Properties

Label 930.2.a.c.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +1.00000 q^{20} +4.00000 q^{21} -2.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -4.00000 q^{28} -8.00000 q^{29} +1.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -4.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +6.00000 q^{46} -12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} +2.00000 q^{52} -2.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} +4.00000 q^{56} +8.00000 q^{58} -12.0000 q^{59} -1.00000 q^{60} -8.00000 q^{61} -1.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -4.00000 q^{67} +6.00000 q^{69} +4.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -8.00000 q^{77} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -16.0000 q^{83} +4.00000 q^{84} -4.00000 q^{86} +8.00000 q^{87} -2.00000 q^{88} -2.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -6.00000 q^{92} -1.00000 q^{93} +12.0000 q^{94} +1.00000 q^{96} -2.00000 q^{97} -9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 4.00000 1.06904
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 4.00000 0.872872
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −4.00000 −0.755929
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 1.00000 0.182574
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −4.00000 −0.617213
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 6.00000 0.884652
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 4.00000 0.534522
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) −1.00000 −0.129099
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −1.00000 −0.127000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 2.00000 0.246183
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 4.00000 0.478091
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −8.00000 −0.911685
\(78\) 2.00000 0.226455
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 8.00000 0.857690
\(88\) −2.00000 −0.213201
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) −1.00000 −0.105409
\(91\) −8.00000 −0.838628
\(92\) −6.00000 −0.625543
\(93\) −1.00000 −0.103695
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −9.00000 −0.909137
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −2.00000 −0.196116
\(105\) 4.00000 0.390360
\(106\) 2.00000 0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) −2.00000 −0.190693
\(111\) 2.00000 0.189832
\(112\) −4.00000 −0.377964
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −8.00000 −0.742781
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) −6.00000 −0.541002
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) 4.00000 0.356348
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) −2.00000 −0.175412
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −6.00000 −0.510754
\(139\) 18.0000 1.52674 0.763370 0.645961i \(-0.223543\pi\)
0.763370 + 0.645961i \(0.223543\pi\)
\(140\) −4.00000 −0.338062
\(141\) 12.0000 1.01058
\(142\) 8.00000 0.671345
\(143\) 4.00000 0.334497
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) −4.00000 −0.331042
\(147\) −9.00000 −0.742307
\(148\) −2.00000 −0.164399
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 1.00000 0.0816497
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) 1.00000 0.0803219
\(156\) −2.00000 −0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −4.00000 −0.318223
\(159\) 2.00000 0.158610
\(160\) −1.00000 −0.0790569
\(161\) 24.0000 1.89146
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 6.00000 0.468521
\(165\) −2.00000 −0.155700
\(166\) 16.0000 1.24184
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) −4.00000 −0.308607
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) −8.00000 −0.606478
\(175\) −4.00000 −0.302372
\(176\) 2.00000 0.150756
\(177\) 12.0000 0.901975
\(178\) 2.00000 0.149906
\(179\) 14.0000 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(180\) 1.00000 0.0745356
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) 8.00000 0.592999
\(183\) 8.00000 0.591377
\(184\) 6.00000 0.442326
\(185\) −2.00000 −0.147043
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) −12.0000 −0.875190
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) 9.00000 0.642857
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −2.00000 −0.142134
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 10.0000 0.703598
\(203\) 32.0000 2.24596
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 4.00000 0.278693
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) −4.00000 −0.276026
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −2.00000 −0.137361
\(213\) 8.00000 0.548151
\(214\) 4.00000 0.273434
\(215\) 4.00000 0.272798
\(216\) 1.00000 0.0680414
\(217\) −4.00000 −0.271538
\(218\) 18.0000 1.21911
\(219\) −4.00000 −0.270295
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) 4.00000 0.267261
\(225\) 1.00000 0.0666667
\(226\) −18.0000 −1.19734
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 6.00000 0.395628
\(231\) 8.00000 0.526361
\(232\) 8.00000 0.525226
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) −2.00000 −0.130744
\(235\) −12.0000 −0.782794
\(236\) −12.0000 −0.781133
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −8.00000 −0.512148
\(245\) 9.00000 0.574989
\(246\) 6.00000 0.382546
\(247\) 0 0
\(248\) −1.00000 −0.0635001
\(249\) 16.0000 1.01396
\(250\) −1.00000 −0.0632456
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) −4.00000 −0.251976
\(253\) −12.0000 −0.754434
\(254\) 6.00000 0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 4.00000 0.249029
\(259\) 8.00000 0.497096
\(260\) 2.00000 0.124035
\(261\) −8.00000 −0.495188
\(262\) 4.00000 0.247121
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 2.00000 0.123091
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 2.00000 0.122398
\(268\) −4.00000 −0.244339
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 1.00000 0.0608581
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 12.0000 0.724947
\(275\) 2.00000 0.120605
\(276\) 6.00000 0.361158
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −18.0000 −1.07957
\(279\) 1.00000 0.0598684
\(280\) 4.00000 0.239046
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) −12.0000 −0.714590
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) −24.0000 −1.41668
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 8.00000 0.469776
\(291\) 2.00000 0.117242
\(292\) 4.00000 0.234082
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 9.00000 0.524891
\(295\) −12.0000 −0.698667
\(296\) 2.00000 0.116248
\(297\) −2.00000 −0.116052
\(298\) −14.0000 −0.810998
\(299\) −12.0000 −0.693978
\(300\) −1.00000 −0.0577350
\(301\) −16.0000 −0.922225
\(302\) −20.0000 −1.15087
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −8.00000 −0.455842
\(309\) 4.00000 0.227552
\(310\) −1.00000 −0.0567962
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 2.00000 0.113228
\(313\) −4.00000 −0.226093 −0.113047 0.993590i \(-0.536061\pi\)
−0.113047 + 0.993590i \(0.536061\pi\)
\(314\) −18.0000 −1.01580
\(315\) −4.00000 −0.225374
\(316\) 4.00000 0.225018
\(317\) −22.0000 −1.23564 −0.617822 0.786318i \(-0.711985\pi\)
−0.617822 + 0.786318i \(0.711985\pi\)
\(318\) −2.00000 −0.112154
\(319\) −16.0000 −0.895828
\(320\) 1.00000 0.0559017
\(321\) 4.00000 0.223258
\(322\) −24.0000 −1.33747
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 2.00000 0.110940
\(326\) −12.0000 −0.664619
\(327\) 18.0000 0.995402
\(328\) −6.00000 −0.331295
\(329\) 48.0000 2.64633
\(330\) 2.00000 0.110096
\(331\) 26.0000 1.42909 0.714545 0.699590i \(-0.246634\pi\)
0.714545 + 0.699590i \(0.246634\pi\)
\(332\) −16.0000 −0.878114
\(333\) −2.00000 −0.109599
\(334\) 18.0000 0.984916
\(335\) −4.00000 −0.218543
\(336\) 4.00000 0.218218
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) 9.00000 0.489535
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 2.00000 0.108306
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) −4.00000 −0.215666
\(345\) 6.00000 0.323029
\(346\) −18.0000 −0.967686
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) 8.00000 0.428845
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 4.00000 0.213809
\(351\) −2.00000 −0.106752
\(352\) −2.00000 −0.106600
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) −12.0000 −0.637793
\(355\) −8.00000 −0.424596
\(356\) −2.00000 −0.106000
\(357\) 0 0
\(358\) −14.0000 −0.739923
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) 12.0000 0.630706
\(363\) 7.00000 0.367405
\(364\) −8.00000 −0.419314
\(365\) 4.00000 0.209370
\(366\) −8.00000 −0.418167
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −6.00000 −0.312772
\(369\) 6.00000 0.312348
\(370\) 2.00000 0.103975
\(371\) 8.00000 0.415339
\(372\) −1.00000 −0.0518476
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 12.0000 0.618853
\(377\) −16.0000 −0.824042
\(378\) −4.00000 −0.205738
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) −8.00000 −0.409316
\(383\) 6.00000 0.306586 0.153293 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 1.00000 0.0510310
\(385\) −8.00000 −0.407718
\(386\) −22.0000 −1.11977
\(387\) 4.00000 0.203331
\(388\) −2.00000 −0.101535
\(389\) −36.0000 −1.82527 −0.912636 0.408773i \(-0.865957\pi\)
−0.912636 + 0.408773i \(0.865957\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) −9.00000 −0.454569
\(393\) 4.00000 0.201773
\(394\) −2.00000 −0.100759
\(395\) 4.00000 0.201262
\(396\) 2.00000 0.100504
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 12.0000 0.601506
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −4.00000 −0.199502
\(403\) 2.00000 0.0996271
\(404\) −10.0000 −0.497519
\(405\) 1.00000 0.0496904
\(406\) −32.0000 −1.58813
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 30.0000 1.48340 0.741702 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −6.00000 −0.296319
\(411\) 12.0000 0.591916
\(412\) −4.00000 −0.197066
\(413\) 48.0000 2.36193
\(414\) 6.00000 0.294884
\(415\) −16.0000 −0.785409
\(416\) −2.00000 −0.0980581
\(417\) −18.0000 −0.881464
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 4.00000 0.195180
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 20.0000 0.973585
\(423\) −12.0000 −0.583460
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 32.0000 1.54859
\(428\) −4.00000 −0.193347
\(429\) −4.00000 −0.193122
\(430\) −4.00000 −0.192897
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 4.00000 0.192006
\(435\) 8.00000 0.383571
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 4.00000 0.191127
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 2.00000 0.0949158
\(445\) −2.00000 −0.0948091
\(446\) 2.00000 0.0947027
\(447\) −14.0000 −0.662177
\(448\) −4.00000 −0.188982
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) 18.0000 0.846649
\(453\) −20.0000 −0.939682
\(454\) 4.00000 0.187729
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) −4.00000 −0.186908
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 24.0000 1.11779 0.558896 0.829238i \(-0.311225\pi\)
0.558896 + 0.829238i \(0.311225\pi\)
\(462\) −8.00000 −0.372194
\(463\) −42.0000 −1.95191 −0.975953 0.217982i \(-0.930053\pi\)
−0.975953 + 0.217982i \(0.930053\pi\)
\(464\) −8.00000 −0.371391
\(465\) −1.00000 −0.0463739
\(466\) 6.00000 0.277945
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 2.00000 0.0924500
\(469\) 16.0000 0.738811
\(470\) 12.0000 0.553519
\(471\) −18.0000 −0.829396
\(472\) 12.0000 0.552345
\(473\) 8.00000 0.367840
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −2.00000 −0.0915737
\(478\) −8.00000 −0.365911
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 1.00000 0.0456435
\(481\) −4.00000 −0.182384
\(482\) −10.0000 −0.455488
\(483\) −24.0000 −1.09204
\(484\) −7.00000 −0.318182
\(485\) −2.00000 −0.0908153
\(486\) 1.00000 0.0453609
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 8.00000 0.362143
\(489\) −12.0000 −0.542659
\(490\) −9.00000 −0.406579
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) −6.00000 −0.270501
\(493\) 0 0
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 1.00000 0.0449013
\(497\) 32.0000 1.43540
\(498\) −16.0000 −0.716977
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 1.00000 0.0447214
\(501\) 18.0000 0.804181
\(502\) −2.00000 −0.0892644
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 4.00000 0.178174
\(505\) −10.0000 −0.444994
\(506\) 12.0000 0.533465
\(507\) 9.00000 0.399704
\(508\) −6.00000 −0.266207
\(509\) −12.0000 −0.531891 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) −4.00000 −0.176261
\(516\) −4.00000 −0.176090
\(517\) −24.0000 −1.05552
\(518\) −8.00000 −0.351500
\(519\) −18.0000 −0.790112
\(520\) −2.00000 −0.0877058
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 8.00000 0.350150
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) −4.00000 −0.174741
\(525\) 4.00000 0.174574
\(526\) 6.00000 0.261612
\(527\) 0 0
\(528\) −2.00000 −0.0870388
\(529\) 13.0000 0.565217
\(530\) 2.00000 0.0868744
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) −2.00000 −0.0865485
\(535\) −4.00000 −0.172935
\(536\) 4.00000 0.172774
\(537\) −14.0000 −0.604145
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) −1.00000 −0.0430331
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 0 0
\(543\) 12.0000 0.514969
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) −8.00000 −0.342368
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −12.0000 −0.512615
\(549\) −8.00000 −0.341432
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) −6.00000 −0.255377
\(553\) −16.0000 −0.680389
\(554\) 2.00000 0.0849719
\(555\) 2.00000 0.0848953
\(556\) 18.0000 0.763370
\(557\) 42.0000 1.77960 0.889799 0.456354i \(-0.150845\pi\)
0.889799 + 0.456354i \(0.150845\pi\)
\(558\) −1.00000 −0.0423334
\(559\) 8.00000 0.338364
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) 12.0000 0.505291
\(565\) 18.0000 0.757266
\(566\) −20.0000 −0.840663
\(567\) −4.00000 −0.167984
\(568\) 8.00000 0.335673
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −34.0000 −1.42286 −0.711428 0.702759i \(-0.751951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(572\) 4.00000 0.167248
\(573\) −8.00000 −0.334205
\(574\) 24.0000 1.00174
\(575\) −6.00000 −0.250217
\(576\) 1.00000 0.0416667
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) 17.0000 0.707107
\(579\) −22.0000 −0.914289
\(580\) −8.00000 −0.332182
\(581\) 64.0000 2.65517
\(582\) −2.00000 −0.0829027
\(583\) −4.00000 −0.165663
\(584\) −4.00000 −0.165521
\(585\) 2.00000 0.0826898
\(586\) −14.0000 −0.578335
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −9.00000 −0.371154
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) −2.00000 −0.0822690
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 12.0000 0.491127
\(598\) 12.0000 0.490716
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 1.00000 0.0408248
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 16.0000 0.652111
\(603\) −4.00000 −0.162893
\(604\) 20.0000 0.813788
\(605\) −7.00000 −0.284590
\(606\) −10.0000 −0.406222
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) −32.0000 −1.29671
\(610\) 8.00000 0.323911
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 4.00000 0.161427
\(615\) −6.00000 −0.241943
\(616\) 8.00000 0.322329
\(617\) −34.0000 −1.36879 −0.684394 0.729112i \(-0.739933\pi\)
−0.684394 + 0.729112i \(0.739933\pi\)
\(618\) −4.00000 −0.160904
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 1.00000 0.0401610
\(621\) 6.00000 0.240772
\(622\) −24.0000 −0.962312
\(623\) 8.00000 0.320513
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 4.00000 0.159872
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) 0 0
\(630\) 4.00000 0.159364
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −4.00000 −0.159111
\(633\) 20.0000 0.794929
\(634\) 22.0000 0.873732
\(635\) −6.00000 −0.238103
\(636\) 2.00000 0.0793052
\(637\) 18.0000 0.713186
\(638\) 16.0000 0.633446
\(639\) −8.00000 −0.316475
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −4.00000 −0.157867
\(643\) −44.0000 −1.73519 −0.867595 0.497271i \(-0.834335\pi\)
−0.867595 + 0.497271i \(0.834335\pi\)
\(644\) 24.0000 0.945732
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 14.0000 0.550397 0.275198 0.961387i \(-0.411256\pi\)
0.275198 + 0.961387i \(0.411256\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.0000 −0.942082
\(650\) −2.00000 −0.0784465
\(651\) 4.00000 0.156772
\(652\) 12.0000 0.469956
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) −18.0000 −0.703856
\(655\) −4.00000 −0.156293
\(656\) 6.00000 0.234261
\(657\) 4.00000 0.156055
\(658\) −48.0000 −1.87123
\(659\) 44.0000 1.71400 0.856998 0.515319i \(-0.172327\pi\)
0.856998 + 0.515319i \(0.172327\pi\)
\(660\) −2.00000 −0.0778499
\(661\) 30.0000 1.16686 0.583432 0.812162i \(-0.301709\pi\)
0.583432 + 0.812162i \(0.301709\pi\)
\(662\) −26.0000 −1.01052
\(663\) 0 0
\(664\) 16.0000 0.620920
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 48.0000 1.85857
\(668\) −18.0000 −0.696441
\(669\) 2.00000 0.0773245
\(670\) 4.00000 0.154533
\(671\) −16.0000 −0.617673
\(672\) −4.00000 −0.154303
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 16.0000 0.616297
\(675\) −1.00000 −0.0384900
\(676\) −9.00000 −0.346154
\(677\) 50.0000 1.92166 0.960828 0.277145i \(-0.0893883\pi\)
0.960828 + 0.277145i \(0.0893883\pi\)
\(678\) 18.0000 0.691286
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) −2.00000 −0.0765840
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 8.00000 0.305441
\(687\) −4.00000 −0.152610
\(688\) 4.00000 0.152499
\(689\) −4.00000 −0.152388
\(690\) −6.00000 −0.228416
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 18.0000 0.684257
\(693\) −8.00000 −0.303895
\(694\) −16.0000 −0.607352
\(695\) 18.0000 0.682779
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) −2.00000 −0.0757011
\(699\) 6.00000 0.226941
\(700\) −4.00000 −0.151186
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 12.0000 0.451946
\(706\) −4.00000 −0.150542
\(707\) 40.0000 1.50435
\(708\) 12.0000 0.450988
\(709\) −40.0000 −1.50223 −0.751116 0.660171i \(-0.770484\pi\)
−0.751116 + 0.660171i \(0.770484\pi\)
\(710\) 8.00000 0.300235
\(711\) 4.00000 0.150012
\(712\) 2.00000 0.0749532
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) 14.0000 0.523205
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 1.00000 0.0372678
\(721\) 16.0000 0.595871
\(722\) 19.0000 0.707107
\(723\) −10.0000 −0.371904
\(724\) −12.0000 −0.445976
\(725\) −8.00000 −0.297113
\(726\) −7.00000 −0.259794
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) 8.00000 0.296500
\(729\) 1.00000 0.0370370
\(730\) −4.00000 −0.148047
\(731\) 0 0
\(732\) 8.00000 0.295689
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) −18.0000 −0.664392
\(735\) −9.00000 −0.331970
\(736\) 6.00000 0.221163
\(737\) −8.00000 −0.294684
\(738\) −6.00000 −0.220863
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) −2.00000 −0.0735215
\(741\) 0 0
\(742\) −8.00000 −0.293689
\(743\) 26.0000 0.953847 0.476924 0.878945i \(-0.341752\pi\)
0.476924 + 0.878945i \(0.341752\pi\)
\(744\) 1.00000 0.0366618
\(745\) 14.0000 0.512920
\(746\) −6.00000 −0.219676
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) 1.00000 0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −12.0000 −0.437595
\(753\) −2.00000 −0.0728841
\(754\) 16.0000 0.582686
\(755\) 20.0000 0.727875
\(756\) 4.00000 0.145479
\(757\) −18.0000 −0.654221 −0.327111 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) −16.0000 −0.581146
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) −6.00000 −0.217357
\(763\) 72.0000 2.60658
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) −24.0000 −0.866590
\(768\) −1.00000 −0.0360844
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 8.00000 0.288300
\(771\) 14.0000 0.504198
\(772\) 22.0000 0.791797
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −4.00000 −0.143777
\(775\) 1.00000 0.0359211
\(776\) 2.00000 0.0717958
\(777\) −8.00000 −0.286998
\(778\) 36.0000 1.29066
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 8.00000 0.285897
\(784\) 9.00000 0.321429
\(785\) 18.0000 0.642448
\(786\) −4.00000 −0.142675
\(787\) −16.0000 −0.570338 −0.285169 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(788\) 2.00000 0.0712470
\(789\) 6.00000 0.213606
\(790\) −4.00000 −0.142314
\(791\) −72.0000 −2.56003
\(792\) −2.00000 −0.0710669
\(793\) −16.0000 −0.568177
\(794\) −18.0000 −0.638796
\(795\) 2.00000 0.0709327
\(796\) −12.0000 −0.425329
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −2.00000 −0.0706665
\(802\) 10.0000 0.353112
\(803\) 8.00000 0.282314
\(804\) 4.00000 0.141069
\(805\) 24.0000 0.845889
\(806\) −2.00000 −0.0704470
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 32.0000 1.12367 0.561836 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 32.0000 1.12298
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) 0 0
\(818\) −30.0000 −1.04893
\(819\) −8.00000 −0.279543
\(820\) 6.00000 0.209529
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −12.0000 −0.418548
\(823\) −22.0000 −0.766872 −0.383436 0.923567i \(-0.625259\pi\)
−0.383436 + 0.923567i \(0.625259\pi\)
\(824\) 4.00000 0.139347
\(825\) −2.00000 −0.0696311
\(826\) −48.0000 −1.67013
\(827\) −16.0000 −0.556375 −0.278187 0.960527i \(-0.589734\pi\)
−0.278187 + 0.960527i \(0.589734\pi\)
\(828\) −6.00000 −0.208514
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 16.0000 0.555368
\(831\) 2.00000 0.0693792
\(832\) 2.00000 0.0693375
\(833\) 0 0
\(834\) 18.0000 0.623289
\(835\) −18.0000 −0.622916
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) −12.0000 −0.414533
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) −4.00000 −0.138013
\(841\) 35.0000 1.20690
\(842\) −18.0000 −0.620321
\(843\) −6.00000 −0.206651
\(844\) −20.0000 −0.688428
\(845\) −9.00000 −0.309609
\(846\) 12.0000 0.412568
\(847\) 28.0000 0.962091
\(848\) −2.00000 −0.0686803
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 8.00000 0.274075
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) −32.0000 −1.09502
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 4.00000 0.136558
\(859\) 22.0000 0.750630 0.375315 0.926897i \(-0.377534\pi\)
0.375315 + 0.926897i \(0.377534\pi\)
\(860\) 4.00000 0.136399
\(861\) 24.0000 0.817918
\(862\) 16.0000 0.544962
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 1.00000 0.0340207
\(865\) 18.0000 0.612018
\(866\) 4.00000 0.135926
\(867\) 17.0000 0.577350
\(868\) −4.00000 −0.135769
\(869\) 8.00000 0.271381
\(870\) −8.00000 −0.271225
\(871\) −8.00000 −0.271070
\(872\) 18.0000 0.609557
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −4.00000 −0.135225
\(876\) −4.00000 −0.135147
\(877\) 18.0000 0.607817 0.303908 0.952701i \(-0.401708\pi\)
0.303908 + 0.952701i \(0.401708\pi\)
\(878\) 24.0000 0.809961
\(879\) −14.0000 −0.472208
\(880\) 2.00000 0.0674200
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) −9.00000 −0.303046
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) −12.0000 −0.403148
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 24.0000 0.804934
\(890\) 2.00000 0.0670402
\(891\) 2.00000 0.0670025
\(892\) −2.00000 −0.0669650
\(893\) 0 0
\(894\) 14.0000 0.468230
\(895\) 14.0000 0.467968
\(896\) 4.00000 0.133631
\(897\) 12.0000 0.400668
\(898\) 34.0000 1.13459
\(899\) −8.00000 −0.266815
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) −12.0000 −0.399556
\(903\) 16.0000 0.532447
\(904\) −18.0000 −0.598671
\(905\) −12.0000 −0.398893
\(906\) 20.0000 0.664455
\(907\) 52.0000 1.72663 0.863316 0.504664i \(-0.168384\pi\)
0.863316 + 0.504664i \(0.168384\pi\)
\(908\) −4.00000 −0.132745
\(909\) −10.0000 −0.331679
\(910\) 8.00000 0.265197
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) −8.00000 −0.264616
\(915\) 8.00000 0.264472
\(916\) 4.00000 0.132164
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 6.00000 0.197814
\(921\) 4.00000 0.131804
\(922\) −24.0000 −0.790398
\(923\) −16.0000 −0.526646
\(924\) 8.00000 0.263181
\(925\) −2.00000 −0.0657596
\(926\) 42.0000 1.38021
\(927\) −4.00000 −0.131377
\(928\) 8.00000 0.262613
\(929\) −42.0000 −1.37798 −0.688988 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 1.00000 0.0327913
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) −24.0000 −0.785725
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −16.0000 −0.522419
\(939\) 4.00000 0.130535
\(940\) −12.0000 −0.391397
\(941\) 36.0000 1.17357 0.586783 0.809744i \(-0.300394\pi\)
0.586783 + 0.809744i \(0.300394\pi\)
\(942\) 18.0000 0.586472
\(943\) −36.0000 −1.17232
\(944\) −12.0000 −0.390567
\(945\) 4.00000 0.130120
\(946\) −8.00000 −0.260102
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) −4.00000 −0.129914
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 22.0000 0.713399
\(952\) 0 0
\(953\) 4.00000 0.129573 0.0647864 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) 2.00000 0.0647524
\(955\) 8.00000 0.258874
\(956\) 8.00000 0.258738
\(957\) 16.0000 0.517207
\(958\) 8.00000 0.258468
\(959\) 48.0000 1.55000
\(960\) −1.00000 −0.0322749
\(961\) 1.00000 0.0322581
\(962\) 4.00000 0.128965
\(963\) −4.00000 −0.128898
\(964\) 10.0000 0.322078
\(965\) 22.0000 0.708205
\(966\) 24.0000 0.772187
\(967\) 2.00000 0.0643157 0.0321578 0.999483i \(-0.489762\pi\)
0.0321578 + 0.999483i \(0.489762\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −72.0000 −2.30821
\(974\) −2.00000 −0.0640841
\(975\) −2.00000 −0.0640513
\(976\) −8.00000 −0.256074
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 12.0000 0.383718
\(979\) −4.00000 −0.127841
\(980\) 9.00000 0.287494
\(981\) −18.0000 −0.574696
\(982\) 42.0000 1.34027
\(983\) −54.0000 −1.72233 −0.861166 0.508323i \(-0.830265\pi\)
−0.861166 + 0.508323i \(0.830265\pi\)
\(984\) 6.00000 0.191273
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) −48.0000 −1.52786
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) −2.00000 −0.0635642
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) −1.00000 −0.0317500
\(993\) −26.0000 −0.825085
\(994\) −32.0000 −1.01498
\(995\) −12.0000 −0.380426
\(996\) 16.0000 0.506979
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 34.0000 1.07625
\(999\) 2.00000 0.0632772
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 930.2.a.c.1.1 1
3.2 odd 2 2790.2.a.o.1.1 1
4.3 odd 2 7440.2.a.bb.1.1 1
5.2 odd 4 4650.2.d.v.3349.1 2
5.3 odd 4 4650.2.d.v.3349.2 2
5.4 even 2 4650.2.a.bw.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.c.1.1 1 1.1 even 1 trivial
2790.2.a.o.1.1 1 3.2 odd 2
4650.2.a.bw.1.1 1 5.4 even 2
4650.2.d.v.3349.1 2 5.2 odd 4
4650.2.d.v.3349.2 2 5.3 odd 4
7440.2.a.bb.1.1 1 4.3 odd 2