Properties

Label 2070.2.a.e.1.1
Level $2070$
Weight $2$
Character 2070.1
Self dual yes
Analytic conductor $16.529$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.5290332184\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -6.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{20} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} +6.00000 q^{26} -2.00000 q^{28} -2.00000 q^{29} -1.00000 q^{32} -4.00000 q^{34} -2.00000 q^{35} -8.00000 q^{37} -1.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{46} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{52} -6.00000 q^{53} +2.00000 q^{55} +2.00000 q^{56} +2.00000 q^{58} -8.00000 q^{61} +1.00000 q^{64} -6.00000 q^{65} -4.00000 q^{67} +4.00000 q^{68} +2.00000 q^{70} -16.0000 q^{71} +6.00000 q^{73} +8.00000 q^{74} -4.00000 q^{77} +14.0000 q^{79} +1.00000 q^{80} -6.00000 q^{82} -14.0000 q^{83} +4.00000 q^{85} +4.00000 q^{86} -2.00000 q^{88} +8.00000 q^{89} +12.0000 q^{91} -1.00000 q^{92} -6.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −6.00000 −0.832050
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 2.00000 0.267261
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) 14.0000 1.57512 0.787562 0.616236i \(-0.211343\pi\)
0.787562 + 0.616236i \(0.211343\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −2.00000 −0.197066 −0.0985329 0.995134i \(-0.531415\pi\)
−0.0985329 + 0.995134i \(0.531415\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) −2.00000 −0.190693
\(111\) 0 0
\(112\) −2.00000 −0.188982
\(113\) −12.0000 −1.12887 −0.564433 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 0 0
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 8.00000 0.724286
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 6.00000 0.526235
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 4.00000 0.322329
\(155\) 0 0
\(156\) 0 0
\(157\) −16.0000 −1.27694 −0.638470 0.769647i \(-0.720432\pi\)
−0.638470 + 0.769647i \(0.720432\pi\)
\(158\) −14.0000 −1.11378
\(159\) 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 \(0.550070\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) −8.00000 −0.599625
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) −12.0000 −0.891953 −0.445976 0.895045i \(-0.647144\pi\)
−0.445976 + 0.895045i \(0.647144\pi\)
\(182\) −12.0000 −0.889499
\(183\) 0 0
\(184\) 1.00000 0.0737210
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 4.00000 0.280745
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) −6.00000 −0.416025
\(209\) 0 0
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) −24.0000 −1.61441
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 1.00000 0.0659380
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 8.00000 0.518563
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 7.00000 0.449977
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) −6.00000 −0.372104
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −8.00000 −0.483298
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −16.0000 −0.949425
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 2.00000 0.117444
\(291\) 0 0
\(292\) 6.00000 0.351123
\(293\) 10.0000 0.584206 0.292103 0.956387i \(-0.405645\pi\)
0.292103 + 0.956387i \(0.405645\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 6.00000 0.346989
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −4.00000 −0.227921
\(309\) 0 0
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) 0 0
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 16.0000 0.902932
\(315\) 0 0
\(316\) 14.0000 0.787562
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) 0 0
\(324\) 0 0
\(325\) −6.00000 −0.332820
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) 8.00000 0.429463 0.214731 0.976673i \(-0.431112\pi\)
0.214731 + 0.976673i \(0.431112\pi\)
\(348\) 0 0
\(349\) −6.00000 −0.321173 −0.160586 0.987022i \(-0.551338\pi\)
−0.160586 + 0.987022i \(0.551338\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 12.0000 0.630706
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 32.0000 1.65690 0.828449 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 0.613171 0.306586 0.951843i \(-0.400813\pi\)
0.306586 + 0.951843i \(0.400813\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 14.0000 0.704416
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 2.00000 0.100251
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) −16.0000 −0.793091
\(408\) 0 0
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) −2.00000 −0.0985329
\(413\) 0 0
\(414\) 0 0
\(415\) −14.0000 −0.687233
\(416\) 6.00000 0.294174
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 20.0000 0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 4.00000 0.194029
\(426\) 0 0
\(427\) 16.0000 0.774294
\(428\) 18.0000 0.870063
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 30.0000 1.44171 0.720854 0.693087i \(-0.243750\pi\)
0.720854 + 0.693087i \(0.243750\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) 0 0
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 24.0000 1.14156
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 8.00000 0.379236
\(446\) 16.0000 0.757622
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) −12.0000 −0.564433
\(453\) 0 0
\(454\) −18.0000 −0.844782
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 20.0000 0.934539
\(459\) 0 0
\(460\) −1.00000 −0.0466252
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −36.0000 −1.67306 −0.836531 0.547920i \(-0.815420\pi\)
−0.836531 + 0.547920i \(0.815420\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 30.0000 1.38823 0.694117 0.719862i \(-0.255795\pi\)
0.694117 + 0.719862i \(0.255795\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 8.00000 0.362143
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 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
\(502\) 6.00000 0.267793
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 2.00000 0.0889108
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) −12.0000 −0.530849
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 14.0000 0.617514
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) 0 0
\(518\) −16.0000 −0.703000
\(519\) 0 0
\(520\) 6.00000 0.263117
\(521\) 4.00000 0.175243 0.0876216 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 0 0
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) −36.0000 −1.55933
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −18.0000 −0.776035
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 8.00000 0.341743
\(549\) 0 0
\(550\) −2.00000 −0.0852803
\(551\) 0 0
\(552\) 0 0
\(553\) −28.0000 −1.19068
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 22.0000 0.932170 0.466085 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) 24.0000 1.01509
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 8.00000 0.337460
\(563\) 10.0000 0.421450 0.210725 0.977545i \(-0.432418\pi\)
0.210725 + 0.977545i \(0.432418\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) −2.00000 −0.0830455
\(581\) 28.0000 1.16164
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) −8.00000 −0.330195 −0.165098 0.986277i \(-0.552794\pi\)
−0.165098 + 0.986277i \(0.552794\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −30.0000 −1.23195 −0.615976 0.787765i \(-0.711238\pi\)
−0.615976 + 0.787765i \(0.711238\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) −8.00000 −0.326056
\(603\) 0 0
\(604\) 12.0000 0.488273
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) 0 0
\(613\) −4.00000 −0.161558 −0.0807792 0.996732i \(-0.525741\pi\)
−0.0807792 + 0.996732i \(0.525741\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 4.00000 0.161165
\(617\) −28.0000 −1.12724 −0.563619 0.826035i \(-0.690591\pi\)
−0.563619 + 0.826035i \(0.690591\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −32.0000 −1.28308
\(623\) −16.0000 −0.641026
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 34.0000 1.35891
\(627\) 0 0
\(628\) −16.0000 −0.638470
\(629\) −32.0000 −1.27592
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) −14.0000 −0.556890
\(633\) 0 0
\(634\) −22.0000 −0.873732
\(635\) 0 0
\(636\) 0 0
\(637\) 18.0000 0.713186
\(638\) 4.00000 0.158362
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −22.0000 −0.860927 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) 0 0
\(667\) 2.00000 0.0774403
\(668\) 0 0
\(669\) 0 0
\(670\) 4.00000 0.154533
\(671\) −16.0000 −0.617673
\(672\) 0 0
\(673\) 42.0000 1.61898 0.809491 0.587133i \(-0.199743\pi\)
0.809491 + 0.587133i \(0.199743\pi\)
\(674\) 34.0000 1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −2.00000 −0.0768662 −0.0384331 0.999261i \(-0.512237\pi\)
−0.0384331 + 0.999261i \(0.512237\pi\)
\(678\) 0 0
\(679\) 12.0000 0.460518
\(680\) −4.00000 −0.153393
\(681\) 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
\(685\) 8.00000 0.305664
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) −8.00000 −0.303676
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 24.0000 0.909065
\(698\) 6.00000 0.227103
\(699\) 0 0
\(700\) −2.00000 −0.0755929
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 26.0000 0.978523
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) 16.0000 0.600469
\(711\) 0 0
\(712\) −8.00000 −0.299813
\(713\) 0 0
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 19.0000 0.707107
\(723\) 0 0
\(724\) −12.0000 −0.445976
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 38.0000 1.40934 0.704671 0.709534i \(-0.251095\pi\)
0.704671 + 0.709534i \(0.251095\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) −6.00000 −0.222070
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) −8.00000 −0.294086
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) −10.0000 −0.366372
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −50.0000 −1.82453 −0.912263 0.409605i \(-0.865667\pi\)
−0.912263 + 0.409605i \(0.865667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) 0 0
\(769\) 6.00000 0.216366 0.108183 0.994131i \(-0.465497\pi\)
0.108183 + 0.994131i \(0.465497\pi\)
\(770\) 4.00000 0.144150
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 18.0000 0.647415 0.323708 0.946157i \(-0.395071\pi\)
0.323708 + 0.946157i \(0.395071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) 0 0
\(780\) 0 0
\(781\) −32.0000 −1.14505
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) −16.0000 −0.571064
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) −6.00000 −0.213741
\(789\) 0 0
\(790\) −14.0000 −0.498098
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 2.00000 0.0704907
\(806\) 0 0
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) 4.00000 0.140372
\(813\) 0 0
\(814\) 16.0000 0.560800
\(815\) −4.00000 −0.140114
\(816\) 0 0
\(817\) 0 0
\(818\) −38.0000 −1.32864
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 0 0
\(827\) 22.0000 0.765015 0.382507 0.923952i \(-0.375061\pi\)
0.382507 + 0.923952i \(0.375061\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 14.0000 0.485947
\(831\) 0 0
\(832\) −6.00000 −0.208013
\(833\) −12.0000 −0.415775
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −14.0000 −0.483622
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −16.0000 −0.547509
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) −30.0000 −1.01944
\(867\) 0 0
\(868\) 0 0
\(869\) 28.0000 0.949835
\(870\) 0 0
\(871\) 24.0000 0.813209
\(872\) 4.00000 0.135457
\(873\) 0 0
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) −32.0000 −1.07995
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 56.0000 1.88669 0.943344 0.331816i \(-0.107661\pi\)
0.943344 + 0.331816i \(0.107661\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −8.00000 −0.268161
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) 0 0
\(895\) 20.0000 0.668526
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −22.0000 −0.734150
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) −12.0000 −0.399556
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) −12.0000 −0.398893
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 18.0000 0.597351
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) 0 0
\(913\) −28.0000 −0.926665
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) 24.0000 0.792550
\(918\) 0 0
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 1.00000 0.0329690
\(921\) 0 0
\(922\) −10.0000 −0.329332
\(923\) 96.0000 3.15988
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 36.0000 1.18303
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) 0 0
\(934\) −30.0000 −0.981630
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 0 0
\(941\) 22.0000 0.717180 0.358590 0.933495i \(-0.383258\pi\)
0.358590 + 0.933495i \(0.383258\pi\)
\(942\) 0 0
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −8.00000 −0.259965 −0.129983 0.991516i \(-0.541492\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) 8.00000 0.258468
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −48.0000 −1.54758
\(963\) 0 0
\(964\) −14.0000 −0.450910
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 46.0000 1.47621 0.738105 0.674686i \(-0.235721\pi\)
0.738105 + 0.674686i \(0.235721\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) 0 0
\(979\) 16.0000 0.511362
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 8.00000 0.254772
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −32.0000 −1.01651 −0.508257 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −32.0000 −1.01498
\(995\) −2.00000 −0.0634043
\(996\) 0 0
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) 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 2070.2.a.e.1.1 1
3.2 odd 2 690.2.a.g.1.1 1
12.11 even 2 5520.2.a.z.1.1 1
15.2 even 4 3450.2.d.d.2899.2 2
15.8 even 4 3450.2.d.d.2899.1 2
15.14 odd 2 3450.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.g.1.1 1 3.2 odd 2
2070.2.a.e.1.1 1 1.1 even 1 trivial
3450.2.a.l.1.1 1 15.14 odd 2
3450.2.d.d.2899.1 2 15.8 even 4
3450.2.d.d.2899.2 2 15.2 even 4
5520.2.a.z.1.1 1 12.11 even 2