Properties

Label 4928.2.a.g.1.1
Level $4928$
Weight $2$
Character 4928.1
Self dual yes
Analytic conductor $39.350$
Analytic rank $1$
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] = [4928,2,Mod(1,4928)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4928, 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("4928.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4928 = 2^{6} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4928.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: \(39.3502781161\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4928.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{3} +2.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -4.00000 q^{13} -4.00000 q^{15} +4.00000 q^{17} +2.00000 q^{21} -4.00000 q^{23} -1.00000 q^{25} +4.00000 q^{27} +6.00000 q^{29} +10.0000 q^{31} +2.00000 q^{33} -2.00000 q^{35} +6.00000 q^{37} +8.00000 q^{39} +4.00000 q^{41} -12.0000 q^{43} +2.00000 q^{45} -10.0000 q^{47} +1.00000 q^{49} -8.00000 q^{51} +6.00000 q^{53} -2.00000 q^{55} -2.00000 q^{59} -1.00000 q^{63} -8.00000 q^{65} -8.00000 q^{67} +8.00000 q^{69} -12.0000 q^{71} -8.00000 q^{73} +2.00000 q^{75} +1.00000 q^{77} +8.00000 q^{79} -11.0000 q^{81} +8.00000 q^{85} -12.0000 q^{87} -6.00000 q^{89} +4.00000 q^{91} -20.0000 q^{93} -10.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 0 0
\(47\) −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
\(50\) 0 0
\(51\) −8.00000 −1.12022
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 8.00000 0.867722
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 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
\(93\) −20.0000 −2.07390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 4.00000 0.398015 0.199007 0.979998i \(-0.436228\pi\)
0.199007 + 0.979998i \(0.436228\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 24.0000 2.11308
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 20.0000 1.68430
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 20.0000 1.60644
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −12.0000 −0.951662
\(160\) 0 0
\(161\) 4.00000 0.315244
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 4.00000 0.311400
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.0000 −0.912343 −0.456172 0.889892i \(-0.650780\pi\)
−0.456172 + 0.889892i \(0.650780\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 16.0000 1.14578
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) −18.0000 −1.27599 −0.637993 0.770042i \(-0.720235\pi\)
−0.637993 + 0.770042i \(0.720235\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) 0 0
\(203\) −6.00000 −0.421117
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) 16.0000 1.08118
\(220\) 0 0
\(221\) −16.0000 −1.07628
\(222\) 0 0
\(223\) 22.0000 1.47323 0.736614 0.676313i \(-0.236423\pi\)
0.736614 + 0.676313i \(0.236423\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) −18.0000 −1.18947 −0.594737 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) −20.0000 −1.30466
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −20.0000 −1.28831 −0.644157 0.764894i \(-0.722792\pi\)
−0.644157 + 0.764894i \(0.722792\pi\)
\(242\) 0 0
\(243\) 10.0000 0.641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.00000 0.126239 0.0631194 0.998006i \(-0.479895\pi\)
0.0631194 + 0.998006i \(0.479895\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) 0 0
\(273\) −8.00000 −0.484182
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 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
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) −4.00000 −0.232104
\(298\) 0 0
\(299\) 16.0000 0.925304
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) −8.00000 −0.459588
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) −28.0000 −1.59286
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 24.0000 1.33955
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) −28.0000 −1.54840
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 6.00000 0.328798
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −36.0000 −1.95525
\(340\) 0 0
\(341\) −10.0000 −0.541530
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) 0 0
\(355\) −24.0000 −1.27379
\(356\) 0 0
\(357\) 8.00000 0.423405
\(358\) 0 0
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −2.00000 −0.104973
\(364\) 0 0
\(365\) −16.0000 −0.837478
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 0 0
\(369\) 4.00000 0.208232
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) 2.00000 0.102195 0.0510976 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) −12.0000 −0.609994
\(388\) 0 0
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) 16.0000 0.805047
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 0 0
\(403\) −40.0000 −1.99254
\(404\) 0 0
\(405\) −22.0000 −1.09319
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) 24.0000 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(410\) 0 0
\(411\) 20.0000 0.986527
\(412\) 0 0
\(413\) 2.00000 0.0984136
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) −2.00000 −0.0977064 −0.0488532 0.998806i \(-0.515557\pi\)
−0.0488532 + 0.998806i \(0.515557\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) 0 0
\(423\) −10.0000 −0.486217
\(424\) 0 0
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) −24.0000 −1.15071
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −12.0000 −0.568855
\(446\) 0 0
\(447\) −20.0000 −0.945968
\(448\) 0 0
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) 32.0000 1.50349
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 0 0
\(459\) 16.0000 0.746816
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 0 0
\(465\) −40.0000 −1.85496
\(466\) 0 0
\(467\) −30.0000 −1.38823 −0.694117 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 28.0000 1.29017
\(472\) 0 0
\(473\) 12.0000 0.551761
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 0 0
\(483\) −8.00000 −0.364013
\(484\) 0 0
\(485\) −20.0000 −0.908153
\(486\) 0 0
\(487\) −28.0000 −1.26880 −0.634401 0.773004i \(-0.718753\pi\)
−0.634401 + 0.773004i \(0.718753\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) −2.00000 −0.0898933
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.00000 0.178351 0.0891756 0.996016i \(-0.471577\pi\)
0.0891756 + 0.996016i \(0.471577\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) −6.00000 −0.266469
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 8.00000 0.353899
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 28.0000 1.23383
\(516\) 0 0
\(517\) 10.0000 0.439799
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 40.0000 1.74243
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −2.00000 −0.0867926
\(532\) 0 0
\(533\) −16.0000 −0.693037
\(534\) 0 0
\(535\) −24.0000 −1.03761
\(536\) 0 0
\(537\) 24.0000 1.03568
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) 0 0
\(543\) 20.0000 0.858282
\(544\) 0 0
\(545\) 28.0000 1.19939
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −24.0000 −1.01874
\(556\) 0 0
\(557\) −22.0000 −0.932170 −0.466085 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 32.0000 1.34864 0.674320 0.738440i \(-0.264437\pi\)
0.674320 + 0.738440i \(0.264437\pi\)
\(564\) 0 0
\(565\) 36.0000 1.51453
\(566\) 0 0
\(567\) 11.0000 0.461957
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 28.0000 1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 0 0
\(587\) 2.00000 0.0825488 0.0412744 0.999148i \(-0.486858\pi\)
0.0412744 + 0.999148i \(0.486858\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 44.0000 1.80992
\(592\) 0 0
\(593\) 32.0000 1.31408 0.657041 0.753855i \(-0.271808\pi\)
0.657041 + 0.753855i \(0.271808\pi\)
\(594\) 0 0
\(595\) −8.00000 −0.327968
\(596\) 0 0
\(597\) 36.0000 1.47338
\(598\) 0 0
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) −16.0000 −0.645182
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) −14.0000 −0.562708 −0.281354 0.959604i \(-0.590783\pi\)
−0.281354 + 0.959604i \(0.590783\pi\)
\(620\) 0 0
\(621\) −16.0000 −0.642058
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) −24.0000 −0.953914
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 48.0000 1.89000
\(646\) 0 0
\(647\) −22.0000 −0.864909 −0.432455 0.901656i \(-0.642352\pi\)
−0.432455 + 0.901656i \(0.642352\pi\)
\(648\) 0 0
\(649\) 2.00000 0.0785069
\(650\) 0 0
\(651\) 20.0000 0.783862
\(652\) 0 0
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) 0 0
\(657\) −8.00000 −0.312110
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 32.0000 1.24278
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 0 0
\(669\) −44.0000 −1.70114
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 12.0000 0.461197 0.230599 0.973049i \(-0.425932\pi\)
0.230599 + 0.973049i \(0.425932\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) 4.00000 0.153056 0.0765279 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −20.0000 −0.764161
\(686\) 0 0
\(687\) 36.0000 1.37349
\(688\) 0 0
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) 46.0000 1.74992 0.874961 0.484193i \(-0.160887\pi\)
0.874961 + 0.484193i \(0.160887\pi\)
\(692\) 0 0
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 40.0000 1.50649
\(706\) 0 0
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) 0 0
\(717\) 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
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 40.0000 1.48762
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 18.0000 0.667583 0.333792 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −20.0000 −0.729810 −0.364905 0.931045i \(-0.618899\pi\)
−0.364905 + 0.931045i \(0.618899\pi\)
\(752\) 0 0
\(753\) −4.00000 −0.145768
\(754\) 0 0
\(755\) −32.0000 −1.16460
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 48.0000 1.74000 0.869999 0.493053i \(-0.164119\pi\)
0.869999 + 0.493053i \(0.164119\pi\)
\(762\) 0 0
\(763\) −14.0000 −0.506834
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) 28.0000 1.00840
\(772\) 0 0
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) −28.0000 −0.999363
\(786\) 0 0
\(787\) −16.0000 −0.570338 −0.285169 0.958477i \(-0.592050\pi\)
−0.285169 + 0.958477i \(0.592050\pi\)
\(788\) 0 0
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −24.0000 −0.851192
\(796\) 0 0
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) −40.0000 −1.41510
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 8.00000 0.282314
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 20.0000 0.704033
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 0 0
\(818\) 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
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 44.0000 1.52634
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 40.0000 1.38260
\(838\) 0 0
\(839\) 34.0000 1.17381 0.586905 0.809656i \(-0.300346\pi\)
0.586905 + 0.809656i \(0.300346\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −12.0000 −0.413302
\(844\) 0 0
\(845\) 6.00000 0.206406
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) −24.0000 −0.822709
\(852\) 0 0
\(853\) −44.0000 −1.50653 −0.753266 0.657716i \(-0.771523\pi\)
−0.753266 + 0.657716i \(0.771523\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 56.0000 1.91292 0.956462 0.291858i \(-0.0942733\pi\)
0.956462 + 0.291858i \(0.0942733\pi\)
\(858\) 0 0
\(859\) −6.00000 −0.204717 −0.102359 0.994748i \(-0.532639\pi\)
−0.102359 + 0.994748i \(0.532639\pi\)
\(860\) 0 0
\(861\) 8.00000 0.272639
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −24.0000 −0.816024
\(866\) 0 0
\(867\) 2.00000 0.0679236
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 0 0
\(873\) −10.0000 −0.338449
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −42.0000 −1.41824 −0.709120 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) 0 0
\(879\) −48.0000 −1.61900
\(880\) 0 0
\(881\) −34.0000 −1.14549 −0.572745 0.819734i \(-0.694121\pi\)
−0.572745 + 0.819734i \(0.694121\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 11.0000 0.368514
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −24.0000 −0.802232
\(896\) 0 0
\(897\) −32.0000 −1.06845
\(898\) 0 0
\(899\) 60.0000 2.00111
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 0 0
\(903\) −24.0000 −0.798670
\(904\) 0 0
\(905\) −20.0000 −0.664822
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 4.00000 0.132672
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) 40.0000 1.31804
\(922\) 0 0
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 14.0000 0.459820
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 36.0000 1.17859
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −16.0000 −0.522697 −0.261349 0.965244i \(-0.584167\pi\)
−0.261349 + 0.965244i \(0.584167\pi\)
\(938\) 0 0
\(939\) −4.00000 −0.130535
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) 0 0
\(945\) −8.00000 −0.260240
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 32.0000 1.03876
\(950\) 0 0
\(951\) −4.00000 −0.129709
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) 40.0000 1.28631 0.643157 0.765735i \(-0.277624\pi\)
0.643157 + 0.765735i \(0.277624\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.0000 −0.449281 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) −8.00000 −0.256205
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 0 0
\(983\) 54.0000 1.72233 0.861166 0.508323i \(-0.169735\pi\)
0.861166 + 0.508323i \(0.169735\pi\)
\(984\) 0 0
\(985\) −44.0000 −1.40196
\(986\) 0 0
\(987\) −20.0000 −0.636607
\(988\) 0 0
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) −40.0000 −1.26936
\(994\) 0 0
\(995\) −36.0000 −1.14128
\(996\) 0 0
\(997\) −20.0000 −0.633406 −0.316703 0.948525i \(-0.602576\pi\)
−0.316703 + 0.948525i \(0.602576\pi\)
\(998\) 0 0
\(999\) 24.0000 0.759326
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 4928.2.a.g.1.1 1
4.3 odd 2 4928.2.a.bi.1.1 1
8.3 odd 2 1232.2.a.a.1.1 1
8.5 even 2 77.2.a.c.1.1 1
24.5 odd 2 693.2.a.a.1.1 1
40.13 odd 4 1925.2.b.d.1849.1 2
40.29 even 2 1925.2.a.c.1.1 1
40.37 odd 4 1925.2.b.d.1849.2 2
56.5 odd 6 539.2.e.b.67.1 2
56.13 odd 2 539.2.a.d.1.1 1
56.27 even 2 8624.2.a.bc.1.1 1
56.37 even 6 539.2.e.a.67.1 2
56.45 odd 6 539.2.e.b.177.1 2
56.53 even 6 539.2.e.a.177.1 2
88.5 even 10 847.2.f.e.729.1 4
88.13 odd 10 847.2.f.k.323.1 4
88.21 odd 2 847.2.a.a.1.1 1
88.29 odd 10 847.2.f.k.148.1 4
88.37 even 10 847.2.f.e.148.1 4
88.53 even 10 847.2.f.e.323.1 4
88.61 odd 10 847.2.f.k.729.1 4
88.69 even 10 847.2.f.e.372.1 4
88.85 odd 10 847.2.f.k.372.1 4
168.125 even 2 4851.2.a.a.1.1 1
264.197 even 2 7623.2.a.n.1.1 1
616.461 even 2 5929.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.c.1.1 1 8.5 even 2
539.2.a.d.1.1 1 56.13 odd 2
539.2.e.a.67.1 2 56.37 even 6
539.2.e.a.177.1 2 56.53 even 6
539.2.e.b.67.1 2 56.5 odd 6
539.2.e.b.177.1 2 56.45 odd 6
693.2.a.a.1.1 1 24.5 odd 2
847.2.a.a.1.1 1 88.21 odd 2
847.2.f.e.148.1 4 88.37 even 10
847.2.f.e.323.1 4 88.53 even 10
847.2.f.e.372.1 4 88.69 even 10
847.2.f.e.729.1 4 88.5 even 10
847.2.f.k.148.1 4 88.29 odd 10
847.2.f.k.323.1 4 88.13 odd 10
847.2.f.k.372.1 4 88.85 odd 10
847.2.f.k.729.1 4 88.61 odd 10
1232.2.a.a.1.1 1 8.3 odd 2
1925.2.a.c.1.1 1 40.29 even 2
1925.2.b.d.1849.1 2 40.13 odd 4
1925.2.b.d.1849.2 2 40.37 odd 4
4851.2.a.a.1.1 1 168.125 even 2
4928.2.a.g.1.1 1 1.1 even 1 trivial
4928.2.a.bi.1.1 1 4.3 odd 2
5929.2.a.b.1.1 1 616.461 even 2
7623.2.a.n.1.1 1 264.197 even 2
8624.2.a.bc.1.1 1 56.27 even 2