Properties

Label 1386.2.a.f.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0672657201\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1386.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{22} -4.00000 q^{23} -1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} -10.0000 q^{31} +1.00000 q^{32} +2.00000 q^{35} -6.00000 q^{37} +4.00000 q^{38} -2.00000 q^{40} -4.00000 q^{43} -1.00000 q^{44} -4.00000 q^{46} -10.0000 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.00000 q^{52} +14.0000 q^{53} +2.00000 q^{55} -1.00000 q^{56} -2.00000 q^{58} -10.0000 q^{59} -8.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +8.00000 q^{65} +8.00000 q^{67} +2.00000 q^{70} +4.00000 q^{71} +4.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} +1.00000 q^{77} +16.0000 q^{79} -2.00000 q^{80} -4.00000 q^{83} -4.00000 q^{86} -1.00000 q^{88} -10.0000 q^{89} +4.00000 q^{91} -4.00000 q^{92} -10.0000 q^{94} -8.00000 q^{95} +6.00000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) −10.0000 −1.30189 −0.650945 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) −2.00000 −0.223607
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −10.0000 −1.03142
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 0 0
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) −10.0000 −0.898027
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 4.00000 0.335673
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −22.0000 −1.80231 −0.901155 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 20.0000 1.60644
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 16.0000 1.27289
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −4.00000 −0.304114 −0.152057 0.988372i \(-0.548590\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −10.0000 −0.749532
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 0 0
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −6.00000 −0.431889 −0.215945 0.976406i \(-0.569283\pi\)
−0.215945 + 0.976406i \(0.569283\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −12.0000 −0.844317
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 2.00000 0.139347
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 14.0000 0.961524
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) 10.0000 0.678844
\(218\) −14.0000 −0.948200
\(219\) 0 0
\(220\) 2.00000 0.134840
\(221\) 0 0
\(222\) 0 0
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 20.0000 1.30466
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −16.0000 −1.01806
\(248\) −10.0000 −0.635001
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 26.0000 1.64111 0.820553 0.571571i \(-0.193666\pi\)
0.820553 + 0.571571i \(0.193666\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) −16.0000 −1.00393
\(255\) 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
\(259\) 6.00000 0.372822
\(260\) 8.00000 0.496139
\(261\) 0 0
\(262\) −8.00000 −0.494242
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) −28.0000 −1.72003
\(266\) −4.00000 −0.245256
\(267\) 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 \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 6.00000 0.360505 0.180253 0.983620i \(-0.442309\pi\)
0.180253 + 0.983620i \(0.442309\pi\)
\(278\) 20.0000 1.19952
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) 0 0
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) 4.00000 0.234082
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 20.0000 1.16445
\(296\) −6.00000 −0.348743
\(297\) 0 0
\(298\) −22.0000 −1.27443
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 16.0000 0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 16.0000 0.916157
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 20.0000 1.13592
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) 16.0000 0.900070
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 3.00000 0.163178
\(339\) 0 0
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −4.00000 −0.215041
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 32.0000 1.71292 0.856460 0.516213i \(-0.172659\pi\)
0.856460 + 0.516213i \(0.172659\pi\)
\(350\) 1.00000 0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −2.00000 −0.106449 −0.0532246 0.998583i \(-0.516950\pi\)
−0.0532246 + 0.998583i \(0.516950\pi\)
\(354\) 0 0
\(355\) −8.00000 −0.424596
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 14.0000 0.735824
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 18.0000 0.939592 0.469796 0.882775i \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) 12.0000 0.623850
\(371\) −14.0000 −0.726844
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) 6.00000 0.304604
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) −32.0000 −1.61009
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 40.0000 1.99254
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 6.00000 0.297409
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.00000 0.0985329
\(413\) 10.0000 0.492068
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) −4.00000 −0.196116
\(417\) 0 0
\(418\) −4.00000 −0.195646
\(419\) 30.0000 1.46560 0.732798 0.680446i \(-0.238214\pi\)
0.732798 + 0.680446i \(0.238214\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 14.0000 0.679900
\(425\) 0 0
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 10.0000 0.480015
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 20.0000 0.948091
\(446\) −14.0000 −0.662919
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 14.0000 0.658505
\(453\) 0 0
\(454\) −8.00000 −0.375459
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) −10.0000 −0.467269
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −32.0000 −1.49039 −0.745194 0.666847i \(-0.767643\pi\)
−0.745194 + 0.666847i \(0.767643\pi\)
\(462\) 0 0
\(463\) 12.0000 0.557687 0.278844 0.960337i \(-0.410049\pi\)
0.278844 + 0.960337i \(0.410049\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) −14.0000 −0.647843 −0.323921 0.946084i \(-0.605001\pi\)
−0.323921 + 0.946084i \(0.605001\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 20.0000 0.922531
\(471\) 0 0
\(472\) −10.0000 −0.460287
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 0 0
\(478\) 8.00000 0.365911
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 24.0000 1.09431
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −12.0000 −0.544892
\(486\) 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
\(490\) −2.00000 −0.0903508
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 12.0000 0.536656
\(501\) 0 0
\(502\) 26.0000 1.16044
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 4.00000 0.177822
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) −38.0000 −1.68432 −0.842160 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) −4.00000 −0.176950
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −2.00000 −0.0882162
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 6.00000 0.263625
\(519\) 0 0
\(520\) 8.00000 0.350823
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −28.0000 −1.21624
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) −14.0000 −0.603583
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) −28.0000 −1.20270
\(543\) 0 0
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) −8.00000 −0.340811
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 2.00000 0.0845154
\(561\) 0 0
\(562\) −30.0000 −1.26547
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 0 0
\(565\) −28.0000 −1.17797
\(566\) 0 0
\(567\) 0 0
\(568\) 4.00000 0.167836
\(569\) −14.0000 −0.586911 −0.293455 0.955973i \(-0.594805\pi\)
−0.293455 + 0.955973i \(0.594805\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −42.0000 −1.74848 −0.874241 0.485491i \(-0.838641\pi\)
−0.874241 + 0.485491i \(0.838641\pi\)
\(578\) −17.0000 −0.707107
\(579\) 0 0
\(580\) 4.00000 0.166091
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) −14.0000 −0.579821
\(584\) 4.00000 0.165521
\(585\) 0 0
\(586\) 0 0
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 0 0
\(589\) −40.0000 −1.64817
\(590\) 20.0000 0.823387
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −12.0000 −0.492781 −0.246390 0.969171i \(-0.579245\pi\)
−0.246390 + 0.969171i \(0.579245\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −22.0000 −0.901155
\(597\) 0 0
\(598\) 16.0000 0.654289
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −24.0000 −0.978980 −0.489490 0.872009i \(-0.662817\pi\)
−0.489490 + 0.872009i \(0.662817\pi\)
\(602\) 4.00000 0.163028
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −24.0000 −0.974130 −0.487065 0.873366i \(-0.661933\pi\)
−0.487065 + 0.873366i \(0.661933\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 16.0000 0.647821
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 16.0000 0.645707
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) −2.00000 −0.0803868 −0.0401934 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) 20.0000 0.803219
\(621\) 0 0
\(622\) 6.00000 0.240578
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 0 0
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 16.0000 0.636446
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 32.0000 1.26988
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 2.00000 0.0791808
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 22.0000 0.867595 0.433798 0.901010i \(-0.357173\pi\)
0.433798 + 0.901010i \(0.357173\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) 10.0000 0.392534
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) 24.0000 0.939913
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 0 0
\(658\) 10.0000 0.389841
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) 8.00000 0.308837
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −12.0000 −0.461197 −0.230599 0.973049i \(-0.574068\pi\)
−0.230599 + 0.973049i \(0.574068\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 10.0000 0.382920
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) 42.0000 1.59776 0.798878 0.601494i \(-0.205427\pi\)
0.798878 + 0.601494i \(0.205427\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) −40.0000 −1.51729
\(696\) 0 0
\(697\) 0 0
\(698\) 32.0000 1.21122
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −2.00000 −0.0752710
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) −8.00000 −0.300235
\(711\) 0 0
\(712\) −10.0000 −0.374766
\(713\) 40.0000 1.49801
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) −2.00000 −0.0744839
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) 46.0000 1.70605 0.853023 0.521874i \(-0.174767\pi\)
0.853023 + 0.521874i \(0.174767\pi\)
\(728\) 4.00000 0.148250
\(729\) 0 0
\(730\) −8.00000 −0.296093
\(731\) 0 0
\(732\) 0 0
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) 18.0000 0.664392
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) −14.0000 −0.513956
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 44.0000 1.61204
\(746\) −34.0000 −1.24483
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) −10.0000 −0.364662
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 8.00000 0.290573
\(759\) 0 0
\(760\) −8.00000 −0.290191
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) 34.0000 1.22290 0.611448 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) 0 0
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 20.0000 0.712923 0.356462 0.934310i \(-0.383983\pi\)
0.356462 + 0.934310i \(0.383983\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) −32.0000 −1.13851
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 32.0000 1.13635
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) −10.0000 −0.353112
\(803\) −4.00000 −0.141157
\(804\) 0 0
\(805\) −8.00000 −0.281963
\(806\) 40.0000 1.40894
\(807\) 0 0
\(808\) −12.0000 −0.422159
\(809\) −26.0000 −0.914111 −0.457056 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 0 0
\(811\) −40.0000 −1.40459 −0.702295 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) −48.0000 −1.68137
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −4.00000 −0.139857
\(819\) 0 0
\(820\) 0 0
\(821\) 50.0000 1.74501 0.872506 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 0 0
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) 2.00000 0.0696733
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 30.0000 1.03633
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 10.0000 0.344623
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) −6.00000 −0.206406
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 14.0000 0.480762
\(849\) 0 0
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) −12.0000 −0.409912 −0.204956 0.978771i \(-0.565705\pi\)
−0.204956 + 0.978771i \(0.565705\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 16.0000 0.544962
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 0 0
\(865\) 8.00000 0.272008
\(866\) −10.0000 −0.339814
\(867\) 0 0
\(868\) 10.0000 0.339422
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) −14.0000 −0.474100
\(873\) 0 0
\(874\) −16.0000 −0.541208
\(875\) −12.0000 −0.405674
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) −28.0000 −0.944954
\(879\) 0 0
\(880\) 2.00000 0.0674200
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 0 0
\(883\) −4.00000 −0.134611 −0.0673054 0.997732i \(-0.521440\pi\)
−0.0673054 + 0.997732i \(0.521440\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 28.0000 0.940148 0.470074 0.882627i \(-0.344227\pi\)
0.470074 + 0.882627i \(0.344227\pi\)
\(888\) 0 0
\(889\) 16.0000 0.536623
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) −14.0000 −0.468755
\(893\) −40.0000 −1.33855
\(894\) 0 0
\(895\) 24.0000 0.802232
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −28.0000 −0.930751
\(906\) 0 0
\(907\) 48.0000 1.59381 0.796907 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) −8.00000 −0.265489
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 0 0
\(913\) 4.00000 0.132381
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 8.00000 0.264183
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 8.00000 0.263752
\(921\) 0 0
\(922\) −32.0000 −1.05386
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) 10.0000 0.328089 0.164045 0.986453i \(-0.447546\pi\)
0.164045 + 0.986453i \(0.447546\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −6.00000 −0.196537
\(933\) 0 0
\(934\) −14.0000 −0.458094
\(935\) 0 0
\(936\) 0 0
\(937\) −52.0000 −1.69877 −0.849383 0.527777i \(-0.823026\pi\)
−0.849383 + 0.527777i \(0.823026\pi\)
\(938\) −8.00000 −0.261209
\(939\) 0 0
\(940\) 20.0000 0.652328
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) −16.0000 −0.519382
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) 0 0
\(953\) −34.0000 −1.10137 −0.550684 0.834714i \(-0.685633\pi\)
−0.550684 + 0.834714i \(0.685633\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 8.00000 0.258738
\(957\) 0 0
\(958\) −12.0000 −0.387702
\(959\) 6.00000 0.193750
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 24.0000 0.773791
\(963\) 0 0
\(964\) 8.00000 0.257663
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) −12.0000 −0.385297
\(971\) −30.0000 −0.962746 −0.481373 0.876516i \(-0.659862\pi\)
−0.481373 + 0.876516i \(0.659862\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) 12.0000 0.384505
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) 22.0000 0.703842 0.351921 0.936030i \(-0.385529\pi\)
0.351921 + 0.936030i \(0.385529\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) 28.0000 0.893516
\(983\) −26.0000 −0.829271 −0.414636 0.909988i \(-0.636091\pi\)
−0.414636 + 0.909988i \(0.636091\pi\)
\(984\) 0 0
\(985\) −36.0000 −1.14706
\(986\) 0 0
\(987\) 0 0
\(988\) −16.0000 −0.509028
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −4.00000 −0.127064 −0.0635321 0.997980i \(-0.520237\pi\)
−0.0635321 + 0.997980i \(0.520237\pi\)
\(992\) −10.0000 −0.317500
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) 28.0000 0.887660
\(996\) 0 0
\(997\) 36.0000 1.14013 0.570066 0.821599i \(-0.306918\pi\)
0.570066 + 0.821599i \(0.306918\pi\)
\(998\) −16.0000 −0.506471
\(999\) 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 1386.2.a.f.1.1 1
3.2 odd 2 154.2.a.b.1.1 1
7.6 odd 2 9702.2.a.bz.1.1 1
12.11 even 2 1232.2.a.c.1.1 1
15.2 even 4 3850.2.c.d.1849.1 2
15.8 even 4 3850.2.c.d.1849.2 2
15.14 odd 2 3850.2.a.o.1.1 1
21.2 odd 6 1078.2.e.h.67.1 2
21.5 even 6 1078.2.e.l.67.1 2
21.11 odd 6 1078.2.e.h.177.1 2
21.17 even 6 1078.2.e.l.177.1 2
21.20 even 2 1078.2.a.b.1.1 1
24.5 odd 2 4928.2.a.d.1.1 1
24.11 even 2 4928.2.a.bf.1.1 1
33.32 even 2 1694.2.a.i.1.1 1
84.83 odd 2 8624.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.a.b.1.1 1 3.2 odd 2
1078.2.a.b.1.1 1 21.20 even 2
1078.2.e.h.67.1 2 21.2 odd 6
1078.2.e.h.177.1 2 21.11 odd 6
1078.2.e.l.67.1 2 21.5 even 6
1078.2.e.l.177.1 2 21.17 even 6
1232.2.a.c.1.1 1 12.11 even 2
1386.2.a.f.1.1 1 1.1 even 1 trivial
1694.2.a.i.1.1 1 33.32 even 2
3850.2.a.o.1.1 1 15.14 odd 2
3850.2.c.d.1849.1 2 15.2 even 4
3850.2.c.d.1849.2 2 15.8 even 4
4928.2.a.d.1.1 1 24.5 odd 2
4928.2.a.bf.1.1 1 24.11 even 2
8624.2.a.z.1.1 1 84.83 odd 2
9702.2.a.bz.1.1 1 7.6 odd 2