Properties

Label 576.8.a.bl.1.1
Level $576$
Weight $8$
Character 576.1
Self dual yes
Analytic conductor $179.934$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{235}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 235 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 96)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-15.3297\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-155.275 q^{5} -1742.38 q^{7} +O(q^{10})\) \(q-155.275 q^{5} -1742.38 q^{7} +51.6521 q^{11} +10471.6 q^{13} -24021.5 q^{17} -1149.27 q^{19} +86508.9 q^{23} -54014.6 q^{25} +153332. q^{29} +132760. q^{31} +270548. q^{35} -504967. q^{37} -99416.4 q^{41} +84318.7 q^{43} +998165. q^{47} +2.21233e6 q^{49} +1.24958e6 q^{53} -8020.30 q^{55} +1.67870e6 q^{59} +405326. q^{61} -1.62598e6 q^{65} -2.16414e6 q^{67} +1.57024e6 q^{71} -2.88524e6 q^{73} -89997.5 q^{77} -3.87170e6 q^{79} -796558. q^{83} +3.72994e6 q^{85} -1.41571e6 q^{89} -1.82454e7 q^{91} +178454. q^{95} -238942. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 180 q^{5} - 1032 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 180 q^{5} - 1032 q^{7} - 2840 q^{11} + 340 q^{13} - 9780 q^{17} + 32040 q^{19} + 11136 q^{23} - 19730 q^{25} + 304212 q^{29} + 77640 q^{31} + 508720 q^{35} - 1015820 q^{37} - 704100 q^{41} - 395496 q^{43} + 1157488 q^{47} + 1893426 q^{49} + 1568580 q^{53} - 977520 q^{55} - 139240 q^{59} - 2603580 q^{61} - 5022840 q^{65} - 5289768 q^{67} + 5721760 q^{71} - 1190700 q^{73} - 2144160 q^{77} + 398280 q^{79} + 6986616 q^{83} + 8504760 q^{85} + 8166732 q^{89} - 25442640 q^{91} + 11306000 q^{95} - 10361500 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −155.275 −0.555530 −0.277765 0.960649i \(-0.589594\pi\)
−0.277765 + 0.960649i \(0.589594\pi\)
\(6\) 0 0
\(7\) −1742.38 −1.91999 −0.959995 0.280017i \(-0.909660\pi\)
−0.959995 + 0.280017i \(0.909660\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 51.6521 0.0117008 0.00585038 0.999983i \(-0.498138\pi\)
0.00585038 + 0.999983i \(0.498138\pi\)
\(12\) 0 0
\(13\) 10471.6 1.32193 0.660967 0.750415i \(-0.270146\pi\)
0.660967 + 0.750415i \(0.270146\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −24021.5 −1.18585 −0.592923 0.805259i \(-0.702026\pi\)
−0.592923 + 0.805259i \(0.702026\pi\)
\(18\) 0 0
\(19\) −1149.27 −0.0384403 −0.0192201 0.999815i \(-0.506118\pi\)
−0.0192201 + 0.999815i \(0.506118\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 86508.9 1.48256 0.741282 0.671194i \(-0.234218\pi\)
0.741282 + 0.671194i \(0.234218\pi\)
\(24\) 0 0
\(25\) −54014.6 −0.691386
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 153332. 1.16746 0.583728 0.811949i \(-0.301593\pi\)
0.583728 + 0.811949i \(0.301593\pi\)
\(30\) 0 0
\(31\) 132760. 0.800392 0.400196 0.916430i \(-0.368942\pi\)
0.400196 + 0.916430i \(0.368942\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 270548. 1.06661
\(36\) 0 0
\(37\) −504967. −1.63892 −0.819458 0.573139i \(-0.805725\pi\)
−0.819458 + 0.573139i \(0.805725\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −99416.4 −0.225276 −0.112638 0.993636i \(-0.535930\pi\)
−0.112638 + 0.993636i \(0.535930\pi\)
\(42\) 0 0
\(43\) 84318.7 0.161728 0.0808638 0.996725i \(-0.474232\pi\)
0.0808638 + 0.996725i \(0.474232\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 998165. 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(48\) 0 0
\(49\) 2.21233e6 2.68636
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.24958e6 1.15292 0.576458 0.817127i \(-0.304434\pi\)
0.576458 + 0.817127i \(0.304434\pi\)
\(54\) 0 0
\(55\) −8020.30 −0.00650012
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.67870e6 1.06412 0.532061 0.846706i \(-0.321418\pi\)
0.532061 + 0.846706i \(0.321418\pi\)
\(60\) 0 0
\(61\) 405326. 0.228639 0.114320 0.993444i \(-0.463531\pi\)
0.114320 + 0.993444i \(0.463531\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.62598e6 −0.734374
\(66\) 0 0
\(67\) −2.16414e6 −0.879072 −0.439536 0.898225i \(-0.644857\pi\)
−0.439536 + 0.898225i \(0.644857\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.57024e6 0.520669 0.260335 0.965518i \(-0.416167\pi\)
0.260335 + 0.965518i \(0.416167\pi\)
\(72\) 0 0
\(73\) −2.88524e6 −0.868065 −0.434032 0.900897i \(-0.642910\pi\)
−0.434032 + 0.900897i \(0.642910\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −89997.5 −0.0224653
\(78\) 0 0
\(79\) −3.87170e6 −0.883500 −0.441750 0.897138i \(-0.645642\pi\)
−0.441750 + 0.897138i \(0.645642\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −796558. −0.152913 −0.0764564 0.997073i \(-0.524361\pi\)
−0.0764564 + 0.997073i \(0.524361\pi\)
\(84\) 0 0
\(85\) 3.72994e6 0.658773
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.41571e6 −0.212867 −0.106434 0.994320i \(-0.533943\pi\)
−0.106434 + 0.994320i \(0.533943\pi\)
\(90\) 0 0
\(91\) −1.82454e7 −2.53810
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 178454. 0.0213547
\(96\) 0 0
\(97\) −238942. −0.0265822 −0.0132911 0.999912i \(-0.504231\pi\)
−0.0132911 + 0.999912i \(0.504231\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.87275e7 −1.80865 −0.904324 0.426847i \(-0.859624\pi\)
−0.904324 + 0.426847i \(0.859624\pi\)
\(102\) 0 0
\(103\) 4.67587e6 0.421630 0.210815 0.977526i \(-0.432388\pi\)
0.210815 + 0.977526i \(0.432388\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.38526e7 −1.09317 −0.546585 0.837404i \(-0.684072\pi\)
−0.546585 + 0.837404i \(0.684072\pi\)
\(108\) 0 0
\(109\) −5.05778e6 −0.374083 −0.187041 0.982352i \(-0.559890\pi\)
−0.187041 + 0.982352i \(0.559890\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.79817e7 1.82431 0.912157 0.409841i \(-0.134416\pi\)
0.912157 + 0.409841i \(0.134416\pi\)
\(114\) 0 0
\(115\) −1.34327e7 −0.823608
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.18545e7 2.27681
\(120\) 0 0
\(121\) −1.94845e7 −0.999863
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.05180e7 0.939616
\(126\) 0 0
\(127\) 4.22490e6 0.183022 0.0915109 0.995804i \(-0.470830\pi\)
0.0915109 + 0.995804i \(0.470830\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.16738e7 1.61962 0.809812 0.586690i \(-0.199569\pi\)
0.809812 + 0.586690i \(0.199569\pi\)
\(132\) 0 0
\(133\) 2.00247e6 0.0738049
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.58796e7 −0.859876 −0.429938 0.902858i \(-0.641465\pi\)
−0.429938 + 0.902858i \(0.641465\pi\)
\(138\) 0 0
\(139\) −5.07048e7 −1.60139 −0.800696 0.599071i \(-0.795537\pi\)
−0.800696 + 0.599071i \(0.795537\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 540879. 0.0154676
\(144\) 0 0
\(145\) −2.38087e7 −0.648557
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.36811e7 −1.08179 −0.540893 0.841091i \(-0.681914\pi\)
−0.540893 + 0.841091i \(0.681914\pi\)
\(150\) 0 0
\(151\) 6.81915e7 1.61180 0.805900 0.592052i \(-0.201682\pi\)
0.805900 + 0.592052i \(0.201682\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.06144e7 −0.444642
\(156\) 0 0
\(157\) −9.99359e6 −0.206098 −0.103049 0.994676i \(-0.532860\pi\)
−0.103049 + 0.994676i \(0.532860\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.50731e8 −2.84651
\(162\) 0 0
\(163\) 4.77375e6 0.0863382 0.0431691 0.999068i \(-0.486255\pi\)
0.0431691 + 0.999068i \(0.486255\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.90961e7 1.48030 0.740152 0.672440i \(-0.234754\pi\)
0.740152 + 0.672440i \(0.234754\pi\)
\(168\) 0 0
\(169\) 4.69052e7 0.747510
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.58721e7 −0.379901 −0.189950 0.981794i \(-0.560833\pi\)
−0.189950 + 0.981794i \(0.560833\pi\)
\(174\) 0 0
\(175\) 9.41137e7 1.32745
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.29009e7 −0.559089 −0.279545 0.960133i \(-0.590183\pi\)
−0.279545 + 0.960133i \(0.590183\pi\)
\(180\) 0 0
\(181\) −1.28747e8 −1.61384 −0.806920 0.590661i \(-0.798867\pi\)
−0.806920 + 0.590661i \(0.798867\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.84089e7 0.910467
\(186\) 0 0
\(187\) −1.24076e6 −0.0138753
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.60958e8 −1.67146 −0.835730 0.549140i \(-0.814955\pi\)
−0.835730 + 0.549140i \(0.814955\pi\)
\(192\) 0 0
\(193\) 1.61920e8 1.62125 0.810625 0.585566i \(-0.199128\pi\)
0.810625 + 0.585566i \(0.199128\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.27020e8 1.18370 0.591848 0.806049i \(-0.298398\pi\)
0.591848 + 0.806049i \(0.298398\pi\)
\(198\) 0 0
\(199\) 1.74928e7 0.157353 0.0786763 0.996900i \(-0.474931\pi\)
0.0786763 + 0.996900i \(0.474931\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.67163e8 −2.24151
\(204\) 0 0
\(205\) 1.54369e7 0.125147
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −59362.5 −0.000449780 0
\(210\) 0 0
\(211\) −4.40594e7 −0.322887 −0.161443 0.986882i \(-0.551615\pi\)
−0.161443 + 0.986882i \(0.551615\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.30926e7 −0.0898445
\(216\) 0 0
\(217\) −2.31319e8 −1.53674
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.51542e8 −1.56761
\(222\) 0 0
\(223\) 1.25971e8 0.760685 0.380343 0.924846i \(-0.375806\pi\)
0.380343 + 0.924846i \(0.375806\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.13624e8 −0.644733 −0.322366 0.946615i \(-0.604478\pi\)
−0.322366 + 0.946615i \(0.604478\pi\)
\(228\) 0 0
\(229\) 1.65608e8 0.911289 0.455645 0.890162i \(-0.349409\pi\)
0.455645 + 0.890162i \(0.349409\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.77584e8 −1.43764 −0.718818 0.695199i \(-0.755316\pi\)
−0.718818 + 0.695199i \(0.755316\pi\)
\(234\) 0 0
\(235\) −1.54990e8 −0.779054
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.00082e8 −1.42183 −0.710915 0.703278i \(-0.751719\pi\)
−0.710915 + 0.703278i \(0.751719\pi\)
\(240\) 0 0
\(241\) −3.72418e8 −1.71384 −0.856922 0.515446i \(-0.827626\pi\)
−0.856922 + 0.515446i \(0.827626\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.43521e8 −1.49235
\(246\) 0 0
\(247\) −1.20347e7 −0.0508155
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.44996e8 0.977916 0.488958 0.872307i \(-0.337377\pi\)
0.488958 + 0.872307i \(0.337377\pi\)
\(252\) 0 0
\(253\) 4.46837e6 0.0173471
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.71767e7 0.210113 0.105057 0.994466i \(-0.466498\pi\)
0.105057 + 0.994466i \(0.466498\pi\)
\(258\) 0 0
\(259\) 8.79842e8 3.14670
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.63071e7 0.292551 0.146275 0.989244i \(-0.453271\pi\)
0.146275 + 0.989244i \(0.453271\pi\)
\(264\) 0 0
\(265\) −1.94029e8 −0.640479
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.72851e8 −0.541425 −0.270712 0.962660i \(-0.587259\pi\)
−0.270712 + 0.962660i \(0.587259\pi\)
\(270\) 0 0
\(271\) −2.93066e8 −0.894484 −0.447242 0.894413i \(-0.647594\pi\)
−0.447242 + 0.894413i \(0.647594\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.78997e6 −0.00808974
\(276\) 0 0
\(277\) −4.02094e8 −1.13671 −0.568353 0.822785i \(-0.692419\pi\)
−0.568353 + 0.822785i \(0.692419\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −2.56183e8 −0.688777 −0.344388 0.938827i \(-0.611914\pi\)
−0.344388 + 0.938827i \(0.611914\pi\)
\(282\) 0 0
\(283\) −7.55378e7 −0.198112 −0.0990562 0.995082i \(-0.531582\pi\)
−0.0990562 + 0.995082i \(0.531582\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.73221e8 0.432527
\(288\) 0 0
\(289\) 1.66693e8 0.406232
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.87808e8 −1.59746 −0.798731 0.601688i \(-0.794495\pi\)
−0.798731 + 0.601688i \(0.794495\pi\)
\(294\) 0 0
\(295\) −2.60661e8 −0.591152
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.05883e8 1.95985
\(300\) 0 0
\(301\) −1.46915e8 −0.310515
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.29372e7 −0.127016
\(306\) 0 0
\(307\) −2.58303e8 −0.509501 −0.254751 0.967007i \(-0.581993\pi\)
−0.254751 + 0.967007i \(0.581993\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.14264e8 0.403913 0.201957 0.979394i \(-0.435270\pi\)
0.201957 + 0.979394i \(0.435270\pi\)
\(312\) 0 0
\(313\) 4.31251e8 0.794923 0.397461 0.917619i \(-0.369891\pi\)
0.397461 + 0.917619i \(0.369891\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.13425e8 0.199986 0.0999932 0.994988i \(-0.468118\pi\)
0.0999932 + 0.994988i \(0.468118\pi\)
\(318\) 0 0
\(319\) 7.91994e6 0.0136601
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.76073e7 0.0455842
\(324\) 0 0
\(325\) −5.65617e8 −0.913967
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.73918e9 −2.69252
\(330\) 0 0
\(331\) 7.46930e8 1.13209 0.566046 0.824374i \(-0.308472\pi\)
0.566046 + 0.824374i \(0.308472\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.36038e8 0.488351
\(336\) 0 0
\(337\) −5.55138e8 −0.790126 −0.395063 0.918654i \(-0.629277\pi\)
−0.395063 + 0.918654i \(0.629277\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.85736e6 0.00936519
\(342\) 0 0
\(343\) −2.41980e9 −3.23780
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.45452e9 1.86881 0.934407 0.356206i \(-0.115930\pi\)
0.934407 + 0.356206i \(0.115930\pi\)
\(348\) 0 0
\(349\) 7.79394e7 0.0981450 0.0490725 0.998795i \(-0.484373\pi\)
0.0490725 + 0.998795i \(0.484373\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.54961e8 −1.03451 −0.517255 0.855831i \(-0.673046\pi\)
−0.517255 + 0.855831i \(0.673046\pi\)
\(354\) 0 0
\(355\) −2.43820e8 −0.289248
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.81451e8 −0.549189 −0.274595 0.961560i \(-0.588544\pi\)
−0.274595 + 0.961560i \(0.588544\pi\)
\(360\) 0 0
\(361\) −8.92551e8 −0.998522
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.48007e8 0.482236
\(366\) 0 0
\(367\) 1.01203e9 1.06872 0.534359 0.845257i \(-0.320553\pi\)
0.534359 + 0.845257i \(0.320553\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.17723e9 −2.21359
\(372\) 0 0
\(373\) −1.31785e9 −1.31487 −0.657437 0.753510i \(-0.728359\pi\)
−0.657437 + 0.753510i \(0.728359\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.60563e9 1.54330
\(378\) 0 0
\(379\) −1.26791e9 −1.19633 −0.598167 0.801372i \(-0.704104\pi\)
−0.598167 + 0.801372i \(0.704104\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.44181e8 −0.676835 −0.338417 0.940996i \(-0.609892\pi\)
−0.338417 + 0.940996i \(0.609892\pi\)
\(384\) 0 0
\(385\) 1.39744e7 0.0124802
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.14243e8 −0.356806 −0.178403 0.983958i \(-0.557093\pi\)
−0.178403 + 0.983958i \(0.557093\pi\)
\(390\) 0 0
\(391\) −2.07807e9 −1.75809
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.01179e8 0.490810
\(396\) 0 0
\(397\) 2.05816e8 0.165087 0.0825436 0.996587i \(-0.473696\pi\)
0.0825436 + 0.996587i \(0.473696\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.86415e9 −1.44369 −0.721846 0.692054i \(-0.756706\pi\)
−0.721846 + 0.692054i \(0.756706\pi\)
\(402\) 0 0
\(403\) 1.39021e9 1.05807
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.60826e7 −0.0191766
\(408\) 0 0
\(409\) 9.07532e8 0.655889 0.327944 0.944697i \(-0.393644\pi\)
0.327944 + 0.944697i \(0.393644\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −2.92493e9 −2.04311
\(414\) 0 0
\(415\) 1.23686e8 0.0849477
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.53137e8 0.300941 0.150470 0.988615i \(-0.451921\pi\)
0.150470 + 0.988615i \(0.451921\pi\)
\(420\) 0 0
\(421\) 1.62727e9 1.06285 0.531425 0.847106i \(-0.321657\pi\)
0.531425 + 0.847106i \(0.321657\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.29751e9 0.819878
\(426\) 0 0
\(427\) −7.06231e8 −0.438985
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.84387e8 −0.592237 −0.296118 0.955151i \(-0.595692\pi\)
−0.296118 + 0.955151i \(0.595692\pi\)
\(432\) 0 0
\(433\) 1.67333e9 0.990545 0.495273 0.868738i \(-0.335068\pi\)
0.495273 + 0.868738i \(0.335068\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.94225e7 −0.0569901
\(438\) 0 0
\(439\) 2.78714e9 1.57229 0.786147 0.618040i \(-0.212073\pi\)
0.786147 + 0.618040i \(0.212073\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.72313e9 0.941683 0.470841 0.882218i \(-0.343950\pi\)
0.470841 + 0.882218i \(0.343950\pi\)
\(444\) 0 0
\(445\) 2.19824e8 0.118254
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.13732e9 1.11431 0.557157 0.830407i \(-0.311892\pi\)
0.557157 + 0.830407i \(0.311892\pi\)
\(450\) 0 0
\(451\) −5.13507e6 −0.00263590
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.83306e9 1.40999
\(456\) 0 0
\(457\) −2.56149e9 −1.25541 −0.627706 0.778451i \(-0.716006\pi\)
−0.627706 + 0.778451i \(0.716006\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.96595e8 0.473768 0.236884 0.971538i \(-0.423874\pi\)
0.236884 + 0.971538i \(0.423874\pi\)
\(462\) 0 0
\(463\) 2.61394e9 1.22395 0.611973 0.790879i \(-0.290376\pi\)
0.611973 + 0.790879i \(0.290376\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.57019e9 −1.16777 −0.583883 0.811838i \(-0.698467\pi\)
−0.583883 + 0.811838i \(0.698467\pi\)
\(468\) 0 0
\(469\) 3.77075e9 1.68781
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 4.35524e6 0.00189233
\(474\) 0 0
\(475\) 6.20776e7 0.0265771
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.93225e8 0.329778 0.164889 0.986312i \(-0.447273\pi\)
0.164889 + 0.986312i \(0.447273\pi\)
\(480\) 0 0
\(481\) −5.28779e9 −2.16654
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.71018e7 0.0147672
\(486\) 0 0
\(487\) 1.26410e9 0.495940 0.247970 0.968768i \(-0.420237\pi\)
0.247970 + 0.968768i \(0.420237\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.37678e9 −1.28741 −0.643706 0.765273i \(-0.722604\pi\)
−0.643706 + 0.765273i \(0.722604\pi\)
\(492\) 0 0
\(493\) −3.68327e9 −1.38442
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.73595e9 −0.999680
\(498\) 0 0
\(499\) 9.76331e8 0.351759 0.175879 0.984412i \(-0.443723\pi\)
0.175879 + 0.984412i \(0.443723\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.35477e9 0.474654 0.237327 0.971430i \(-0.423729\pi\)
0.237327 + 0.971430i \(0.423729\pi\)
\(504\) 0 0
\(505\) 2.90791e9 1.00476
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.02597e9 0.680959 0.340479 0.940252i \(-0.389411\pi\)
0.340479 + 0.940252i \(0.389411\pi\)
\(510\) 0 0
\(511\) 5.02718e9 1.66668
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.26047e8 −0.234228
\(516\) 0 0
\(517\) 5.15573e7 0.0164087
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.99361e8 0.0617601 0.0308800 0.999523i \(-0.490169\pi\)
0.0308800 + 0.999523i \(0.490169\pi\)
\(522\) 0 0
\(523\) −4.02783e9 −1.23116 −0.615581 0.788074i \(-0.711079\pi\)
−0.615581 + 0.788074i \(0.711079\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.18910e9 −0.949142
\(528\) 0 0
\(529\) 4.07896e9 1.19799
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.04105e9 −0.297800
\(534\) 0 0
\(535\) 2.15096e9 0.607288
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.14272e8 0.0314325
\(540\) 0 0
\(541\) −1.55841e9 −0.423148 −0.211574 0.977362i \(-0.567859\pi\)
−0.211574 + 0.977362i \(0.567859\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.85349e8 0.207814
\(546\) 0 0
\(547\) 3.01460e9 0.787542 0.393771 0.919209i \(-0.371170\pi\)
0.393771 + 0.919209i \(0.371170\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.76221e8 −0.0448773
\(552\) 0 0
\(553\) 6.74595e9 1.69631
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.97397e9 1.70996 0.854981 0.518659i \(-0.173569\pi\)
0.854981 + 0.518659i \(0.173569\pi\)
\(558\) 0 0
\(559\) 8.82948e8 0.213793
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.91575e9 −1.16094 −0.580471 0.814281i \(-0.697131\pi\)
−0.580471 + 0.814281i \(0.697131\pi\)
\(564\) 0 0
\(565\) −4.34487e9 −1.01346
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.91661e9 0.663720 0.331860 0.943329i \(-0.392324\pi\)
0.331860 + 0.943329i \(0.392324\pi\)
\(570\) 0 0
\(571\) −2.78071e9 −0.625071 −0.312536 0.949906i \(-0.601178\pi\)
−0.312536 + 0.949906i \(0.601178\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.67274e9 −1.02502
\(576\) 0 0
\(577\) −1.95173e9 −0.422965 −0.211482 0.977382i \(-0.567829\pi\)
−0.211482 + 0.977382i \(0.567829\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.38790e9 0.293591
\(582\) 0 0
\(583\) 6.45433e7 0.0134900
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.31559e9 −1.08472 −0.542361 0.840146i \(-0.682469\pi\)
−0.542361 + 0.840146i \(0.682469\pi\)
\(588\) 0 0
\(589\) −1.52578e8 −0.0307673
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.17508e9 0.231407 0.115703 0.993284i \(-0.463088\pi\)
0.115703 + 0.993284i \(0.463088\pi\)
\(594\) 0 0
\(595\) −6.49897e9 −1.26484
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.88790e8 0.0929242 0.0464621 0.998920i \(-0.485205\pi\)
0.0464621 + 0.998920i \(0.485205\pi\)
\(600\) 0 0
\(601\) −1.92744e9 −0.362176 −0.181088 0.983467i \(-0.557962\pi\)
−0.181088 + 0.983467i \(0.557962\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.02546e9 0.555454
\(606\) 0 0
\(607\) −6.50300e9 −1.18019 −0.590097 0.807332i \(-0.700911\pi\)
−0.590097 + 0.807332i \(0.700911\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.04523e10 1.85383
\(612\) 0 0
\(613\) −2.40924e9 −0.422444 −0.211222 0.977438i \(-0.567744\pi\)
−0.211222 + 0.977438i \(0.567744\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.02270e9 1.37506 0.687531 0.726155i \(-0.258694\pi\)
0.687531 + 0.726155i \(0.258694\pi\)
\(618\) 0 0
\(619\) −7.38409e9 −1.25135 −0.625677 0.780083i \(-0.715177\pi\)
−0.625677 + 0.780083i \(0.715177\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.46670e9 0.408703
\(624\) 0 0
\(625\) 1.03395e9 0.169402
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.21300e10 1.94350
\(630\) 0 0
\(631\) −7.86728e8 −0.124659 −0.0623293 0.998056i \(-0.519853\pi\)
−0.0623293 + 0.998056i \(0.519853\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.56022e8 −0.101674
\(636\) 0 0
\(637\) 2.31666e10 3.55119
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −5.81312e8 −0.0871779 −0.0435890 0.999050i \(-0.513879\pi\)
−0.0435890 + 0.999050i \(0.513879\pi\)
\(642\) 0 0
\(643\) 8.39677e9 1.24559 0.622793 0.782386i \(-0.285998\pi\)
0.622793 + 0.782386i \(0.285998\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.27887e9 1.20173 0.600864 0.799351i \(-0.294823\pi\)
0.600864 + 0.799351i \(0.294823\pi\)
\(648\) 0 0
\(649\) 8.67086e7 0.0124510
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.05982e9 −0.992195 −0.496098 0.868267i \(-0.665234\pi\)
−0.496098 + 0.868267i \(0.665234\pi\)
\(654\) 0 0
\(655\) −6.47092e9 −0.899749
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.61653e9 −0.356145 −0.178072 0.984017i \(-0.556986\pi\)
−0.178072 + 0.984017i \(0.556986\pi\)
\(660\) 0 0
\(661\) 8.77438e9 1.18171 0.590855 0.806778i \(-0.298790\pi\)
0.590855 + 0.806778i \(0.298790\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.10934e8 −0.0410008
\(666\) 0 0
\(667\) 1.32646e10 1.73083
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.09360e7 0.00267525
\(672\) 0 0
\(673\) 4.93417e9 0.623967 0.311983 0.950088i \(-0.399007\pi\)
0.311983 + 0.950088i \(0.399007\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.15194e10 −1.42682 −0.713411 0.700746i \(-0.752851\pi\)
−0.713411 + 0.700746i \(0.752851\pi\)
\(678\) 0 0
\(679\) 4.16327e8 0.0510376
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.83610e9 0.700891 0.350445 0.936583i \(-0.386030\pi\)
0.350445 + 0.936583i \(0.386030\pi\)
\(684\) 0 0
\(685\) 4.01847e9 0.477687
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.30850e10 1.52408
\(690\) 0 0
\(691\) −1.34502e10 −1.55080 −0.775402 0.631468i \(-0.782453\pi\)
−0.775402 + 0.631468i \(0.782453\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.87321e9 0.889621
\(696\) 0 0
\(697\) 2.38813e9 0.267142
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.35211e9 1.02541 0.512704 0.858565i \(-0.328644\pi\)
0.512704 + 0.858565i \(0.328644\pi\)
\(702\) 0 0
\(703\) 5.80346e8 0.0630004
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.26303e10 3.47259
\(708\) 0 0
\(709\) −1.07639e10 −1.13425 −0.567126 0.823631i \(-0.691945\pi\)
−0.567126 + 0.823631i \(0.691945\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.14850e10 1.18663
\(714\) 0 0
\(715\) −8.39851e7 −0.00859273
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.95823e8 0.0296811 0.0148406 0.999890i \(-0.495276\pi\)
0.0148406 + 0.999890i \(0.495276\pi\)
\(720\) 0 0
\(721\) −8.14712e9 −0.809526
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.28218e9 −0.807164
\(726\) 0 0
\(727\) −7.92934e9 −0.765362 −0.382681 0.923881i \(-0.624999\pi\)
−0.382681 + 0.923881i \(0.624999\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.02546e9 −0.191784
\(732\) 0 0
\(733\) −1.06379e10 −0.997681 −0.498841 0.866694i \(-0.666241\pi\)
−0.498841 + 0.866694i \(0.666241\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.11783e8 −0.0102858
\(738\) 0 0
\(739\) 1.21560e10 1.10799 0.553993 0.832522i \(-0.313104\pi\)
0.553993 + 0.832522i \(0.313104\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.04486e9 −0.451220 −0.225610 0.974218i \(-0.572438\pi\)
−0.225610 + 0.974218i \(0.572438\pi\)
\(744\) 0 0
\(745\) 6.78260e9 0.600965
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.41364e10 2.09887
\(750\) 0 0
\(751\) −1.58691e10 −1.36714 −0.683568 0.729887i \(-0.739573\pi\)
−0.683568 + 0.729887i \(0.739573\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.05885e10 −0.895403
\(756\) 0 0
\(757\) −3.76028e9 −0.315054 −0.157527 0.987515i \(-0.550352\pi\)
−0.157527 + 0.987515i \(0.550352\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.38635e9 0.196285 0.0981427 0.995172i \(-0.468710\pi\)
0.0981427 + 0.995172i \(0.468710\pi\)
\(762\) 0 0
\(763\) 8.81256e9 0.718235
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.75786e10 1.40670
\(768\) 0 0
\(769\) −1.63901e9 −0.129969 −0.0649845 0.997886i \(-0.520700\pi\)
−0.0649845 + 0.997886i \(0.520700\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.01355e9 0.624017 0.312009 0.950079i \(-0.398998\pi\)
0.312009 + 0.950079i \(0.398998\pi\)
\(774\) 0 0
\(775\) −7.17100e9 −0.553380
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.14257e8 0.00865966
\(780\) 0 0
\(781\) 8.11063e7 0.00609223
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.55176e9 0.114493
\(786\) 0 0
\(787\) −1.21205e9 −0.0886361 −0.0443181 0.999017i \(-0.514111\pi\)
−0.0443181 + 0.999017i \(0.514111\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.87547e10 −3.50267
\(792\) 0 0
\(793\) 4.24440e9 0.302246
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.31165e8 0.0651512 0.0325756 0.999469i \(-0.489629\pi\)
0.0325756 + 0.999469i \(0.489629\pi\)
\(798\) 0 0
\(799\) −2.39774e10 −1.66298
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.49029e8 −0.0101570
\(804\) 0 0
\(805\) 2.34048e10 1.58132
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −7.48310e9 −0.496891 −0.248446 0.968646i \(-0.579920\pi\)
−0.248446 + 0.968646i \(0.579920\pi\)
\(810\) 0 0
\(811\) 2.60208e10 1.71296 0.856479 0.516181i \(-0.172647\pi\)
0.856479 + 0.516181i \(0.172647\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −7.41246e8 −0.0479635
\(816\) 0 0
\(817\) −9.69053e7 −0.00621685
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.24111e9 0.141339 0.0706695 0.997500i \(-0.477486\pi\)
0.0706695 + 0.997500i \(0.477486\pi\)
\(822\) 0 0
\(823\) 2.16414e10 1.35327 0.676636 0.736317i \(-0.263437\pi\)
0.676636 + 0.736317i \(0.263437\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.18759e10 0.730125 0.365062 0.930983i \(-0.381048\pi\)
0.365062 + 0.930983i \(0.381048\pi\)
\(828\) 0 0
\(829\) −9.14265e9 −0.557354 −0.278677 0.960385i \(-0.589896\pi\)
−0.278677 + 0.960385i \(0.589896\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −5.31435e10 −3.18561
\(834\) 0 0
\(835\) −1.38344e10 −0.822353
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.27491e10 −0.745267 −0.372633 0.927979i \(-0.621545\pi\)
−0.372633 + 0.927979i \(0.621545\pi\)
\(840\) 0 0
\(841\) 6.26094e9 0.362956
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.28321e9 −0.415264
\(846\) 0 0
\(847\) 3.39493e10 1.91973
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.36841e10 −2.42980
\(852\) 0 0
\(853\) −1.72779e10 −0.953167 −0.476583 0.879129i \(-0.658125\pi\)
−0.476583 + 0.879129i \(0.658125\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.14152e10 0.619514 0.309757 0.950816i \(-0.399752\pi\)
0.309757 + 0.950816i \(0.399752\pi\)
\(858\) 0 0
\(859\) 1.27618e10 0.686965 0.343483 0.939159i \(-0.388393\pi\)
0.343483 + 0.939159i \(0.388393\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.19302e8 0.0169108 0.00845540 0.999964i \(-0.497309\pi\)
0.00845540 + 0.999964i \(0.497309\pi\)
\(864\) 0 0
\(865\) 4.01730e9 0.211046
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.99981e8 −0.0103376
\(870\) 0 0
\(871\) −2.26620e10 −1.16208
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.57501e10 −1.80405
\(876\) 0 0
\(877\) −8.51426e9 −0.426234 −0.213117 0.977027i \(-0.568362\pi\)
−0.213117 + 0.977027i \(0.568362\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 8.20262e9 0.404145 0.202072 0.979371i \(-0.435232\pi\)
0.202072 + 0.979371i \(0.435232\pi\)
\(882\) 0 0
\(883\) −1.57333e10 −0.769054 −0.384527 0.923114i \(-0.625635\pi\)
−0.384527 + 0.923114i \(0.625635\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.84173e10 −0.886124 −0.443062 0.896491i \(-0.646108\pi\)
−0.443062 + 0.896491i \(0.646108\pi\)
\(888\) 0 0
\(889\) −7.36136e9 −0.351400
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.14717e9 −0.0539071
\(894\) 0 0
\(895\) 6.66146e9 0.310591
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2.03565e10 0.934423
\(900\) 0 0
\(901\) −3.00167e10 −1.36718
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.99912e10 0.896537
\(906\) 0 0
\(907\) 4.44837e9 0.197959 0.0989796 0.995089i \(-0.468442\pi\)
0.0989796 + 0.995089i \(0.468442\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.83330e10 0.803377 0.401688 0.915776i \(-0.368424\pi\)
0.401688 + 0.915776i \(0.368424\pi\)
\(912\) 0 0
\(913\) −4.11439e7 −0.00178920
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.26115e10 −3.10966
\(918\) 0 0
\(919\) 3.38356e10 1.43803 0.719017 0.694993i \(-0.244592\pi\)
0.719017 + 0.694993i \(0.244592\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.64429e10 0.688291
\(924\) 0 0
\(925\) 2.72756e10 1.13312
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.27055e10 −1.33834 −0.669169 0.743110i \(-0.733350\pi\)
−0.669169 + 0.743110i \(0.733350\pi\)
\(930\) 0 0
\(931\) −2.54258e9 −0.103264
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.92660e8 0.00770814
\(936\) 0 0
\(937\) −2.13238e10 −0.846791 −0.423395 0.905945i \(-0.639162\pi\)
−0.423395 + 0.905945i \(0.639162\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.37346e10 0.928578 0.464289 0.885684i \(-0.346310\pi\)
0.464289 + 0.885684i \(0.346310\pi\)
\(942\) 0 0
\(943\) −8.60040e9 −0.333986
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.23219e9 0.0471470 0.0235735 0.999722i \(-0.492496\pi\)
0.0235735 + 0.999722i \(0.492496\pi\)
\(948\) 0 0
\(949\) −3.02130e10 −1.14752
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.42965e10 0.909325 0.454662 0.890664i \(-0.349760\pi\)
0.454662 + 0.890664i \(0.349760\pi\)
\(954\) 0 0
\(955\) 2.49928e10 0.928547
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.50921e10 1.65095
\(960\) 0 0
\(961\) −9.88727e9 −0.359372
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.51422e10 −0.900653
\(966\) 0 0
\(967\) 4.54395e10 1.61600 0.807998 0.589185i \(-0.200551\pi\)
0.807998 + 0.589185i \(0.200551\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.55196e9 0.0894555 0.0447277 0.998999i \(-0.485758\pi\)
0.0447277 + 0.998999i \(0.485758\pi\)
\(972\) 0 0
\(973\) 8.83469e10 3.07466
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.12078e9 −0.175673 −0.0878365 0.996135i \(-0.527995\pi\)
−0.0878365 + 0.996135i \(0.527995\pi\)
\(978\) 0 0
\(979\) −7.31243e7 −0.00249071
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.49695e10 0.838442 0.419221 0.907884i \(-0.362303\pi\)
0.419221 + 0.907884i \(0.362303\pi\)
\(984\) 0 0
\(985\) −1.97231e10 −0.657579
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.29431e9 0.239771
\(990\) 0 0
\(991\) 2.79851e9 0.0913417 0.0456709 0.998957i \(-0.485457\pi\)
0.0456709 + 0.998957i \(0.485457\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.71620e9 −0.0874141
\(996\) 0 0
\(997\) 2.32195e10 0.742026 0.371013 0.928628i \(-0.379010\pi\)
0.371013 + 0.928628i \(0.379010\pi\)
\(998\) 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.bl.1.1 2
3.2 odd 2 192.8.a.u.1.2 2
4.3 odd 2 576.8.a.bm.1.1 2
8.3 odd 2 288.8.a.j.1.2 2
8.5 even 2 288.8.a.i.1.2 2
12.11 even 2 192.8.a.r.1.2 2
24.5 odd 2 96.8.a.d.1.1 2
24.11 even 2 96.8.a.g.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.8.a.d.1.1 2 24.5 odd 2
96.8.a.g.1.1 yes 2 24.11 even 2
192.8.a.r.1.2 2 12.11 even 2
192.8.a.u.1.2 2 3.2 odd 2
288.8.a.i.1.2 2 8.5 even 2
288.8.a.j.1.2 2 8.3 odd 2
576.8.a.bl.1.1 2 1.1 even 1 trivial
576.8.a.bm.1.1 2 4.3 odd 2