Properties

Label 6930.2.a.w.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2310)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -8.00000 q^{19} -1.00000 q^{20} +1.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} -6.00000 q^{37} -8.00000 q^{38} -1.00000 q^{40} +2.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +8.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +14.0000 q^{53} -1.00000 q^{55} -1.00000 q^{56} +6.00000 q^{58} -4.00000 q^{59} -10.0000 q^{61} +1.00000 q^{64} -2.00000 q^{65} +8.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} -10.0000 q^{73} -6.00000 q^{74} -8.00000 q^{76} -1.00000 q^{77} +4.00000 q^{79} -1.00000 q^{80} +2.00000 q^{82} +4.00000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +1.00000 q^{88} +14.0000 q^{89} -2.00000 q^{91} +8.00000 q^{92} -12.0000 q^{94} +8.00000 q^{95} +14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 8.00000 1.66812 0.834058 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) −8.00000 −1.29777
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 8.00000 1.17954
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 14.0000 1.48400 0.741999 0.670402i \(-0.233878\pi\)
0.741999 + 0.670402i \(0.233878\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.0000 −0.905357
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 0 0
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −10.0000 −0.827606
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −8.00000 −0.648886
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 4.00000 0.318223
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) −8.00000 −0.630488
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 22.0000 1.67263 0.836315 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 14.0000 1.04934
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 8.00000 0.580381
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 0 0
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 10.0000 0.703598
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 4.00000 0.269069
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) −4.00000 −0.260378
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.0000 0.623783 0.311891 0.950118i \(-0.399037\pi\)
0.311891 + 0.950118i \(0.399037\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −14.0000 −0.860013
\(266\) 8.00000 0.490511
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 16.0000 0.959616
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) −10.0000 −0.585206
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 0 0
\(295\) 4.00000 0.232889
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) 6.00000 0.347571
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) −18.0000 −1.01742 −0.508710 0.860938i \(-0.669877\pi\)
−0.508710 + 0.860938i \(0.669877\pi\)
\(314\) 6.00000 0.338600
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 22.0000 1.18273
\(347\) −28.0000 −1.50312 −0.751559 0.659665i \(-0.770698\pi\)
−0.751559 + 0.659665i \(0.770698\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −22.0000 −1.17094 −0.585471 0.810693i \(-0.699090\pi\)
−0.585471 + 0.810693i \(0.699090\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 22.0000 1.15629
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 6.00000 0.311925
\(371\) −14.0000 −0.726844
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 8.00000 0.410391
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 22.0000 1.11977
\(387\) 0 0
\(388\) 14.0000 0.710742
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 12.0000 0.591198
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −8.00000 −0.391293
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) −28.0000 −1.36302
\(423\) 0 0
\(424\) 14.0000 0.679900
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −4.00000 −0.192897
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −64.0000 −3.06154
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) 4.00000 0.190261
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 6.00000 0.280668 0.140334 0.990104i \(-0.455182\pi\)
0.140334 + 0.990104i \(0.455182\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) −8.00000 −0.373002
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 10.0000 0.463241
\(467\) −28.0000 −1.29569 −0.647843 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 12.0000 0.553519
\(471\) 0 0
\(472\) −4.00000 −0.184115
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −10.0000 −0.452679
\(489\) 0 0
\(490\) −1.00000 −0.0451754
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 4.00000 0.178529
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 10.0000 0.441081
\(515\) −12.0000 −0.528783
\(516\) 0 0
\(517\) −12.0000 −0.527759
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −36.0000 −1.57417 −0.787085 0.616844i \(-0.788411\pi\)
−0.787085 + 0.616844i \(0.788411\pi\)
\(524\) 8.00000 0.349482
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −14.0000 −0.608121
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 12.0000 0.518805
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −48.0000 −2.04487
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 16.0000 0.678551
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) 0 0
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) 14.0000 0.579821
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −20.0000 −0.825488 −0.412744 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 4.00000 0.164677
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) 16.0000 0.654289
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) −8.00000 −0.324443
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) −26.0000 −1.04672 −0.523360 0.852111i \(-0.675322\pi\)
−0.523360 + 0.852111i \(0.675322\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −18.0000 −0.719425
\(627\) 0 0
\(628\) 6.00000 0.239426
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 4.00000 0.159111
\(633\) 0 0
\(634\) 22.0000 0.873732
\(635\) −8.00000 −0.317470
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) −8.00000 −0.315244
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) −4.00000 −0.157014
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) −8.00000 −0.310227
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) 0 0
\(683\) −48.0000 −1.83667 −0.918334 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 28.0000 1.06672
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) −28.0000 −1.06287
\(695\) −16.0000 −0.606915
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) −1.00000 −0.0377964
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) 0 0
\(703\) 48.0000 1.81035
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −22.0000 −0.827981
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14.0000 0.524672
\(713\) 0 0
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 20.0000 0.746393
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 45.0000 1.67473
\(723\) 0 0
\(724\) 22.0000 0.817624
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) −20.0000 −0.741759 −0.370879 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 10.0000 0.370117
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 26.0000 0.960332 0.480166 0.877178i \(-0.340576\pi\)
0.480166 + 0.877178i \(0.340576\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) −14.0000 −0.513956
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 2.00000 0.0731272
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 8.00000 0.290191
\(761\) −38.0000 −1.37750 −0.688749 0.724999i \(-0.741840\pi\)
−0.688749 + 0.724999i \(0.741840\pi\)
\(762\) 0 0
\(763\) −2.00000 −0.0724049
\(764\) −24.0000 −0.868290
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 22.0000 0.791797
\(773\) −14.0000 −0.503545 −0.251773 0.967786i \(-0.581013\pi\)
−0.251773 + 0.967786i \(0.581013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) 12.0000 0.427754 0.213877 0.976861i \(-0.431391\pi\)
0.213877 + 0.976861i \(0.431391\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) −4.00000 −0.142314
\(791\) −6.00000 −0.213335
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −22.0000 −0.779280 −0.389640 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 14.0000 0.494357
\(803\) −10.0000 −0.352892
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 0 0
\(808\) 10.0000 0.351799
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) −6.00000 −0.210300
\(815\) 8.00000 0.280228
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −2.00000 −0.0698430
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 4.00000 0.139178
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 0 0
\(838\) 12.0000 0.414533
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 22.0000 0.758170
\(843\) 0 0
\(844\) −28.0000 −0.963800
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −22.0000 −0.748022
\(866\) −26.0000 −0.883516
\(867\) 0 0
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) −64.0000 −2.16483
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 4.00000 0.134535
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) 96.0000 3.21252
\(894\) 0 0
\(895\) −20.0000 −0.668526
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 0 0
\(901\) 28.0000 0.932815
\(902\) 2.00000 0.0665927
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 40.0000 1.32818 0.664089 0.747653i \(-0.268820\pi\)
0.664089 + 0.747653i \(0.268820\pi\)
\(908\) 12.0000 0.398234
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) −8.00000 −0.263752
\(921\) 0 0
\(922\) 10.0000 0.329332
\(923\) 0 0
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −26.0000 −0.853032 −0.426516 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 10.0000 0.327561
\(933\) 0 0
\(934\) −28.0000 −0.916188
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) −30.0000 −0.977972 −0.488986 0.872292i \(-0.662633\pi\)
−0.488986 + 0.872292i \(0.662633\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −32.0000 −1.03986 −0.519930 0.854209i \(-0.674042\pi\)
−0.519930 + 0.854209i \(0.674042\pi\)
\(948\) 0 0
\(949\) −20.0000 −0.649227
\(950\) −8.00000 −0.259554
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) 0 0
\(955\) 24.0000 0.776622
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −12.0000 −0.386896
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) −22.0000 −0.708205
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −14.0000 −0.449513
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 0 0
\(973\) −16.0000 −0.512936
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 14.0000 0.447442
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −62.0000 −1.96356 −0.981780 0.190022i \(-0.939144\pi\)
−0.981780 + 0.190022i \(0.939144\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
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 6930.2.a.w.1.1 1
3.2 odd 2 2310.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2310.2.a.k.1.1 1 3.2 odd 2
6930.2.a.w.1.1 1 1.1 even 1 trivial