Properties

Label 6930.2.a.bk.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 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{19} +1.00000 q^{20} +1.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{28} -4.00000 q^{29} +1.00000 q^{32} -2.00000 q^{34} +1.00000 q^{35} +8.00000 q^{37} +6.00000 q^{38} +1.00000 q^{40} +4.00000 q^{43} +1.00000 q^{44} -6.00000 q^{46} +4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{52} +12.0000 q^{53} +1.00000 q^{55} +1.00000 q^{56} -4.00000 q^{58} +2.00000 q^{61} +1.00000 q^{64} +2.00000 q^{65} -8.00000 q^{67} -2.00000 q^{68} +1.00000 q^{70} +12.0000 q^{71} -6.00000 q^{73} +8.00000 q^{74} +6.00000 q^{76} +1.00000 q^{77} +10.0000 q^{79} +1.00000 q^{80} +12.0000 q^{83} -2.00000 q^{85} +4.00000 q^{86} +1.00000 q^{88} -14.0000 q^{89} +2.00000 q^{91} -6.00000 q^{92} +4.00000 q^{94} +6.00000 q^{95} +4.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 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
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 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
\(46\) −6.00000 −0.884652
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −4.00000 −0.525226
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 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 \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) 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 \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 0 0
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 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
\(115\) −6.00000 −0.559503
\(116\) −4.00000 −0.371391
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 2.00000 0.181071
\(123\) 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
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 0 0
\(139\) −6.00000 −0.508913 −0.254457 0.967084i \(-0.581897\pi\)
−0.254457 + 0.967084i \(0.581897\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 12.0000 1.00702
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −6.00000 −0.488273 −0.244137 0.969741i \(-0.578505\pi\)
−0.244137 + 0.969741i \(0.578505\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −24.0000 −1.87983 −0.939913 0.341415i \(-0.889094\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) 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
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −14.0000 −1.04934
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 2.00000 0.148250
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 6.00000 0.435286
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 4.00000 0.287183
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 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
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 1.00000 0.0674200
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −6.00000 −0.395628
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) −2.00000 −0.129641
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.0000 1.74659 0.873296 0.487190i \(-0.161978\pi\)
0.873296 + 0.487190i \(0.161978\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 2.00000 0.124035
\(261\) 0 0
\(262\) 6.00000 0.370681
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 6.00000 0.367884
\(267\) 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
\(271\) 28.0000 1.70088 0.850439 0.526073i \(-0.176336\pi\)
0.850439 + 0.526073i \(0.176336\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 22.0000 1.32907
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) −6.00000 −0.359856
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 0 0
\(292\) −6.00000 −0.351123
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) −4.00000 −0.231714
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) −6.00000 −0.345261
\(303\) 0 0
\(304\) 6.00000 0.344124
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) −12.0000 −0.678280 −0.339140 0.940736i \(-0.610136\pi\)
−0.339140 + 0.940736i \(0.610136\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −6.00000 −0.334367
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) −24.0000 −1.32924
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00000 0.220527
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −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
\(346\) 18.0000 0.967686
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 0 0
\(349\) 6.00000 0.321173 0.160586 0.987022i \(-0.448662\pi\)
0.160586 + 0.987022i \(0.448662\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −20.0000 −1.06449 −0.532246 0.846590i \(-0.678652\pi\)
−0.532246 + 0.846590i \(0.678652\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −6.00000 −0.312772
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) 4.00000 0.203069
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 10.0000 0.503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 16.0000 0.802008
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 22.0000 1.05970 0.529851 0.848091i \(-0.322248\pi\)
0.529851 + 0.848091i \(0.322248\pi\)
\(432\) 0 0
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) −36.0000 −1.72211
\(438\) 0 0
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −40.0000 −1.90046 −0.950229 0.311553i \(-0.899151\pi\)
−0.950229 + 0.311553i \(0.899151\pi\)
\(444\) 0 0
\(445\) −14.0000 −0.663664
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.00000 0.282216
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −6.00000 −0.279751
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) −10.0000 −0.464739 −0.232370 0.972628i \(-0.574648\pi\)
−0.232370 + 0.972628i \(0.574648\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −38.0000 −1.75843 −0.879215 0.476425i \(-0.841932\pi\)
−0.879215 + 0.476425i \(0.841932\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 4.00000 0.184506
\(471\) 0 0
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −6.00000 −0.274434
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) −8.00000 −0.364390
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 4.00000 0.181631
\(486\) 0 0
\(487\) −18.0000 −0.815658 −0.407829 0.913058i \(-0.633714\pi\)
−0.407829 + 0.913058i \(0.633714\pi\)
\(488\) 2.00000 0.0905357
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 8.00000 0.360302
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) −6.00000 −0.266733
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 28.0000 1.23503
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 8.00000 0.351500
\(519\) 0 0
\(520\) 2.00000 0.0877058
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 0 0
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) −8.00000 −0.345547
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) −16.0000 −0.687894 −0.343947 0.938989i \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 28.0000 1.20270
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 22.0000 0.939793
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −24.0000 −1.02243
\(552\) 0 0
\(553\) 10.0000 0.425243
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −6.00000 −0.254457
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 32.0000 1.34506
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) −2.00000 −0.0819920
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 44.0000 1.79480 0.897399 0.441221i \(-0.145454\pi\)
0.897399 + 0.441221i \(0.145454\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) −6.00000 −0.244137
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 16.0000 0.649420 0.324710 0.945814i \(-0.394733\pi\)
0.324710 + 0.945814i \(0.394733\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 2.00000 0.0809776
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 42.0000 1.69086 0.845428 0.534089i \(-0.179345\pi\)
0.845428 + 0.534089i \(0.179345\pi\)
\(618\) 0 0
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −16.0000 −0.641542
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.0000 −0.479616
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) 34.0000 1.34083 0.670415 0.741987i \(-0.266116\pi\)
0.670415 + 0.741987i \(0.266116\pi\)
\(644\) −6.00000 −0.236433
\(645\) 0 0
\(646\) −12.0000 −0.472134
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 36.0000 1.40879 0.704394 0.709809i \(-0.251219\pi\)
0.704394 + 0.709809i \(0.251219\pi\)
\(654\) 0 0
\(655\) 6.00000 0.234439
\(656\) 0 0
\(657\) 0 0
\(658\) 4.00000 0.155936
\(659\) −8.00000 −0.311636 −0.155818 0.987786i \(-0.549801\pi\)
−0.155818 + 0.987786i \(0.549801\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 0 0
\(670\) −8.00000 −0.309067
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −42.0000 −1.61898 −0.809491 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −38.0000 −1.46046 −0.730229 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −2.00000 −0.0766965
\(681\) 0 0
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) 22.0000 0.840577
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) −20.0000 −0.759190
\(695\) −6.00000 −0.227593
\(696\) 0 0
\(697\) 0 0
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 8.00000 0.302156 0.151078 0.988522i \(-0.451726\pi\)
0.151078 + 0.988522i \(0.451726\pi\)
\(702\) 0 0
\(703\) 48.0000 1.81035
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −20.0000 −0.752710
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 12.0000 0.450352
\(711\) 0 0
\(712\) −14.0000 −0.524672
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 6.00000 0.223918
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 17.0000 0.632674
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) −4.00000 −0.148556
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 2.00000 0.0741249
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −2.00000 −0.0738717 −0.0369358 0.999318i \(-0.511760\pi\)
−0.0369358 + 0.999318i \(0.511760\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 12.0000 0.440534
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) 0 0
\(745\) −4.00000 −0.146549
\(746\) −26.0000 −0.951928
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) −20.0000 −0.730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) −6.00000 −0.218362
\(756\) 0 0
\(757\) 52.0000 1.88997 0.944986 0.327111i \(-0.106075\pi\)
0.944986 + 0.327111i \(0.106075\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 6.00000 0.217643
\(761\) 24.0000 0.869999 0.435000 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 0 0
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 4.00000 0.143592
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 12.0000 0.429119
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −48.0000 −1.71102 −0.855508 0.517790i \(-0.826755\pi\)
−0.855508 + 0.517790i \(0.826755\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) 10.0000 0.355784
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 34.0000 1.20058
\(803\) −6.00000 −0.211735
\(804\) 0 0
\(805\) −6.00000 −0.211472
\(806\) 0 0
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) −30.0000 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 12.0000 0.416526
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 16.0000 0.552711
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −26.0000 −0.896019
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −2.00000 −0.0685994
\(851\) −48.0000 −1.64542
\(852\) 0 0
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) −20.0000 −0.683586
\(857\) 30.0000 1.02478 0.512390 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 48.0000 1.63774 0.818869 0.573980i \(-0.194601\pi\)
0.818869 + 0.573980i \(0.194601\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 22.0000 0.749323
\(863\) −30.0000 −1.02121 −0.510606 0.859815i \(-0.670579\pi\)
−0.510606 + 0.859815i \(0.670579\pi\)
\(864\) 0 0
\(865\) 18.0000 0.612018
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 0 0
\(869\) 10.0000 0.339227
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) −36.0000 −1.21772
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 12.0000 0.404980
\(879\) 0 0
\(880\) 1.00000 0.0337100
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) −40.0000 −1.34383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −14.0000 −0.469281
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) 4.00000 0.133705
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −14.0000 −0.467186
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 0 0
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 2.00000 0.0662994
\(911\) 12.0000 0.397578 0.198789 0.980042i \(-0.436299\pi\)
0.198789 + 0.980042i \(0.436299\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 26.0000 0.857661 0.428830 0.903385i \(-0.358926\pi\)
0.428830 + 0.903385i \(0.358926\pi\)
\(920\) −6.00000 −0.197814
\(921\) 0 0
\(922\) 26.0000 0.856264
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −10.0000 −0.328620
\(927\) 0 0
\(928\) −4.00000 −0.131306
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) −38.0000 −1.24340
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 6.00000 0.194666
\(951\) 0 0
\(952\) −2.00000 −0.0648204
\(953\) −18.0000 −0.583077 −0.291539 0.956559i \(-0.594167\pi\)
−0.291539 + 0.956559i \(0.594167\pi\)
\(954\) 0 0
\(955\) −8.00000 −0.258874
\(956\) −6.00000 −0.194054
\(957\) 0 0
\(958\) −8.00000 −0.258468
\(959\) 22.0000 0.710417
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 16.0000 0.515861
\(963\) 0 0
\(964\) −8.00000 −0.257663
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −6.00000 −0.192351
\(974\) −18.0000 −0.576757
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 54.0000 1.72761 0.863807 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −14.0000 −0.447442
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 12.0000 0.380617
\(995\) 16.0000 0.507234
\(996\) 0 0
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) −28.0000 −0.886325
\(999\) 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.bk.1.1 1
3.2 odd 2 770.2.a.e.1.1 1
12.11 even 2 6160.2.a.a.1.1 1
15.2 even 4 3850.2.c.c.1849.1 2
15.8 even 4 3850.2.c.c.1849.2 2
15.14 odd 2 3850.2.a.m.1.1 1
21.20 even 2 5390.2.a.c.1.1 1
33.32 even 2 8470.2.a.bg.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.e.1.1 1 3.2 odd 2
3850.2.a.m.1.1 1 15.14 odd 2
3850.2.c.c.1849.1 2 15.2 even 4
3850.2.c.c.1849.2 2 15.8 even 4
5390.2.a.c.1.1 1 21.20 even 2
6160.2.a.a.1.1 1 12.11 even 2
6930.2.a.bk.1.1 1 1.1 even 1 trivial
8470.2.a.bg.1.1 1 33.32 even 2