Properties

Label 5070.2.a.n.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} -2.00000 q^{21} -4.00000 q^{22} +2.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} +8.00000 q^{29} +1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +4.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} -10.0000 q^{41} -2.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} +2.00000 q^{46} -1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -4.00000 q^{51} +6.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +2.00000 q^{56} -2.00000 q^{57} +8.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} +8.00000 q^{67} +4.00000 q^{68} -2.00000 q^{69} -2.00000 q^{70} +1.00000 q^{72} -6.00000 q^{74} -1.00000 q^{75} +2.00000 q^{76} -8.00000 q^{77} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000 q^{83} -2.00000 q^{84} -4.00000 q^{85} +4.00000 q^{86} -8.00000 q^{87} -4.00000 q^{88} +10.0000 q^{89} -1.00000 q^{90} +2.00000 q^{92} +4.00000 q^{93} -2.00000 q^{95} -1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 2.00000 0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) −4.00000 −0.852803
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.00000 0.377964
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.00000 0.696311
\(34\) 4.00000 0.685994
\(35\) −2.00000 −0.338062
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) −1.00000 −0.149071
\(46\) 2.00000 0.294884
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 1.00000 0.141421
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.00000 0.539360
\(56\) 2.00000 0.267261
\(57\) −2.00000 −0.264906
\(58\) 8.00000 1.05045
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 4.00000 0.485071
\(69\) −2.00000 −0.240772
\(70\) −2.00000 −0.239046
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −6.00000 −0.697486
\(75\) −1.00000 −0.115470
\(76\) 2.00000 0.229416
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −10.0000 −1.10432
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.00000 −0.218218
\(85\) −4.00000 −0.433861
\(86\) 4.00000 0.431331
\(87\) −8.00000 −0.857690
\(88\) −4.00000 −0.426401
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 2.00000 0.208514
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) −1.00000 −0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −3.00000 −0.303046
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) 20.0000 1.99007 0.995037 0.0995037i \(-0.0317255\pi\)
0.995037 + 0.0995037i \(0.0317255\pi\)
\(102\) −4.00000 −0.396059
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 6.00000 0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 4.00000 0.381385
\(111\) 6.00000 0.569495
\(112\) 2.00000 0.188982
\(113\) −16.0000 −1.50515 −0.752577 0.658505i \(-0.771189\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −2.00000 −0.187317
\(115\) −2.00000 −0.186501
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 8.00000 0.733359
\(120\) 1.00000 0.0912871
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 10.0000 0.901670
\(124\) −4.00000 −0.359211
\(125\) −1.00000 −0.0894427
\(126\) 2.00000 0.178174
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 4.00000 0.348155
\(133\) 4.00000 0.346844
\(134\) 8.00000 0.691095
\(135\) 1.00000 0.0860663
\(136\) 4.00000 0.342997
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) −2.00000 −0.170251
\(139\) −16.0000 −1.35710 −0.678551 0.734553i \(-0.737392\pi\)
−0.678551 + 0.734553i \(0.737392\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) −6.00000 −0.493197
\(149\) 22.0000 1.80231 0.901155 0.433497i \(-0.142720\pi\)
0.901155 + 0.433497i \(0.142720\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 2.00000 0.162221
\(153\) 4.00000 0.323381
\(154\) −8.00000 −0.644658
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −8.00000 −0.636446
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 4.00000 0.315244
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −10.0000 −0.780869
\(165\) −4.00000 −0.311400
\(166\) 12.0000 0.931381
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) −2.00000 −0.154303
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) 2.00000 0.152944
\(172\) 4.00000 0.304997
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −8.00000 −0.606478
\(175\) 2.00000 0.151186
\(176\) −4.00000 −0.301511
\(177\) −12.0000 −0.901975
\(178\) 10.0000 0.749532
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 2.00000 0.147442
\(185\) 6.00000 0.441129
\(186\) 4.00000 0.293294
\(187\) −16.0000 −1.17004
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) −2.00000 −0.145095
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −12.0000 −0.863779 −0.431889 0.901927i \(-0.642153\pi\)
−0.431889 + 0.901927i \(0.642153\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) −4.00000 −0.284268
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 1.00000 0.0707107
\(201\) −8.00000 −0.564276
\(202\) 20.0000 1.40720
\(203\) 16.0000 1.12298
\(204\) −4.00000 −0.280056
\(205\) 10.0000 0.698430
\(206\) 4.00000 0.278693
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 2.00000 0.138013
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) −8.00000 −0.543075
\(218\) −4.00000 −0.270914
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 2.00000 0.133631
\(225\) 1.00000 0.0666667
\(226\) −16.0000 −1.06430
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −2.00000 −0.132453
\(229\) −28.0000 −1.85029 −0.925146 0.379611i \(-0.876058\pi\)
−0.925146 + 0.379611i \(0.876058\pi\)
\(230\) −2.00000 −0.131876
\(231\) 8.00000 0.526361
\(232\) 8.00000 0.525226
\(233\) 28.0000 1.83434 0.917170 0.398495i \(-0.130467\pi\)
0.917170 + 0.398495i \(0.130467\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 8.00000 0.519656
\(238\) 8.00000 0.518563
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 1.00000 0.0645497
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 5.00000 0.321412
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 3.00000 0.191663
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) −4.00000 −0.254000
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) 10.0000 0.631194 0.315597 0.948893i \(-0.397795\pi\)
0.315597 + 0.948893i \(0.397795\pi\)
\(252\) 2.00000 0.125988
\(253\) −8.00000 −0.502956
\(254\) 20.0000 1.25491
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) −4.00000 −0.249029
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 14.0000 0.864923
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 4.00000 0.246183
\(265\) −6.00000 −0.368577
\(266\) 4.00000 0.245256
\(267\) −10.0000 −0.611990
\(268\) 8.00000 0.488678
\(269\) −4.00000 −0.243884 −0.121942 0.992537i \(-0.538912\pi\)
−0.121942 + 0.992537i \(0.538912\pi\)
\(270\) 1.00000 0.0608581
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) −4.00000 −0.241209
\(276\) −2.00000 −0.120386
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −16.0000 −0.959616
\(279\) −4.00000 −0.239474
\(280\) −2.00000 −0.119523
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 0 0
\(287\) −20.0000 −1.18056
\(288\) 1.00000 0.0589256
\(289\) −1.00000 −0.0588235
\(290\) −8.00000 −0.469776
\(291\) −8.00000 −0.468968
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 3.00000 0.174964
\(295\) −12.0000 −0.698667
\(296\) −6.00000 −0.348743
\(297\) 4.00000 0.232104
\(298\) 22.0000 1.27443
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.00000 0.461112
\(302\) 4.00000 0.230174
\(303\) −20.0000 −1.14897
\(304\) 2.00000 0.114708
\(305\) 2.00000 0.114520
\(306\) 4.00000 0.228665
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −8.00000 −0.455842
\(309\) −4.00000 −0.227552
\(310\) 4.00000 0.227185
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −10.0000 −0.564333
\(315\) −2.00000 −0.112687
\(316\) −8.00000 −0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −6.00000 −0.336463
\(319\) −32.0000 −1.79166
\(320\) −1.00000 −0.0559017
\(321\) −4.00000 −0.223258
\(322\) 4.00000 0.222911
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) 4.00000 0.221201
\(328\) −10.0000 −0.552158
\(329\) 0 0
\(330\) −4.00000 −0.220193
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) 12.0000 0.658586
\(333\) −6.00000 −0.328798
\(334\) 12.0000 0.656611
\(335\) −8.00000 −0.437087
\(336\) −2.00000 −0.109109
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) −4.00000 −0.216930
\(341\) 16.0000 0.866449
\(342\) 2.00000 0.108148
\(343\) −20.0000 −1.07990
\(344\) 4.00000 0.215666
\(345\) 2.00000 0.107676
\(346\) 14.0000 0.752645
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) −8.00000 −0.428845
\(349\) 28.0000 1.49881 0.749403 0.662114i \(-0.230341\pi\)
0.749403 + 0.662114i \(0.230341\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −8.00000 −0.423405
\(358\) 2.00000 0.105703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −15.0000 −0.789474
\(362\) −22.0000 −1.15629
\(363\) −5.00000 −0.262432
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −4.00000 −0.208798 −0.104399 0.994535i \(-0.533292\pi\)
−0.104399 + 0.994535i \(0.533292\pi\)
\(368\) 2.00000 0.104257
\(369\) −10.0000 −0.520579
\(370\) 6.00000 0.311925
\(371\) 12.0000 0.623009
\(372\) 4.00000 0.207390
\(373\) 14.0000 0.724893 0.362446 0.932005i \(-0.381942\pi\)
0.362446 + 0.932005i \(0.381942\pi\)
\(374\) −16.0000 −0.827340
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 14.0000 0.719132 0.359566 0.933120i \(-0.382925\pi\)
0.359566 + 0.933120i \(0.382925\pi\)
\(380\) −2.00000 −0.102598
\(381\) −20.0000 −1.02463
\(382\) −8.00000 −0.409316
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.00000 0.407718
\(386\) −12.0000 −0.610784
\(387\) 4.00000 0.203331
\(388\) 8.00000 0.406138
\(389\) 20.0000 1.01404 0.507020 0.861934i \(-0.330747\pi\)
0.507020 + 0.861934i \(0.330747\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) −3.00000 −0.151523
\(393\) −14.0000 −0.706207
\(394\) −10.0000 −0.503793
\(395\) 8.00000 0.402524
\(396\) −4.00000 −0.201008
\(397\) −6.00000 −0.301131 −0.150566 0.988600i \(-0.548110\pi\)
−0.150566 + 0.988600i \(0.548110\pi\)
\(398\) 24.0000 1.20301
\(399\) −4.00000 −0.200250
\(400\) 1.00000 0.0500000
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) −8.00000 −0.399004
\(403\) 0 0
\(404\) 20.0000 0.995037
\(405\) −1.00000 −0.0496904
\(406\) 16.0000 0.794067
\(407\) 24.0000 1.18964
\(408\) −4.00000 −0.198030
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 10.0000 0.493865
\(411\) −2.00000 −0.0986527
\(412\) 4.00000 0.197066
\(413\) 24.0000 1.18096
\(414\) 2.00000 0.0982946
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 16.0000 0.783523
\(418\) −8.00000 −0.391293
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 2.00000 0.0975900
\(421\) 16.0000 0.779792 0.389896 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) −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
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) −8.00000 −0.384012
\(435\) 8.00000 0.383571
\(436\) −4.00000 −0.191565
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 4.00000 0.190693
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 6.00000 0.284747
\(445\) −10.0000 −0.474045
\(446\) 22.0000 1.04173
\(447\) −22.0000 −1.04056
\(448\) 2.00000 0.0944911
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) −16.0000 −0.752577
\(453\) −4.00000 −0.187936
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 16.0000 0.748448 0.374224 0.927338i \(-0.377909\pi\)
0.374224 + 0.927338i \(0.377909\pi\)
\(458\) −28.0000 −1.30835
\(459\) −4.00000 −0.186704
\(460\) −2.00000 −0.0932505
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 8.00000 0.372194
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) 8.00000 0.371391
\(465\) −4.00000 −0.185496
\(466\) 28.0000 1.29707
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) 12.0000 0.552345
\(473\) −16.0000 −0.735681
\(474\) 8.00000 0.367452
\(475\) 2.00000 0.0917663
\(476\) 8.00000 0.366679
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 22.0000 1.00207
\(483\) −4.00000 −0.182006
\(484\) 5.00000 0.227273
\(485\) −8.00000 −0.363261
\(486\) −1.00000 −0.0453609
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 12.0000 0.542659
\(490\) 3.00000 0.135526
\(491\) 38.0000 1.71492 0.857458 0.514554i \(-0.172042\pi\)
0.857458 + 0.514554i \(0.172042\pi\)
\(492\) 10.0000 0.450835
\(493\) 32.0000 1.44121
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) −4.00000 −0.179605
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −12.0000 −0.536120
\(502\) 10.0000 0.446322
\(503\) −42.0000 −1.87269 −0.936344 0.351085i \(-0.885813\pi\)
−0.936344 + 0.351085i \(0.885813\pi\)
\(504\) 2.00000 0.0890871
\(505\) −20.0000 −0.889988
\(506\) −8.00000 −0.355643
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −8.00000 −0.352865
\(515\) −4.00000 −0.176261
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) −12.0000 −0.527250
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 8.00000 0.350150
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 14.0000 0.611593
\(525\) −2.00000 −0.0872872
\(526\) −18.0000 −0.784837
\(527\) −16.0000 −0.696971
\(528\) 4.00000 0.174078
\(529\) −19.0000 −0.826087
\(530\) −6.00000 −0.260623
\(531\) 12.0000 0.520756
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) −10.0000 −0.432742
\(535\) −4.00000 −0.172935
\(536\) 8.00000 0.345547
\(537\) −2.00000 −0.0863064
\(538\) −4.00000 −0.172452
\(539\) 12.0000 0.516877
\(540\) 1.00000 0.0430331
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 28.0000 1.20270
\(543\) 22.0000 0.944110
\(544\) 4.00000 0.171499
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 2.00000 0.0854358
\(549\) −2.00000 −0.0853579
\(550\) −4.00000 −0.170561
\(551\) 16.0000 0.681623
\(552\) −2.00000 −0.0851257
\(553\) −16.0000 −0.680389
\(554\) 22.0000 0.934690
\(555\) −6.00000 −0.254686
\(556\) −16.0000 −0.678551
\(557\) 10.0000 0.423714 0.211857 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 16.0000 0.675521
\(562\) 10.0000 0.421825
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 16.0000 0.673125
\(566\) 28.0000 1.17693
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 2.00000 0.0837708
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) −20.0000 −0.834784
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 12.0000 0.498703
\(580\) −8.00000 −0.332182
\(581\) 24.0000 0.995688
\(582\) −8.00000 −0.331611
\(583\) −24.0000 −0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 3.00000 0.123718
\(589\) −8.00000 −0.329634
\(590\) −12.0000 −0.494032
\(591\) 10.0000 0.411345
\(592\) −6.00000 −0.246598
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 4.00000 0.164122
\(595\) −8.00000 −0.327968
\(596\) 22.0000 0.901155
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) 8.00000 0.326056
\(603\) 8.00000 0.325785
\(604\) 4.00000 0.162758
\(605\) −5.00000 −0.203279
\(606\) −20.0000 −0.812444
\(607\) −12.0000 −0.487065 −0.243532 0.969893i \(-0.578306\pi\)
−0.243532 + 0.969893i \(0.578306\pi\)
\(608\) 2.00000 0.0811107
\(609\) −16.0000 −0.648353
\(610\) 2.00000 0.0809776
\(611\) 0 0
\(612\) 4.00000 0.161690
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 0 0
\(615\) −10.0000 −0.403239
\(616\) −8.00000 −0.322329
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −4.00000 −0.160904
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 4.00000 0.160644
\(621\) −2.00000 −0.0802572
\(622\) −16.0000 −0.641542
\(623\) 20.0000 0.801283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 8.00000 0.319489
\(628\) −10.0000 −0.399043
\(629\) −24.0000 −0.956943
\(630\) −2.00000 −0.0796819
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) 2.00000 0.0794301
\(635\) −20.0000 −0.793676
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) −32.0000 −1.26689
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −4.00000 −0.157867
\(643\) 40.0000 1.57745 0.788723 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 4.00000 0.157622
\(645\) 4.00000 0.157500
\(646\) 8.00000 0.314756
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 1.00000 0.0392837
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) −12.0000 −0.469956
\(653\) −38.0000 −1.48705 −0.743527 0.668705i \(-0.766849\pi\)
−0.743527 + 0.668705i \(0.766849\pi\)
\(654\) 4.00000 0.156412
\(655\) −14.0000 −0.547025
\(656\) −10.0000 −0.390434
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) −4.00000 −0.155700
\(661\) 16.0000 0.622328 0.311164 0.950356i \(-0.399281\pi\)
0.311164 + 0.950356i \(0.399281\pi\)
\(662\) 10.0000 0.388661
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) −4.00000 −0.155113
\(666\) −6.00000 −0.232495
\(667\) 16.0000 0.619522
\(668\) 12.0000 0.464294
\(669\) −22.0000 −0.850569
\(670\) −8.00000 −0.309067
\(671\) 8.00000 0.308837
\(672\) −2.00000 −0.0771517
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −26.0000 −1.00148
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 16.0000 0.614476
\(679\) 16.0000 0.614024
\(680\) −4.00000 −0.153393
\(681\) 12.0000 0.459841
\(682\) 16.0000 0.612672
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 2.00000 0.0764719
\(685\) −2.00000 −0.0764161
\(686\) −20.0000 −0.763604
\(687\) 28.0000 1.06827
\(688\) 4.00000 0.152499
\(689\) 0 0
\(690\) 2.00000 0.0761387
\(691\) 18.0000 0.684752 0.342376 0.939563i \(-0.388768\pi\)
0.342376 + 0.939563i \(0.388768\pi\)
\(692\) 14.0000 0.532200
\(693\) −8.00000 −0.303895
\(694\) 16.0000 0.607352
\(695\) 16.0000 0.606915
\(696\) −8.00000 −0.303239
\(697\) −40.0000 −1.51511
\(698\) 28.0000 1.05982
\(699\) −28.0000 −1.05906
\(700\) 2.00000 0.0755929
\(701\) −20.0000 −0.755390 −0.377695 0.925930i \(-0.623283\pi\)
−0.377695 + 0.925930i \(0.623283\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) 40.0000 1.50435
\(708\) −12.0000 −0.450988
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 10.0000 0.374766
\(713\) −8.00000 −0.299602
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) −8.00000 −0.298765
\(718\) −24.0000 −0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 8.00000 0.297936
\(722\) −15.0000 −0.558242
\(723\) −22.0000 −0.818189
\(724\) −22.0000 −0.817624
\(725\) 8.00000 0.297113
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) 2.00000 0.0739221
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −4.00000 −0.147643
\(735\) −3.00000 −0.110657
\(736\) 2.00000 0.0737210
\(737\) −32.0000 −1.17874
\(738\) −10.0000 −0.368105
\(739\) −50.0000 −1.83928 −0.919640 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 4.00000 0.146647
\(745\) −22.0000 −0.806018
\(746\) 14.0000 0.512576
\(747\) 12.0000 0.439057
\(748\) −16.0000 −0.585018
\(749\) 8.00000 0.292314
\(750\) 1.00000 0.0365148
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) −10.0000 −0.364420
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) −2.00000 −0.0727393
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 14.0000 0.508503
\(759\) 8.00000 0.290382
\(760\) −2.00000 −0.0725476
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) −20.0000 −0.724524
\(763\) −8.00000 −0.289619
\(764\) −8.00000 −0.289430
\(765\) −4.00000 −0.144620
\(766\) 20.0000 0.722629
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −10.0000 −0.360609 −0.180305 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 8.00000 0.288300
\(771\) 8.00000 0.288113
\(772\) −12.0000 −0.431889
\(773\) −26.0000 −0.935155 −0.467578 0.883952i \(-0.654873\pi\)
−0.467578 + 0.883952i \(0.654873\pi\)
\(774\) 4.00000 0.143777
\(775\) −4.00000 −0.143684
\(776\) 8.00000 0.287183
\(777\) 12.0000 0.430498
\(778\) 20.0000 0.717035
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 8.00000 0.286079
\(783\) −8.00000 −0.285897
\(784\) −3.00000 −0.107143
\(785\) 10.0000 0.356915
\(786\) −14.0000 −0.499363
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) −10.0000 −0.356235
\(789\) 18.0000 0.640817
\(790\) 8.00000 0.284627
\(791\) −32.0000 −1.13779
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) −6.00000 −0.212932
\(795\) 6.00000 0.212798
\(796\) 24.0000 0.850657
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) −4.00000 −0.141598
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 10.0000 0.353333
\(802\) 30.0000 1.05934
\(803\) 0 0
\(804\) −8.00000 −0.282138
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 4.00000 0.140807
\(808\) 20.0000 0.703598
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 34.0000 1.19390 0.596951 0.802278i \(-0.296379\pi\)
0.596951 + 0.802278i \(0.296379\pi\)
\(812\) 16.0000 0.561490
\(813\) −28.0000 −0.982003
\(814\) 24.0000 0.841200
\(815\) 12.0000 0.420342
\(816\) −4.00000 −0.140028
\(817\) 8.00000 0.279885
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −2.00000 −0.0697580
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 4.00000 0.139347
\(825\) 4.00000 0.139262
\(826\) 24.0000 0.835067
\(827\) 44.0000 1.53003 0.765015 0.644013i \(-0.222732\pi\)
0.765015 + 0.644013i \(0.222732\pi\)
\(828\) 2.00000 0.0695048
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) −12.0000 −0.416526
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) −12.0000 −0.415775
\(834\) 16.0000 0.554035
\(835\) −12.0000 −0.415277
\(836\) −8.00000 −0.276686
\(837\) 4.00000 0.138260
\(838\) 14.0000 0.483622
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 2.00000 0.0690066
\(841\) 35.0000 1.20690
\(842\) 16.0000 0.551396
\(843\) −10.0000 −0.344418
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) 6.00000 0.206041
\(849\) −28.0000 −0.960958
\(850\) 4.00000 0.137199
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) −4.00000 −0.136877
\(855\) −2.00000 −0.0683986
\(856\) 4.00000 0.136717
\(857\) −16.0000 −0.546550 −0.273275 0.961936i \(-0.588107\pi\)
−0.273275 + 0.961936i \(0.588107\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −4.00000 −0.136399
\(861\) 20.0000 0.681598
\(862\) −16.0000 −0.544962
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −14.0000 −0.476014
\(866\) −26.0000 −0.883516
\(867\) 1.00000 0.0339618
\(868\) −8.00000 −0.271538
\(869\) 32.0000 1.08553
\(870\) 8.00000 0.271225
\(871\) 0 0
\(872\) −4.00000 −0.135457
\(873\) 8.00000 0.270759
\(874\) 4.00000 0.135302
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 8.00000 0.269987
\(879\) −6.00000 −0.202375
\(880\) 4.00000 0.134840
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −3.00000 −0.101015
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 12.0000 0.403376
\(886\) −16.0000 −0.537531
\(887\) −6.00000 −0.201460 −0.100730 0.994914i \(-0.532118\pi\)
−0.100730 + 0.994914i \(0.532118\pi\)
\(888\) 6.00000 0.201347
\(889\) 40.0000 1.34156
\(890\) −10.0000 −0.335201
\(891\) −4.00000 −0.134005
\(892\) 22.0000 0.736614
\(893\) 0 0
\(894\) −22.0000 −0.735790
\(895\) −2.00000 −0.0668526
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −32.0000 −1.06726
\(900\) 1.00000 0.0333333
\(901\) 24.0000 0.799556
\(902\) 40.0000 1.33185
\(903\) −8.00000 −0.266223
\(904\) −16.0000 −0.532152
\(905\) 22.0000 0.731305
\(906\) −4.00000 −0.132891
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) −12.0000 −0.398234
\(909\) 20.0000 0.663358
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −2.00000 −0.0662266
\(913\) −48.0000 −1.58857
\(914\) 16.0000 0.529233
\(915\) −2.00000 −0.0661180
\(916\) −28.0000 −0.925146
\(917\) 28.0000 0.924641
\(918\) −4.00000 −0.132020
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) −2.00000 −0.0659380
\(921\) 0 0
\(922\) −30.0000 −0.987997
\(923\) 0 0
\(924\) 8.00000 0.263181
\(925\) −6.00000 −0.197279
\(926\) −2.00000 −0.0657241
\(927\) 4.00000 0.131377
\(928\) 8.00000 0.262613
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) −4.00000 −0.131165
\(931\) −6.00000 −0.196642
\(932\) 28.0000 0.917170
\(933\) 16.0000 0.523816
\(934\) 12.0000 0.392652
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 16.0000 0.522419
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 10.0000 0.325818
\(943\) −20.0000 −0.651290
\(944\) 12.0000 0.390567
\(945\) 2.00000 0.0650600
\(946\) −16.0000 −0.520205
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 2.00000 0.0648886
\(951\) −2.00000 −0.0648544
\(952\) 8.00000 0.259281
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 6.00000 0.194257
\(955\) 8.00000 0.258874
\(956\) 8.00000 0.258738
\(957\) 32.0000 1.03441
\(958\) −24.0000 −0.775405
\(959\) 4.00000 0.129167
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) 22.0000 0.708572
\(965\) 12.0000 0.386294
\(966\) −4.00000 −0.128698
\(967\) −14.0000 −0.450210 −0.225105 0.974335i \(-0.572272\pi\)
−0.225105 + 0.974335i \(0.572272\pi\)
\(968\) 5.00000 0.160706
\(969\) −8.00000 −0.256997
\(970\) −8.00000 −0.256865
\(971\) 42.0000 1.34784 0.673922 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −32.0000 −1.02587
\(974\) −38.0000 −1.21760
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 12.0000 0.383718
\(979\) −40.0000 −1.27841
\(980\) 3.00000 0.0958315
\(981\) −4.00000 −0.127710
\(982\) 38.0000 1.21263
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 10.0000 0.318788
\(985\) 10.0000 0.318626
\(986\) 32.0000 1.01909
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 4.00000 0.127128
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) −4.00000 −0.127000
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) −24.0000 −0.760851
\(996\) −12.0000 −0.380235
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 14.0000 0.443162
\(999\) 6.00000 0.189832
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 5070.2.a.n.1.1 1
13.5 odd 4 5070.2.b.f.1351.1 2
13.8 odd 4 5070.2.b.f.1351.2 2
13.12 even 2 390.2.a.b.1.1 1
39.38 odd 2 1170.2.a.j.1.1 1
52.51 odd 2 3120.2.a.y.1.1 1
65.12 odd 4 1950.2.e.m.1249.1 2
65.38 odd 4 1950.2.e.m.1249.2 2
65.64 even 2 1950.2.a.ba.1.1 1
156.155 even 2 9360.2.a.v.1.1 1
195.38 even 4 5850.2.e.h.5149.1 2
195.77 even 4 5850.2.e.h.5149.2 2
195.194 odd 2 5850.2.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.b.1.1 1 13.12 even 2
1170.2.a.j.1.1 1 39.38 odd 2
1950.2.a.ba.1.1 1 65.64 even 2
1950.2.e.m.1249.1 2 65.12 odd 4
1950.2.e.m.1249.2 2 65.38 odd 4
3120.2.a.y.1.1 1 52.51 odd 2
5070.2.a.n.1.1 1 1.1 even 1 trivial
5070.2.b.f.1351.1 2 13.5 odd 4
5070.2.b.f.1351.2 2 13.8 odd 4
5850.2.a.s.1.1 1 195.194 odd 2
5850.2.e.h.5149.1 2 195.38 even 4
5850.2.e.h.5149.2 2 195.77 even 4
9360.2.a.v.1.1 1 156.155 even 2