Properties

Label 966.2.a.f.1.1
Level $966$
Weight $2$
Character 966.1
Self dual yes
Analytic conductor $7.714$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 966.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.71354883526\)
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 \(\chi\) \(=\) 966.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} +1.00000 q^{21} -6.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -1.00000 q^{42} +2.00000 q^{43} +6.00000 q^{44} -1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} +5.00000 q^{50} -6.00000 q^{51} +2.00000 q^{52} -12.0000 q^{53} -1.00000 q^{54} -1.00000 q^{56} +2.00000 q^{57} +6.00000 q^{58} +8.00000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -6.00000 q^{66} -10.0000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -1.00000 q^{72} +14.0000 q^{73} -8.00000 q^{74} -5.00000 q^{75} +2.00000 q^{76} +6.00000 q^{77} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +6.00000 q^{83} +1.00000 q^{84} -2.00000 q^{86} -6.00000 q^{87} -6.00000 q^{88} +6.00000 q^{89} +2.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} -1.00000 q^{96} -10.0000 q^{97} -1.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 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
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −6.00000 −1.27920
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) −5.00000 −1.00000
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.00000 −0.324443
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −1.00000 −0.154303
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) −6.00000 −0.840168
\(52\) 2.00000 0.277350
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 2.00000 0.264906
\(58\) 6.00000 0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −8.00000 −0.929981
\(75\) −5.00000 −0.577350
\(76\) 2.00000 0.229416
\(77\) 6.00000 0.683763
\(78\) −2.00000 −0.226455
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) 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
\(86\) −2.00000 −0.215666
\(87\) −6.00000 −0.643268
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 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 \(0.669494\pi\)
\(98\) −1.00000 −0.101015
\(99\) 6.00000 0.603023
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 6.00000 0.594089
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −8.00000 −0.724286
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000 0.522233
\(133\) 2.00000 0.173422
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) 12.0000 0.983078 0.491539 0.870855i \(-0.336434\pi\)
0.491539 + 0.870855i \(0.336434\pi\)
\(150\) 5.00000 0.408248
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −2.00000 −0.162221
\(153\) −6.00000 −0.485071
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −8.00000 −0.636446
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 24.0000 1.85718 0.928588 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 2.00000 0.152499
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 6.00000 0.454859
\(175\) −5.00000 −0.377964
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) −2.00000 −0.148250
\(183\) 8.00000 0.591377
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) −36.0000 −2.63258
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) −6.00000 −0.426401
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 5.00000 0.353553
\(201\) −10.0000 −0.705346
\(202\) 6.00000 0.422159
\(203\) −6.00000 −0.421117
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 1.00000 0.0695048
\(208\) 2.00000 0.138675
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −12.0000 −0.824163
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) 4.00000 0.270914
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −8.00000 −0.536925
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −5.00000 −0.333333
\(226\) −6.00000 −0.399114
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 2.00000 0.132453
\(229\) 8.00000 0.528655 0.264327 0.964433i \(-0.414850\pi\)
0.264327 + 0.964433i \(0.414850\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 6.00000 0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 6.00000 0.388922
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −25.0000 −1.60706
\(243\) 1.00000 0.0641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 4.00000 0.254514
\(248\) −8.00000 −0.508001
\(249\) 6.00000 0.380235
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 1.00000 0.0629941
\(253\) 6.00000 0.377217
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −2.00000 −0.124515
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) −6.00000 −0.369274
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 6.00000 0.367194
\(268\) −10.0000 −0.610847
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 2.00000 0.121046
\(274\) 6.00000 0.362473
\(275\) −30.0000 −1.80907
\(276\) 1.00000 0.0601929
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 16.0000 0.959616
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −10.0000 −0.594438 −0.297219 0.954809i \(-0.596059\pi\)
−0.297219 + 0.954809i \(0.596059\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −12.0000 −0.709575
\(287\) 6.00000 0.354169
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 14.0000 0.819288
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 6.00000 0.348155
\(298\) −12.0000 −0.695141
\(299\) 2.00000 0.115663
\(300\) −5.00000 −0.288675
\(301\) 2.00000 0.115278
\(302\) −8.00000 −0.460348
\(303\) −6.00000 −0.344691
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 6.00000 0.341882
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −2.00000 −0.113228
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 12.0000 0.672927
\(319\) −36.0000 −2.01561
\(320\) 0 0
\(321\) −6.00000 −0.334887
\(322\) −1.00000 −0.0557278
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) −10.0000 −0.554700
\(326\) 16.0000 0.886158
\(327\) −4.00000 −0.221201
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 6.00000 0.329293
\(333\) 8.00000 0.438397
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 48.0000 2.59935
\(342\) −2.00000 −0.108148
\(343\) 1.00000 0.0539949
\(344\) −2.00000 −0.107833
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) −6.00000 −0.321634
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 5.00000 0.267261
\(351\) 2.00000 0.106752
\(352\) −6.00000 −0.319801
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −6.00000 −0.317554
\(358\) 24.0000 1.26844
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −8.00000 −0.420471
\(363\) 25.0000 1.31216
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) −8.00000 −0.418167
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −12.0000 −0.623009
\(372\) 8.00000 0.414781
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) −1.00000 −0.0514344
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 12.0000 0.613973
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 2.00000 0.101666
\(388\) −10.0000 −0.507673
\(389\) 24.0000 1.21685 0.608424 0.793612i \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) 16.0000 0.802008
\(399\) 2.00000 0.100125
\(400\) −5.00000 −0.250000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 10.0000 0.498755
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 48.0000 2.37927
\(408\) 6.00000 0.297044
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −16.0000 −0.783523
\(418\) −12.0000 −0.586939
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) 30.0000 1.45521
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) −6.00000 −0.290021
\(429\) 12.0000 0.579365
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 1.00000 0.0481125
\(433\) 38.0000 1.82616 0.913082 0.407777i \(-0.133696\pi\)
0.913082 + 0.407777i \(0.133696\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 2.00000 0.0956730
\(438\) −14.0000 −0.668946
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 8.00000 0.379663
\(445\) 0 0
\(446\) −8.00000 −0.378811
\(447\) 12.0000 0.567581
\(448\) 1.00000 0.0472456
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 5.00000 0.235702
\(451\) 36.0000 1.69517
\(452\) 6.00000 0.282216
\(453\) 8.00000 0.375873
\(454\) −18.0000 −0.844782
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −34.0000 −1.59045 −0.795226 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) −8.00000 −0.373815
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) −6.00000 −0.279145
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 2.00000 0.0924500
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) −8.00000 −0.367452
\(475\) −10.0000 −0.458831
\(476\) −6.00000 −0.275010
\(477\) −12.0000 −0.549442
\(478\) 24.0000 1.09773
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 10.0000 0.455488
\(483\) 1.00000 0.0455016
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −40.0000 −1.81257 −0.906287 0.422664i \(-0.861095\pi\)
−0.906287 + 0.422664i \(0.861095\pi\)
\(488\) −8.00000 −0.362143
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 6.00000 0.270501
\(493\) 36.0000 1.62136
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −6.00000 −0.268866
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) −6.00000 −0.267793
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) 2.00000 0.0883022
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 6.00000 0.262613
\(523\) 26.0000 1.13690 0.568450 0.822718i \(-0.307543\pi\)
0.568450 + 0.822718i \(0.307543\pi\)
\(524\) 0 0
\(525\) −5.00000 −0.218218
\(526\) 24.0000 1.04645
\(527\) −48.0000 −2.09091
\(528\) 6.00000 0.261116
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) 12.0000 0.519778
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) −24.0000 −1.03568
\(538\) 18.0000 0.776035
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 16.0000 0.687259
\(543\) 8.00000 0.343313
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) 8.00000 0.341432
\(550\) 30.0000 1.27920
\(551\) −12.0000 −0.511217
\(552\) −1.00000 −0.0425628
\(553\) 8.00000 0.340195
\(554\) −26.0000 −1.10463
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) −8.00000 −0.338667
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) −36.0000 −1.51992
\(562\) −6.00000 −0.253095
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 10.0000 0.420331
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 12.0000 0.501745
\(573\) −12.0000 −0.501307
\(574\) −6.00000 −0.250435
\(575\) −5.00000 −0.208514
\(576\) 1.00000 0.0416667
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) −19.0000 −0.790296
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 6.00000 0.248922
\(582\) 10.0000 0.414513
\(583\) −72.0000 −2.98194
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −24.0000 −0.991431
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.00000 0.0412393
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 8.00000 0.328798
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) −16.0000 −0.654836
\(598\) −2.00000 −0.0817861
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 5.00000 0.204124
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −10.0000 −0.407231
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 6.00000 0.243733
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −2.00000 −0.0811107
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) 20.0000 0.807792 0.403896 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 4.00000 0.160904
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 24.0000 0.962312
\(623\) 6.00000 0.240385
\(624\) 2.00000 0.0800641
\(625\) 25.0000 1.00000
\(626\) 34.0000 1.35891
\(627\) 12.0000 0.479234
\(628\) −4.00000 −0.159617
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −8.00000 −0.318223
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 2.00000 0.0792429
\(638\) 36.0000 1.42525
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 6.00000 0.236801
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 1.00000 0.0394055
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 10.0000 0.392232
\(651\) 8.00000 0.313545
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 18.0000 0.701180 0.350590 0.936529i \(-0.385981\pi\)
0.350590 + 0.936529i \(0.385981\pi\)
\(660\) 0 0
\(661\) 8.00000 0.311164 0.155582 0.987823i \(-0.450275\pi\)
0.155582 + 0.987823i \(0.450275\pi\)
\(662\) −20.0000 −0.777322
\(663\) −12.0000 −0.466041
\(664\) −6.00000 −0.232845
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) −6.00000 −0.232321
\(668\) 24.0000 0.928588
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 48.0000 1.85302
\(672\) −1.00000 −0.0385758
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 22.0000 0.847408
\(675\) −5.00000 −0.192450
\(676\) −9.00000 −0.346154
\(677\) 24.0000 0.922395 0.461197 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) −6.00000 −0.230429
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) −48.0000 −1.83801
\(683\) −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
\(686\) −1.00000 −0.0381802
\(687\) 8.00000 0.305219
\(688\) 2.00000 0.0762493
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −18.0000 −0.684257
\(693\) 6.00000 0.227921
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) −36.0000 −1.36360
\(698\) 10.0000 0.378506
\(699\) −6.00000 −0.226941
\(700\) −5.00000 −0.188982
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 16.0000 0.603451
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) −6.00000 −0.224860
\(713\) 8.00000 0.299602
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) −24.0000 −0.896296
\(718\) −36.0000 −1.34351
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 15.0000 0.558242
\(723\) −10.0000 −0.371904
\(724\) 8.00000 0.297318
\(725\) 30.0000 1.11417
\(726\) −25.0000 −0.927837
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 8.00000 0.295689
\(733\) 20.0000 0.738717 0.369358 0.929287i \(-0.379577\pi\)
0.369358 + 0.929287i \(0.379577\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −60.0000 −2.21013
\(738\) −6.00000 −0.220863
\(739\) 8.00000 0.294285 0.147142 0.989115i \(-0.452992\pi\)
0.147142 + 0.989115i \(0.452992\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 12.0000 0.440534
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) 16.0000 0.585802
\(747\) 6.00000 0.219529
\(748\) −36.0000 −1.31629
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −52.0000 −1.88997 −0.944986 0.327111i \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 10.0000 0.363216
\(759\) 6.00000 0.217786
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −8.00000 −0.289809
\(763\) −4.00000 −0.144810
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) −22.0000 −0.791797
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) −2.00000 −0.0718885
\(775\) −40.0000 −1.43684
\(776\) 10.0000 0.358979
\(777\) 8.00000 0.286998
\(778\) −24.0000 −0.860442
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) −6.00000 −0.214423
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 14.0000 0.499046 0.249523 0.968369i \(-0.419726\pi\)
0.249523 + 0.968369i \(0.419726\pi\)
\(788\) −6.00000 −0.213741
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) −6.00000 −0.213201
\(793\) 16.0000 0.568177
\(794\) −26.0000 −0.922705
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) −2.00000 −0.0707992
\(799\) 0 0
\(800\) 5.00000 0.176777
\(801\) 6.00000 0.212000
\(802\) 30.0000 1.05934
\(803\) 84.0000 2.96430
\(804\) −10.0000 −0.352673
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −6.00000 −0.210559
\(813\) −16.0000 −0.561144
\(814\) −48.0000 −1.68240
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 4.00000 0.139942
\(818\) 22.0000 0.769212
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 6.00000 0.209274
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 4.00000 0.139347
\(825\) −30.0000 −1.04447
\(826\) 0 0
\(827\) −30.0000 −1.04320 −0.521601 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) 1.00000 0.0347524
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 2.00000 0.0693375
\(833\) −6.00000 −0.207888
\(834\) 16.0000 0.554035
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 8.00000 0.276520
\(838\) 18.0000 0.621800
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) 6.00000 0.206651
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 25.0000 0.859010
\(848\) −12.0000 −0.412082
\(849\) −10.0000 −0.343199
\(850\) −30.0000 −1.02899
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 6.00000 0.205076
\(857\) −54.0000 −1.84460 −0.922302 0.386469i \(-0.873695\pi\)
−0.922302 + 0.386469i \(0.873695\pi\)
\(858\) −12.0000 −0.409673
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) −12.0000 −0.408722
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −38.0000 −1.29129
\(867\) 19.0000 0.645274
\(868\) 8.00000 0.271538
\(869\) 48.0000 1.62829
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) 4.00000 0.135457
\(873\) −10.0000 −0.338449
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) −8.00000 −0.269987
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −52.0000 −1.74994 −0.874970 0.484178i \(-0.839119\pi\)
−0.874970 + 0.484178i \(0.839119\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 0 0
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) −8.00000 −0.268462
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −12.0000 −0.401340
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 2.00000 0.0667781
\(898\) −6.00000 −0.200223
\(899\) −48.0000 −1.60089
\(900\) −5.00000 −0.166667
\(901\) 72.0000 2.39867
\(902\) −36.0000 −1.19867
\(903\) 2.00000 0.0665558
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 18.0000 0.597351
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 2.00000 0.0662266
\(913\) 36.0000 1.19143
\(914\) 34.0000 1.12462
\(915\) 0 0
\(916\) 8.00000 0.264327
\(917\) 0 0
\(918\) 6.00000 0.198030
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 6.00000 0.197386
\(925\) −40.0000 −1.31519
\(926\) 16.0000 0.525793
\(927\) −4.00000 −0.131377
\(928\) 6.00000 0.196960
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) −30.0000 −0.981630
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −46.0000 −1.50275 −0.751377 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) 10.0000 0.326512
\(939\) −34.0000 −1.10955
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 4.00000 0.130327
\(943\) 6.00000 0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 8.00000 0.259828
\(949\) 28.0000 0.908918
\(950\) 10.0000 0.324443
\(951\) 18.0000 0.583690
\(952\) 6.00000 0.194461
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 12.0000 0.388514
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −36.0000 −1.16371
\(958\) −24.0000 −0.775405
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −16.0000 −0.515861
\(963\) −6.00000 −0.193347
\(964\) −10.0000 −0.322078
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −25.0000 −0.803530
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 1.00000 0.0320750
\(973\) −16.0000 −0.512936
\(974\) 40.0000 1.28168
\(975\) −10.0000 −0.320256
\(976\) 8.00000 0.256074
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) 16.0000 0.511624
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) 60.0000 1.91370 0.956851 0.290578i \(-0.0938475\pi\)
0.956851 + 0.290578i \(0.0938475\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −8.00000 −0.254000
\(993\) 20.0000 0.634681
\(994\) 0 0
\(995\) 0 0
\(996\) 6.00000 0.190117
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 16.0000 0.506471
\(999\) 8.00000 0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 966.2.a.f.1.1 1
3.2 odd 2 2898.2.a.o.1.1 1
4.3 odd 2 7728.2.a.c.1.1 1
7.6 odd 2 6762.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.f.1.1 1 1.1 even 1 trivial
2898.2.a.o.1.1 1 3.2 odd 2
6762.2.a.h.1.1 1 7.6 odd 2
7728.2.a.c.1.1 1 4.3 odd 2