Properties

Label 4830.2.a.o.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
Analytic rank: \(1\)
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
Character \(\chi\) \(=\) 4830.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} +1.00000 q^{21} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{30} +8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} -1.00000 q^{56} +2.00000 q^{58} +8.00000 q^{59} +1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} -1.00000 q^{70} -12.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -2.00000 q^{74} +1.00000 q^{75} -4.00000 q^{77} +2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -12.0000 q^{83} +1.00000 q^{84} -6.00000 q^{85} +4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 1.00000 0.258199
\(16\) 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
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) 4.00000 0.852803
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) −1.00000 −0.182574
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 6.00000 1.02899
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) −1.00000 −0.154303
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −6.00000 −0.840168
\(52\) −2.00000 −0.277350
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.00000 −0.539360
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 1.00000 0.129099
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −8.00000 −1.01600
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −6.00000 −0.727607
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −2.00000 −0.232495
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 2.00000 0.226455
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 1.00000 0.109109
\(85\) −6.00000 −0.650791
\(86\) 4.00000 0.431331
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −1.00000 −0.105409
\(91\) −2.00000 −0.209657
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
\(100\) 1.00000 0.100000
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) 6.00000 0.594089
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 2.00000 0.196116
\(105\) 1.00000 0.0975900
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 4.00000 0.381385
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) −2.00000 −0.185695
\(117\) −2.00000 −0.184900
\(118\) −8.00000 −0.736460
\(119\) −6.00000 −0.550019
\(120\) −1.00000 −0.0912871
\(121\) 5.00000 0.454545
\(122\) 2.00000 0.181071
\(123\) 2.00000 0.180334
\(124\) 8.00000 0.718421
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.00000 −0.352180
\(130\) 2.00000 0.175412
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 1.00000 0.0860663
\(136\) 6.00000 0.514496
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 1.00000 0.0845154
\(141\) −8.00000 −0.673722
\(142\) 12.0000 1.00702
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) 14.0000 1.15865
\(147\) 1.00000 0.0824786
\(148\) 2.00000 0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 4.00000 0.322329
\(155\) 8.00000 0.642575
\(156\) −2.00000 −0.160128
\(157\) −6.00000 −0.478852 −0.239426 0.970915i \(-0.576959\pi\)
−0.239426 + 0.970915i \(0.576959\pi\)
\(158\) 12.0000 0.954669
\(159\) 6.00000 0.475831
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 2.00000 0.156174
\(165\) −4.00000 −0.311400
\(166\) 12.0000 0.931381
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 8.00000 0.601317
\(178\) 6.00000 0.449719
\(179\) 8.00000 0.597948 0.298974 0.954261i \(-0.403356\pi\)
0.298974 + 0.954261i \(0.403356\pi\)
\(180\) 1.00000 0.0745356
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 2.00000 0.148250
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) 2.00000 0.147043
\(186\) −8.00000 −0.586588
\(187\) 24.0000 1.75505
\(188\) −8.00000 −0.583460
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −14.0000 −1.00514
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 4.00000 0.284268
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 4.00000 0.282138
\(202\) 18.0000 1.26648
\(203\) −2.00000 −0.140372
\(204\) −6.00000 −0.420084
\(205\) 2.00000 0.139686
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) 0 0
\(210\) −1.00000 −0.0690066
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000 0.412082
\(213\) −12.0000 −0.822226
\(214\) 12.0000 0.820303
\(215\) −4.00000 −0.272798
\(216\) −1.00000 −0.0680414
\(217\) 8.00000 0.543075
\(218\) 10.0000 0.677285
\(219\) −14.0000 −0.946032
\(220\) −4.00000 −0.269680
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 1.00000 0.0666667
\(226\) 6.00000 0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) −1.00000 −0.0659380
\(231\) −4.00000 −0.263181
\(232\) 2.00000 0.131306
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) 2.00000 0.130744
\(235\) −8.00000 −0.521862
\(236\) 8.00000 0.520756
\(237\) −12.0000 −0.779484
\(238\) 6.00000 0.388922
\(239\) −20.0000 −1.29369 −0.646846 0.762620i \(-0.723912\pi\)
−0.646846 + 0.762620i \(0.723912\pi\)
\(240\) 1.00000 0.0645497
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 1.00000 0.0638877
\(246\) −2.00000 −0.127515
\(247\) 0 0
\(248\) −8.00000 −0.508001
\(249\) −12.0000 −0.760469
\(250\) −1.00000 −0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) −6.00000 −0.375735
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 4.00000 0.249029
\(259\) 2.00000 0.124274
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) −8.00000 −0.494242
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 4.00000 0.246183
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 4.00000 0.244339
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) −18.0000 −1.08742
\(275\) −4.00000 −0.241209
\(276\) 1.00000 0.0601929
\(277\) −26.0000 −1.56219 −0.781094 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 20.0000 1.19952
\(279\) 8.00000 0.478947
\(280\) −1.00000 −0.0597614
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 8.00000 0.476393
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) 2.00000 0.118056
\(288\) −1.00000 −0.0589256
\(289\) 19.0000 1.11765
\(290\) 2.00000 0.117444
\(291\) 14.0000 0.820695
\(292\) −14.0000 −0.819288
\(293\) 30.0000 1.75262 0.876309 0.481749i \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 8.00000 0.465778
\(296\) −2.00000 −0.116248
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) −2.00000 −0.115663
\(300\) 1.00000 0.0577350
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −18.0000 −1.03407
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 6.00000 0.342997
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) −8.00000 −0.454369
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 2.00000 0.113228
\(313\) 30.0000 1.69570 0.847850 0.530236i \(-0.177897\pi\)
0.847850 + 0.530236i \(0.177897\pi\)
\(314\) 6.00000 0.338600
\(315\) 1.00000 0.0563436
\(316\) −12.0000 −0.675053
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.00000 0.447914
\(320\) 1.00000 0.0559017
\(321\) −12.0000 −0.669775
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) 20.0000 1.10770
\(327\) −10.0000 −0.553001
\(328\) −2.00000 −0.110432
\(329\) −8.00000 −0.441054
\(330\) 4.00000 0.220193
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) 8.00000 0.437741
\(335\) 4.00000 0.218543
\(336\) 1.00000 0.0545545
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) −6.00000 −0.325396
\(341\) −32.0000 −1.73290
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 1.00000 0.0538382
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) 4.00000 0.213201
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) −8.00000 −0.425195
\(355\) −12.0000 −0.636894
\(356\) −6.00000 −0.317999
\(357\) −6.00000 −0.317554
\(358\) −8.00000 −0.422813
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −19.0000 −1.00000
\(362\) −14.0000 −0.735824
\(363\) 5.00000 0.262432
\(364\) −2.00000 −0.104828
\(365\) −14.0000 −0.732793
\(366\) 2.00000 0.104542
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) 1.00000 0.0521286
\(369\) 2.00000 0.104116
\(370\) −2.00000 −0.103975
\(371\) 6.00000 0.311504
\(372\) 8.00000 0.414781
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −24.0000 −1.24101
\(375\) 1.00000 0.0516398
\(376\) 8.00000 0.412568
\(377\) 4.00000 0.206010
\(378\) −1.00000 −0.0514344
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 8.00000 0.409316
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −4.00000 −0.203859
\(386\) −10.0000 −0.508987
\(387\) −4.00000 −0.203331
\(388\) 14.0000 0.710742
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 2.00000 0.101274
\(391\) −6.00000 −0.303433
\(392\) −1.00000 −0.0505076
\(393\) 8.00000 0.403547
\(394\) 6.00000 0.302276
\(395\) −12.0000 −0.603786
\(396\) −4.00000 −0.201008
\(397\) −26.0000 −1.30490 −0.652451 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) −4.00000 −0.199502
\(403\) −16.0000 −0.797017
\(404\) −18.0000 −0.895533
\(405\) 1.00000 0.0496904
\(406\) 2.00000 0.0992583
\(407\) −8.00000 −0.396545
\(408\) 6.00000 0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) −1.00000 −0.0491473
\(415\) −12.0000 −0.589057
\(416\) 2.00000 0.0980581
\(417\) −20.0000 −0.979404
\(418\) 0 0
\(419\) 36.0000 1.75872 0.879358 0.476162i \(-0.157972\pi\)
0.879358 + 0.476162i \(0.157972\pi\)
\(420\) 1.00000 0.0487950
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −20.0000 −0.973585
\(423\) −8.00000 −0.388973
\(424\) −6.00000 −0.291386
\(425\) −6.00000 −0.291043
\(426\) 12.0000 0.581402
\(427\) −2.00000 −0.0967868
\(428\) −12.0000 −0.580042
\(429\) 8.00000 0.386244
\(430\) 4.00000 0.192897
\(431\) 40.0000 1.92673 0.963366 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 1.00000 0.0481125
\(433\) −18.0000 −0.865025 −0.432512 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) −8.00000 −0.384012
\(435\) −2.00000 −0.0958927
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 14.0000 0.668946
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 4.00000 0.190693
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) 2.00000 0.0949158
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) 34.0000 1.60456 0.802280 0.596948i \(-0.203620\pi\)
0.802280 + 0.596948i \(0.203620\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −8.00000 −0.376705
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 10.0000 0.467269
\(459\) −6.00000 −0.280056
\(460\) 1.00000 0.0466252
\(461\) −26.0000 −1.21094 −0.605470 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) 4.00000 0.186097
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 8.00000 0.370991
\(466\) 26.0000 1.20443
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 4.00000 0.184703
\(470\) 8.00000 0.369012
\(471\) −6.00000 −0.276465
\(472\) −8.00000 −0.368230
\(473\) 16.0000 0.735681
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) −6.00000 −0.275010
\(477\) 6.00000 0.274721
\(478\) 20.0000 0.914779
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) −1.00000 −0.0456435
\(481\) −4.00000 −0.182384
\(482\) 22.0000 1.00207
\(483\) 1.00000 0.0455016
\(484\) 5.00000 0.227273
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 2.00000 0.0905357
\(489\) −20.0000 −0.904431
\(490\) −1.00000 −0.0451754
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 2.00000 0.0901670
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) −4.00000 −0.179787
\(496\) 8.00000 0.359211
\(497\) −12.0000 −0.538274
\(498\) 12.0000 0.537733
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 1.00000 0.0447214
\(501\) −8.00000 −0.357414
\(502\) −12.0000 −0.535586
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −18.0000 −0.800989
\(506\) 4.00000 0.177822
\(507\) −9.00000 −0.399704
\(508\) −8.00000 −0.354943
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 6.00000 0.265684
\(511\) −14.0000 −0.619324
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 32.0000 1.40736
\(518\) −2.00000 −0.0878750
\(519\) −6.00000 −0.263371
\(520\) 2.00000 0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 2.00000 0.0875376
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 8.00000 0.349482
\(525\) 1.00000 0.0436436
\(526\) 8.00000 0.348817
\(527\) −48.0000 −2.09091
\(528\) −4.00000 −0.174078
\(529\) 1.00000 0.0434783
\(530\) −6.00000 −0.260623
\(531\) 8.00000 0.347170
\(532\) 0 0
\(533\) −4.00000 −0.173259
\(534\) 6.00000 0.259645
\(535\) −12.0000 −0.518805
\(536\) −4.00000 −0.172774
\(537\) 8.00000 0.345225
\(538\) −6.00000 −0.258678
\(539\) −4.00000 −0.172292
\(540\) 1.00000 0.0430331
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −16.0000 −0.687259
\(543\) 14.0000 0.600798
\(544\) 6.00000 0.257248
\(545\) −10.0000 −0.428353
\(546\) 2.00000 0.0855921
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 18.0000 0.768922
\(549\) −2.00000 −0.0853579
\(550\) 4.00000 0.170561
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) −12.0000 −0.510292
\(554\) 26.0000 1.10463
\(555\) 2.00000 0.0848953
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −8.00000 −0.338667
\(559\) 8.00000 0.338364
\(560\) 1.00000 0.0422577
\(561\) 24.0000 1.01328
\(562\) −26.0000 −1.09674
\(563\) 20.0000 0.842900 0.421450 0.906852i \(-0.361521\pi\)
0.421450 + 0.906852i \(0.361521\pi\)
\(564\) −8.00000 −0.336861
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 1.00000 0.0419961
\(568\) 12.0000 0.503509
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 16.0000 0.669579 0.334790 0.942293i \(-0.391335\pi\)
0.334790 + 0.942293i \(0.391335\pi\)
\(572\) 8.00000 0.334497
\(573\) −8.00000 −0.334205
\(574\) −2.00000 −0.0834784
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) −19.0000 −0.790296
\(579\) 10.0000 0.415586
\(580\) −2.00000 −0.0830455
\(581\) −12.0000 −0.497844
\(582\) −14.0000 −0.580319
\(583\) −24.0000 −0.993978
\(584\) 14.0000 0.579324
\(585\) −2.00000 −0.0826898
\(586\) −30.0000 −1.23929
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) −8.00000 −0.329355
\(591\) −6.00000 −0.246807
\(592\) 2.00000 0.0821995
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 4.00000 0.164122
\(595\) −6.00000 −0.245976
\(596\) 6.00000 0.245770
\(597\) 20.0000 0.818546
\(598\) 2.00000 0.0817861
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 5.00000 0.203279
\(606\) 18.0000 0.731200
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 2.00000 0.0809776
\(611\) 16.0000 0.647291
\(612\) −6.00000 −0.242536
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −28.0000 −1.12999
\(615\) 2.00000 0.0806478
\(616\) 4.00000 0.161165
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 8.00000 0.321288
\(621\) 1.00000 0.0401286
\(622\) 12.0000 0.481156
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −30.0000 −1.19904
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −12.0000 −0.478471
\(630\) −1.00000 −0.0398410
\(631\) 12.0000 0.477712 0.238856 0.971055i \(-0.423228\pi\)
0.238856 + 0.971055i \(0.423228\pi\)
\(632\) 12.0000 0.477334
\(633\) 20.0000 0.794929
\(634\) 6.00000 0.238290
\(635\) −8.00000 −0.317470
\(636\) 6.00000 0.237915
\(637\) −2.00000 −0.0792429
\(638\) −8.00000 −0.316723
\(639\) −12.0000 −0.474713
\(640\) −1.00000 −0.0395285
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) 12.0000 0.473602
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 1.00000 0.0394055
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −32.0000 −1.25611
\(650\) 2.00000 0.0784465
\(651\) 8.00000 0.313545
\(652\) −20.0000 −0.783260
\(653\) 34.0000 1.33052 0.665261 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 10.0000 0.391031
\(655\) 8.00000 0.312586
\(656\) 2.00000 0.0780869
\(657\) −14.0000 −0.546192
\(658\) 8.00000 0.311872
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −4.00000 −0.155700
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −20.0000 −0.777322
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −2.00000 −0.0774403
\(668\) −8.00000 −0.309529
\(669\) 8.00000 0.309298
\(670\) −4.00000 −0.154533
\(671\) 8.00000 0.308837
\(672\) −1.00000 −0.0385758
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) −22.0000 −0.847408
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 6.00000 0.230429
\(679\) 14.0000 0.537271
\(680\) 6.00000 0.230089
\(681\) −20.0000 −0.766402
\(682\) 32.0000 1.22534
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) −1.00000 −0.0381802
\(687\) −10.0000 −0.381524
\(688\) −4.00000 −0.152499
\(689\) −12.0000 −0.457164
\(690\) −1.00000 −0.0380693
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −6.00000 −0.228086
\(693\) −4.00000 −0.151947
\(694\) −12.0000 −0.455514
\(695\) −20.0000 −0.758643
\(696\) 2.00000 0.0758098
\(697\) −12.0000 −0.454532
\(698\) 2.00000 0.0757011
\(699\) −26.0000 −0.983410
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 2.00000 0.0754851
\(703\) 0 0
\(704\) −4.00000 −0.150756
\(705\) −8.00000 −0.301297
\(706\) 18.0000 0.677439
\(707\) −18.0000 −0.676960
\(708\) 8.00000 0.300658
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 12.0000 0.450352
\(711\) −12.0000 −0.450035
\(712\) 6.00000 0.224860
\(713\) 8.00000 0.299602
\(714\) 6.00000 0.224544
\(715\) 8.00000 0.299183
\(716\) 8.00000 0.298974
\(717\) −20.0000 −0.746914
\(718\) −24.0000 −0.895672
\(719\) 28.0000 1.04422 0.522112 0.852877i \(-0.325144\pi\)
0.522112 + 0.852877i \(0.325144\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) −22.0000 −0.818189
\(724\) 14.0000 0.520306
\(725\) −2.00000 −0.0742781
\(726\) −5.00000 −0.185567
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 14.0000 0.518163
\(731\) 24.0000 0.887672
\(732\) −2.00000 −0.0739221
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 24.0000 0.885856
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −16.0000 −0.589368
\(738\) −2.00000 −0.0736210
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 2.00000 0.0735215
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −8.00000 −0.293294
\(745\) 6.00000 0.219823
\(746\) −10.0000 −0.366126
\(747\) −12.0000 −0.439057
\(748\) 24.0000 0.877527
\(749\) −12.0000 −0.438470
\(750\) −1.00000 −0.0365148
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) −8.00000 −0.291730
\(753\) 12.0000 0.437304
\(754\) −4.00000 −0.145671
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −14.0000 −0.508839 −0.254419 0.967094i \(-0.581884\pi\)
−0.254419 + 0.967094i \(0.581884\pi\)
\(758\) −16.0000 −0.581146
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 8.00000 0.289809
\(763\) −10.0000 −0.362024
\(764\) −8.00000 −0.289430
\(765\) −6.00000 −0.216930
\(766\) −8.00000 −0.289052
\(767\) −16.0000 −0.577727
\(768\) 1.00000 0.0360844
\(769\) 42.0000 1.51456 0.757279 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 4.00000 0.144150
\(771\) −18.0000 −0.648254
\(772\) 10.0000 0.359908
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 4.00000 0.143777
\(775\) 8.00000 0.287368
\(776\) −14.0000 −0.502571
\(777\) 2.00000 0.0717496
\(778\) 26.0000 0.932145
\(779\) 0 0
\(780\) −2.00000 −0.0716115
\(781\) 48.0000 1.71758
\(782\) 6.00000 0.214560
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) −8.00000 −0.285351
\(787\) −44.0000 −1.56843 −0.784215 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −6.00000 −0.213741
\(789\) −8.00000 −0.284808
\(790\) 12.0000 0.426941
\(791\) −6.00000 −0.213335
\(792\) 4.00000 0.142134
\(793\) 4.00000 0.142044
\(794\) 26.0000 0.922705
\(795\) 6.00000 0.212798
\(796\) 20.0000 0.708881
\(797\) 38.0000 1.34603 0.673015 0.739629i \(-0.264999\pi\)
0.673015 + 0.739629i \(0.264999\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 30.0000 1.05934
\(803\) 56.0000 1.97620
\(804\) 4.00000 0.141069
\(805\) 1.00000 0.0352454
\(806\) 16.0000 0.563576
\(807\) 6.00000 0.211210
\(808\) 18.0000 0.633238
\(809\) 34.0000 1.19538 0.597688 0.801729i \(-0.296086\pi\)
0.597688 + 0.801729i \(0.296086\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 44.0000 1.54505 0.772524 0.634985i \(-0.218994\pi\)
0.772524 + 0.634985i \(0.218994\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 16.0000 0.561144
\(814\) 8.00000 0.280400
\(815\) −20.0000 −0.700569
\(816\) −6.00000 −0.210042
\(817\) 0 0
\(818\) −10.0000 −0.349642
\(819\) −2.00000 −0.0698857
\(820\) 2.00000 0.0698430
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −18.0000 −0.627822
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) −8.00000 −0.278356
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 1.00000 0.0347524
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 12.0000 0.416526
\(831\) −26.0000 −0.901930
\(832\) −2.00000 −0.0693375
\(833\) −6.00000 −0.207888
\(834\) 20.0000 0.692543
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 8.00000 0.276520
\(838\) −36.0000 −1.24360
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −25.0000 −0.862069
\(842\) −22.0000 −0.758170
\(843\) 26.0000 0.895488
\(844\) 20.0000 0.688428
\(845\) −9.00000 −0.309609
\(846\) 8.00000 0.275046
\(847\) 5.00000 0.171802
\(848\) 6.00000 0.206041
\(849\) 4.00000 0.137280
\(850\) 6.00000 0.205798
\(851\) 2.00000 0.0685591
\(852\) −12.0000 −0.411113
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) −8.00000 −0.273115
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −4.00000 −0.136399
\(861\) 2.00000 0.0681598
\(862\) −40.0000 −1.36241
\(863\) 32.0000 1.08929 0.544646 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −6.00000 −0.204006
\(866\) 18.0000 0.611665
\(867\) 19.0000 0.645274
\(868\) 8.00000 0.271538
\(869\) 48.0000 1.62829
\(870\) 2.00000 0.0678064
\(871\) −8.00000 −0.271070
\(872\) 10.0000 0.338643
\(873\) 14.0000 0.473828
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) −14.0000 −0.473016
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −8.00000 −0.269987
\(879\) 30.0000 1.01187
\(880\) −4.00000 −0.134840
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 12.0000 0.403604
\(885\) 8.00000 0.268917
\(886\) −4.00000 −0.134383
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −2.00000 −0.0671156
\(889\) −8.00000 −0.268311
\(890\) 6.00000 0.201120
\(891\) −4.00000 −0.134005
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 8.00000 0.267411
\(896\) −1.00000 −0.0334077
\(897\) −2.00000 −0.0667781
\(898\) −34.0000 −1.13459
\(899\) −16.0000 −0.533630
\(900\) 1.00000 0.0333333
\(901\) −36.0000 −1.19933
\(902\) 8.00000 0.266371
\(903\) −4.00000 −0.133112
\(904\) 6.00000 0.199557
\(905\) 14.0000 0.465376
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −20.0000 −0.663723
\(909\) −18.0000 −0.597022
\(910\) 2.00000 0.0662994
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 48.0000 1.58857
\(914\) 18.0000 0.595387
\(915\) −2.00000 −0.0661180
\(916\) −10.0000 −0.330409
\(917\) 8.00000 0.264183
\(918\) 6.00000 0.198030
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 28.0000 0.922631
\(922\) 26.0000 0.856264
\(923\) 24.0000 0.789970
\(924\) −4.00000 −0.131590
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −8.00000 −0.262330
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) −12.0000 −0.392862
\(934\) 36.0000 1.17796
\(935\) 24.0000 0.784884
\(936\) 2.00000 0.0653720
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) −4.00000 −0.130605
\(939\) 30.0000 0.979013
\(940\) −8.00000 −0.260931
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 6.00000 0.195491
\(943\) 2.00000 0.0651290
\(944\) 8.00000 0.260378
\(945\) 1.00000 0.0325300
\(946\) −16.0000 −0.520205
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) −12.0000 −0.389742
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 6.00000 0.194461
\(953\) −22.0000 −0.712650 −0.356325 0.934362i \(-0.615970\pi\)
−0.356325 + 0.934362i \(0.615970\pi\)
\(954\) −6.00000 −0.194257
\(955\) −8.00000 −0.258874
\(956\) −20.0000 −0.646846
\(957\) 8.00000 0.258603
\(958\) 8.00000 0.258468
\(959\) 18.0000 0.581250
\(960\) 1.00000 0.0322749
\(961\) 33.0000 1.06452
\(962\) 4.00000 0.128965
\(963\) −12.0000 −0.386695
\(964\) −22.0000 −0.708572
\(965\) 10.0000 0.321911
\(966\) −1.00000 −0.0321745
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) −20.0000 −0.641171
\(974\) 24.0000 0.769010
\(975\) −2.00000 −0.0640513
\(976\) −2.00000 −0.0640184
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) 20.0000 0.639529
\(979\) 24.0000 0.767043
\(980\) 1.00000 0.0319438
\(981\) −10.0000 −0.319275
\(982\) −32.0000 −1.02116
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) −2.00000 −0.0637577
\(985\) −6.00000 −0.191176
\(986\) −12.0000 −0.382158
\(987\) −8.00000 −0.254643
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 4.00000 0.127128
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) −8.00000 −0.254000
\(993\) 20.0000 0.634681
\(994\) 12.0000 0.380617
\(995\) 20.0000 0.634043
\(996\) −12.0000 −0.380235
\(997\) 22.0000 0.696747 0.348373 0.937356i \(-0.386734\pi\)
0.348373 + 0.937356i \(0.386734\pi\)
\(998\) −12.0000 −0.379853
\(999\) 2.00000 0.0632772
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 4830.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.o.1.1 1 1.1 even 1 trivial