Properties

Label 4830.2.a.q.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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} +1.00000 q^{20} +1.00000 q^{21} -2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -8.00000 q^{29} -1.00000 q^{30} -10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +1.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} +6.00000 q^{38} -2.00000 q^{39} -1.00000 q^{40} -10.0000 q^{41} -1.00000 q^{42} +2.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{55} -1.00000 q^{56} -6.00000 q^{57} +8.00000 q^{58} -4.00000 q^{59} +1.00000 q^{60} +10.0000 q^{61} +10.0000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} -2.00000 q^{66} -14.0000 q^{67} +1.00000 q^{69} -1.00000 q^{70} +6.00000 q^{71} -1.00000 q^{72} +10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} -6.00000 q^{76} +2.00000 q^{77} +2.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +10.0000 q^{82} -18.0000 q^{83} +1.00000 q^{84} -2.00000 q^{86} -8.00000 q^{87} -2.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} -2.00000 q^{91} +1.00000 q^{92} -10.0000 q^{93} +2.00000 q^{94} -6.00000 q^{95} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 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
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 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
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.00000 0.218218
\(22\) −2.00000 −0.426401
\(23\) 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
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −1.00000 −0.182574
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 6.00000 0.973329
\(39\) −2.00000 −0.320256
\(40\) −1.00000 −0.158114
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) −1.00000 −0.154303
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) −1.00000 −0.147442
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) −1.00000 −0.133631
\(57\) −6.00000 −0.794719
\(58\) 8.00000 1.05045
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 1.00000 0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 10.0000 1.27000
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) −2.00000 −0.246183
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) −1.00000 −0.117851
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) −6.00000 −0.688247
\(77\) 2.00000 0.227921
\(78\) 2.00000 0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −2.00000 −0.215666
\(87\) −8.00000 −0.857690
\(88\) −2.00000 −0.213201
\(89\) −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
\(93\) −10.0000 −1.03695
\(94\) 2.00000 0.206284
\(95\) −6.00000 −0.615587
\(96\) −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
\(99\) 2.00000 0.201008
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 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
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −2.00000 −0.190693
\(111\) −10.0000 −0.949158
\(112\) 1.00000 0.0944911
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 6.00000 0.561951
\(115\) 1.00000 0.0932505
\(116\) −8.00000 −0.742781
\(117\) −2.00000 −0.184900
\(118\) 4.00000 0.368230
\(119\) 0 0
\(120\) −1.00000 −0.0912871
\(121\) −7.00000 −0.636364
\(122\) −10.0000 −0.905357
\(123\) −10.0000 −0.901670
\(124\) −10.0000 −0.898027
\(125\) 1.00000 0.0894427
\(126\) −1.00000 −0.0890871
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.00000 0.176090
\(130\) 2.00000 0.175412
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 2.00000 0.174078
\(133\) −6.00000 −0.520266
\(134\) 14.0000 1.20942
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 1.00000 0.0845154
\(141\) −2.00000 −0.168430
\(142\) −6.00000 −0.503509
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) −8.00000 −0.664364
\(146\) −10.0000 −0.827606
\(147\) 1.00000 0.0824786
\(148\) −10.0000 −0.821995
\(149\) 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 \(0.5\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) −2.00000 −0.161165
\(155\) −10.0000 −0.803219
\(156\) −2.00000 −0.160128
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) −1.00000 −0.0790569
\(161\) 1.00000 0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −10.0000 −0.780869
\(165\) 2.00000 0.155700
\(166\) 18.0000 1.39707
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) −1.00000 −0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 2.00000 0.152499
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) 8.00000 0.606478
\(175\) 1.00000 0.0755929
\(176\) 2.00000 0.150756
\(177\) −4.00000 −0.300658
\(178\) 6.00000 0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 1.00000 0.0745356
\(181\) 26.0000 1.93256 0.966282 0.257485i \(-0.0828937\pi\)
0.966282 + 0.257485i \(0.0828937\pi\)
\(182\) 2.00000 0.148250
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) −10.0000 −0.735215
\(186\) 10.0000 0.733236
\(187\) 0 0
\(188\) −2.00000 −0.145865
\(189\) 1.00000 0.0727393
\(190\) 6.00000 0.435286
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −14.0000 −1.00514
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) −2.00000 −0.142134
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −14.0000 −0.987484
\(202\) 6.00000 0.422159
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −10.0000 −0.698430
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) −2.00000 −0.138675
\(209\) −12.0000 −0.830057
\(210\) −1.00000 −0.0690066
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) −6.00000 −0.412082
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) −1.00000 −0.0680414
\(217\) −10.0000 −0.678844
\(218\) −14.0000 −0.948200
\(219\) 10.0000 0.675737
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 10.0000 0.671156
\(223\) 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
\(227\) −26.0000 −1.72568 −0.862840 0.505477i \(-0.831317\pi\)
−0.862840 + 0.505477i \(0.831317\pi\)
\(228\) −6.00000 −0.397360
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 2.00000 0.131590
\(232\) 8.00000 0.525226
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 2.00000 0.130744
\(235\) −2.00000 −0.130466
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) 0 0
\(239\) −2.00000 −0.129369 −0.0646846 0.997906i \(-0.520604\pi\)
−0.0646846 + 0.997906i \(0.520604\pi\)
\(240\) 1.00000 0.0645497
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 1.00000 0.0638877
\(246\) 10.0000 0.637577
\(247\) 12.0000 0.763542
\(248\) 10.0000 0.635001
\(249\) −18.0000 −1.14070
\(250\) −1.00000 −0.0632456
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 1.00000 0.0629941
\(253\) 2.00000 0.125739
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −2.00000 −0.124515
\(259\) −10.0000 −0.621370
\(260\) −2.00000 −0.124035
\(261\) −8.00000 −0.495188
\(262\) −20.0000 −1.23560
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −2.00000 −0.123091
\(265\) −6.00000 −0.368577
\(266\) 6.00000 0.367884
\(267\) −6.00000 −0.367194
\(268\) −14.0000 −0.855186
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 0 0
\(273\) −2.00000 −0.121046
\(274\) 18.0000 1.08742
\(275\) 2.00000 0.120605
\(276\) 1.00000 0.0601929
\(277\) 4.00000 0.240337 0.120168 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) −16.0000 −0.959616
\(279\) −10.0000 −0.598684
\(280\) −1.00000 −0.0597614
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 2.00000 0.119098
\(283\) −32.0000 −1.90220 −0.951101 0.308879i \(-0.900046\pi\)
−0.951101 + 0.308879i \(0.900046\pi\)
\(284\) 6.00000 0.356034
\(285\) −6.00000 −0.355409
\(286\) 4.00000 0.236525
\(287\) −10.0000 −0.590281
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 8.00000 0.469776
\(291\) 14.0000 0.820695
\(292\) 10.0000 0.585206
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −4.00000 −0.232889
\(296\) 10.0000 0.581238
\(297\) 2.00000 0.116052
\(298\) −6.00000 −0.347571
\(299\) −2.00000 −0.115663
\(300\) 1.00000 0.0577350
\(301\) 2.00000 0.115278
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) −6.00000 −0.344124
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 2.00000 0.113961
\(309\) 0 0
\(310\) 10.0000 0.567962
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 2.00000 0.113228
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −18.0000 −1.01580
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 6.00000 0.336463
\(319\) −16.0000 −0.895828
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −1.00000 −0.0557278
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 14.0000 0.774202
\(328\) 10.0000 0.552158
\(329\) −2.00000 −0.110264
\(330\) −2.00000 −0.110096
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −18.0000 −0.987878
\(333\) −10.0000 −0.547997
\(334\) 2.00000 0.109435
\(335\) −14.0000 −0.764902
\(336\) 1.00000 0.0545545
\(337\) −8.00000 −0.435788 −0.217894 0.975972i \(-0.569919\pi\)
−0.217894 + 0.975972i \(0.569919\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) 6.00000 0.324443
\(343\) 1.00000 0.0539949
\(344\) −2.00000 −0.107833
\(345\) 1.00000 0.0538382
\(346\) −12.0000 −0.645124
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −8.00000 −0.428845
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) −1.00000 −0.0534522
\(351\) −2.00000 −0.106752
\(352\) −2.00000 −0.106600
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 4.00000 0.212598
\(355\) 6.00000 0.318447
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) −26.0000 −1.36653
\(363\) −7.00000 −0.367405
\(364\) −2.00000 −0.104828
\(365\) 10.0000 0.523424
\(366\) −10.0000 −0.522708
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 1.00000 0.0521286
\(369\) −10.0000 −0.520579
\(370\) 10.0000 0.519875
\(371\) −6.00000 −0.311504
\(372\) −10.0000 −0.518476
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 2.00000 0.103142
\(377\) 16.0000 0.824042
\(378\) −1.00000 −0.0514344
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) −6.00000 −0.307794
\(381\) −2.00000 −0.102463
\(382\) 8.00000 0.409316
\(383\) −4.00000 −0.204390 −0.102195 0.994764i \(-0.532587\pi\)
−0.102195 + 0.994764i \(0.532587\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 2.00000 0.101929
\(386\) 14.0000 0.712581
\(387\) 2.00000 0.101666
\(388\) 14.0000 0.710742
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 2.00000 0.101274
\(391\) 0 0
\(392\) −1.00000 −0.0505076
\(393\) 20.0000 1.00887
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 4.00000 0.200502
\(399\) −6.00000 −0.300376
\(400\) 1.00000 0.0500000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 14.0000 0.698257
\(403\) 20.0000 0.996271
\(404\) −6.00000 −0.298511
\(405\) 1.00000 0.0496904
\(406\) 8.00000 0.397033
\(407\) −20.0000 −0.991363
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 10.0000 0.493865
\(411\) −18.0000 −0.887875
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) −1.00000 −0.0491473
\(415\) −18.0000 −0.883585
\(416\) 2.00000 0.0980581
\(417\) 16.0000 0.783523
\(418\) 12.0000 0.586939
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 1.00000 0.0487950
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 4.00000 0.194717
\(423\) −2.00000 −0.0972433
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) −2.00000 −0.0964486
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 1.00000 0.0481125
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 10.0000 0.480015
\(435\) −8.00000 −0.383571
\(436\) 14.0000 0.670478
\(437\) −6.00000 −0.287019
\(438\) −10.0000 −0.477818
\(439\) 38.0000 1.81364 0.906821 0.421517i \(-0.138502\pi\)
0.906821 + 0.421517i \(0.138502\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 4.00000 0.190046 0.0950229 0.995475i \(-0.469708\pi\)
0.0950229 + 0.995475i \(0.469708\pi\)
\(444\) −10.0000 −0.474579
\(445\) −6.00000 −0.284427
\(446\) −8.00000 −0.378811
\(447\) 6.00000 0.283790
\(448\) 1.00000 0.0472456
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −20.0000 −0.941763
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 26.0000 1.22024
\(455\) −2.00000 −0.0937614
\(456\) 6.00000 0.280976
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) −2.00000 −0.0934539
\(459\) 0 0
\(460\) 1.00000 0.0466252
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 6.00000 0.278844 0.139422 0.990233i \(-0.455476\pi\)
0.139422 + 0.990233i \(0.455476\pi\)
\(464\) −8.00000 −0.371391
\(465\) −10.0000 −0.463739
\(466\) −10.0000 −0.463241
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −14.0000 −0.646460
\(470\) 2.00000 0.0922531
\(471\) 18.0000 0.829396
\(472\) 4.00000 0.184115
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 2.00000 0.0914779
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 20.0000 0.911922
\(482\) −20.0000 −0.910975
\(483\) 1.00000 0.0455016
\(484\) −7.00000 −0.318182
\(485\) 14.0000 0.635707
\(486\) −1.00000 −0.0453609
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) −10.0000 −0.452679
\(489\) 4.00000 0.180886
\(490\) −1.00000 −0.0451754
\(491\) −4.00000 −0.180517 −0.0902587 0.995918i \(-0.528769\pi\)
−0.0902587 + 0.995918i \(0.528769\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) −12.0000 −0.539906
\(495\) 2.00000 0.0898933
\(496\) −10.0000 −0.449013
\(497\) 6.00000 0.269137
\(498\) 18.0000 0.806599
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 1.00000 0.0447214
\(501\) −2.00000 −0.0893534
\(502\) −24.0000 −1.07117
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −6.00000 −0.266996
\(506\) −2.00000 −0.0889108
\(507\) −9.00000 −0.399704
\(508\) −2.00000 −0.0887357
\(509\) −22.0000 −0.975133 −0.487566 0.873086i \(-0.662115\pi\)
−0.487566 + 0.873086i \(0.662115\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 2.00000 0.0880451
\(517\) −4.00000 −0.175920
\(518\) 10.0000 0.439375
\(519\) 12.0000 0.526742
\(520\) 2.00000 0.0877058
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 8.00000 0.350150
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 20.0000 0.873704
\(525\) 1.00000 0.0436436
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) −4.00000 −0.173585
\(532\) −6.00000 −0.260133
\(533\) 20.0000 0.866296
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 14.0000 0.604708
\(537\) 20.0000 0.863064
\(538\) 18.0000 0.776035
\(539\) 2.00000 0.0861461
\(540\) 1.00000 0.0430331
\(541\) −38.0000 −1.63375 −0.816874 0.576816i \(-0.804295\pi\)
−0.816874 + 0.576816i \(0.804295\pi\)
\(542\) −22.0000 −0.944981
\(543\) 26.0000 1.11577
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 2.00000 0.0855921
\(547\) −36.0000 −1.53925 −0.769624 0.638497i \(-0.779557\pi\)
−0.769624 + 0.638497i \(0.779557\pi\)
\(548\) −18.0000 −0.768922
\(549\) 10.0000 0.426790
\(550\) −2.00000 −0.0852803
\(551\) 48.0000 2.04487
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −4.00000 −0.169944
\(555\) −10.0000 −0.424476
\(556\) 16.0000 0.678551
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 10.0000 0.423334
\(559\) −4.00000 −0.169182
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 16.0000 0.674919
\(563\) −34.0000 −1.43293 −0.716465 0.697623i \(-0.754241\pi\)
−0.716465 + 0.697623i \(0.754241\pi\)
\(564\) −2.00000 −0.0842152
\(565\) −6.00000 −0.252422
\(566\) 32.0000 1.34506
\(567\) 1.00000 0.0419961
\(568\) −6.00000 −0.251754
\(569\) −36.0000 −1.50920 −0.754599 0.656186i \(-0.772169\pi\)
−0.754599 + 0.656186i \(0.772169\pi\)
\(570\) 6.00000 0.251312
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) −4.00000 −0.167248
\(573\) −8.00000 −0.334205
\(574\) 10.0000 0.417392
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 17.0000 0.707107
\(579\) −14.0000 −0.581820
\(580\) −8.00000 −0.332182
\(581\) −18.0000 −0.746766
\(582\) −14.0000 −0.580319
\(583\) −12.0000 −0.496989
\(584\) −10.0000 −0.413803
\(585\) −2.00000 −0.0826898
\(586\) −6.00000 −0.247858
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) 60.0000 2.47226
\(590\) 4.00000 0.164677
\(591\) −18.0000 −0.740421
\(592\) −10.0000 −0.410997
\(593\) 26.0000 1.06769 0.533846 0.845582i \(-0.320746\pi\)
0.533846 + 0.845582i \(0.320746\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 6.00000 0.245770
\(597\) −4.00000 −0.163709
\(598\) 2.00000 0.0817861
\(599\) 38.0000 1.55264 0.776319 0.630340i \(-0.217085\pi\)
0.776319 + 0.630340i \(0.217085\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −42.0000 −1.71322 −0.856608 0.515968i \(-0.827432\pi\)
−0.856608 + 0.515968i \(0.827432\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −14.0000 −0.570124
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 6.00000 0.243733
\(607\) −28.0000 −1.13648 −0.568242 0.822861i \(-0.692376\pi\)
−0.568242 + 0.822861i \(0.692376\pi\)
\(608\) 6.00000 0.243332
\(609\) −8.00000 −0.324176
\(610\) −10.0000 −0.404888
\(611\) 4.00000 0.161823
\(612\) 0 0
\(613\) −22.0000 −0.888572 −0.444286 0.895885i \(-0.646543\pi\)
−0.444286 + 0.895885i \(0.646543\pi\)
\(614\) −4.00000 −0.161427
\(615\) −10.0000 −0.403239
\(616\) −2.00000 −0.0805823
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) −10.0000 −0.401610
\(621\) 1.00000 0.0401286
\(622\) −12.0000 −0.481156
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) −6.00000 −0.239808
\(627\) −12.0000 −0.479234
\(628\) 18.0000 0.718278
\(629\) 0 0
\(630\) −1.00000 −0.0398410
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) −18.0000 −0.714871
\(635\) −2.00000 −0.0793676
\(636\) −6.00000 −0.237915
\(637\) −2.00000 −0.0792429
\(638\) 16.0000 0.633446
\(639\) 6.00000 0.237356
\(640\) −1.00000 −0.0395285
\(641\) −8.00000 −0.315981 −0.157991 0.987441i \(-0.550502\pi\)
−0.157991 + 0.987441i \(0.550502\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 1.00000 0.0394055
\(645\) 2.00000 0.0787499
\(646\) 0 0
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.00000 −0.314027
\(650\) 2.00000 0.0784465
\(651\) −10.0000 −0.391931
\(652\) 4.00000 0.156652
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −14.0000 −0.547443
\(655\) 20.0000 0.781465
\(656\) −10.0000 −0.390434
\(657\) 10.0000 0.390137
\(658\) 2.00000 0.0779681
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 2.00000 0.0778499
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 18.0000 0.698535
\(665\) −6.00000 −0.232670
\(666\) 10.0000 0.387492
\(667\) −8.00000 −0.309761
\(668\) −2.00000 −0.0773823
\(669\) 8.00000 0.309298
\(670\) 14.0000 0.540867
\(671\) 20.0000 0.772091
\(672\) −1.00000 −0.0385758
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 8.00000 0.308148
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −26.0000 −0.999261 −0.499631 0.866239i \(-0.666531\pi\)
−0.499631 + 0.866239i \(0.666531\pi\)
\(678\) 6.00000 0.230429
\(679\) 14.0000 0.537271
\(680\) 0 0
\(681\) −26.0000 −0.996322
\(682\) 20.0000 0.765840
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −6.00000 −0.229416
\(685\) −18.0000 −0.687745
\(686\) −1.00000 −0.0381802
\(687\) 2.00000 0.0763048
\(688\) 2.00000 0.0762493
\(689\) 12.0000 0.457164
\(690\) −1.00000 −0.0380693
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 12.0000 0.456172
\(693\) 2.00000 0.0759737
\(694\) −12.0000 −0.455514
\(695\) 16.0000 0.606915
\(696\) 8.00000 0.303239
\(697\) 0 0
\(698\) 8.00000 0.302804
\(699\) 10.0000 0.378235
\(700\) 1.00000 0.0377964
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 2.00000 0.0754851
\(703\) 60.0000 2.26294
\(704\) 2.00000 0.0753778
\(705\) −2.00000 −0.0753244
\(706\) −6.00000 −0.225813
\(707\) −6.00000 −0.225653
\(708\) −4.00000 −0.150329
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) 6.00000 0.224860
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 20.0000 0.747435
\(717\) −2.00000 −0.0746914
\(718\) 24.0000 0.895672
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 1.00000 0.0372678
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 20.0000 0.743808
\(724\) 26.0000 0.966282
\(725\) −8.00000 −0.297113
\(726\) 7.00000 0.259794
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) −10.0000 −0.370117
\(731\) 0 0
\(732\) 10.0000 0.369611
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −28.0000 −1.03139
\(738\) 10.0000 0.368105
\(739\) 44.0000 1.61857 0.809283 0.587419i \(-0.199856\pi\)
0.809283 + 0.587419i \(0.199856\pi\)
\(740\) −10.0000 −0.367607
\(741\) 12.0000 0.440831
\(742\) 6.00000 0.220267
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 10.0000 0.366618
\(745\) 6.00000 0.219823
\(746\) 2.00000 0.0732252
\(747\) −18.0000 −0.658586
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −0.0365148
\(751\) 48.0000 1.75154 0.875772 0.482724i \(-0.160353\pi\)
0.875772 + 0.482724i \(0.160353\pi\)
\(752\) −2.00000 −0.0729325
\(753\) 24.0000 0.874609
\(754\) −16.0000 −0.582686
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 34.0000 1.23575 0.617876 0.786276i \(-0.287994\pi\)
0.617876 + 0.786276i \(0.287994\pi\)
\(758\) 8.00000 0.290573
\(759\) 2.00000 0.0725954
\(760\) 6.00000 0.217643
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 2.00000 0.0724524
\(763\) 14.0000 0.506834
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 4.00000 0.144526
\(767\) 8.00000 0.288863
\(768\) 1.00000 0.0360844
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) −2.00000 −0.0720750
\(771\) −6.00000 −0.216085
\(772\) −14.0000 −0.503871
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) −2.00000 −0.0718885
\(775\) −10.0000 −0.359211
\(776\) −14.0000 −0.502571
\(777\) −10.0000 −0.358748
\(778\) −10.0000 −0.358517
\(779\) 60.0000 2.14972
\(780\) −2.00000 −0.0716115
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −8.00000 −0.285897
\(784\) 1.00000 0.0357143
\(785\) 18.0000 0.642448
\(786\) −20.0000 −0.713376
\(787\) 52.0000 1.85360 0.926800 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) −18.0000 −0.641223
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) −2.00000 −0.0710669
\(793\) −20.0000 −0.710221
\(794\) 2.00000 0.0709773
\(795\) −6.00000 −0.212798
\(796\) −4.00000 −0.141776
\(797\) 26.0000 0.920967 0.460484 0.887668i \(-0.347676\pi\)
0.460484 + 0.887668i \(0.347676\pi\)
\(798\) 6.00000 0.212398
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) −6.00000 −0.212000
\(802\) 12.0000 0.423735
\(803\) 20.0000 0.705785
\(804\) −14.0000 −0.493742
\(805\) 1.00000 0.0352454
\(806\) −20.0000 −0.704470
\(807\) −18.0000 −0.633630
\(808\) 6.00000 0.211079
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −8.00000 −0.280745
\(813\) 22.0000 0.771574
\(814\) 20.0000 0.701000
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 14.0000 0.489499
\(819\) −2.00000 −0.0698857
\(820\) −10.0000 −0.349215
\(821\) 24.0000 0.837606 0.418803 0.908077i \(-0.362450\pi\)
0.418803 + 0.908077i \(0.362450\pi\)
\(822\) 18.0000 0.627822
\(823\) −18.0000 −0.627441 −0.313720 0.949515i \(-0.601575\pi\)
−0.313720 + 0.949515i \(0.601575\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 4.00000 0.139178
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 1.00000 0.0347524
\(829\) 28.0000 0.972480 0.486240 0.873825i \(-0.338368\pi\)
0.486240 + 0.873825i \(0.338368\pi\)
\(830\) 18.0000 0.624789
\(831\) 4.00000 0.138758
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −16.0000 −0.554035
\(835\) −2.00000 −0.0692129
\(836\) −12.0000 −0.415029
\(837\) −10.0000 −0.345651
\(838\) −12.0000 −0.414533
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 35.0000 1.20690
\(842\) −34.0000 −1.17172
\(843\) −16.0000 −0.551069
\(844\) −4.00000 −0.137686
\(845\) −9.00000 −0.309609
\(846\) 2.00000 0.0687614
\(847\) −7.00000 −0.240523
\(848\) −6.00000 −0.206041
\(849\) −32.0000 −1.09824
\(850\) 0 0
\(851\) −10.0000 −0.342796
\(852\) 6.00000 0.205557
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −10.0000 −0.342193
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 4.00000 0.136558
\(859\) 28.0000 0.955348 0.477674 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 2.00000 0.0681994
\(861\) −10.0000 −0.340799
\(862\) 20.0000 0.681203
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 12.0000 0.408012
\(866\) −30.0000 −1.01944
\(867\) −17.0000 −0.577350
\(868\) −10.0000 −0.339422
\(869\) 0 0
\(870\) 8.00000 0.271225
\(871\) 28.0000 0.948744
\(872\) −14.0000 −0.474100
\(873\) 14.0000 0.473828
\(874\) 6.00000 0.202953
\(875\) 1.00000 0.0338062
\(876\) 10.0000 0.337869
\(877\) −48.0000 −1.62084 −0.810422 0.585846i \(-0.800762\pi\)
−0.810422 + 0.585846i \(0.800762\pi\)
\(878\) −38.0000 −1.28244
\(879\) 6.00000 0.202375
\(880\) 2.00000 0.0674200
\(881\) 54.0000 1.81931 0.909653 0.415369i \(-0.136347\pi\)
0.909653 + 0.415369i \(0.136347\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) −4.00000 −0.134383
\(887\) −46.0000 −1.54453 −0.772264 0.635301i \(-0.780876\pi\)
−0.772264 + 0.635301i \(0.780876\pi\)
\(888\) 10.0000 0.335578
\(889\) −2.00000 −0.0670778
\(890\) 6.00000 0.201120
\(891\) 2.00000 0.0670025
\(892\) 8.00000 0.267860
\(893\) 12.0000 0.401565
\(894\) −6.00000 −0.200670
\(895\) 20.0000 0.668526
\(896\) −1.00000 −0.0334077
\(897\) −2.00000 −0.0667781
\(898\) 2.00000 0.0667409
\(899\) 80.0000 2.66815
\(900\) 1.00000 0.0333333
\(901\) 0 0
\(902\) 20.0000 0.665927
\(903\) 2.00000 0.0665558
\(904\) 6.00000 0.199557
\(905\) 26.0000 0.864269
\(906\) 0 0
\(907\) 38.0000 1.26177 0.630885 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) −26.0000 −0.862840
\(909\) −6.00000 −0.199007
\(910\) 2.00000 0.0662994
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) −6.00000 −0.198680
\(913\) −36.0000 −1.19143
\(914\) 24.0000 0.793849
\(915\) 10.0000 0.330590
\(916\) 2.00000 0.0660819
\(917\) 20.0000 0.660458
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 4.00000 0.131804
\(922\) −10.0000 −0.329332
\(923\) −12.0000 −0.394985
\(924\) 2.00000 0.0657952
\(925\) −10.0000 −0.328798
\(926\) −6.00000 −0.197172
\(927\) 0 0
\(928\) 8.00000 0.262613
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 10.0000 0.327913
\(931\) −6.00000 −0.196642
\(932\) 10.0000 0.327561
\(933\) 12.0000 0.392862
\(934\) 30.0000 0.981630
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 14.0000 0.457116
\(939\) 6.00000 0.195803
\(940\) −2.00000 −0.0652328
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) −18.0000 −0.586472
\(943\) −10.0000 −0.325645
\(944\) −4.00000 −0.130189
\(945\) 1.00000 0.0325300
\(946\) −4.00000 −0.130051
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) 6.00000 0.194666
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −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
\(956\) −2.00000 −0.0646846
\(957\) −16.0000 −0.517207
\(958\) 32.0000 1.03387
\(959\) −18.0000 −0.581250
\(960\) 1.00000 0.0322749
\(961\) 69.0000 2.22581
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) −14.0000 −0.450676
\(966\) −1.00000 −0.0321745
\(967\) 6.00000 0.192947 0.0964735 0.995336i \(-0.469244\pi\)
0.0964735 + 0.995336i \(0.469244\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000 0.0320750
\(973\) 16.0000 0.512936
\(974\) 18.0000 0.576757
\(975\) −2.00000 −0.0640513
\(976\) 10.0000 0.320092
\(977\) −38.0000 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −4.00000 −0.127906
\(979\) −12.0000 −0.383522
\(980\) 1.00000 0.0319438
\(981\) 14.0000 0.446986
\(982\) 4.00000 0.127645
\(983\) 60.0000 1.91370 0.956851 0.290578i \(-0.0938475\pi\)
0.956851 + 0.290578i \(0.0938475\pi\)
\(984\) 10.0000 0.318788
\(985\) −18.0000 −0.573528
\(986\) 0 0
\(987\) −2.00000 −0.0636607
\(988\) 12.0000 0.381771
\(989\) 2.00000 0.0635963
\(990\) −2.00000 −0.0635642
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 10.0000 0.317500
\(993\) 20.0000 0.634681
\(994\) −6.00000 −0.190308
\(995\) −4.00000 −0.126809
\(996\) −18.0000 −0.570352
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 12.0000 0.379853
\(999\) −10.0000 −0.316386
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.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.q.1.1 1 1.1 even 1 trivial