Properties

Label 8820.2.d.b.881.2
Level $8820$
Weight $2$
Character 8820.881
Analytic conductor $70.428$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8820,2,Mod(881,8820)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8820, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("8820.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8820 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8820.d (of order \(2\), degree \(1\), 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: \(70.4280545828\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 9 x^{10} + 58 x^{9} - 78 x^{8} - 298 x^{7} + 1341 x^{6} - 2086 x^{5} - 3822 x^{4} + 19894 x^{3} - 21609 x^{2} - 33614 x + 117649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1260)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.2
Root \(0.260926 + 2.63285i\) of defining polynomial
Character \(\chi\) \(=\) 8820.881
Dual form 8820.2.d.b.881.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +O(q^{10})\) \(q+1.00000 q^{5} -3.53366i q^{11} +0.599777i q^{13} -1.69803 q^{17} +7.78350i q^{19} +3.31319i q^{23} +1.00000 q^{25} -0.456069i q^{29} -1.05585i q^{31} -0.386249 q^{37} -0.478149 q^{41} -4.21237 q^{43} +8.55804 q^{47} -3.98973i q^{53} -3.53366i q^{55} -2.60613 q^{59} +11.0895i q^{61} +0.599777i q^{65} +2.87422 q^{67} +0.336875i q^{71} +3.08479i q^{73} -13.6089 q^{79} -16.2901 q^{83} -1.69803 q^{85} -10.6964 q^{89} +7.78350i q^{95} +2.18092i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{5} + 12 q^{25} + 4 q^{37} - 8 q^{41} + 36 q^{43} + 32 q^{47} - 4 q^{67} - 28 q^{79} + 40 q^{83} + 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8820\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(4411\) \(7057\) \(7841\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.53366i − 1.06544i −0.846292 0.532719i \(-0.821170\pi\)
0.846292 0.532719i \(-0.178830\pi\)
\(12\) 0 0
\(13\) 0.599777i 0.166348i 0.996535 + 0.0831742i \(0.0265058\pi\)
−0.996535 + 0.0831742i \(0.973494\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.69803 −0.411834 −0.205917 0.978569i \(-0.566018\pi\)
−0.205917 + 0.978569i \(0.566018\pi\)
\(18\) 0 0
\(19\) 7.78350i 1.78566i 0.450397 + 0.892829i \(0.351283\pi\)
−0.450397 + 0.892829i \(0.648717\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31319i 0.690848i 0.938447 + 0.345424i \(0.112265\pi\)
−0.938447 + 0.345424i \(0.887735\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.456069i − 0.0846898i −0.999103 0.0423449i \(-0.986517\pi\)
0.999103 0.0423449i \(-0.0134828\pi\)
\(30\) 0 0
\(31\) − 1.05585i − 0.189636i −0.995495 0.0948178i \(-0.969773\pi\)
0.995495 0.0948178i \(-0.0302268\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.386249 −0.0634989 −0.0317494 0.999496i \(-0.510108\pi\)
−0.0317494 + 0.999496i \(0.510108\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.478149 −0.0746743 −0.0373372 0.999303i \(-0.511888\pi\)
−0.0373372 + 0.999303i \(0.511888\pi\)
\(42\) 0 0
\(43\) −4.21237 −0.642381 −0.321190 0.947015i \(-0.604083\pi\)
−0.321190 + 0.947015i \(0.604083\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.55804 1.24832 0.624159 0.781297i \(-0.285442\pi\)
0.624159 + 0.781297i \(0.285442\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 3.98973i − 0.548031i −0.961725 0.274015i \(-0.911648\pi\)
0.961725 0.274015i \(-0.0883519\pi\)
\(54\) 0 0
\(55\) − 3.53366i − 0.476478i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.60613 −0.339290 −0.169645 0.985505i \(-0.554262\pi\)
−0.169645 + 0.985505i \(0.554262\pi\)
\(60\) 0 0
\(61\) 11.0895i 1.41986i 0.704271 + 0.709932i \(0.251274\pi\)
−0.704271 + 0.709932i \(0.748726\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.599777i 0.0743932i
\(66\) 0 0
\(67\) 2.87422 0.351142 0.175571 0.984467i \(-0.443823\pi\)
0.175571 + 0.984467i \(0.443823\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.336875i 0.0399797i 0.999800 + 0.0199898i \(0.00636339\pi\)
−0.999800 + 0.0199898i \(0.993637\pi\)
\(72\) 0 0
\(73\) 3.08479i 0.361047i 0.983571 + 0.180524i \(0.0577792\pi\)
−0.983571 + 0.180524i \(0.942221\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.6089 −1.53112 −0.765562 0.643362i \(-0.777539\pi\)
−0.765562 + 0.643362i \(0.777539\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −16.2901 −1.78807 −0.894036 0.447996i \(-0.852138\pi\)
−0.894036 + 0.447996i \(0.852138\pi\)
\(84\) 0 0
\(85\) −1.69803 −0.184178
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.6964 −1.13382 −0.566910 0.823779i \(-0.691861\pi\)
−0.566910 + 0.823779i \(0.691861\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.78350i 0.798570i
\(96\) 0 0
\(97\) 2.18092i 0.221439i 0.993852 + 0.110719i \(0.0353155\pi\)
−0.993852 + 0.110719i \(0.964685\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.70835 0.767009 0.383505 0.923539i \(-0.374717\pi\)
0.383505 + 0.923539i \(0.374717\pi\)
\(102\) 0 0
\(103\) 0.734562i 0.0723785i 0.999345 + 0.0361893i \(0.0115219\pi\)
−0.999345 + 0.0361893i \(0.988478\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.9774i 1.93129i 0.259867 + 0.965645i \(0.416321\pi\)
−0.259867 + 0.965645i \(0.583679\pi\)
\(108\) 0 0
\(109\) −0.443025 −0.0424341 −0.0212170 0.999775i \(-0.506754\pi\)
−0.0212170 + 0.999775i \(0.506754\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.6235i 1.37566i 0.725871 + 0.687830i \(0.241437\pi\)
−0.725871 + 0.687830i \(0.758563\pi\)
\(114\) 0 0
\(115\) 3.31319i 0.308957i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.48673 −0.135157
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 0.961155 0.0852887 0.0426444 0.999090i \(-0.486422\pi\)
0.0426444 + 0.999090i \(0.486422\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.9947 1.39746 0.698731 0.715384i \(-0.253748\pi\)
0.698731 + 0.715384i \(0.253748\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.59969i 0.820157i 0.912050 + 0.410078i \(0.134499\pi\)
−0.912050 + 0.410078i \(0.865501\pi\)
\(138\) 0 0
\(139\) − 9.13902i − 0.775162i −0.921836 0.387581i \(-0.873311\pi\)
0.921836 0.387581i \(-0.126689\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.11941 0.177234
\(144\) 0 0
\(145\) − 0.456069i − 0.0378745i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 20.2697i − 1.66056i −0.557347 0.830279i \(-0.688181\pi\)
0.557347 0.830279i \(-0.311819\pi\)
\(150\) 0 0
\(151\) −8.01031 −0.651870 −0.325935 0.945392i \(-0.605679\pi\)
−0.325935 + 0.945392i \(0.605679\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.05585i − 0.0848076i
\(156\) 0 0
\(157\) − 9.33186i − 0.744763i −0.928080 0.372382i \(-0.878541\pi\)
0.928080 0.372382i \(-0.121459\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −13.6537 −1.06944 −0.534720 0.845029i \(-0.679583\pi\)
−0.534720 + 0.845029i \(0.679583\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.94429 0.769512 0.384756 0.923018i \(-0.374286\pi\)
0.384756 + 0.923018i \(0.374286\pi\)
\(168\) 0 0
\(169\) 12.6403 0.972328
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.93099 0.755039 0.377520 0.926002i \(-0.376777\pi\)
0.377520 + 0.926002i \(0.376777\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 18.8485i 1.40880i 0.709803 + 0.704400i \(0.248784\pi\)
−0.709803 + 0.704400i \(0.751216\pi\)
\(180\) 0 0
\(181\) 20.0424i 1.48974i 0.667211 + 0.744869i \(0.267488\pi\)
−0.667211 + 0.744869i \(0.732512\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.386249 −0.0283976
\(186\) 0 0
\(187\) 6.00027i 0.438783i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.22725i 0.667660i 0.942633 + 0.333830i \(0.108341\pi\)
−0.942633 + 0.333830i \(0.891659\pi\)
\(192\) 0 0
\(193\) −26.0346 −1.87401 −0.937007 0.349310i \(-0.886416\pi\)
−0.937007 + 0.349310i \(0.886416\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 15.6156i − 1.11256i −0.830994 0.556282i \(-0.812228\pi\)
0.830994 0.556282i \(-0.187772\pi\)
\(198\) 0 0
\(199\) 6.11946i 0.433797i 0.976194 + 0.216899i \(0.0695941\pi\)
−0.976194 + 0.216899i \(0.930406\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.478149 −0.0333954
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 27.5042 1.90251
\(210\) 0 0
\(211\) 26.7224 1.83964 0.919822 0.392337i \(-0.128333\pi\)
0.919822 + 0.392337i \(0.128333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.21237 −0.287281
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 1.01844i − 0.0685078i
\(222\) 0 0
\(223\) 20.3668i 1.36386i 0.731416 + 0.681932i \(0.238860\pi\)
−0.731416 + 0.681932i \(0.761140\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19.9390 1.32340 0.661698 0.749770i \(-0.269836\pi\)
0.661698 + 0.749770i \(0.269836\pi\)
\(228\) 0 0
\(229\) 18.9389i 1.25152i 0.780017 + 0.625758i \(0.215210\pi\)
−0.780017 + 0.625758i \(0.784790\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.5531i 1.73955i 0.493448 + 0.869775i \(0.335736\pi\)
−0.493448 + 0.869775i \(0.664264\pi\)
\(234\) 0 0
\(235\) 8.55804 0.558265
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 6.76076i − 0.437317i −0.975801 0.218659i \(-0.929832\pi\)
0.975801 0.218659i \(-0.0701681\pi\)
\(240\) 0 0
\(241\) − 12.0844i − 0.778423i −0.921148 0.389211i \(-0.872748\pi\)
0.921148 0.389211i \(-0.127252\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.66837 −0.297041
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.3410 1.09455 0.547276 0.836952i \(-0.315665\pi\)
0.547276 + 0.836952i \(0.315665\pi\)
\(252\) 0 0
\(253\) 11.7077 0.736055
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.34255 0.520394 0.260197 0.965556i \(-0.416213\pi\)
0.260197 + 0.965556i \(0.416213\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.4187i 0.827433i 0.910406 + 0.413717i \(0.135770\pi\)
−0.910406 + 0.413717i \(0.864230\pi\)
\(264\) 0 0
\(265\) − 3.98973i − 0.245087i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.96289 −0.485506 −0.242753 0.970088i \(-0.578050\pi\)
−0.242753 + 0.970088i \(0.578050\pi\)
\(270\) 0 0
\(271\) 7.46942i 0.453735i 0.973926 + 0.226867i \(0.0728484\pi\)
−0.973926 + 0.226867i \(0.927152\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 3.53366i − 0.213087i
\(276\) 0 0
\(277\) −31.2617 −1.87833 −0.939166 0.343463i \(-0.888400\pi\)
−0.939166 + 0.343463i \(0.888400\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 23.4956i − 1.40163i −0.713343 0.700815i \(-0.752820\pi\)
0.713343 0.700815i \(-0.247180\pi\)
\(282\) 0 0
\(283\) 18.5127i 1.10046i 0.835012 + 0.550232i \(0.185461\pi\)
−0.835012 + 0.550232i \(0.814539\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −14.1167 −0.830393
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −14.7388 −0.861048 −0.430524 0.902579i \(-0.641671\pi\)
−0.430524 + 0.902579i \(0.641671\pi\)
\(294\) 0 0
\(295\) −2.60613 −0.151735
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.98718 −0.114921
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.0895i 0.634982i
\(306\) 0 0
\(307\) 17.6583i 1.00781i 0.863758 + 0.503906i \(0.168104\pi\)
−0.863758 + 0.503906i \(0.831896\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.87563 0.163062 0.0815310 0.996671i \(-0.474019\pi\)
0.0815310 + 0.996671i \(0.474019\pi\)
\(312\) 0 0
\(313\) − 20.0386i − 1.13265i −0.824183 0.566324i \(-0.808365\pi\)
0.824183 0.566324i \(-0.191635\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 20.5910i − 1.15650i −0.815858 0.578252i \(-0.803735\pi\)
0.815858 0.578252i \(-0.196265\pi\)
\(318\) 0 0
\(319\) −1.61159 −0.0902317
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 13.2166i − 0.735394i
\(324\) 0 0
\(325\) 0.599777i 0.0332697i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −22.8995 −1.25867 −0.629335 0.777134i \(-0.716673\pi\)
−0.629335 + 0.777134i \(0.716673\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.87422 0.157035
\(336\) 0 0
\(337\) 17.7305 0.965840 0.482920 0.875664i \(-0.339576\pi\)
0.482920 + 0.875664i \(0.339576\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3.73100 −0.202045
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 18.1899i − 0.976486i −0.872708 0.488243i \(-0.837638\pi\)
0.872708 0.488243i \(-0.162362\pi\)
\(348\) 0 0
\(349\) − 4.44213i − 0.237782i −0.992907 0.118891i \(-0.962066\pi\)
0.992907 0.118891i \(-0.0379339\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.18270 −0.275848 −0.137924 0.990443i \(-0.544043\pi\)
−0.137924 + 0.990443i \(0.544043\pi\)
\(354\) 0 0
\(355\) 0.336875i 0.0178795i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.2319i 0.962244i 0.876654 + 0.481122i \(0.159771\pi\)
−0.876654 + 0.481122i \(0.840229\pi\)
\(360\) 0 0
\(361\) −41.5829 −2.18857
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.08479i 0.161465i
\(366\) 0 0
\(367\) 20.0597i 1.04711i 0.851993 + 0.523553i \(0.175394\pi\)
−0.851993 + 0.523553i \(0.824606\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.5889 1.37672 0.688361 0.725368i \(-0.258331\pi\)
0.688361 + 0.725368i \(0.258331\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.273540 0.0140880
\(378\) 0 0
\(379\) −18.3623 −0.943210 −0.471605 0.881810i \(-0.656325\pi\)
−0.471605 + 0.881810i \(0.656325\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 36.8099 1.88090 0.940449 0.339935i \(-0.110405\pi\)
0.940449 + 0.339935i \(0.110405\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 31.6392i 1.60417i 0.597211 + 0.802084i \(0.296276\pi\)
−0.597211 + 0.802084i \(0.703724\pi\)
\(390\) 0 0
\(391\) − 5.62591i − 0.284515i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −13.6089 −0.684740
\(396\) 0 0
\(397\) 26.9858i 1.35438i 0.735808 + 0.677190i \(0.236802\pi\)
−0.735808 + 0.677190i \(0.763198\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.80307i 0.489542i 0.969581 + 0.244771i \(0.0787127\pi\)
−0.969581 + 0.244771i \(0.921287\pi\)
\(402\) 0 0
\(403\) 0.633273 0.0315456
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.36487i 0.0676541i
\(408\) 0 0
\(409\) 8.57996i 0.424252i 0.977242 + 0.212126i \(0.0680387\pi\)
−0.977242 + 0.212126i \(0.931961\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −16.2901 −0.799650
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.7882 1.06443 0.532213 0.846611i \(-0.321361\pi\)
0.532213 + 0.846611i \(0.321361\pi\)
\(420\) 0 0
\(421\) −24.2280 −1.18080 −0.590400 0.807111i \(-0.701030\pi\)
−0.590400 + 0.807111i \(0.701030\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.69803 −0.0823668
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 23.8330i − 1.14800i −0.818857 0.573998i \(-0.805392\pi\)
0.818857 0.573998i \(-0.194608\pi\)
\(432\) 0 0
\(433\) − 1.39596i − 0.0670855i −0.999437 0.0335427i \(-0.989321\pi\)
0.999437 0.0335427i \(-0.0106790\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.7882 −1.23362
\(438\) 0 0
\(439\) 11.8879i 0.567381i 0.958916 + 0.283690i \(0.0915588\pi\)
−0.958916 + 0.283690i \(0.908441\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 26.7403i 1.27047i 0.772319 + 0.635235i \(0.219097\pi\)
−0.772319 + 0.635235i \(0.780903\pi\)
\(444\) 0 0
\(445\) −10.6964 −0.507060
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 32.8394i − 1.54979i −0.632091 0.774895i \(-0.717803\pi\)
0.632091 0.774895i \(-0.282197\pi\)
\(450\) 0 0
\(451\) 1.68961i 0.0795608i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.0454 −1.63935 −0.819677 0.572826i \(-0.805847\pi\)
−0.819677 + 0.572826i \(0.805847\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −8.90991 −0.414976 −0.207488 0.978238i \(-0.566529\pi\)
−0.207488 + 0.978238i \(0.566529\pi\)
\(462\) 0 0
\(463\) −10.2000 −0.474036 −0.237018 0.971505i \(-0.576170\pi\)
−0.237018 + 0.971505i \(0.576170\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.8383 −0.779185 −0.389592 0.920987i \(-0.627384\pi\)
−0.389592 + 0.920987i \(0.627384\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14.8851i 0.684417i
\(474\) 0 0
\(475\) 7.78350i 0.357131i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.3703 0.839360 0.419680 0.907672i \(-0.362142\pi\)
0.419680 + 0.907672i \(0.362142\pi\)
\(480\) 0 0
\(481\) − 0.231663i − 0.0105629i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.18092i 0.0990304i
\(486\) 0 0
\(487\) −11.6639 −0.528540 −0.264270 0.964449i \(-0.585131\pi\)
−0.264270 + 0.964449i \(0.585131\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.4694i 0.968902i 0.874818 + 0.484451i \(0.160981\pi\)
−0.874818 + 0.484451i \(0.839019\pi\)
\(492\) 0 0
\(493\) 0.774420i 0.0348781i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −14.8218 −0.663515 −0.331757 0.943365i \(-0.607642\pi\)
−0.331757 + 0.943365i \(0.607642\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −28.9563 −1.29110 −0.645548 0.763719i \(-0.723371\pi\)
−0.645548 + 0.763719i \(0.723371\pi\)
\(504\) 0 0
\(505\) 7.70835 0.343017
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 29.1793 1.29335 0.646675 0.762766i \(-0.276159\pi\)
0.646675 + 0.762766i \(0.276159\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.734562i 0.0323687i
\(516\) 0 0
\(517\) − 30.2412i − 1.33001i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.6208 1.34152 0.670761 0.741674i \(-0.265968\pi\)
0.670761 + 0.741674i \(0.265968\pi\)
\(522\) 0 0
\(523\) 12.0824i 0.528324i 0.964478 + 0.264162i \(0.0850954\pi\)
−0.964478 + 0.264162i \(0.914905\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.79286i 0.0780983i
\(528\) 0 0
\(529\) 12.0228 0.522729
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 0.286783i − 0.0124219i
\(534\) 0 0
\(535\) 19.9774i 0.863699i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.54879 0.238561 0.119281 0.992861i \(-0.461941\pi\)
0.119281 + 0.992861i \(0.461941\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.443025 −0.0189771
\(546\) 0 0
\(547\) 4.60440 0.196870 0.0984349 0.995143i \(-0.468616\pi\)
0.0984349 + 0.995143i \(0.468616\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.54981 0.151227
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 15.7009i − 0.665269i −0.943056 0.332634i \(-0.892063\pi\)
0.943056 0.332634i \(-0.107937\pi\)
\(558\) 0 0
\(559\) − 2.52649i − 0.106859i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −32.6851 −1.37751 −0.688756 0.724993i \(-0.741843\pi\)
−0.688756 + 0.724993i \(0.741843\pi\)
\(564\) 0 0
\(565\) 14.6235i 0.615214i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.1594i 0.970892i 0.874267 + 0.485446i \(0.161343\pi\)
−0.874267 + 0.485446i \(0.838657\pi\)
\(570\) 0 0
\(571\) −3.90494 −0.163417 −0.0817083 0.996656i \(-0.526038\pi\)
−0.0817083 + 0.996656i \(0.526038\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.31319i 0.138170i
\(576\) 0 0
\(577\) − 7.67922i − 0.319690i −0.987142 0.159845i \(-0.948901\pi\)
0.987142 0.159845i \(-0.0510995\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −14.0983 −0.583893
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −0.375586 −0.0155021 −0.00775105 0.999970i \(-0.502467\pi\)
−0.00775105 + 0.999970i \(0.502467\pi\)
\(588\) 0 0
\(589\) 8.21818 0.338624
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.0404 0.946157 0.473078 0.881020i \(-0.343143\pi\)
0.473078 + 0.881020i \(0.343143\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.61675i 0.352071i 0.984384 + 0.176036i \(0.0563274\pi\)
−0.984384 + 0.176036i \(0.943673\pi\)
\(600\) 0 0
\(601\) − 27.4954i − 1.12156i −0.827965 0.560780i \(-0.810501\pi\)
0.827965 0.560780i \(-0.189499\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.48673 −0.0604440
\(606\) 0 0
\(607\) 35.9277i 1.45826i 0.684375 + 0.729130i \(0.260075\pi\)
−0.684375 + 0.729130i \(0.739925\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 5.13292i 0.207656i
\(612\) 0 0
\(613\) 41.8542 1.69048 0.845238 0.534389i \(-0.179458\pi\)
0.845238 + 0.534389i \(0.179458\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 27.6411i 1.11279i 0.830918 + 0.556395i \(0.187816\pi\)
−0.830918 + 0.556395i \(0.812184\pi\)
\(618\) 0 0
\(619\) − 13.2580i − 0.532885i −0.963851 0.266442i \(-0.914152\pi\)
0.963851 0.266442i \(-0.0858482\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.655863 0.0261510
\(630\) 0 0
\(631\) −12.2455 −0.487485 −0.243742 0.969840i \(-0.578375\pi\)
−0.243742 + 0.969840i \(0.578375\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.961155 0.0381423
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 12.3916i − 0.489441i −0.969594 0.244720i \(-0.921304\pi\)
0.969594 0.244720i \(-0.0786962\pi\)
\(642\) 0 0
\(643\) 42.8847i 1.69121i 0.533811 + 0.845604i \(0.320759\pi\)
−0.533811 + 0.845604i \(0.679241\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 26.8700 1.05637 0.528185 0.849130i \(-0.322873\pi\)
0.528185 + 0.849130i \(0.322873\pi\)
\(648\) 0 0
\(649\) 9.20918i 0.361492i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.622092i 0.0243444i 0.999926 + 0.0121722i \(0.00387462\pi\)
−0.999926 + 0.0121722i \(0.996125\pi\)
\(654\) 0 0
\(655\) 15.9947 0.624964
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.9996i 0.701167i 0.936532 + 0.350583i \(0.114017\pi\)
−0.936532 + 0.350583i \(0.885983\pi\)
\(660\) 0 0
\(661\) − 30.0737i − 1.16973i −0.811130 0.584866i \(-0.801147\pi\)
0.811130 0.584866i \(-0.198853\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.51104 0.0585078
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 39.1864 1.51278
\(672\) 0 0
\(673\) 39.4637 1.52121 0.760606 0.649213i \(-0.224902\pi\)
0.760606 + 0.649213i \(0.224902\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.6775 0.756267 0.378134 0.925751i \(-0.376566\pi\)
0.378134 + 0.925751i \(0.376566\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0223i 0.689603i 0.938676 + 0.344801i \(0.112054\pi\)
−0.938676 + 0.344801i \(0.887946\pi\)
\(684\) 0 0
\(685\) 9.59969i 0.366785i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.39295 0.0911640
\(690\) 0 0
\(691\) 27.2057i 1.03496i 0.855697 + 0.517478i \(0.173129\pi\)
−0.855697 + 0.517478i \(0.826871\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 9.13902i − 0.346663i
\(696\) 0 0
\(697\) 0.811913 0.0307534
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 33.1556i − 1.25227i −0.779715 0.626134i \(-0.784636\pi\)
0.779715 0.626134i \(-0.215364\pi\)
\(702\) 0 0
\(703\) − 3.00637i − 0.113387i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.97919 0.0743301 0.0371650 0.999309i \(-0.488167\pi\)
0.0371650 + 0.999309i \(0.488167\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.49822 0.131009
\(714\) 0 0
\(715\) 2.11941 0.0792613
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.3589 1.13220 0.566098 0.824338i \(-0.308452\pi\)
0.566098 + 0.824338i \(0.308452\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 0.456069i − 0.0169380i
\(726\) 0 0
\(727\) 17.5883i 0.652313i 0.945316 + 0.326156i \(0.105754\pi\)
−0.945316 + 0.326156i \(0.894246\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.15275 0.264554
\(732\) 0 0
\(733\) 36.9445i 1.36457i 0.731084 + 0.682287i \(0.239015\pi\)
−0.731084 + 0.682287i \(0.760985\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 10.1565i − 0.374119i
\(738\) 0 0
\(739\) −5.05915 −0.186104 −0.0930520 0.995661i \(-0.529662\pi\)
−0.0930520 + 0.995661i \(0.529662\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.19725i 0.227355i 0.993518 + 0.113678i \(0.0362631\pi\)
−0.993518 + 0.113678i \(0.963737\pi\)
\(744\) 0 0
\(745\) − 20.2697i − 0.742624i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −37.8201 −1.38007 −0.690037 0.723774i \(-0.742406\pi\)
−0.690037 + 0.723774i \(0.742406\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.01031 −0.291525
\(756\) 0 0
\(757\) −13.7436 −0.499519 −0.249760 0.968308i \(-0.580352\pi\)
−0.249760 + 0.968308i \(0.580352\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.39947 0.0869807 0.0434904 0.999054i \(-0.486152\pi\)
0.0434904 + 0.999054i \(0.486152\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1.56310i − 0.0564403i
\(768\) 0 0
\(769\) − 9.42300i − 0.339802i −0.985461 0.169901i \(-0.945655\pi\)
0.985461 0.169901i \(-0.0543448\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −22.5676 −0.811700 −0.405850 0.913940i \(-0.633025\pi\)
−0.405850 + 0.913940i \(0.633025\pi\)
\(774\) 0 0
\(775\) − 1.05585i − 0.0379271i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 3.72167i − 0.133343i
\(780\) 0 0
\(781\) 1.19040 0.0425959
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 9.33186i − 0.333068i
\(786\) 0 0
\(787\) − 25.9146i − 0.923757i −0.886943 0.461878i \(-0.847176\pi\)
0.886943 0.461878i \(-0.152824\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.65122 −0.236192
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.94772 −0.246101 −0.123050 0.992400i \(-0.539268\pi\)
−0.123050 + 0.992400i \(0.539268\pi\)
\(798\) 0 0
\(799\) −14.5318 −0.514100
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.9006 0.384673
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 33.3973i − 1.17419i −0.809520 0.587093i \(-0.800272\pi\)
0.809520 0.587093i \(-0.199728\pi\)
\(810\) 0 0
\(811\) − 28.9701i − 1.01728i −0.860979 0.508640i \(-0.830149\pi\)
0.860979 0.508640i \(-0.169851\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13.6537 −0.478268
\(816\) 0 0
\(817\) − 32.7870i − 1.14707i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.60759i 0.0910056i 0.998964 + 0.0455028i \(0.0144890\pi\)
−0.998964 + 0.0455028i \(0.985511\pi\)
\(822\) 0 0
\(823\) 8.45695 0.294791 0.147395 0.989078i \(-0.452911\pi\)
0.147395 + 0.989078i \(0.452911\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 13.3362i − 0.463744i −0.972746 0.231872i \(-0.925515\pi\)
0.972746 0.231872i \(-0.0744851\pi\)
\(828\) 0 0
\(829\) 11.6374i 0.404183i 0.979367 + 0.202091i \(0.0647738\pi\)
−0.979367 + 0.202091i \(0.935226\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 9.94429 0.344136
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −19.9335 −0.688181 −0.344090 0.938937i \(-0.611813\pi\)
−0.344090 + 0.938937i \(0.611813\pi\)
\(840\) 0 0
\(841\) 28.7920 0.992828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.6403 0.434838
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1.27972i − 0.0438681i
\(852\) 0 0
\(853\) 2.08233i 0.0712975i 0.999364 + 0.0356487i \(0.0113498\pi\)
−0.999364 + 0.0356487i \(0.988650\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 46.5482 1.59006 0.795028 0.606573i \(-0.207456\pi\)
0.795028 + 0.606573i \(0.207456\pi\)
\(858\) 0 0
\(859\) − 5.34497i − 0.182368i −0.995834 0.0911840i \(-0.970935\pi\)
0.995834 0.0911840i \(-0.0290651\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 13.3070i − 0.452976i −0.974014 0.226488i \(-0.927276\pi\)
0.974014 0.226488i \(-0.0727243\pi\)
\(864\) 0 0
\(865\) 9.93099 0.337664
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 48.0893i 1.63132i
\(870\) 0 0
\(871\) 1.72389i 0.0584118i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −19.7845 −0.668075 −0.334037 0.942560i \(-0.608411\pi\)
−0.334037 + 0.942560i \(0.608411\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −34.3203 −1.15628 −0.578141 0.815937i \(-0.696222\pi\)
−0.578141 + 0.815937i \(0.696222\pi\)
\(882\) 0 0
\(883\) 28.1109 0.946006 0.473003 0.881061i \(-0.343170\pi\)
0.473003 + 0.881061i \(0.343170\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −41.7652 −1.40234 −0.701169 0.712995i \(-0.747338\pi\)
−0.701169 + 0.712995i \(0.747338\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 66.6115i 2.22907i
\(894\) 0 0
\(895\) 18.8485i 0.630035i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.481538 −0.0160602
\(900\) 0 0
\(901\) 6.77469i 0.225698i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 20.0424i 0.666231i
\(906\) 0 0
\(907\) −7.92101 −0.263013 −0.131506 0.991315i \(-0.541981\pi\)
−0.131506 + 0.991315i \(0.541981\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 49.9301i 1.65426i 0.562012 + 0.827129i \(0.310027\pi\)
−0.562012 + 0.827129i \(0.689973\pi\)
\(912\) 0 0
\(913\) 57.5636i 1.90508i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −0.264145 −0.00871335 −0.00435668 0.999991i \(-0.501387\pi\)
−0.00435668 + 0.999991i \(0.501387\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −0.202050 −0.00665055
\(924\) 0 0
\(925\) −0.386249 −0.0126998
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −56.8692 −1.86582 −0.932910 0.360110i \(-0.882739\pi\)
−0.932910 + 0.360110i \(0.882739\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 6.00027i 0.196230i
\(936\) 0 0
\(937\) − 37.3673i − 1.22074i −0.792118 0.610369i \(-0.791021\pi\)
0.792118 0.610369i \(-0.208979\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 47.7334 1.55607 0.778033 0.628224i \(-0.216218\pi\)
0.778033 + 0.628224i \(0.216218\pi\)
\(942\) 0 0
\(943\) − 1.58420i − 0.0515886i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 33.1741i 1.07801i 0.842302 + 0.539006i \(0.181200\pi\)
−0.842302 + 0.539006i \(0.818800\pi\)
\(948\) 0 0
\(949\) −1.85019 −0.0600596
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 28.1591i − 0.912164i −0.889938 0.456082i \(-0.849252\pi\)
0.889938 0.456082i \(-0.150748\pi\)
\(954\) 0 0
\(955\) 9.22725i 0.298587i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 29.8852 0.964038
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −26.0346 −0.838085
\(966\) 0 0
\(967\) −53.8983 −1.73325 −0.866626 0.498959i \(-0.833716\pi\)
−0.866626 + 0.498959i \(0.833716\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −41.0531 −1.31746 −0.658729 0.752380i \(-0.728906\pi\)
−0.658729 + 0.752380i \(0.728906\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.9189i 0.925197i 0.886568 + 0.462598i \(0.153083\pi\)
−0.886568 + 0.462598i \(0.846917\pi\)
\(978\) 0 0
\(979\) 37.7976i 1.20802i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −36.2752 −1.15700 −0.578499 0.815683i \(-0.696361\pi\)
−0.578499 + 0.815683i \(0.696361\pi\)
\(984\) 0 0
\(985\) − 15.6156i − 0.497554i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 13.9564i − 0.443787i
\(990\) 0 0
\(991\) −39.2201 −1.24587 −0.622934 0.782274i \(-0.714060\pi\)
−0.622934 + 0.782274i \(0.714060\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.11946i 0.194000i
\(996\) 0 0
\(997\) − 12.4136i − 0.393142i −0.980490 0.196571i \(-0.937019\pi\)
0.980490 0.196571i \(-0.0629806\pi\)
\(998\) 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 8820.2.d.b.881.2 12
3.2 odd 2 8820.2.d.a.881.11 12
7.4 even 3 1260.2.cg.a.341.3 12
7.5 odd 6 1260.2.cg.b.521.3 yes 12
7.6 odd 2 8820.2.d.a.881.2 12
21.5 even 6 1260.2.cg.a.521.3 yes 12
21.11 odd 6 1260.2.cg.b.341.3 yes 12
21.20 even 2 inner 8820.2.d.b.881.11 12
35.4 even 6 6300.2.ch.c.1601.4 12
35.12 even 12 6300.2.dd.b.4049.1 24
35.18 odd 12 6300.2.dd.c.1349.1 24
35.19 odd 6 6300.2.ch.b.4301.4 12
35.32 odd 12 6300.2.dd.c.1349.12 24
35.33 even 12 6300.2.dd.b.4049.12 24
105.32 even 12 6300.2.dd.b.1349.12 24
105.47 odd 12 6300.2.dd.c.4049.1 24
105.53 even 12 6300.2.dd.b.1349.1 24
105.68 odd 12 6300.2.dd.c.4049.12 24
105.74 odd 6 6300.2.ch.b.1601.4 12
105.89 even 6 6300.2.ch.c.4301.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1260.2.cg.a.341.3 12 7.4 even 3
1260.2.cg.a.521.3 yes 12 21.5 even 6
1260.2.cg.b.341.3 yes 12 21.11 odd 6
1260.2.cg.b.521.3 yes 12 7.5 odd 6
6300.2.ch.b.1601.4 12 105.74 odd 6
6300.2.ch.b.4301.4 12 35.19 odd 6
6300.2.ch.c.1601.4 12 35.4 even 6
6300.2.ch.c.4301.4 12 105.89 even 6
6300.2.dd.b.1349.1 24 105.53 even 12
6300.2.dd.b.1349.12 24 105.32 even 12
6300.2.dd.b.4049.1 24 35.12 even 12
6300.2.dd.b.4049.12 24 35.33 even 12
6300.2.dd.c.1349.1 24 35.18 odd 12
6300.2.dd.c.1349.12 24 35.32 odd 12
6300.2.dd.c.4049.1 24 105.47 odd 12
6300.2.dd.c.4049.12 24 105.68 odd 12
8820.2.d.a.881.2 12 7.6 odd 2
8820.2.d.a.881.11 12 3.2 odd 2
8820.2.d.b.881.2 12 1.1 even 1 trivial
8820.2.d.b.881.11 12 21.20 even 2 inner