Properties

Label 6422.2.a.h.1.1
Level 6422
Weight 2
Character 6422.1
Self dual yes
Analytic conductor 51.280
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 6422.1

$q$-expansion

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

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 \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 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
\(26\) 0 0
\(27\) −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 \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 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 \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 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
\(52\) 0 0
\(53\) −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 \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 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 \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 3.00000 0.363803
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −2.00000 −0.235702
\(73\) 7.00000 0.819288 0.409644 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) −2.00000 −0.232495
\(75\) −5.00000 −0.577350
\(76\) −1.00000 −0.114708
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) −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 \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 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 \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 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 \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −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 \(0.257729\pi\)
\(104\) 0 0
\(105\) 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 \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 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
\(117\) 0 0
\(118\) −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 \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 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
\(143\) 0 0
\(144\) −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 \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −6.00000 −0.485071
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) −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 \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 6.00000 0.465690
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 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 \(0.476318\pi\)
\(182\) 0 0
\(183\) −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 \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 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
\(208\) 0 0
\(209\) −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
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 1.00000 0.0668153
\(225\) 10.0000 0.666667
\(226\) 6.00000 0.399114
\(227\) 15.0000 0.995585 0.497792 0.867296i \(-0.334144\pi\)
0.497792 + 0.867296i \(0.334144\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 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 \(0.562969\pi\)
\(234\) 0 0
\(235\) 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 \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 25.0000 1.60706
\(243\) 16.0000 1.02640
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 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 \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) 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
\(286\) 0 0
\(287\) 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 \(-0.289795\pi\)
0.613417 + 0.789760i \(0.289795\pi\)
\(294\) −6.00000 −0.349927
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −30.0000 −1.74078
\(298\) 0 0
\(299\) 0 0
\(300\) −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 \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 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 \(0.703007\pi\)
\(312\) 0 0
\(313\) −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 \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) −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
\(325\) 0 0
\(326\) −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 \(-0.491251\pi\)
0.0274825 + 0.999622i \(0.491251\pi\)
\(332\) 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 \(0.534748\pi\)
\(338\) 0 0
\(339\) 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 \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) 6.00000 0.319801
\(353\) 15.0000 0.798369 0.399185 0.916871i \(-0.369293\pi\)
0.399185 + 0.916871i \(0.369293\pi\)
\(354\) −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 \(-0.686964\pi\)
−0.554169 + 0.832404i \(0.686964\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.00000 0.105118
\(363\) 25.0000 1.31216
\(364\) 0 0
\(365\) 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
\(377\) 0 0
\(378\) −5.00000 −0.257172
\(379\) 7.00000 0.359566 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 3.00000 0.153493
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 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 \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 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
\(403\) 0 0
\(404\) 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 \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 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
\(416\) 0 0
\(417\) −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 \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 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
\(429\) 0 0
\(430\) 0 0
\(431\) −6.00000 −0.289010 −0.144505 0.989504i \(-0.546159\pi\)
−0.144505 + 0.989504i \(0.546159\pi\)
\(432\) −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
\(442\) 0 0
\(443\) −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 \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 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 \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 22.0000 1.02799
\(459\) −15.0000 −0.700140
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 6.00000 0.279145
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 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 \(0.363267\pi\)
\(468\) 0 0
\(469\) −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 \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −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 \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −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
\(494\) 0 0
\(495\) 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 \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 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
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 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 \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −11.0000 −0.472490
\(543\) 2.00000 0.0858282
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 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 \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) −8.00000 −0.338667
\(559\) 0 0
\(560\) 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 \(0.526673\pi\)
\(572\) 0 0
\(573\) 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 \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) −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 \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −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 \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −30.0000 −1.23091
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0000 0.450200
\(598\) 0 0
\(599\) −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 \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) 6.00000 0.241747
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 14.0000 0.563163
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) −21.0000 −0.842023
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) 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 \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) −10.0000 −0.397779
\(633\) 5.00000 0.198732
\(634\) 9.00000 0.357436
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) 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 \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) 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
\(650\) 0 0
\(651\) 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 \(-0.418642\pi\)
0.252821 + 0.967513i \(0.418642\pi\)
\(662\) 1.00000 0.0388661
\(663\) 0 0
\(664\) 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
\(676\) 0 0
\(677\) −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 \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 22.0000 0.839352
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 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 \(0.427233\pi\)
\(702\) 0 0
\(703\) 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 \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 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 \(0.740691\pi\)
\(728\) 0 0
\(729\) 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 \(-0.701256\pi\)
−0.590973 + 0.806691i \(0.701256\pi\)
\(734\) −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 \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −3.00000 −0.110133
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 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
\(754\) 0 0
\(755\) 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 \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 2.00000 0.0724524
\(763\) −11.0000 −0.398227
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) −14.0000 −0.503871
\(773\) −51.0000 −1.83434 −0.917171 0.398493i \(-0.869533\pi\)
−0.917171 + 0.398493i \(0.869533\pi\)
\(774\) −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 \(-0.313668\pi\)
0.552515 + 0.833503i \(0.313668\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) −12.0000 −0.426401
\(793\) 0 0
\(794\) −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
\(806\) 0 0
\(807\) −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 \(-0.561864\pi\)
−0.193131 + 0.981173i \(0.561864\pi\)
\(812\) 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
\(819\) 0 0
\(820\) 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 \(-0.694516\pi\)
−0.573761 + 0.819023i \(0.694516\pi\)
\(828\) −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
\(832\) 0 0
\(833\) −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 \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 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 \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −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 \(0.565705\pi\)
\(858\) 0 0
\(859\) 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 \(-0.400888\pi\)
0.306364 + 0.951915i \(0.400888\pi\)
\(864\) −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
\(871\) 0 0
\(872\) −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 \(-0.626947\pi\)
−0.388327 + 0.921521i \(0.626947\pi\)
\(878\) −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 \(0.693869\pi\)
\(884\) 0 0
\(885\) 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
\(897\) 0 0
\(898\) 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
\(923\) 0 0
\(924\) 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 \(-0.682086\pi\)
−0.541347 + 0.840799i \(0.682086\pi\)
\(930\) 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
\(936\) 0 0
\(937\) −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 \(-0.611203\pi\)
−0.342290 + 0.939594i \(0.611203\pi\)
\(942\) −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 \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −10.0000 −0.324785
\(949\) 0 0
\(950\) 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
\(962\) 0 0
\(963\) 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 \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 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
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) −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 \(-0.341208\pi\)
0.478426 + 0.878128i \(0.341208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 27.0000 0.859855
\(987\) 0 0
\(988\) 0 0
\(989\) 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 6422.2.a.h.1.1 1
13.12 even 2 38.2.a.a.1.1 1
39.38 odd 2 342.2.a.e.1.1 1
52.51 odd 2 304.2.a.c.1.1 1
65.12 odd 4 950.2.b.b.799.1 2
65.38 odd 4 950.2.b.b.799.2 2
65.64 even 2 950.2.a.d.1.1 1
91.90 odd 2 1862.2.a.b.1.1 1
104.51 odd 2 1216.2.a.m.1.1 1
104.77 even 2 1216.2.a.e.1.1 1
143.142 odd 2 4598.2.a.p.1.1 1
156.155 even 2 2736.2.a.n.1.1 1
195.194 odd 2 8550.2.a.m.1.1 1
247.12 odd 6 722.2.c.c.429.1 2
247.25 even 18 722.2.e.f.245.1 6
247.51 odd 18 722.2.e.e.245.1 6
247.64 even 6 722.2.c.e.429.1 2
247.90 odd 18 722.2.e.e.595.1 6
247.103 odd 6 722.2.c.c.653.1 2
247.116 odd 18 722.2.e.e.99.1 6
247.129 odd 18 722.2.e.e.415.1 6
247.142 even 18 722.2.e.f.423.1 6
247.155 odd 18 722.2.e.e.389.1 6
247.168 even 18 722.2.e.f.389.1 6
247.181 odd 18 722.2.e.e.423.1 6
247.194 even 18 722.2.e.f.415.1 6
247.207 even 18 722.2.e.f.99.1 6
247.220 even 6 722.2.c.e.653.1 2
247.233 even 18 722.2.e.f.595.1 6
247.246 odd 2 722.2.a.e.1.1 1
260.259 odd 2 7600.2.a.n.1.1 1
741.740 even 2 6498.2.a.f.1.1 1
988.987 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 13.12 even 2
304.2.a.c.1.1 1 52.51 odd 2
342.2.a.e.1.1 1 39.38 odd 2
722.2.a.e.1.1 1 247.246 odd 2
722.2.c.c.429.1 2 247.12 odd 6
722.2.c.c.653.1 2 247.103 odd 6
722.2.c.e.429.1 2 247.64 even 6
722.2.c.e.653.1 2 247.220 even 6
722.2.e.e.99.1 6 247.116 odd 18
722.2.e.e.245.1 6 247.51 odd 18
722.2.e.e.389.1 6 247.155 odd 18
722.2.e.e.415.1 6 247.129 odd 18
722.2.e.e.423.1 6 247.181 odd 18
722.2.e.e.595.1 6 247.90 odd 18
722.2.e.f.99.1 6 247.207 even 18
722.2.e.f.245.1 6 247.25 even 18
722.2.e.f.389.1 6 247.168 even 18
722.2.e.f.415.1 6 247.194 even 18
722.2.e.f.423.1 6 247.142 even 18
722.2.e.f.595.1 6 247.233 even 18
950.2.a.d.1.1 1 65.64 even 2
950.2.b.b.799.1 2 65.12 odd 4
950.2.b.b.799.2 2 65.38 odd 4
1216.2.a.e.1.1 1 104.77 even 2
1216.2.a.m.1.1 1 104.51 odd 2
1862.2.a.b.1.1 1 91.90 odd 2
2736.2.a.n.1.1 1 156.155 even 2
4598.2.a.p.1.1 1 143.142 odd 2
5776.2.a.m.1.1 1 988.987 even 2
6422.2.a.h.1.1 1 1.1 even 1 trivial
6498.2.a.f.1.1 1 741.740 even 2
7600.2.a.n.1.1 1 260.259 odd 2
8550.2.a.m.1.1 1 195.194 odd 2