Properties

Label 864.2.f.a.431.6
Level $864$
Weight $2$
Character 864.431
Analytic conductor $6.899$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [864,2,Mod(431,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.f (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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.23123460096.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{6} + 6x^{4} + 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 431.6
Root \(-1.08766 + 0.903873i\) of defining polynomial
Character \(\chi\) \(=\) 864.431
Dual form 864.2.f.a.431.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.59245 q^{5} +1.43937i q^{7} +O(q^{10})\) \(q+1.59245 q^{5} +1.43937i q^{7} -4.93886i q^{11} -5.37182i q^{13} -1.32336i q^{17} -0.267949 q^{19} +5.94311 q^{23} -2.46410 q^{25} +8.70131 q^{29} +7.86488i q^{31} +2.29213i q^{35} +2.49307i q^{37} -9.87771i q^{41} -2.00000 q^{43} +9.12801 q^{47} +4.92820 q^{49} -5.51641 q^{53} -7.86488i q^{55} -4.93886i q^{59} +5.37182i q^{61} -8.55435i q^{65} -1.19615 q^{67} -5.51641 q^{71} -7.92820 q^{73} +7.10886 q^{77} -12.1830i q^{79} -7.23099i q^{83} -2.10739i q^{85} +15.7853i q^{89} +7.73205 q^{91} -0.426696 q^{95} +9.39230 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{19} + 8 q^{25} - 16 q^{43} - 16 q^{49} + 32 q^{67} - 8 q^{73} + 48 q^{91} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 1.59245 0.712165 0.356083 0.934454i \(-0.384112\pi\)
0.356083 + 0.934454i \(0.384112\pi\)
\(6\) 0 0
\(7\) 1.43937i 0.544032i 0.962293 + 0.272016i \(0.0876904\pi\)
−0.962293 + 0.272016i \(0.912310\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.93886i − 1.48912i −0.667555 0.744561i \(-0.732659\pi\)
0.667555 0.744561i \(-0.267341\pi\)
\(12\) 0 0
\(13\) − 5.37182i − 1.48987i −0.667135 0.744937i \(-0.732480\pi\)
0.667135 0.744937i \(-0.267520\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.32336i − 0.320963i −0.987039 0.160481i \(-0.948695\pi\)
0.987039 0.160481i \(-0.0513046\pi\)
\(18\) 0 0
\(19\) −0.267949 −0.0614718 −0.0307359 0.999528i \(-0.509785\pi\)
−0.0307359 + 0.999528i \(0.509785\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.94311 1.23922 0.619612 0.784909i \(-0.287290\pi\)
0.619612 + 0.784909i \(0.287290\pi\)
\(24\) 0 0
\(25\) −2.46410 −0.492820
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.70131 1.61579 0.807896 0.589324i \(-0.200606\pi\)
0.807896 + 0.589324i \(0.200606\pi\)
\(30\) 0 0
\(31\) 7.86488i 1.41257i 0.707925 + 0.706287i \(0.249631\pi\)
−0.707925 + 0.706287i \(0.750369\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.29213i 0.387441i
\(36\) 0 0
\(37\) 2.49307i 0.409858i 0.978777 + 0.204929i \(0.0656963\pi\)
−0.978777 + 0.204929i \(0.934304\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 9.87771i − 1.54264i −0.636448 0.771320i \(-0.719597\pi\)
0.636448 0.771320i \(-0.280403\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.12801 1.33146 0.665728 0.746194i \(-0.268121\pi\)
0.665728 + 0.746194i \(0.268121\pi\)
\(48\) 0 0
\(49\) 4.92820 0.704029
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.51641 −0.757737 −0.378869 0.925450i \(-0.623687\pi\)
−0.378869 + 0.925450i \(0.623687\pi\)
\(54\) 0 0
\(55\) − 7.86488i − 1.06050i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4.93886i − 0.642984i −0.946912 0.321492i \(-0.895816\pi\)
0.946912 0.321492i \(-0.104184\pi\)
\(60\) 0 0
\(61\) 5.37182i 0.687791i 0.939008 + 0.343895i \(0.111747\pi\)
−0.939008 + 0.343895i \(0.888253\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 8.55435i − 1.06104i
\(66\) 0 0
\(67\) −1.19615 −0.146133 −0.0730666 0.997327i \(-0.523279\pi\)
−0.0730666 + 0.997327i \(0.523279\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.51641 −0.654677 −0.327339 0.944907i \(-0.606152\pi\)
−0.327339 + 0.944907i \(0.606152\pi\)
\(72\) 0 0
\(73\) −7.92820 −0.927926 −0.463963 0.885855i \(-0.653573\pi\)
−0.463963 + 0.885855i \(0.653573\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.10886 0.810130
\(78\) 0 0
\(79\) − 12.1830i − 1.37070i −0.728216 0.685348i \(-0.759650\pi\)
0.728216 0.685348i \(-0.240350\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7.23099i − 0.793704i −0.917883 0.396852i \(-0.870103\pi\)
0.917883 0.396852i \(-0.129897\pi\)
\(84\) 0 0
\(85\) − 2.10739i − 0.228578i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.7853i 1.67324i 0.547782 + 0.836621i \(0.315472\pi\)
−0.547782 + 0.836621i \(0.684528\pi\)
\(90\) 0 0
\(91\) 7.73205 0.810539
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.426696 −0.0437781
\(96\) 0 0
\(97\) 9.39230 0.953644 0.476822 0.879000i \(-0.341789\pi\)
0.476822 + 0.879000i \(0.341789\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.18490 0.316909 0.158455 0.987366i \(-0.449349\pi\)
0.158455 + 0.987366i \(0.449349\pi\)
\(102\) 0 0
\(103\) − 1.43937i − 0.141826i −0.997483 0.0709129i \(-0.977409\pi\)
0.997483 0.0709129i \(-0.0225912\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.1698i 1.17650i 0.808678 + 0.588252i \(0.200184\pi\)
−0.808678 + 0.588252i \(0.799816\pi\)
\(108\) 0 0
\(109\) 13.6224i 1.30479i 0.757880 + 0.652394i \(0.226235\pi\)
−0.757880 + 0.652394i \(0.773765\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.2011i 1.05371i 0.849956 + 0.526854i \(0.176629\pi\)
−0.849956 + 0.526854i \(0.823371\pi\)
\(114\) 0 0
\(115\) 9.46410 0.882532
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.90481 0.174614
\(120\) 0 0
\(121\) −13.3923 −1.21748
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.8862 −1.06314
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 7.23099i − 0.631774i −0.948797 0.315887i \(-0.897698\pi\)
0.948797 0.315887i \(-0.102302\pi\)
\(132\) 0 0
\(133\) − 0.385679i − 0.0334426i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 5.90763i − 0.504722i −0.967633 0.252361i \(-0.918793\pi\)
0.967633 0.252361i \(-0.0812071\pi\)
\(138\) 0 0
\(139\) −7.19615 −0.610370 −0.305185 0.952293i \(-0.598718\pi\)
−0.305185 + 0.952293i \(0.598718\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −26.5306 −2.21860
\(144\) 0 0
\(145\) 13.8564 1.15071
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.18490 −0.260917 −0.130459 0.991454i \(-0.541645\pi\)
−0.130459 + 0.991454i \(0.541645\pi\)
\(150\) 0 0
\(151\) 6.42551i 0.522901i 0.965217 + 0.261450i \(0.0842008\pi\)
−0.965217 + 0.261450i \(0.915799\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.5244i 1.00599i
\(156\) 0 0
\(157\) 2.10739i 0.168188i 0.996458 + 0.0840940i \(0.0267996\pi\)
−0.996458 + 0.0840940i \(0.973200\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.55435i 0.674177i
\(162\) 0 0
\(163\) −19.1962 −1.50356 −0.751779 0.659415i \(-0.770804\pi\)
−0.751779 + 0.659415i \(0.770804\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.08971 −0.393854 −0.196927 0.980418i \(-0.563096\pi\)
−0.196927 + 0.980418i \(0.563096\pi\)
\(168\) 0 0
\(169\) −15.8564 −1.21972
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.8862 −0.903692 −0.451846 0.892096i \(-0.649234\pi\)
−0.451846 + 0.892096i \(0.649234\pi\)
\(174\) 0 0
\(175\) − 3.54676i − 0.268110i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.87771i 0.738295i 0.929371 + 0.369147i \(0.120350\pi\)
−0.929371 + 0.369147i \(0.879650\pi\)
\(180\) 0 0
\(181\) − 16.1154i − 1.19785i −0.800804 0.598926i \(-0.795594\pi\)
0.800804 0.598926i \(-0.204406\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.97009i 0.291887i
\(186\) 0 0
\(187\) −6.53590 −0.477952
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.12801 0.660479 0.330240 0.943897i \(-0.392870\pi\)
0.330240 + 0.943897i \(0.392870\pi\)
\(192\) 0 0
\(193\) 15.3923 1.10796 0.553981 0.832529i \(-0.313108\pi\)
0.553981 + 0.832529i \(0.313108\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −19.8485 −1.41414 −0.707072 0.707141i \(-0.749984\pi\)
−0.707072 + 0.707141i \(0.749984\pi\)
\(198\) 0 0
\(199\) 14.2904i 1.01302i 0.862234 + 0.506510i \(0.169065\pi\)
−0.862234 + 0.506510i \(0.830935\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.5244i 0.879043i
\(204\) 0 0
\(205\) − 15.7298i − 1.09861i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.32336i 0.0915389i
\(210\) 0 0
\(211\) 17.0526 1.17395 0.586973 0.809606i \(-0.300319\pi\)
0.586973 + 0.809606i \(0.300319\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.18490 −0.217208
\(216\) 0 0
\(217\) −11.3205 −0.768486
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.10886 −0.478194
\(222\) 0 0
\(223\) 19.3799i 1.29777i 0.760885 + 0.648886i \(0.224765\pi\)
−0.760885 + 0.648886i \(0.775235\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.87771i 0.655607i 0.944746 + 0.327803i \(0.106308\pi\)
−0.944746 + 0.327803i \(0.893692\pi\)
\(228\) 0 0
\(229\) 15.7298i 1.03945i 0.854333 + 0.519726i \(0.173966\pi\)
−0.854333 + 0.519726i \(0.826034\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.64673i 0.173393i 0.996235 + 0.0866964i \(0.0276310\pi\)
−0.996235 + 0.0866964i \(0.972369\pi\)
\(234\) 0 0
\(235\) 14.5359 0.948217
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.36980 −0.412028 −0.206014 0.978549i \(-0.566049\pi\)
−0.206014 + 0.978549i \(0.566049\pi\)
\(240\) 0 0
\(241\) 15.3923 0.991506 0.495753 0.868464i \(-0.334892\pi\)
0.495753 + 0.868464i \(0.334892\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.84792 0.501385
\(246\) 0 0
\(247\) 1.43937i 0.0915852i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 2.64673i − 0.167060i −0.996505 0.0835299i \(-0.973381\pi\)
0.996505 0.0835299i \(-0.0266194\pi\)
\(252\) 0 0
\(253\) − 29.3521i − 1.84535i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.87771i − 0.616155i −0.951361 0.308077i \(-0.900314\pi\)
0.951361 0.308077i \(-0.0996856\pi\)
\(258\) 0 0
\(259\) −3.58846 −0.222976
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −23.7724 −1.46587 −0.732935 0.680298i \(-0.761850\pi\)
−0.732935 + 0.680298i \(0.761850\pi\)
\(264\) 0 0
\(265\) −8.78461 −0.539634
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.739059 −0.0450612 −0.0225306 0.999746i \(-0.507172\pi\)
−0.0225306 + 0.999746i \(0.507172\pi\)
\(270\) 0 0
\(271\) 4.31812i 0.262307i 0.991362 + 0.131154i \(0.0418681\pi\)
−0.991362 + 0.131154i \(0.958132\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.1698i 0.733869i
\(276\) 0 0
\(277\) − 2.10739i − 0.126621i −0.997994 0.0633104i \(-0.979834\pi\)
0.997994 0.0633104i \(-0.0201658\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.23099i 0.431365i 0.976464 + 0.215682i \(0.0691975\pi\)
−0.976464 + 0.215682i \(0.930802\pi\)
\(282\) 0 0
\(283\) −0.143594 −0.00853575 −0.00426787 0.999991i \(-0.501359\pi\)
−0.00426787 + 0.999991i \(0.501359\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.2177 0.839246
\(288\) 0 0
\(289\) 15.2487 0.896983
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.10886 0.415304 0.207652 0.978203i \(-0.433418\pi\)
0.207652 + 0.978203i \(0.433418\pi\)
\(294\) 0 0
\(295\) − 7.86488i − 0.457911i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 31.9253i − 1.84629i
\(300\) 0 0
\(301\) − 2.87875i − 0.165928i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.55435i 0.489821i
\(306\) 0 0
\(307\) −7.07180 −0.403609 −0.201804 0.979426i \(-0.564681\pi\)
−0.201804 + 0.979426i \(0.564681\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.9759 0.962616 0.481308 0.876551i \(-0.340162\pi\)
0.481308 + 0.876551i \(0.340162\pi\)
\(312\) 0 0
\(313\) −11.3923 −0.643931 −0.321966 0.946751i \(-0.604344\pi\)
−0.321966 + 0.946751i \(0.604344\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.33151 −0.130951 −0.0654753 0.997854i \(-0.520856\pi\)
−0.0654753 + 0.997854i \(0.520856\pi\)
\(318\) 0 0
\(319\) − 42.9745i − 2.40611i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.354594i 0.0197301i
\(324\) 0 0
\(325\) 13.2367i 0.734240i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 13.1386i 0.724355i
\(330\) 0 0
\(331\) −26.1244 −1.43592 −0.717962 0.696082i \(-0.754925\pi\)
−0.717962 + 0.696082i \(0.754925\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.90481 −0.104071
\(336\) 0 0
\(337\) 9.39230 0.511631 0.255816 0.966726i \(-0.417656\pi\)
0.255816 + 0.966726i \(0.417656\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 38.8435 2.10350
\(342\) 0 0
\(343\) 17.1691i 0.927047i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 19.7554i − 1.06053i −0.847833 0.530263i \(-0.822093\pi\)
0.847833 0.530263i \(-0.177907\pi\)
\(348\) 0 0
\(349\) 23.9803i 1.28364i 0.766856 + 0.641819i \(0.221820\pi\)
−0.766856 + 0.641819i \(0.778180\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.7554i 1.05148i 0.850647 + 0.525738i \(0.176211\pi\)
−0.850647 + 0.525738i \(0.823789\pi\)
\(354\) 0 0
\(355\) −8.78461 −0.466239
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.79650 0.358705 0.179353 0.983785i \(-0.442600\pi\)
0.179353 + 0.983785i \(0.442600\pi\)
\(360\) 0 0
\(361\) −18.9282 −0.996221
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −12.6253 −0.660837
\(366\) 0 0
\(367\) 16.3978i 0.855957i 0.903789 + 0.427979i \(0.140774\pi\)
−0.903789 + 0.427979i \(0.859226\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 7.94018i − 0.412233i
\(372\) 0 0
\(373\) 18.2228i 0.943543i 0.881721 + 0.471771i \(0.156385\pi\)
−0.881721 + 0.471771i \(0.843615\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 46.7418i − 2.40733i
\(378\) 0 0
\(379\) 29.7321 1.52723 0.763616 0.645670i \(-0.223422\pi\)
0.763616 + 0.645670i \(0.223422\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.8862 −0.607357 −0.303679 0.952775i \(-0.598215\pi\)
−0.303679 + 0.952775i \(0.598215\pi\)
\(384\) 0 0
\(385\) 11.3205 0.576947
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.96225 0.403702 0.201851 0.979416i \(-0.435304\pi\)
0.201851 + 0.979416i \(0.435304\pi\)
\(390\) 0 0
\(391\) − 7.86488i − 0.397744i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 19.4008i − 0.976162i
\(396\) 0 0
\(397\) 13.6224i 0.683688i 0.939757 + 0.341844i \(0.111051\pi\)
−0.939757 + 0.341844i \(0.888949\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 39.5109i − 1.97308i −0.163527 0.986539i \(-0.552287\pi\)
0.163527 0.986539i \(-0.447713\pi\)
\(402\) 0 0
\(403\) 42.2487 2.10456
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.3129 0.610328
\(408\) 0 0
\(409\) −29.3923 −1.45336 −0.726678 0.686978i \(-0.758937\pi\)
−0.726678 + 0.686978i \(0.758937\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.10886 0.349804
\(414\) 0 0
\(415\) − 11.5150i − 0.565249i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 0.354594i − 0.0173230i −0.999962 0.00866152i \(-0.997243\pi\)
0.999962 0.00866152i \(-0.00275708\pi\)
\(420\) 0 0
\(421\) − 13.2367i − 0.645117i −0.946549 0.322559i \(-0.895457\pi\)
0.946549 0.322559i \(-0.104543\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.26090i 0.158177i
\(426\) 0 0
\(427\) −7.73205 −0.374180
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.4168 1.85047 0.925237 0.379390i \(-0.123866\pi\)
0.925237 + 0.379390i \(0.123866\pi\)
\(432\) 0 0
\(433\) 11.4641 0.550930 0.275465 0.961311i \(-0.411168\pi\)
0.275465 + 0.961311i \(0.411168\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.59245 −0.0761772
\(438\) 0 0
\(439\) − 35.1096i − 1.67569i −0.545907 0.837846i \(-0.683815\pi\)
0.545907 0.837846i \(-0.316185\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 31.5707i 1.49997i 0.661456 + 0.749984i \(0.269939\pi\)
−0.661456 + 0.749984i \(0.730061\pi\)
\(444\) 0 0
\(445\) 25.1374i 1.19163i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 23.0163i − 1.08621i −0.839666 0.543104i \(-0.817249\pi\)
0.839666 0.543104i \(-0.182751\pi\)
\(450\) 0 0
\(451\) −48.7846 −2.29718
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 12.3129 0.577238
\(456\) 0 0
\(457\) −14.3923 −0.673244 −0.336622 0.941640i \(-0.609284\pi\)
−0.336622 + 0.941640i \(0.609284\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.9951 0.884689 0.442344 0.896845i \(-0.354147\pi\)
0.442344 + 0.896845i \(0.354147\pi\)
\(462\) 0 0
\(463\) 30.7915i 1.43100i 0.698611 + 0.715502i \(0.253802\pi\)
−0.698611 + 0.715502i \(0.746198\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17.4633i − 0.808105i −0.914736 0.404052i \(-0.867601\pi\)
0.914736 0.404052i \(-0.132399\pi\)
\(468\) 0 0
\(469\) − 1.72171i − 0.0795012i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.87771i 0.454178i
\(474\) 0 0
\(475\) 0.660254 0.0302945
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.18490 −0.145522 −0.0727609 0.997349i \(-0.523181\pi\)
−0.0727609 + 0.997349i \(0.523181\pi\)
\(480\) 0 0
\(481\) 13.3923 0.610637
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 14.9568 0.679152
\(486\) 0 0
\(487\) − 37.8850i − 1.71674i −0.513035 0.858368i \(-0.671479\pi\)
0.513035 0.858368i \(-0.328521\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 16.7541i 0.756102i 0.925785 + 0.378051i \(0.123406\pi\)
−0.925785 + 0.378051i \(0.876594\pi\)
\(492\) 0 0
\(493\) − 11.5150i − 0.518609i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 7.94018i − 0.356166i
\(498\) 0 0
\(499\) 26.3923 1.18148 0.590741 0.806861i \(-0.298836\pi\)
0.590741 + 0.806861i \(0.298836\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 34.3785 1.53286 0.766432 0.642326i \(-0.222030\pi\)
0.766432 + 0.642326i \(0.222030\pi\)
\(504\) 0 0
\(505\) 5.07180 0.225692
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 22.1800 0.983110 0.491555 0.870847i \(-0.336429\pi\)
0.491555 + 0.870847i \(0.336429\pi\)
\(510\) 0 0
\(511\) − 11.4116i − 0.504822i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 2.29213i − 0.101003i
\(516\) 0 0
\(517\) − 45.0819i − 1.98270i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 1.32336i − 0.0579776i −0.999580 0.0289888i \(-0.990771\pi\)
0.999580 0.0289888i \(-0.00922871\pi\)
\(522\) 0 0
\(523\) 6.41154 0.280357 0.140179 0.990126i \(-0.455232\pi\)
0.140179 + 0.990126i \(0.455232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.4081 0.453384
\(528\) 0 0
\(529\) 12.3205 0.535674
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −53.0613 −2.29834
\(534\) 0 0
\(535\) 19.3799i 0.837865i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 24.3397i − 1.04838i
\(540\) 0 0
\(541\) 12.4653i 0.535927i 0.963429 + 0.267963i \(0.0863506\pi\)
−0.963429 + 0.267963i \(0.913649\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 21.6930i 0.929224i
\(546\) 0 0
\(547\) 0.411543 0.0175963 0.00879815 0.999961i \(-0.497199\pi\)
0.00879815 + 0.999961i \(0.497199\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.33151 −0.0993256
\(552\) 0 0
\(553\) 17.5359 0.745702
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 36.3977 1.54222 0.771110 0.636702i \(-0.219702\pi\)
0.771110 + 0.636702i \(0.219702\pi\)
\(558\) 0 0
\(559\) 10.7436i 0.454407i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 11.8153i − 0.497953i −0.968509 0.248977i \(-0.919906\pi\)
0.968509 0.248977i \(-0.0800943\pi\)
\(564\) 0 0
\(565\) 17.8372i 0.750415i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 35.5408i − 1.48995i −0.667094 0.744973i \(-0.732462\pi\)
0.667094 0.744973i \(-0.267538\pi\)
\(570\) 0 0
\(571\) −7.19615 −0.301150 −0.150575 0.988599i \(-0.548112\pi\)
−0.150575 + 0.988599i \(0.548112\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −14.6444 −0.610714
\(576\) 0 0
\(577\) −1.92820 −0.0802722 −0.0401361 0.999194i \(-0.512779\pi\)
−0.0401361 + 0.999194i \(0.512779\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.4081 0.431801
\(582\) 0 0
\(583\) 27.2448i 1.12836i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.7541i 0.691516i 0.938324 + 0.345758i \(0.112378\pi\)
−0.938324 + 0.345758i \(0.887622\pi\)
\(588\) 0 0
\(589\) − 2.10739i − 0.0868335i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.93754i − 0.0795651i −0.999208 0.0397826i \(-0.987333\pi\)
0.999208 0.0397826i \(-0.0126665\pi\)
\(594\) 0 0
\(595\) 3.03332 0.124354
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −43.5066 −1.77763 −0.888815 0.458267i \(-0.848471\pi\)
−0.888815 + 0.458267i \(0.848471\pi\)
\(600\) 0 0
\(601\) 9.60770 0.391906 0.195953 0.980613i \(-0.437220\pi\)
0.195953 + 0.980613i \(0.437220\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −21.3266 −0.867049
\(606\) 0 0
\(607\) 27.1414i 1.10164i 0.834625 + 0.550818i \(0.185684\pi\)
−0.834625 + 0.550818i \(0.814316\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 49.0340i − 1.98370i
\(612\) 0 0
\(613\) − 2.49307i − 0.100694i −0.998732 0.0503470i \(-0.983967\pi\)
0.998732 0.0503470i \(-0.0160327\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 23.7255i 0.955153i 0.878590 + 0.477577i \(0.158485\pi\)
−0.878590 + 0.477577i \(0.841515\pi\)
\(618\) 0 0
\(619\) −26.8038 −1.07734 −0.538669 0.842518i \(-0.681073\pi\)
−0.538669 + 0.842518i \(0.681073\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −22.7210 −0.910298
\(624\) 0 0
\(625\) −6.60770 −0.264308
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.29923 0.131549
\(630\) 0 0
\(631\) 17.9405i 0.714200i 0.934066 + 0.357100i \(0.116234\pi\)
−0.934066 + 0.357100i \(0.883766\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 26.4734i − 1.04891i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.8153i 0.466674i 0.972396 + 0.233337i \(0.0749646\pi\)
−0.972396 + 0.233337i \(0.925035\pi\)
\(642\) 0 0
\(643\) −22.7846 −0.898537 −0.449269 0.893397i \(-0.648315\pi\)
−0.449269 + 0.893397i \(0.648315\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.47812 0.0581108 0.0290554 0.999578i \(-0.490750\pi\)
0.0290554 + 0.999578i \(0.490750\pi\)
\(648\) 0 0
\(649\) −24.3923 −0.957482
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −27.8107 −1.08832 −0.544159 0.838982i \(-0.683151\pi\)
−0.544159 + 0.838982i \(0.683151\pi\)
\(654\) 0 0
\(655\) − 11.5150i − 0.449928i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 22.4022i 0.872664i 0.899786 + 0.436332i \(0.143723\pi\)
−0.899786 + 0.436332i \(0.856277\pi\)
\(660\) 0 0
\(661\) − 41.2528i − 1.60455i −0.596956 0.802274i \(-0.703623\pi\)
0.596956 0.802274i \(-0.296377\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 0.614175i − 0.0238167i
\(666\) 0 0
\(667\) 51.7128 2.00233
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 26.5306 1.02420
\(672\) 0 0
\(673\) −13.2487 −0.510700 −0.255350 0.966849i \(-0.582191\pi\)
−0.255350 + 0.966849i \(0.582191\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.0005 0.461218 0.230609 0.973046i \(-0.425928\pi\)
0.230609 + 0.973046i \(0.425928\pi\)
\(678\) 0 0
\(679\) 13.5190i 0.518813i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.9266i 1.33643i 0.743969 + 0.668214i \(0.232941\pi\)
−0.743969 + 0.668214i \(0.767059\pi\)
\(684\) 0 0
\(685\) − 9.40760i − 0.359446i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 29.6331i 1.12893i
\(690\) 0 0
\(691\) 23.8564 0.907540 0.453770 0.891119i \(-0.350079\pi\)
0.453770 + 0.891119i \(0.350079\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.4595 −0.434684
\(696\) 0 0
\(697\) −13.0718 −0.495130
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.2883 0.652970 0.326485 0.945202i \(-0.394136\pi\)
0.326485 + 0.945202i \(0.394136\pi\)
\(702\) 0 0
\(703\) − 0.668016i − 0.0251947i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.58426i 0.172409i
\(708\) 0 0
\(709\) 27.6304i 1.03768i 0.854870 + 0.518841i \(0.173636\pi\)
−0.854870 + 0.518841i \(0.826364\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 46.7418i 1.75050i
\(714\) 0 0
\(715\) −42.2487 −1.58001
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 15.0711 0.562058 0.281029 0.959699i \(-0.409324\pi\)
0.281029 + 0.959699i \(0.409324\pi\)
\(720\) 0 0
\(721\) 2.07180 0.0771577
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −21.4409 −0.796296
\(726\) 0 0
\(727\) 19.3799i 0.718760i 0.933191 + 0.359380i \(0.117012\pi\)
−0.933191 + 0.359380i \(0.882988\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.64673i 0.0978927i
\(732\) 0 0
\(733\) − 45.0819i − 1.66514i −0.553921 0.832569i \(-0.686869\pi\)
0.553921 0.832569i \(-0.313131\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.90763i 0.217610i
\(738\) 0 0
\(739\) −48.6410 −1.78929 −0.894644 0.446779i \(-0.852571\pi\)
−0.894644 + 0.446779i \(0.852571\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −43.3085 −1.58884 −0.794418 0.607372i \(-0.792224\pi\)
−0.794418 + 0.607372i \(0.792224\pi\)
\(744\) 0 0
\(745\) −5.07180 −0.185816
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17.5170 −0.640056
\(750\) 0 0
\(751\) 7.19687i 0.262617i 0.991342 + 0.131309i \(0.0419179\pi\)
−0.991342 + 0.131309i \(0.958082\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 10.2323i 0.372392i
\(756\) 0 0
\(757\) 36.0600i 1.31062i 0.755359 + 0.655311i \(0.227463\pi\)
−0.755359 + 0.655311i \(0.772537\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 26.3722i − 0.955993i −0.878362 0.477996i \(-0.841363\pi\)
0.878362 0.477996i \(-0.158637\pi\)
\(762\) 0 0
\(763\) −19.6077 −0.709846
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −26.5306 −0.957965
\(768\) 0 0
\(769\) 17.9282 0.646508 0.323254 0.946312i \(-0.395223\pi\)
0.323254 + 0.946312i \(0.395223\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −34.1805 −1.22939 −0.614694 0.788766i \(-0.710720\pi\)
−0.614694 + 0.788766i \(0.710720\pi\)
\(774\) 0 0
\(775\) − 19.3799i − 0.696146i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.64673i 0.0948288i
\(780\) 0 0
\(781\) 27.2448i 0.974894i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 3.35591i 0.119778i
\(786\) 0 0
\(787\) 51.4449 1.83381 0.916906 0.399104i \(-0.130679\pi\)
0.916906 + 0.399104i \(0.130679\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −16.1225 −0.573251
\(792\) 0 0
\(793\) 28.8564 1.02472
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.84792 0.277988 0.138994 0.990293i \(-0.455613\pi\)
0.138994 + 0.990293i \(0.455613\pi\)
\(798\) 0 0
\(799\) − 12.0797i − 0.427348i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 39.1563i 1.38179i
\(804\) 0 0
\(805\) 13.6224i 0.480126i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.64673i 0.0930539i 0.998917 + 0.0465270i \(0.0148153\pi\)
−0.998917 + 0.0465270i \(0.985185\pi\)
\(810\) 0 0
\(811\) 40.9282 1.43718 0.718592 0.695432i \(-0.244787\pi\)
0.718592 + 0.695432i \(0.244787\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −30.5689 −1.07078
\(816\) 0 0
\(817\) 0.535898 0.0187487
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16.6636 −0.581562 −0.290781 0.956790i \(-0.593915\pi\)
−0.290781 + 0.956790i \(0.593915\pi\)
\(822\) 0 0
\(823\) − 28.6841i − 0.999866i −0.866064 0.499933i \(-0.833358\pi\)
0.866064 0.499933i \(-0.166642\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.00132i 0.104366i 0.998638 + 0.0521830i \(0.0166179\pi\)
−0.998638 + 0.0521830i \(0.983382\pi\)
\(828\) 0 0
\(829\) − 12.4653i − 0.432939i −0.976289 0.216470i \(-0.930546\pi\)
0.976289 0.216470i \(-0.0694542\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 6.52180i − 0.225967i
\(834\) 0 0
\(835\) −8.10512 −0.280489
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 57.0996 1.97130 0.985648 0.168815i \(-0.0539942\pi\)
0.985648 + 0.168815i \(0.0539942\pi\)
\(840\) 0 0
\(841\) 46.7128 1.61079
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −25.2505 −0.868645
\(846\) 0 0
\(847\) − 19.2765i − 0.662349i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 14.8166i 0.507905i
\(852\) 0 0
\(853\) 0.385679i 0.0132054i 0.999978 + 0.00660270i \(0.00210172\pi\)
−0.999978 + 0.00660270i \(0.997898\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 36.8641i 1.25925i 0.776897 + 0.629627i \(0.216792\pi\)
−0.776897 + 0.629627i \(0.783208\pi\)
\(858\) 0 0
\(859\) 19.5885 0.668350 0.334175 0.942511i \(-0.391542\pi\)
0.334175 + 0.942511i \(0.391542\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.27462 −0.281671 −0.140836 0.990033i \(-0.544979\pi\)
−0.140836 + 0.990033i \(0.544979\pi\)
\(864\) 0 0
\(865\) −18.9282 −0.643578
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −60.1701 −2.04113
\(870\) 0 0
\(871\) 6.42551i 0.217720i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 17.1087i − 0.578380i
\(876\) 0 0
\(877\) 21.8729i 0.738597i 0.929311 + 0.369298i \(0.120402\pi\)
−0.929311 + 0.369298i \(0.879598\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 35.5408i − 1.19740i −0.800974 0.598699i \(-0.795684\pi\)
0.800974 0.598699i \(-0.204316\pi\)
\(882\) 0 0
\(883\) −19.8756 −0.668869 −0.334434 0.942419i \(-0.608545\pi\)
−0.334434 + 0.942419i \(0.608545\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.7341 0.662607 0.331304 0.943524i \(-0.392512\pi\)
0.331304 + 0.943524i \(0.392512\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.44584 −0.0818470
\(894\) 0 0
\(895\) 15.7298i 0.525788i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 68.4348i 2.28243i
\(900\) 0 0
\(901\) 7.30021i 0.243205i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 25.6631i − 0.853069i
\(906\) 0 0
\(907\) −26.3731 −0.875703 −0.437852 0.899047i \(-0.644260\pi\)
−0.437852 + 0.899047i \(0.644260\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −44.3599 −1.46971 −0.734855 0.678224i \(-0.762750\pi\)
−0.734855 + 0.678224i \(0.762750\pi\)
\(912\) 0 0
\(913\) −35.7128 −1.18192
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 10.4081 0.343706
\(918\) 0 0
\(919\) − 30.8949i − 1.01913i −0.860433 0.509564i \(-0.829807\pi\)
0.860433 0.509564i \(-0.170193\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 29.6331i 0.975387i
\(924\) 0 0
\(925\) − 6.14317i − 0.201986i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 14.4620i − 0.474482i −0.971451 0.237241i \(-0.923757\pi\)
0.971451 0.237241i \(-0.0762431\pi\)
\(930\) 0 0
\(931\) −1.32051 −0.0432779
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10.4081 −0.340381
\(936\) 0 0
\(937\) −2.60770 −0.0851897 −0.0425948 0.999092i \(-0.513562\pi\)
−0.0425948 + 0.999092i \(0.513562\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 31.7347 1.03452 0.517260 0.855828i \(-0.326952\pi\)
0.517260 + 0.855828i \(0.326952\pi\)
\(942\) 0 0
\(943\) − 58.7043i − 1.91167i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 26.6318i − 0.865418i −0.901534 0.432709i \(-0.857558\pi\)
0.901534 0.432709i \(-0.142442\pi\)
\(948\) 0 0
\(949\) 42.5888i 1.38249i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 57.9429i 1.87696i 0.345340 + 0.938478i \(0.387764\pi\)
−0.345340 + 0.938478i \(0.612236\pi\)
\(954\) 0 0
\(955\) 14.5359 0.470371
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 8.50328 0.274585
\(960\) 0 0
\(961\) −30.8564 −0.995368
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 24.5115 0.789053
\(966\) 0 0
\(967\) − 6.42551i − 0.206630i −0.994649 0.103315i \(-0.967055\pi\)
0.994649 0.103315i \(-0.0329451\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 42.5122i − 1.36428i −0.731221 0.682140i \(-0.761049\pi\)
0.731221 0.682140i \(-0.238951\pi\)
\(972\) 0 0
\(973\) − 10.3580i − 0.332061i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41.4484i 1.32605i 0.748597 + 0.663026i \(0.230728\pi\)
−0.748597 + 0.663026i \(0.769272\pi\)
\(978\) 0 0
\(979\) 77.9615 2.49166
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −54.1127 −1.72593 −0.862963 0.505267i \(-0.831394\pi\)
−0.862963 + 0.505267i \(0.831394\pi\)
\(984\) 0 0
\(985\) −31.6077 −1.00710
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −11.8862 −0.377960
\(990\) 0 0
\(991\) 0.668016i 0.0212202i 0.999944 + 0.0106101i \(0.00337737\pi\)
−0.999944 + 0.0106101i \(0.996623\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 22.7567i 0.721437i
\(996\) 0 0
\(997\) − 29.3521i − 0.929592i −0.885418 0.464796i \(-0.846128\pi\)
0.885418 0.464796i \(-0.153872\pi\)
\(998\) 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 864.2.f.a.431.6 8
3.2 odd 2 inner 864.2.f.a.431.4 8
4.3 odd 2 216.2.f.a.107.8 yes 8
8.3 odd 2 inner 864.2.f.a.431.3 8
8.5 even 2 216.2.f.a.107.2 yes 8
9.2 odd 6 2592.2.p.f.2159.6 16
9.4 even 3 2592.2.p.f.431.3 16
9.5 odd 6 2592.2.p.f.431.5 16
9.7 even 3 2592.2.p.f.2159.4 16
12.11 even 2 216.2.f.a.107.1 8
24.5 odd 2 216.2.f.a.107.7 yes 8
24.11 even 2 inner 864.2.f.a.431.5 8
36.7 odd 6 648.2.l.f.539.5 16
36.11 even 6 648.2.l.f.539.4 16
36.23 even 6 648.2.l.f.107.7 16
36.31 odd 6 648.2.l.f.107.2 16
72.5 odd 6 648.2.l.f.107.5 16
72.11 even 6 2592.2.p.f.2159.3 16
72.13 even 6 648.2.l.f.107.4 16
72.29 odd 6 648.2.l.f.539.2 16
72.43 odd 6 2592.2.p.f.2159.5 16
72.59 even 6 2592.2.p.f.431.4 16
72.61 even 6 648.2.l.f.539.7 16
72.67 odd 6 2592.2.p.f.431.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.f.a.107.1 8 12.11 even 2
216.2.f.a.107.2 yes 8 8.5 even 2
216.2.f.a.107.7 yes 8 24.5 odd 2
216.2.f.a.107.8 yes 8 4.3 odd 2
648.2.l.f.107.2 16 36.31 odd 6
648.2.l.f.107.4 16 72.13 even 6
648.2.l.f.107.5 16 72.5 odd 6
648.2.l.f.107.7 16 36.23 even 6
648.2.l.f.539.2 16 72.29 odd 6
648.2.l.f.539.4 16 36.11 even 6
648.2.l.f.539.5 16 36.7 odd 6
648.2.l.f.539.7 16 72.61 even 6
864.2.f.a.431.3 8 8.3 odd 2 inner
864.2.f.a.431.4 8 3.2 odd 2 inner
864.2.f.a.431.5 8 24.11 even 2 inner
864.2.f.a.431.6 8 1.1 even 1 trivial
2592.2.p.f.431.3 16 9.4 even 3
2592.2.p.f.431.4 16 72.59 even 6
2592.2.p.f.431.5 16 9.5 odd 6
2592.2.p.f.431.6 16 72.67 odd 6
2592.2.p.f.2159.3 16 72.11 even 6
2592.2.p.f.2159.4 16 9.7 even 3
2592.2.p.f.2159.5 16 72.43 odd 6
2592.2.p.f.2159.6 16 9.2 odd 6