Properties

Label 6768.2.a.h.1.1
Level $6768$
Weight $2$
Character 6768.1
Self dual yes
Analytic conductor $54.043$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6768,2,Mod(1,6768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6768.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6768 = 2^{4} \cdot 3^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6768.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.0427520880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 141)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6768.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{7} +O(q^{10})\) \(q-4.00000 q^{7} +6.00000 q^{13} +6.00000 q^{17} -2.00000 q^{19} +4.00000 q^{23} -5.00000 q^{25} -8.00000 q^{29} -6.00000 q^{31} -6.00000 q^{37} +8.00000 q^{41} +6.00000 q^{43} +1.00000 q^{47} +9.00000 q^{49} -2.00000 q^{53} +12.0000 q^{59} +2.00000 q^{61} +2.00000 q^{67} -10.0000 q^{73} +4.00000 q^{79} +4.00000 q^{83} +10.0000 q^{89} -24.0000 q^{91} -18.0000 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.00000 0.145865
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 16.0000 1.36697 0.683486 0.729964i \(-0.260463\pi\)
0.683486 + 0.729964i \(0.260463\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 20.0000 1.51186
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.0000 −0.997459 −0.498729 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 32.0000 2.24596
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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 \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.0000 1.04819 0.524097 0.851658i \(-0.324403\pi\)
0.524097 + 0.851658i \(0.324403\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 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
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.0000 0.748539 0.374270 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −16.0000 −0.954480 −0.477240 0.878773i \(-0.658363\pi\)
−0.477240 + 0.878773i \(0.658363\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −32.0000 −1.88890
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.00000 0.467365 0.233682 0.972313i \(-0.424922\pi\)
0.233682 + 0.972313i \(0.424922\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) 0 0
\(313\) 34.0000 1.92179 0.960897 0.276907i \(-0.0893093\pi\)
0.960897 + 0.276907i \(0.0893093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −30.0000 −1.66410
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.00000 −0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 18.0000 0.958043 0.479022 0.877803i \(-0.340992\pi\)
0.479022 + 0.877803i \(0.340992\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.0000 0.521996 0.260998 0.965339i \(-0.415948\pi\)
0.260998 + 0.965339i \(0.415948\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −48.0000 −2.47213
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.0000 −1.01404 −0.507020 0.861934i \(-0.669253\pi\)
−0.507020 + 0.861934i \(0.669253\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.0000 1.10415 0.552074 0.833795i \(-0.313837\pi\)
0.552074 + 0.833795i \(0.313837\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) −36.0000 −1.79329
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −48.0000 −2.36193
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −30.0000 −1.45521
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.00000 −0.382692
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.0000 1.88772 0.943858 0.330350i \(-0.107167\pi\)
0.943858 + 0.330350i \(0.107167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −24.0000 −1.11779 −0.558896 0.829238i \(-0.688775\pi\)
−0.558896 + 0.829238i \(0.688775\pi\)
\(462\) 0 0
\(463\) 18.0000 0.836531 0.418265 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.0000 1.46212 0.731059 0.682315i \(-0.239027\pi\)
0.731059 + 0.682315i \(0.239027\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 0 0
\(493\) −48.0000 −2.16181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.00000 −0.356702 −0.178351 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) 40.0000 1.76950
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −36.0000 −1.56818
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 48.0000 2.07911
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −24.0000 −1.01691 −0.508456 0.861088i \(-0.669784\pi\)
−0.508456 + 0.861088i \(0.669784\pi\)
\(558\) 0 0
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.0000 −1.00613 −0.503066 0.864248i \(-0.667795\pi\)
−0.503066 + 0.864248i \(0.667795\pi\)
\(570\) 0 0
\(571\) −44.0000 −1.84134 −0.920671 0.390339i \(-0.872358\pi\)
−0.920671 + 0.390339i \(0.872358\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −20.0000 −0.834058
\(576\) 0 0
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 48.0000 1.97112 0.985562 0.169316i \(-0.0541557\pi\)
0.985562 + 0.169316i \(0.0541557\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −32.0000 −1.30748 −0.653742 0.756717i \(-0.726802\pi\)
−0.653742 + 0.756717i \(0.726802\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.0000 −0.892952 −0.446476 0.894795i \(-0.647321\pi\)
−0.446476 + 0.894795i \(0.647321\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.00000 0.242734
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) −24.0000 −0.964641 −0.482321 0.875995i \(-0.660206\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −40.0000 −1.60257
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −36.0000 −1.43541
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 54.0000 2.13956
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.0000 0.473972 0.236986 0.971513i \(-0.423841\pi\)
0.236986 + 0.971513i \(0.423841\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −34.0000 −1.33052 −0.665261 0.746611i \(-0.731680\pi\)
−0.665261 + 0.746611i \(0.731680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.0000 0.922395 0.461197 0.887298i \(-0.347420\pi\)
0.461197 + 0.887298i \(0.347420\pi\)
\(678\) 0 0
\(679\) 72.0000 2.76311
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0000 1.81813
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −40.0000 −1.50435
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 40.0000 1.48556
\(726\) 0 0
\(727\) −50.0000 −1.85440 −0.927199 0.374570i \(-0.877790\pi\)
−0.927199 + 0.374570i \(0.877790\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 36.0000 1.33151
\(732\) 0 0
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −20.0000 −0.733729 −0.366864 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 48.0000 1.75388
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −42.0000 −1.52652 −0.763258 0.646094i \(-0.776401\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 72.0000 2.59977
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) 0 0
\(775\) 30.0000 1.07763
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 32.0000 1.13779
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 6.00000 0.212265
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −12.0000 −0.421898 −0.210949 0.977497i \(-0.567655\pi\)
−0.210949 + 0.977497i \(0.567655\pi\)
\(810\) 0 0
\(811\) −24.0000 −0.842754 −0.421377 0.906886i \(-0.638453\pi\)
−0.421377 + 0.906886i \(0.638453\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 54.0000 1.87099
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 32.0000 1.10476 0.552381 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000 1.51186
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 32.0000 1.09310 0.546550 0.837427i \(-0.315941\pi\)
0.546550 + 0.837427i \(0.315941\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 12.0000 0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.00000 −0.0669274
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −80.0000 −2.64183
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 30.0000 0.986394
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 10.0000 0.326686 0.163343 0.986569i \(-0.447772\pi\)
0.163343 + 0.986569i \(0.447772\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) −60.0000 −1.94768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −64.0000 −2.06667
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.00000 −0.0639857 −0.0319928 0.999488i \(-0.510185\pi\)
−0.0319928 + 0.999488i \(0.510185\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −48.0000 −1.52477 −0.762385 0.647124i \(-0.775972\pi\)
−0.762385 + 0.647124i \(0.775972\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.0000 −0.443384 −0.221692 0.975117i \(-0.571158\pi\)
−0.221692 + 0.975117i \(0.571158\pi\)
\(998\) 0 0
\(999\) 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 6768.2.a.h.1.1 1
3.2 odd 2 2256.2.a.l.1.1 1
4.3 odd 2 423.2.a.e.1.1 1
12.11 even 2 141.2.a.b.1.1 1
24.5 odd 2 9024.2.a.i.1.1 1
24.11 even 2 9024.2.a.bk.1.1 1
60.59 even 2 3525.2.a.k.1.1 1
84.83 odd 2 6909.2.a.e.1.1 1
564.563 odd 2 6627.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
141.2.a.b.1.1 1 12.11 even 2
423.2.a.e.1.1 1 4.3 odd 2
2256.2.a.l.1.1 1 3.2 odd 2
3525.2.a.k.1.1 1 60.59 even 2
6627.2.a.b.1.1 1 564.563 odd 2
6768.2.a.h.1.1 1 1.1 even 1 trivial
6909.2.a.e.1.1 1 84.83 odd 2
9024.2.a.i.1.1 1 24.5 odd 2
9024.2.a.bk.1.1 1 24.11 even 2