Properties

Label 6720.2.a.bu.1.1
Level $6720$
Weight $2$
Character 6720.1
Self dual yes
Analytic conductor $53.659$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6720.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.6594701583\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3360)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6720.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +4.00000 q^{13} -1.00000 q^{15} -2.00000 q^{17} -6.00000 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +10.0000 q^{29} -2.00000 q^{31} +2.00000 q^{33} -1.00000 q^{35} +2.00000 q^{37} +4.00000 q^{39} -10.0000 q^{41} +4.00000 q^{43} -1.00000 q^{45} +8.00000 q^{47} +1.00000 q^{49} -2.00000 q^{51} -4.00000 q^{53} -2.00000 q^{55} -6.00000 q^{57} +4.00000 q^{59} +2.00000 q^{61} +1.00000 q^{63} -4.00000 q^{65} +4.00000 q^{67} +4.00000 q^{69} -6.00000 q^{71} +1.00000 q^{75} +2.00000 q^{77} +4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.00000 q^{85} +10.0000 q^{87} -6.00000 q^{89} +4.00000 q^{91} -2.00000 q^{93} +6.00000 q^{95} +4.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 6.00000 0.615587
\(96\) 0 0
\(97\) 4.00000 0.406138 0.203069 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −10.0000 −0.901670
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) −22.0000 −1.86602 −0.933008 0.359856i \(-0.882826\pi\)
−0.933008 + 0.359856i \(0.882826\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 0 0
\(173\) 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
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −18.0000 −1.30243 −0.651217 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) 0 0
\(193\) −22.0000 −1.58359 −0.791797 0.610784i \(-0.790854\pi\)
−0.791797 + 0.610784i \(0.790854\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) −10.0000 −0.708881 −0.354441 0.935079i \(-0.615329\pi\)
−0.354441 + 0.935079i \(0.615329\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 10.0000 0.701862
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) 4.00000 0.278019
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 12.0000 0.803579 0.401790 0.915732i \(-0.368388\pi\)
0.401790 + 0.915732i \(0.368388\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) −12.0000 −0.786146 −0.393073 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) −14.0000 −0.850439 −0.425220 0.905090i \(-0.639803\pi\)
−0.425220 + 0.905090i \(0.639803\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 0 0
\(285\) 6.00000 0.355409
\(286\) 0 0
\(287\) −10.0000 −0.590281
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) −2.00000 −0.116841 −0.0584206 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −8.00000 −0.456584 −0.228292 0.973593i \(-0.573314\pi\)
−0.228292 + 0.973593i \(0.573314\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 28.0000 1.58265 0.791327 0.611393i \(-0.209391\pi\)
0.791327 + 0.611393i \(0.209391\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 0 0
\(317\) 28.0000 1.57264 0.786318 0.617822i \(-0.211985\pi\)
0.786318 + 0.617822i \(0.211985\pi\)
\(318\) 0 0
\(319\) 20.0000 1.11979
\(320\) 0 0
\(321\) 20.0000 1.11629
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 14.0000 0.774202
\(328\) 0 0
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 0 0
\(357\) −2.00000 −0.105851
\(358\) 0 0
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) 0 0
\(373\) −18.0000 −0.932005 −0.466002 0.884783i \(-0.654306\pi\)
−0.466002 + 0.884783i \(0.654306\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 40.0000 2.06010
\(378\) 0 0
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 0 0
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −8.00000 −0.398508
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 0 0
\(413\) 4.00000 0.196827
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) −22.0000 −1.07734
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 8.00000 0.388973
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) 8.00000 0.386244
\(430\) 0 0
\(431\) 10.0000 0.481683 0.240842 0.970564i \(-0.422577\pi\)
0.240842 + 0.970564i \(0.422577\pi\)
\(432\) 0 0
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) −10.0000 −0.479463
\(436\) 0 0
\(437\) −24.0000 −1.14808
\(438\) 0 0
\(439\) 6.00000 0.286364 0.143182 0.989696i \(-0.454267\pi\)
0.143182 + 0.989696i \(0.454267\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) −20.0000 −0.941763
\(452\) 0 0
\(453\) 16.0000 0.751746
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 26.0000 1.21094 0.605470 0.795868i \(-0.292985\pi\)
0.605470 + 0.795868i \(0.292985\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 2.00000 0.0927478
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) 0 0
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.00000 −0.264906
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 16.0000 0.703679
\(518\) 0 0
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −32.0000 −1.39926 −0.699631 0.714504i \(-0.746652\pi\)
−0.699631 + 0.714504i \(0.746652\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) −40.0000 −1.73259
\(534\) 0 0
\(535\) −20.0000 −0.864675
\(536\) 0 0
\(537\) 18.0000 0.776757
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0 0
\(543\) 6.00000 0.257485
\(544\) 0 0
\(545\) −14.0000 −0.599694
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −60.0000 −2.55609
\(552\) 0 0
\(553\) 4.00000 0.170097
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) −36.0000 −1.52537 −0.762684 0.646771i \(-0.776119\pi\)
−0.762684 + 0.646771i \(0.776119\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 0 0
\(565\) −12.0000 −0.504844
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) 0 0
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −8.00000 −0.331326
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 0 0
\(597\) −10.0000 −0.409273
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) −36.0000 −1.46119 −0.730597 0.682808i \(-0.760758\pi\)
−0.730597 + 0.682808i \(0.760758\pi\)
\(608\) 0 0
\(609\) 10.0000 0.405220
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) −38.0000 −1.53481 −0.767403 0.641165i \(-0.778451\pi\)
−0.767403 + 0.641165i \(0.778451\pi\)
\(614\) 0 0
\(615\) 10.0000 0.403239
\(616\) 0 0
\(617\) −20.0000 −0.805170 −0.402585 0.915383i \(-0.631888\pi\)
−0.402585 + 0.915383i \(0.631888\pi\)
\(618\) 0 0
\(619\) −46.0000 −1.84890 −0.924448 0.381308i \(-0.875474\pi\)
−0.924448 + 0.381308i \(0.875474\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −12.0000 −0.479234
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −44.0000 −1.75161 −0.875806 0.482663i \(-0.839670\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 20.0000 0.794929
\(634\) 0 0
\(635\) 8.00000 0.317470
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 38.0000 1.50091 0.750455 0.660922i \(-0.229834\pi\)
0.750455 + 0.660922i \(0.229834\pi\)
\(642\) 0 0
\(643\) −40.0000 −1.57745 −0.788723 0.614749i \(-0.789257\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) 0 0
\(645\) −4.00000 −0.157500
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 0 0
\(653\) −16.0000 −0.626128 −0.313064 0.949732i \(-0.601356\pi\)
−0.313064 + 0.949732i \(0.601356\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 42.0000 1.63361 0.816805 0.576913i \(-0.195743\pi\)
0.816805 + 0.576913i \(0.195743\pi\)
\(662\) 0 0
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 0 0
\(669\) 12.0000 0.463947
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 0 0
\(687\) 2.00000 0.0763048
\(688\) 0 0
\(689\) −16.0000 −0.609551
\(690\) 0 0
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) 22.0000 0.834508
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 0 0
\(705\) −8.00000 −0.301297
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) −6.00000 −0.224074
\(718\) 0 0
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 14.0000 0.520666
\(724\) 0 0
\(725\) 10.0000 0.371391
\(726\) 0 0
\(727\) 44.0000 1.63187 0.815935 0.578144i \(-0.196223\pi\)
0.815935 + 0.578144i \(0.196223\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) −44.0000 −1.62518 −0.812589 0.582838i \(-0.801942\pi\)
−0.812589 + 0.582838i \(0.801942\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) 20.0000 0.730784
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 14.0000 0.506834
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 16.0000 0.577727
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) 0 0
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 0 0
\(779\) 60.0000 2.14972
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) 0 0
\(789\) −8.00000 −0.284808
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 8.00000 0.284088
\(794\) 0 0
\(795\) 4.00000 0.141865
\(796\) 0 0
\(797\) 46.0000 1.62940 0.814702 0.579880i \(-0.196901\pi\)
0.814702 + 0.579880i \(0.196901\pi\)
\(798\) 0 0
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) 0 0
\(811\) −38.0000 −1.33436 −0.667180 0.744896i \(-0.732499\pi\)
−0.667180 + 0.744896i \(0.732499\pi\)
\(812\) 0 0
\(813\) −14.0000 −0.491001
\(814\) 0 0
\(815\) 20.0000 0.700569
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −46.0000 −1.60541 −0.802706 0.596376i \(-0.796607\pi\)
−0.802706 + 0.596376i \(0.796607\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −2.00000 −0.0691301
\(838\) 0 0
\(839\) 52.0000 1.79524 0.897620 0.440771i \(-0.145295\pi\)
0.897620 + 0.440771i \(0.145295\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −36.0000 −1.23262 −0.616308 0.787505i \(-0.711372\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) −50.0000 −1.70598 −0.852989 0.521929i \(-0.825213\pi\)
−0.852989 + 0.521929i \(0.825213\pi\)
\(860\) 0 0
\(861\) −10.0000 −0.340799
\(862\) 0 0
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) −4.00000 −0.134459
\(886\) 0 0
\(887\) −40.0000 −1.34307 −0.671534 0.740973i \(-0.734364\pi\)
−0.671534 + 0.740973i \(0.734364\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) −48.0000 −1.60626
\(894\) 0 0
\(895\) −18.0000 −0.601674
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) 0 0
\(899\) −20.0000 −0.667037
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −34.0000 −1.12647 −0.563235 0.826297i \(-0.690443\pi\)
−0.563235 + 0.826297i \(0.690443\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) −2.00000 −0.0661180
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) −8.00000 −0.263609
\(922\) 0 0
\(923\) −24.0000 −0.789970
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) 0 0
\(941\) 34.0000 1.10837 0.554184 0.832394i \(-0.313030\pi\)
0.554184 + 0.832394i \(0.313030\pi\)
\(942\) 0 0
\(943\) −40.0000 −1.30258
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −16.0000 −0.519930 −0.259965 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 28.0000 0.907962
\(952\) 0 0
\(953\) −28.0000 −0.907009 −0.453504 0.891254i \(-0.649826\pi\)
−0.453504 + 0.891254i \(0.649826\pi\)
\(954\) 0 0
\(955\) 18.0000 0.582466
\(956\) 0 0
\(957\) 20.0000 0.646508
\(958\) 0 0
\(959\) 16.0000 0.516667
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 0 0
\(965\) 22.0000 0.708205
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) −22.0000 −0.705288
\(974\) 0 0
\(975\) 4.00000 0.128103
\(976\) 0 0
\(977\) −28.0000 −0.895799 −0.447900 0.894084i \(-0.647828\pi\)
−0.447900 + 0.894084i \(0.647828\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 8.00000 0.254643
\(988\) 0 0
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) 10.0000 0.317021
\(996\) 0 0
\(997\) 28.0000 0.886769 0.443384 0.896332i \(-0.353778\pi\)
0.443384 + 0.896332i \(0.353778\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 6720.2.a.bu.1.1 1
4.3 odd 2 6720.2.a.f.1.1 1
8.3 odd 2 3360.2.a.w.1.1 yes 1
8.5 even 2 3360.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.j.1.1 1 8.5 even 2
3360.2.a.w.1.1 yes 1 8.3 odd 2
6720.2.a.f.1.1 1 4.3 odd 2
6720.2.a.bu.1.1 1 1.1 even 1 trivial