Properties

Label 490.2.a.d.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 \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −2.00000 −0.816497
\(7\) 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 \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −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 \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −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 \(-0.575267\pi\)
−0.234261 + 0.972174i \(0.575267\pi\)
\(42\) 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 \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) 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 \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 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 \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 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 \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 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 \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 1.00000 0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) −12.0000 −1.18818
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −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 \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) 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 \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 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 \(-0.870036\pi\)
−0.917800 + 0.397043i \(0.870036\pi\)
\(158\) 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 \(-0.537031\pi\)
−0.116073 + 0.993241i \(0.537031\pi\)
\(168\) 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 \(-0.611148\pi\)
−0.342129 + 0.939653i \(0.611148\pi\)
\(174\) −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 \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 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 \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) −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 \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 2.00000 0.132453
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 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 \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) 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 \(-0.343028\pi\)
0.473396 + 0.880850i \(0.343028\pi\)
\(252\) 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 \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 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 \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 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 \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 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 \(-0.630820\pi\)
−0.399511 + 0.916728i \(0.630820\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) −8.00000 −0.454369
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −2.00000 −0.113228
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 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 \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −3.00000 −0.159901
\(353\) −12.0000 −0.638696 −0.319348 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 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 \(-0.334840\pi\)
0.495896 + 0.868382i \(0.334840\pi\)
\(368\) 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 \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) −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 \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) −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 \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) −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 \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 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 \(-0.0890309\pi\)
0.961139 + 0.276066i \(0.0890309\pi\)
\(434\) 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 \(-0.713053\pi\)
−0.620456 + 0.784241i \(0.713053\pi\)
\(440\) 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 \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 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 \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 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 \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 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 \(-0.119104\pi\)
0.930809 + 0.365507i \(0.119104\pi\)
\(510\) 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 \(-0.393430\pi\)
0.328581 + 0.944476i \(0.393430\pi\)
\(522\) −6.00000 −0.262613
\(523\) 28.0000 1.22435 0.612177 0.790721i \(-0.290294\pi\)
0.612177 + 0.790721i \(0.290294\pi\)
\(524\) −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 \(-0.845872\pi\)
−0.885044 + 0.465506i \(0.845872\pi\)
\(564\) −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 \(-0.868499\pi\)
−0.915872 + 0.401470i \(0.868499\pi\)
\(578\) −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 \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 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 \(-0.664020\pi\)
−0.492781 + 0.870153i \(0.664020\pi\)
\(594\) 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 \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 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 \(-0.493540\pi\)
0.0202944 + 0.999794i \(0.493540\pi\)
\(608\) −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 \(-0.652948\pi\)
−0.462224 + 0.886763i \(0.652948\pi\)
\(620\) 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 \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) −6.00000 −0.236067
\(647\) −33.0000 −1.29736 −0.648682 0.761060i \(-0.724679\pi\)
−0.648682 + 0.761060i \(0.724679\pi\)
\(648\) 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 \(-0.316705\pi\)
0.544537 + 0.838737i \(0.316705\pi\)
\(662\) 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 \(-0.444670\pi\)
0.172949 + 0.984931i \(0.444670\pi\)
\(678\) 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 \(-0.708296\pi\)
−0.608669 + 0.793424i \(0.708296\pi\)
\(692\) −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 \(-0.494097\pi\)
0.0185440 + 0.999828i \(0.494097\pi\)
\(728\) 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 \(-0.207932\pi\)
0.794121 + 0.607760i \(0.207932\pi\)
\(734\) −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 \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) 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 \(-0.636114\pi\)
−0.414701 + 0.909958i \(0.636114\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −16.0000 −0.575853
\(773\) 51.0000 1.83434 0.917171 0.398493i \(-0.130467\pi\)
0.917171 + 0.398493i \(0.130467\pi\)
\(774\) −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 \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 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 \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 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 \(-0.355356\pi\)
0.438934 + 0.898519i \(0.355356\pi\)
\(812\) 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 \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 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 \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 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 \(-0.394544\pi\)
0.325274 + 0.945620i \(0.394544\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 12.0000 0.410152
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −6.00000 −0.204837
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) −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 \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) 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 \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 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 \(-0.884650\pi\)
−0.935055 + 0.354504i \(0.884650\pi\)
\(930\) −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 \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 56.0000 1.82749
\(940\) 9.00000 0.293548
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) 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 \(-0.243196\pi\)
0.722059 + 0.691831i \(0.243196\pi\)
\(972\) −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 \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 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 \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) 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.d.1.1 1
3.2 odd 2 4410.2.a.bg.1.1 1
4.3 odd 2 3920.2.a.e.1.1 1
5.2 odd 4 2450.2.c.e.99.1 2
5.3 odd 4 2450.2.c.e.99.2 2
5.4 even 2 2450.2.a.v.1.1 1
7.2 even 3 490.2.e.g.361.1 2
7.3 odd 6 70.2.e.d.51.1 yes 2
7.4 even 3 490.2.e.g.471.1 2
7.5 odd 6 70.2.e.d.11.1 2
7.6 odd 2 490.2.a.a.1.1 1
21.5 even 6 630.2.k.d.361.1 2
21.17 even 6 630.2.k.d.541.1 2
21.20 even 2 4410.2.a.x.1.1 1
28.3 even 6 560.2.q.b.401.1 2
28.19 even 6 560.2.q.b.81.1 2
28.27 even 2 3920.2.a.bh.1.1 1
35.3 even 12 350.2.j.d.149.1 4
35.12 even 12 350.2.j.d.249.1 4
35.13 even 4 2450.2.c.q.99.2 2
35.17 even 12 350.2.j.d.149.2 4
35.19 odd 6 350.2.e.b.151.1 2
35.24 odd 6 350.2.e.b.51.1 2
35.27 even 4 2450.2.c.q.99.1 2
35.33 even 12 350.2.j.d.249.2 4
35.34 odd 2 2450.2.a.bf.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 7.5 odd 6
70.2.e.d.51.1 yes 2 7.3 odd 6
350.2.e.b.51.1 2 35.24 odd 6
350.2.e.b.151.1 2 35.19 odd 6
350.2.j.d.149.1 4 35.3 even 12
350.2.j.d.149.2 4 35.17 even 12
350.2.j.d.249.1 4 35.12 even 12
350.2.j.d.249.2 4 35.33 even 12
490.2.a.a.1.1 1 7.6 odd 2
490.2.a.d.1.1 1 1.1 even 1 trivial
490.2.e.g.361.1 2 7.2 even 3
490.2.e.g.471.1 2 7.4 even 3
560.2.q.b.81.1 2 28.19 even 6
560.2.q.b.401.1 2 28.3 even 6
630.2.k.d.361.1 2 21.5 even 6
630.2.k.d.541.1 2 21.17 even 6
2450.2.a.v.1.1 1 5.4 even 2
2450.2.a.bf.1.1 1 35.34 odd 2
2450.2.c.e.99.1 2 5.2 odd 4
2450.2.c.e.99.2 2 5.3 odd 4
2450.2.c.q.99.1 2 35.27 even 4
2450.2.c.q.99.2 2 35.13 even 4
3920.2.a.e.1.1 1 4.3 odd 2
3920.2.a.bh.1.1 1 28.27 even 2
4410.2.a.x.1.1 1 21.20 even 2
4410.2.a.bg.1.1 1 3.2 odd 2