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Properties

Label 576.8.a.i.1.1
Level $576$
Weight $8$
Character 576.1
Self dual yes
Analytic conductor $179.934$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,8,Mod(1,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 576.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(179.933774679\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 576.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-114.000 q^{5} +1576.00 q^{7} +O(q^{10})\) \(q-114.000 q^{5} +1576.00 q^{7} -7332.00 q^{11} +3802.00 q^{13} +6606.00 q^{17} +24860.0 q^{19} +41448.0 q^{23} -65129.0 q^{25} -41610.0 q^{29} -33152.0 q^{31} -179664. q^{35} +36466.0 q^{37} +639078. q^{41} -156412. q^{43} -433776. q^{47} +1.66023e6 q^{49} +786078. q^{53} +835848. q^{55} -745140. q^{59} +1.66062e6 q^{61} -433428. q^{65} -3.29084e6 q^{67} +5.71615e6 q^{71} +2.65990e6 q^{73} -1.15552e7 q^{77} -3.80744e6 q^{79} -2.22947e6 q^{83} -753084. q^{85} -5.99121e6 q^{89} +5.99195e6 q^{91} -2.83404e6 q^{95} -4.06013e6 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −114.000 −0.407859 −0.203929 − 0.978986i \(-0.565371\pi\)
−0.203929 + 0.978986i \(0.565371\pi\)
\(6\) 0 0
\(7\) 1576.00 1.73665 0.868327 − 0.495993i \(-0.165196\pi\)
0.868327 + 0.495993i \(0.165196\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −7332.00 −1.66092 −0.830459 − 0.557080i \(-0.811922\pi\)
−0.830459 + 0.557080i \(0.811922\pi\)
\(12\) 0 0
\(13\) 3802.00 0.479966 0.239983 − 0.970777i \(-0.422858\pi\)
0.239983 + 0.970777i \(0.422858\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6606.00 0.326112 0.163056 − 0.986617i \(-0.447865\pi\)
0.163056 + 0.986617i \(0.447865\pi\)
\(18\) 0 0
\(19\) 24860.0 0.831502 0.415751 − 0.909478i \(-0.363519\pi\)
0.415751 + 0.909478i \(0.363519\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 41448.0 0.710323 0.355162 − 0.934805i \(-0.384426\pi\)
0.355162 + 0.934805i \(0.384426\pi\)
\(24\) 0 0
\(25\) −65129.0 −0.833651
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −41610.0 −0.316814 −0.158407 − 0.987374i \(-0.550636\pi\)
−0.158407 + 0.987374i \(0.550636\pi\)
\(30\) 0 0
\(31\) −33152.0 −0.199868 −0.0999341 − 0.994994i \(-0.531863\pi\)
−0.0999341 + 0.994994i \(0.531863\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −179664. −0.708309
\(36\) 0 0
\(37\) 36466.0 0.118354 0.0591769 − 0.998248i \(-0.481152\pi\)
0.0591769 + 0.998248i \(0.481152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 639078. 1.44814 0.724070 − 0.689727i \(-0.242269\pi\)
0.724070 + 0.689727i \(0.242269\pi\)
\(42\) 0 0
\(43\) −156412. −0.300006 −0.150003 − 0.988686i \(-0.547928\pi\)
−0.150003 + 0.988686i \(0.547928\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −433776. −0.609429 −0.304714 − 0.952444i \(-0.598561\pi\)
−0.304714 + 0.952444i \(0.598561\pi\)
\(48\) 0 0
\(49\) 1.66023e6 2.01596
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 786078. 0.725271 0.362635 − 0.931931i \(-0.381877\pi\)
0.362635 + 0.931931i \(0.381877\pi\)
\(54\) 0 0
\(55\) 835848. 0.677420
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −745140. −0.472341 −0.236171 − 0.971712i \(-0.575892\pi\)
−0.236171 + 0.971712i \(0.575892\pi\)
\(60\) 0 0
\(61\) 1.66062e6 0.936732 0.468366 − 0.883535i \(-0.344843\pi\)
0.468366 + 0.883535i \(0.344843\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −433428. −0.195758
\(66\) 0 0
\(67\) −3.29084e6 −1.33673 −0.668366 − 0.743832i \(-0.733006\pi\)
−0.668366 + 0.743832i \(0.733006\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.71615e6 1.89539 0.947697 − 0.319171i \(-0.103404\pi\)
0.947697 + 0.319171i \(0.103404\pi\)
\(72\) 0 0
\(73\) 2.65990e6 0.800267 0.400134 − 0.916457i \(-0.368964\pi\)
0.400134 + 0.916457i \(0.368964\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.15552e7 −2.88444
\(78\) 0 0
\(79\) −3.80744e6 −0.868837 −0.434418 − 0.900711i \(-0.643046\pi\)
−0.434418 + 0.900711i \(0.643046\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.22947e6 −0.427984 −0.213992 − 0.976835i \(-0.568647\pi\)
−0.213992 + 0.976835i \(0.568647\pi\)
\(84\) 0 0
\(85\) −753084. −0.133008
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.99121e6 −0.900844 −0.450422 − 0.892816i \(-0.648726\pi\)
−0.450422 + 0.892816i \(0.648726\pi\)
\(90\) 0 0
\(91\) 5.99195e6 0.833534
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.83404e6 −0.339136
\(96\) 0 0
\(97\) −4.06013e6 −0.451688 −0.225844 − 0.974163i \(-0.572514\pi\)
−0.225844 + 0.974163i \(0.572514\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.72819e7 −1.66904 −0.834522 − 0.550975i \(-0.814256\pi\)
−0.834522 + 0.550975i \(0.814256\pi\)
\(102\) 0 0
\(103\) 1.43623e7 1.29507 0.647536 − 0.762035i \(-0.275799\pi\)
0.647536 + 0.762035i \(0.275799\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −6.45440e6 −0.509346 −0.254673 − 0.967027i \(-0.581968\pi\)
−0.254673 + 0.967027i \(0.581968\pi\)
\(108\) 0 0
\(109\) 884410. 0.0654125 0.0327063 − 0.999465i \(-0.489587\pi\)
0.0327063 + 0.999465i \(0.489587\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.21325e7 −0.790999 −0.395499 − 0.918466i \(-0.629428\pi\)
−0.395499 + 0.918466i \(0.629428\pi\)
\(114\) 0 0
\(115\) −4.72507e6 −0.289712
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.04111e7 0.566344
\(120\) 0 0
\(121\) 3.42711e7 1.75865
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.63310e7 0.747871
\(126\) 0 0
\(127\) −6.86806e6 −0.297524 −0.148762 − 0.988873i \(-0.547529\pi\)
−0.148762 + 0.988873i \(0.547529\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.95208e7 1.53595 0.767973 − 0.640482i \(-0.221265\pi\)
0.767973 + 0.640482i \(0.221265\pi\)
\(132\) 0 0
\(133\) 3.91794e7 1.44403
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.91741e7 −0.637078 −0.318539 − 0.947910i \(-0.603192\pi\)
−0.318539 + 0.947910i \(0.603192\pi\)
\(138\) 0 0
\(139\) 1.32449e7 0.418309 0.209154 − 0.977883i \(-0.432929\pi\)
0.209154 + 0.977883i \(0.432929\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2.78763e7 −0.797184
\(144\) 0 0
\(145\) 4.74354e6 0.129215
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.73624e7 1.42061 0.710306 − 0.703893i \(-0.248556\pi\)
0.710306 + 0.703893i \(0.248556\pi\)
\(150\) 0 0
\(151\) 3.10873e7 0.734790 0.367395 − 0.930065i \(-0.380250\pi\)
0.367395 + 0.930065i \(0.380250\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.77933e6 0.0815180
\(156\) 0 0
\(157\) 3.37835e7 0.696715 0.348358 − 0.937362i \(-0.386739\pi\)
0.348358 + 0.937362i \(0.386739\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.53220e7 1.23359
\(162\) 0 0
\(163\) 6.26659e7 1.13338 0.566689 − 0.823932i \(-0.308224\pi\)
0.566689 + 0.823932i \(0.308224\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.27072e7 1.04186 0.520931 − 0.853599i \(-0.325585\pi\)
0.520931 + 0.853599i \(0.325585\pi\)
\(168\) 0 0
\(169\) −4.82933e7 −0.769633
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.70521e7 −0.397228 −0.198614 − 0.980078i \(-0.563644\pi\)
−0.198614 + 0.980078i \(0.563644\pi\)
\(174\) 0 0
\(175\) −1.02643e8 −1.44776
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.34281e8 1.74996 0.874981 − 0.484157i \(-0.160874\pi\)
0.874981 + 0.484157i \(0.160874\pi\)
\(180\) 0 0
\(181\) −1.14661e8 −1.43727 −0.718636 − 0.695386i \(-0.755233\pi\)
−0.718636 + 0.695386i \(0.755233\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.15712e6 −0.0482716
\(186\) 0 0
\(187\) −4.84352e7 −0.541646
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.63605e7 0.169895 0.0849474 − 0.996385i \(-0.472928\pi\)
0.0849474 + 0.996385i \(0.472928\pi\)
\(192\) 0 0
\(193\) −1.54198e8 −1.54394 −0.771968 − 0.635661i \(-0.780728\pi\)
−0.771968 + 0.635661i \(0.780728\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.32288e7 0.775607 0.387804 − 0.921742i \(-0.373234\pi\)
0.387804 + 0.921742i \(0.373234\pi\)
\(198\) 0 0
\(199\) 7.61722e7 0.685190 0.342595 − 0.939483i \(-0.388694\pi\)
0.342595 + 0.939483i \(0.388694\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.55774e7 −0.550196
\(204\) 0 0
\(205\) −7.28549e7 −0.590636
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.82274e8 −1.38106
\(210\) 0 0
\(211\) 3.52446e7 0.258288 0.129144 − 0.991626i \(-0.458777\pi\)
0.129144 + 0.991626i \(0.458777\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.78310e7 0.122360
\(216\) 0 0
\(217\) −5.22476e7 −0.347102
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.51160e7 0.156523
\(222\) 0 0
\(223\) 1.89131e8 1.14208 0.571040 − 0.820922i \(-0.306540\pi\)
0.571040 + 0.820922i \(0.306540\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.76100e8 0.999239 0.499620 − 0.866245i \(-0.333473\pi\)
0.499620 + 0.866245i \(0.333473\pi\)
\(228\) 0 0
\(229\) −6.50396e7 −0.357894 −0.178947 − 0.983859i \(-0.557269\pi\)
−0.178947 + 0.983859i \(0.557269\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.51319e8 1.30160 0.650802 − 0.759248i \(-0.274433\pi\)
0.650802 + 0.759248i \(0.274433\pi\)
\(234\) 0 0
\(235\) 4.94505e7 0.248561
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.13079e8 1.00960 0.504799 − 0.863237i \(-0.331566\pi\)
0.504799 + 0.863237i \(0.331566\pi\)
\(240\) 0 0
\(241\) 2.57284e8 1.18400 0.592001 − 0.805937i \(-0.298338\pi\)
0.592001 + 0.805937i \(0.298338\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.89267e8 −0.822229
\(246\) 0 0
\(247\) 9.45177e7 0.399093
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.23058e8 −0.491193 −0.245596 − 0.969372i \(-0.578984\pi\)
−0.245596 + 0.969372i \(0.578984\pi\)
\(252\) 0 0
\(253\) −3.03897e8 −1.17979
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.43334e8 1.62916 0.814582 − 0.580048i \(-0.196966\pi\)
0.814582 + 0.580048i \(0.196966\pi\)
\(258\) 0 0
\(259\) 5.74704e7 0.205539
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.98925e8 1.01325 0.506625 − 0.862166i \(-0.330893\pi\)
0.506625 + 0.862166i \(0.330893\pi\)
\(264\) 0 0
\(265\) −8.96129e7 −0.295808
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.08908e8 0.654368 0.327184 − 0.944961i \(-0.393900\pi\)
0.327184 + 0.944961i \(0.393900\pi\)
\(270\) 0 0
\(271\) 1.12749e7 0.0344129 0.0172064 − 0.999852i \(-0.494523\pi\)
0.0172064 + 0.999852i \(0.494523\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.77526e8 1.38463
\(276\) 0 0
\(277\) 6.58964e8 1.86287 0.931435 − 0.363907i \(-0.118557\pi\)
0.931435 + 0.363907i \(0.118557\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.05123e8 0.282634 0.141317 − 0.989964i \(-0.454866\pi\)
0.141317 + 0.989964i \(0.454866\pi\)
\(282\) 0 0
\(283\) 3.30161e8 0.865911 0.432956 − 0.901415i \(-0.357471\pi\)
0.432956 + 0.901415i \(0.357471\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00719e9 2.51492
\(288\) 0 0
\(289\) −3.66699e8 −0.893651
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.71002e7 −0.202294 −0.101147 − 0.994871i \(-0.532251\pi\)
−0.101147 + 0.994871i \(0.532251\pi\)
\(294\) 0 0
\(295\) 8.49460e7 0.192649
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.57585e8 0.340931
\(300\) 0 0
\(301\) −2.46505e8 −0.521007
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.89310e8 −0.382054
\(306\) 0 0
\(307\) −3.91709e8 −0.772644 −0.386322 − 0.922364i \(-0.626255\pi\)
−0.386322 + 0.922364i \(0.626255\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.04936e8 −0.386328 −0.193164 − 0.981166i \(-0.561875\pi\)
−0.193164 + 0.981166i \(0.561875\pi\)
\(312\) 0 0
\(313\) 8.77202e8 1.61694 0.808471 − 0.588536i \(-0.200295\pi\)
0.808471 + 0.588536i \(0.200295\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.40831e8 −0.777256 −0.388628 − 0.921395i \(-0.627051\pi\)
−0.388628 + 0.921395i \(0.627051\pi\)
\(318\) 0 0
\(319\) 3.05085e8 0.526202
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.64225e8 0.271163
\(324\) 0 0
\(325\) −2.47620e8 −0.400124
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.83631e8 −1.05837
\(330\) 0 0
\(331\) 1.11223e9 1.68576 0.842882 − 0.538099i \(-0.180857\pi\)
0.842882 + 0.538099i \(0.180857\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.75155e8 0.545198
\(336\) 0 0
\(337\) 2.88198e8 0.410191 0.205096 − 0.978742i \(-0.434249\pi\)
0.205096 + 0.978742i \(0.434249\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.43070e8 0.331965
\(342\) 0 0
\(343\) 1.31862e9 1.76438
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.10601e9 1.42103 0.710517 − 0.703680i \(-0.248461\pi\)
0.710517 + 0.703680i \(0.248461\pi\)
\(348\) 0 0
\(349\) 1.32184e9 1.66453 0.832264 − 0.554379i \(-0.187044\pi\)
0.832264 + 0.554379i \(0.187044\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.20395e9 −1.45679 −0.728396 − 0.685157i \(-0.759734\pi\)
−0.728396 + 0.685157i \(0.759734\pi\)
\(354\) 0 0
\(355\) −6.51641e8 −0.773053
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.32057e9 −1.50637 −0.753185 − 0.657809i \(-0.771484\pi\)
−0.753185 + 0.657809i \(0.771484\pi\)
\(360\) 0 0
\(361\) −2.75852e8 −0.308604
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.03228e8 −0.326396
\(366\) 0 0
\(367\) −1.75107e9 −1.84915 −0.924575 − 0.381000i \(-0.875580\pi\)
−0.924575 + 0.381000i \(0.875580\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.23886e9 1.25954
\(372\) 0 0
\(373\) 4.87945e8 0.486844 0.243422 − 0.969920i \(-0.421730\pi\)
0.243422 + 0.969920i \(0.421730\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.58201e8 −0.152060
\(378\) 0 0
\(379\) 1.11007e9 1.04740 0.523700 − 0.851903i \(-0.324551\pi\)
0.523700 + 0.851903i \(0.324551\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.86912e9 1.69997 0.849983 − 0.526810i \(-0.176612\pi\)
0.849983 + 0.526810i \(0.176612\pi\)
\(384\) 0 0
\(385\) 1.31730e9 1.17644
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.73895e8 −0.235918 −0.117959 − 0.993018i \(-0.537635\pi\)
−0.117959 + 0.993018i \(0.537635\pi\)
\(390\) 0 0
\(391\) 2.73805e8 0.231645
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.34048e8 0.354363
\(396\) 0 0
\(397\) −6.24552e8 −0.500958 −0.250479 − 0.968122i \(-0.580588\pi\)
−0.250479 + 0.968122i \(0.580588\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.55500e8 −0.430208 −0.215104 − 0.976591i \(-0.569009\pi\)
−0.215104 + 0.976591i \(0.569009\pi\)
\(402\) 0 0
\(403\) −1.26044e8 −0.0959299
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.67369e8 −0.196576
\(408\) 0 0
\(409\) −2.15770e9 −1.55941 −0.779704 − 0.626149i \(-0.784630\pi\)
−0.779704 + 0.626149i \(0.784630\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.17434e9 −0.820293
\(414\) 0 0
\(415\) 2.54159e8 0.174557
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.67797e9 −1.11438 −0.557191 − 0.830384i \(-0.688121\pi\)
−0.557191 + 0.830384i \(0.688121\pi\)
\(420\) 0 0
\(421\) 5.25233e8 0.343056 0.171528 − 0.985179i \(-0.445130\pi\)
0.171528 + 0.985179i \(0.445130\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.30242e8 −0.271864
\(426\) 0 0
\(427\) 2.61713e9 1.62678
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.70593e8 0.102634 0.0513169 − 0.998682i \(-0.483658\pi\)
0.0513169 + 0.998682i \(0.483658\pi\)
\(432\) 0 0
\(433\) −1.68797e9 −0.999210 −0.499605 − 0.866253i \(-0.666522\pi\)
−0.499605 + 0.866253i \(0.666522\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.03040e9 0.590636
\(438\) 0 0
\(439\) 1.17850e9 0.664817 0.332409 − 0.943135i \(-0.392139\pi\)
0.332409 + 0.943135i \(0.392139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.15755e8 −0.391157 −0.195579 − 0.980688i \(-0.562658\pi\)
−0.195579 + 0.980688i \(0.562658\pi\)
\(444\) 0 0
\(445\) 6.82998e8 0.367417
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.37358e9 0.716132 0.358066 − 0.933696i \(-0.383436\pi\)
0.358066 + 0.933696i \(0.383436\pi\)
\(450\) 0 0
\(451\) −4.68572e9 −2.40524
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.83083e8 −0.339964
\(456\) 0 0
\(457\) 1.84752e9 0.905488 0.452744 − 0.891641i \(-0.350445\pi\)
0.452744 + 0.891641i \(0.350445\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.09414e9 1.47091 0.735455 − 0.677573i \(-0.236968\pi\)
0.735455 + 0.677573i \(0.236968\pi\)
\(462\) 0 0
\(463\) −3.00451e9 −1.40682 −0.703412 − 0.710782i \(-0.748341\pi\)
−0.703412 + 0.710782i \(0.748341\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.99252e9 1.35965 0.679825 − 0.733374i \(-0.262056\pi\)
0.679825 + 0.733374i \(0.262056\pi\)
\(468\) 0 0
\(469\) −5.18636e9 −2.32144
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.14681e9 0.498286
\(474\) 0 0
\(475\) −1.61911e9 −0.693183
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.84041e9 0.765141 0.382570 − 0.923926i \(-0.375039\pi\)
0.382570 + 0.923926i \(0.375039\pi\)
\(480\) 0 0
\(481\) 1.38644e8 0.0568058
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.62854e8 0.184225
\(486\) 0 0
\(487\) 4.26676e8 0.167397 0.0836983 − 0.996491i \(-0.473327\pi\)
0.0836983 + 0.996491i \(0.473327\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.07547e7 −0.0231630 −0.0115815 − 0.999933i \(-0.503687\pi\)
−0.0115815 + 0.999933i \(0.503687\pi\)
\(492\) 0 0
\(493\) −2.74876e8 −0.103317
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.00866e9 3.29164
\(498\) 0 0
\(499\) 3.24588e9 1.16945 0.584723 − 0.811233i \(-0.301203\pi\)
0.584723 + 0.811233i \(0.301203\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.44381e8 −0.260800 −0.130400 − 0.991461i \(-0.541626\pi\)
−0.130400 + 0.991461i \(0.541626\pi\)
\(504\) 0 0
\(505\) 1.97014e9 0.680734
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.44155e8 −0.149287 −0.0746436 − 0.997210i \(-0.523782\pi\)
−0.0746436 + 0.997210i \(0.523782\pi\)
\(510\) 0 0
\(511\) 4.19200e9 1.38979
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.63730e9 −0.528207
\(516\) 0 0
\(517\) 3.18045e9 1.01221
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.04963e9 −0.944745 −0.472372 − 0.881399i \(-0.656602\pi\)
−0.472372 + 0.881399i \(0.656602\pi\)
\(522\) 0 0
\(523\) −1.40306e9 −0.428866 −0.214433 − 0.976739i \(-0.568790\pi\)
−0.214433 + 0.976739i \(0.568790\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.19002e8 −0.0651795
\(528\) 0 0
\(529\) −1.68689e9 −0.495441
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.42977e9 0.695058
\(534\) 0 0
\(535\) 7.35802e8 0.207741
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.21728e10 −3.34835
\(540\) 0 0
\(541\) −4.21106e9 −1.14341 −0.571704 − 0.820460i \(-0.693717\pi\)
−0.571704 + 0.820460i \(0.693717\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.00823e8 −0.0266791
\(546\) 0 0
\(547\) 1.99956e9 0.522371 0.261185 − 0.965289i \(-0.415887\pi\)
0.261185 + 0.965289i \(0.415887\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.03442e9 −0.263432
\(552\) 0 0
\(553\) −6.00053e9 −1.50887
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.37403e9 −0.827287 −0.413643 − 0.910439i \(-0.635744\pi\)
−0.413643 + 0.910439i \(0.635744\pi\)
\(558\) 0 0
\(559\) −5.94678e8 −0.143993
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.58021e9 1.31787 0.658933 − 0.752201i \(-0.271008\pi\)
0.658933 + 0.752201i \(0.271008\pi\)
\(564\) 0 0
\(565\) 1.38310e9 0.322616
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.88310e8 0.202149 0.101074 − 0.994879i \(-0.467772\pi\)
0.101074 + 0.994879i \(0.467772\pi\)
\(570\) 0 0
\(571\) −1.79171e9 −0.402755 −0.201377 − 0.979514i \(-0.564542\pi\)
−0.201377 + 0.979514i \(0.564542\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.69947e9 −0.592162
\(576\) 0 0
\(577\) 3.82103e9 0.828066 0.414033 − 0.910262i \(-0.364120\pi\)
0.414033 + 0.910262i \(0.364120\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.51364e9 −0.743260
\(582\) 0 0
\(583\) −5.76352e9 −1.20461
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.36219e9 −0.890166 −0.445083 − 0.895489i \(-0.646826\pi\)
−0.445083 + 0.895489i \(0.646826\pi\)
\(588\) 0 0
\(589\) −8.24159e8 −0.166191
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.38531e9 −1.25745 −0.628724 − 0.777628i \(-0.716423\pi\)
−0.628724 + 0.777628i \(0.716423\pi\)
\(594\) 0 0
\(595\) −1.18686e9 −0.230988
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.04297e8 0.152905 0.0764527 − 0.997073i \(-0.475641\pi\)
0.0764527 + 0.997073i \(0.475641\pi\)
\(600\) 0 0
\(601\) −4.87162e9 −0.915403 −0.457702 − 0.889106i \(-0.651327\pi\)
−0.457702 + 0.889106i \(0.651327\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.90690e9 −0.717280
\(606\) 0 0
\(607\) −7.17517e9 −1.30218 −0.651091 − 0.759000i \(-0.725688\pi\)
−0.651091 + 0.759000i \(0.725688\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.64922e9 −0.292505
\(612\) 0 0
\(613\) −3.47891e9 −0.610002 −0.305001 − 0.952352i \(-0.598657\pi\)
−0.305001 + 0.952352i \(0.598657\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.39378e8 0.0410286 0.0205143 − 0.999790i \(-0.493470\pi\)
0.0205143 + 0.999790i \(0.493470\pi\)
\(618\) 0 0
\(619\) −5.52959e9 −0.937078 −0.468539 − 0.883443i \(-0.655219\pi\)
−0.468539 + 0.883443i \(0.655219\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.44215e9 −1.56445
\(624\) 0 0
\(625\) 3.22647e9 0.528626
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.40894e8 0.0385966
\(630\) 0 0
\(631\) 6.13683e9 0.972392 0.486196 − 0.873850i \(-0.338384\pi\)
0.486196 + 0.873850i \(0.338384\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 7.82959e8 0.121348
\(636\) 0 0
\(637\) 6.31221e9 0.967594
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.07038e10 −1.60522 −0.802611 − 0.596503i \(-0.796556\pi\)
−0.802611 + 0.596503i \(0.796556\pi\)
\(642\) 0 0
\(643\) −1.39803e9 −0.207385 −0.103692 − 0.994609i \(-0.533066\pi\)
−0.103692 + 0.994609i \(0.533066\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.31605e9 −0.771656 −0.385828 − 0.922571i \(-0.626084\pi\)
−0.385828 + 0.922571i \(0.626084\pi\)
\(648\) 0 0
\(649\) 5.46337e9 0.784520
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.24403e9 0.455921 0.227960 − 0.973670i \(-0.426794\pi\)
0.227960 + 0.973670i \(0.426794\pi\)
\(654\) 0 0
\(655\) −4.50537e9 −0.626449
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.16506e9 0.703034 0.351517 − 0.936181i \(-0.385666\pi\)
0.351517 + 0.936181i \(0.385666\pi\)
\(660\) 0 0
\(661\) 3.22515e9 0.434356 0.217178 − 0.976132i \(-0.430315\pi\)
0.217178 + 0.976132i \(0.430315\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.46645e9 −0.588961
\(666\) 0 0
\(667\) −1.72465e9 −0.225041
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.21757e10 −1.55583
\(672\) 0 0
\(673\) −2.00633e9 −0.253718 −0.126859 − 0.991921i \(-0.540490\pi\)
−0.126859 + 0.991921i \(0.540490\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.00211e10 1.24124 0.620619 − 0.784112i \(-0.286881\pi\)
0.620619 + 0.784112i \(0.286881\pi\)
\(678\) 0 0
\(679\) −6.39876e9 −0.784425
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.84861e9 0.702393 0.351196 − 0.936302i \(-0.385775\pi\)
0.351196 + 0.936302i \(0.385775\pi\)
\(684\) 0 0
\(685\) 2.18584e9 0.259838
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.98867e9 0.348105
\(690\) 0 0
\(691\) 2.58686e9 0.298263 0.149131 − 0.988817i \(-0.452352\pi\)
0.149131 + 0.988817i \(0.452352\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.50992e9 −0.170611
\(696\) 0 0
\(697\) 4.22175e9 0.472256
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.74460e9 −0.191286 −0.0956429 − 0.995416i \(-0.530491\pi\)
−0.0956429 + 0.995416i \(0.530491\pi\)
\(702\) 0 0
\(703\) 9.06545e8 0.0984114
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.72363e10 −2.89855
\(708\) 0 0
\(709\) 1.12051e10 1.18074 0.590368 − 0.807134i \(-0.298983\pi\)
0.590368 + 0.807134i \(0.298983\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.37408e9 −0.141971
\(714\) 0 0
\(715\) 3.17789e9 0.325138
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −9.36568e8 −0.0939698 −0.0469849 − 0.998896i \(-0.514961\pi\)
−0.0469849 + 0.998896i \(0.514961\pi\)
\(720\) 0 0
\(721\) 2.26350e10 2.24909
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.71002e9 0.264113
\(726\) 0 0
\(727\) −4.20445e9 −0.405825 −0.202913 − 0.979197i \(-0.565041\pi\)
−0.202913 + 0.979197i \(0.565041\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.03326e9 −0.0978358
\(732\) 0 0
\(733\) 1.15491e10 1.08314 0.541571 − 0.840655i \(-0.317830\pi\)
0.541571 + 0.840655i \(0.317830\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.41284e10 2.22020
\(738\) 0 0
\(739\) −1.39655e10 −1.27292 −0.636460 − 0.771310i \(-0.719602\pi\)
−0.636460 + 0.771310i \(0.719602\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.43832e10 1.28646 0.643230 − 0.765673i \(-0.277594\pi\)
0.643230 + 0.765673i \(0.277594\pi\)
\(744\) 0 0
\(745\) −6.53932e9 −0.579409
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.01721e10 −0.884557
\(750\) 0 0
\(751\) 6.70841e8 0.0577936 0.0288968 − 0.999582i \(-0.490801\pi\)
0.0288968 + 0.999582i \(0.490801\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −3.54395e9 −0.299691
\(756\) 0 0
\(757\) 1.91569e10 1.60506 0.802529 − 0.596613i \(-0.203487\pi\)
0.802529 + 0.596613i \(0.203487\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.79120e10 −1.47332 −0.736660 − 0.676263i \(-0.763598\pi\)
−0.736660 + 0.676263i \(0.763598\pi\)
\(762\) 0 0
\(763\) 1.39383e9 0.113599
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.83302e9 −0.226708
\(768\) 0 0
\(769\) −2.14072e10 −1.69753 −0.848765 − 0.528771i \(-0.822653\pi\)
−0.848765 + 0.528771i \(0.822653\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.55163e8 −0.0588047 −0.0294024 − 0.999568i \(-0.509360\pi\)
−0.0294024 + 0.999568i \(0.509360\pi\)
\(774\) 0 0
\(775\) 2.15916e9 0.166620
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.58875e10 1.20413
\(780\) 0 0
\(781\) −4.19108e10 −3.14809
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.85132e9 −0.284162
\(786\) 0 0
\(787\) 2.04665e10 1.49669 0.748347 − 0.663308i \(-0.230848\pi\)
0.748347 + 0.663308i \(0.230848\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.91208e10 −1.37369
\(792\) 0 0
\(793\) 6.31367e9 0.449599
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.03098e10 −0.721348 −0.360674 − 0.932692i \(-0.617453\pi\)
−0.360674 + 0.932692i \(0.617453\pi\)
\(798\) 0 0
\(799\) −2.86552e9 −0.198742
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.95024e10 −1.32918
\(804\) 0 0
\(805\) −7.44671e9 −0.503129
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.16428e8 0.0276516 0.0138258 − 0.999904i \(-0.495599\pi\)
0.0138258 + 0.999904i \(0.495599\pi\)
\(810\) 0 0
\(811\) −5.82687e9 −0.383586 −0.191793 − 0.981435i \(-0.561430\pi\)
−0.191793 + 0.981435i \(0.561430\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.14391e9 −0.462258
\(816\) 0 0
\(817\) −3.88840e9 −0.249456
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.08333e10 −1.31388 −0.656941 − 0.753942i \(-0.728150\pi\)
−0.656941 + 0.753942i \(0.728150\pi\)
\(822\) 0 0
\(823\) −4.23403e9 −0.264761 −0.132381 − 0.991199i \(-0.542262\pi\)
−0.132381 + 0.991199i \(0.542262\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.70597e9 −0.350800 −0.175400 − 0.984497i \(-0.556122\pi\)
−0.175400 + 0.984497i \(0.556122\pi\)
\(828\) 0 0
\(829\) −2.51612e10 −1.53388 −0.766940 − 0.641719i \(-0.778221\pi\)
−0.766940 + 0.641719i \(0.778221\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.09675e10 0.657431
\(834\) 0 0
\(835\) −7.14862e9 −0.424932
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.27048e10 1.32724 0.663622 − 0.748068i \(-0.269018\pi\)
0.663622 + 0.748068i \(0.269018\pi\)
\(840\) 0 0
\(841\) −1.55185e10 −0.899629
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.50544e9 0.313901
\(846\) 0 0
\(847\) 5.40112e10 3.05416
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.51144e9 0.0840695
\(852\) 0 0
\(853\) −2.86872e10 −1.58258 −0.791292 − 0.611439i \(-0.790591\pi\)
−0.791292 + 0.611439i \(0.790591\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.34950e9 −0.290322 −0.145161 − 0.989408i \(-0.546370\pi\)
−0.145161 + 0.989408i \(0.546370\pi\)
\(858\) 0 0
\(859\) −1.40330e10 −0.755394 −0.377697 − 0.925929i \(-0.623284\pi\)
−0.377697 + 0.925929i \(0.623284\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.24994e10 1.72122 0.860612 − 0.509261i \(-0.170081\pi\)
0.860612 + 0.509261i \(0.170081\pi\)
\(864\) 0 0
\(865\) 3.08394e9 0.162013
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.79162e10 1.44307
\(870\) 0 0
\(871\) −1.25118e10 −0.641586
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.57376e10 1.29879
\(876\) 0 0
\(877\) 3.38694e10 1.69554 0.847772 − 0.530361i \(-0.177944\pi\)
0.847772 + 0.530361i \(0.177944\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.52708e10 0.752397 0.376198 − 0.926539i \(-0.377231\pi\)
0.376198 + 0.926539i \(0.377231\pi\)
\(882\) 0 0
\(883\) −1.12045e10 −0.547685 −0.273842 − 0.961775i \(-0.588295\pi\)
−0.273842 + 0.961775i \(0.588295\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.97232e9 0.335463 0.167732 − 0.985833i \(-0.446356\pi\)
0.167732 + 0.985833i \(0.446356\pi\)
\(888\) 0 0
\(889\) −1.08241e10 −0.516695
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.07837e10 −0.506742
\(894\) 0 0
\(895\) −1.53080e10 −0.713737
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.37945e9 0.0633211
\(900\) 0 0
\(901\) 5.19283e9 0.236520
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.30713e10 0.586204
\(906\) 0 0
\(907\) 1.18095e9 0.0525539 0.0262770 − 0.999655i \(-0.491635\pi\)
0.0262770 + 0.999655i \(0.491635\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.27915e10 0.560541 0.280271 − 0.959921i \(-0.409576\pi\)
0.280271 + 0.959921i \(0.409576\pi\)
\(912\) 0 0
\(913\) 1.63465e10 0.710847
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.22848e10 2.66741
\(918\) 0 0
\(919\) −3.52353e10 −1.49752 −0.748761 − 0.662840i \(-0.769351\pi\)
−0.748761 + 0.662840i \(0.769351\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.17328e10 0.909725
\(924\) 0 0
\(925\) −2.37499e9 −0.0986658
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.17764e10 0.891111 0.445556 − 0.895254i \(-0.353006\pi\)
0.445556 + 0.895254i \(0.353006\pi\)
\(930\) 0 0
\(931\) 4.12734e10 1.67628
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5.52161e9 0.220915
\(936\) 0 0
\(937\) 1.15795e10 0.459833 0.229916 − 0.973210i \(-0.426155\pi\)
0.229916 + 0.973210i \(0.426155\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.83930e10 −1.50207 −0.751033 − 0.660265i \(-0.770444\pi\)
−0.751033 + 0.660265i \(0.770444\pi\)
\(942\) 0 0
\(943\) 2.64885e10 1.02865
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.11841e10 −0.427933 −0.213966 − 0.976841i \(-0.568638\pi\)
−0.213966 + 0.976841i \(0.568638\pi\)
\(948\) 0 0
\(949\) 1.01129e10 0.384101
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.19835e9 0.194554 0.0972770 − 0.995257i \(-0.468987\pi\)
0.0972770 + 0.995257i \(0.468987\pi\)
\(954\) 0 0
\(955\) −1.86510e9 −0.0692931
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.02183e10 −1.10638
\(960\) 0 0
\(961\) −2.64136e10 −0.960053
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.75786e10 0.629708
\(966\) 0 0
\(967\) 2.69243e8 0.00957529 0.00478765 − 0.999989i \(-0.498476\pi\)
0.00478765 + 0.999989i \(0.498476\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.37283e9 −0.153284 −0.0766418 − 0.997059i \(-0.524420\pi\)
−0.0766418 + 0.997059i \(0.524420\pi\)
\(972\) 0 0
\(973\) 2.08740e10 0.726457
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.74991e10 1.28644 0.643220 − 0.765681i \(-0.277598\pi\)
0.643220 + 0.765681i \(0.277598\pi\)
\(978\) 0 0
\(979\) 4.39276e10 1.49623
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.06190e10 −1.02814 −0.514071 − 0.857748i \(-0.671863\pi\)
−0.514071 + 0.857748i \(0.671863\pi\)
\(984\) 0 0
\(985\) −9.48808e9 −0.316338
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.48296e9 −0.213102
\(990\) 0 0
\(991\) 1.72703e10 0.563693 0.281847 − 0.959459i \(-0.409053\pi\)
0.281847 + 0.959459i \(0.409053\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −8.68363e9 −0.279461
\(996\) 0 0
\(997\) 3.75077e9 0.119864 0.0599319 − 0.998202i \(-0.480912\pi\)
0.0599319 + 0.998202i \(0.480912\pi\)
\(998\) 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 576.8.a.i.1.1 1
3.2 odd 2 192.8.a.n.1.1 1
4.3 odd 2 576.8.a.h.1.1 1
8.3 odd 2 18.8.a.a.1.1 1
8.5 even 2 144.8.a.h.1.1 1
12.11 even 2 192.8.a.f.1.1 1
24.5 odd 2 48.8.a.b.1.1 1
24.11 even 2 6.8.a.a.1.1 ✓ 1
40.3 even 4 450.8.c.a.199.2 2
40.19 odd 2 450.8.a.ba.1.1 1
40.27 even 4 450.8.c.a.199.1 2
72.11 even 6 162.8.c.d.109.1 2
72.43 odd 6 162.8.c.i.109.1 2
72.59 even 6 162.8.c.d.55.1 2
72.67 odd 6 162.8.c.i.55.1 2
120.59 even 2 150.8.a.e.1.1 1
120.83 odd 4 150.8.c.k.49.1 2
120.107 odd 4 150.8.c.k.49.2 2
168.11 even 6 294.8.e.c.79.1 2
168.59 odd 6 294.8.e.d.79.1 2
168.83 odd 2 294.8.a.l.1.1 1
168.107 even 6 294.8.e.c.67.1 2
168.131 odd 6 294.8.e.d.67.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.8.a.a.1.1 ✓ 1 24.11 even 2
18.8.a.a.1.1 1 8.3 odd 2
48.8.a.b.1.1 1 24.5 odd 2
144.8.a.h.1.1 1 8.5 even 2
150.8.a.e.1.1 1 120.59 even 2
150.8.c.k.49.1 2 120.83 odd 4
150.8.c.k.49.2 2 120.107 odd 4
162.8.c.d.55.1 2 72.59 even 6
162.8.c.d.109.1 2 72.11 even 6
162.8.c.i.55.1 2 72.67 odd 6
162.8.c.i.109.1 2 72.43 odd 6
192.8.a.f.1.1 1 12.11 even 2
192.8.a.n.1.1 1 3.2 odd 2
294.8.a.l.1.1 1 168.83 odd 2
294.8.e.c.67.1 2 168.107 even 6
294.8.e.c.79.1 2 168.11 even 6
294.8.e.d.67.1 2 168.131 odd 6
294.8.e.d.79.1 2 168.59 odd 6
450.8.a.ba.1.1 1 40.19 odd 2
450.8.c.a.199.1 2 40.27 even 4
450.8.c.a.199.2 2 40.3 even 4
576.8.a.h.1.1 1 4.3 odd 2
576.8.a.i.1.1 1 1.1 even 1 trivial