) −9.00000 −0.414259
\(473\) −48.0000 −2.20704
\(474\) 10.0000 0.459315
\(475\) −5.00000 −0.229416
\(476\) −3.00000 −0.137505
\(477\) 6.00000 0.274721
\(478\) 21.0000 0.960518
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) −8.00000 −0.364390
\(483\) −3.00000 −0.136505
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 10.0000 0.452679
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 27.0000 1.21602
\(494\) −5.00000 −0.224961
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) 6.00000 0.269137
\(498\) 6.00000 0.268866
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) −6.00000 −0.267793
\(503\) −21.0000 −0.936344 −0.468172 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 18.0000 0.800198
\(507\) 12.0000 0.532939
\(508\) 2.00000 0.0887357
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) −1.00000 −0.0441942
\(513\) −5.00000 −0.220755
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −36.0000 −1.57719 −0.788594 0.614914i \(-0.789191\pi\)
−0.788594 + 0.614914i \(0.789191\pi\)
\(522\) 18.0000 0.787839
\(523\) 11.0000 0.480996 0.240498 0.970650i \(-0.422689\pi\)
0.240498 + 0.970650i \(0.422689\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) −24.0000 −1.04645
\(527\) −12.0000 −0.522728
\(528\) −6.00000 −0.261116
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −18.0000 −0.781133
\(532\) −1.00000 −0.0433555
\(533\) 0 0
\(534\) 12.0000 0.519291
\(535\) 0 0
\(536\) −5.00000 −0.215967
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −11.0000 −0.472490
\(543\) 2.00000 0.0858282
\(544\) −3.00000 −0.128624
\(545\) 0 0
\(546\) 5.00000 0.213980
\(547\) 44.0000 1.88130 0.940652 0.339372i \(-0.110215\pi\)
0.940652 + 0.339372i \(0.110215\pi\)
\(548\) −9.00000 −0.384461
\(549\) 20.0000 0.853579
\(550\) −30.0000 −1.27920
\(551\) 9.00000 0.383413
\(552\) −3.00000 −0.127688
\(553\) 10.0000 0.425243
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 24.0000 1.01691 0.508456 0.861088i \(-0.330216\pi\)
0.508456 + 0.861088i \(0.330216\pi\)
\(558\) −8.00000 −0.338667
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) −1.00000 −0.0419961
\(568\) 6.00000 0.251754
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −30.0000 −1.25436
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) −15.0000 −0.625543
\(576\) −2.00000 −0.0833333
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 8.00000 0.332756
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 10.0000 0.414513
\(583\) 18.0000 0.745484
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −6.00000 −0.247436
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 2.00000 0.0821995
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) −30.0000 −1.23091
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0000 0.450200
\(598\) −15.0000 −0.613396
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 5.00000 0.204124
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 8.00000 0.326056
\(603\) −10.0000 −0.407231
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −14.0000 −0.563163
\(619\) −10.0000 −0.401934 −0.200967 0.979598i \(-0.564408\pi\)
−0.200967 + 0.979598i \(0.564408\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) 21.0000 0.842023
\(623\) 12.0000 0.480770
\(624\) 5.00000 0.200160
\(625\) 25.0000 1.00000
\(626\) 19.0000 0.759393
\(627\) −6.00000 −0.239617
\(628\) −22.0000 −0.877896
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 10.0000 0.397779
\(633\) 5.00000 0.198732
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) −30.0000 −1.18864
\(638\) 54.0000 2.13788
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 9.00000 0.355202
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −54.0000 −2.11969
\(650\) 25.0000 0.980581
\(651\) 4.00000 0.156772
\(652\) 20.0000 0.783260
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) −11.0000 −0.430134
\(655\) 0 0
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) −45.0000 −1.75295 −0.876476 0.481446i \(-0.840112\pi\)
−0.876476 + 0.481446i \(0.840112\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) 1.00000 0.0388661
\(663\) 15.0000 0.582552
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 27.0000 1.04544
\(668\) 12.0000 0.464294
\(669\) 26.0000 1.00522
\(670\) 0 0
\(671\) 60.0000 2.31627
\(672\) 1.00000 0.0385758
\(673\) 44.0000 1.69608 0.848038 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) 4.00000 0.154074
\(675\) 25.0000 0.962250
\(676\) 12.0000 0.461538
\(677\) −33.0000 −1.26829 −0.634147 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) −6.00000 −0.230429
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −15.0000 −0.574801
\(682\) −24.0000 −0.919007
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) −22.0000 −0.839352
\(688\) 8.00000 0.304997
\(689\) −15.0000 −0.571454
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) 6.00000 0.228086
\(693\) −12.0000 −0.455842
\(694\) −18.0000 −0.683271
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) −6.00000 −0.226941
\(700\) 5.00000 0.188982
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 25.0000 0.943564
\(703\) 2.00000 0.0754314
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) 15.0000 0.564532
\(707\) −18.0000 −0.676960
\(708\) 9.00000 0.338241
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 20.0000 0.750059
\(712\) 12.0000 0.449719
\(713\) −12.0000 −0.449404
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) 0 0
\(717\) −21.0000 −0.784259
\(718\) −21.0000 −0.783713
\(719\) 39.0000 1.45445 0.727227 0.686397i \(-0.240809\pi\)
0.727227 + 0.686397i \(0.240809\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) −1.00000 −0.0372161
\(723\) 8.00000 0.297523
\(724\) 2.00000 0.0743294
\(725\) −45.0000 −1.67126
\(726\) −25.0000 −0.927837
\(727\) −37.0000 −1.37225 −0.686127 0.727482i \(-0.740691\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 5.00000 0.185312
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) 32.0000 1.18195 0.590973 0.806691i \(-0.298744\pi\)
0.590973 + 0.806691i \(0.298744\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −3.00000 −0.110581
\(737\) −30.0000 −1.10506
\(738\) 0 0
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 0 0
\(741\) 5.00000 0.183680
\(742\) −3.00000 −0.110133
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) −23.0000 −0.842090
\(747\) 12.0000 0.439057
\(748\) −18.0000 −0.658145
\(749\) 9.00000 0.328853
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) −45.0000 −1.63880
\(755\) 0 0
\(756\) 5.00000 0.181848
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 7.00000 0.254251
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) −2.00000 −0.0724524
\(763\) −11.0000 −0.398227
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 45.0000 1.62486
\(768\) 1.00000 0.0360844
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 14.0000 0.503871
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) 16.0000 0.575108
\(775\) 20.0000 0.718421
\(776\) 10.0000 0.358979
\(777\) −2.00000 −0.0717496
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) −9.00000 −0.321839
\(783\) −45.0000 −1.60817
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 0 0
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −12.0000 −0.426401
\(793\) −50.0000 −1.77555
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) −39.0000 −1.38145 −0.690725 0.723117i \(-0.742709\pi\)
−0.690725 + 0.723117i \(0.742709\pi\)
\(798\) 1.00000 0.0353996
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 24.0000 0.847998
\(802\) 0 0
\(803\) 42.0000 1.48215
\(804\) 5.00000 0.176336
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) −6.00000 −0.211210
\(808\) −18.0000 −0.633238
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) 11.0000 0.386262 0.193131 0.981173i \(-0.438136\pi\)
0.193131 + 0.981173i \(0.438136\pi\)
\(812\) −9.00000 −0.315838
\(813\) 11.0000 0.385787
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 8.00000 0.279885
\(818\) −32.0000 −1.11885
\(819\) 10.0000 0.349428
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 9.00000 0.313911
\(823\) 41.0000 1.42917 0.714585 0.699549i \(-0.246616\pi\)
0.714585 + 0.699549i \(0.246616\pi\)
\(824\) −14.0000 −0.487713
\(825\) 30.0000 1.04447
\(826\) 9.00000 0.313150
\(827\) 33.0000 1.14752 0.573761 0.819023i \(-0.305484\pi\)
0.573761 + 0.819023i \(0.305484\pi\)
\(828\) −6.00000 −0.208514
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) 8.00000 0.277517
\(832\) 5.00000 0.173344
\(833\) −18.0000 −0.623663
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) −6.00000 −0.207514
\(837\) 20.0000 0.691301
\(838\) 12.0000 0.414533
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −17.0000 −0.585859
\(843\) 0 0
\(844\) 5.00000 0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) −25.0000 −0.859010
\(848\) −3.00000 −0.103020
\(849\) −22.0000 −0.755038
\(850\) 15.0000 0.514496
\(851\) 6.00000 0.205677
\(852\) −6.00000 −0.205557
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) 9.00000 0.307614
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 30.0000 1.02418
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −6.00000 −0.204361
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −8.00000 −0.271694
\(868\) 4.00000 0.135769
\(869\) 60.0000 2.03536
\(870\) 0 0
\(871\) 25.0000 0.847093
\(872\) −11.0000 −0.372507
\(873\) 20.0000 0.676897
\(874\) −3.00000 −0.101477
\(875\) 0 0
\(876\) −7.00000 −0.236508
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) 28.0000 0.944954
\(879\) −21.0000 −0.708312
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −12.0000 −0.404061
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) 15.0000 0.504505
\(885\) 0 0
\(886\) 18.0000 0.604722
\(887\) −42.0000 −1.41022 −0.705111 0.709097i \(-0.749103\pi\)
−0.705111 + 0.709097i \(0.749103\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) −6.00000 −0.201008
\(892\) 26.0000 0.870544
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) 15.0000 0.500835
\(898\) 18.0000 0.600668
\(899\) −36.0000 −1.20067
\(900\) 10.0000 0.333333
\(901\) −9.00000 −0.299833
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) −37.0000 −1.22856 −0.614282 0.789086i \(-0.710554\pi\)
−0.614282 + 0.789086i \(0.710554\pi\)
\(908\) −15.0000 −0.497792
\(909\) −36.0000 −1.19404
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 1.00000 0.0331133
\(913\) 36.0000 1.19143
\(914\) −17.0000 −0.562310
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 15.0000 0.495074
\(919\) −7.00000 −0.230909 −0.115454 0.993313i \(-0.536832\pi\)
−0.115454 + 0.993313i \(0.536832\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 12.0000 0.395199
\(923\) −30.0000 −0.987462
\(924\) 6.00000 0.197386
\(925\) −10.0000 −0.328798
\(926\) 4.00000 0.131448
\(927\) −28.0000 −0.919641
\(928\) −9.00000 −0.295439
\(929\) 33.0000 1.08269 0.541347 0.840799i \(-0.317914\pi\)
0.541347 + 0.840799i \(0.317914\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −6.00000 −0.196537
\(933\) −21.0000 −0.687509
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) 10.0000 0.326860
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 5.00000 0.163256
\(939\) −19.0000 −0.620042
\(940\) 0 0
\(941\) 21.0000 0.684580 0.342290 0.939594i \(-0.388797\pi\)
0.342290 + 0.939594i \(0.388797\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) 9.00000 0.292925
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) −10.0000 −0.324785
\(949\) −35.0000 −1.13615
\(950\) 5.00000 0.162221
\(951\) −9.00000 −0.291845
\(952\) 3.00000 0.0972306
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) −21.0000 −0.679189
\(957\) −54.0000 −1.74557
\(958\) −36.0000 −1.16311
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −10.0000 −0.322413
\(963\) 18.0000 0.580042
\(964\) 8.00000 0.257663
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) −25.0000 −0.803530
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 16.0000 0.513200
\(973\) 4.00000 0.128234
\(974\) −2.00000 −0.0640841
\(975\) −25.0000 −0.800641
\(976\) −10.0000 −0.320092
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) −20.0000 −0.639529
\(979\) 72.0000 2.30113
\(980\) 0 0
\(981\) −22.0000 −0.702406
\(982\) 36.0000 1.14881
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −27.0000 −0.859855
\(987\) 0 0
\(988\) 5.00000 0.159071
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 4.00000 0.127000
\(993\) −1.00000 −0.0317340
\(994\) −6.00000 −0.190308
\(995\) 0 0
\(996\) −6.00000 −0.190117
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 4.00000 0.126618
\(999\) −10.0000 −0.316386
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 38.2.a.a.1.1 1
3.2 odd 2 342.2.a.e.1.1 1
4.3 odd 2 304.2.a.c.1.1 1
5.2 odd 4 950.2.b.b.799.1 2
5.3 odd 4 950.2.b.b.799.2 2
5.4 even 2 950.2.a.d.1.1 1
7.6 odd 2 1862.2.a.b.1.1 1
8.3 odd 2 1216.2.a.m.1.1 1
8.5 even 2 1216.2.a.e.1.1 1
11.10 odd 2 4598.2.a.p.1.1 1
12.11 even 2 2736.2.a.n.1.1 1
13.12 even 2 6422.2.a.h.1.1 1
15.14 odd 2 8550.2.a.m.1.1 1
19.2 odd 18 722.2.e.e.99.1 6
19.3 odd 18 722.2.e.e.389.1 6
19.4 even 9 722.2.e.f.415.1 6
19.5 even 9 722.2.e.f.595.1 6
19.6 even 9 722.2.e.f.245.1 6
19.7 even 3 722.2.c.e.429.1 2
19.8 odd 6 722.2.c.c.653.1 2
19.9 even 9 722.2.e.f.423.1 6
19.10 odd 18 722.2.e.e.423.1 6
19.11 even 3 722.2.c.e.653.1 2
19.12 odd 6 722.2.c.c.429.1 2
19.13 odd 18 722.2.e.e.245.1 6
19.14 odd 18 722.2.e.e.595.1 6
19.15 odd 18 722.2.e.e.415.1 6
19.16 even 9 722.2.e.f.389.1 6
19.17 even 9 722.2.e.f.99.1 6
19.18 odd 2 722.2.a.e.1.1 1
20.19 odd 2 7600.2.a.n.1.1 1
57.56 even 2 6498.2.a.f.1.1 1
76.75 even 2 5776.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.2.a.a.1.1 1 1.1 even 1 trivial
304.2.a.c.1.1 1 4.3 odd 2
342.2.a.e.1.1 1 3.2 odd 2
722.2.a.e.1.1 1 19.18 odd 2
722.2.c.c.429.1 2 19.12 odd 6
722.2.c.c.653.1 2 19.8 odd 6
722.2.c.e.429.1 2 19.7 even 3
722.2.c.e.653.1 2 19.11 even 3
722.2.e.e.99.1 6 19.2 odd 18
722.2.e.e.245.1 6 19.13 odd 18
722.2.e.e.389.1 6 19.3 odd 18
722.2.e.e.415.1 6 19.15 odd 18
722.2.e.e.423.1 6 19.10 odd 18
722.2.e.e.595.1 6 19.14 odd 18
722.2.e.f.99.1 6 19.17 even 9
722.2.e.f.245.1 6 19.6 even 9
722.2.e.f.389.1 6 19.16 even 9
722.2.e.f.415.1 6 19.4 even 9
722.2.e.f.423.1 6 19.9 even 9
722.2.e.f.595.1 6 19.5 even 9
950.2.a.d.1.1 1 5.4 even 2
950.2.b.b.799.1 2 5.2 odd 4
950.2.b.b.799.2 2 5.3 odd 4
1216.2.a.e.1.1 1 8.5 even 2
1216.2.a.m.1.1 1 8.3 odd 2
1862.2.a.b.1.1 1 7.6 odd 2
2736.2.a.n.1.1 1 12.11 even 2
4598.2.a.p.1.1 1 11.10 odd 2
5776.2.a.m.1.1 1 76.75 even 2
6422.2.a.h.1.1 1 13.12 even 2
6498.2.a.f.1.1 1 57.56 even 2
7600.2.a.n.1.1 1 20.19 odd 2
8550.2.a.m.1.1 1 15.14 odd 2