Properties

Label 9802.2.a.a.1.1
Level $9802$
Weight $2$
Character 9802.1
Self dual yes
Analytic conductor $78.269$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 9802 = 2 \cdot 13^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9802.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.2693640613\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9802.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +8.00000 q^{17} +2.00000 q^{18} -1.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +5.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} -1.00000 q^{30} +3.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -8.00000 q^{34} -2.00000 q^{35} -2.00000 q^{36} -8.00000 q^{37} +1.00000 q^{40} -2.00000 q^{41} +2.00000 q^{42} -11.0000 q^{43} +3.00000 q^{44} +2.00000 q^{45} -4.00000 q^{46} -13.0000 q^{47} -1.00000 q^{48} -3.00000 q^{49} +4.00000 q^{50} -8.00000 q^{51} -11.0000 q^{53} -5.00000 q^{54} -3.00000 q^{55} -2.00000 q^{56} +1.00000 q^{58} +1.00000 q^{60} -8.00000 q^{61} -3.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +12.0000 q^{67} +8.00000 q^{68} -4.00000 q^{69} +2.00000 q^{70} -2.00000 q^{71} +2.00000 q^{72} -4.00000 q^{73} +8.00000 q^{74} +4.00000 q^{75} +6.00000 q^{77} +15.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -4.00000 q^{83} -2.00000 q^{84} -8.00000 q^{85} +11.0000 q^{86} +1.00000 q^{87} -3.00000 q^{88} +10.0000 q^{89} -2.00000 q^{90} +4.00000 q^{92} -3.00000 q^{93} +13.0000 q^{94} +1.00000 q^{96} +2.00000 q^{97} +3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 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
\(9\) −2.00000 −0.666667
\(10\) 1.00000 0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 2.00000 0.471405
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −1.00000 −0.223607
\(21\) −2.00000 −0.436436
\(22\) −3.00000 −0.639602
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 2.00000 0.377964
\(29\) −1.00000 −0.185695
\(30\) −1.00000 −0.182574
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) −8.00000 −1.37199
\(35\) −2.00000 −0.338062
\(36\) −2.00000 −0.333333
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 2.00000 0.308607
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 3.00000 0.452267
\(45\) 2.00000 0.298142
\(46\) −4.00000 −0.589768
\(47\) −13.0000 −1.89624 −0.948122 0.317905i \(-0.897021\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 4.00000 0.565685
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) −5.00000 −0.680414
\(55\) −3.00000 −0.404520
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 1.00000 0.131306
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 1.00000 0.129099
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −3.00000 −0.381000
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 8.00000 0.970143
\(69\) −4.00000 −0.481543
\(70\) 2.00000 0.239046
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 2.00000 0.235702
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 8.00000 0.929981
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 15.0000 1.68763 0.843816 0.536633i \(-0.180304\pi\)
0.843816 + 0.536633i \(0.180304\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) −2.00000 −0.218218
\(85\) −8.00000 −0.867722
\(86\) 11.0000 1.18616
\(87\) 1.00000 0.107211
\(88\) −3.00000 −0.319801
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 4.00000 0.417029
\(93\) −3.00000 −0.311086
\(94\) 13.0000 1.34085
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 3.00000 0.303046
\(99\) −6.00000 −0.603023
\(100\) −4.00000 −0.400000
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 8.00000 0.792118
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 11.0000 1.06841
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) 5.00000 0.481125
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 3.00000 0.286039
\(111\) 8.00000 0.759326
\(112\) 2.00000 0.188982
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −1.00000 −0.0928477
\(117\) 0 0
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) −1.00000 −0.0912871
\(121\) −2.00000 −0.181818
\(122\) 8.00000 0.724286
\(123\) 2.00000 0.180334
\(124\) 3.00000 0.269408
\(125\) 9.00000 0.804984
\(126\) 4.00000 0.356348
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.0000 0.968496
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −3.00000 −0.261116
\(133\) 0 0
\(134\) −12.0000 −1.03664
\(135\) −5.00000 −0.430331
\(136\) −8.00000 −0.685994
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 4.00000 0.340503
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −2.00000 −0.169031
\(141\) 13.0000 1.09480
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) −2.00000 −0.166667
\(145\) 1.00000 0.0830455
\(146\) 4.00000 0.331042
\(147\) 3.00000 0.247436
\(148\) −8.00000 −0.657596
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) −4.00000 −0.326599
\(151\) −2.00000 −0.162758 −0.0813788 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) 0 0
\(153\) −16.0000 −1.29352
\(154\) −6.00000 −0.483494
\(155\) −3.00000 −0.240966
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) −15.0000 −1.19334
\(159\) 11.0000 0.872357
\(160\) 1.00000 0.0790569
\(161\) 8.00000 0.630488
\(162\) −1.00000 −0.0785674
\(163\) −9.00000 −0.704934 −0.352467 0.935824i \(-0.614657\pi\)
−0.352467 + 0.935824i \(0.614657\pi\)
\(164\) −2.00000 −0.156174
\(165\) 3.00000 0.233550
\(166\) 4.00000 0.310460
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 8.00000 0.613572
\(171\) 0 0
\(172\) −11.0000 −0.838742
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) −1.00000 −0.0758098
\(175\) −8.00000 −0.604743
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 2.00000 0.149071
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) −4.00000 −0.294884
\(185\) 8.00000 0.588172
\(186\) 3.00000 0.219971
\(187\) 24.0000 1.75505
\(188\) −13.0000 −0.948122
\(189\) 10.0000 0.727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 6.00000 0.426401
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 4.00000 0.282843
\(201\) −12.0000 −0.846415
\(202\) 8.00000 0.562878
\(203\) −2.00000 −0.140372
\(204\) −8.00000 −0.560112
\(205\) 2.00000 0.139686
\(206\) −14.0000 −0.975426
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 0 0
\(210\) −2.00000 −0.138013
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) −11.0000 −0.755483
\(213\) 2.00000 0.137038
\(214\) 2.00000 0.136717
\(215\) 11.0000 0.750194
\(216\) −5.00000 −0.340207
\(217\) 6.00000 0.407307
\(218\) 5.00000 0.338643
\(219\) 4.00000 0.270295
\(220\) −3.00000 −0.202260
\(221\) 0 0
\(222\) −8.00000 −0.536925
\(223\) 26.0000 1.74109 0.870544 0.492090i \(-0.163767\pi\)
0.870544 + 0.492090i \(0.163767\pi\)
\(224\) −2.00000 −0.133631
\(225\) 8.00000 0.533333
\(226\) 6.00000 0.399114
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 4.00000 0.263752
\(231\) −6.00000 −0.394771
\(232\) 1.00000 0.0656532
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 0 0
\(235\) 13.0000 0.848026
\(236\) 0 0
\(237\) −15.0000 −0.974355
\(238\) −16.0000 −1.03713
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000 0.0645497
\(241\) −17.0000 −1.09507 −0.547533 0.836784i \(-0.684433\pi\)
−0.547533 + 0.836784i \(0.684433\pi\)
\(242\) 2.00000 0.128565
\(243\) −16.0000 −1.02640
\(244\) −8.00000 −0.512148
\(245\) 3.00000 0.191663
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −3.00000 −0.190500
\(249\) 4.00000 0.253490
\(250\) −9.00000 −0.569210
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) −4.00000 −0.251976
\(253\) 12.0000 0.754434
\(254\) −8.00000 −0.501965
\(255\) 8.00000 0.500979
\(256\) 1.00000 0.0625000
\(257\) 13.0000 0.810918 0.405459 0.914113i \(-0.367112\pi\)
0.405459 + 0.914113i \(0.367112\pi\)
\(258\) −11.0000 −0.684830
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) −12.0000 −0.741362
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 3.00000 0.184637
\(265\) 11.0000 0.675725
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 5.00000 0.304290
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 8.00000 0.485071
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −12.0000 −0.723627
\(276\) −4.00000 −0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) −6.00000 −0.359211
\(280\) 2.00000 0.119523
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) −13.0000 −0.774139
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 2.00000 0.117851
\(289\) 47.0000 2.76471
\(290\) −1.00000 −0.0587220
\(291\) −2.00000 −0.117242
\(292\) −4.00000 −0.234082
\(293\) −14.0000 −0.817889 −0.408944 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) −3.00000 −0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 15.0000 0.870388
\(298\) 15.0000 0.868927
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) −22.0000 −1.26806
\(302\) 2.00000 0.115087
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) 8.00000 0.458079
\(306\) 16.0000 0.914659
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 6.00000 0.341882
\(309\) −14.0000 −0.796432
\(310\) 3.00000 0.170389
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) 9.00000 0.508710 0.254355 0.967111i \(-0.418137\pi\)
0.254355 + 0.967111i \(0.418137\pi\)
\(314\) −18.0000 −1.01580
\(315\) 4.00000 0.225374
\(316\) 15.0000 0.843816
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) −11.0000 −0.616849
\(319\) −3.00000 −0.167968
\(320\) −1.00000 −0.0559017
\(321\) 2.00000 0.111629
\(322\) −8.00000 −0.445823
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 9.00000 0.498464
\(327\) 5.00000 0.276501
\(328\) 2.00000 0.110432
\(329\) −26.0000 −1.43343
\(330\) −3.00000 −0.165145
\(331\) 23.0000 1.26419 0.632097 0.774889i \(-0.282194\pi\)
0.632097 + 0.774889i \(0.282194\pi\)
\(332\) −4.00000 −0.219529
\(333\) 16.0000 0.876795
\(334\) −2.00000 −0.109435
\(335\) −12.0000 −0.655630
\(336\) −2.00000 −0.109109
\(337\) −32.0000 −1.74315 −0.871576 0.490261i \(-0.836901\pi\)
−0.871576 + 0.490261i \(0.836901\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) −8.00000 −0.433861
\(341\) 9.00000 0.487377
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 11.0000 0.593080
\(345\) 4.00000 0.215353
\(346\) 6.00000 0.322562
\(347\) −2.00000 −0.107366 −0.0536828 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 1.00000 0.0536056
\(349\) 15.0000 0.802932 0.401466 0.915874i \(-0.368501\pi\)
0.401466 + 0.915874i \(0.368501\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 2.00000 0.106149
\(356\) 10.0000 0.529999
\(357\) −16.0000 −0.846810
\(358\) 10.0000 0.528516
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) −2.00000 −0.105409
\(361\) −19.0000 −1.00000
\(362\) −7.00000 −0.367912
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) −8.00000 −0.418167
\(367\) −32.0000 −1.67039 −0.835193 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) 4.00000 0.208514
\(369\) 4.00000 0.208232
\(370\) −8.00000 −0.415900
\(371\) −22.0000 −1.14218
\(372\) −3.00000 −0.155543
\(373\) −21.0000 −1.08734 −0.543669 0.839299i \(-0.682965\pi\)
−0.543669 + 0.839299i \(0.682965\pi\)
\(374\) −24.0000 −1.24101
\(375\) −9.00000 −0.464758
\(376\) 13.0000 0.670424
\(377\) 0 0
\(378\) −10.0000 −0.514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 8.00000 0.409316
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 1.00000 0.0510310
\(385\) −6.00000 −0.305788
\(386\) 14.0000 0.712581
\(387\) 22.0000 1.11832
\(388\) 2.00000 0.101535
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) 3.00000 0.151523
\(393\) −12.0000 −0.605320
\(394\) 18.0000 0.906827
\(395\) −15.0000 −0.754732
\(396\) −6.00000 −0.301511
\(397\) 17.0000 0.853206 0.426603 0.904439i \(-0.359710\pi\)
0.426603 + 0.904439i \(0.359710\pi\)
\(398\) 10.0000 0.501255
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) −8.00000 −0.398015
\(405\) −1.00000 −0.0496904
\(406\) 2.00000 0.0992583
\(407\) −24.0000 −1.18964
\(408\) 8.00000 0.396059
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) −2.00000 −0.0987730
\(411\) −12.0000 −0.591916
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 20.0000 0.979404
\(418\) 0 0
\(419\) −10.0000 −0.488532 −0.244266 0.969708i \(-0.578547\pi\)
−0.244266 + 0.969708i \(0.578547\pi\)
\(420\) 2.00000 0.0975900
\(421\) −32.0000 −1.55958 −0.779792 0.626038i \(-0.784675\pi\)
−0.779792 + 0.626038i \(0.784675\pi\)
\(422\) 3.00000 0.146038
\(423\) 26.0000 1.26416
\(424\) 11.0000 0.534207
\(425\) −32.0000 −1.55223
\(426\) −2.00000 −0.0969003
\(427\) −16.0000 −0.774294
\(428\) −2.00000 −0.0966736
\(429\) 0 0
\(430\) −11.0000 −0.530467
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 5.00000 0.240563
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −6.00000 −0.288009
\(435\) −1.00000 −0.0479463
\(436\) −5.00000 −0.239457
\(437\) 0 0
\(438\) −4.00000 −0.191127
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 3.00000 0.143019
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 8.00000 0.379663
\(445\) −10.0000 −0.474045
\(446\) −26.0000 −1.23114
\(447\) 15.0000 0.709476
\(448\) 2.00000 0.0944911
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) −8.00000 −0.377124
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) 2.00000 0.0939682
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 10.0000 0.467269
\(459\) 40.0000 1.86704
\(460\) −4.00000 −0.186501
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 6.00000 0.279145
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 3.00000 0.139122
\(466\) 1.00000 0.0463241
\(467\) −27.0000 −1.24941 −0.624705 0.780860i \(-0.714781\pi\)
−0.624705 + 0.780860i \(0.714781\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) −13.0000 −0.599645
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) −33.0000 −1.51734
\(474\) 15.0000 0.688973
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) 22.0000 1.00731
\(478\) 0 0
\(479\) 5.00000 0.228456 0.114228 0.993455i \(-0.463561\pi\)
0.114228 + 0.993455i \(0.463561\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 17.0000 0.774329
\(483\) −8.00000 −0.364013
\(484\) −2.00000 −0.0909091
\(485\) −2.00000 −0.0908153
\(486\) 16.0000 0.725775
\(487\) 22.0000 0.996915 0.498458 0.866914i \(-0.333900\pi\)
0.498458 + 0.866914i \(0.333900\pi\)
\(488\) 8.00000 0.362143
\(489\) 9.00000 0.406994
\(490\) −3.00000 −0.135526
\(491\) −33.0000 −1.48927 −0.744635 0.667472i \(-0.767376\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) 2.00000 0.0901670
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 3.00000 0.134704
\(497\) −4.00000 −0.179425
\(498\) −4.00000 −0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 9.00000 0.402492
\(501\) −2.00000 −0.0893534
\(502\) −27.0000 −1.20507
\(503\) 19.0000 0.847168 0.423584 0.905857i \(-0.360772\pi\)
0.423584 + 0.905857i \(0.360772\pi\)
\(504\) 4.00000 0.178174
\(505\) 8.00000 0.355995
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 15.0000 0.664863 0.332432 0.943127i \(-0.392131\pi\)
0.332432 + 0.943127i \(0.392131\pi\)
\(510\) −8.00000 −0.354246
\(511\) −8.00000 −0.353899
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −13.0000 −0.573405
\(515\) −14.0000 −0.616914
\(516\) 11.0000 0.484248
\(517\) −39.0000 −1.71522
\(518\) 16.0000 0.703000
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −13.0000 −0.569540 −0.284770 0.958596i \(-0.591917\pi\)
−0.284770 + 0.958596i \(0.591917\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 12.0000 0.524222
\(525\) 8.00000 0.349149
\(526\) −9.00000 −0.392419
\(527\) 24.0000 1.04546
\(528\) −3.00000 −0.130558
\(529\) −7.00000 −0.304348
\(530\) −11.0000 −0.477809
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 10.0000 0.432742
\(535\) 2.00000 0.0864675
\(536\) −12.0000 −0.518321
\(537\) 10.0000 0.431532
\(538\) 0 0
\(539\) −9.00000 −0.387657
\(540\) −5.00000 −0.215166
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −13.0000 −0.558398
\(543\) −7.00000 −0.300399
\(544\) −8.00000 −0.342997
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) 38.0000 1.62476 0.812381 0.583127i \(-0.198171\pi\)
0.812381 + 0.583127i \(0.198171\pi\)
\(548\) 12.0000 0.512615
\(549\) 16.0000 0.682863
\(550\) 12.0000 0.511682
\(551\) 0 0
\(552\) 4.00000 0.170251
\(553\) 30.0000 1.27573
\(554\) 2.00000 0.0849719
\(555\) −8.00000 −0.339581
\(556\) −20.0000 −0.848189
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 6.00000 0.254000
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) −24.0000 −1.01328
\(562\) 27.0000 1.13893
\(563\) −11.0000 −0.463595 −0.231797 0.972764i \(-0.574461\pi\)
−0.231797 + 0.972764i \(0.574461\pi\)
\(564\) 13.0000 0.547399
\(565\) 6.00000 0.252422
\(566\) −4.00000 −0.168133
\(567\) 2.00000 0.0839921
\(568\) 2.00000 0.0839181
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 8.00000 0.334205
\(574\) 4.00000 0.166957
\(575\) −16.0000 −0.667246
\(576\) −2.00000 −0.0833333
\(577\) −8.00000 −0.333044 −0.166522 0.986038i \(-0.553254\pi\)
−0.166522 + 0.986038i \(0.553254\pi\)
\(578\) −47.0000 −1.95494
\(579\) 14.0000 0.581820
\(580\) 1.00000 0.0415227
\(581\) −8.00000 −0.331896
\(582\) 2.00000 0.0829027
\(583\) −33.0000 −1.36672
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 14.0000 0.578335
\(587\) −28.0000 −1.15568 −0.577842 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 3.00000 0.123718
\(589\) 0 0
\(590\) 0 0
\(591\) 18.0000 0.740421
\(592\) −8.00000 −0.328798
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) −15.0000 −0.615457
\(595\) −16.0000 −0.655936
\(596\) −15.0000 −0.614424
\(597\) 10.0000 0.409273
\(598\) 0 0
\(599\) −5.00000 −0.204294 −0.102147 0.994769i \(-0.532571\pi\)
−0.102147 + 0.994769i \(0.532571\pi\)
\(600\) −4.00000 −0.163299
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 22.0000 0.896653
\(603\) −24.0000 −0.977356
\(604\) −2.00000 −0.0813788
\(605\) 2.00000 0.0813116
\(606\) −8.00000 −0.324978
\(607\) 3.00000 0.121766 0.0608831 0.998145i \(-0.480608\pi\)
0.0608831 + 0.998145i \(0.480608\pi\)
\(608\) 0 0
\(609\) 2.00000 0.0810441
\(610\) −8.00000 −0.323911
\(611\) 0 0
\(612\) −16.0000 −0.646762
\(613\) 31.0000 1.25208 0.626039 0.779792i \(-0.284675\pi\)
0.626039 + 0.779792i \(0.284675\pi\)
\(614\) −7.00000 −0.282497
\(615\) −2.00000 −0.0806478
\(616\) −6.00000 −0.241747
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 14.0000 0.563163
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) −3.00000 −0.120483
\(621\) 20.0000 0.802572
\(622\) 8.00000 0.320771
\(623\) 20.0000 0.801283
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) −9.00000 −0.359712
\(627\) 0 0
\(628\) 18.0000 0.718278
\(629\) −64.0000 −2.55185
\(630\) −4.00000 −0.159364
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) −15.0000 −0.596668
\(633\) 3.00000 0.119239
\(634\) −12.0000 −0.476581
\(635\) −8.00000 −0.317470
\(636\) 11.0000 0.436178
\(637\) 0 0
\(638\) 3.00000 0.118771
\(639\) 4.00000 0.158238
\(640\) 1.00000 0.0395285
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) −2.00000 −0.0789337
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 8.00000 0.315244
\(645\) −11.0000 −0.433125
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) −9.00000 −0.352467
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −5.00000 −0.195515
\(655\) −12.0000 −0.468879
\(656\) −2.00000 −0.0780869
\(657\) 8.00000 0.312110
\(658\) 26.0000 1.01359
\(659\) 15.0000 0.584317 0.292159 0.956370i \(-0.405627\pi\)
0.292159 + 0.956370i \(0.405627\pi\)
\(660\) 3.00000 0.116775
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) −23.0000 −0.893920
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) −4.00000 −0.154881
\(668\) 2.00000 0.0773823
\(669\) −26.0000 −1.00522
\(670\) 12.0000 0.463600
\(671\) −24.0000 −0.926510
\(672\) 2.00000 0.0771517
\(673\) 9.00000 0.346925 0.173462 0.984841i \(-0.444505\pi\)
0.173462 + 0.984841i \(0.444505\pi\)
\(674\) 32.0000 1.23259
\(675\) −20.0000 −0.769800
\(676\) 0 0
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) −6.00000 −0.230429
\(679\) 4.00000 0.153506
\(680\) 8.00000 0.306786
\(681\) 18.0000 0.689761
\(682\) −9.00000 −0.344628
\(683\) −24.0000 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 20.0000 0.763604
\(687\) 10.0000 0.381524
\(688\) −11.0000 −0.419371
\(689\) 0 0
\(690\) −4.00000 −0.152277
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −6.00000 −0.228086
\(693\) −12.0000 −0.455842
\(694\) 2.00000 0.0759190
\(695\) 20.0000 0.758643
\(696\) −1.00000 −0.0379049
\(697\) −16.0000 −0.606043
\(698\) −15.0000 −0.567758
\(699\) 1.00000 0.0378235
\(700\) −8.00000 −0.302372
\(701\) 27.0000 1.01978 0.509888 0.860241i \(-0.329687\pi\)
0.509888 + 0.860241i \(0.329687\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 3.00000 0.113067
\(705\) −13.0000 −0.489608
\(706\) −26.0000 −0.978523
\(707\) −16.0000 −0.601742
\(708\) 0 0
\(709\) −15.0000 −0.563337 −0.281668 0.959512i \(-0.590888\pi\)
−0.281668 + 0.959512i \(0.590888\pi\)
\(710\) −2.00000 −0.0750587
\(711\) −30.0000 −1.12509
\(712\) −10.0000 −0.374766
\(713\) 12.0000 0.449404
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) −25.0000 −0.932992
\(719\) −50.0000 −1.86469 −0.932343 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 2.00000 0.0745356
\(721\) 28.0000 1.04277
\(722\) 19.0000 0.707107
\(723\) 17.0000 0.632237
\(724\) 7.00000 0.260153
\(725\) 4.00000 0.148556
\(726\) −2.00000 −0.0742270
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −4.00000 −0.148047
\(731\) −88.0000 −3.25480
\(732\) 8.00000 0.295689
\(733\) −24.0000 −0.886460 −0.443230 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 32.0000 1.18114
\(735\) −3.00000 −0.110657
\(736\) −4.00000 −0.147442
\(737\) 36.0000 1.32608
\(738\) −4.00000 −0.147242
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 22.0000 0.807645
\(743\) −44.0000 −1.61420 −0.807102 0.590412i \(-0.798965\pi\)
−0.807102 + 0.590412i \(0.798965\pi\)
\(744\) 3.00000 0.109985
\(745\) 15.0000 0.549557
\(746\) 21.0000 0.768865
\(747\) 8.00000 0.292705
\(748\) 24.0000 0.877527
\(749\) −4.00000 −0.146157
\(750\) 9.00000 0.328634
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −13.0000 −0.474061
\(753\) −27.0000 −0.983935
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) 10.0000 0.363696
\(757\) 8.00000 0.290765 0.145382 0.989376i \(-0.453559\pi\)
0.145382 + 0.989376i \(0.453559\pi\)
\(758\) 20.0000 0.726433
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 8.00000 0.289809
\(763\) −10.0000 −0.362024
\(764\) −8.00000 −0.289430
\(765\) 16.0000 0.578481
\(766\) 14.0000 0.505841
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 20.0000 0.721218 0.360609 0.932717i \(-0.382569\pi\)
0.360609 + 0.932717i \(0.382569\pi\)
\(770\) 6.00000 0.216225
\(771\) −13.0000 −0.468184
\(772\) −14.0000 −0.503871
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) −22.0000 −0.790774
\(775\) −12.0000 −0.431053
\(776\) −2.00000 −0.0717958
\(777\) 16.0000 0.573997
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) −32.0000 −1.14432
\(783\) −5.00000 −0.178685
\(784\) −3.00000 −0.107143
\(785\) −18.0000 −0.642448
\(786\) 12.0000 0.428026
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) −18.0000 −0.641223
\(789\) −9.00000 −0.320408
\(790\) 15.0000 0.533676
\(791\) −12.0000 −0.426671
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −17.0000 −0.603307
\(795\) −11.0000 −0.390130
\(796\) −10.0000 −0.354441
\(797\) −32.0000 −1.13350 −0.566749 0.823890i \(-0.691799\pi\)
−0.566749 + 0.823890i \(0.691799\pi\)
\(798\) 0 0
\(799\) −104.000 −3.67926
\(800\) 4.00000 0.141421
\(801\) −20.0000 −0.706665
\(802\) 27.0000 0.953403
\(803\) −12.0000 −0.423471
\(804\) −12.0000 −0.423207
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 8.00000 0.281439
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 1.00000 0.0351364
\(811\) 18.0000 0.632065 0.316033 0.948748i \(-0.397649\pi\)
0.316033 + 0.948748i \(0.397649\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −13.0000 −0.455930
\(814\) 24.0000 0.841200
\(815\) 9.00000 0.315256
\(816\) −8.00000 −0.280056
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 12.0000 0.418548
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −14.0000 −0.487713
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) −13.0000 −0.452054 −0.226027 0.974121i \(-0.572574\pi\)
−0.226027 + 0.974121i \(0.572574\pi\)
\(828\) −8.00000 −0.278019
\(829\) 40.0000 1.38926 0.694629 0.719368i \(-0.255569\pi\)
0.694629 + 0.719368i \(0.255569\pi\)
\(830\) −4.00000 −0.138842
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) −24.0000 −0.831551
\(834\) −20.0000 −0.692543
\(835\) −2.00000 −0.0692129
\(836\) 0 0
\(837\) 15.0000 0.518476
\(838\) 10.0000 0.345444
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) −2.00000 −0.0690066
\(841\) 1.00000 0.0344828
\(842\) 32.0000 1.10279
\(843\) 27.0000 0.929929
\(844\) −3.00000 −0.103264
\(845\) 0 0
\(846\) −26.0000 −0.893898
\(847\) −4.00000 −0.137442
\(848\) −11.0000 −0.377742
\(849\) −4.00000 −0.137280
\(850\) 32.0000 1.09759
\(851\) −32.0000 −1.09695
\(852\) 2.00000 0.0685189
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 16.0000 0.547509
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) −27.0000 −0.922302 −0.461151 0.887322i \(-0.652563\pi\)
−0.461151 + 0.887322i \(0.652563\pi\)
\(858\) 0 0
\(859\) −25.0000 −0.852989 −0.426494 0.904490i \(-0.640252\pi\)
−0.426494 + 0.904490i \(0.640252\pi\)
\(860\) 11.0000 0.375097
\(861\) 4.00000 0.136320
\(862\) 32.0000 1.08992
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) −5.00000 −0.170103
\(865\) 6.00000 0.204006
\(866\) 16.0000 0.543702
\(867\) −47.0000 −1.59620
\(868\) 6.00000 0.203653
\(869\) 45.0000 1.52652
\(870\) 1.00000 0.0339032
\(871\) 0 0
\(872\) 5.00000 0.169321
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) 18.0000 0.608511
\(876\) 4.00000 0.135147
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −20.0000 −0.674967
\(879\) 14.0000 0.472208
\(880\) −3.00000 −0.101130
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −6.00000 −0.202031
\(883\) −26.0000 −0.874970 −0.437485 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 33.0000 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) −8.00000 −0.268462
\(889\) 16.0000 0.536623
\(890\) 10.0000 0.335201
\(891\) 3.00000 0.100504
\(892\) 26.0000 0.870544
\(893\) 0 0
\(894\) −15.0000 −0.501675
\(895\) 10.0000 0.334263
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) −10.0000 −0.333704
\(899\) −3.00000 −0.100056
\(900\) 8.00000 0.266667
\(901\) −88.0000 −2.93171
\(902\) 6.00000 0.199778
\(903\) 22.0000 0.732114
\(904\) 6.00000 0.199557
\(905\) −7.00000 −0.232688
\(906\) −2.00000 −0.0664455
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −18.0000 −0.597351
\(909\) 16.0000 0.530687
\(910\) 0 0
\(911\) −13.0000 −0.430709 −0.215355 0.976536i \(-0.569091\pi\)
−0.215355 + 0.976536i \(0.569091\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) −2.00000 −0.0661541
\(915\) −8.00000 −0.264472
\(916\) −10.0000 −0.330409
\(917\) 24.0000 0.792550
\(918\) −40.0000 −1.32020
\(919\) 30.0000 0.989609 0.494804 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) 4.00000 0.131876
\(921\) −7.00000 −0.230658
\(922\) 2.00000 0.0658665
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) 32.0000 1.05215
\(926\) 4.00000 0.131448
\(927\) −28.0000 −0.919641
\(928\) 1.00000 0.0328266
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) −3.00000 −0.0983739
\(931\) 0 0
\(932\) −1.00000 −0.0327561
\(933\) 8.00000 0.261908
\(934\) 27.0000 0.883467
\(935\) −24.0000 −0.784884
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) −24.0000 −0.783628
\(939\) −9.00000 −0.293704
\(940\) 13.0000 0.424013
\(941\) −37.0000 −1.20617 −0.603083 0.797679i \(-0.706061\pi\)
−0.603083 + 0.797679i \(0.706061\pi\)
\(942\) 18.0000 0.586472
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) −10.0000 −0.325300
\(946\) 33.0000 1.07292
\(947\) −33.0000 −1.07236 −0.536178 0.844105i \(-0.680132\pi\)
−0.536178 + 0.844105i \(0.680132\pi\)
\(948\) −15.0000 −0.487177
\(949\) 0 0
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) −16.0000 −0.518563
\(953\) −1.00000 −0.0323932 −0.0161966 0.999869i \(-0.505156\pi\)
−0.0161966 + 0.999869i \(0.505156\pi\)
\(954\) −22.0000 −0.712276
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) 3.00000 0.0969762
\(958\) −5.00000 −0.161543
\(959\) 24.0000 0.775000
\(960\) 1.00000 0.0322749
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 4.00000 0.128898
\(964\) −17.0000 −0.547533
\(965\) 14.0000 0.450676
\(966\) 8.00000 0.257396
\(967\) −13.0000 −0.418052 −0.209026 0.977910i \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 2.00000 0.0642161
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) −16.0000 −0.513200
\(973\) −40.0000 −1.28234
\(974\) −22.0000 −0.704925
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −13.0000 −0.415907 −0.207953 0.978139i \(-0.566680\pi\)
−0.207953 + 0.978139i \(0.566680\pi\)
\(978\) −9.00000 −0.287788
\(979\) 30.0000 0.958804
\(980\) 3.00000 0.0958315
\(981\) 10.0000 0.319275
\(982\) 33.0000 1.05307
\(983\) −49.0000 −1.56286 −0.781429 0.623995i \(-0.785509\pi\)
−0.781429 + 0.623995i \(0.785509\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 18.0000 0.573528
\(986\) 8.00000 0.254772
\(987\) 26.0000 0.827589
\(988\) 0 0
\(989\) −44.0000 −1.39912
\(990\) −6.00000 −0.190693
\(991\) 22.0000 0.698853 0.349427 0.936964i \(-0.386376\pi\)
0.349427 + 0.936964i \(0.386376\pi\)
\(992\) −3.00000 −0.0952501
\(993\) −23.0000 −0.729883
\(994\) 4.00000 0.126872
\(995\) 10.0000 0.317021
\(996\) 4.00000 0.126745
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) −20.0000 −0.633089
\(999\) −40.0000 −1.26554
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 9802.2.a.a.1.1 1
13.12 even 2 58.2.a.b.1.1 1
39.38 odd 2 522.2.a.b.1.1 1
52.51 odd 2 464.2.a.e.1.1 1
65.12 odd 4 1450.2.b.b.349.2 2
65.38 odd 4 1450.2.b.b.349.1 2
65.64 even 2 1450.2.a.c.1.1 1
91.90 odd 2 2842.2.a.e.1.1 1
104.51 odd 2 1856.2.a.f.1.1 1
104.77 even 2 1856.2.a.k.1.1 1
143.142 odd 2 7018.2.a.a.1.1 1
156.155 even 2 4176.2.a.n.1.1 1
377.12 odd 4 1682.2.b.a.1681.2 2
377.220 odd 4 1682.2.b.a.1681.1 2
377.376 even 2 1682.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.b.1.1 1 13.12 even 2
464.2.a.e.1.1 1 52.51 odd 2
522.2.a.b.1.1 1 39.38 odd 2
1450.2.a.c.1.1 1 65.64 even 2
1450.2.b.b.349.1 2 65.38 odd 4
1450.2.b.b.349.2 2 65.12 odd 4
1682.2.a.d.1.1 1 377.376 even 2
1682.2.b.a.1681.1 2 377.220 odd 4
1682.2.b.a.1681.2 2 377.12 odd 4
1856.2.a.f.1.1 1 104.51 odd 2
1856.2.a.k.1.1 1 104.77 even 2
2842.2.a.e.1.1 1 91.90 odd 2
4176.2.a.n.1.1 1 156.155 even 2
7018.2.a.a.1.1 1 143.142 odd 2
9802.2.a.a.1.1 1 1.1 even 1 trivial