Properties

Label 1050.2.a.e.1.1
Level 1050
Weight 2
Character 1050.1
Self dual yes
Analytic conductor 8.384
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0\)
Character \(\chi\) = 1050.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{21} -2.00000 q^{22} -7.00000 q^{23} +1.00000 q^{24} +1.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +1.00000 q^{29} +3.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} +1.00000 q^{39} -3.00000 q^{41} +1.00000 q^{42} +1.00000 q^{43} +2.00000 q^{44} +7.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{51} -1.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} -1.00000 q^{58} -3.00000 q^{59} +5.00000 q^{61} -3.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} +12.0000 q^{67} +1.00000 q^{68} +7.00000 q^{69} +4.00000 q^{71} -1.00000 q^{72} +14.0000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.00000 q^{77} -1.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} +3.00000 q^{82} +3.00000 q^{83} -1.00000 q^{84} -1.00000 q^{86} -1.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} -1.00000 q^{91} -7.00000 q^{92} -3.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) −2.00000 −0.426401
\(23\) −7.00000 −1.45960 −0.729800 0.683660i \(-0.760387\pi\)
−0.729800 + 0.683660i \(0.760387\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) 1.00000 0.185695 0.0928477 0.995680i \(-0.470403\pi\)
0.0928477 + 0.995680i \(0.470403\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) −4.00000 −0.648886
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −3.00000 −0.468521 −0.234261 0.972174i \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 1.00000 0.154303
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 7.00000 1.03209
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −1.00000 −0.140028
\(52\) −1.00000 −0.138675
\(53\) −11.0000 −1.51097 −0.755483 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −4.00000 −0.529813
\(58\) −1.00000 −0.131306
\(59\) −3.00000 −0.390567 −0.195283 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) −3.00000 −0.381000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 12.0000 1.46603 0.733017 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 1.00000 0.121268
\(69\) 7.00000 0.842701
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) −1.00000 −0.117851
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 2.00000 0.227921
\(78\) −1.00000 −0.113228
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.00000 0.331295
\(83\) 3.00000 0.329293 0.164646 0.986353i \(-0.447352\pi\)
0.164646 + 0.986353i \(0.447352\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) −1.00000 −0.107211
\(88\) −2.00000 −0.213201
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) −7.00000 −0.729800
\(93\) −3.00000 −0.311086
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −1.00000 −0.101015
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 1.00000 0.0990148
\(103\) 17.0000 1.67506 0.837530 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) 11.0000 1.06841
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) −1.00000 −0.0924500
\(118\) 3.00000 0.276172
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) −5.00000 −0.452679
\(123\) 3.00000 0.270501
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −14.0000 −1.24230 −0.621150 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −2.00000 −0.174078
\(133\) 4.00000 0.346844
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −4.00000 −0.341743 −0.170872 0.985293i \(-0.554658\pi\)
−0.170872 + 0.985293i \(0.554658\pi\)
\(138\) −7.00000 −0.595880
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −4.00000 −0.335673
\(143\) −2.00000 −0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) −1.00000 −0.0824786
\(148\) 6.00000 0.493197
\(149\) 5.00000 0.409616 0.204808 0.978802i \(-0.434343\pi\)
0.204808 + 0.978802i \(0.434343\pi\)
\(150\) 0 0
\(151\) 22.0000 1.79033 0.895167 0.445730i \(-0.147056\pi\)
0.895167 + 0.445730i \(0.147056\pi\)
\(152\) −4.00000 −0.324443
\(153\) 1.00000 0.0808452
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 2.00000 0.159111
\(159\) 11.0000 0.872357
\(160\) 0 0
\(161\) −7.00000 −0.551677
\(162\) −1.00000 −0.0785674
\(163\) 19.0000 1.48819 0.744097 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 1.00000 0.0771517
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 1.00000 0.0762493
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 3.00000 0.225494
\(178\) −10.0000 −0.749532
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 1.00000 0.0741249
\(183\) −5.00000 −0.369611
\(184\) 7.00000 0.516047
\(185\) 0 0
\(186\) 3.00000 0.219971
\(187\) 2.00000 0.146254
\(188\) 12.0000 0.875190
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −13.0000 −0.940647 −0.470323 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) −2.00000 −0.142134
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 1.00000 0.0701862
\(204\) −1.00000 −0.0700140
\(205\) 0 0
\(206\) −17.0000 −1.18445
\(207\) −7.00000 −0.486534
\(208\) −1.00000 −0.0693375
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 15.0000 1.03264 0.516321 0.856395i \(-0.327301\pi\)
0.516321 + 0.856395i \(0.327301\pi\)
\(212\) −11.0000 −0.755483
\(213\) −4.00000 −0.274075
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 3.00000 0.203653
\(218\) 4.00000 0.270914
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −1.00000 −0.0672673
\(222\) 6.00000 0.402694
\(223\) 13.0000 0.870544 0.435272 0.900299i \(-0.356652\pi\)
0.435272 + 0.900299i \(0.356652\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) −7.00000 −0.464606 −0.232303 0.972643i \(-0.574626\pi\)
−0.232303 + 0.972643i \(0.574626\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) −1.00000 −0.0656532
\(233\) −20.0000 −1.31024 −0.655122 0.755523i \(-0.727383\pi\)
−0.655122 + 0.755523i \(0.727383\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) −3.00000 −0.195283
\(237\) 2.00000 0.129914
\(238\) −1.00000 −0.0648204
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −3.00000 −0.191273
\(247\) −4.00000 −0.254514
\(248\) −3.00000 −0.190500
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 1.00000 0.0629941
\(253\) −14.0000 −0.880172
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.00000 0.311891 0.155946 0.987766i \(-0.450158\pi\)
0.155946 + 0.987766i \(0.450158\pi\)
\(258\) 1.00000 0.0622573
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) −8.00000 −0.494242
\(263\) −11.0000 −0.678289 −0.339145 0.940734i \(-0.610138\pi\)
−0.339145 + 0.940734i \(0.610138\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) −10.0000 −0.611990
\(268\) 12.0000 0.733017
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 1.00000 0.0606339
\(273\) 1.00000 0.0605228
\(274\) 4.00000 0.241649
\(275\) 0 0
\(276\) 7.00000 0.421350
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 4.00000 0.239904
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 12.0000 0.714590
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −3.00000 −0.177084
\(288\) −1.00000 −0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 14.0000 0.819288
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) −2.00000 −0.116052
\(298\) −5.00000 −0.289642
\(299\) 7.00000 0.404820
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) −22.0000 −1.26596
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 2.00000 0.113961
\(309\) −17.0000 −0.967096
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) 17.0000 0.954815 0.477408 0.878682i \(-0.341577\pi\)
0.477408 + 0.878682i \(0.341577\pi\)
\(318\) −11.0000 −0.616849
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 7.00000 0.390095
\(323\) 4.00000 0.222566
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −19.0000 −1.05231
\(327\) 4.00000 0.221201
\(328\) 3.00000 0.165647
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −3.00000 −0.164895 −0.0824475 0.996595i \(-0.526274\pi\)
−0.0824475 + 0.996595i \(0.526274\pi\)
\(332\) 3.00000 0.164646
\(333\) 6.00000 0.328798
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) −27.0000 −1.47078 −0.735392 0.677642i \(-0.763002\pi\)
−0.735392 + 0.677642i \(0.763002\pi\)
\(338\) 12.0000 0.652714
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) −4.00000 −0.216295
\(343\) 1.00000 0.0539949
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −2.00000 −0.106600
\(353\) −10.0000 −0.532246 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(354\) −3.00000 −0.159448
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) −1.00000 −0.0529256
\(358\) −10.0000 −0.528516
\(359\) −9.00000 −0.475002 −0.237501 0.971387i \(-0.576328\pi\)
−0.237501 + 0.971387i \(0.576328\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 18.0000 0.946059
\(363\) 7.00000 0.367405
\(364\) −1.00000 −0.0524142
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) −13.0000 −0.678594 −0.339297 0.940679i \(-0.610189\pi\)
−0.339297 + 0.940679i \(0.610189\pi\)
\(368\) −7.00000 −0.364900
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) −11.0000 −0.571092
\(372\) −3.00000 −0.155543
\(373\) 4.00000 0.207112 0.103556 0.994624i \(-0.466978\pi\)
0.103556 + 0.994624i \(0.466978\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) −1.00000 −0.0515026
\(378\) 1.00000 0.0514344
\(379\) −5.00000 −0.256833 −0.128416 0.991720i \(-0.540989\pi\)
−0.128416 + 0.991720i \(0.540989\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) 13.0000 0.665138
\(383\) 20.0000 1.02195 0.510976 0.859595i \(-0.329284\pi\)
0.510976 + 0.859595i \(0.329284\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) 1.00000 0.0508329
\(388\) 10.0000 0.507673
\(389\) 22.0000 1.11544 0.557722 0.830028i \(-0.311675\pi\)
0.557722 + 0.830028i \(0.311675\pi\)
\(390\) 0 0
\(391\) −7.00000 −0.354005
\(392\) −1.00000 −0.0505076
\(393\) −8.00000 −0.403547
\(394\) 27.0000 1.36024
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −3.00000 −0.150566 −0.0752828 0.997162i \(-0.523986\pi\)
−0.0752828 + 0.997162i \(0.523986\pi\)
\(398\) 24.0000 1.20301
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 12.0000 0.598506
\(403\) −3.00000 −0.149441
\(404\) 0 0
\(405\) 0 0
\(406\) −1.00000 −0.0496292
\(407\) 12.0000 0.594818
\(408\) 1.00000 0.0495074
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 17.0000 0.837530
\(413\) −3.00000 −0.147620
\(414\) 7.00000 0.344031
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 4.00000 0.195881
\(418\) −8.00000 −0.391293
\(419\) −25.0000 −1.22133 −0.610665 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) −15.0000 −0.730189
\(423\) 12.0000 0.583460
\(424\) 11.0000 0.534207
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 5.00000 0.241967
\(428\) 18.0000 0.870063
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 27.0000 1.30054 0.650272 0.759701i \(-0.274655\pi\)
0.650272 + 0.759701i \(0.274655\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −28.0000 −1.33942
\(438\) 14.0000 0.668946
\(439\) 7.00000 0.334092 0.167046 0.985949i \(-0.446577\pi\)
0.167046 + 0.985949i \(0.446577\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 1.00000 0.0475651
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) −6.00000 −0.284747
\(445\) 0 0
\(446\) −13.0000 −0.615568
\(447\) −5.00000 −0.236492
\(448\) 1.00000 0.0472456
\(449\) −16.0000 −0.755087 −0.377543 0.925992i \(-0.623231\pi\)
−0.377543 + 0.925992i \(0.623231\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −6.00000 −0.282216
\(453\) −22.0000 −1.03365
\(454\) 7.00000 0.328526
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 17.0000 0.795226 0.397613 0.917553i \(-0.369839\pi\)
0.397613 + 0.917553i \(0.369839\pi\)
\(458\) −14.0000 −0.654177
\(459\) −1.00000 −0.0466760
\(460\) 0 0
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 2.00000 0.0930484
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) 20.0000 0.926482
\(467\) 39.0000 1.80470 0.902352 0.430999i \(-0.141839\pi\)
0.902352 + 0.430999i \(0.141839\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 3.00000 0.138086
\(473\) 2.00000 0.0919601
\(474\) −2.00000 −0.0918630
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) −11.0000 −0.503655
\(478\) 24.0000 1.09773
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −6.00000 −0.273576
\(482\) 26.0000 1.18427
\(483\) 7.00000 0.318511
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −22.0000 −0.996915 −0.498458 0.866914i \(-0.666100\pi\)
−0.498458 + 0.866914i \(0.666100\pi\)
\(488\) −5.00000 −0.226339
\(489\) −19.0000 −0.859210
\(490\) 0 0
\(491\) 30.0000 1.35388 0.676941 0.736038i \(-0.263305\pi\)
0.676941 + 0.736038i \(0.263305\pi\)
\(492\) 3.00000 0.135250
\(493\) 1.00000 0.0450377
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) 4.00000 0.179425
\(498\) 3.00000 0.134433
\(499\) −27.0000 −1.20869 −0.604343 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 15.0000 0.669483
\(503\) 2.00000 0.0891756 0.0445878 0.999005i \(-0.485803\pi\)
0.0445878 + 0.999005i \(0.485803\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 14.0000 0.622376
\(507\) 12.0000 0.532939
\(508\) −14.0000 −0.621150
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) −5.00000 −0.220541
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) 24.0000 1.05552
\(518\) −6.00000 −0.263625
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 39.0000 1.70862 0.854311 0.519763i \(-0.173980\pi\)
0.854311 + 0.519763i \(0.173980\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) 3.00000 0.130682
\(528\) −2.00000 −0.0870388
\(529\) 26.0000 1.13043
\(530\) 0 0
\(531\) −3.00000 −0.130189
\(532\) 4.00000 0.173422
\(533\) 3.00000 0.129944
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) −10.0000 −0.431532
\(538\) −18.0000 −0.776035
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) 20.0000 0.859074
\(543\) 18.0000 0.772454
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) −1.00000 −0.0427960
\(547\) 15.0000 0.641354 0.320677 0.947189i \(-0.396090\pi\)
0.320677 + 0.947189i \(0.396090\pi\)
\(548\) −4.00000 −0.170872
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 4.00000 0.170406
\(552\) −7.00000 −0.297940
\(553\) −2.00000 −0.0850487
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −3.00000 −0.127000
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −2.00000 −0.0844401
\(562\) 0 0
\(563\) −41.0000 −1.72794 −0.863972 0.503540i \(-0.832031\pi\)
−0.863972 + 0.503540i \(0.832031\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) −20.0000 −0.840663
\(567\) 1.00000 0.0419961
\(568\) −4.00000 −0.167836
\(569\) 14.0000 0.586911 0.293455 0.955973i \(-0.405195\pi\)
0.293455 + 0.955973i \(0.405195\pi\)
\(570\) 0 0
\(571\) 19.0000 0.795125 0.397563 0.917575i \(-0.369856\pi\)
0.397563 + 0.917575i \(0.369856\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 13.0000 0.543083
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −34.0000 −1.41544 −0.707719 0.706494i \(-0.750276\pi\)
−0.707719 + 0.706494i \(0.750276\pi\)
\(578\) 16.0000 0.665512
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 3.00000 0.124461
\(582\) 10.0000 0.414513
\(583\) −22.0000 −0.911147
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) 0 0
\(587\) 27.0000 1.11441 0.557205 0.830375i \(-0.311874\pi\)
0.557205 + 0.830375i \(0.311874\pi\)
\(588\) −1.00000 −0.0412393
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 27.0000 1.11063
\(592\) 6.00000 0.246598
\(593\) 18.0000 0.739171 0.369586 0.929197i \(-0.379500\pi\)
0.369586 + 0.929197i \(0.379500\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 5.00000 0.204808
\(597\) 24.0000 0.982255
\(598\) −7.00000 −0.286251
\(599\) −45.0000 −1.83865 −0.919325 0.393499i \(-0.871265\pi\)
−0.919325 + 0.393499i \(0.871265\pi\)
\(600\) 0 0
\(601\) 28.0000 1.14214 0.571072 0.820900i \(-0.306528\pi\)
0.571072 + 0.820900i \(0.306528\pi\)
\(602\) −1.00000 −0.0407570
\(603\) 12.0000 0.488678
\(604\) 22.0000 0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) −4.00000 −0.162221
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 1.00000 0.0404226
\(613\) −18.0000 −0.727013 −0.363507 0.931592i \(-0.618421\pi\)
−0.363507 + 0.931592i \(0.618421\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −14.0000 −0.563619 −0.281809 0.959470i \(-0.590935\pi\)
−0.281809 + 0.959470i \(0.590935\pi\)
\(618\) 17.0000 0.683840
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 7.00000 0.280900
\(622\) −34.0000 −1.36328
\(623\) 10.0000 0.400642
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) −8.00000 −0.319489
\(628\) −18.0000 −0.718278
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 26.0000 1.03504 0.517522 0.855670i \(-0.326855\pi\)
0.517522 + 0.855670i \(0.326855\pi\)
\(632\) 2.00000 0.0795557
\(633\) −15.0000 −0.596196
\(634\) −17.0000 −0.675156
\(635\) 0 0
\(636\) 11.0000 0.436178
\(637\) −1.00000 −0.0396214
\(638\) −2.00000 −0.0791808
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −26.0000 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(642\) 18.0000 0.710403
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) −7.00000 −0.275839
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) −3.00000 −0.117579
\(652\) 19.0000 0.744097
\(653\) −14.0000 −0.547862 −0.273931 0.961749i \(-0.588324\pi\)
−0.273931 + 0.961749i \(0.588324\pi\)
\(654\) −4.00000 −0.156412
\(655\) 0 0
\(656\) −3.00000 −0.117130
\(657\) 14.0000 0.546192
\(658\) −12.0000 −0.467809
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 3.00000 0.116598
\(663\) 1.00000 0.0388368
\(664\) −3.00000 −0.116423
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) −7.00000 −0.271041
\(668\) 2.00000 0.0773823
\(669\) −13.0000 −0.502609
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 1.00000 0.0385758
\(673\) −27.0000 −1.04077 −0.520387 0.853931i \(-0.674212\pi\)
−0.520387 + 0.853931i \(0.674212\pi\)
\(674\) 27.0000 1.04000
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 20.0000 0.768662 0.384331 0.923195i \(-0.374432\pi\)
0.384331 + 0.923195i \(0.374432\pi\)
\(678\) −6.00000 −0.230429
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 7.00000 0.268241
\(682\) −6.00000 −0.229752
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −14.0000 −0.534133
\(688\) 1.00000 0.0381246
\(689\) 11.0000 0.419067
\(690\) 0 0
\(691\) −42.0000 −1.59776 −0.798878 0.601494i \(-0.794573\pi\)
−0.798878 + 0.601494i \(0.794573\pi\)
\(692\) 12.0000 0.456172
\(693\) 2.00000 0.0759737
\(694\) 26.0000 0.986947
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) −3.00000 −0.113633
\(698\) 1.00000 0.0378506
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) −39.0000 −1.47301 −0.736505 0.676432i \(-0.763525\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 24.0000 0.905177
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 10.0000 0.376355
\(707\) 0 0
\(708\) 3.00000 0.112747
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) −2.00000 −0.0750059
\(712\) −10.0000 −0.374766
\(713\) −21.0000 −0.786456
\(714\) 1.00000 0.0374241
\(715\) 0 0
\(716\) 10.0000 0.373718
\(717\) 24.0000 0.896296
\(718\) 9.00000 0.335877
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 17.0000 0.633113
\(722\) 3.00000 0.111648
\(723\) 26.0000 0.966950
\(724\) −18.0000 −0.668965
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 1.00000 0.0370879 0.0185440 0.999828i \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 1.00000 0.0370625
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.00000 0.0369863
\(732\) −5.00000 −0.184805
\(733\) −19.0000 −0.701781 −0.350891 0.936416i \(-0.614121\pi\)
−0.350891 + 0.936416i \(0.614121\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) 7.00000 0.258023
\(737\) 24.0000 0.884051
\(738\) 3.00000 0.110432
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 11.0000 0.403823
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 3.00000 0.109985
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 3.00000 0.109764
\(748\) 2.00000 0.0731272
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 12.0000 0.437595
\(753\) 15.0000 0.546630
\(754\) 1.00000 0.0364179
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 5.00000 0.181608
\(759\) 14.0000 0.508168
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) −14.0000 −0.507166
\(763\) −4.00000 −0.144810
\(764\) −13.0000 −0.470323
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 3.00000 0.108324
\(768\) −1.00000 −0.0360844
\(769\) 40.0000 1.44244 0.721218 0.692708i \(-0.243582\pi\)
0.721218 + 0.692708i \(0.243582\pi\)
\(770\) 0 0
\(771\) −5.00000 −0.180071
\(772\) −2.00000 −0.0719816
\(773\) 20.0000 0.719350 0.359675 0.933078i \(-0.382888\pi\)
0.359675 + 0.933078i \(0.382888\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) −6.00000 −0.215249
\(778\) −22.0000 −0.788738
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) 7.00000 0.250319
\(783\) −1.00000 −0.0357371
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) −38.0000 −1.35455 −0.677277 0.735728i \(-0.736840\pi\)
−0.677277 + 0.735728i \(0.736840\pi\)
\(788\) −27.0000 −0.961835
\(789\) 11.0000 0.391610
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −2.00000 −0.0710669
\(793\) −5.00000 −0.177555
\(794\) 3.00000 0.106466
\(795\) 0 0
\(796\) −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
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −16.0000 −0.564980
\(803\) 28.0000 0.988099
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 3.00000 0.105670
\(807\) −18.0000 −0.633630
\(808\) 0 0
\(809\) 20.0000 0.703163 0.351581 0.936157i \(-0.385644\pi\)
0.351581 + 0.936157i \(0.385644\pi\)
\(810\) 0 0
\(811\) −14.0000 −0.491606 −0.245803 0.969320i \(-0.579052\pi\)
−0.245803 + 0.969320i \(0.579052\pi\)
\(812\) 1.00000 0.0350931
\(813\) 20.0000 0.701431
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) −1.00000 −0.0350070
\(817\) 4.00000 0.139942
\(818\) 14.0000 0.489499
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −38.0000 −1.32621 −0.663105 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −4.00000 −0.139516
\(823\) 18.0000 0.627441 0.313720 0.949515i \(-0.398425\pi\)
0.313720 + 0.949515i \(0.398425\pi\)
\(824\) −17.0000 −0.592223
\(825\) 0 0
\(826\) 3.00000 0.104383
\(827\) 50.0000 1.73867 0.869335 0.494223i \(-0.164547\pi\)
0.869335 + 0.494223i \(0.164547\pi\)
\(828\) −7.00000 −0.243267
\(829\) 3.00000 0.104194 0.0520972 0.998642i \(-0.483409\pi\)
0.0520972 + 0.998642i \(0.483409\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) −1.00000 −0.0346688
\(833\) 1.00000 0.0346479
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) −3.00000 −0.103695
\(838\) 25.0000 0.863611
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) −34.0000 −1.17172
\(843\) 0 0
\(844\) 15.0000 0.516321
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) −7.00000 −0.240523
\(848\) −11.0000 −0.377742
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) −42.0000 −1.43974
\(852\) −4.00000 −0.137038
\(853\) −43.0000 −1.47229 −0.736146 0.676823i \(-0.763356\pi\)
−0.736146 + 0.676823i \(0.763356\pi\)
\(854\) −5.00000 −0.171096
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 46.0000 1.57133 0.785665 0.618652i \(-0.212321\pi\)
0.785665 + 0.618652i \(0.212321\pi\)
\(858\) −2.00000 −0.0682789
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 3.00000 0.102240
\(862\) −27.0000 −0.919624
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 16.0000 0.543388
\(868\) 3.00000 0.101827
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 4.00000 0.135457
\(873\) 10.0000 0.338449
\(874\) 28.0000 0.947114
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) −7.00000 −0.236239
\(879\) 0 0
\(880\) 0 0
\(881\) 5.00000 0.168454 0.0842271 0.996447i \(-0.473158\pi\)
0.0842271 + 0.996447i \(0.473158\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −29.0000 −0.975928 −0.487964 0.872864i \(-0.662260\pi\)
−0.487964 + 0.872864i \(0.662260\pi\)
\(884\) −1.00000 −0.0336336
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −50.0000 −1.67884 −0.839418 0.543487i \(-0.817104\pi\)
−0.839418 + 0.543487i \(0.817104\pi\)
\(888\) 6.00000 0.201347
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 13.0000 0.435272
\(893\) 48.0000 1.60626
\(894\) 5.00000 0.167225
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −7.00000 −0.233723
\(898\) 16.0000 0.533927
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −11.0000 −0.366463
\(902\) 6.00000 0.199778
\(903\) −1.00000 −0.0332779
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 11.0000 0.365249 0.182625 0.983183i \(-0.441541\pi\)
0.182625 + 0.983183i \(0.441541\pi\)
\(908\) −7.00000 −0.232303
\(909\) 0 0
\(910\) 0 0
\(911\) −19.0000 −0.629498 −0.314749 0.949175i \(-0.601920\pi\)
−0.314749 + 0.949175i \(0.601920\pi\)
\(912\) −4.00000 −0.132453
\(913\) 6.00000 0.198571
\(914\) −17.0000 −0.562310
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 8.00000 0.264183
\(918\) 1.00000 0.0330049
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) −22.0000 −0.724925
\(922\) −32.0000 −1.05386
\(923\) −4.00000 −0.131662
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 17.0000 0.558353
\(928\) −1.00000 −0.0328266
\(929\) 25.0000 0.820223 0.410112 0.912035i \(-0.365490\pi\)
0.410112 + 0.912035i \(0.365490\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −20.0000 −0.655122
\(933\) −34.0000 −1.11311
\(934\) −39.0000 −1.27612
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) −12.0000 −0.391814
\(939\) 18.0000 0.587408
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) −18.0000 −0.586472
\(943\) 21.0000 0.683854
\(944\) −3.00000 −0.0976417
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) 32.0000 1.03986 0.519930 0.854209i \(-0.325958\pi\)
0.519930 + 0.854209i \(0.325958\pi\)
\(948\) 2.00000 0.0649570
\(949\) −14.0000 −0.454459
\(950\) 0 0
\(951\) −17.0000 −0.551263
\(952\) −1.00000 −0.0324102
\(953\) 40.0000 1.29573 0.647864 0.761756i \(-0.275663\pi\)
0.647864 + 0.761756i \(0.275663\pi\)
\(954\) 11.0000 0.356138
\(955\) 0 0
\(956\) −24.0000 −0.776215
\(957\) −2.00000 −0.0646508
\(958\) −8.00000 −0.258468
\(959\) −4.00000 −0.129167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 6.00000 0.193448
\(963\) 18.0000 0.580042
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) −7.00000 −0.225221
\(967\) 44.0000 1.41494 0.707472 0.706741i \(-0.249835\pi\)
0.707472 + 0.706741i \(0.249835\pi\)
\(968\) 7.00000 0.224989
\(969\) −4.00000 −0.128499
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −4.00000 −0.128234
\(974\) 22.0000 0.704925
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 19.0000 0.607553
\(979\) 20.0000 0.639203
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) −30.0000 −0.957338
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −3.00000 −0.0956365
\(985\) 0 0
\(986\) −1.00000 −0.0318465
\(987\) −12.0000 −0.381964
\(988\) −4.00000 −0.127257
\(989\) −7.00000 −0.222587
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) −3.00000 −0.0952501
\(993\) 3.00000 0.0952021
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) −3.00000 −0.0950586
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 27.0000 0.854670
\(999\) −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 1050.2.a.e.1.1 1
3.2 odd 2 3150.2.a.bl.1.1 1
4.3 odd 2 8400.2.a.bt.1.1 1
5.2 odd 4 1050.2.g.j.799.1 2
5.3 odd 4 1050.2.g.j.799.2 2
5.4 even 2 1050.2.a.o.1.1 yes 1
7.6 odd 2 7350.2.a.bj.1.1 1
15.2 even 4 3150.2.g.g.2899.2 2
15.8 even 4 3150.2.g.g.2899.1 2
15.14 odd 2 3150.2.a.c.1.1 1
20.19 odd 2 8400.2.a.t.1.1 1
35.34 odd 2 7350.2.a.ca.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.e.1.1 1 1.1 even 1 trivial
1050.2.a.o.1.1 yes 1 5.4 even 2
1050.2.g.j.799.1 2 5.2 odd 4
1050.2.g.j.799.2 2 5.3 odd 4
3150.2.a.c.1.1 1 15.14 odd 2
3150.2.a.bl.1.1 1 3.2 odd 2
3150.2.g.g.2899.1 2 15.8 even 4
3150.2.g.g.2899.2 2 15.2 even 4
7350.2.a.bj.1.1 1 7.6 odd 2
7350.2.a.ca.1.1 1 35.34 odd 2
8400.2.a.t.1.1 1 20.19 odd 2
8400.2.a.bt.1.1 1 4.3 odd 2