↑

Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
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)
 
\(f(q)\) \(=\) \(q+110.000 q^{5} +504.000 q^{7} +O(q^{10})\) \(q+110.000 q^{5} +504.000 q^{7} +3812.00 q^{11} -9574.00 q^{13} -26098.0 q^{17} +38308.0 q^{19} +71128.0 q^{23} -66025.0 q^{25} +74262.0 q^{29} -275680. q^{31} +55440.0 q^{35} +266610. q^{37} -684762. q^{41} -245956. q^{43} -478800. q^{47} -569527. q^{49} -569410. q^{53} +419320. q^{55} -1.52532e6 q^{59} +2.64046e6 q^{61} -1.05314e6 q^{65} -1.41624e6 q^{67} +3.51130e6 q^{71} +4.73862e6 q^{73} +1.92125e6 q^{77} +4.66149e6 q^{79} -5.72925e6 q^{83} -2.87078e6 q^{85} -1.19935e7 q^{89} -4.82530e6 q^{91} +4.21388e6 q^{95} +7.15075e6 q^{97} +O(q^{100})\)

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
\(5\) 110.000 0.393548 0.196774 − 0.980449i \(-0.436953\pi\)
0.196774 + 0.980449i \(0.436953\pi\)
\(6\) 0 0
\(7\) 504.000 0.555376 0.277688 − 0.960671i \(-0.410432\pi\)
0.277688 + 0.960671i \(0.410432\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3812.00 0.863532 0.431766 − 0.901986i \(-0.357891\pi\)
0.431766 + 0.901986i \(0.357891\pi\)
\(12\) 0 0
\(13\) −9574.00 −1.20863 −0.604313 − 0.796747i \(-0.706552\pi\)
−0.604313 + 0.796747i \(0.706552\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −26098.0 −1.28836 −0.644178 − 0.764875i \(-0.722800\pi\)
−0.644178 + 0.764875i \(0.722800\pi\)
\(18\) 0 0
\(19\) 38308.0 1.28130 0.640652 − 0.767832i \(-0.278664\pi\)
0.640652 + 0.767832i \(0.278664\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 71128.0 1.21897 0.609485 − 0.792797i \(-0.291376\pi\)
0.609485 + 0.792797i \(0.291376\pi\)
\(24\) 0 0
\(25\) −66025.0 −0.845120
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 74262.0 0.565423 0.282712 − 0.959205i \(-0.408766\pi\)
0.282712 + 0.959205i \(0.408766\pi\)
\(30\) 0 0
\(31\) −275680. −1.66203 −0.831016 − 0.556249i \(-0.812240\pi\)
−0.831016 + 0.556249i \(0.812240\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 55440.0 0.218567
\(36\) 0 0
\(37\) 266610. 0.865307 0.432654 − 0.901560i \(-0.357577\pi\)
0.432654 + 0.901560i \(0.357577\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −684762. −1.55166 −0.775829 − 0.630943i \(-0.782668\pi\)
−0.775829 + 0.630943i \(0.782668\pi\)
\(42\) 0 0
\(43\) −245956. −0.471756 −0.235878 − 0.971783i \(-0.575797\pi\)
−0.235878 + 0.971783i \(0.575797\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −478800. −0.672685 −0.336342 − 0.941740i \(-0.609190\pi\)
−0.336342 + 0.941740i \(0.609190\pi\)
\(48\) 0 0
\(49\) −569527. −0.691557
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −569410. −0.525363 −0.262682 − 0.964883i \(-0.584607\pi\)
−0.262682 + 0.964883i \(0.584607\pi\)
\(54\) 0 0
\(55\) 419320. 0.339841
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.52532e6 −0.966897 −0.483448 − 0.875373i \(-0.660616\pi\)
−0.483448 + 0.875373i \(0.660616\pi\)
\(60\) 0 0
\(61\) 2.64046e6 1.48945 0.744723 − 0.667374i \(-0.232582\pi\)
0.744723 + 0.667374i \(0.232582\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.05314e6 −0.475652
\(66\) 0 0
\(67\) −1.41624e6 −0.575273 −0.287636 − 0.957740i \(-0.592869\pi\)
−0.287636 + 0.957740i \(0.592869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.51130e6 1.16430 0.582149 − 0.813082i \(-0.302212\pi\)
0.582149 + 0.813082i \(0.302212\pi\)
\(72\) 0 0
\(73\) 4.73862e6 1.42568 0.712839 − 0.701327i \(-0.247409\pi\)
0.712839 + 0.701327i \(0.247409\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.92125e6 0.479585
\(78\) 0 0
\(79\) 4.66149e6 1.06373 0.531863 − 0.846830i \(-0.321492\pi\)
0.531863 + 0.846830i \(0.321492\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.72925e6 −1.09983 −0.549914 − 0.835221i \(-0.685339\pi\)
−0.549914 + 0.835221i \(0.685339\pi\)
\(84\) 0 0
\(85\) −2.87078e6 −0.507030
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.19935e7 −1.80336 −0.901678 − 0.432408i \(-0.857664\pi\)
−0.901678 + 0.432408i \(0.857664\pi\)
\(90\) 0 0
\(91\) −4.82530e6 −0.671242
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.21388e6 0.504254
\(96\) 0 0
\(97\) 7.15075e6 0.795519 0.397760 − 0.917490i \(-0.369788\pi\)
0.397760 + 0.917490i \(0.369788\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −8.78373e6 −0.848309 −0.424155 − 0.905590i \(-0.639429\pi\)
−0.424155 + 0.905590i \(0.639429\pi\)
\(102\) 0 0
\(103\) −8.01610e6 −0.722825 −0.361412 − 0.932406i \(-0.617705\pi\)
−0.361412 + 0.932406i \(0.617705\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.15123e6 −0.406507 −0.203253 − 0.979126i \(-0.565151\pi\)
−0.203253 + 0.979126i \(0.565151\pi\)
\(108\) 0 0
\(109\) 2.41280e7 1.78455 0.892274 − 0.451493i \(-0.149109\pi\)
0.892274 + 0.451493i \(0.149109\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.04827e6 −0.133541 −0.0667703 − 0.997768i \(-0.521269\pi\)
−0.0667703 + 0.997768i \(0.521269\pi\)
\(114\) 0 0
\(115\) 7.82408e6 0.479723
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.31534e7 −0.715523
\(120\) 0 0
\(121\) −4.95583e6 −0.254312
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.58565e7 −0.726143
\(126\) 0 0
\(127\) 1.36634e6 0.0591895 0.0295947 − 0.999562i \(-0.490578\pi\)
0.0295947 + 0.999562i \(0.490578\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.84645e7 1.49489 0.747447 − 0.664321i \(-0.231279\pi\)
0.747447 + 0.664321i \(0.231279\pi\)
\(132\) 0 0
\(133\) 1.93072e7 0.711605
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.62585e6 −0.253376 −0.126688 − 0.991943i \(-0.540435\pi\)
−0.126688 + 0.991943i \(0.540435\pi\)
\(138\) 0 0
\(139\) −5.32324e6 −0.168122 −0.0840609 − 0.996461i \(-0.526789\pi\)
−0.0840609 + 0.996461i \(0.526789\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.64961e7 −1.04369
\(144\) 0 0
\(145\) 8.16882e6 0.222521
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.61366e6 0.188557 0.0942783 − 0.995546i \(-0.469946\pi\)
0.0942783 + 0.995546i \(0.469946\pi\)
\(150\) 0 0
\(151\) −2.50221e7 −0.591432 −0.295716 − 0.955276i \(-0.595558\pi\)
−0.295716 + 0.955276i \(0.595558\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.03248e7 −0.654089
\(156\) 0 0
\(157\) −3.93145e7 −0.810782 −0.405391 − 0.914143i \(-0.632865\pi\)
−0.405391 + 0.914143i \(0.632865\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.58485e7 0.676987
\(162\) 0 0
\(163\) 6.28387e7 1.13650 0.568252 − 0.822855i \(-0.307620\pi\)
0.568252 + 0.822855i \(0.307620\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.04133e7 −0.173014 −0.0865072 − 0.996251i \(-0.527571\pi\)
−0.0865072 + 0.996251i \(0.527571\pi\)
\(168\) 0 0
\(169\) 2.89130e7 0.460775
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.03551e7 −1.17992 −0.589959 − 0.807433i \(-0.700856\pi\)
−0.589959 + 0.807433i \(0.700856\pi\)
\(174\) 0 0
\(175\) −3.32766e7 −0.469360
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.40084e7 −1.09481 −0.547403 − 0.836869i \(-0.684383\pi\)
−0.547403 + 0.836869i \(0.684383\pi\)
\(180\) 0 0
\(181\) −1.15469e8 −1.44741 −0.723703 − 0.690112i \(-0.757561\pi\)
−0.723703 + 0.690112i \(0.757561\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.93271e7 0.340540
\(186\) 0 0
\(187\) −9.94856e7 −1.11254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.97154e7 −1.03549 −0.517744 − 0.855535i \(-0.673228\pi\)
−0.517744 + 0.855535i \(0.673228\pi\)
\(192\) 0 0
\(193\) −1.86157e7 −0.186393 −0.0931965 − 0.995648i \(-0.529708\pi\)
−0.0931965 + 0.995648i \(0.529708\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.30384e7 0.867022 0.433511 − 0.901148i \(-0.357274\pi\)
0.433511 + 0.901148i \(0.357274\pi\)
\(198\) 0 0
\(199\) 7.39686e7 0.665367 0.332684 − 0.943038i \(-0.392046\pi\)
0.332684 + 0.943038i \(0.392046\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.74280e7 0.314023
\(204\) 0 0
\(205\) −7.53238e7 −0.610652
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.46030e8 1.10645
\(210\) 0 0
\(211\) −1.85163e8 −1.35695 −0.678476 − 0.734623i \(-0.737359\pi\)
−0.678476 + 0.734623i \(0.737359\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.70552e7 −0.185659
\(216\) 0 0
\(217\) −1.38943e8 −0.923053
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.49862e8 1.55714
\(222\) 0 0
\(223\) −1.20862e8 −0.729830 −0.364915 − 0.931041i \(-0.618902\pi\)
−0.364915 + 0.931041i \(0.618902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.82315e8 −1.60193 −0.800966 − 0.598710i \(-0.795680\pi\)
−0.800966 + 0.598710i \(0.795680\pi\)
\(228\) 0 0
\(229\) 8.91913e7 0.490793 0.245397 − 0.969423i \(-0.421082\pi\)
0.245397 + 0.969423i \(0.421082\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.32240e8 1.20279 0.601396 − 0.798951i \(-0.294611\pi\)
0.601396 + 0.798951i \(0.294611\pi\)
\(234\) 0 0
\(235\) −5.26680e7 −0.264734
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3.21986e8 −1.52561 −0.762807 − 0.646626i \(-0.776179\pi\)
−0.762807 + 0.646626i \(0.776179\pi\)
\(240\) 0 0
\(241\) −2.00366e8 −0.922072 −0.461036 − 0.887381i \(-0.652522\pi\)
−0.461036 + 0.887381i \(0.652522\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.26480e7 −0.272161
\(246\) 0 0
\(247\) −3.66761e8 −1.54862
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −8.70560e7 −0.347489 −0.173744 − 0.984791i \(-0.555587\pi\)
−0.173744 + 0.984791i \(0.555587\pi\)
\(252\) 0 0
\(253\) 2.71140e8 1.05262
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.22879e8 −1.92148 −0.960738 − 0.277457i \(-0.910509\pi\)
−0.960738 + 0.277457i \(0.910509\pi\)
\(258\) 0 0
\(259\) 1.34371e8 0.480571
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.06215e8 1.37693 0.688464 − 0.725270i \(-0.258285\pi\)
0.688464 + 0.725270i \(0.258285\pi\)
\(264\) 0 0
\(265\) −6.26351e7 −0.206756
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.82347e8 1.19764 0.598818 − 0.800885i \(-0.295637\pi\)
0.598818 + 0.800885i \(0.295637\pi\)
\(270\) 0 0
\(271\) 2.84165e8 0.867317 0.433658 − 0.901077i \(-0.357222\pi\)
0.433658 + 0.901077i \(0.357222\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.51687e8 −0.729788
\(276\) 0 0
\(277\) −2.93752e8 −0.830427 −0.415213 − 0.909724i \(-0.636293\pi\)
−0.415213 + 0.909724i \(0.636293\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4.15399e8 −1.11685 −0.558424 − 0.829556i \(-0.688594\pi\)
−0.558424 + 0.829556i \(0.688594\pi\)
\(282\) 0 0
\(283\) −5.06429e7 −0.132821 −0.0664104 − 0.997792i \(-0.521155\pi\)
−0.0664104 + 0.997792i \(0.521155\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.45120e8 −0.861754
\(288\) 0 0
\(289\) 2.70767e8 0.659862
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.47714e7 0.173660 0.0868298 − 0.996223i \(-0.472326\pi\)
0.0868298 + 0.996223i \(0.472326\pi\)
\(294\) 0 0
\(295\) −1.67786e8 −0.380520
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.80979e8 −1.47328
\(300\) 0 0
\(301\) −1.23962e8 −0.262002
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.90450e8 0.586169
\(306\) 0 0
\(307\) −8.52577e7 −0.168170 −0.0840851 − 0.996459i \(-0.526797\pi\)
−0.0840851 + 0.996459i \(0.526797\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −9.39129e8 −1.77037 −0.885184 − 0.465240i \(-0.845968\pi\)
−0.885184 + 0.465240i \(0.845968\pi\)
\(312\) 0 0
\(313\) −3.43040e8 −0.632323 −0.316162 − 0.948705i \(-0.602394\pi\)
−0.316162 + 0.948705i \(0.602394\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.03960e9 −1.83298 −0.916492 − 0.400054i \(-0.868991\pi\)
−0.916492 + 0.400054i \(0.868991\pi\)
\(318\) 0 0
\(319\) 2.83087e8 0.488261
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.99762e8 −1.65077
\(324\) 0 0
\(325\) 6.32123e8 1.02143
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.41315e8 −0.373593
\(330\) 0 0
\(331\) −1.10022e9 −1.66756 −0.833779 − 0.552098i \(-0.813827\pi\)
−0.833779 + 0.552098i \(0.813827\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.55786e8 −0.226397
\(336\) 0 0
\(337\) 1.28272e9 1.82569 0.912847 − 0.408302i \(-0.133879\pi\)
0.912847 + 0.408302i \(0.133879\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.05089e9 −1.43522
\(342\) 0 0
\(343\) −7.02107e8 −0.939451
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.25822e9 1.61660 0.808301 − 0.588770i \(-0.200388\pi\)
0.808301 + 0.588770i \(0.200388\pi\)
\(348\) 0 0
\(349\) −1.35371e8 −0.170465 −0.0852327 − 0.996361i \(-0.527163\pi\)
−0.0852327 + 0.996361i \(0.527163\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.49221e8 1.02756 0.513782 − 0.857921i \(-0.328244\pi\)
0.513782 + 0.857921i \(0.328244\pi\)
\(354\) 0 0
\(355\) 3.86243e8 0.458207
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.20121e8 −0.251091 −0.125546 − 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(360\) 0 0
\(361\) 5.73631e8 0.641738
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.21248e8 0.561073
\(366\) 0 0
\(367\) 4.66505e8 0.492635 0.246317 − 0.969189i \(-0.420779\pi\)
0.246317 + 0.969189i \(0.420779\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.86983e8 −0.291774
\(372\) 0 0
\(373\) 2.98453e8 0.297780 0.148890 − 0.988854i \(-0.452430\pi\)
0.148890 + 0.988854i \(0.452430\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.10984e8 −0.683385
\(378\) 0 0
\(379\) −1.46218e9 −1.37964 −0.689818 − 0.723983i \(-0.742309\pi\)
−0.689818 + 0.723983i \(0.742309\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.58702e9 −1.44340 −0.721700 − 0.692206i \(-0.756639\pi\)
−0.721700 + 0.692206i \(0.756639\pi\)
\(384\) 0 0
\(385\) 2.11337e8 0.188740
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.14439e8 −0.270840 −0.135420 − 0.990788i \(-0.543238\pi\)
−0.135420 + 0.990788i \(0.543238\pi\)
\(390\) 0 0
\(391\) −1.85630e9 −1.57047
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5.12764e8 0.418627
\(396\) 0 0
\(397\) 8.52757e8 0.684004 0.342002 − 0.939699i \(-0.388895\pi\)
0.342002 + 0.939699i \(0.388895\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.92522e8 −0.536325 −0.268163 − 0.963374i \(-0.586416\pi\)
−0.268163 + 0.963374i \(0.586416\pi\)
\(402\) 0 0
\(403\) 2.63936e9 2.00877
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.01632e9 0.747221
\(408\) 0 0
\(409\) 6.17357e8 0.446174 0.223087 − 0.974799i \(-0.428387\pi\)
0.223087 + 0.974799i \(0.428387\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.68763e8 −0.536992
\(414\) 0 0
\(415\) −6.30218e8 −0.432835
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.65512e9 −1.09921 −0.549604 − 0.835425i \(-0.685221\pi\)
−0.549604 + 0.835425i \(0.685221\pi\)
\(420\) 0 0
\(421\) −7.01472e7 −0.0458166 −0.0229083 − 0.999738i \(-0.507293\pi\)
−0.0229083 + 0.999738i \(0.507293\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.72312e9 1.08882
\(426\) 0 0
\(427\) 1.33079e9 0.827203
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.81387e9 −1.09128 −0.545640 − 0.838020i \(-0.683713\pi\)
−0.545640 + 0.838020i \(0.683713\pi\)
\(432\) 0 0
\(433\) −2.59970e9 −1.53892 −0.769460 − 0.638695i \(-0.779475\pi\)
−0.769460 + 0.638695i \(0.779475\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.72477e9 1.56187
\(438\) 0 0
\(439\) −1.67431e9 −0.944517 −0.472258 − 0.881460i \(-0.656561\pi\)
−0.472258 + 0.881460i \(0.656561\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.52711e8 −0.138105 −0.0690527 − 0.997613i \(-0.521998\pi\)
−0.0690527 + 0.997613i \(0.521998\pi\)
\(444\) 0 0
\(445\) −1.31929e9 −0.709707
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.55311e8 −0.393789 −0.196895 − 0.980425i \(-0.563086\pi\)
−0.196895 + 0.980425i \(0.563086\pi\)
\(450\) 0 0
\(451\) −2.61031e9 −1.33991
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.30783e8 −0.264166
\(456\) 0 0
\(457\) −1.51584e8 −0.0742928 −0.0371464 − 0.999310i \(-0.511827\pi\)
−0.0371464 + 0.999310i \(0.511827\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −7.78405e8 −0.370043 −0.185022 − 0.982734i \(-0.559236\pi\)
−0.185022 + 0.982734i \(0.559236\pi\)
\(462\) 0 0
\(463\) 2.41052e9 1.12870 0.564349 − 0.825536i \(-0.309127\pi\)
0.564349 + 0.825536i \(0.309127\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.76192e9 −0.800527 −0.400264 − 0.916400i \(-0.631081\pi\)
−0.400264 + 0.916400i \(0.631081\pi\)
\(468\) 0 0
\(469\) −7.13783e8 −0.319493
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −9.37584e8 −0.407377
\(474\) 0 0
\(475\) −2.52929e9 −1.08285
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.43811e8 −0.267661 −0.133830 − 0.991004i \(-0.542728\pi\)
−0.133830 + 0.991004i \(0.542728\pi\)
\(480\) 0 0
\(481\) −2.55252e9 −1.04583
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.86583e8 0.313075
\(486\) 0 0
\(487\) 3.16421e9 1.24141 0.620704 − 0.784045i \(-0.286847\pi\)
0.620704 + 0.784045i \(0.286847\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.62406e9 1.38169 0.690844 − 0.723004i \(-0.257239\pi\)
0.690844 + 0.723004i \(0.257239\pi\)
\(492\) 0 0
\(493\) −1.93809e9 −0.728467
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.76970e9 0.646624
\(498\) 0 0
\(499\) 1.35483e9 0.488128 0.244064 − 0.969759i \(-0.421519\pi\)
0.244064 + 0.969759i \(0.421519\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.66389e9 1.63403 0.817015 − 0.576616i \(-0.195627\pi\)
0.817015 + 0.576616i \(0.195627\pi\)
\(504\) 0 0
\(505\) −9.66210e8 −0.333850
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.34292e9 −0.451376 −0.225688 − 0.974200i \(-0.572463\pi\)
−0.225688 + 0.974200i \(0.572463\pi\)
\(510\) 0 0
\(511\) 2.38826e9 0.791788
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.81771e8 −0.284466
\(516\) 0 0
\(517\) −1.82519e9 −0.580885
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.45400e9 0.450435 0.225217 − 0.974309i \(-0.427691\pi\)
0.225217 + 0.974309i \(0.427691\pi\)
\(522\) 0 0
\(523\) −4.90309e8 −0.149870 −0.0749349 − 0.997188i \(-0.523875\pi\)
−0.0749349 + 0.997188i \(0.523875\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.19470e9 2.14129
\(528\) 0 0
\(529\) 1.65437e9 0.485889
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.55591e9 1.87537
\(534\) 0 0
\(535\) −5.66635e8 −0.159980
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.17104e9 −0.597182
\(540\) 0 0
\(541\) 4.82889e9 1.31116 0.655582 − 0.755124i \(-0.272423\pi\)
0.655582 + 0.755124i \(0.272423\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.65408e9 0.702306
\(546\) 0 0
\(547\) 3.08793e9 0.806698 0.403349 − 0.915046i \(-0.367846\pi\)
0.403349 + 0.915046i \(0.367846\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.84483e9 0.724479
\(552\) 0 0
\(553\) 2.34939e9 0.590768
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 4.09889e9 1.00502 0.502508 − 0.864573i \(-0.332411\pi\)
0.502508 + 0.864573i \(0.332411\pi\)
\(558\) 0 0
\(559\) 2.35478e9 0.570177
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.97105e9 1.17400 0.587001 − 0.809587i \(-0.300309\pi\)
0.587001 + 0.809587i \(0.300309\pi\)
\(564\) 0 0
\(565\) −2.25310e8 −0.0525547
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.71316e9 −0.844988 −0.422494 − 0.906366i \(-0.638845\pi\)
−0.422494 + 0.906366i \(0.638845\pi\)
\(570\) 0 0
\(571\) 2.36205e9 0.530961 0.265481 − 0.964116i \(-0.414469\pi\)
0.265481 + 0.964116i \(0.414469\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.69623e9 −1.03018
\(576\) 0 0
\(577\) −1.81146e8 −0.0392566 −0.0196283 − 0.999807i \(-0.506248\pi\)
−0.0196283 + 0.999807i \(0.506248\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.88754e9 −0.610818
\(582\) 0 0
\(583\) −2.17059e9 −0.453668
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.31976e9 0.473380 0.236690 − 0.971585i \(-0.423937\pi\)
0.236690 + 0.971585i \(0.423937\pi\)
\(588\) 0 0
\(589\) −1.05607e10 −2.12957
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.27806e9 −0.448615 −0.224308 − 0.974518i \(-0.572012\pi\)
−0.224308 + 0.974518i \(0.572012\pi\)
\(594\) 0 0
\(595\) −1.44687e9 −0.281592
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.88253e9 0.928220 0.464110 − 0.885778i \(-0.346374\pi\)
0.464110 + 0.885778i \(0.346374\pi\)
\(600\) 0 0
\(601\) −6.74758e9 −1.26791 −0.633954 − 0.773371i \(-0.718569\pi\)
−0.633954 + 0.773371i \(0.718569\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.45141e8 −0.100084
\(606\) 0 0
\(607\) −9.05928e9 −1.64412 −0.822060 − 0.569401i \(-0.807175\pi\)
−0.822060 + 0.569401i \(0.807175\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.58403e9 0.813024
\(612\) 0 0
\(613\) 8.48777e9 1.48827 0.744135 − 0.668029i \(-0.232862\pi\)
0.744135 + 0.668029i \(0.232862\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.34407e9 −0.401766 −0.200883 − 0.979615i \(-0.564381\pi\)
−0.200883 + 0.979615i \(0.564381\pi\)
\(618\) 0 0
\(619\) 1.01541e9 0.172077 0.0860384 − 0.996292i \(-0.472579\pi\)
0.0860384 + 0.996292i \(0.472579\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.04473e9 −1.00154
\(624\) 0 0
\(625\) 3.41399e9 0.559348
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6.95799e9 −1.11482
\(630\) 0 0
\(631\) 7.01911e9 1.11219 0.556095 − 0.831119i \(-0.312299\pi\)
0.556095 + 0.831119i \(0.312299\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.50297e8 0.0232939
\(636\) 0 0
\(637\) 5.45265e9 0.835833
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.52776e9 0.679016 0.339508 − 0.940603i \(-0.389739\pi\)
0.339508 + 0.940603i \(0.389739\pi\)
\(642\) 0 0
\(643\) 8.63094e9 1.28032 0.640162 − 0.768240i \(-0.278867\pi\)
0.640162 + 0.768240i \(0.278867\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.57401e9 −0.373632 −0.186816 − 0.982395i \(-0.559817\pi\)
−0.186816 + 0.982395i \(0.559817\pi\)
\(648\) 0 0
\(649\) −5.81454e9 −0.834946
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.31827e9 −1.30960 −0.654800 − 0.755802i \(-0.727247\pi\)
−0.654800 + 0.755802i \(0.727247\pi\)
\(654\) 0 0
\(655\) 4.23109e9 0.588313
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.04422e10 1.42133 0.710663 − 0.703532i \(-0.248395\pi\)
0.710663 + 0.703532i \(0.248395\pi\)
\(660\) 0 0
\(661\) 1.04761e10 1.41090 0.705449 − 0.708761i \(-0.250746\pi\)
0.705449 + 0.708761i \(0.250746\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.12380e9 0.280051
\(666\) 0 0
\(667\) 5.28211e9 0.689234
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.00654e10 1.28618
\(672\) 0 0
\(673\) 1.38891e10 1.75639 0.878197 − 0.478299i \(-0.158746\pi\)
0.878197 + 0.478299i \(0.158746\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −7.48893e8 −0.0927598 −0.0463799 − 0.998924i \(-0.514768\pi\)
−0.0463799 + 0.998924i \(0.514768\pi\)
\(678\) 0 0
\(679\) 3.60398e9 0.441813
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.15581e10 1.38808 0.694038 − 0.719938i \(-0.255830\pi\)
0.694038 + 0.719938i \(0.255830\pi\)
\(684\) 0 0
\(685\) −8.38844e8 −0.0997158
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.45153e9 0.634967
\(690\) 0 0
\(691\) −3.34337e8 −0.0385489 −0.0192744 − 0.999814i \(-0.506136\pi\)
−0.0192744 + 0.999814i \(0.506136\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −5.85556e8 −0.0661640
\(696\) 0 0
\(697\) 1.78709e10 1.99909
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.55383e9 −0.608948 −0.304474 − 0.952521i \(-0.598481\pi\)
−0.304474 + 0.952521i \(0.598481\pi\)
\(702\) 0 0
\(703\) 1.02133e10 1.10872
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.42700e9 −0.471131
\(708\) 0 0
\(709\) −1.30817e10 −1.37849 −0.689243 − 0.724530i \(-0.742057\pi\)
−0.689243 + 0.724530i \(0.742057\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.96086e10 −2.02597
\(714\) 0 0
\(715\) −4.01457e9 −0.410741
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.10847e10 1.11217 0.556085 − 0.831125i \(-0.312303\pi\)
0.556085 + 0.831125i \(0.312303\pi\)
\(720\) 0 0
\(721\) −4.04012e9 −0.401440
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.90315e9 −0.477850
\(726\) 0 0
\(727\) −7.79416e9 −0.752314 −0.376157 − 0.926556i \(-0.622755\pi\)
−0.376157 + 0.926556i \(0.622755\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 6.41896e9 0.607790
\(732\) 0 0
\(733\) −6.83552e9 −0.641073 −0.320537 − 0.947236i \(-0.603863\pi\)
−0.320537 + 0.947236i \(0.603863\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.39869e9 −0.496767
\(738\) 0 0
\(739\) −1.73862e10 −1.58471 −0.792356 − 0.610059i \(-0.791146\pi\)
−0.792356 + 0.610059i \(0.791146\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −2.25537e9 −0.201724 −0.100862 − 0.994900i \(-0.532160\pi\)
−0.100862 + 0.994900i \(0.532160\pi\)
\(744\) 0 0
\(745\) 8.37503e8 0.0742061
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.59622e9 −0.225764
\(750\) 0 0
\(751\) −2.05027e10 −1.76632 −0.883162 − 0.469068i \(-0.844590\pi\)
−0.883162 + 0.469068i \(0.844590\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.75243e9 −0.232757
\(756\) 0 0
\(757\) 2.57872e8 0.0216057 0.0108029 − 0.999942i \(-0.496561\pi\)
0.0108029 + 0.999942i \(0.496561\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.34452e10 1.10591 0.552957 − 0.833210i \(-0.313499\pi\)
0.552957 + 0.833210i \(0.313499\pi\)
\(762\) 0 0
\(763\) 1.21605e10 0.991096
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.46035e10 1.16862
\(768\) 0 0
\(769\) 8.28541e9 0.657009 0.328505 − 0.944502i \(-0.393455\pi\)
0.328505 + 0.944502i \(0.393455\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.43430e10 1.11689 0.558447 − 0.829540i \(-0.311397\pi\)
0.558447 + 0.829540i \(0.311397\pi\)
\(774\) 0 0
\(775\) 1.82018e10 1.40462
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.62319e10 −1.98814
\(780\) 0 0
\(781\) 1.33851e10 1.00541
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −4.32460e9 −0.319082
\(786\) 0 0
\(787\) 3.83137e9 0.280184 0.140092 − 0.990139i \(-0.455260\pi\)
0.140092 + 0.990139i \(0.455260\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.03233e9 −0.0741653
\(792\) 0 0
\(793\) −2.52797e10 −1.80018
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.95859e9 −0.137038 −0.0685188 − 0.997650i \(-0.521827\pi\)
−0.0685188 + 0.997650i \(0.521827\pi\)
\(798\) 0 0
\(799\) 1.24957e10 0.866658
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.80636e10 1.23112
\(804\) 0 0
\(805\) 3.94334e9 0.266427
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.07415e9 −0.137727 −0.0688637 − 0.997626i \(-0.521937\pi\)
−0.0688637 + 0.997626i \(0.521937\pi\)
\(810\) 0 0
\(811\) 5.71508e9 0.376227 0.188113 − 0.982147i \(-0.439763\pi\)
0.188113 + 0.982147i \(0.439763\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.91226e9 0.447268
\(816\) 0 0
\(817\) −9.42208e9 −0.604463
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.82748e10 1.78319 0.891596 − 0.452832i \(-0.149586\pi\)
0.891596 + 0.452832i \(0.149586\pi\)
\(822\) 0 0
\(823\) −2.09283e9 −0.130868 −0.0654342 − 0.997857i \(-0.520843\pi\)
−0.0654342 + 0.997857i \(0.520843\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.71453e9 −0.228368 −0.114184 − 0.993460i \(-0.536425\pi\)
−0.114184 + 0.993460i \(0.536425\pi\)
\(828\) 0 0
\(829\) −3.37924e9 −0.206005 −0.103003 − 0.994681i \(-0.532845\pi\)
−0.103003 + 0.994681i \(0.532845\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.48635e10 0.890972
\(834\) 0 0
\(835\) −1.14547e9 −0.0680895
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.64907e10 −0.963990 −0.481995 − 0.876174i \(-0.660088\pi\)
−0.481995 + 0.876174i \(0.660088\pi\)
\(840\) 0 0
\(841\) −1.17350e10 −0.680297
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.18043e9 0.181337
\(846\) 0 0
\(847\) −2.49774e9 −0.141239
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.89634e10 1.05478
\(852\) 0 0
\(853\) 4.77028e9 0.263162 0.131581 − 0.991305i \(-0.457995\pi\)
0.131581 + 0.991305i \(0.457995\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.61514e10 −0.876554 −0.438277 − 0.898840i \(-0.644411\pi\)
−0.438277 + 0.898840i \(0.644411\pi\)
\(858\) 0 0
\(859\) 3.41593e8 0.0183879 0.00919397 − 0.999958i \(-0.497073\pi\)
0.00919397 + 0.999958i \(0.497073\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.07878e9 −0.321943 −0.160971 − 0.986959i \(-0.551463\pi\)
−0.160971 + 0.986959i \(0.551463\pi\)
\(864\) 0 0
\(865\) −8.83906e9 −0.464354
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1.77696e10 0.918562
\(870\) 0 0
\(871\) 1.35590e10 0.695289
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.99168e9 −0.403283
\(876\) 0 0
\(877\) −1.23852e10 −0.620020 −0.310010 − 0.950733i \(-0.600332\pi\)
−0.310010 + 0.950733i \(0.600332\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.37801e10 0.678949 0.339475 − 0.940615i \(-0.389751\pi\)
0.339475 + 0.940615i \(0.389751\pi\)
\(882\) 0 0
\(883\) −1.89296e9 −0.0925292 −0.0462646 − 0.998929i \(-0.514732\pi\)
−0.0462646 + 0.998929i \(0.514732\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.23912e9 −0.203959 −0.101979 − 0.994787i \(-0.532518\pi\)
−0.101979 + 0.994787i \(0.532518\pi\)
\(888\) 0 0
\(889\) 6.88633e8 0.0328724
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1.83419e10 −0.861913
\(894\) 0 0
\(895\) −9.24092e9 −0.430859
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.04725e10 −0.939751
\(900\) 0 0
\(901\) 1.48605e10 0.676855
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.27016e10 −0.569623
\(906\) 0 0
\(907\) 9.51367e9 0.423372 0.211686 − 0.977338i \(-0.432105\pi\)
0.211686 + 0.977338i \(0.432105\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.16235e10 0.509359 0.254680 − 0.967025i \(-0.418030\pi\)
0.254680 + 0.967025i \(0.418030\pi\)
\(912\) 0 0
\(913\) −2.18399e10 −0.949736
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.93861e10 0.830229
\(918\) 0 0
\(919\) −9.22943e9 −0.392257 −0.196128 − 0.980578i \(-0.562837\pi\)
−0.196128 + 0.980578i \(0.562837\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3.36172e10 −1.40720
\(924\) 0 0
\(925\) −1.76029e10 −0.731289
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.09353e10 1.67511 0.837555 − 0.546353i \(-0.183984\pi\)
0.837555 + 0.546353i \(0.183984\pi\)
\(930\) 0 0
\(931\) −2.18174e10 −0.886094
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.09434e10 −0.437837
\(936\) 0 0
\(937\) −1.75085e9 −0.0695281 −0.0347641 − 0.999396i \(-0.511068\pi\)
−0.0347641 + 0.999396i \(0.511068\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.91102e9 0.231259 0.115630 − 0.993292i \(-0.463111\pi\)
0.115630 + 0.993292i \(0.463111\pi\)
\(942\) 0 0
\(943\) −4.87058e10 −1.89143
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.26089e10 −0.865077 −0.432538 − 0.901616i \(-0.642382\pi\)
−0.432538 + 0.901616i \(0.642382\pi\)
\(948\) 0 0
\(949\) −4.53675e10 −1.72311
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.11773e10 −0.418322 −0.209161 − 0.977881i \(-0.567073\pi\)
−0.209161 + 0.977881i \(0.567073\pi\)
\(954\) 0 0
\(955\) −1.09687e10 −0.407515
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.84343e9 −0.140719
\(960\) 0 0
\(961\) 4.84868e10 1.76235
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −2.04773e9 −0.0733546
\(966\) 0 0
\(967\) 1.55518e10 0.553078 0.276539 − 0.961003i \(-0.410812\pi\)
0.276539 + 0.961003i \(0.410812\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.34508e9 0.292525 0.146263 − 0.989246i \(-0.453276\pi\)
0.146263 + 0.989246i \(0.453276\pi\)
\(972\) 0 0
\(973\) −2.68291e9 −0.0933709
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.85180e9 0.337975 0.168988 − 0.985618i \(-0.445950\pi\)
0.168988 + 0.985618i \(0.445950\pi\)
\(978\) 0 0
\(979\) −4.57193e10 −1.55726
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.70884e10 −1.24538 −0.622688 − 0.782470i \(-0.713959\pi\)
−0.622688 + 0.782470i \(0.713959\pi\)
\(984\) 0 0
\(985\) 1.02342e10 0.341215
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −1.74944e10 −0.575057
\(990\) 0 0
\(991\) 6.43526e9 0.210043 0.105022 − 0.994470i \(-0.466509\pi\)
0.105022 + 0.994470i \(0.466509\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.13654e9 0.261854
\(996\) 0 0
\(997\) 1.23071e10 0.393299 0.196650 − 0.980474i \(-0.436994\pi\)
0.196650 + 0.980474i \(0.436994\pi\)
\(998\) 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.t.1.1 1
3.2 odd 2 192.8.a.c.1.1 1
4.3 odd 2 576.8.a.s.1.1 1
8.3 odd 2 144.8.a.d.1.1 1
8.5 even 2 72.8.a.b.1.1 1
12.11 even 2 192.8.a.k.1.1 1
24.5 odd 2 24.8.a.c.1.1 ✓ 1
24.11 even 2 48.8.a.c.1.1 1
120.29 odd 2 600.8.a.a.1.1 1
120.53 even 4 600.8.f.d.49.2 2
120.77 even 4 600.8.f.d.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.8.a.c.1.1 ✓ 1 24.5 odd 2
48.8.a.c.1.1 1 24.11 even 2
72.8.a.b.1.1 1 8.5 even 2
144.8.a.d.1.1 1 8.3 odd 2
192.8.a.c.1.1 1 3.2 odd 2
192.8.a.k.1.1 1 12.11 even 2
576.8.a.s.1.1 1 4.3 odd 2
576.8.a.t.1.1 1 1.1 even 1 trivial
600.8.a.a.1.1 1 120.29 odd 2
600.8.f.d.49.1 2 120.77 even 4
600.8.f.d.49.2 2 120.53 even 4