Properties

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

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -4.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} +2.00000 q^{23} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{28} -6.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +10.0000 q^{37} -4.00000 q^{38} +1.00000 q^{40} +8.00000 q^{41} +4.00000 q^{43} -1.00000 q^{44} +2.00000 q^{46} -4.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{55} +1.00000 q^{56} -6.00000 q^{58} +12.0000 q^{59} +10.0000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{67} +2.00000 q^{68} +1.00000 q^{70} +8.00000 q^{71} +14.0000 q^{73} +10.0000 q^{74} -4.00000 q^{76} -1.00000 q^{77} -2.00000 q^{79} +1.00000 q^{80} +8.00000 q^{82} +2.00000 q^{83} +2.00000 q^{85} +4.00000 q^{86} -1.00000 q^{88} +10.0000 q^{89} -2.00000 q^{91} +2.00000 q^{92} -4.00000 q^{94} -4.00000 q^{95} -8.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 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
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 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 \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 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
\(46\) 2.00000 0.294884
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 8.00000 0.883452
\(83\) 2.00000 0.219529 0.109764 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 2.00000 0.208514
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 12.0000 1.19404 0.597022 0.802225i \(-0.296350\pi\)
0.597022 + 0.802225i \(0.296350\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) 2.00000 0.186501
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 2.00000 0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −2.00000 −0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) −4.00000 −0.345547
\(135\) 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
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 1.00000 0.0845154
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) 2.00000 0.167248
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) 10.0000 0.821995
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −12.0000 −0.957704 −0.478852 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 8.00000 0.624695
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) 10.0000 0.735215
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −4.00000 −0.291730
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) −16.0000 −1.11477
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 6.00000 0.413057 0.206529 0.978441i \(-0.433783\pi\)
0.206529 + 0.978441i \(0.433783\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −6.00000 −0.393919
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) −4.00000 −0.260931
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 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
\(247\) 8.00000 0.509028
\(248\) 2.00000 0.127000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) 10.0000 0.621370
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 28.0000 1.72655 0.863277 0.504730i \(-0.168408\pi\)
0.863277 + 0.504730i \(0.168408\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −10.0000 −0.596550 −0.298275 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) −6.00000 −0.352332
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 22.0000 1.28525 0.642627 0.766179i \(-0.277845\pi\)
0.642627 + 0.766179i \(0.277845\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 6.00000 0.345261
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) −1.00000 −0.0569803
\(309\) 0 0
\(310\) 2.00000 0.113592
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) −12.0000 −0.677199
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 2.00000 0.111456
\(323\) −8.00000 −0.445132
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 8.00000 0.441726
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −16.0000 −0.871576 −0.435788 0.900049i \(-0.643530\pi\)
−0.435788 + 0.900049i \(0.643530\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) −2.00000 −0.108306
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 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
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 14.0000 0.732793
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 2.00000 0.104257
\(369\) 0 0
\(370\) 10.0000 0.519875
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) 36.0000 1.86401 0.932005 0.362446i \(-0.118058\pi\)
0.932005 + 0.362446i \(0.118058\pi\)
\(374\) −2.00000 −0.103418
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −32.0000 −1.64373 −0.821865 0.569683i \(-0.807066\pi\)
−0.821865 + 0.569683i \(0.807066\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) −8.00000 −0.406138
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) −14.0000 −0.705310
\(395\) −2.00000 −0.100631
\(396\) 0 0
\(397\) 24.0000 1.20453 0.602263 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) −6.00000 −0.300753
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) −6.00000 −0.297775
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 8.00000 0.395092
\(411\) 0 0
\(412\) −16.0000 −0.788263
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) 2.00000 0.0981761
\(416\) −2.00000 −0.0980581
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 6.00000 0.292075
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 12.0000 0.568216
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) −2.00000 −0.0940721
\(453\) 0 0
\(454\) 2.00000 0.0938647
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 2.00000 0.0932505
\(461\) 32.0000 1.49039 0.745194 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 2.00000 0.0916698
\(477\) 0 0
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 8.00000 0.359937
\(495\) 0 0
\(496\) 2.00000 0.0898027
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) 18.0000 0.797836 0.398918 0.916987i \(-0.369386\pi\)
0.398918 + 0.916987i \(0.369386\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 10.0000 0.439375
\(519\) 0 0
\(520\) −2.00000 −0.0877058
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 28.0000 1.22086
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) −16.0000 −0.693037
\(534\) 0 0
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) 0 0
\(538\) 10.0000 0.431131
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) 2.00000 0.0857493
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 40.0000 1.71028 0.855138 0.518400i \(-0.173472\pi\)
0.855138 + 0.518400i \(0.173472\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) −2.00000 −0.0850487
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −14.0000 −0.593199 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 1.00000 0.0422577
\(561\) 0 0
\(562\) −10.0000 −0.421825
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −42.0000 −1.75765 −0.878823 0.477149i \(-0.841670\pi\)
−0.878823 + 0.477149i \(0.841670\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 8.00000 0.333914
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) 12.0000 0.499567 0.249783 0.968302i \(-0.419641\pi\)
0.249783 + 0.968302i \(0.419641\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) −6.00000 −0.249136
\(581\) 2.00000 0.0829740
\(582\) 0 0
\(583\) −2.00000 −0.0828315
\(584\) 14.0000 0.579324
\(585\) 0 0
\(586\) 22.0000 0.908812
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 12.0000 0.494032
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) −4.00000 −0.163572
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 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
\(604\) 6.00000 0.244137
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −4.00000 −0.162221
\(609\) 0 0
\(610\) 10.0000 0.404888
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 4.00000 0.161427
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 2.00000 0.0803219
\(621\) 0 0
\(622\) 0 0
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) −12.0000 −0.478852
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −2.00000 −0.0795557
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) −8.00000 −0.314756
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) −2.00000 −0.0784465
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 8.00000 0.312348
\(657\) 0 0
\(658\) −4.00000 −0.155936
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) −32.0000 −1.24372
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −14.0000 −0.538064 −0.269032 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −2.00000 −0.0765840
\(683\) −18.0000 −0.688751 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(684\) 0 0
\(685\) −2.00000 −0.0764161
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 24.0000 0.911028
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) −2.00000 −0.0757011
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) −40.0000 −1.50863
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 8.00000 0.300235
\(711\) 0 0
\(712\) 10.0000 0.374766
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) 2.00000 0.0747958
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) −16.0000 −0.597115
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 24.0000 0.890111 0.445055 0.895503i \(-0.353184\pi\)
0.445055 + 0.895503i \(0.353184\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 14.0000 0.518163
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) 10.0000 0.369358 0.184679 0.982799i \(-0.440875\pi\)
0.184679 + 0.982799i \(0.440875\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 10.0000 0.367607
\(741\) 0 0
\(742\) 2.00000 0.0734223
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 36.0000 1.31805
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) 0 0
\(751\) −52.0000 −1.89751 −0.948753 0.316017i \(-0.897654\pi\)
−0.948753 + 0.316017i \(0.897654\pi\)
\(752\) −4.00000 −0.145865
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −32.0000 −1.16229
\(759\) 0 0
\(760\) −4.00000 −0.145095
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) −10.0000 −0.362024
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −1.00000 −0.0360375
\(771\) 0 0
\(772\) 4.00000 0.143963
\(773\) 38.0000 1.36677 0.683383 0.730061i \(-0.260508\pi\)
0.683383 + 0.730061i \(0.260508\pi\)
\(774\) 0 0
\(775\) 2.00000 0.0718421
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) −32.0000 −1.14652
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −12.0000 −0.428298
\(786\) 0 0
\(787\) 36.0000 1.28326 0.641631 0.767014i \(-0.278258\pi\)
0.641631 + 0.767014i \(0.278258\pi\)
\(788\) −14.0000 −0.498729
\(789\) 0 0
\(790\) −2.00000 −0.0711568
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 24.0000 0.851728
\(795\) 0 0
\(796\) −6.00000 −0.212664
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) −14.0000 −0.494049
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 12.0000 0.422159
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −6.00000 −0.210559
\(813\) 0 0
\(814\) −10.0000 −0.350500
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 2.00000 0.0699284
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) −16.0000 −0.557386
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 2.00000 0.0694210
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 4.00000 0.138343
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) 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
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) 6.00000 0.206529
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 20.0000 0.685591
\(852\) 0 0
\(853\) 10.0000 0.342393 0.171197 0.985237i \(-0.445237\pi\)
0.171197 + 0.985237i \(0.445237\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 0 0
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −46.0000 −1.56950 −0.784750 0.619813i \(-0.787209\pi\)
−0.784750 + 0.619813i \(0.787209\pi\)
\(860\) 4.00000 0.136399
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) 46.0000 1.56586 0.782929 0.622111i \(-0.213725\pi\)
0.782929 + 0.622111i \(0.213725\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 4.00000 0.135926
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 2.00000 0.0678454
\(870\) 0 0
\(871\) 8.00000 0.271070
\(872\) −10.0000 −0.338643
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) 32.0000 1.08056 0.540282 0.841484i \(-0.318318\pi\)
0.540282 + 0.841484i \(0.318318\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 10.0000 0.335201
\(891\) 0 0
\(892\) 12.0000 0.401790
\(893\) 16.0000 0.535420
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) −12.0000 −0.400222
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) −8.00000 −0.266371
\(903\) 0 0
\(904\) −2.00000 −0.0665190
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 2.00000 0.0663723
\(909\) 0 0
\(910\) −2.00000 −0.0662994
\(911\) −16.0000 −0.530104 −0.265052 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) −2.00000 −0.0661903
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −54.0000 −1.78130 −0.890648 0.454694i \(-0.849749\pi\)
−0.890648 + 0.454694i \(0.849749\pi\)
\(920\) 2.00000 0.0659380
\(921\) 0 0
\(922\) 32.0000 1.05386
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) −32.0000 −1.05159
\(927\) 0 0
\(928\) −6.00000 −0.196960
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 22.0000 0.720634
\(933\) 0 0
\(934\) 24.0000 0.785304
\(935\) −2.00000 −0.0654070
\(936\) 0 0
\(937\) 50.0000 1.63343 0.816714 0.577042i \(-0.195793\pi\)
0.816714 + 0.577042i \(0.195793\pi\)
\(938\) −4.00000 −0.130605
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −56.0000 −1.82555 −0.912774 0.408465i \(-0.866064\pi\)
−0.912774 + 0.408465i \(0.866064\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) −38.0000 −1.23483 −0.617417 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 0 0
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −20.0000 −0.644826
\(963\) 0 0
\(964\) −10.0000 −0.322078
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −8.00000 −0.256865
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 12.0000 0.384702
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 0 0
\(979\) −10.0000 −0.319601
\(980\) 1.00000 0.0319438
\(981\) 0 0
\(982\) 20.0000 0.638226
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −14.0000 −0.446077
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 2.00000 0.0635001
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) −6.00000 −0.190213
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 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.bh.1.1 yes 1
3.2 odd 2 6930.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6930.2.a.e.1.1 1 3.2 odd 2
6930.2.a.bh.1.1 yes 1 1.1 even 1 trivial