↑

Properties

Label 72.8.a.c.1.1
Level $72$
Weight $8$
Character 72.1
Self dual yes
Analytic conductor $22.492$
Analytic rank $0$
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] = [72,8,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, 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("72.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(22.4917218349\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+26.0000 q^{5} +1056.00 q^{7} +O(q^{10})\) \(q+26.0000 q^{5} +1056.00 q^{7} -6412.00 q^{11} +5206.00 q^{13} +6238.00 q^{17} +41492.0 q^{19} +29432.0 q^{23} -77449.0 q^{25} +210498. q^{29} +185240. q^{31} +27456.0 q^{35} +507630. q^{37} -360042. q^{41} +620044. q^{43} +847680. q^{47} +291593. q^{49} -1.42375e6 q^{53} -166712. q^{55} +2.54872e6 q^{59} -706058. q^{61} +135356. q^{65} -2.41880e6 q^{67} -265976. q^{71} -5.79124e6 q^{73} -6.77107e6 q^{77} +2.95569e6 q^{79} -3.46293e6 q^{83} +162188. q^{85} +2.21113e6 q^{89} +5.49754e6 q^{91} +1.07879e6 q^{95} -1.55948e7 q^{97} +O(q^{100})\)

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
\(5\) 26.0000 0.0930204 0.0465102 − 0.998918i \(-0.485190\pi\)
0.0465102 + 0.998918i \(0.485190\pi\)
\(6\) 0 0
\(7\) 1056.00 1.16365 0.581823 − 0.813316i \(-0.302340\pi\)
0.581823 + 0.813316i \(0.302340\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6412.00 −1.45251 −0.726255 − 0.687425i \(-0.758741\pi\)
−0.726255 + 0.687425i \(0.758741\pi\)
\(12\) 0 0
\(13\) 5206.00 0.657207 0.328604 − 0.944468i \(-0.393422\pi\)
0.328604 + 0.944468i \(0.393422\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6238.00 0.307946 0.153973 − 0.988075i \(-0.450793\pi\)
0.153973 + 0.988075i \(0.450793\pi\)
\(18\) 0 0
\(19\) 41492.0 1.38780 0.693900 − 0.720072i \(-0.255891\pi\)
0.693900 + 0.720072i \(0.255891\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 29432.0 0.504397 0.252198 − 0.967676i \(-0.418846\pi\)
0.252198 + 0.967676i \(0.418846\pi\)
\(24\) 0 0
\(25\) −77449.0 −0.991347
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 210498. 1.60271 0.801355 − 0.598189i \(-0.204113\pi\)
0.801355 + 0.598189i \(0.204113\pi\)
\(30\) 0 0
\(31\) 185240. 1.11678 0.558392 − 0.829578i \(-0.311419\pi\)
0.558392 + 0.829578i \(0.311419\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 27456.0 0.108243
\(36\) 0 0
\(37\) 507630. 1.64756 0.823780 − 0.566910i \(-0.191861\pi\)
0.823780 + 0.566910i \(0.191861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −360042. −0.815849 −0.407924 − 0.913016i \(-0.633747\pi\)
−0.407924 + 0.913016i \(0.633747\pi\)
\(42\) 0 0
\(43\) 620044. 1.18928 0.594638 − 0.803993i \(-0.297295\pi\)
0.594638 + 0.803993i \(0.297295\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 847680. 1.19094 0.595469 − 0.803378i \(-0.296966\pi\)
0.595469 + 0.803378i \(0.296966\pi\)
\(48\) 0 0
\(49\) 291593. 0.354071
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.42375e6 −1.31362 −0.656808 − 0.754058i \(-0.728094\pi\)
−0.656808 + 0.754058i \(0.728094\pi\)
\(54\) 0 0
\(55\) −166712. −0.135113
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.54872e6 1.61563 0.807813 − 0.589439i \(-0.200651\pi\)
0.807813 + 0.589439i \(0.200651\pi\)
\(60\) 0 0
\(61\) −706058. −0.398278 −0.199139 − 0.979971i \(-0.563814\pi\)
−0.199139 + 0.979971i \(0.563814\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 135356. 0.0611337
\(66\) 0 0
\(67\) −2.41880e6 −0.982511 −0.491256 − 0.871016i \(-0.663462\pi\)
−0.491256 + 0.871016i \(0.663462\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −265976. −0.0881938 −0.0440969 − 0.999027i \(-0.514041\pi\)
−0.0440969 + 0.999027i \(0.514041\pi\)
\(72\) 0 0
\(73\) −5.79124e6 −1.74237 −0.871187 − 0.490951i \(-0.836649\pi\)
−0.871187 + 0.490951i \(0.836649\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.77107e6 −1.69021
\(78\) 0 0
\(79\) 2.95569e6 0.674472 0.337236 − 0.941420i \(-0.390508\pi\)
0.337236 + 0.941420i \(0.390508\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.46293e6 −0.664769 −0.332384 − 0.943144i \(-0.607853\pi\)
−0.332384 + 0.943144i \(0.607853\pi\)
\(84\) 0 0
\(85\) 162188. 0.0286452
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.21113e6 0.332467 0.166233 − 0.986086i \(-0.446839\pi\)
0.166233 + 0.986086i \(0.446839\pi\)
\(90\) 0 0
\(91\) 5.49754e6 0.764757
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.07879e6 0.129094
\(96\) 0 0
\(97\) −1.55948e7 −1.73492 −0.867459 − 0.497508i \(-0.834248\pi\)
−0.867459 + 0.497508i \(0.834248\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.29563e7 −1.25129 −0.625643 − 0.780110i \(-0.715163\pi\)
−0.625643 + 0.780110i \(0.715163\pi\)
\(102\) 0 0
\(103\) −6.82002e6 −0.614972 −0.307486 − 0.951553i \(-0.599488\pi\)
−0.307486 + 0.951553i \(0.599488\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.37785e7 1.08733 0.543663 − 0.839304i \(-0.317037\pi\)
0.543663 + 0.839304i \(0.317037\pi\)
\(108\) 0 0
\(109\) 9.67721e6 0.715743 0.357872 − 0.933771i \(-0.383503\pi\)
0.357872 + 0.933771i \(0.383503\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.19184e7 1.42901 0.714504 − 0.699632i \(-0.246652\pi\)
0.714504 + 0.699632i \(0.246652\pi\)
\(114\) 0 0
\(115\) 765232. 0.0469192
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.58733e6 0.358340
\(120\) 0 0
\(121\) 2.16266e7 1.10979
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −4.04492e6 −0.185236
\(126\) 0 0
\(127\) −661016. −0.0286351 −0.0143176 − 0.999897i \(-0.504558\pi\)
−0.0143176 + 0.999897i \(0.504558\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.20309e6 −0.0467573 −0.0233786 − 0.999727i \(-0.507442\pi\)
−0.0233786 + 0.999727i \(0.507442\pi\)
\(132\) 0 0
\(133\) 4.38156e7 1.61491
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.20046e7 −0.398866 −0.199433 − 0.979911i \(-0.563910\pi\)
−0.199433 + 0.979911i \(0.563910\pi\)
\(138\) 0 0
\(139\) 2.22914e7 0.704019 0.352010 − 0.935996i \(-0.385498\pi\)
0.352010 + 0.935996i \(0.385498\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.33809e7 −0.954600
\(144\) 0 0
\(145\) 5.47295e6 0.149085
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.92355e7 0.971689 0.485844 − 0.874045i \(-0.338512\pi\)
0.485844 + 0.874045i \(0.338512\pi\)
\(150\) 0 0
\(151\) 1.13013e7 0.267122 0.133561 − 0.991041i \(-0.457359\pi\)
0.133561 + 0.991041i \(0.457359\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.81624e6 0.103884
\(156\) 0 0
\(157\) −2.22302e7 −0.458453 −0.229227 − 0.973373i \(-0.573620\pi\)
−0.229227 + 0.973373i \(0.573620\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.10802e7 0.586939
\(162\) 0 0
\(163\) −3.23811e7 −0.585645 −0.292822 − 0.956167i \(-0.594594\pi\)
−0.292822 + 0.956167i \(0.594594\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.15095e7 −0.855815 −0.427908 − 0.903822i \(-0.640749\pi\)
−0.427908 + 0.903822i \(0.640749\pi\)
\(168\) 0 0
\(169\) −3.56461e7 −0.568078
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.00284e8 1.47255 0.736274 − 0.676683i \(-0.236583\pi\)
0.736274 + 0.676683i \(0.236583\pi\)
\(174\) 0 0
\(175\) −8.17861e7 −1.15358
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.93179e7 −0.903358 −0.451679 − 0.892181i \(-0.649175\pi\)
−0.451679 + 0.892181i \(0.649175\pi\)
\(180\) 0 0
\(181\) −7.46718e7 −0.936012 −0.468006 − 0.883725i \(-0.655028\pi\)
−0.468006 + 0.883725i \(0.655028\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.31984e7 0.153257
\(186\) 0 0
\(187\) −3.99981e7 −0.447294
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.70910e8 −1.77480 −0.887402 − 0.460996i \(-0.847492\pi\)
−0.887402 + 0.460996i \(0.847492\pi\)
\(192\) 0 0
\(193\) 1.15066e7 0.115211 0.0576056 − 0.998339i \(-0.481653\pi\)
0.0576056 + 0.998339i \(0.481653\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.93036e7 −0.739029 −0.369514 − 0.929225i \(-0.620476\pi\)
−0.369514 + 0.929225i \(0.620476\pi\)
\(198\) 0 0
\(199\) 2.38620e7 0.214645 0.107322 − 0.994224i \(-0.465772\pi\)
0.107322 + 0.994224i \(0.465772\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.22286e8 1.86499
\(204\) 0 0
\(205\) −9.36109e6 −0.0758906
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.66047e8 −2.01579
\(210\) 0 0
\(211\) 1.09739e8 0.804216 0.402108 − 0.915592i \(-0.368278\pi\)
0.402108 + 0.915592i \(0.368278\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.61211e7 0.110627
\(216\) 0 0
\(217\) 1.95613e8 1.29954
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.24750e7 0.202384
\(222\) 0 0
\(223\) −2.71085e8 −1.63696 −0.818481 − 0.574534i \(-0.805183\pi\)
−0.818481 + 0.574534i \(0.805183\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.21029e7 0.125418 0.0627089 − 0.998032i \(-0.480026\pi\)
0.0627089 + 0.998032i \(0.480026\pi\)
\(228\) 0 0
\(229\) −1.51778e8 −0.835190 −0.417595 − 0.908633i \(-0.637127\pi\)
−0.417595 + 0.908633i \(0.637127\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.94412e7 0.463225 0.231613 − 0.972808i \(-0.425600\pi\)
0.231613 + 0.972808i \(0.425600\pi\)
\(234\) 0 0
\(235\) 2.20397e7 0.110782
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.69285e8 1.27591 0.637954 − 0.770075i \(-0.279781\pi\)
0.637954 + 0.770075i \(0.279781\pi\)
\(240\) 0 0
\(241\) 1.98807e8 0.914897 0.457449 − 0.889236i \(-0.348763\pi\)
0.457449 + 0.889236i \(0.348763\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.58142e6 0.0329359
\(246\) 0 0
\(247\) 2.16007e8 0.912072
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.38049e8 1.34934 0.674671 − 0.738118i \(-0.264285\pi\)
0.674671 + 0.738118i \(0.264285\pi\)
\(252\) 0 0
\(253\) −1.88718e8 −0.732641
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.64489e8 0.604464 0.302232 − 0.953234i \(-0.402268\pi\)
0.302232 + 0.953234i \(0.402268\pi\)
\(258\) 0 0
\(259\) 5.36057e8 1.91718
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −4.21664e8 −1.42929 −0.714646 − 0.699486i \(-0.753412\pi\)
−0.714646 + 0.699486i \(0.753412\pi\)
\(264\) 0 0
\(265\) −3.70175e7 −0.122193
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.13468e8 −0.668651 −0.334325 − 0.942458i \(-0.608508\pi\)
−0.334325 + 0.942458i \(0.608508\pi\)
\(270\) 0 0
\(271\) 2.23317e8 0.681601 0.340800 − 0.940136i \(-0.389302\pi\)
0.340800 + 0.940136i \(0.389302\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.96603e8 1.43994
\(276\) 0 0
\(277\) 1.72179e8 0.486743 0.243372 − 0.969933i \(-0.421747\pi\)
0.243372 + 0.969933i \(0.421747\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.84836e8 −1.84126 −0.920628 − 0.390441i \(-0.872323\pi\)
−0.920628 + 0.390441i \(0.872323\pi\)
\(282\) 0 0
\(283\) −7.80003e7 −0.204571 −0.102285 − 0.994755i \(-0.532616\pi\)
−0.102285 + 0.994755i \(0.532616\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.80204e8 −0.949359
\(288\) 0 0
\(289\) −3.71426e8 −0.905169
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.50814e8 1.51154 0.755771 − 0.654836i \(-0.227262\pi\)
0.755771 + 0.654836i \(0.227262\pi\)
\(294\) 0 0
\(295\) 6.62668e7 0.150286
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.53223e8 0.331493
\(300\) 0 0
\(301\) 6.54766e8 1.38390
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.83575e7 −0.0370480
\(306\) 0 0
\(307\) 3.67913e8 0.725707 0.362853 − 0.931846i \(-0.381803\pi\)
0.362853 + 0.931846i \(0.381803\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.42351e8 −0.833883 −0.416942 − 0.908933i \(-0.636898\pi\)
−0.416942 + 0.908933i \(0.636898\pi\)
\(312\) 0 0
\(313\) −1.06172e9 −1.95705 −0.978527 − 0.206118i \(-0.933917\pi\)
−0.978527 + 0.206118i \(0.933917\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.30645e8 −0.759298 −0.379649 − 0.925131i \(-0.623955\pi\)
−0.379649 + 0.925131i \(0.623955\pi\)
\(318\) 0 0
\(319\) −1.34971e9 −2.32795
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.58827e8 0.427367
\(324\) 0 0
\(325\) −4.03199e8 −0.651521
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 8.95150e8 1.38583
\(330\) 0 0
\(331\) −7.77479e8 −1.17839 −0.589197 − 0.807989i \(-0.700556\pi\)
−0.589197 + 0.807989i \(0.700556\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.28887e7 −0.0913936
\(336\) 0 0
\(337\) 2.38365e8 0.339265 0.169632 − 0.985507i \(-0.445742\pi\)
0.169632 + 0.985507i \(0.445742\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.18776e9 −1.62214
\(342\) 0 0
\(343\) −5.61739e8 −0.751632
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.17421e8 1.17873 0.589367 − 0.807866i \(-0.299377\pi\)
0.589367 + 0.807866i \(0.299377\pi\)
\(348\) 0 0
\(349\) 3.96693e8 0.499535 0.249768 − 0.968306i \(-0.419646\pi\)
0.249768 + 0.968306i \(0.419646\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.27149e8 0.274852 0.137426 − 0.990512i \(-0.456117\pi\)
0.137426 + 0.990512i \(0.456117\pi\)
\(354\) 0 0
\(355\) −6.91538e6 −0.00820383
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.59175e9 −1.81570 −0.907848 − 0.419300i \(-0.862276\pi\)
−0.907848 + 0.419300i \(0.862276\pi\)
\(360\) 0 0
\(361\) 8.27714e8 0.925988
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.50572e8 −0.162076
\(366\) 0 0
\(367\) 2.90259e8 0.306517 0.153258 − 0.988186i \(-0.451023\pi\)
0.153258 + 0.988186i \(0.451023\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.50348e9 −1.52858
\(372\) 0 0
\(373\) 6.51931e8 0.650460 0.325230 − 0.945635i \(-0.394558\pi\)
0.325230 + 0.945635i \(0.394558\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.09585e9 1.05331
\(378\) 0 0
\(379\) 8.58738e8 0.810258 0.405129 − 0.914259i \(-0.367227\pi\)
0.405129 + 0.914259i \(0.367227\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.24441e9 1.13179 0.565897 − 0.824476i \(-0.308530\pi\)
0.565897 + 0.824476i \(0.308530\pi\)
\(384\) 0 0
\(385\) −1.76048e8 −0.157224
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.35597e8 −0.375198 −0.187599 − 0.982246i \(-0.560071\pi\)
−0.187599 + 0.982246i \(0.560071\pi\)
\(390\) 0 0
\(391\) 1.83597e8 0.155327
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 7.68479e7 0.0627397
\(396\) 0 0
\(397\) −1.07940e9 −0.865796 −0.432898 − 0.901443i \(-0.642509\pi\)
−0.432898 + 0.901443i \(0.642509\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.22249e9 1.72122 0.860608 − 0.509269i \(-0.170084\pi\)
0.860608 + 0.509269i \(0.170084\pi\)
\(402\) 0 0
\(403\) 9.64359e8 0.733958
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.25492e9 −2.39310
\(408\) 0 0
\(409\) 3.70641e8 0.267869 0.133934 − 0.990990i \(-0.457239\pi\)
0.133934 + 0.990990i \(0.457239\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.69145e9 1.88002
\(414\) 0 0
\(415\) −9.00362e7 −0.0618371
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.05917e9 0.703422 0.351711 − 0.936109i \(-0.385600\pi\)
0.351711 + 0.936109i \(0.385600\pi\)
\(420\) 0 0
\(421\) 2.15633e9 1.40840 0.704202 − 0.709999i \(-0.251305\pi\)
0.704202 + 0.709999i \(0.251305\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.83127e8 −0.305281
\(426\) 0 0
\(427\) −7.45597e8 −0.463454
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.94009e9 1.16722 0.583608 − 0.812036i \(-0.301641\pi\)
0.583608 + 0.812036i \(0.301641\pi\)
\(432\) 0 0
\(433\) −4.84393e8 −0.286741 −0.143371 − 0.989669i \(-0.545794\pi\)
−0.143371 + 0.989669i \(0.545794\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.22119e9 0.700002
\(438\) 0 0
\(439\) 3.68469e8 0.207862 0.103931 − 0.994585i \(-0.466858\pi\)
0.103931 + 0.994585i \(0.466858\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.75727e9 0.960340 0.480170 − 0.877175i \(-0.340575\pi\)
0.480170 + 0.877175i \(0.340575\pi\)
\(444\) 0 0
\(445\) 5.74893e7 0.0309262
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.93704e9 −1.53126 −0.765628 − 0.643284i \(-0.777571\pi\)
−0.765628 + 0.643284i \(0.777571\pi\)
\(450\) 0 0
\(451\) 2.30859e9 1.18503
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.42936e8 0.0711380
\(456\) 0 0
\(457\) −4.11305e8 −0.201585 −0.100792 − 0.994907i \(-0.532138\pi\)
−0.100792 + 0.994907i \(0.532138\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.92967e9 −0.917341 −0.458670 − 0.888606i \(-0.651674\pi\)
−0.458670 + 0.888606i \(0.651674\pi\)
\(462\) 0 0
\(463\) 3.07870e8 0.144157 0.0720783 − 0.997399i \(-0.477037\pi\)
0.0720783 + 0.997399i \(0.477037\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.50298e9 −0.682878 −0.341439 − 0.939904i \(-0.610914\pi\)
−0.341439 + 0.939904i \(0.610914\pi\)
\(468\) 0 0
\(469\) −2.55425e9 −1.14329
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.97572e9 −1.72744
\(474\) 0 0
\(475\) −3.21351e9 −1.37579
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.92209e7 0.0163058 0.00815292 − 0.999967i \(-0.497405\pi\)
0.00815292 + 0.999967i \(0.497405\pi\)
\(480\) 0 0
\(481\) 2.64272e9 1.08279
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.05465e8 −0.161383
\(486\) 0 0
\(487\) 2.80262e9 1.09955 0.549773 − 0.835314i \(-0.314714\pi\)
0.549773 + 0.835314i \(0.314714\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.71459e9 0.653694 0.326847 − 0.945077i \(-0.394014\pi\)
0.326847 + 0.945077i \(0.394014\pi\)
\(492\) 0 0
\(493\) 1.31309e9 0.493548
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.80871e8 −0.102626
\(498\) 0 0
\(499\) −2.46873e9 −0.889449 −0.444724 − 0.895667i \(-0.646698\pi\)
−0.444724 + 0.895667i \(0.646698\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.68007e9 0.588626 0.294313 − 0.955709i \(-0.404909\pi\)
0.294313 + 0.955709i \(0.404909\pi\)
\(504\) 0 0
\(505\) −3.36864e8 −0.116395
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.51153e9 0.844162 0.422081 − 0.906558i \(-0.361300\pi\)
0.422081 + 0.906558i \(0.361300\pi\)
\(510\) 0 0
\(511\) −6.11555e9 −2.02751
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.77320e8 −0.0572049
\(516\) 0 0
\(517\) −5.43532e9 −1.72985
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.17452e9 −1.29323 −0.646614 − 0.762818i \(-0.723815\pi\)
−0.646614 + 0.762818i \(0.723815\pi\)
\(522\) 0 0
\(523\) 2.83172e8 0.0865554 0.0432777 − 0.999063i \(-0.486220\pi\)
0.0432777 + 0.999063i \(0.486220\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.15553e9 0.343909
\(528\) 0 0
\(529\) −2.53858e9 −0.745584
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.87438e9 −0.536182
\(534\) 0 0
\(535\) 3.58242e8 0.101144
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.86969e9 −0.514292
\(540\) 0 0
\(541\) −6.66923e9 −1.81086 −0.905431 − 0.424493i \(-0.860452\pi\)
−0.905431 + 0.424493i \(0.860452\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.51607e8 0.0665787
\(546\) 0 0
\(547\) 5.12201e9 1.33809 0.669045 − 0.743222i \(-0.266703\pi\)
0.669045 + 0.743222i \(0.266703\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.73398e9 2.22424
\(552\) 0 0
\(553\) 3.12121e9 0.784846
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.45491e9 0.356732 0.178366 − 0.983964i \(-0.442919\pi\)
0.178366 + 0.983964i \(0.442919\pi\)
\(558\) 0 0
\(559\) 3.22795e9 0.781601
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.50227e9 −1.77179 −0.885897 − 0.463882i \(-0.846456\pi\)
−0.885897 + 0.463882i \(0.846456\pi\)
\(564\) 0 0
\(565\) 5.69879e8 0.132927
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.23661e9 0.964108 0.482054 − 0.876142i \(-0.339891\pi\)
0.482054 + 0.876142i \(0.339891\pi\)
\(570\) 0 0
\(571\) −1.48301e9 −0.333364 −0.166682 − 0.986011i \(-0.553305\pi\)
−0.166682 + 0.986011i \(0.553305\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.27948e9 −0.500032
\(576\) 0 0
\(577\) −2.36381e8 −0.0512268 −0.0256134 − 0.999672i \(-0.508154\pi\)
−0.0256134 + 0.999672i \(0.508154\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.65686e9 −0.773555
\(582\) 0 0
\(583\) 9.12908e9 1.90804
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.92918e9 −0.393675 −0.196838 − 0.980436i \(-0.563067\pi\)
−0.196838 + 0.980436i \(0.563067\pi\)
\(588\) 0 0
\(589\) 7.68598e9 1.54987
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.58212e9 0.902350 0.451175 − 0.892436i \(-0.351005\pi\)
0.451175 + 0.892436i \(0.351005\pi\)
\(594\) 0 0
\(595\) 1.71271e8 0.0333329
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.35564e9 −0.828053 −0.414027 − 0.910265i \(-0.635878\pi\)
−0.414027 + 0.910265i \(0.635878\pi\)
\(600\) 0 0
\(601\) 1.60073e9 0.300786 0.150393 − 0.988626i \(-0.451946\pi\)
0.150393 + 0.988626i \(0.451946\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.62291e8 0.103233
\(606\) 0 0
\(607\) −3.16856e9 −0.575044 −0.287522 − 0.957774i \(-0.592831\pi\)
−0.287522 + 0.957774i \(0.592831\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.41302e9 0.782694
\(612\) 0 0
\(613\) 5.29300e9 0.928090 0.464045 − 0.885811i \(-0.346398\pi\)
0.464045 + 0.885811i \(0.346398\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.19316e8 0.0375899 0.0187950 − 0.999823i \(-0.494017\pi\)
0.0187950 + 0.999823i \(0.494017\pi\)
\(618\) 0 0
\(619\) −4.66552e9 −0.790648 −0.395324 − 0.918542i \(-0.629368\pi\)
−0.395324 + 0.918542i \(0.629368\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.33495e9 0.386874
\(624\) 0 0
\(625\) 5.94554e9 0.974116
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.16660e9 0.507359
\(630\) 0 0
\(631\) 3.93963e9 0.624241 0.312121 − 0.950042i \(-0.398961\pi\)
0.312121 + 0.950042i \(0.398961\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.71864e7 −0.00266365
\(636\) 0 0
\(637\) 1.51803e9 0.232698
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.74548e9 1.01160 0.505801 − 0.862650i \(-0.331197\pi\)
0.505801 + 0.862650i \(0.331197\pi\)
\(642\) 0 0
\(643\) −7.70175e9 −1.14249 −0.571243 − 0.820781i \(-0.693539\pi\)
−0.571243 + 0.820781i \(0.693539\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.65642e9 1.40169 0.700844 − 0.713315i \(-0.252807\pi\)
0.700844 + 0.713315i \(0.252807\pi\)
\(648\) 0 0
\(649\) −1.63424e10 −2.34671
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.95427e9 −0.415197 −0.207599 − 0.978214i \(-0.566565\pi\)
−0.207599 + 0.978214i \(0.566565\pi\)
\(654\) 0 0
\(655\) −3.12804e7 −0.00434938
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.90170e9 −1.21164 −0.605821 − 0.795601i \(-0.707155\pi\)
−0.605821 + 0.795601i \(0.707155\pi\)
\(660\) 0 0
\(661\) 1.26579e10 1.70474 0.852369 − 0.522941i \(-0.175165\pi\)
0.852369 + 0.522941i \(0.175165\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.13920e9 0.150219
\(666\) 0 0
\(667\) 6.19538e9 0.808402
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.52724e9 0.578502
\(672\) 0 0
\(673\) 7.30112e9 0.923287 0.461644 − 0.887065i \(-0.347260\pi\)
0.461644 + 0.887065i \(0.347260\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.15717e9 0.762642 0.381321 − 0.924443i \(-0.375469\pi\)
0.381321 + 0.924443i \(0.375469\pi\)
\(678\) 0 0
\(679\) −1.64681e10 −2.01883
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.07209e10 −1.28753 −0.643766 − 0.765222i \(-0.722629\pi\)
−0.643766 + 0.765222i \(0.722629\pi\)
\(684\) 0 0
\(685\) −3.12121e8 −0.0371027
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −7.41204e9 −0.863318
\(690\) 0 0
\(691\) −1.40404e10 −1.61885 −0.809425 − 0.587223i \(-0.800221\pi\)
−0.809425 + 0.587223i \(0.800221\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.79575e8 0.0654882
\(696\) 0 0
\(697\) −2.24594e9 −0.251237
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −3.68495e9 −0.404035 −0.202018 − 0.979382i \(-0.564750\pi\)
−0.202018 + 0.979382i \(0.564750\pi\)
\(702\) 0 0
\(703\) 2.10626e10 2.28648
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.36819e10 −1.45605
\(708\) 0 0
\(709\) −1.55532e10 −1.63892 −0.819461 − 0.573135i \(-0.805727\pi\)
−0.819461 + 0.573135i \(0.805727\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.45198e9 0.563302
\(714\) 0 0
\(715\) −8.67903e8 −0.0887973
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.47108e9 −0.749605 −0.374802 − 0.927105i \(-0.622289\pi\)
−0.374802 + 0.927105i \(0.622289\pi\)
\(720\) 0 0
\(721\) −7.20194e9 −0.715609
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.63029e10 −1.58884
\(726\) 0 0
\(727\) −1.77786e9 −0.171604 −0.0858020 − 0.996312i \(-0.527345\pi\)
−0.0858020 + 0.996312i \(0.527345\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.86783e9 0.366233
\(732\) 0 0
\(733\) 1.33140e10 1.24866 0.624331 − 0.781160i \(-0.285372\pi\)
0.624331 + 0.781160i \(0.285372\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.55093e10 1.42711
\(738\) 0 0
\(739\) −3.06064e9 −0.278970 −0.139485 − 0.990224i \(-0.544545\pi\)
−0.139485 + 0.990224i \(0.544545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.03506e10 −0.925774 −0.462887 − 0.886417i \(-0.653186\pi\)
−0.462887 + 0.886417i \(0.653186\pi\)
\(744\) 0 0
\(745\) 1.02012e9 0.0903869
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.45501e10 1.26526
\(750\) 0 0
\(751\) −1.79720e9 −0.154831 −0.0774153 − 0.996999i \(-0.524667\pi\)
−0.0774153 + 0.996999i \(0.524667\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.93834e8 0.0248478
\(756\) 0 0
\(757\) −1.61244e10 −1.35098 −0.675488 − 0.737371i \(-0.736067\pi\)
−0.675488 + 0.737371i \(0.736067\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.65172e9 −0.629380 −0.314690 − 0.949195i \(-0.601901\pi\)
−0.314690 + 0.949195i \(0.601901\pi\)
\(762\) 0 0
\(763\) 1.02191e10 0.832872
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.32687e10 1.06180
\(768\) 0 0
\(769\) −1.90645e10 −1.51176 −0.755882 − 0.654708i \(-0.772792\pi\)
−0.755882 + 0.654708i \(0.772792\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.37260e9 −0.496236 −0.248118 − 0.968730i \(-0.579812\pi\)
−0.248118 + 0.968730i \(0.579812\pi\)
\(774\) 0 0
\(775\) −1.43467e10 −1.10712
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.49389e10 −1.13223
\(780\) 0 0
\(781\) 1.70544e9 0.128102
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.77986e8 −0.0426455
\(786\) 0 0
\(787\) −2.68873e9 −0.196623 −0.0983116 − 0.995156i \(-0.531344\pi\)
−0.0983116 + 0.995156i \(0.531344\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 2.31458e10 1.66286
\(792\) 0 0
\(793\) −3.67574e9 −0.261751
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.10359e9 0.357085 0.178542 − 0.983932i \(-0.442862\pi\)
0.178542 + 0.983932i \(0.442862\pi\)
\(798\) 0 0
\(799\) 5.28783e9 0.366744
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.71334e10 2.53082
\(804\) 0 0
\(805\) 8.08085e8 0.0545973
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.37206e10 −1.57509 −0.787547 − 0.616254i \(-0.788649\pi\)
−0.787547 + 0.616254i \(0.788649\pi\)
\(810\) 0 0
\(811\) 1.48151e10 0.975287 0.487644 − 0.873043i \(-0.337857\pi\)
0.487644 + 0.873043i \(0.337857\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.41907e8 −0.0544769
\(816\) 0 0
\(817\) 2.57269e10 1.65048
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.38885e9 −0.465989 −0.232994 − 0.972478i \(-0.574852\pi\)
−0.232994 + 0.972478i \(0.574852\pi\)
\(822\) 0 0
\(823\) 1.23070e10 0.769580 0.384790 − 0.923004i \(-0.374274\pi\)
0.384790 + 0.923004i \(0.374274\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.19040e9 −0.380583 −0.190292 − 0.981728i \(-0.560943\pi\)
−0.190292 + 0.981728i \(0.560943\pi\)
\(828\) 0 0
\(829\) 6.54741e9 0.399143 0.199572 − 0.979883i \(-0.436045\pi\)
0.199572 + 0.979883i \(0.436045\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.81896e9 0.109035
\(834\) 0 0
\(835\) −1.33925e9 −0.0796083
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.98560e9 0.466810 0.233405 − 0.972380i \(-0.425013\pi\)
0.233405 + 0.972380i \(0.425013\pi\)
\(840\) 0 0
\(841\) 2.70595e10 1.56868
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.26798e8 −0.0528429
\(846\) 0 0
\(847\) 2.28377e10 1.29140
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.49406e10 0.831024
\(852\) 0 0
\(853\) 4.32266e9 0.238467 0.119234 − 0.992866i \(-0.461956\pi\)
0.119234 + 0.992866i \(0.461956\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.25925e10 −1.22612 −0.613058 − 0.790038i \(-0.710061\pi\)
−0.613058 + 0.790038i \(0.710061\pi\)
\(858\) 0 0
\(859\) −4.20339e9 −0.226268 −0.113134 − 0.993580i \(-0.536089\pi\)
−0.113134 + 0.993580i \(0.536089\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.09766e10 −0.581338 −0.290669 − 0.956824i \(-0.593878\pi\)
−0.290669 + 0.956824i \(0.593878\pi\)
\(864\) 0 0
\(865\) 2.60738e9 0.136977
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.89519e10 −0.979677
\(870\) 0 0
\(871\) −1.25923e10 −0.645713
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.27144e9 −0.215549
\(876\) 0 0
\(877\) −1.43474e10 −0.718246 −0.359123 − 0.933290i \(-0.616924\pi\)
−0.359123 + 0.933290i \(0.616924\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.65119e10 0.813546 0.406773 − 0.913529i \(-0.366654\pi\)
0.406773 + 0.913529i \(0.366654\pi\)
\(882\) 0 0
\(883\) 8.58509e9 0.419645 0.209823 − 0.977739i \(-0.432711\pi\)
0.209823 + 0.977739i \(0.432711\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.55541e9 −0.171063 −0.0855316 − 0.996335i \(-0.527259\pi\)
−0.0855316 + 0.996335i \(0.527259\pi\)
\(888\) 0 0
\(889\) −6.98033e8 −0.0333211
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.51719e10 1.65278
\(894\) 0 0
\(895\) −1.80227e9 −0.0840307
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.89926e10 1.78988
\(900\) 0 0
\(901\) −8.88135e9 −0.404522
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.94147e9 −0.0870683
\(906\) 0 0
\(907\) −2.50681e9 −0.111557 −0.0557783 − 0.998443i \(-0.517764\pi\)
−0.0557783 + 0.998443i \(0.517764\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.82057e10 −0.797799 −0.398899 − 0.916995i \(-0.630608\pi\)
−0.398899 + 0.916995i \(0.630608\pi\)
\(912\) 0 0
\(913\) 2.22043e10 0.965583
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.27047e9 −0.0544089
\(918\) 0 0
\(919\) 2.41571e9 0.102669 0.0513347 − 0.998682i \(-0.483652\pi\)
0.0513347 + 0.998682i \(0.483652\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.38467e9 −0.0579616
\(924\) 0 0
\(925\) −3.93154e10 −1.63330
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.61139e10 −0.659396 −0.329698 − 0.944087i \(-0.606947\pi\)
−0.329698 + 0.944087i \(0.606947\pi\)
\(930\) 0 0
\(931\) 1.20988e10 0.491380
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.03995e9 −0.0416075
\(936\) 0 0
\(937\) −3.64699e10 −1.44826 −0.724129 − 0.689664i \(-0.757758\pi\)
−0.724129 + 0.689664i \(0.757758\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.19999e10 0.860711 0.430356 − 0.902659i \(-0.358388\pi\)
0.430356 + 0.902659i \(0.358388\pi\)
\(942\) 0 0
\(943\) −1.05968e10 −0.411512
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.52734e10 −0.584402 −0.292201 − 0.956357i \(-0.594388\pi\)
−0.292201 + 0.956357i \(0.594388\pi\)
\(948\) 0 0
\(949\) −3.01492e10 −1.14510
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.79256e9 0.329071 0.164536 − 0.986371i \(-0.447387\pi\)
0.164536 + 0.986371i \(0.447387\pi\)
\(954\) 0 0
\(955\) −4.44366e9 −0.165093
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.26769e10 −0.464139
\(960\) 0 0
\(961\) 6.80124e9 0.247205
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.99171e8 0.0107170
\(966\) 0 0
\(967\) −4.31656e10 −1.53513 −0.767566 − 0.640970i \(-0.778532\pi\)
−0.767566 + 0.640970i \(0.778532\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2.84776e10 −0.998241 −0.499121 − 0.866533i \(-0.666344\pi\)
−0.499121 + 0.866533i \(0.666344\pi\)
\(972\) 0 0
\(973\) 2.35397e10 0.819229
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.49908e9 0.0857334 0.0428667 − 0.999081i \(-0.486351\pi\)
0.0428667 + 0.999081i \(0.486351\pi\)
\(978\) 0 0
\(979\) −1.41777e10 −0.482912
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.08680e10 −1.37229 −0.686145 − 0.727465i \(-0.740698\pi\)
−0.686145 + 0.727465i \(0.740698\pi\)
\(984\) 0 0
\(985\) −2.06189e9 −0.0687448
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.82491e10 0.599867
\(990\) 0 0
\(991\) −8.31493e9 −0.271394 −0.135697 − 0.990750i \(-0.543327\pi\)
−0.135697 + 0.990750i \(0.543327\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 6.20411e8 0.0199664
\(996\) 0 0
\(997\) 4.11316e10 1.31445 0.657223 − 0.753696i \(-0.271731\pi\)
0.657223 + 0.753696i \(0.271731\pi\)
\(998\) 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 72.8.a.c.1.1 1
3.2 odd 2 24.8.a.a.1.1 ✓ 1
4.3 odd 2 144.8.a.f.1.1 1
8.3 odd 2 576.8.a.o.1.1 1
8.5 even 2 576.8.a.p.1.1 1
12.11 even 2 48.8.a.f.1.1 1
15.2 even 4 600.8.f.e.49.2 2
15.8 even 4 600.8.f.e.49.1 2
15.14 odd 2 600.8.a.e.1.1 1
24.5 odd 2 192.8.a.l.1.1 1
24.11 even 2 192.8.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.8.a.a.1.1 ✓ 1 3.2 odd 2
48.8.a.f.1.1 1 12.11 even 2
72.8.a.c.1.1 1 1.1 even 1 trivial
144.8.a.f.1.1 1 4.3 odd 2
192.8.a.d.1.1 1 24.11 even 2
192.8.a.l.1.1 1 24.5 odd 2
576.8.a.o.1.1 1 8.3 odd 2
576.8.a.p.1.1 1 8.5 even 2
600.8.a.e.1.1 1 15.14 odd 2
600.8.f.e.49.1 2 15.8 even 4
600.8.f.e.49.2 2 15.2 even 4