Properties

Label 490.2.a.a.1.1
Level $490$
Weight $2$
Character 490.1
Self dual yes
Analytic conductor $3.913$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.91266969904\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 490.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.00000 q^{11} -2.00000 q^{12} -1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} +9.00000 q^{23} +2.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +4.00000 q^{27} +6.00000 q^{29} +2.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -7.00000 q^{37} +1.00000 q^{38} +2.00000 q^{39} -1.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} +3.00000 q^{44} +1.00000 q^{45} -9.00000 q^{46} +9.00000 q^{47} -2.00000 q^{48} -1.00000 q^{50} +12.0000 q^{51} -1.00000 q^{52} +9.00000 q^{53} -4.00000 q^{54} +3.00000 q^{55} +2.00000 q^{57} -6.00000 q^{58} -2.00000 q^{60} +8.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} +6.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} -18.0000 q^{69} -1.00000 q^{72} -4.00000 q^{73} +7.00000 q^{74} -2.00000 q^{75} -1.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} -11.0000 q^{81} -3.00000 q^{82} -6.00000 q^{85} -2.00000 q^{86} -12.0000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +9.00000 q^{92} -16.0000 q^{93} -9.00000 q^{94} -1.00000 q^{95} +2.00000 q^{96} -10.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) −2.00000 −0.577350
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 2.00000 0.408248
\(25\) 1.00000 0.200000
\(26\) 1.00000 0.196116
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 2.00000 0.365148
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) −6.00000 −1.04447
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 1.00000 0.162221
\(39\) 2.00000 0.320256
\(40\) −1.00000 −0.158114
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 3.00000 0.452267
\(45\) 1.00000 0.149071
\(46\) −9.00000 −1.32698
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −2.00000 −0.288675
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) 12.0000 1.68034
\(52\) −1.00000 −0.138675
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −4.00000 −0.544331
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) 2.00000 0.264906
\(58\) −6.00000 −0.787839
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −8.00000 −1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 6.00000 0.738549
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) −18.0000 −2.16695
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 7.00000 0.813733
\(75\) −2.00000 −0.230940
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 1.00000 0.111803
\(81\) −11.0000 −1.22222
\(82\) −3.00000 −0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −2.00000 −0.215666
\(87\) −12.0000 −1.28654
\(88\) −3.00000 −0.319801
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 9.00000 0.938315
\(93\) −16.0000 −1.65912
\(94\) −9.00000 −0.928279
\(95\) −1.00000 −0.102598
\(96\) 2.00000 0.204124
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) −12.0000 −1.18818
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 1.00000 0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 4.00000 0.384900
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) −3.00000 −0.286039
\(111\) 14.0000 1.32882
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) −2.00000 −0.187317
\(115\) 9.00000 0.839254
\(116\) 6.00000 0.557086
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) −6.00000 −0.541002
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 1.00000 0.0877058
\(131\) 3.00000 0.262111 0.131056 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) −6.00000 −0.522233
\(133\) 0 0
\(134\) −8.00000 −0.691095
\(135\) 4.00000 0.344265
\(136\) 6.00000 0.514496
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 18.0000 1.53226
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −18.0000 −1.51587
\(142\) 0 0
\(143\) −3.00000 −0.250873
\(144\) 1.00000 0.0833333
\(145\) 6.00000 0.498273
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 2.00000 0.163299
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000 0.0811107
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 2.00000 0.160128
\(157\) 23.0000 1.83560 0.917800 0.397043i \(-0.129964\pi\)
0.917800 + 0.397043i \(0.129964\pi\)
\(158\) 10.0000 0.795557
\(159\) −18.0000 −1.42749
\(160\) −1.00000 −0.0790569
\(161\) 0 0
\(162\) 11.0000 0.864242
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) 3.00000 0.234261
\(165\) −6.00000 −0.467099
\(166\) 0 0
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 6.00000 0.460179
\(171\) −1.00000 −0.0764719
\(172\) 2.00000 0.152499
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 12.0000 0.909718
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −6.00000 −0.449719
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 1.00000 0.0745356
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −16.0000 −1.18275
\(184\) −9.00000 −0.663489
\(185\) −7.00000 −0.514650
\(186\) 16.0000 1.17318
\(187\) −18.0000 −1.31629
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 1.00000 0.0725476
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −2.00000 −0.144338
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 10.0000 0.717958
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) −3.00000 −0.213201
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −16.0000 −1.12855
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 12.0000 0.840168
\(205\) 3.00000 0.209529
\(206\) 4.00000 0.278693
\(207\) 9.00000 0.625543
\(208\) −1.00000 −0.0693375
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 23.0000 1.58339 0.791693 0.610920i \(-0.209200\pi\)
0.791693 + 0.610920i \(0.209200\pi\)
\(212\) 9.00000 0.618123
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 2.00000 0.136399
\(216\) −4.00000 −0.272166
\(217\) 0 0
\(218\) 16.0000 1.08366
\(219\) 8.00000 0.540590
\(220\) 3.00000 0.202260
\(221\) 6.00000 0.403604
\(222\) −14.0000 −0.939618
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 2.00000 0.132453
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 1.00000 0.0653720
\(235\) 9.00000 0.587095
\(236\) 0 0
\(237\) 20.0000 1.29914
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −2.00000 −0.129099
\(241\) −1.00000 −0.0644157 −0.0322078 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) 2.00000 0.128565
\(243\) 10.0000 0.641500
\(244\) 8.00000 0.512148
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 1.00000 0.0636285
\(248\) −8.00000 −0.508001
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 27.0000 1.69748
\(254\) 1.00000 0.0627456
\(255\) 12.0000 0.751469
\(256\) 1.00000 0.0625000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 4.00000 0.249029
\(259\) 0 0
\(260\) −1.00000 −0.0620174
\(261\) 6.00000 0.371391
\(262\) −3.00000 −0.185341
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 6.00000 0.369274
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 8.00000 0.488678
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −4.00000 −0.243432
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 3.00000 0.180907
\(276\) −18.0000 −1.08347
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −27.0000 −1.61068 −0.805342 0.592810i \(-0.798019\pi\)
−0.805342 + 0.592810i \(0.798019\pi\)
\(282\) 18.0000 1.07188
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 0 0
\(285\) 2.00000 0.118470
\(286\) 3.00000 0.177394
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) −6.00000 −0.352332
\(291\) 20.0000 1.17242
\(292\) −4.00000 −0.234082
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 7.00000 0.406867
\(297\) 12.0000 0.696311
\(298\) 6.00000 0.347571
\(299\) −9.00000 −0.520483
\(300\) −2.00000 −0.115470
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) −24.0000 −1.37876
\(304\) −1.00000 −0.0573539
\(305\) 8.00000 0.458079
\(306\) 6.00000 0.342997
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) −8.00000 −0.454369
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −2.00000 −0.113228
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) −23.0000 −1.29797
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 18.0000 1.00939
\(319\) 18.0000 1.00781
\(320\) 1.00000 0.0559017
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) −11.0000 −0.611111
\(325\) −1.00000 −0.0554700
\(326\) −20.0000 −1.10770
\(327\) 32.0000 1.76960
\(328\) −3.00000 −0.165647
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) −7.00000 −0.383598
\(334\) −3.00000 −0.164153
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 12.0000 0.652714
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 24.0000 1.29967
\(342\) 1.00000 0.0540738
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) −18.0000 −0.969087
\(346\) −9.00000 −0.483843
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) −12.0000 −0.643268
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −3.00000 −0.159901
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 3.00000 0.158555
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.0000 −0.947368
\(362\) −2.00000 −0.105118
\(363\) 4.00000 0.209946
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 16.0000 0.836333
\(367\) −19.0000 −0.991792 −0.495896 0.868382i \(-0.665160\pi\)
−0.495896 + 0.868382i \(0.665160\pi\)
\(368\) 9.00000 0.469157
\(369\) 3.00000 0.156174
\(370\) 7.00000 0.363913
\(371\) 0 0
\(372\) −16.0000 −0.829561
\(373\) 2.00000 0.103556 0.0517780 0.998659i \(-0.483511\pi\)
0.0517780 + 0.998659i \(0.483511\pi\)
\(374\) 18.0000 0.930758
\(375\) −2.00000 −0.103280
\(376\) −9.00000 −0.464140
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) −1.00000 −0.0512989
\(381\) 2.00000 0.102463
\(382\) −12.0000 −0.613973
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 16.0000 0.814379
\(387\) 2.00000 0.101666
\(388\) −10.0000 −0.507673
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) −2.00000 −0.101274
\(391\) −54.0000 −2.73090
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) −15.0000 −0.755689
\(395\) −10.0000 −0.503155
\(396\) 3.00000 0.150756
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 16.0000 0.798007
\(403\) −8.00000 −0.398508
\(404\) 12.0000 0.597022
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) −21.0000 −1.04093
\(408\) −12.0000 −0.594089
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) −3.00000 −0.148159
\(411\) −24.0000 −1.18383
\(412\) −4.00000 −0.197066
\(413\) 0 0
\(414\) −9.00000 −0.442326
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) 8.00000 0.391762
\(418\) 3.00000 0.146735
\(419\) −9.00000 −0.439679 −0.219839 0.975536i \(-0.570553\pi\)
−0.219839 + 0.975536i \(0.570553\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −23.0000 −1.11962
\(423\) 9.00000 0.437595
\(424\) −9.00000 −0.437079
\(425\) −6.00000 −0.291043
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 6.00000 0.289683
\(430\) −2.00000 −0.0964486
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 4.00000 0.192450
\(433\) −40.0000 −1.92228 −0.961139 0.276066i \(-0.910969\pi\)
−0.961139 + 0.276066i \(0.910969\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) −16.0000 −0.766261
\(437\) −9.00000 −0.430528
\(438\) −8.00000 −0.382255
\(439\) 26.0000 1.24091 0.620456 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) −3.00000 −0.143019
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 14.0000 0.664411
\(445\) 6.00000 0.284427
\(446\) −8.00000 −0.378811
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) 20.0000 0.939682
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 4.00000 0.186908
\(459\) −24.0000 −1.12022
\(460\) 9.00000 0.419627
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −1.00000 −0.0464739 −0.0232370 0.999730i \(-0.507397\pi\)
−0.0232370 + 0.999730i \(0.507397\pi\)
\(464\) 6.00000 0.278543
\(465\) −16.0000 −0.741982
\(466\) 6.00000 0.277945
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) −9.00000 −0.415139
\(471\) −46.0000 −2.11957
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) −20.0000 −0.918630
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 6.00000 0.274434
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 2.00000 0.0912871
\(481\) 7.00000 0.319173
\(482\) 1.00000 0.0455488
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −10.0000 −0.454077
\(486\) −10.0000 −0.453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −8.00000 −0.362143
\(489\) −40.0000 −1.80886
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) −6.00000 −0.270501
\(493\) −36.0000 −1.62136
\(494\) −1.00000 −0.0449921
\(495\) 3.00000 0.134840
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 1.00000 0.0447214
\(501\) −6.00000 −0.268060
\(502\) 15.0000 0.669483
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −27.0000 −1.20030
\(507\) 24.0000 1.06588
\(508\) −1.00000 −0.0443678
\(509\) −42.0000 −1.86162 −0.930809 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) −12.0000 −0.531369
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −4.00000 −0.176261
\(516\) −4.00000 −0.176090
\(517\) 27.0000 1.18746
\(518\) 0 0
\(519\) −18.0000 −0.790112
\(520\) 1.00000 0.0438529
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) −6.00000 −0.262613
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 3.00000 0.131056
\(525\) 0 0
\(526\) 0 0
\(527\) −48.0000 −2.09091
\(528\) −6.00000 −0.261116
\(529\) 58.0000 2.52174
\(530\) −9.00000 −0.390935
\(531\) 0 0
\(532\) 0 0
\(533\) −3.00000 −0.129944
\(534\) 12.0000 0.519291
\(535\) −12.0000 −0.518805
\(536\) −8.00000 −0.345547
\(537\) 6.00000 0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 4.00000 0.172133
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) 16.0000 0.687259
\(543\) −4.00000 −0.171656
\(544\) 6.00000 0.257248
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 12.0000 0.512615
\(549\) 8.00000 0.341432
\(550\) −3.00000 −0.127920
\(551\) −6.00000 −0.255609
\(552\) 18.0000 0.766131
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 14.0000 0.594267
\(556\) −4.00000 −0.169638
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) −8.00000 −0.338667
\(559\) −2.00000 −0.0845910
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 27.0000 1.13893
\(563\) 42.0000 1.77009 0.885044 0.465506i \(-0.154128\pi\)
0.885044 + 0.465506i \(0.154128\pi\)
\(564\) −18.0000 −0.757937
\(565\) 0 0
\(566\) −14.0000 −0.588464
\(567\) 0 0
\(568\) 0 0
\(569\) 21.0000 0.880366 0.440183 0.897908i \(-0.354914\pi\)
0.440183 + 0.897908i \(0.354914\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −3.00000 −0.125436
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 9.00000 0.375326
\(576\) 1.00000 0.0416667
\(577\) 44.0000 1.83174 0.915872 0.401470i \(-0.131501\pi\)
0.915872 + 0.401470i \(0.131501\pi\)
\(578\) −19.0000 −0.790296
\(579\) 32.0000 1.32987
\(580\) 6.00000 0.249136
\(581\) 0 0
\(582\) −20.0000 −0.829027
\(583\) 27.0000 1.11823
\(584\) 4.00000 0.165521
\(585\) −1.00000 −0.0413449
\(586\) 9.00000 0.371787
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −30.0000 −1.23404
\(592\) −7.00000 −0.287698
\(593\) 24.0000 0.985562 0.492781 0.870153i \(-0.335980\pi\)
0.492781 + 0.870153i \(0.335980\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 32.0000 1.30967
\(598\) 9.00000 0.368037
\(599\) −42.0000 −1.71607 −0.858037 0.513588i \(-0.828316\pi\)
−0.858037 + 0.513588i \(0.828316\pi\)
\(600\) 2.00000 0.0816497
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) −10.0000 −0.406894
\(605\) −2.00000 −0.0813116
\(606\) 24.0000 0.974933
\(607\) −1.00000 −0.0405887 −0.0202944 0.999794i \(-0.506460\pi\)
−0.0202944 + 0.999794i \(0.506460\pi\)
\(608\) 1.00000 0.0405554
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) −9.00000 −0.364101
\(612\) −6.00000 −0.242536
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) −14.0000 −0.564994
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −8.00000 −0.321807
\(619\) 23.0000 0.924448 0.462224 0.886763i \(-0.347052\pi\)
0.462224 + 0.886763i \(0.347052\pi\)
\(620\) 8.00000 0.321288
\(621\) 36.0000 1.44463
\(622\) 24.0000 0.962312
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 1.00000 0.0400000
\(626\) 28.0000 1.11911
\(627\) 6.00000 0.239617
\(628\) 23.0000 0.917800
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 10.0000 0.397779
\(633\) −46.0000 −1.82834
\(634\) −6.00000 −0.238290
\(635\) −1.00000 −0.0396838
\(636\) −18.0000 −0.713746
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) −24.0000 −0.947204
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −6.00000 −0.236067
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 11.0000 0.432121
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 20.0000 0.783260
\(653\) −9.00000 −0.352197 −0.176099 0.984373i \(-0.556348\pi\)
−0.176099 + 0.984373i \(0.556348\pi\)
\(654\) −32.0000 −1.25130
\(655\) 3.00000 0.117220
\(656\) 3.00000 0.117130
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) −6.00000 −0.233550
\(661\) −28.0000 −1.08907 −0.544537 0.838737i \(-0.683295\pi\)
−0.544537 + 0.838737i \(0.683295\pi\)
\(662\) 7.00000 0.272063
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 0 0
\(666\) 7.00000 0.271244
\(667\) 54.0000 2.09089
\(668\) 3.00000 0.116073
\(669\) −16.0000 −0.618596
\(670\) −8.00000 −0.309067
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 22.0000 0.847408
\(675\) 4.00000 0.153960
\(676\) −12.0000 −0.461538
\(677\) −9.00000 −0.345898 −0.172949 0.984931i \(-0.555330\pi\)
−0.172949 + 0.984931i \(0.555330\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 6.00000 0.230089
\(681\) 24.0000 0.919682
\(682\) −24.0000 −0.919007
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 8.00000 0.305219
\(688\) 2.00000 0.0762493
\(689\) −9.00000 −0.342873
\(690\) 18.0000 0.685248
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) 9.00000 0.342129
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 12.0000 0.454859
\(697\) −18.0000 −0.681799
\(698\) −26.0000 −0.984115
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 4.00000 0.150970
\(703\) 7.00000 0.264010
\(704\) 3.00000 0.113067
\(705\) −18.0000 −0.677919
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) −6.00000 −0.224860
\(713\) 72.0000 2.69642
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) −3.00000 −0.112115
\(717\) 12.0000 0.448148
\(718\) −18.0000 −0.671754
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 18.0000 0.669891
\(723\) 2.00000 0.0743808
\(724\) 2.00000 0.0743294
\(725\) 6.00000 0.222834
\(726\) −4.00000 −0.148454
\(727\) −1.00000 −0.0370879 −0.0185440 0.999828i \(-0.505903\pi\)
−0.0185440 + 0.999828i \(0.505903\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 4.00000 0.148047
\(731\) −12.0000 −0.443836
\(732\) −16.0000 −0.591377
\(733\) −43.0000 −1.58824 −0.794121 0.607760i \(-0.792068\pi\)
−0.794121 + 0.607760i \(0.792068\pi\)
\(734\) 19.0000 0.701303
\(735\) 0 0
\(736\) −9.00000 −0.331744
\(737\) 24.0000 0.884051
\(738\) −3.00000 −0.110432
\(739\) 35.0000 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(740\) −7.00000 −0.257325
\(741\) −2.00000 −0.0734718
\(742\) 0 0
\(743\) −45.0000 −1.65089 −0.825445 0.564483i \(-0.809076\pi\)
−0.825445 + 0.564483i \(0.809076\pi\)
\(744\) 16.0000 0.586588
\(745\) −6.00000 −0.219823
\(746\) −2.00000 −0.0732252
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 0 0
\(750\) 2.00000 0.0730297
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) 9.00000 0.328196
\(753\) 30.0000 1.09326
\(754\) 6.00000 0.218507
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −23.0000 −0.835398
\(759\) −54.0000 −1.96008
\(760\) 1.00000 0.0362738
\(761\) −27.0000 −0.978749 −0.489375 0.872074i \(-0.662775\pi\)
−0.489375 + 0.872074i \(0.662775\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) −6.00000 −0.216930
\(766\) −21.0000 −0.758761
\(767\) 0 0
\(768\) −2.00000 −0.0721688
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) −51.0000 −1.83434 −0.917171 0.398493i \(-0.869533\pi\)
−0.917171 + 0.398493i \(0.869533\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 8.00000 0.287368
\(776\) 10.0000 0.358979
\(777\) 0 0
\(778\) 12.0000 0.430221
\(779\) −3.00000 −0.107486
\(780\) 2.00000 0.0716115
\(781\) 0 0
\(782\) 54.0000 1.93104
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) 23.0000 0.820905
\(786\) 6.00000 0.214013
\(787\) −22.0000 −0.784215 −0.392108 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 15.0000 0.534353
\(789\) 0 0
\(790\) 10.0000 0.355784
\(791\) 0 0
\(792\) −3.00000 −0.106600
\(793\) −8.00000 −0.284088
\(794\) −14.0000 −0.496841
\(795\) −18.0000 −0.638394
\(796\) −16.0000 −0.567105
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 0 0
\(799\) −54.0000 −1.91038
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) 27.0000 0.953403
\(803\) −12.0000 −0.423471
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) −9.00000 −0.316423 −0.158212 0.987405i \(-0.550573\pi\)
−0.158212 + 0.987405i \(0.550573\pi\)
\(810\) 11.0000 0.386501
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 21.0000 0.736050
\(815\) 20.0000 0.700569
\(816\) 12.0000 0.420084
\(817\) −2.00000 −0.0699711
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 3.00000 0.104765
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 24.0000 0.837096
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 4.00000 0.139347
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 9.00000 0.312772
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 20.0000 0.693792
\(832\) −1.00000 −0.0346688
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 3.00000 0.103819
\(836\) −3.00000 −0.103757
\(837\) 32.0000 1.10608
\(838\) 9.00000 0.310900
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −2.00000 −0.0689246
\(843\) 54.0000 1.85986
\(844\) 23.0000 0.791693
\(845\) −12.0000 −0.412813
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) 9.00000 0.309061
\(849\) −28.0000 −0.960958
\(850\) 6.00000 0.205798
\(851\) −63.0000 −2.15961
\(852\) 0 0
\(853\) −19.0000 −0.650548 −0.325274 0.945620i \(-0.605456\pi\)
−0.325274 + 0.945620i \(0.605456\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) −6.00000 −0.204837
\(859\) 32.0000 1.09183 0.545913 0.837842i \(-0.316183\pi\)
0.545913 + 0.837842i \(0.316183\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) −3.00000 −0.102121 −0.0510606 0.998696i \(-0.516260\pi\)
−0.0510606 + 0.998696i \(0.516260\pi\)
\(864\) −4.00000 −0.136083
\(865\) 9.00000 0.306009
\(866\) 40.0000 1.35926
\(867\) −38.0000 −1.29055
\(868\) 0 0
\(869\) −30.0000 −1.01768
\(870\) 12.0000 0.406838
\(871\) −8.00000 −0.271070
\(872\) 16.0000 0.541828
\(873\) −10.0000 −0.338449
\(874\) 9.00000 0.304430
\(875\) 0 0
\(876\) 8.00000 0.270295
\(877\) −13.0000 −0.438979 −0.219489 0.975615i \(-0.570439\pi\)
−0.219489 + 0.975615i \(0.570439\pi\)
\(878\) −26.0000 −0.877457
\(879\) 18.0000 0.607125
\(880\) 3.00000 0.101130
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −14.0000 −0.469809
\(889\) 0 0
\(890\) −6.00000 −0.201120
\(891\) −33.0000 −1.10554
\(892\) 8.00000 0.267860
\(893\) −9.00000 −0.301174
\(894\) −12.0000 −0.401340
\(895\) −3.00000 −0.100279
\(896\) 0 0
\(897\) 18.0000 0.601003
\(898\) −21.0000 −0.700779
\(899\) 48.0000 1.60089
\(900\) 1.00000 0.0333333
\(901\) −54.0000 −1.79900
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) −20.0000 −0.664455
\(907\) −10.0000 −0.332045 −0.166022 0.986122i \(-0.553092\pi\)
−0.166022 + 0.986122i \(0.553092\pi\)
\(908\) −12.0000 −0.398234
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) −14.0000 −0.463079
\(915\) −16.0000 −0.528944
\(916\) −4.00000 −0.132164
\(917\) 0 0
\(918\) 24.0000 0.792118
\(919\) −22.0000 −0.725713 −0.362857 0.931845i \(-0.618198\pi\)
−0.362857 + 0.931845i \(0.618198\pi\)
\(920\) −9.00000 −0.296721
\(921\) −28.0000 −0.922631
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) 1.00000 0.0328620
\(927\) −4.00000 −0.131377
\(928\) −6.00000 −0.196960
\(929\) 57.0000 1.87011 0.935055 0.354504i \(-0.115350\pi\)
0.935055 + 0.354504i \(0.115350\pi\)
\(930\) 16.0000 0.524661
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 48.0000 1.57145
\(934\) 6.00000 0.196326
\(935\) −18.0000 −0.588663
\(936\) 1.00000 0.0326860
\(937\) −10.0000 −0.326686 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(938\) 0 0
\(939\) 56.0000 1.82749
\(940\) 9.00000 0.293548
\(941\) 48.0000 1.56476 0.782378 0.622804i \(-0.214007\pi\)
0.782378 + 0.622804i \(0.214007\pi\)
\(942\) 46.0000 1.49876
\(943\) 27.0000 0.879241
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 6.00000 0.194974 0.0974869 0.995237i \(-0.468920\pi\)
0.0974869 + 0.995237i \(0.468920\pi\)
\(948\) 20.0000 0.649570
\(949\) 4.00000 0.129845
\(950\) 1.00000 0.0324443
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) −9.00000 −0.291386
\(955\) 12.0000 0.388311
\(956\) −6.00000 −0.194054
\(957\) −36.0000 −1.16371
\(958\) 0 0
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 33.0000 1.06452
\(962\) −7.00000 −0.225689
\(963\) −12.0000 −0.386695
\(964\) −1.00000 −0.0322078
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 2.00000 0.0642824
\(969\) −12.0000 −0.385496
\(970\) 10.0000 0.321081
\(971\) −45.0000 −1.44412 −0.722059 0.691831i \(-0.756804\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 2.00000 0.0640513
\(976\) 8.00000 0.256074
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 40.0000 1.27906
\(979\) 18.0000 0.575282
\(980\) 0 0
\(981\) −16.0000 −0.510841
\(982\) −36.0000 −1.14881
\(983\) 3.00000 0.0956851 0.0478426 0.998855i \(-0.484765\pi\)
0.0478426 + 0.998855i \(0.484765\pi\)
\(984\) 6.00000 0.191273
\(985\) 15.0000 0.477940
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 1.00000 0.0318142
\(989\) 18.0000 0.572367
\(990\) −3.00000 −0.0953463
\(991\) 44.0000 1.39771 0.698853 0.715265i \(-0.253694\pi\)
0.698853 + 0.715265i \(0.253694\pi\)
\(992\) −8.00000 −0.254000
\(993\) 14.0000 0.444277
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −22.0000 −0.696747 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(998\) 4.00000 0.126618
\(999\) −28.0000 −0.885881
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 490.2.a.a.1.1 1
3.2 odd 2 4410.2.a.x.1.1 1
4.3 odd 2 3920.2.a.bh.1.1 1
5.2 odd 4 2450.2.c.q.99.1 2
5.3 odd 4 2450.2.c.q.99.2 2
5.4 even 2 2450.2.a.bf.1.1 1
7.2 even 3 70.2.e.d.11.1 2
7.3 odd 6 490.2.e.g.471.1 2
7.4 even 3 70.2.e.d.51.1 yes 2
7.5 odd 6 490.2.e.g.361.1 2
7.6 odd 2 490.2.a.d.1.1 1
21.2 odd 6 630.2.k.d.361.1 2
21.11 odd 6 630.2.k.d.541.1 2
21.20 even 2 4410.2.a.bg.1.1 1
28.11 odd 6 560.2.q.b.401.1 2
28.23 odd 6 560.2.q.b.81.1 2
28.27 even 2 3920.2.a.e.1.1 1
35.2 odd 12 350.2.j.d.249.1 4
35.4 even 6 350.2.e.b.51.1 2
35.9 even 6 350.2.e.b.151.1 2
35.13 even 4 2450.2.c.e.99.2 2
35.18 odd 12 350.2.j.d.149.1 4
35.23 odd 12 350.2.j.d.249.2 4
35.27 even 4 2450.2.c.e.99.1 2
35.32 odd 12 350.2.j.d.149.2 4
35.34 odd 2 2450.2.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 7.2 even 3
70.2.e.d.51.1 yes 2 7.4 even 3
350.2.e.b.51.1 2 35.4 even 6
350.2.e.b.151.1 2 35.9 even 6
350.2.j.d.149.1 4 35.18 odd 12
350.2.j.d.149.2 4 35.32 odd 12
350.2.j.d.249.1 4 35.2 odd 12
350.2.j.d.249.2 4 35.23 odd 12
490.2.a.a.1.1 1 1.1 even 1 trivial
490.2.a.d.1.1 1 7.6 odd 2
490.2.e.g.361.1 2 7.5 odd 6
490.2.e.g.471.1 2 7.3 odd 6
560.2.q.b.81.1 2 28.23 odd 6
560.2.q.b.401.1 2 28.11 odd 6
630.2.k.d.361.1 2 21.2 odd 6
630.2.k.d.541.1 2 21.11 odd 6
2450.2.a.v.1.1 1 35.34 odd 2
2450.2.a.bf.1.1 1 5.4 even 2
2450.2.c.e.99.1 2 35.27 even 4
2450.2.c.e.99.2 2 35.13 even 4
2450.2.c.q.99.1 2 5.2 odd 4
2450.2.c.q.99.2 2 5.3 odd 4
3920.2.a.e.1.1 1 28.27 even 2
3920.2.a.bh.1.1 1 4.3 odd 2
4410.2.a.x.1.1 1 3.2 odd 2
4410.2.a.bg.1.1 1 21.20 even 2