Properties

Label 816.2.c.c.577.1
Level $816$
Weight $2$
Character 816.577
Analytic conductor $6.516$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [816,2,Mod(577,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 816.577
Dual form 816.2.c.c.577.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +4.00000i q^{7} -1.00000 q^{9} -4.00000i q^{11} +2.00000 q^{13} +(1.00000 + 4.00000i) q^{17} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{23} +5.00000 q^{25} +1.00000i q^{27} +4.00000i q^{31} -4.00000 q^{33} +8.00000i q^{37} -2.00000i q^{39} -8.00000i q^{41} +4.00000 q^{43} +8.00000 q^{47} -9.00000 q^{49} +(4.00000 - 1.00000i) q^{51} +6.00000 q^{53} -4.00000i q^{57} +12.0000 q^{59} -8.00000i q^{61} -4.00000i q^{63} -12.0000 q^{67} +4.00000 q^{69} -12.0000i q^{71} -5.00000i q^{75} +16.0000 q^{77} +4.00000i q^{79} +1.00000 q^{81} -12.0000 q^{83} -10.0000 q^{89} +8.00000i q^{91} +4.00000 q^{93} +16.0000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 4 q^{13} + 2 q^{17} + 8 q^{19} + 8 q^{21} + 10 q^{25} - 8 q^{33} + 8 q^{43} + 16 q^{47} - 18 q^{49} + 8 q^{51} + 12 q^{53} + 24 q^{59} - 24 q^{67} + 8 q^{69} + 32 q^{77} + 2 q^{81} - 24 q^{83} - 20 q^{89} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/816\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(545\) \(613\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000i 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000 + 4.00000i 0.242536 + 0.970143i
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.00000i 0.718421i 0.933257 + 0.359211i \(0.116954\pi\)
−0.933257 + 0.359211i \(0.883046\pi\)
\(32\) 0 0
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 2.00000i 0.320256i
\(40\) 0 0
\(41\) 8.00000i 1.24939i −0.780869 0.624695i \(-0.785223\pi\)
0.780869 0.624695i \(-0.214777\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 4.00000 1.00000i 0.560112 0.140028i
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i −0.858898 0.512148i \(-0.828850\pi\)
0.858898 0.512148i \(-0.171150\pi\)
\(62\) 0 0
\(63\) 4.00000i 0.503953i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 12.0000i 1.42414i −0.702109 0.712069i \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 5.00000i 0.577350i
\(76\) 0 0
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 4.00000i 0.450035i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 8.00000i 0.838628i
\(92\) 0 0
\(93\) 4.00000 0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) 0 0
\(99\) 4.00000i 0.402015i
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 0 0
\(109\) 8.00000i 0.766261i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 8.00000i 0.752577i −0.926503 0.376288i \(-0.877200\pi\)
0.926503 0.376288i \(-0.122800\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −16.0000 + 4.00000i −1.46672 + 0.366679i
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 4.00000i 0.352180i
\(130\) 0 0
\(131\) 4.00000i 0.349482i −0.984614 0.174741i \(-0.944091\pi\)
0.984614 0.174741i \(-0.0559088\pi\)
\(132\) 0 0
\(133\) 16.0000i 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) 8.00000i 0.673722i
\(142\) 0 0
\(143\) 8.00000i 0.668994i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.00000i 0.742307i
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −1.00000 4.00000i −0.0808452 0.323381i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 6.00000i 0.475831i
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 20.0000i 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) 16.0000i 1.21646i −0.793762 0.608229i \(-0.791880\pi\)
0.793762 0.608229i \(-0.208120\pi\)
\(174\) 0 0
\(175\) 20.0000i 1.51186i
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) 8.00000i 0.594635i −0.954779 0.297318i \(-0.903908\pi\)
0.954779 0.297318i \(-0.0960920\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.0000 4.00000i 1.17004 0.292509i
\(188\) 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
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.0000i 1.13995i 0.821661 + 0.569976i \(0.193048\pi\)
−0.821661 + 0.569976i \(0.806952\pi\)
\(198\) 0 0
\(199\) 20.0000i 1.41776i 0.705328 + 0.708881i \(0.250800\pi\)
−0.705328 + 0.708881i \(0.749200\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 4.00000i 0.278019i
\(208\) 0 0
\(209\) 16.0000i 1.10674i
\(210\) 0 0
\(211\) 4.00000i 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.00000 + 8.00000i 0.134535 + 0.538138i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 12.0000i 0.796468i 0.917284 + 0.398234i \(0.130377\pi\)
−0.917284 + 0.398234i \(0.869623\pi\)
\(228\) 0 0
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 16.0000i 1.05272i
\(232\) 0 0
\(233\) 8.00000i 0.524097i 0.965055 + 0.262049i \(0.0843981\pi\)
−0.965055 + 0.262049i \(0.915602\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 16.0000i 1.03065i −0.856995 0.515325i \(-0.827671\pi\)
0.856995 0.515325i \(-0.172329\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.00000 0.509028
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 0 0
\(259\) −32.0000 −1.98838
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000i 0.611990i
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) 20.0000i 1.20605i
\(276\) 0 0
\(277\) 8.00000i 0.480673i −0.970690 0.240337i \(-0.922742\pi\)
0.970690 0.240337i \(-0.0772579\pi\)
\(278\) 0 0
\(279\) 4.00000i 0.239474i
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 32.0000 1.88890
\(288\) 0 0
\(289\) −15.0000 + 8.00000i −0.882353 + 0.470588i
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 0 0
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.00000 0.232104
\(298\) 0 0
\(299\) 8.00000i 0.462652i
\(300\) 0 0
\(301\) 16.0000i 0.922225i
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.0000i 0.680458i −0.940343 0.340229i \(-0.889495\pi\)
0.940343 0.340229i \(-0.110505\pi\)
\(312\) 0 0
\(313\) 16.0000i 0.904373i −0.891923 0.452187i \(-0.850644\pi\)
0.891923 0.452187i \(-0.149356\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.0000i 1.79730i −0.438667 0.898650i \(-0.644549\pi\)
0.438667 0.898650i \(-0.355451\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 4.00000 + 16.0000i 0.222566 + 0.890264i
\(324\) 0 0
\(325\) 10.0000 0.554700
\(326\) 0 0
\(327\) 8.00000 0.442401
\(328\) 0 0
\(329\) 32.0000i 1.76422i
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 8.00000i 0.438397i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 16.0000i 0.871576i −0.900049 0.435788i \(-0.856470\pi\)
0.900049 0.435788i \(-0.143530\pi\)
\(338\) 0 0
\(339\) −8.00000 −0.434500
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) 0 0
\(343\) 8.00000i 0.431959i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000i 0.214731i −0.994220 0.107366i \(-0.965758\pi\)
0.994220 0.107366i \(-0.0342415\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 2.00000i 0.106752i
\(352\) 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
\(357\) 4.00000 + 16.0000i 0.211702 + 0.846810i
\(358\) 0 0
\(359\) −32.0000 −1.68890 −0.844448 0.535638i \(-0.820071\pi\)
−0.844448 + 0.535638i \(0.820071\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 12.0000i 0.626395i −0.949688 0.313197i \(-0.898600\pi\)
0.949688 0.313197i \(-0.101400\pi\)
\(368\) 0 0
\(369\) 8.00000i 0.416463i
\(370\) 0 0
\(371\) 24.0000i 1.24602i
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 20.0000i 1.02733i −0.857991 0.513665i \(-0.828287\pi\)
0.857991 0.513665i \(-0.171713\pi\)
\(380\) 0 0
\(381\) 8.00000i 0.409852i
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 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 + 4.00000i −0.809155 + 0.202289i
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000i 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 24.0000i 1.19850i −0.800561 0.599251i \(-0.795465\pi\)
0.800561 0.599251i \(-0.204535\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0000 1.58618
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 10.0000i 0.493264i
\(412\) 0 0
\(413\) 48.0000i 2.36193i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 4.00000i 0.195413i −0.995215 0.0977064i \(-0.968849\pi\)
0.995215 0.0977064i \(-0.0311506\pi\)
\(420\) 0 0
\(421\) −22.0000 −1.07221 −0.536107 0.844150i \(-0.680106\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) −8.00000 −0.388973
\(424\) 0 0
\(425\) 5.00000 + 20.0000i 0.242536 + 0.970143i
\(426\) 0 0
\(427\) 32.0000 1.54859
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 20.0000i 0.963366i 0.876346 + 0.481683i \(0.159974\pi\)
−0.876346 + 0.481683i \(0.840026\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.0000i 0.765384i
\(438\) 0 0
\(439\) 36.0000i 1.71819i 0.511819 + 0.859093i \(0.328972\pi\)
−0.511819 + 0.859093i \(0.671028\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000i 0.283790i
\(448\) 0 0
\(449\) 8.00000i 0.377543i 0.982021 + 0.188772i \(0.0604506\pi\)
−0.982021 + 0.188772i \(0.939549\pi\)
\(450\) 0 0
\(451\) −32.0000 −1.50682
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) 0 0
\(459\) −4.00000 + 1.00000i −0.186704 + 0.0466760i
\(460\) 0 0
\(461\) −34.0000 −1.58354 −0.791769 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 0 0
\(469\) 48.0000i 2.21643i
\(470\) 0 0
\(471\) 2.00000i 0.0921551i
\(472\) 0 0
\(473\) 16.0000i 0.735681i
\(474\) 0 0
\(475\) 20.0000 0.917663
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 12.0000i 0.548294i −0.961688 0.274147i \(-0.911605\pi\)
0.961688 0.274147i \(-0.0883955\pi\)
\(480\) 0 0
\(481\) 16.0000i 0.729537i
\(482\) 0 0
\(483\) 16.0000i 0.728025i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4.00000i 0.181257i 0.995885 + 0.0906287i \(0.0288876\pi\)
−0.995885 + 0.0906287i \(0.971112\pi\)
\(488\) 0 0
\(489\) −20.0000 −0.904431
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 48.0000 2.15309
\(498\) 0 0
\(499\) 36.0000i 1.61158i −0.592200 0.805791i \(-0.701741\pi\)
0.592200 0.805791i \(-0.298259\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 0 0
\(503\) 12.0000i 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.00000i 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) 24.0000i 1.05146i 0.850652 + 0.525730i \(0.176208\pi\)
−0.850652 + 0.525730i \(0.823792\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) 20.0000 0.872872
\(526\) 0 0
\(527\) −16.0000 + 4.00000i −0.696971 + 0.174243i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) 16.0000i 0.693037i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.00000i 0.172613i
\(538\) 0 0
\(539\) 36.0000i 1.55063i
\(540\) 0 0
\(541\) 40.0000i 1.71973i 0.510518 + 0.859867i \(0.329454\pi\)
−0.510518 + 0.859867i \(0.670546\pi\)
\(542\) 0 0
\(543\) −8.00000 −0.343313
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 36.0000i 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 0 0
\(549\) 8.00000i 0.341432i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −16.0000 −0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) −4.00000 16.0000i −0.168880 0.675521i
\(562\) 0 0
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 4.00000i 0.167984i
\(568\) 0 0
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 20.0000i 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 8.00000i 0.334205i
\(574\) 0 0
\(575\) 20.0000i 0.834058i
\(576\) 0 0
\(577\) 30.0000 1.24892 0.624458 0.781058i \(-0.285320\pi\)
0.624458 + 0.781058i \(0.285320\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 48.0000i 1.99138i
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 16.0000i 0.659269i
\(590\) 0 0
\(591\) 16.0000 0.658152
\(592\) 0 0
\(593\) −46.0000 −1.88899 −0.944497 0.328521i \(-0.893450\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000 0.818546
\(598\) 0 0
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 0 0
\(601\) 16.0000i 0.652654i 0.945257 + 0.326327i \(0.105811\pi\)
−0.945257 + 0.326327i \(0.894189\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.00000i 0.162355i 0.996700 + 0.0811775i \(0.0258681\pi\)
−0.996700 + 0.0811775i \(0.974132\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 16.0000 0.647291
\(612\) 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
\(617\) 24.0000i 0.966204i −0.875564 0.483102i \(-0.839510\pi\)
0.875564 0.483102i \(-0.160490\pi\)
\(618\) 0 0
\(619\) 36.0000i 1.44696i −0.690344 0.723481i \(-0.742541\pi\)
0.690344 0.723481i \(-0.257459\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) 40.0000i 1.60257i
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) −16.0000 −0.638978
\(628\) 0 0
\(629\) −32.0000 + 8.00000i −1.27592 + 0.318981i
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 0 0
\(639\) 12.0000i 0.474713i
\(640\) 0 0
\(641\) 8.00000i 0.315981i 0.987441 + 0.157991i \(0.0505015\pi\)
−0.987441 + 0.157991i \(0.949498\pi\)
\(642\) 0 0
\(643\) 4.00000i 0.157745i −0.996885 0.0788723i \(-0.974868\pi\)
0.996885 0.0788723i \(-0.0251319\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 48.0000i 1.88416i
\(650\) 0 0
\(651\) 16.0000i 0.627089i
\(652\) 0 0
\(653\) 32.0000i 1.25226i −0.779720 0.626128i \(-0.784639\pi\)
0.779720 0.626128i \(-0.215361\pi\)
\(654\) 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
\(661\) −42.0000 −1.63361 −0.816805 0.576913i \(-0.804257\pi\)
−0.816805 + 0.576913i \(0.804257\pi\)
\(662\) 0 0
\(663\) 8.00000 2.00000i 0.310694 0.0776736i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 16.0000i 0.618596i
\(670\) 0 0
\(671\) −32.0000 −1.23535
\(672\) 0 0
\(673\) 32.0000i 1.23351i 0.787155 + 0.616755i \(0.211553\pi\)
−0.787155 + 0.616755i \(0.788447\pi\)
\(674\) 0 0
\(675\) 5.00000i 0.192450i
\(676\) 0 0
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 0 0
\(679\) −64.0000 −2.45609
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 0 0
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 20.0000i 0.760836i −0.924815 0.380418i \(-0.875780\pi\)
0.924815 0.380418i \(-0.124220\pi\)
\(692\) 0 0
\(693\) −16.0000 −0.607790
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.0000 8.00000i 1.21209 0.303022i
\(698\) 0 0
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 32.0000i 1.20690i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 24.0000i 0.902613i
\(708\) 0 0
\(709\) 40.0000i 1.50223i −0.660171 0.751116i \(-0.729516\pi\)
0.660171 0.751116i \(-0.270484\pi\)
\(710\) 0 0
\(711\) 4.00000i 0.150012i
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.00000i 0.298765i
\(718\) 0 0
\(719\) 36.0000i 1.34257i 0.741198 + 0.671287i \(0.234258\pi\)
−0.741198 + 0.671287i \(0.765742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −16.0000 −0.595046
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 4.00000 + 16.0000i 0.147945 + 0.591781i
\(732\) 0 0
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000i 1.76810i
\(738\) 0 0
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 8.00000i 0.293887i
\(742\) 0 0
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 12.0000 0.439057
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) 20.0000i 0.729810i 0.931045 + 0.364905i \(0.118899\pi\)
−0.931045 + 0.364905i \(0.881101\pi\)
\(752\) 0 0
\(753\) 12.0000i 0.437304i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 16.0000i 0.580763i
\(760\) 0 0
\(761\) 6.00000 0.217500 0.108750 0.994069i \(-0.465315\pi\)
0.108750 + 0.994069i \(0.465315\pi\)
\(762\) 0 0
\(763\) −32.0000 −1.15848
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 20.0000i 0.718421i
\(776\) 0 0
\(777\) 32.0000i 1.14799i
\(778\) 0 0
\(779\) 32.0000i 1.14652i
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 12.0000i 0.427754i 0.976861 + 0.213877i \(0.0686091\pi\)
−0.976861 + 0.213877i \(0.931391\pi\)
\(788\) 0 0
\(789\) 24.0000i 0.854423i
\(790\) 0 0
\(791\) 32.0000 1.13779
\(792\) 0 0
\(793\) 16.0000i 0.568177i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 8.00000 + 32.0000i 0.283020 + 1.13208i
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 40.0000i 1.40633i −0.711029 0.703163i \(-0.751771\pi\)
0.711029 0.703163i \(-0.248229\pi\)
\(810\) 0 0
\(811\) 4.00000i 0.140459i −0.997531 0.0702295i \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 8.00000i 0.279543i
\(820\) 0 0
\(821\) 32.0000i 1.11681i 0.829569 + 0.558404i \(0.188586\pi\)
−0.829569 + 0.558404i \(0.811414\pi\)
\(822\) 0 0
\(823\) 44.0000i 1.53374i −0.641800 0.766872i \(-0.721812\pi\)
0.641800 0.766872i \(-0.278188\pi\)
\(824\) 0 0
\(825\) −20.0000 −0.696311
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) 0 0
\(833\) −9.00000 36.0000i −0.311832 1.24733i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.00000 −0.138260
\(838\) 0 0
\(839\) 44.0000i 1.51905i −0.650479 0.759524i \(-0.725432\pi\)
0.650479 0.759524i \(-0.274568\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 10.0000i 0.344418i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20.0000i 0.687208i
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 24.0000i 0.821744i −0.911693 0.410872i \(-0.865224\pi\)
0.911693 0.410872i \(-0.134776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8.00000i 0.273275i 0.990621 + 0.136637i \(0.0436295\pi\)
−0.990621 + 0.136637i \(0.956370\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 32.0000i 1.09056i
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.00000 + 15.0000i 0.271694 + 0.509427i
\(868\) 0 0
\(869\) 16.0000 0.542763
\(870\) 0 0
\(871\) −24.0000 −0.813209
\(872\) 0 0
\(873\) 16.0000i 0.541518i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 40.0000i 1.35070i −0.737496 0.675352i \(-0.763992\pi\)
0.737496 0.675352i \(-0.236008\pi\)
\(878\) 0 0
\(879\) 10.0000i 0.337292i
\(880\) 0 0
\(881\) 24.0000i 0.808581i 0.914631 + 0.404290i \(0.132481\pi\)
−0.914631 + 0.404290i \(0.867519\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.00000i 0.134307i 0.997743 + 0.0671534i \(0.0213917\pi\)
−0.997743 + 0.0671534i \(0.978608\pi\)
\(888\) 0 0
\(889\) 32.0000i 1.07325i
\(890\) 0 0
\(891\) 4.00000i 0.134005i
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 6.00000 + 24.0000i 0.199889 + 0.799556i
\(902\) 0 0
\(903\) 16.0000 0.532447
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 28.0000i 0.927681i −0.885919 0.463841i \(-0.846471\pi\)
0.885919 0.463841i \(-0.153529\pi\)
\(912\) 0 0
\(913\) 48.0000i 1.58857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 12.0000i 0.395413i
\(922\) 0 0
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) 40.0000i 1.31519i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 24.0000i 0.787414i 0.919236 + 0.393707i \(0.128808\pi\)
−0.919236 + 0.393707i \(0.871192\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) −16.0000 −0.522140
\(940\) 0 0
\(941\) 48.0000i 1.56476i −0.622804 0.782378i \(-0.714007\pi\)
0.622804 0.782378i \(-0.285993\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −32.0000 −1.03767
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 40.0000i 1.29167i
\(960\) 0 0
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) 0 0
\(969\) 16.0000 4.00000i 0.513994 0.128499i
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 0 0
\(975\) 10.0000i 0.320256i
\(976\) 0 0
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) 0 0
\(979\) 40.0000i 1.27841i
\(980\) 0 0
\(981\) 8.00000i 0.255420i
\(982\) 0 0
\(983\) 20.0000i 0.637901i 0.947771 + 0.318950i \(0.103330\pi\)
−0.947771 + 0.318950i \(0.896670\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 32.0000 1.01857
\(988\) 0 0
\(989\) 16.0000i 0.508770i
\(990\) 0 0
\(991\) 12.0000i 0.381193i −0.981669 0.190596i \(-0.938958\pi\)
0.981669 0.190596i \(-0.0610421\pi\)
\(992\) 0 0
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 8.00000i 0.253363i 0.991943 + 0.126681i \(0.0404325\pi\)
−0.991943 + 0.126681i \(0.959567\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 816.2.c.c.577.1 2
3.2 odd 2 2448.2.c.j.577.2 2
4.3 odd 2 51.2.d.b.16.2 yes 2
8.3 odd 2 3264.2.c.e.577.1 2
8.5 even 2 3264.2.c.d.577.2 2
12.11 even 2 153.2.d.a.118.1 2
17.16 even 2 inner 816.2.c.c.577.2 2
20.3 even 4 1275.2.d.b.424.1 2
20.7 even 4 1275.2.d.d.424.2 2
20.19 odd 2 1275.2.g.a.526.1 2
51.50 odd 2 2448.2.c.j.577.1 2
68.3 even 16 867.2.h.d.688.1 8
68.7 even 16 867.2.h.d.733.2 8
68.11 even 16 867.2.h.d.712.1 8
68.15 odd 8 867.2.e.d.829.1 4
68.19 odd 8 867.2.e.d.829.2 4
68.23 even 16 867.2.h.d.712.2 8
68.27 even 16 867.2.h.d.733.1 8
68.31 even 16 867.2.h.d.688.2 8
68.39 even 16 867.2.h.d.757.1 8
68.43 odd 8 867.2.e.d.616.1 4
68.47 odd 4 867.2.a.a.1.1 1
68.55 odd 4 867.2.a.b.1.1 1
68.59 odd 8 867.2.e.d.616.2 4
68.63 even 16 867.2.h.d.757.2 8
68.67 odd 2 51.2.d.b.16.1 2
136.67 odd 2 3264.2.c.e.577.2 2
136.101 even 2 3264.2.c.d.577.1 2
204.47 even 4 2601.2.a.i.1.1 1
204.191 even 4 2601.2.a.j.1.1 1
204.203 even 2 153.2.d.a.118.2 2
340.67 even 4 1275.2.d.b.424.2 2
340.203 even 4 1275.2.d.d.424.1 2
340.339 odd 2 1275.2.g.a.526.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
51.2.d.b.16.1 2 68.67 odd 2
51.2.d.b.16.2 yes 2 4.3 odd 2
153.2.d.a.118.1 2 12.11 even 2
153.2.d.a.118.2 2 204.203 even 2
816.2.c.c.577.1 2 1.1 even 1 trivial
816.2.c.c.577.2 2 17.16 even 2 inner
867.2.a.a.1.1 1 68.47 odd 4
867.2.a.b.1.1 1 68.55 odd 4
867.2.e.d.616.1 4 68.43 odd 8
867.2.e.d.616.2 4 68.59 odd 8
867.2.e.d.829.1 4 68.15 odd 8
867.2.e.d.829.2 4 68.19 odd 8
867.2.h.d.688.1 8 68.3 even 16
867.2.h.d.688.2 8 68.31 even 16
867.2.h.d.712.1 8 68.11 even 16
867.2.h.d.712.2 8 68.23 even 16
867.2.h.d.733.1 8 68.27 even 16
867.2.h.d.733.2 8 68.7 even 16
867.2.h.d.757.1 8 68.39 even 16
867.2.h.d.757.2 8 68.63 even 16
1275.2.d.b.424.1 2 20.3 even 4
1275.2.d.b.424.2 2 340.67 even 4
1275.2.d.d.424.1 2 340.203 even 4
1275.2.d.d.424.2 2 20.7 even 4
1275.2.g.a.526.1 2 20.19 odd 2
1275.2.g.a.526.2 2 340.339 odd 2
2448.2.c.j.577.1 2 51.50 odd 2
2448.2.c.j.577.2 2 3.2 odd 2
2601.2.a.i.1.1 1 204.47 even 4
2601.2.a.j.1.1 1 204.191 even 4
3264.2.c.d.577.1 2 136.101 even 2
3264.2.c.d.577.2 2 8.5 even 2
3264.2.c.e.577.1 2 8.3 odd 2
3264.2.c.e.577.2 2 136.67 odd 2