Properties

Label 1728.4.f.g.863.6
Level $1728$
Weight $4$
Character 1728.863
Analytic conductor $101.955$
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] = [1728,4,Mod(863,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.863");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1728.f (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: \(101.955300490\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.58594980096.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 863.6
Root \(-3.20565 + 1.85078i\) of defining polynomial
Character \(\chi\) \(=\) 1728.863
Dual form 1728.4.f.g.863.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.35849 q^{5} +21.2094i q^{7} +O(q^{10})\) \(q+9.35849 q^{5} +21.2094i q^{7} -50.9277i q^{11} +33.2716i q^{13} -120.628i q^{17} -82.1317 q^{19} +95.3719 q^{23} -37.4187 q^{25} +83.1384 q^{29} +36.4187i q^{31} +198.488i q^{35} +201.724i q^{37} -291.769i q^{41} -457.424 q^{43} -628.397 q^{47} -106.837 q^{49} -659.013 q^{53} -476.606i q^{55} +44.8330i q^{59} -149.600i q^{61} +311.372i q^{65} +134.256 q^{67} -291.769 q^{71} +888.212 q^{73} +1080.14 q^{77} +714.653i q^{79} -827.466i q^{83} -1128.90i q^{85} +19.6032i q^{89} -705.670 q^{91} -768.628 q^{95} -805.469 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 9.35849 0.837048 0.418524 0.908206i \(-0.362548\pi\)
0.418524 + 0.908206i \(0.362548\pi\)
\(6\) 0 0
\(7\) 21.2094i 1.14520i 0.819835 + 0.572599i \(0.194065\pi\)
−0.819835 + 0.572599i \(0.805935\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 50.9277i − 1.39593i −0.716130 0.697967i \(-0.754088\pi\)
0.716130 0.697967i \(-0.245912\pi\)
\(12\) 0 0
\(13\) 33.2716i 0.709837i 0.934897 + 0.354919i \(0.115491\pi\)
−0.934897 + 0.354919i \(0.884509\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 120.628i − 1.72098i −0.509470 0.860489i \(-0.670158\pi\)
0.509470 0.860489i \(-0.329842\pi\)
\(18\) 0 0
\(19\) −82.1317 −0.991700 −0.495850 0.868408i \(-0.665143\pi\)
−0.495850 + 0.868408i \(0.665143\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 95.3719 0.864627 0.432313 0.901723i \(-0.357697\pi\)
0.432313 + 0.901723i \(0.357697\pi\)
\(24\) 0 0
\(25\) −37.4187 −0.299350
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 83.1384 0.532359 0.266180 0.963923i \(-0.414239\pi\)
0.266180 + 0.963923i \(0.414239\pi\)
\(30\) 0 0
\(31\) 36.4187i 0.211000i 0.994419 + 0.105500i \(0.0336443\pi\)
−0.994419 + 0.105500i \(0.966356\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 198.488i 0.958587i
\(36\) 0 0
\(37\) 201.724i 0.896305i 0.893957 + 0.448152i \(0.147918\pi\)
−0.893957 + 0.448152i \(0.852082\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 291.769i − 1.11138i −0.831389 0.555690i \(-0.812454\pi\)
0.831389 0.555690i \(-0.187546\pi\)
\(42\) 0 0
\(43\) −457.424 −1.62224 −0.811122 0.584877i \(-0.801143\pi\)
−0.811122 + 0.584877i \(0.801143\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −628.397 −1.95024 −0.975118 0.221686i \(-0.928844\pi\)
−0.975118 + 0.221686i \(0.928844\pi\)
\(48\) 0 0
\(49\) −106.837 −0.311480
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −659.013 −1.70797 −0.853985 0.520298i \(-0.825821\pi\)
−0.853985 + 0.520298i \(0.825821\pi\)
\(54\) 0 0
\(55\) − 476.606i − 1.16846i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 44.8330i 0.0989282i 0.998776 + 0.0494641i \(0.0157513\pi\)
−0.998776 + 0.0494641i \(0.984249\pi\)
\(60\) 0 0
\(61\) − 149.600i − 0.314006i −0.987598 0.157003i \(-0.949817\pi\)
0.987598 0.157003i \(-0.0501833\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 311.372i 0.594168i
\(66\) 0 0
\(67\) 134.256 0.244805 0.122402 0.992481i \(-0.460940\pi\)
0.122402 + 0.992481i \(0.460940\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −291.769 −0.487698 −0.243849 0.969813i \(-0.578410\pi\)
−0.243849 + 0.969813i \(0.578410\pi\)
\(72\) 0 0
\(73\) 888.212 1.42407 0.712037 0.702142i \(-0.247773\pi\)
0.712037 + 0.702142i \(0.247773\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1080.14 1.59862
\(78\) 0 0
\(79\) 714.653i 1.01778i 0.860831 + 0.508891i \(0.169944\pi\)
−0.860831 + 0.508891i \(0.830056\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 827.466i − 1.09429i −0.837038 0.547145i \(-0.815714\pi\)
0.837038 0.547145i \(-0.184286\pi\)
\(84\) 0 0
\(85\) − 1128.90i − 1.44054i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 19.6032i 0.0233476i 0.999932 + 0.0116738i \(0.00371596\pi\)
−0.999932 + 0.0116738i \(0.996284\pi\)
\(90\) 0 0
\(91\) −705.670 −0.812905
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −768.628 −0.830101
\(96\) 0 0
\(97\) −805.469 −0.843123 −0.421562 0.906800i \(-0.638518\pi\)
−0.421562 + 0.906800i \(0.638518\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 421.787 0.415538 0.207769 0.978178i \(-0.433380\pi\)
0.207769 + 0.978178i \(0.433380\pi\)
\(102\) 0 0
\(103\) 372.653i 0.356491i 0.983986 + 0.178246i \(0.0570422\pi\)
−0.983986 + 0.178246i \(0.942958\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 40.9142i − 0.0369656i −0.999829 0.0184828i \(-0.994116\pi\)
0.999829 0.0184828i \(-0.00588360\pi\)
\(108\) 0 0
\(109\) − 2080.96i − 1.82862i −0.405011 0.914312i \(-0.632732\pi\)
0.405011 0.914312i \(-0.367268\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 614.447i 0.511525i 0.966740 + 0.255762i \(0.0823265\pi\)
−0.966740 + 0.255762i \(0.917674\pi\)
\(114\) 0 0
\(115\) 892.536 0.723734
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2558.45 1.97086
\(120\) 0 0
\(121\) −1262.63 −0.948633
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1519.99 −1.08762
\(126\) 0 0
\(127\) − 1682.24i − 1.17539i −0.809083 0.587695i \(-0.800036\pi\)
0.809083 0.587695i \(-0.199964\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2008.81i 1.33978i 0.742462 + 0.669888i \(0.233658\pi\)
−0.742462 + 0.669888i \(0.766342\pi\)
\(132\) 0 0
\(133\) − 1741.96i − 1.13569i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2140.40i − 1.33479i −0.744703 0.667396i \(-0.767409\pi\)
0.744703 0.667396i \(-0.232591\pi\)
\(138\) 0 0
\(139\) −2626.56 −1.60275 −0.801373 0.598165i \(-0.795897\pi\)
−0.801373 + 0.598165i \(0.795897\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1694.45 0.990886
\(144\) 0 0
\(145\) 778.050 0.445611
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 260.728 0.143353 0.0716766 0.997428i \(-0.477165\pi\)
0.0716766 + 0.997428i \(0.477165\pi\)
\(150\) 0 0
\(151\) 1825.26i 0.983692i 0.870682 + 0.491846i \(0.163678\pi\)
−0.870682 + 0.491846i \(0.836322\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 340.824i 0.176617i
\(156\) 0 0
\(157\) − 1731.00i − 0.879929i −0.898015 0.439965i \(-0.854991\pi\)
0.898015 0.439965i \(-0.145009\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2022.78i 0.990169i
\(162\) 0 0
\(163\) −1066.74 −0.512597 −0.256299 0.966598i \(-0.582503\pi\)
−0.256299 + 0.966598i \(0.582503\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1643.93 0.761745 0.380873 0.924628i \(-0.375624\pi\)
0.380873 + 0.924628i \(0.375624\pi\)
\(168\) 0 0
\(169\) 1090.00 0.496131
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3182.76 −1.39873 −0.699367 0.714763i \(-0.746535\pi\)
−0.699367 + 0.714763i \(0.746535\pi\)
\(174\) 0 0
\(175\) − 793.628i − 0.342815i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 3669.40i − 1.53220i −0.642720 0.766101i \(-0.722194\pi\)
0.642720 0.766101i \(-0.277806\pi\)
\(180\) 0 0
\(181\) − 1363.16i − 0.559796i −0.960030 0.279898i \(-0.909699\pi\)
0.960030 0.279898i \(-0.0903006\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1887.83i 0.750251i
\(186\) 0 0
\(187\) −6143.31 −2.40237
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −807.834 −0.306036 −0.153018 0.988223i \(-0.548899\pi\)
−0.153018 + 0.988223i \(0.548899\pi\)
\(192\) 0 0
\(193\) 898.681 0.335173 0.167587 0.985857i \(-0.446403\pi\)
0.167587 + 0.985857i \(0.446403\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4394.80 1.58943 0.794713 0.606986i \(-0.207622\pi\)
0.794713 + 0.606986i \(0.207622\pi\)
\(198\) 0 0
\(199\) − 878.234i − 0.312846i −0.987690 0.156423i \(-0.950004\pi\)
0.987690 0.156423i \(-0.0499963\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1763.31i 0.609657i
\(204\) 0 0
\(205\) − 2730.51i − 0.930280i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 4182.78i 1.38435i
\(210\) 0 0
\(211\) −5015.10 −1.63627 −0.818137 0.575024i \(-0.804993\pi\)
−0.818137 + 0.575024i \(0.804993\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4280.79 −1.35790
\(216\) 0 0
\(217\) −772.419 −0.241637
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4013.49 1.22161
\(222\) 0 0
\(223\) 1304.69i 0.391788i 0.980625 + 0.195894i \(0.0627609\pi\)
−0.980625 + 0.195894i \(0.937239\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1393.77i 0.407523i 0.979021 + 0.203762i \(0.0653168\pi\)
−0.979021 + 0.203762i \(0.934683\pi\)
\(228\) 0 0
\(229\) 1153.16i 0.332763i 0.986061 + 0.166381i \(0.0532083\pi\)
−0.986061 + 0.166381i \(0.946792\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 3540.43i − 0.995456i −0.867333 0.497728i \(-0.834168\pi\)
0.867333 0.497728i \(-0.165832\pi\)
\(234\) 0 0
\(235\) −5880.84 −1.63244
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 78.4127 0.0212222 0.0106111 0.999944i \(-0.496622\pi\)
0.0106111 + 0.999944i \(0.496622\pi\)
\(240\) 0 0
\(241\) 3906.31 1.04410 0.522048 0.852916i \(-0.325168\pi\)
0.522048 + 0.852916i \(0.325168\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −999.837 −0.260723
\(246\) 0 0
\(247\) − 2732.65i − 0.703946i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 5007.89i − 1.25934i −0.776861 0.629672i \(-0.783189\pi\)
0.776861 0.629672i \(-0.216811\pi\)
\(252\) 0 0
\(253\) − 4857.07i − 1.20696i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2019.77i − 0.490232i −0.969494 0.245116i \(-0.921174\pi\)
0.969494 0.245116i \(-0.0788261\pi\)
\(258\) 0 0
\(259\) −4278.45 −1.02645
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6575.74 −1.54174 −0.770869 0.636993i \(-0.780178\pi\)
−0.770869 + 0.636993i \(0.780178\pi\)
\(264\) 0 0
\(265\) −6167.36 −1.42965
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3661.36 −0.829876 −0.414938 0.909850i \(-0.636197\pi\)
−0.414938 + 0.909850i \(0.636197\pi\)
\(270\) 0 0
\(271\) 3645.23i 0.817092i 0.912738 + 0.408546i \(0.133964\pi\)
−0.912738 + 0.408546i \(0.866036\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1905.65i 0.417873i
\(276\) 0 0
\(277\) − 5103.76i − 1.10706i −0.832830 0.553529i \(-0.813281\pi\)
0.832830 0.553529i \(-0.186719\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 487.800i 0.103558i 0.998659 + 0.0517789i \(0.0164891\pi\)
−0.998659 + 0.0517789i \(0.983511\pi\)
\(282\) 0 0
\(283\) −339.147 −0.0712374 −0.0356187 0.999365i \(-0.511340\pi\)
−0.0356187 + 0.999365i \(0.511340\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6188.23 1.27275
\(288\) 0 0
\(289\) −9638.14 −1.96176
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9014.85 −1.79745 −0.898726 0.438511i \(-0.855506\pi\)
−0.898726 + 0.438511i \(0.855506\pi\)
\(294\) 0 0
\(295\) 419.569i 0.0828077i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3173.18i 0.613744i
\(300\) 0 0
\(301\) − 9701.67i − 1.85779i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 1400.03i − 0.262838i
\(306\) 0 0
\(307\) −4127.92 −0.767404 −0.383702 0.923457i \(-0.625351\pi\)
−0.383702 + 0.923457i \(0.625351\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2597.65 −0.473631 −0.236816 0.971555i \(-0.576104\pi\)
−0.236816 + 0.971555i \(0.576104\pi\)
\(312\) 0 0
\(313\) 8571.57 1.54790 0.773952 0.633245i \(-0.218277\pi\)
0.773952 + 0.633245i \(0.218277\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3321.62 0.588520 0.294260 0.955725i \(-0.404927\pi\)
0.294260 + 0.955725i \(0.404927\pi\)
\(318\) 0 0
\(319\) − 4234.05i − 0.743139i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 9907.39i 1.70669i
\(324\) 0 0
\(325\) − 1244.98i − 0.212490i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 13327.9i − 2.23341i
\(330\) 0 0
\(331\) −184.431 −0.0306262 −0.0153131 0.999883i \(-0.504874\pi\)
−0.0153131 + 0.999883i \(0.504874\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1256.43 0.204914
\(336\) 0 0
\(337\) 6322.76 1.02202 0.511012 0.859573i \(-0.329271\pi\)
0.511012 + 0.859573i \(0.329271\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1854.72 0.294542
\(342\) 0 0
\(343\) 5008.86i 0.788493i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 6318.50i − 0.977507i −0.872422 0.488753i \(-0.837452\pi\)
0.872422 0.488753i \(-0.162548\pi\)
\(348\) 0 0
\(349\) − 8503.65i − 1.30427i −0.758103 0.652135i \(-0.773874\pi\)
0.758103 0.652135i \(-0.226126\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 7697.59i − 1.16063i −0.814393 0.580313i \(-0.802930\pi\)
0.814393 0.580313i \(-0.197070\pi\)
\(354\) 0 0
\(355\) −2730.51 −0.408227
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −903.571 −0.132838 −0.0664188 0.997792i \(-0.521157\pi\)
−0.0664188 + 0.997792i \(0.521157\pi\)
\(360\) 0 0
\(361\) −113.388 −0.0165312
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 8312.32 1.19202
\(366\) 0 0
\(367\) 6619.95i 0.941576i 0.882246 + 0.470788i \(0.156030\pi\)
−0.882246 + 0.470788i \(0.843970\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 13977.2i − 1.95596i
\(372\) 0 0
\(373\) − 11830.0i − 1.64219i −0.570794 0.821093i \(-0.693364\pi\)
0.570794 0.821093i \(-0.306636\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2766.15i 0.377888i
\(378\) 0 0
\(379\) 3711.81 0.503068 0.251534 0.967849i \(-0.419065\pi\)
0.251534 + 0.967849i \(0.419065\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1761.92 0.235065 0.117532 0.993069i \(-0.462502\pi\)
0.117532 + 0.993069i \(0.462502\pi\)
\(384\) 0 0
\(385\) 10108.5 1.33812
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4849.66 0.632102 0.316051 0.948742i \(-0.397643\pi\)
0.316051 + 0.948742i \(0.397643\pi\)
\(390\) 0 0
\(391\) − 11504.5i − 1.48800i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6688.07i 0.851933i
\(396\) 0 0
\(397\) − 8066.93i − 1.01982i −0.860228 0.509909i \(-0.829679\pi\)
0.860228 0.509909i \(-0.170321\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 14632.2i 1.82219i 0.412199 + 0.911094i \(0.364761\pi\)
−0.412199 + 0.911094i \(0.635239\pi\)
\(402\) 0 0
\(403\) −1211.71 −0.149776
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10273.4 1.25118
\(408\) 0 0
\(409\) −12812.9 −1.54904 −0.774519 0.632551i \(-0.782008\pi\)
−0.774519 + 0.632551i \(0.782008\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −950.880 −0.113292
\(414\) 0 0
\(415\) − 7743.82i − 0.915974i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 9701.95i − 1.13120i −0.824681 0.565598i \(-0.808645\pi\)
0.824681 0.565598i \(-0.191355\pi\)
\(420\) 0 0
\(421\) 16265.0i 1.88292i 0.337123 + 0.941460i \(0.390546\pi\)
−0.337123 + 0.941460i \(0.609454\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4513.75i 0.515175i
\(426\) 0 0
\(427\) 3172.93 0.359599
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −16472.2 −1.84092 −0.920460 0.390837i \(-0.872185\pi\)
−0.920460 + 0.390837i \(0.872185\pi\)
\(432\) 0 0
\(433\) −14696.1 −1.63106 −0.815532 0.578712i \(-0.803556\pi\)
−0.815532 + 0.578712i \(0.803556\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7833.05 −0.857450
\(438\) 0 0
\(439\) 7192.32i 0.781938i 0.920404 + 0.390969i \(0.127860\pi\)
−0.920404 + 0.390969i \(0.872140\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12096.0i 1.29728i 0.761094 + 0.648642i \(0.224663\pi\)
−0.761094 + 0.648642i \(0.775337\pi\)
\(444\) 0 0
\(445\) 183.456i 0.0195430i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2061.98i 0.216728i 0.994111 + 0.108364i \(0.0345613\pi\)
−0.994111 + 0.108364i \(0.965439\pi\)
\(450\) 0 0
\(451\) −14859.1 −1.55142
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6604.00 −0.680440
\(456\) 0 0
\(457\) 18568.8 1.90068 0.950340 0.311215i \(-0.100736\pi\)
0.950340 + 0.311215i \(0.100736\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2231.88 0.225486 0.112743 0.993624i \(-0.464036\pi\)
0.112743 + 0.993624i \(0.464036\pi\)
\(462\) 0 0
\(463\) − 3414.82i − 0.342765i −0.985205 0.171383i \(-0.945177\pi\)
0.985205 0.171383i \(-0.0548234\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8313.82i 0.823806i 0.911228 + 0.411903i \(0.135136\pi\)
−0.911228 + 0.411903i \(0.864864\pi\)
\(468\) 0 0
\(469\) 2847.48i 0.280350i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23295.5i 2.26455i
\(474\) 0 0
\(475\) 3073.26 0.296865
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3888.73 0.370941 0.185470 0.982650i \(-0.440619\pi\)
0.185470 + 0.982650i \(0.440619\pi\)
\(480\) 0 0
\(481\) −6711.69 −0.636231
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7537.97 −0.705735
\(486\) 0 0
\(487\) − 1477.35i − 0.137464i −0.997635 0.0687320i \(-0.978105\pi\)
0.997635 0.0687320i \(-0.0218953\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 11772.5i − 1.08205i −0.841006 0.541026i \(-0.818036\pi\)
0.841006 0.541026i \(-0.181964\pi\)
\(492\) 0 0
\(493\) − 10028.8i − 0.916178i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 6188.23i − 0.558511i
\(498\) 0 0
\(499\) −9378.01 −0.841318 −0.420659 0.907219i \(-0.638201\pi\)
−0.420659 + 0.907219i \(0.638201\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22078.3 1.95710 0.978552 0.205998i \(-0.0660442\pi\)
0.978552 + 0.205998i \(0.0660442\pi\)
\(504\) 0 0
\(505\) 3947.29 0.347826
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18081.3 −1.57453 −0.787267 0.616612i \(-0.788505\pi\)
−0.787267 + 0.616612i \(0.788505\pi\)
\(510\) 0 0
\(511\) 18838.4i 1.63085i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3487.47i 0.298400i
\(516\) 0 0
\(517\) 32002.8i 2.72240i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8655.05i 0.727801i 0.931438 + 0.363901i \(0.118555\pi\)
−0.931438 + 0.363901i \(0.881445\pi\)
\(522\) 0 0
\(523\) 9920.20 0.829407 0.414704 0.909956i \(-0.363885\pi\)
0.414704 + 0.909956i \(0.363885\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4393.12 0.363126
\(528\) 0 0
\(529\) −3071.20 −0.252421
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9707.61 0.788900
\(534\) 0 0
\(535\) − 382.895i − 0.0309420i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5440.99i 0.434805i
\(540\) 0 0
\(541\) 12337.7i 0.980481i 0.871587 + 0.490241i \(0.163091\pi\)
−0.871587 + 0.490241i \(0.836909\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 19474.6i − 1.53065i
\(546\) 0 0
\(547\) −7446.60 −0.582073 −0.291036 0.956712i \(-0.594000\pi\)
−0.291036 + 0.956712i \(0.594000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6828.30 −0.527941
\(552\) 0 0
\(553\) −15157.3 −1.16556
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10322.9 0.785268 0.392634 0.919695i \(-0.371564\pi\)
0.392634 + 0.919695i \(0.371564\pi\)
\(558\) 0 0
\(559\) − 15219.2i − 1.15153i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18218.6i 1.36381i 0.731442 + 0.681904i \(0.238848\pi\)
−0.731442 + 0.681904i \(0.761152\pi\)
\(564\) 0 0
\(565\) 5750.29i 0.428171i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6399.31i 0.471481i 0.971816 + 0.235741i \(0.0757516\pi\)
−0.971816 + 0.235741i \(0.924248\pi\)
\(570\) 0 0
\(571\) −1999.51 −0.146545 −0.0732723 0.997312i \(-0.523344\pi\)
−0.0732723 + 0.997312i \(0.523344\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3568.70 −0.258826
\(576\) 0 0
\(577\) 10659.3 0.769067 0.384534 0.923111i \(-0.374362\pi\)
0.384534 + 0.923111i \(0.374362\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17550.0 1.25318
\(582\) 0 0
\(583\) 33562.0i 2.38421i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 8379.15i − 0.589173i −0.955625 0.294586i \(-0.904818\pi\)
0.955625 0.294586i \(-0.0951819\pi\)
\(588\) 0 0
\(589\) − 2991.13i − 0.209249i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 7052.96i − 0.488416i −0.969723 0.244208i \(-0.921472\pi\)
0.969723 0.244208i \(-0.0785279\pi\)
\(594\) 0 0
\(595\) 23943.2 1.64971
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 21730.4 1.48227 0.741135 0.671356i \(-0.234288\pi\)
0.741135 + 0.671356i \(0.234288\pi\)
\(600\) 0 0
\(601\) 18068.1 1.22631 0.613156 0.789962i \(-0.289900\pi\)
0.613156 + 0.789962i \(0.289900\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11816.3 −0.794052
\(606\) 0 0
\(607\) − 21545.7i − 1.44071i −0.693603 0.720357i \(-0.743978\pi\)
0.693603 0.720357i \(-0.256022\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 20907.8i − 1.38435i
\(612\) 0 0
\(613\) 19035.8i 1.25424i 0.778923 + 0.627120i \(0.215766\pi\)
−0.778923 + 0.627120i \(0.784234\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 17984.2i − 1.17344i −0.809788 0.586722i \(-0.800418\pi\)
0.809788 0.586722i \(-0.199582\pi\)
\(618\) 0 0
\(619\) 10517.1 0.682907 0.341454 0.939899i \(-0.389081\pi\)
0.341454 + 0.939899i \(0.389081\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −415.771 −0.0267376
\(624\) 0 0
\(625\) −9547.49 −0.611040
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 24333.6 1.54252
\(630\) 0 0
\(631\) − 23831.9i − 1.50354i −0.659425 0.751771i \(-0.729200\pi\)
0.659425 0.751771i \(-0.270800\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 15743.2i − 0.983858i
\(636\) 0 0
\(637\) − 3554.66i − 0.221100i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15647.0i 0.964149i 0.876130 + 0.482075i \(0.160117\pi\)
−0.876130 + 0.482075i \(0.839883\pi\)
\(642\) 0 0
\(643\) 29777.2 1.82628 0.913139 0.407648i \(-0.133651\pi\)
0.913139 + 0.407648i \(0.133651\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9149.68 0.555968 0.277984 0.960586i \(-0.410334\pi\)
0.277984 + 0.960586i \(0.410334\pi\)
\(648\) 0 0
\(649\) 2283.24 0.138097
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −19011.7 −1.13933 −0.569667 0.821875i \(-0.692928\pi\)
−0.569667 + 0.821875i \(0.692928\pi\)
\(654\) 0 0
\(655\) 18799.4i 1.12146i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 299.938i 0.0177298i 0.999961 + 0.00886490i \(0.00282182\pi\)
−0.999961 + 0.00886490i \(0.997178\pi\)
\(660\) 0 0
\(661\) 23512.2i 1.38354i 0.722118 + 0.691769i \(0.243169\pi\)
−0.722118 + 0.691769i \(0.756831\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 16302.1i − 0.950630i
\(666\) 0 0
\(667\) 7929.07 0.460292
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7618.81 −0.438332
\(672\) 0 0
\(673\) 22913.5 1.31241 0.656204 0.754583i \(-0.272161\pi\)
0.656204 + 0.754583i \(0.272161\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16220.1 −0.920809 −0.460405 0.887709i \(-0.652296\pi\)
−0.460405 + 0.887709i \(0.652296\pi\)
\(678\) 0 0
\(679\) − 17083.5i − 0.965543i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 18445.4i − 1.03337i −0.856175 0.516686i \(-0.827165\pi\)
0.856175 0.516686i \(-0.172835\pi\)
\(684\) 0 0
\(685\) − 20030.9i − 1.11729i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 21926.4i − 1.21238i
\(690\) 0 0
\(691\) 18971.8 1.04446 0.522231 0.852804i \(-0.325100\pi\)
0.522231 + 0.852804i \(0.325100\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24580.6 −1.34158
\(696\) 0 0
\(697\) −35195.5 −1.91266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9303.46 0.501265 0.250632 0.968082i \(-0.419361\pi\)
0.250632 + 0.968082i \(0.419361\pi\)
\(702\) 0 0
\(703\) − 16568.0i − 0.888865i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8945.84i 0.475874i
\(708\) 0 0
\(709\) 33456.9i 1.77221i 0.463481 + 0.886107i \(0.346600\pi\)
−0.463481 + 0.886107i \(0.653400\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3473.32i 0.182436i
\(714\) 0 0
\(715\) 15857.5 0.829420
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19474.6 −1.01013 −0.505064 0.863082i \(-0.668531\pi\)
−0.505064 + 0.863082i \(0.668531\pi\)
\(720\) 0 0
\(721\) −7903.74 −0.408253
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3110.94 −0.159362
\(726\) 0 0
\(727\) 7641.59i 0.389836i 0.980820 + 0.194918i \(0.0624441\pi\)
−0.980820 + 0.194918i \(0.937556\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 55178.2i 2.79184i
\(732\) 0 0
\(733\) − 23153.2i − 1.16669i −0.812224 0.583345i \(-0.801743\pi\)
0.812224 0.583345i \(-0.198257\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 6837.33i − 0.341732i
\(738\) 0 0
\(739\) 20627.7 1.02680 0.513398 0.858150i \(-0.328386\pi\)
0.513398 + 0.858150i \(0.328386\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −30112.9 −1.48686 −0.743429 0.668815i \(-0.766802\pi\)
−0.743429 + 0.668815i \(0.766802\pi\)
\(744\) 0 0
\(745\) 2440.02 0.119994
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 867.764 0.0423330
\(750\) 0 0
\(751\) 13806.8i 0.670861i 0.942065 + 0.335431i \(0.108882\pi\)
−0.942065 + 0.335431i \(0.891118\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 17081.7i 0.823398i
\(756\) 0 0
\(757\) 16505.6i 0.792478i 0.918147 + 0.396239i \(0.129685\pi\)
−0.918147 + 0.396239i \(0.870315\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 35472.2i 1.68971i 0.534997 + 0.844854i \(0.320313\pi\)
−0.534997 + 0.844854i \(0.679687\pi\)
\(762\) 0 0
\(763\) 44135.9 2.09414
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1491.67 −0.0702229
\(768\) 0 0
\(769\) 2816.64 0.132081 0.0660406 0.997817i \(-0.478963\pi\)
0.0660406 + 0.997817i \(0.478963\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30378.5 −1.41351 −0.706753 0.707461i \(-0.749841\pi\)
−0.706753 + 0.707461i \(0.749841\pi\)
\(774\) 0 0
\(775\) − 1362.74i − 0.0631628i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23963.5i 1.10216i
\(780\) 0 0
\(781\) 14859.1i 0.680795i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 16199.5i − 0.736543i
\(786\) 0 0
\(787\) 26302.1 1.19132 0.595660 0.803237i \(-0.296891\pi\)
0.595660 + 0.803237i \(0.296891\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −13032.0 −0.585797
\(792\) 0 0
\(793\) 4977.45 0.222893
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 27016.4 1.20072 0.600358 0.799731i \(-0.295025\pi\)
0.600358 + 0.799731i \(0.295025\pi\)
\(798\) 0 0
\(799\) 75802.3i 3.35631i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 45234.6i − 1.98791i
\(804\) 0 0
\(805\) 18930.1i 0.828819i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 33893.5i 1.47297i 0.676455 + 0.736484i \(0.263516\pi\)
−0.676455 + 0.736484i \(0.736484\pi\)
\(810\) 0 0
\(811\) −16820.8 −0.728309 −0.364155 0.931338i \(-0.618642\pi\)
−0.364155 + 0.931338i \(0.618642\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9983.05 −0.429069
\(816\) 0 0
\(817\) 37569.0 1.60878
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27615.2 −1.17391 −0.586953 0.809621i \(-0.699673\pi\)
−0.586953 + 0.809621i \(0.699673\pi\)
\(822\) 0 0
\(823\) − 27645.0i − 1.17089i −0.810712 0.585445i \(-0.800920\pi\)
0.810712 0.585445i \(-0.199080\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 33748.9i − 1.41906i −0.704674 0.709531i \(-0.748907\pi\)
0.704674 0.709531i \(-0.251093\pi\)
\(828\) 0 0
\(829\) 19289.5i 0.808145i 0.914727 + 0.404072i \(0.132406\pi\)
−0.914727 + 0.404072i \(0.867594\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 12887.6i 0.536049i
\(834\) 0 0
\(835\) 15384.7 0.637618
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8017.26 0.329900 0.164950 0.986302i \(-0.447254\pi\)
0.164950 + 0.986302i \(0.447254\pi\)
\(840\) 0 0
\(841\) −17477.0 −0.716594
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 10200.7 0.415286
\(846\) 0 0
\(847\) − 26779.6i − 1.08637i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 19238.8i 0.774969i
\(852\) 0 0
\(853\) 23777.8i 0.954439i 0.878784 + 0.477220i \(0.158355\pi\)
−0.878784 + 0.477220i \(0.841645\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 40669.8i − 1.62107i −0.585692 0.810534i \(-0.699177\pi\)
0.585692 0.810534i \(-0.300823\pi\)
\(858\) 0 0
\(859\) −7166.21 −0.284642 −0.142321 0.989821i \(-0.545457\pi\)
−0.142321 + 0.989821i \(0.545457\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 30914.4 1.21939 0.609697 0.792635i \(-0.291291\pi\)
0.609697 + 0.792635i \(0.291291\pi\)
\(864\) 0 0
\(865\) −29785.8 −1.17081
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 36395.6 1.42076
\(870\) 0 0
\(871\) 4466.90i 0.173772i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 32238.1i − 1.24554i
\(876\) 0 0
\(877\) 5414.60i 0.208481i 0.994552 + 0.104241i \(0.0332412\pi\)
−0.994552 + 0.104241i \(0.966759\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 6604.46i − 0.252565i −0.991994 0.126283i \(-0.959695\pi\)
0.991994 0.126283i \(-0.0403046\pi\)
\(882\) 0 0
\(883\) 35256.0 1.34367 0.671834 0.740701i \(-0.265507\pi\)
0.671834 + 0.740701i \(0.265507\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 17276.2 0.653976 0.326988 0.945028i \(-0.393966\pi\)
0.326988 + 0.945028i \(0.393966\pi\)
\(888\) 0 0
\(889\) 35679.2 1.34605
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 51611.3 1.93405
\(894\) 0 0
\(895\) − 34340.1i − 1.28253i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3027.80i 0.112328i
\(900\) 0 0
\(901\) 79495.5i 2.93938i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 12757.1i − 0.468576i
\(906\) 0 0
\(907\) −34491.5 −1.26270 −0.631351 0.775497i \(-0.717499\pi\)
−0.631351 + 0.775497i \(0.717499\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 38213.5 1.38976 0.694879 0.719127i \(-0.255458\pi\)
0.694879 + 0.719127i \(0.255458\pi\)
\(912\) 0 0
\(913\) −42140.9 −1.52756
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −42605.6 −1.53431
\(918\) 0 0
\(919\) 36899.1i 1.32447i 0.749295 + 0.662236i \(0.230392\pi\)
−0.749295 + 0.662236i \(0.769608\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 9707.61i − 0.346186i
\(924\) 0 0
\(925\) − 7548.27i − 0.268309i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 279.002i − 0.00985335i −0.999988 0.00492668i \(-0.998432\pi\)
0.999988 0.00492668i \(-0.00156822\pi\)
\(930\) 0 0
\(931\) 8774.74 0.308894
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −57492.1 −2.01090
\(936\) 0 0
\(937\) −41546.4 −1.44852 −0.724259 0.689528i \(-0.757818\pi\)
−0.724259 + 0.689528i \(0.757818\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −18490.9 −0.640580 −0.320290 0.947320i \(-0.603780\pi\)
−0.320290 + 0.947320i \(0.603780\pi\)
\(942\) 0 0
\(943\) − 27826.5i − 0.960930i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 7862.91i − 0.269810i −0.990859 0.134905i \(-0.956927\pi\)
0.990859 0.134905i \(-0.0430730\pi\)
\(948\) 0 0
\(949\) 29552.3i 1.01086i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 23650.7i − 0.803904i −0.915661 0.401952i \(-0.868332\pi\)
0.915661 0.401952i \(-0.131668\pi\)
\(954\) 0 0
\(955\) −7560.11 −0.256167
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 45396.5 1.52860
\(960\) 0 0
\(961\) 28464.7 0.955479
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 8410.29 0.280556
\(966\) 0 0
\(967\) − 13515.5i − 0.449460i −0.974421 0.224730i \(-0.927850\pi\)
0.974421 0.224730i \(-0.0721501\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 39505.1i 1.30564i 0.757512 + 0.652821i \(0.226415\pi\)
−0.757512 + 0.652821i \(0.773585\pi\)
\(972\) 0 0
\(973\) − 55707.6i − 1.83546i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 42131.6i 1.37964i 0.723981 + 0.689820i \(0.242310\pi\)
−0.723981 + 0.689820i \(0.757690\pi\)
\(978\) 0 0
\(979\) 998.345 0.0325917
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 48982.9 1.58933 0.794665 0.607049i \(-0.207647\pi\)
0.794665 + 0.607049i \(0.207647\pi\)
\(984\) 0 0
\(985\) 41128.7 1.33043
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −43625.4 −1.40263
\(990\) 0 0
\(991\) − 3555.12i − 0.113958i −0.998375 0.0569788i \(-0.981853\pi\)
0.998375 0.0569788i \(-0.0181468\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 8218.94i − 0.261867i
\(996\) 0 0
\(997\) 43140.0i 1.37037i 0.728370 + 0.685184i \(0.240278\pi\)
−0.728370 + 0.685184i \(0.759722\pi\)
\(998\) 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 1728.4.f.g.863.6 yes 8
3.2 odd 2 1728.4.f.c.863.4 yes 8
4.3 odd 2 1728.4.f.c.863.5 yes 8
8.3 odd 2 1728.4.f.c.863.3 8
8.5 even 2 inner 1728.4.f.g.863.4 yes 8
12.11 even 2 inner 1728.4.f.g.863.3 yes 8
24.5 odd 2 1728.4.f.c.863.6 yes 8
24.11 even 2 inner 1728.4.f.g.863.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.4.f.c.863.3 8 8.3 odd 2
1728.4.f.c.863.4 yes 8 3.2 odd 2
1728.4.f.c.863.5 yes 8 4.3 odd 2
1728.4.f.c.863.6 yes 8 24.5 odd 2
1728.4.f.g.863.3 yes 8 12.11 even 2 inner
1728.4.f.g.863.4 yes 8 8.5 even 2 inner
1728.4.f.g.863.5 yes 8 24.11 even 2 inner
1728.4.f.g.863.6 yes 8 1.1 even 1 trivial