Properties

Label 8470.2.a.bg.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} +6.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} -4.00000 q^{29} -2.00000 q^{30} +1.00000 q^{32} -2.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +8.00000 q^{37} -6.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} +6.00000 q^{46} -4.00000 q^{47} +2.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} -2.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} -1.00000 q^{56} -12.0000 q^{57} -4.00000 q^{58} -2.00000 q^{60} -2.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} -2.00000 q^{68} +12.0000 q^{69} +1.00000 q^{70} -12.0000 q^{71} +1.00000 q^{72} +6.00000 q^{73} +8.00000 q^{74} +2.00000 q^{75} -6.00000 q^{76} -4.00000 q^{78} -10.0000 q^{79} -1.00000 q^{80} -11.0000 q^{81} +12.0000 q^{83} -2.00000 q^{84} +2.00000 q^{85} -4.00000 q^{86} -8.00000 q^{87} +14.0000 q^{89} -1.00000 q^{90} +2.00000 q^{91} +6.00000 q^{92} -4.00000 q^{94} +6.00000 q^{95} +2.00000 q^{96} +4.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) −2.00000 −0.392232
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −2.00000 −0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −6.00000 −0.973329
\(39\) −4.00000 −0.640513
\(40\) −1.00000 −0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 6.00000 0.884652
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) −2.00000 −0.277350
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −12.0000 −1.58944
\(58\) −4.00000 −0.525226
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 12.0000 1.44463
\(70\) 1.00000 0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000 0.117851
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 8.00000 0.929981
\(75\) 2.00000 0.230940
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −1.00000 −0.111803
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) −1.00000 −0.105409
\(91\) 2.00000 0.209657
\(92\) 6.00000 0.625543
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 6.00000 0.615587
\(96\) 2.00000 0.204124
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −4.00000 −0.396059
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) 2.00000 0.195180
\(106\) −12.0000 −1.16554
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) −4.00000 −0.384900
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 16.0000 1.51865
\(112\) −1.00000 −0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −12.0000 −1.12390
\(115\) −6.00000 −0.559503
\(116\) −4.00000 −0.371391
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) −2.00000 −0.182574
\(121\) 0 0
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −8.00000 −0.704361
\(130\) 2.00000 0.175412
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −8.00000 −0.691095
\(135\) 4.00000 0.344265
\(136\) −2.00000 −0.171499
\(137\) −22.0000 −1.87959 −0.939793 0.341743i \(-0.888983\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 12.0000 1.02151
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) 1.00000 0.0845154
\(141\) −8.00000 −0.673722
\(142\) −12.0000 −1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 6.00000 0.496564
\(147\) 2.00000 0.164957
\(148\) 8.00000 0.657596
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 2.00000 0.163299
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) −6.00000 −0.486664
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −10.0000 −0.795557
\(159\) −24.0000 −1.90332
\(160\) −1.00000 −0.0790569
\(161\) −6.00000 −0.472866
\(162\) −11.0000 −0.864242
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) −6.00000 −0.458831
\(172\) −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
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) −4.00000 −0.295689
\(184\) 6.00000 0.442326
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 0 0
\(188\) −4.00000 −0.291730
\(189\) 4.00000 0.290957
\(190\) 6.00000 0.435286
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 2.00000 0.144338
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 4.00000 0.287183
\(195\) 4.00000 0.286446
\(196\) 1.00000 0.0714286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) −16.0000 −1.12855
\(202\) −10.0000 −0.703598
\(203\) 4.00000 0.280745
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 6.00000 0.417029
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) 2.00000 0.138013
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −12.0000 −0.824163
\(213\) −24.0000 −1.64445
\(214\) −20.0000 −1.36717
\(215\) 4.00000 0.272798
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) −4.00000 −0.270914
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) 16.0000 1.07385
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) −6.00000 −0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −12.0000 −0.794719
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) −20.0000 −1.29914
\(238\) 2.00000 0.129641
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −2.00000 −0.129099
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) −2.00000 −0.128037
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 24.0000 1.52094
\(250\) −1.00000 −0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −8.00000 −0.501965
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −28.0000 −1.74659 −0.873296 0.487190i \(-0.838022\pi\)
−0.873296 + 0.487190i \(0.838022\pi\)
\(258\) −8.00000 −0.498058
\(259\) −8.00000 −0.497096
\(260\) 2.00000 0.124035
\(261\) −4.00000 −0.247594
\(262\) 6.00000 0.370681
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 6.00000 0.367884
\(267\) 28.0000 1.71357
\(268\) −8.00000 −0.488678
\(269\) 14.0000 0.853595 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 4.00000 0.243432
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) −2.00000 −0.121268
\(273\) 4.00000 0.242091
\(274\) −22.0000 −1.32907
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 6.00000 0.359856
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −8.00000 −0.476393
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) −12.0000 −0.712069
\(285\) 12.0000 0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) 8.00000 0.468968
\(292\) 6.00000 0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) −12.0000 −0.693978
\(300\) 2.00000 0.115470
\(301\) 4.00000 0.230556
\(302\) 6.00000 0.345261
\(303\) −20.0000 −1.14897
\(304\) −6.00000 −0.344124
\(305\) 2.00000 0.114520
\(306\) −2.00000 −0.114332
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −4.00000 −0.226455
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) −10.0000 −0.564333
\(315\) 1.00000 0.0563436
\(316\) −10.0000 −0.562544
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −24.0000 −1.34585
\(319\) 0 0
\(320\) −1.00000 −0.0559017
\(321\) −40.0000 −2.23258
\(322\) −6.00000 −0.334367
\(323\) 12.0000 0.667698
\(324\) −11.0000 −0.611111
\(325\) −2.00000 −0.110940
\(326\) −24.0000 −1.32924
\(327\) −8.00000 −0.442401
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) −2.00000 −0.109109
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) −9.00000 −0.489535
\(339\) −12.0000 −0.651751
\(340\) 2.00000 0.108465
\(341\) 0 0
\(342\) −6.00000 −0.324443
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) −12.0000 −0.646058
\(346\) 18.0000 0.967686
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) −8.00000 −0.428845
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) 20.0000 1.06449 0.532246 0.846590i \(-0.321348\pi\)
0.532246 + 0.846590i \(0.321348\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) 14.0000 0.741999
\(357\) 4.00000 0.211702
\(358\) −4.00000 −0.211407
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −6.00000 −0.314054
\(366\) −4.00000 −0.209083
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 6.00000 0.312772
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) −2.00000 −0.103280
\(376\) −4.00000 −0.206284
\(377\) 8.00000 0.412021
\(378\) 4.00000 0.205738
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 6.00000 0.307794
\(381\) −16.0000 −0.819705
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −4.00000 −0.203331
\(388\) 4.00000 0.203069
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 4.00000 0.202548
\(391\) −12.0000 −0.606866
\(392\) 1.00000 0.0505076
\(393\) 12.0000 0.605320
\(394\) 2.00000 0.100759
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 12.0000 0.600751
\(400\) 1.00000 0.0500000
\(401\) −34.0000 −1.69788 −0.848939 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) −16.0000 −0.798007
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 11.0000 0.546594
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) −4.00000 −0.198030
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −44.0000 −2.17036
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) −12.0000 −0.589057
\(416\) −2.00000 −0.0980581
\(417\) 12.0000 0.587643
\(418\) 0 0
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 2.00000 0.0975900
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 20.0000 0.973585
\(423\) −4.00000 −0.194487
\(424\) −12.0000 −0.582772
\(425\) −2.00000 −0.0970143
\(426\) −24.0000 −1.16280
\(427\) 2.00000 0.0967868
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 22.0000 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) −4.00000 −0.192450
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 8.00000 0.383571
\(436\) −4.00000 −0.191565
\(437\) −36.0000 −1.72211
\(438\) 12.0000 0.573382
\(439\) −12.0000 −0.572729 −0.286364 0.958121i \(-0.592447\pi\)
−0.286364 + 0.958121i \(0.592447\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) 40.0000 1.90046 0.950229 0.311553i \(-0.100849\pi\)
0.950229 + 0.311553i \(0.100849\pi\)
\(444\) 16.0000 0.759326
\(445\) −14.0000 −0.663664
\(446\) 16.0000 0.757622
\(447\) −8.00000 −0.378387
\(448\) −1.00000 −0.0472456
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 12.0000 0.563809
\(454\) 24.0000 1.12638
\(455\) −2.00000 −0.0937614
\(456\) −12.0000 −0.561951
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 6.00000 0.280362
\(459\) 8.00000 0.373408
\(460\) −6.00000 −0.279751
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 38.0000 1.75843 0.879215 0.476425i \(-0.158068\pi\)
0.879215 + 0.476425i \(0.158068\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 8.00000 0.369406
\(470\) 4.00000 0.184506
\(471\) −20.0000 −0.921551
\(472\) 0 0
\(473\) 0 0
\(474\) −20.0000 −0.918630
\(475\) −6.00000 −0.275299
\(476\) 2.00000 0.0916698
\(477\) −12.0000 −0.549442
\(478\) −6.00000 −0.274434
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −2.00000 −0.0912871
\(481\) −16.0000 −0.729537
\(482\) 8.00000 0.364390
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) −10.0000 −0.453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −48.0000 −2.17064
\(490\) −1.00000 −0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 24.0000 1.07547
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) −8.00000 −0.354943
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 4.00000 0.177123
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 24.0000 1.05963
\(514\) −28.0000 −1.23503
\(515\) 8.00000 0.352522
\(516\) −8.00000 −0.352180
\(517\) 0 0
\(518\) −8.00000 −0.351500
\(519\) 36.0000 1.58022
\(520\) 2.00000 0.0877058
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −4.00000 −0.175075
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 6.00000 0.262111
\(525\) −2.00000 −0.0872872
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 28.0000 1.21168
\(535\) 20.0000 0.864675
\(536\) −8.00000 −0.345547
\(537\) −8.00000 −0.345225
\(538\) 14.0000 0.603583
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) −28.0000 −1.20270
\(543\) 4.00000 0.171656
\(544\) −2.00000 −0.0857493
\(545\) 4.00000 0.171341
\(546\) 4.00000 0.171184
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) −22.0000 −0.939793
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 12.0000 0.510754
\(553\) 10.0000 0.425243
\(554\) 22.0000 0.934690
\(555\) −16.0000 −0.679162
\(556\) 6.00000 0.254457
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.00000 0.252422
\(566\) −32.0000 −1.34506
\(567\) 11.0000 0.461957
\(568\) −12.0000 −0.503509
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 12.0000 0.502625
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 0 0
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −13.0000 −0.540729
\(579\) −4.00000 −0.166234
\(580\) 4.00000 0.166091
\(581\) −12.0000 −0.497844
\(582\) 8.00000 0.331611
\(583\) 0 0
\(584\) 6.00000 0.248282
\(585\) 2.00000 0.0826898
\(586\) 6.00000 0.247858
\(587\) −2.00000 −0.0825488 −0.0412744 0.999148i \(-0.513142\pi\)
−0.0412744 + 0.999148i \(0.513142\pi\)
\(588\) 2.00000 0.0824786
\(589\) 0 0
\(590\) 0 0
\(591\) 4.00000 0.164538
\(592\) 8.00000 0.328798
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −4.00000 −0.163846
\(597\) 32.0000 1.30967
\(598\) −12.0000 −0.490716
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) 2.00000 0.0816497
\(601\) −44.0000 −1.79480 −0.897399 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) 4.00000 0.163028
\(603\) −8.00000 −0.325785
\(604\) 6.00000 0.244137
\(605\) 0 0
\(606\) −20.0000 −0.812444
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −6.00000 −0.243332
\(609\) 8.00000 0.324176
\(610\) 2.00000 0.0809776
\(611\) 8.00000 0.323645
\(612\) −2.00000 −0.0808452
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) −16.0000 −0.643614
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 16.0000 0.641542
\(623\) −14.0000 −0.560898
\(624\) −4.00000 −0.160128
\(625\) 1.00000 0.0400000
\(626\) −12.0000 −0.479616
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −16.0000 −0.637962
\(630\) 1.00000 0.0398410
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −10.0000 −0.397779
\(633\) 40.0000 1.58986
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) −24.0000 −0.951662
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) −40.0000 −1.57867
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −6.00000 −0.236433
\(645\) 8.00000 0.315000
\(646\) 12.0000 0.472134
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −11.0000 −0.432121
\(649\) 0 0
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −8.00000 −0.312825
\(655\) −6.00000 −0.234439
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 4.00000 0.155936
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 12.0000 0.466393
\(663\) 8.00000 0.310694
\(664\) 12.0000 0.465690
\(665\) −6.00000 −0.232670
\(666\) 8.00000 0.309994
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) 32.0000 1.23719
\(670\) 8.00000 0.309067
\(671\) 0 0
\(672\) −2.00000 −0.0771517
\(673\) 42.0000 1.61898 0.809491 0.587133i \(-0.199743\pi\)
0.809491 + 0.587133i \(0.199743\pi\)
\(674\) 2.00000 0.0770371
\(675\) −4.00000 −0.153960
\(676\) −9.00000 −0.346154
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) −12.0000 −0.460857
\(679\) −4.00000 −0.153506
\(680\) 2.00000 0.0766965
\(681\) 48.0000 1.83936
\(682\) 0 0
\(683\) −4.00000 −0.153056 −0.0765279 0.997067i \(-0.524383\pi\)
−0.0765279 + 0.997067i \(0.524383\pi\)
\(684\) −6.00000 −0.229416
\(685\) 22.0000 0.840577
\(686\) −1.00000 −0.0381802
\(687\) 12.0000 0.457829
\(688\) −4.00000 −0.152499
\(689\) 24.0000 0.914327
\(690\) −12.0000 −0.456832
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −6.00000 −0.227593
\(696\) −8.00000 −0.303239
\(697\) 0 0
\(698\) −6.00000 −0.227103
\(699\) 52.0000 1.96682
\(700\) −1.00000 −0.0377964
\(701\) 8.00000 0.302156 0.151078 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) 8.00000 0.301941
\(703\) −48.0000 −1.81035
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 20.0000 0.752710
\(707\) 10.0000 0.376089
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 12.0000 0.450352
\(711\) −10.0000 −0.375029
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) −12.0000 −0.448148
\(718\) 6.00000 0.223918
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) 17.0000 0.632674
\(723\) 16.0000 0.595046
\(724\) 2.00000 0.0743294
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 2.00000 0.0741249
\(729\) 13.0000 0.481481
\(730\) −6.00000 −0.222070
\(731\) 8.00000 0.295891
\(732\) −4.00000 −0.147844
\(733\) 2.00000 0.0738717 0.0369358 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 6.00000 0.221163
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −8.00000 −0.294086
\(741\) 24.0000 0.881662
\(742\) 12.0000 0.440534
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) 4.00000 0.146549
\(746\) 26.0000 0.951928
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) −2.00000 −0.0730297
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) −6.00000 −0.218362
\(756\) 4.00000 0.145479
\(757\) 52.0000 1.88997 0.944986 0.327111i \(-0.106075\pi\)
0.944986 + 0.327111i \(0.106075\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) −16.0000 −0.579619
\(763\) 4.00000 0.144810
\(764\) 8.00000 0.289430
\(765\) 2.00000 0.0723102
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) 2.00000 0.0721688
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) −56.0000 −2.01679
\(772\) −2.00000 −0.0719816
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 4.00000 0.143592
\(777\) −16.0000 −0.573997
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 4.00000 0.143223
\(781\) 0 0
\(782\) −12.0000 −0.429119
\(783\) 16.0000 0.571793
\(784\) 1.00000 0.0357143
\(785\) 10.0000 0.356915
\(786\) 12.0000 0.428026
\(787\) 48.0000 1.71102 0.855508 0.517790i \(-0.173245\pi\)
0.855508 + 0.517790i \(0.173245\pi\)
\(788\) 2.00000 0.0712470
\(789\) −16.0000 −0.569615
\(790\) 10.0000 0.355784
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 14.0000 0.496841
\(795\) 24.0000 0.851192
\(796\) 16.0000 0.567105
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 12.0000 0.424795
\(799\) 8.00000 0.283020
\(800\) 1.00000 0.0353553
\(801\) 14.0000 0.494666
\(802\) −34.0000 −1.20058
\(803\) 0 0
\(804\) −16.0000 −0.564276
\(805\) 6.00000 0.211472
\(806\) 0 0
\(807\) 28.0000 0.985647
\(808\) −10.0000 −0.351799
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 11.0000 0.386501
\(811\) 30.0000 1.05344 0.526721 0.850038i \(-0.323421\pi\)
0.526721 + 0.850038i \(0.323421\pi\)
\(812\) 4.00000 0.140372
\(813\) −56.0000 −1.96401
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) −4.00000 −0.140028
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) −44.0000 −1.53468
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 6.00000 0.208514
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) −12.0000 −0.416526
\(831\) 44.0000 1.52634
\(832\) −2.00000 −0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −16.0000 −0.552711
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 2.00000 0.0690066
\(841\) −13.0000 −0.448276
\(842\) −26.0000 −0.896019
\(843\) 4.00000 0.137767
\(844\) 20.0000 0.688428
\(845\) 9.00000 0.309609
\(846\) −4.00000 −0.137523
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) −64.0000 −2.19647
\(850\) −2.00000 −0.0685994
\(851\) 48.0000 1.64542
\(852\) −24.0000 −0.822226
\(853\) −22.0000 −0.753266 −0.376633 0.926363i \(-0.622918\pi\)
−0.376633 + 0.926363i \(0.622918\pi\)
\(854\) 2.00000 0.0684386
\(855\) 6.00000 0.205196
\(856\) −20.0000 −0.683586
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 22.0000 0.749323
\(863\) 30.0000 1.02121 0.510606 0.859815i \(-0.329421\pi\)
0.510606 + 0.859815i \(0.329421\pi\)
\(864\) −4.00000 −0.136083
\(865\) −18.0000 −0.612018
\(866\) 16.0000 0.543702
\(867\) −26.0000 −0.883006
\(868\) 0 0
\(869\) 0 0
\(870\) 8.00000 0.271225
\(871\) 16.0000 0.542139
\(872\) −4.00000 −0.135457
\(873\) 4.00000 0.135379
\(874\) −36.0000 −1.21772
\(875\) 1.00000 0.0338062
\(876\) 12.0000 0.405442
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −12.0000 −0.404980
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) 1.00000 0.0336718
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 40.0000 1.34383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 16.0000 0.536925
\(889\) 8.00000 0.268311
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 24.0000 0.803129
\(894\) −8.00000 −0.267560
\(895\) 4.00000 0.133705
\(896\) −1.00000 −0.0334077
\(897\) −24.0000 −0.801337
\(898\) 14.0000 0.467186
\(899\) 0 0
\(900\) 1.00000 0.0333333
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) 8.00000 0.266223
\(904\) −6.00000 −0.199557
\(905\) −2.00000 −0.0664822
\(906\) 12.0000 0.398673
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 24.0000 0.796468
\(909\) −10.0000 −0.331679
\(910\) −2.00000 −0.0662994
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −12.0000 −0.397360
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 4.00000 0.132236
\(916\) 6.00000 0.198246
\(917\) −6.00000 −0.198137
\(918\) 8.00000 0.264039
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) −6.00000 −0.197814
\(921\) 24.0000 0.790827
\(922\) 26.0000 0.856264
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −10.0000 −0.328620
\(927\) −8.00000 −0.262754
\(928\) −4.00000 −0.131306
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 26.0000 0.851658
\(933\) 32.0000 1.04763
\(934\) 38.0000 1.24340
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −6.00000 −0.196011 −0.0980057 0.995186i \(-0.531246\pi\)
−0.0980057 + 0.995186i \(0.531246\pi\)
\(938\) 8.00000 0.261209
\(939\) −24.0000 −0.783210
\(940\) 4.00000 0.130466
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −20.0000 −0.651635
\(943\) 0 0
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) −20.0000 −0.649570
\(949\) −12.0000 −0.389536
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) −12.0000 −0.388514
\(955\) −8.00000 −0.258874
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 22.0000 0.710417
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) −16.0000 −0.515861
\(963\) −20.0000 −0.644491
\(964\) 8.00000 0.257663
\(965\) 2.00000 0.0643823
\(966\) −12.0000 −0.386094
\(967\) 56.0000 1.80084 0.900419 0.435023i \(-0.143260\pi\)
0.900419 + 0.435023i \(0.143260\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) −4.00000 −0.128432
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −10.0000 −0.320750
\(973\) −6.00000 −0.192351
\(974\) −18.0000 −0.576757
\(975\) −4.00000 −0.128103
\(976\) −2.00000 −0.0640184
\(977\) −54.0000 −1.72761 −0.863807 0.503824i \(-0.831926\pi\)
−0.863807 + 0.503824i \(0.831926\pi\)
\(978\) −48.0000 −1.53487
\(979\) 0 0
\(980\) −1.00000 −0.0319438
\(981\) −4.00000 −0.127710
\(982\) −12.0000 −0.382935
\(983\) −56.0000 −1.78612 −0.893061 0.449935i \(-0.851447\pi\)
−0.893061 + 0.449935i \(0.851447\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 8.00000 0.254772
\(987\) 8.00000 0.254643
\(988\) 12.0000 0.381771
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 24.0000 0.761617
\(994\) 12.0000 0.380617
\(995\) −16.0000 −0.507234
\(996\) 24.0000 0.760469
\(997\) 42.0000 1.33015 0.665077 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) −28.0000 −0.886325
\(999\) −32.0000 −1.01244
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 8470.2.a.bg.1.1 1
11.10 odd 2 770.2.a.e.1.1 1
33.32 even 2 6930.2.a.bk.1.1 1
44.43 even 2 6160.2.a.a.1.1 1
55.32 even 4 3850.2.c.c.1849.1 2
55.43 even 4 3850.2.c.c.1849.2 2
55.54 odd 2 3850.2.a.m.1.1 1
77.76 even 2 5390.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.e.1.1 1 11.10 odd 2
3850.2.a.m.1.1 1 55.54 odd 2
3850.2.c.c.1849.1 2 55.32 even 4
3850.2.c.c.1849.2 2 55.43 even 4
5390.2.a.c.1.1 1 77.76 even 2
6160.2.a.a.1.1 1 44.43 even 2
6930.2.a.bk.1.1 1 33.32 even 2
8470.2.a.bg.1.1 1 1.1 even 1 trivial