Properties

Label 162.8.a.b.1.1
Level $162$
Weight $8$
Character 162.1
Self dual yes
Analytic conductor $50.606$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,8,Mod(1,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(50.6063741284\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 162.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.00000 q^{2} +64.0000 q^{4} -165.000 q^{5} -508.000 q^{7} +512.000 q^{8} +O(q^{10})\) \(q+8.00000 q^{2} +64.0000 q^{4} -165.000 q^{5} -508.000 q^{7} +512.000 q^{8} -1320.00 q^{10} +3024.00 q^{11} +5039.00 q^{13} -4064.00 q^{14} +4096.00 q^{16} -3189.00 q^{17} +1508.00 q^{19} -10560.0 q^{20} +24192.0 q^{22} -75600.0 q^{23} -50900.0 q^{25} +40312.0 q^{26} -32512.0 q^{28} -82665.0 q^{29} -174892. q^{31} +32768.0 q^{32} -25512.0 q^{34} +83820.0 q^{35} -323569. q^{37} +12064.0 q^{38} -84480.0 q^{40} -308118. q^{41} +336680. q^{43} +193536. q^{44} -604800. q^{46} -383196. q^{47} -565479. q^{49} -407200. q^{50} +322496. q^{52} +760206. q^{53} -498960. q^{55} -260096. q^{56} -661320. q^{58} -2.22566e6 q^{59} +2.24482e6 q^{61} -1.39914e6 q^{62} +262144. q^{64} -831435. q^{65} +1.47319e6 q^{67} -204096. q^{68} +670560. q^{70} -5.00689e6 q^{71} -5.89830e6 q^{73} -2.58855e6 q^{74} +96512.0 q^{76} -1.53619e6 q^{77} +7.02877e6 q^{79} -675840. q^{80} -2.46494e6 q^{82} -2.65120e6 q^{83} +526185. q^{85} +2.69344e6 q^{86} +1.54829e6 q^{88} -6.77090e6 q^{89} -2.55981e6 q^{91} -4.83840e6 q^{92} -3.06557e6 q^{94} -248820. q^{95} +1.61764e7 q^{97} -4.52383e6 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 0 0
\(4\) 64.0000 0.500000
\(5\) −165.000 −0.590322 −0.295161 0.955448i \(-0.595373\pi\)
−0.295161 + 0.955448i \(0.595373\pi\)
\(6\) 0 0
\(7\) −508.000 −0.559784 −0.279892 0.960031i \(-0.590299\pi\)
−0.279892 + 0.960031i \(0.590299\pi\)
\(8\) 512.000 0.353553
\(9\) 0 0
\(10\) −1320.00 −0.417421
\(11\) 3024.00 0.685027 0.342513 0.939513i \(-0.388722\pi\)
0.342513 + 0.939513i \(0.388722\pi\)
\(12\) 0 0
\(13\) 5039.00 0.636125 0.318063 0.948070i \(-0.396968\pi\)
0.318063 + 0.948070i \(0.396968\pi\)
\(14\) −4064.00 −0.395827
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) −3189.00 −0.157428 −0.0787142 0.996897i \(-0.525081\pi\)
−0.0787142 + 0.996897i \(0.525081\pi\)
\(18\) 0 0
\(19\) 1508.00 0.0504387 0.0252193 0.999682i \(-0.491972\pi\)
0.0252193 + 0.999682i \(0.491972\pi\)
\(20\) −10560.0 −0.295161
\(21\) 0 0
\(22\) 24192.0 0.484387
\(23\) −75600.0 −1.29561 −0.647805 0.761806i \(-0.724313\pi\)
−0.647805 + 0.761806i \(0.724313\pi\)
\(24\) 0 0
\(25\) −50900.0 −0.651520
\(26\) 40312.0 0.449808
\(27\) 0 0
\(28\) −32512.0 −0.279892
\(29\) −82665.0 −0.629403 −0.314701 0.949191i \(-0.601904\pi\)
−0.314701 + 0.949191i \(0.601904\pi\)
\(30\) 0 0
\(31\) −174892. −1.05440 −0.527198 0.849742i \(-0.676758\pi\)
−0.527198 + 0.849742i \(0.676758\pi\)
\(32\) 32768.0 0.176777
\(33\) 0 0
\(34\) −25512.0 −0.111319
\(35\) 83820.0 0.330453
\(36\) 0 0
\(37\) −323569. −1.05017 −0.525087 0.851049i \(-0.675967\pi\)
−0.525087 + 0.851049i \(0.675967\pi\)
\(38\) 12064.0 0.0356655
\(39\) 0 0
\(40\) −84480.0 −0.208710
\(41\) −308118. −0.698190 −0.349095 0.937087i \(-0.613511\pi\)
−0.349095 + 0.937087i \(0.613511\pi\)
\(42\) 0 0
\(43\) 336680. 0.645770 0.322885 0.946438i \(-0.395347\pi\)
0.322885 + 0.946438i \(0.395347\pi\)
\(44\) 193536. 0.342513
\(45\) 0 0
\(46\) −604800. −0.916135
\(47\) −383196. −0.538367 −0.269184 0.963089i \(-0.586754\pi\)
−0.269184 + 0.963089i \(0.586754\pi\)
\(48\) 0 0
\(49\) −565479. −0.686642
\(50\) −407200. −0.460694
\(51\) 0 0
\(52\) 322496. 0.318063
\(53\) 760206. 0.701400 0.350700 0.936488i \(-0.385944\pi\)
0.350700 + 0.936488i \(0.385944\pi\)
\(54\) 0 0
\(55\) −498960. −0.404386
\(56\) −260096. −0.197914
\(57\) 0 0
\(58\) −661320. −0.445055
\(59\) −2.22566e6 −1.41084 −0.705420 0.708790i \(-0.749241\pi\)
−0.705420 + 0.708790i \(0.749241\pi\)
\(60\) 0 0
\(61\) 2.24482e6 1.26627 0.633135 0.774042i \(-0.281768\pi\)
0.633135 + 0.774042i \(0.281768\pi\)
\(62\) −1.39914e6 −0.745571
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −831435. −0.375519
\(66\) 0 0
\(67\) 1.47319e6 0.598407 0.299203 0.954189i \(-0.403279\pi\)
0.299203 + 0.954189i \(0.403279\pi\)
\(68\) −204096. −0.0787142
\(69\) 0 0
\(70\) 670560. 0.233665
\(71\) −5.00689e6 −1.66021 −0.830107 0.557604i \(-0.811721\pi\)
−0.830107 + 0.557604i \(0.811721\pi\)
\(72\) 0 0
\(73\) −5.89830e6 −1.77459 −0.887293 0.461207i \(-0.847417\pi\)
−0.887293 + 0.461207i \(0.847417\pi\)
\(74\) −2.58855e6 −0.742584
\(75\) 0 0
\(76\) 96512.0 0.0252193
\(77\) −1.53619e6 −0.383467
\(78\) 0 0
\(79\) 7.02877e6 1.60393 0.801963 0.597374i \(-0.203789\pi\)
0.801963 + 0.597374i \(0.203789\pi\)
\(80\) −675840. −0.147580
\(81\) 0 0
\(82\) −2.46494e6 −0.493695
\(83\) −2.65120e6 −0.508942 −0.254471 0.967080i \(-0.581901\pi\)
−0.254471 + 0.967080i \(0.581901\pi\)
\(84\) 0 0
\(85\) 526185. 0.0929335
\(86\) 2.69344e6 0.456628
\(87\) 0 0
\(88\) 1.54829e6 0.242193
\(89\) −6.77090e6 −1.01808 −0.509039 0.860743i \(-0.669999\pi\)
−0.509039 + 0.860743i \(0.669999\pi\)
\(90\) 0 0
\(91\) −2.55981e6 −0.356093
\(92\) −4.83840e6 −0.647805
\(93\) 0 0
\(94\) −3.06557e6 −0.380683
\(95\) −248820. −0.0297751
\(96\) 0 0
\(97\) 1.61764e7 1.79962 0.899809 0.436284i \(-0.143706\pi\)
0.899809 + 0.436284i \(0.143706\pi\)
\(98\) −4.52383e6 −0.485529
\(99\) 0 0
\(100\) −3.25760e6 −0.325760
\(101\) 2.70161e6 0.260915 0.130457 0.991454i \(-0.458355\pi\)
0.130457 + 0.991454i \(0.458355\pi\)
\(102\) 0 0
\(103\) 2.06892e6 0.186557 0.0932787 0.995640i \(-0.470265\pi\)
0.0932787 + 0.995640i \(0.470265\pi\)
\(104\) 2.57997e6 0.224904
\(105\) 0 0
\(106\) 6.08165e6 0.495965
\(107\) −9.15450e6 −0.722423 −0.361211 0.932484i \(-0.617637\pi\)
−0.361211 + 0.932484i \(0.617637\pi\)
\(108\) 0 0
\(109\) 8.68645e6 0.642465 0.321233 0.947000i \(-0.395903\pi\)
0.321233 + 0.947000i \(0.395903\pi\)
\(110\) −3.99168e6 −0.285944
\(111\) 0 0
\(112\) −2.08077e6 −0.139946
\(113\) −2.49776e7 −1.62845 −0.814227 0.580547i \(-0.802839\pi\)
−0.814227 + 0.580547i \(0.802839\pi\)
\(114\) 0 0
\(115\) 1.24740e7 0.764827
\(116\) −5.29056e6 −0.314701
\(117\) 0 0
\(118\) −1.78053e7 −0.997614
\(119\) 1.62001e6 0.0881260
\(120\) 0 0
\(121\) −1.03426e7 −0.530739
\(122\) 1.79585e7 0.895388
\(123\) 0 0
\(124\) −1.11931e7 −0.527198
\(125\) 2.12891e7 0.974929
\(126\) 0 0
\(127\) −4.98659e6 −0.216018 −0.108009 0.994150i \(-0.534448\pi\)
−0.108009 + 0.994150i \(0.534448\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 0 0
\(130\) −6.65148e6 −0.265532
\(131\) −2.23405e7 −0.868246 −0.434123 0.900854i \(-0.642942\pi\)
−0.434123 + 0.900854i \(0.642942\pi\)
\(132\) 0 0
\(133\) −766064. −0.0282348
\(134\) 1.17855e7 0.423137
\(135\) 0 0
\(136\) −1.63277e6 −0.0556594
\(137\) 1.28631e7 0.427389 0.213694 0.976901i \(-0.431450\pi\)
0.213694 + 0.976901i \(0.431450\pi\)
\(138\) 0 0
\(139\) 4.41902e7 1.39564 0.697822 0.716271i \(-0.254153\pi\)
0.697822 + 0.716271i \(0.254153\pi\)
\(140\) 5.36448e6 0.165226
\(141\) 0 0
\(142\) −4.00551e7 −1.17395
\(143\) 1.52379e7 0.435763
\(144\) 0 0
\(145\) 1.36397e7 0.371550
\(146\) −4.71864e7 −1.25482
\(147\) 0 0
\(148\) −2.07084e7 −0.525087
\(149\) −2.62887e6 −0.0651055 −0.0325527 0.999470i \(-0.510364\pi\)
−0.0325527 + 0.999470i \(0.510364\pi\)
\(150\) 0 0
\(151\) −5.53019e7 −1.30714 −0.653568 0.756868i \(-0.726729\pi\)
−0.653568 + 0.756868i \(0.726729\pi\)
\(152\) 772096. 0.0178328
\(153\) 0 0
\(154\) −1.22895e7 −0.271152
\(155\) 2.88572e7 0.622433
\(156\) 0 0
\(157\) −2.23711e7 −0.461358 −0.230679 0.973030i \(-0.574095\pi\)
−0.230679 + 0.973030i \(0.574095\pi\)
\(158\) 5.62301e7 1.13415
\(159\) 0 0
\(160\) −5.40672e6 −0.104355
\(161\) 3.84048e7 0.725262
\(162\) 0 0
\(163\) 5.23606e7 0.946995 0.473498 0.880795i \(-0.342991\pi\)
0.473498 + 0.880795i \(0.342991\pi\)
\(164\) −1.97196e7 −0.349095
\(165\) 0 0
\(166\) −2.12096e7 −0.359877
\(167\) −5.47957e7 −0.910414 −0.455207 0.890386i \(-0.650435\pi\)
−0.455207 + 0.890386i \(0.650435\pi\)
\(168\) 0 0
\(169\) −3.73570e7 −0.595345
\(170\) 4.20948e6 0.0657139
\(171\) 0 0
\(172\) 2.15475e7 0.322885
\(173\) 1.23645e8 1.81558 0.907791 0.419424i \(-0.137768\pi\)
0.907791 + 0.419424i \(0.137768\pi\)
\(174\) 0 0
\(175\) 2.58572e7 0.364711
\(176\) 1.23863e7 0.171257
\(177\) 0 0
\(178\) −5.41672e7 −0.719891
\(179\) 9.87297e7 1.28666 0.643328 0.765591i \(-0.277553\pi\)
0.643328 + 0.765591i \(0.277553\pi\)
\(180\) 0 0
\(181\) 9.19855e7 1.15304 0.576520 0.817083i \(-0.304410\pi\)
0.576520 + 0.817083i \(0.304410\pi\)
\(182\) −2.04785e7 −0.251796
\(183\) 0 0
\(184\) −3.87072e7 −0.458067
\(185\) 5.33889e7 0.619940
\(186\) 0 0
\(187\) −9.64354e6 −0.107843
\(188\) −2.45245e7 −0.269184
\(189\) 0 0
\(190\) −1.99056e6 −0.0210541
\(191\) 1.31979e8 1.37052 0.685262 0.728296i \(-0.259688\pi\)
0.685262 + 0.728296i \(0.259688\pi\)
\(192\) 0 0
\(193\) 1.47907e8 1.48095 0.740473 0.672086i \(-0.234602\pi\)
0.740473 + 0.672086i \(0.234602\pi\)
\(194\) 1.29411e8 1.27252
\(195\) 0 0
\(196\) −3.61907e7 −0.343321
\(197\) −2.69561e7 −0.251203 −0.125602 0.992081i \(-0.540086\pi\)
−0.125602 + 0.992081i \(0.540086\pi\)
\(198\) 0 0
\(199\) 1.35831e8 1.22183 0.610916 0.791695i \(-0.290801\pi\)
0.610916 + 0.791695i \(0.290801\pi\)
\(200\) −2.60608e7 −0.230347
\(201\) 0 0
\(202\) 2.16129e7 0.184495
\(203\) 4.19938e7 0.352330
\(204\) 0 0
\(205\) 5.08395e7 0.412157
\(206\) 1.65513e7 0.131916
\(207\) 0 0
\(208\) 2.06397e7 0.159031
\(209\) 4.56019e6 0.0345518
\(210\) 0 0
\(211\) −2.19340e6 −0.0160742 −0.00803710 0.999968i \(-0.502558\pi\)
−0.00803710 + 0.999968i \(0.502558\pi\)
\(212\) 4.86532e7 0.350700
\(213\) 0 0
\(214\) −7.32360e7 −0.510830
\(215\) −5.55522e7 −0.381212
\(216\) 0 0
\(217\) 8.88451e7 0.590234
\(218\) 6.94916e7 0.454292
\(219\) 0 0
\(220\) −3.19334e7 −0.202193
\(221\) −1.60694e7 −0.100144
\(222\) 0 0
\(223\) 1.99443e7 0.120435 0.0602174 0.998185i \(-0.480821\pi\)
0.0602174 + 0.998185i \(0.480821\pi\)
\(224\) −1.66461e7 −0.0989568
\(225\) 0 0
\(226\) −1.99820e8 −1.15149
\(227\) −1.25985e8 −0.714872 −0.357436 0.933938i \(-0.616349\pi\)
−0.357436 + 0.933938i \(0.616349\pi\)
\(228\) 0 0
\(229\) 2.35046e7 0.129339 0.0646693 0.997907i \(-0.479401\pi\)
0.0646693 + 0.997907i \(0.479401\pi\)
\(230\) 9.97920e7 0.540814
\(231\) 0 0
\(232\) −4.23245e7 −0.222527
\(233\) −1.27757e8 −0.661665 −0.330833 0.943689i \(-0.607330\pi\)
−0.330833 + 0.943689i \(0.607330\pi\)
\(234\) 0 0
\(235\) 6.32273e7 0.317810
\(236\) −1.42442e8 −0.705420
\(237\) 0 0
\(238\) 1.29601e7 0.0623145
\(239\) −9.34878e7 −0.442958 −0.221479 0.975165i \(-0.571088\pi\)
−0.221479 + 0.975165i \(0.571088\pi\)
\(240\) 0 0
\(241\) −1.34780e8 −0.620250 −0.310125 0.950696i \(-0.600371\pi\)
−0.310125 + 0.950696i \(0.600371\pi\)
\(242\) −8.27408e7 −0.375289
\(243\) 0 0
\(244\) 1.43668e8 0.633135
\(245\) 9.33040e7 0.405340
\(246\) 0 0
\(247\) 7.59881e6 0.0320853
\(248\) −8.95447e7 −0.372786
\(249\) 0 0
\(250\) 1.70313e8 0.689379
\(251\) 1.31470e7 0.0524771 0.0262385 0.999656i \(-0.491647\pi\)
0.0262385 + 0.999656i \(0.491647\pi\)
\(252\) 0 0
\(253\) −2.28614e8 −0.887527
\(254\) −3.98927e7 −0.152748
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) −1.28471e8 −0.472105 −0.236053 0.971740i \(-0.575854\pi\)
−0.236053 + 0.971740i \(0.575854\pi\)
\(258\) 0 0
\(259\) 1.64373e8 0.587870
\(260\) −5.32118e7 −0.187759
\(261\) 0 0
\(262\) −1.78724e8 −0.613942
\(263\) −9.77517e7 −0.331344 −0.165672 0.986181i \(-0.552979\pi\)
−0.165672 + 0.986181i \(0.552979\pi\)
\(264\) 0 0
\(265\) −1.25434e8 −0.414052
\(266\) −6.12851e6 −0.0199650
\(267\) 0 0
\(268\) 9.42840e7 0.299203
\(269\) 3.96851e7 0.124307 0.0621534 0.998067i \(-0.480203\pi\)
0.0621534 + 0.998067i \(0.480203\pi\)
\(270\) 0 0
\(271\) −3.50733e8 −1.07049 −0.535246 0.844696i \(-0.679781\pi\)
−0.535246 + 0.844696i \(0.679781\pi\)
\(272\) −1.30621e7 −0.0393571
\(273\) 0 0
\(274\) 1.02905e8 0.302209
\(275\) −1.53922e8 −0.446308
\(276\) 0 0
\(277\) 5.39602e8 1.52544 0.762719 0.646730i \(-0.223864\pi\)
0.762719 + 0.646730i \(0.223864\pi\)
\(278\) 3.53522e8 0.986869
\(279\) 0 0
\(280\) 4.29158e7 0.116833
\(281\) −1.00852e8 −0.271151 −0.135576 0.990767i \(-0.543288\pi\)
−0.135576 + 0.990767i \(0.543288\pi\)
\(282\) 0 0
\(283\) −2.54112e8 −0.666458 −0.333229 0.942846i \(-0.608138\pi\)
−0.333229 + 0.942846i \(0.608138\pi\)
\(284\) −3.20441e8 −0.830107
\(285\) 0 0
\(286\) 1.21903e8 0.308131
\(287\) 1.56524e8 0.390836
\(288\) 0 0
\(289\) −4.00169e8 −0.975216
\(290\) 1.09118e8 0.262726
\(291\) 0 0
\(292\) −3.77491e8 −0.887293
\(293\) −5.43084e8 −1.26134 −0.630668 0.776053i \(-0.717219\pi\)
−0.630668 + 0.776053i \(0.717219\pi\)
\(294\) 0 0
\(295\) 3.67235e8 0.832849
\(296\) −1.65667e8 −0.371292
\(297\) 0 0
\(298\) −2.10310e7 −0.0460365
\(299\) −3.80948e8 −0.824170
\(300\) 0 0
\(301\) −1.71033e8 −0.361492
\(302\) −4.42415e8 −0.924285
\(303\) 0 0
\(304\) 6.17677e6 0.0126097
\(305\) −3.70394e8 −0.747507
\(306\) 0 0
\(307\) −7.29877e8 −1.43968 −0.719839 0.694141i \(-0.755784\pi\)
−0.719839 + 0.694141i \(0.755784\pi\)
\(308\) −9.83163e7 −0.191733
\(309\) 0 0
\(310\) 2.30857e8 0.440127
\(311\) 6.38770e8 1.20416 0.602078 0.798437i \(-0.294339\pi\)
0.602078 + 0.798437i \(0.294339\pi\)
\(312\) 0 0
\(313\) −6.37761e8 −1.17558 −0.587790 0.809013i \(-0.700002\pi\)
−0.587790 + 0.809013i \(0.700002\pi\)
\(314\) −1.78968e8 −0.326229
\(315\) 0 0
\(316\) 4.49841e8 0.801963
\(317\) −6.54377e8 −1.15377 −0.576887 0.816824i \(-0.695733\pi\)
−0.576887 + 0.816824i \(0.695733\pi\)
\(318\) 0 0
\(319\) −2.49979e8 −0.431158
\(320\) −4.32538e7 −0.0737902
\(321\) 0 0
\(322\) 3.07238e8 0.512838
\(323\) −4.80901e6 −0.00794049
\(324\) 0 0
\(325\) −2.56485e8 −0.414448
\(326\) 4.18885e8 0.669627
\(327\) 0 0
\(328\) −1.57756e8 −0.246847
\(329\) 1.94664e8 0.301369
\(330\) 0 0
\(331\) 8.07689e8 1.22418 0.612091 0.790787i \(-0.290329\pi\)
0.612091 + 0.790787i \(0.290329\pi\)
\(332\) −1.69677e8 −0.254471
\(333\) 0 0
\(334\) −4.38366e8 −0.643760
\(335\) −2.43076e8 −0.353253
\(336\) 0 0
\(337\) −1.87280e8 −0.266555 −0.133278 0.991079i \(-0.542550\pi\)
−0.133278 + 0.991079i \(0.542550\pi\)
\(338\) −2.98856e8 −0.420972
\(339\) 0 0
\(340\) 3.36758e7 0.0464667
\(341\) −5.28873e8 −0.722290
\(342\) 0 0
\(343\) 7.05623e8 0.944155
\(344\) 1.72380e8 0.228314
\(345\) 0 0
\(346\) 9.89161e8 1.28381
\(347\) 2.82556e8 0.363037 0.181518 0.983388i \(-0.441899\pi\)
0.181518 + 0.983388i \(0.441899\pi\)
\(348\) 0 0
\(349\) −7.60612e8 −0.957799 −0.478899 0.877870i \(-0.658964\pi\)
−0.478899 + 0.877870i \(0.658964\pi\)
\(350\) 2.06858e8 0.257889
\(351\) 0 0
\(352\) 9.90904e7 0.121097
\(353\) −4.68893e8 −0.567365 −0.283683 0.958918i \(-0.591556\pi\)
−0.283683 + 0.958918i \(0.591556\pi\)
\(354\) 0 0
\(355\) 8.26137e8 0.980061
\(356\) −4.33338e8 −0.509039
\(357\) 0 0
\(358\) 7.89838e8 0.909803
\(359\) 1.53906e8 0.175560 0.0877802 0.996140i \(-0.472023\pi\)
0.0877802 + 0.996140i \(0.472023\pi\)
\(360\) 0 0
\(361\) −8.91598e8 −0.997456
\(362\) 7.35884e8 0.815323
\(363\) 0 0
\(364\) −1.63828e8 −0.178046
\(365\) 9.73220e8 1.04758
\(366\) 0 0
\(367\) −7.49528e8 −0.791510 −0.395755 0.918356i \(-0.629517\pi\)
−0.395755 + 0.918356i \(0.629517\pi\)
\(368\) −3.09658e8 −0.323903
\(369\) 0 0
\(370\) 4.27111e8 0.438364
\(371\) −3.86185e8 −0.392633
\(372\) 0 0
\(373\) −5.41728e8 −0.540506 −0.270253 0.962789i \(-0.587107\pi\)
−0.270253 + 0.962789i \(0.587107\pi\)
\(374\) −7.71483e7 −0.0762563
\(375\) 0 0
\(376\) −1.96196e8 −0.190341
\(377\) −4.16549e8 −0.400379
\(378\) 0 0
\(379\) 1.32974e9 1.25467 0.627334 0.778750i \(-0.284146\pi\)
0.627334 + 0.778750i \(0.284146\pi\)
\(380\) −1.59245e7 −0.0148875
\(381\) 0 0
\(382\) 1.05583e9 0.969107
\(383\) 7.19949e8 0.654796 0.327398 0.944887i \(-0.393828\pi\)
0.327398 + 0.944887i \(0.393828\pi\)
\(384\) 0 0
\(385\) 2.53472e8 0.226369
\(386\) 1.18326e9 1.04719
\(387\) 0 0
\(388\) 1.03529e9 0.899809
\(389\) 2.32067e9 1.99889 0.999447 0.0332482i \(-0.0105852\pi\)
0.999447 + 0.0332482i \(0.0105852\pi\)
\(390\) 0 0
\(391\) 2.41088e8 0.203966
\(392\) −2.89525e8 −0.242765
\(393\) 0 0
\(394\) −2.15649e8 −0.177628
\(395\) −1.15975e9 −0.946833
\(396\) 0 0
\(397\) 1.18373e9 0.949477 0.474739 0.880127i \(-0.342543\pi\)
0.474739 + 0.880127i \(0.342543\pi\)
\(398\) 1.08664e9 0.863966
\(399\) 0 0
\(400\) −2.08486e8 −0.162880
\(401\) 1.26764e9 0.981726 0.490863 0.871237i \(-0.336681\pi\)
0.490863 + 0.871237i \(0.336681\pi\)
\(402\) 0 0
\(403\) −8.81281e8 −0.670728
\(404\) 1.72903e8 0.130457
\(405\) 0 0
\(406\) 3.35951e8 0.249135
\(407\) −9.78473e8 −0.719396
\(408\) 0 0
\(409\) −1.01949e9 −0.736802 −0.368401 0.929667i \(-0.620095\pi\)
−0.368401 + 0.929667i \(0.620095\pi\)
\(410\) 4.06716e8 0.291439
\(411\) 0 0
\(412\) 1.32411e8 0.0932787
\(413\) 1.13064e9 0.789765
\(414\) 0 0
\(415\) 4.37447e8 0.300440
\(416\) 1.65118e8 0.112452
\(417\) 0 0
\(418\) 3.64815e7 0.0244318
\(419\) 1.04129e9 0.691546 0.345773 0.938318i \(-0.387617\pi\)
0.345773 + 0.938318i \(0.387617\pi\)
\(420\) 0 0
\(421\) −1.95845e9 −1.27916 −0.639580 0.768725i \(-0.720892\pi\)
−0.639580 + 0.768725i \(0.720892\pi\)
\(422\) −1.75472e7 −0.0113662
\(423\) 0 0
\(424\) 3.89225e8 0.247982
\(425\) 1.62320e8 0.102568
\(426\) 0 0
\(427\) −1.14037e9 −0.708838
\(428\) −5.85888e8 −0.361211
\(429\) 0 0
\(430\) −4.44418e8 −0.269558
\(431\) 2.84256e9 1.71017 0.855084 0.518489i \(-0.173505\pi\)
0.855084 + 0.518489i \(0.173505\pi\)
\(432\) 0 0
\(433\) −8.41292e7 −0.0498011 −0.0249006 0.999690i \(-0.507927\pi\)
−0.0249006 + 0.999690i \(0.507927\pi\)
\(434\) 7.10761e8 0.417359
\(435\) 0 0
\(436\) 5.55933e8 0.321233
\(437\) −1.14005e8 −0.0653489
\(438\) 0 0
\(439\) −1.87047e9 −1.05517 −0.527587 0.849501i \(-0.676903\pi\)
−0.527587 + 0.849501i \(0.676903\pi\)
\(440\) −2.55468e8 −0.142972
\(441\) 0 0
\(442\) −1.28555e8 −0.0708127
\(443\) −1.01333e9 −0.553779 −0.276889 0.960902i \(-0.589304\pi\)
−0.276889 + 0.960902i \(0.589304\pi\)
\(444\) 0 0
\(445\) 1.11720e9 0.600994
\(446\) 1.59554e8 0.0851602
\(447\) 0 0
\(448\) −1.33169e8 −0.0699730
\(449\) −1.65756e8 −0.0864183 −0.0432092 0.999066i \(-0.513758\pi\)
−0.0432092 + 0.999066i \(0.513758\pi\)
\(450\) 0 0
\(451\) −9.31749e8 −0.478279
\(452\) −1.59856e9 −0.814227
\(453\) 0 0
\(454\) −1.00788e9 −0.505491
\(455\) 4.22369e8 0.210209
\(456\) 0 0
\(457\) −1.38080e9 −0.676742 −0.338371 0.941013i \(-0.609876\pi\)
−0.338371 + 0.941013i \(0.609876\pi\)
\(458\) 1.88037e8 0.0914562
\(459\) 0 0
\(460\) 7.98336e8 0.382414
\(461\) 2.78491e9 1.32391 0.661955 0.749544i \(-0.269727\pi\)
0.661955 + 0.749544i \(0.269727\pi\)
\(462\) 0 0
\(463\) 2.78919e9 1.30600 0.653001 0.757357i \(-0.273510\pi\)
0.653001 + 0.757357i \(0.273510\pi\)
\(464\) −3.38596e8 −0.157351
\(465\) 0 0
\(466\) −1.02205e9 −0.467868
\(467\) 3.97258e9 1.80494 0.902471 0.430751i \(-0.141751\pi\)
0.902471 + 0.430751i \(0.141751\pi\)
\(468\) 0 0
\(469\) −7.48380e8 −0.334979
\(470\) 5.05819e8 0.224726
\(471\) 0 0
\(472\) −1.13954e9 −0.498807
\(473\) 1.01812e9 0.442369
\(474\) 0 0
\(475\) −7.67572e7 −0.0328618
\(476\) 1.03681e8 0.0440630
\(477\) 0 0
\(478\) −7.47902e8 −0.313218
\(479\) −2.34467e9 −0.974782 −0.487391 0.873184i \(-0.662051\pi\)
−0.487391 + 0.873184i \(0.662051\pi\)
\(480\) 0 0
\(481\) −1.63046e9 −0.668042
\(482\) −1.07824e9 −0.438583
\(483\) 0 0
\(484\) −6.61926e8 −0.265369
\(485\) −2.66910e9 −1.06235
\(486\) 0 0
\(487\) 3.08381e9 1.20986 0.604931 0.796278i \(-0.293201\pi\)
0.604931 + 0.796278i \(0.293201\pi\)
\(488\) 1.14935e9 0.447694
\(489\) 0 0
\(490\) 7.46432e8 0.286618
\(491\) −3.71851e9 −1.41770 −0.708848 0.705361i \(-0.750785\pi\)
−0.708848 + 0.705361i \(0.750785\pi\)
\(492\) 0 0
\(493\) 2.63619e8 0.0990859
\(494\) 6.07905e7 0.0226877
\(495\) 0 0
\(496\) −7.16358e8 −0.263599
\(497\) 2.54350e9 0.929361
\(498\) 0 0
\(499\) 4.36426e8 0.157238 0.0786192 0.996905i \(-0.474949\pi\)
0.0786192 + 0.996905i \(0.474949\pi\)
\(500\) 1.36250e9 0.487464
\(501\) 0 0
\(502\) 1.05176e8 0.0371069
\(503\) −5.04603e8 −0.176792 −0.0883958 0.996085i \(-0.528174\pi\)
−0.0883958 + 0.996085i \(0.528174\pi\)
\(504\) 0 0
\(505\) −4.45766e8 −0.154024
\(506\) −1.82892e9 −0.627577
\(507\) 0 0
\(508\) −3.19142e8 −0.108009
\(509\) −4.60085e9 −1.54642 −0.773208 0.634153i \(-0.781349\pi\)
−0.773208 + 0.634153i \(0.781349\pi\)
\(510\) 0 0
\(511\) 2.99634e9 0.993385
\(512\) 1.34218e8 0.0441942
\(513\) 0 0
\(514\) −1.02777e9 −0.333829
\(515\) −3.41371e8 −0.110129
\(516\) 0 0
\(517\) −1.15878e9 −0.368796
\(518\) 1.31498e9 0.415687
\(519\) 0 0
\(520\) −4.25695e8 −0.132766
\(521\) 7.14937e7 0.0221481 0.0110740 0.999939i \(-0.496475\pi\)
0.0110740 + 0.999939i \(0.496475\pi\)
\(522\) 0 0
\(523\) −2.12182e9 −0.648563 −0.324281 0.945961i \(-0.605122\pi\)
−0.324281 + 0.945961i \(0.605122\pi\)
\(524\) −1.42979e9 −0.434123
\(525\) 0 0
\(526\) −7.82013e8 −0.234296
\(527\) 5.57731e8 0.165992
\(528\) 0 0
\(529\) 2.31053e9 0.678606
\(530\) −1.00347e9 −0.292779
\(531\) 0 0
\(532\) −4.90281e7 −0.0141174
\(533\) −1.55261e9 −0.444136
\(534\) 0 0
\(535\) 1.51049e9 0.426462
\(536\) 7.54272e8 0.211569
\(537\) 0 0
\(538\) 3.17481e8 0.0878981
\(539\) −1.71001e9 −0.470368
\(540\) 0 0
\(541\) −2.11351e9 −0.573870 −0.286935 0.957950i \(-0.592636\pi\)
−0.286935 + 0.957950i \(0.592636\pi\)
\(542\) −2.80586e9 −0.756953
\(543\) 0 0
\(544\) −1.04497e8 −0.0278297
\(545\) −1.43326e9 −0.379261
\(546\) 0 0
\(547\) 6.25553e9 1.63421 0.817107 0.576487i \(-0.195577\pi\)
0.817107 + 0.576487i \(0.195577\pi\)
\(548\) 8.23237e8 0.213694
\(549\) 0 0
\(550\) −1.23137e9 −0.315588
\(551\) −1.24659e8 −0.0317462
\(552\) 0 0
\(553\) −3.57061e9 −0.897852
\(554\) 4.31682e9 1.07865
\(555\) 0 0
\(556\) 2.82818e9 0.697822
\(557\) −6.50846e9 −1.59583 −0.797913 0.602773i \(-0.794062\pi\)
−0.797913 + 0.602773i \(0.794062\pi\)
\(558\) 0 0
\(559\) 1.69653e9 0.410790
\(560\) 3.43327e8 0.0826132
\(561\) 0 0
\(562\) −8.06815e8 −0.191733
\(563\) −1.73249e8 −0.0409158 −0.0204579 0.999791i \(-0.506512\pi\)
−0.0204579 + 0.999791i \(0.506512\pi\)
\(564\) 0 0
\(565\) 4.12130e9 0.961312
\(566\) −2.03290e9 −0.471257
\(567\) 0 0
\(568\) −2.56353e9 −0.586974
\(569\) 7.28881e9 1.65869 0.829343 0.558740i \(-0.188715\pi\)
0.829343 + 0.558740i \(0.188715\pi\)
\(570\) 0 0
\(571\) 4.29036e9 0.964423 0.482212 0.876055i \(-0.339834\pi\)
0.482212 + 0.876055i \(0.339834\pi\)
\(572\) 9.75228e8 0.217881
\(573\) 0 0
\(574\) 1.25219e9 0.276363
\(575\) 3.84804e9 0.844116
\(576\) 0 0
\(577\) 6.28378e8 0.136178 0.0680888 0.997679i \(-0.478310\pi\)
0.0680888 + 0.997679i \(0.478310\pi\)
\(578\) −3.20135e9 −0.689582
\(579\) 0 0
\(580\) 8.72942e8 0.185775
\(581\) 1.34681e9 0.284898
\(582\) 0 0
\(583\) 2.29886e9 0.480478
\(584\) −3.01993e9 −0.627411
\(585\) 0 0
\(586\) −4.34467e9 −0.891899
\(587\) −8.96802e9 −1.83005 −0.915026 0.403395i \(-0.867830\pi\)
−0.915026 + 0.403395i \(0.867830\pi\)
\(588\) 0 0
\(589\) −2.63737e8 −0.0531824
\(590\) 2.93788e9 0.588913
\(591\) 0 0
\(592\) −1.32534e9 −0.262543
\(593\) −7.57484e9 −1.49170 −0.745851 0.666112i \(-0.767957\pi\)
−0.745851 + 0.666112i \(0.767957\pi\)
\(594\) 0 0
\(595\) −2.67302e8 −0.0520227
\(596\) −1.68248e8 −0.0325527
\(597\) 0 0
\(598\) −3.04759e9 −0.582776
\(599\) −6.05946e9 −1.15197 −0.575984 0.817461i \(-0.695381\pi\)
−0.575984 + 0.817461i \(0.695381\pi\)
\(600\) 0 0
\(601\) −6.55797e8 −0.123228 −0.0616139 0.998100i \(-0.519625\pi\)
−0.0616139 + 0.998100i \(0.519625\pi\)
\(602\) −1.36827e9 −0.255613
\(603\) 0 0
\(604\) −3.53932e9 −0.653568
\(605\) 1.70653e9 0.313307
\(606\) 0 0
\(607\) 3.85611e9 0.699825 0.349913 0.936782i \(-0.386211\pi\)
0.349913 + 0.936782i \(0.386211\pi\)
\(608\) 4.94141e7 0.00891638
\(609\) 0 0
\(610\) −2.96316e9 −0.528567
\(611\) −1.93092e9 −0.342469
\(612\) 0 0
\(613\) −4.89889e9 −0.858986 −0.429493 0.903070i \(-0.641308\pi\)
−0.429493 + 0.903070i \(0.641308\pi\)
\(614\) −5.83902e9 −1.01801
\(615\) 0 0
\(616\) −7.86530e8 −0.135576
\(617\) −6.65456e9 −1.14057 −0.570284 0.821448i \(-0.693167\pi\)
−0.570284 + 0.821448i \(0.693167\pi\)
\(618\) 0 0
\(619\) 1.05531e10 1.78840 0.894199 0.447669i \(-0.147746\pi\)
0.894199 + 0.447669i \(0.147746\pi\)
\(620\) 1.84686e9 0.311217
\(621\) 0 0
\(622\) 5.11016e9 0.851468
\(623\) 3.43962e9 0.569904
\(624\) 0 0
\(625\) 4.63857e8 0.0759983
\(626\) −5.10208e9 −0.831261
\(627\) 0 0
\(628\) −1.43175e9 −0.230679
\(629\) 1.03186e9 0.165327
\(630\) 0 0
\(631\) 6.37775e9 1.01057 0.505283 0.862954i \(-0.331388\pi\)
0.505283 + 0.862954i \(0.331388\pi\)
\(632\) 3.59873e9 0.567074
\(633\) 0 0
\(634\) −5.23502e9 −0.815842
\(635\) 8.22788e8 0.127520
\(636\) 0 0
\(637\) −2.84945e9 −0.436790
\(638\) −1.99983e9 −0.304874
\(639\) 0 0
\(640\) −3.46030e8 −0.0521776
\(641\) 6.68703e8 0.100284 0.0501418 0.998742i \(-0.484033\pi\)
0.0501418 + 0.998742i \(0.484033\pi\)
\(642\) 0 0
\(643\) 3.24632e9 0.481564 0.240782 0.970579i \(-0.422596\pi\)
0.240782 + 0.970579i \(0.422596\pi\)
\(644\) 2.45791e9 0.362631
\(645\) 0 0
\(646\) −3.84721e7 −0.00561477
\(647\) 7.46321e9 1.08333 0.541665 0.840595i \(-0.317794\pi\)
0.541665 + 0.840595i \(0.317794\pi\)
\(648\) 0 0
\(649\) −6.73041e9 −0.966462
\(650\) −2.05188e9 −0.293059
\(651\) 0 0
\(652\) 3.35108e9 0.473498
\(653\) −1.01630e10 −1.42832 −0.714160 0.699982i \(-0.753191\pi\)
−0.714160 + 0.699982i \(0.753191\pi\)
\(654\) 0 0
\(655\) 3.68618e9 0.512544
\(656\) −1.26205e9 −0.174547
\(657\) 0 0
\(658\) 1.55731e9 0.213100
\(659\) −2.83068e9 −0.385294 −0.192647 0.981268i \(-0.561707\pi\)
−0.192647 + 0.981268i \(0.561707\pi\)
\(660\) 0 0
\(661\) 3.96275e9 0.533693 0.266846 0.963739i \(-0.414018\pi\)
0.266846 + 0.963739i \(0.414018\pi\)
\(662\) 6.46151e9 0.865628
\(663\) 0 0
\(664\) −1.35741e9 −0.179938
\(665\) 1.26401e8 0.0166676
\(666\) 0 0
\(667\) 6.24947e9 0.815461
\(668\) −3.50693e9 −0.455207
\(669\) 0 0
\(670\) −1.94461e9 −0.249787
\(671\) 6.78832e9 0.867428
\(672\) 0 0
\(673\) −1.18615e10 −1.49999 −0.749995 0.661443i \(-0.769944\pi\)
−0.749995 + 0.661443i \(0.769944\pi\)
\(674\) −1.49824e9 −0.188483
\(675\) 0 0
\(676\) −2.39085e9 −0.297672
\(677\) 9.86836e9 1.22232 0.611159 0.791508i \(-0.290703\pi\)
0.611159 + 0.791508i \(0.290703\pi\)
\(678\) 0 0
\(679\) −8.21760e9 −1.00740
\(680\) 2.69407e8 0.0328569
\(681\) 0 0
\(682\) −4.23099e9 −0.510736
\(683\) 1.36287e10 1.63674 0.818372 0.574689i \(-0.194877\pi\)
0.818372 + 0.574689i \(0.194877\pi\)
\(684\) 0 0
\(685\) −2.12241e9 −0.252297
\(686\) 5.64499e9 0.667619
\(687\) 0 0
\(688\) 1.37904e9 0.161442
\(689\) 3.83068e9 0.446178
\(690\) 0 0
\(691\) −2.16270e9 −0.249357 −0.124679 0.992197i \(-0.539790\pi\)
−0.124679 + 0.992197i \(0.539790\pi\)
\(692\) 7.91329e9 0.907791
\(693\) 0 0
\(694\) 2.26045e9 0.256706
\(695\) −7.29139e9 −0.823879
\(696\) 0 0
\(697\) 9.82588e8 0.109915
\(698\) −6.08490e9 −0.677266
\(699\) 0 0
\(700\) 1.65486e9 0.182355
\(701\) 2.93914e9 0.322260 0.161130 0.986933i \(-0.448486\pi\)
0.161130 + 0.986933i \(0.448486\pi\)
\(702\) 0 0
\(703\) −4.87942e8 −0.0529693
\(704\) 7.92723e8 0.0856283
\(705\) 0 0
\(706\) −3.75115e9 −0.401188
\(707\) −1.37242e9 −0.146056
\(708\) 0 0
\(709\) −1.30914e10 −1.37951 −0.689756 0.724041i \(-0.742282\pi\)
−0.689756 + 0.724041i \(0.742282\pi\)
\(710\) 6.60910e9 0.693007
\(711\) 0 0
\(712\) −3.46670e9 −0.359945
\(713\) 1.32218e10 1.36609
\(714\) 0 0
\(715\) −2.51426e9 −0.257240
\(716\) 6.31870e9 0.643328
\(717\) 0 0
\(718\) 1.23125e9 0.124140
\(719\) −3.03015e9 −0.304027 −0.152014 0.988378i \(-0.548576\pi\)
−0.152014 + 0.988378i \(0.548576\pi\)
\(720\) 0 0
\(721\) −1.05101e9 −0.104432
\(722\) −7.13278e9 −0.705308
\(723\) 0 0
\(724\) 5.88707e9 0.576520
\(725\) 4.20765e9 0.410069
\(726\) 0 0
\(727\) −1.06848e10 −1.03132 −0.515661 0.856793i \(-0.672454\pi\)
−0.515661 + 0.856793i \(0.672454\pi\)
\(728\) −1.31062e9 −0.125898
\(729\) 0 0
\(730\) 7.78576e9 0.740749
\(731\) −1.07367e9 −0.101663
\(732\) 0 0
\(733\) −1.67705e10 −1.57283 −0.786414 0.617699i \(-0.788065\pi\)
−0.786414 + 0.617699i \(0.788065\pi\)
\(734\) −5.99622e9 −0.559682
\(735\) 0 0
\(736\) −2.47726e9 −0.229034
\(737\) 4.45492e9 0.409924
\(738\) 0 0
\(739\) 1.68052e10 1.53175 0.765877 0.642987i \(-0.222305\pi\)
0.765877 + 0.642987i \(0.222305\pi\)
\(740\) 3.41689e9 0.309970
\(741\) 0 0
\(742\) −3.08948e9 −0.277633
\(743\) 2.07834e10 1.85890 0.929448 0.368952i \(-0.120283\pi\)
0.929448 + 0.368952i \(0.120283\pi\)
\(744\) 0 0
\(745\) 4.33764e8 0.0384332
\(746\) −4.33382e9 −0.382195
\(747\) 0 0
\(748\) −6.17186e8 −0.0539213
\(749\) 4.65049e9 0.404401
\(750\) 0 0
\(751\) 1.00601e10 0.866689 0.433344 0.901228i \(-0.357333\pi\)
0.433344 + 0.901228i \(0.357333\pi\)
\(752\) −1.56957e9 −0.134592
\(753\) 0 0
\(754\) −3.33239e9 −0.283111
\(755\) 9.12481e9 0.771631
\(756\) 0 0
\(757\) 6.59893e9 0.552889 0.276445 0.961030i \(-0.410844\pi\)
0.276445 + 0.961030i \(0.410844\pi\)
\(758\) 1.06379e10 0.887185
\(759\) 0 0
\(760\) −1.27396e8 −0.0105271
\(761\) −6.07925e9 −0.500039 −0.250019 0.968241i \(-0.580437\pi\)
−0.250019 + 0.968241i \(0.580437\pi\)
\(762\) 0 0
\(763\) −4.41272e9 −0.359642
\(764\) 8.44663e9 0.685262
\(765\) 0 0
\(766\) 5.75959e9 0.463011
\(767\) −1.12151e10 −0.897471
\(768\) 0 0
\(769\) 4.85049e9 0.384630 0.192315 0.981333i \(-0.438400\pi\)
0.192315 + 0.981333i \(0.438400\pi\)
\(770\) 2.02777e9 0.160067
\(771\) 0 0
\(772\) 9.46607e9 0.740473
\(773\) −2.33466e10 −1.81800 −0.909002 0.416792i \(-0.863154\pi\)
−0.909002 + 0.416792i \(0.863154\pi\)
\(774\) 0 0
\(775\) 8.90200e9 0.686961
\(776\) 8.28231e9 0.636261
\(777\) 0 0
\(778\) 1.85654e10 1.41343
\(779\) −4.64642e8 −0.0352158
\(780\) 0 0
\(781\) −1.51408e10 −1.13729
\(782\) 1.92871e9 0.144226
\(783\) 0 0
\(784\) −2.31620e9 −0.171660
\(785\) 3.69123e9 0.272350
\(786\) 0 0
\(787\) 9.36056e9 0.684526 0.342263 0.939604i \(-0.388807\pi\)
0.342263 + 0.939604i \(0.388807\pi\)
\(788\) −1.72519e9 −0.125602
\(789\) 0 0
\(790\) −9.27797e9 −0.669512
\(791\) 1.26886e10 0.911582
\(792\) 0 0
\(793\) 1.13116e10 0.805506
\(794\) 9.46981e9 0.671382
\(795\) 0 0
\(796\) 8.69315e9 0.610916
\(797\) 2.62373e10 1.83576 0.917878 0.396864i \(-0.129901\pi\)
0.917878 + 0.396864i \(0.129901\pi\)
\(798\) 0 0
\(799\) 1.22201e9 0.0847543
\(800\) −1.66789e9 −0.115174
\(801\) 0 0
\(802\) 1.01411e10 0.694185
\(803\) −1.78365e10 −1.21564
\(804\) 0 0
\(805\) −6.33679e9 −0.428138
\(806\) −7.05025e9 −0.474277
\(807\) 0 0
\(808\) 1.38323e9 0.0922473
\(809\) −1.59304e10 −1.05780 −0.528902 0.848683i \(-0.677396\pi\)
−0.528902 + 0.848683i \(0.677396\pi\)
\(810\) 0 0
\(811\) −1.69281e10 −1.11438 −0.557191 0.830384i \(-0.688121\pi\)
−0.557191 + 0.830384i \(0.688121\pi\)
\(812\) 2.68760e9 0.176165
\(813\) 0 0
\(814\) −7.82778e9 −0.508690
\(815\) −8.63949e9 −0.559032
\(816\) 0 0
\(817\) 5.07713e8 0.0325718
\(818\) −8.15591e9 −0.520997
\(819\) 0 0
\(820\) 3.25373e9 0.206078
\(821\) 8.73592e9 0.550944 0.275472 0.961309i \(-0.411166\pi\)
0.275472 + 0.961309i \(0.411166\pi\)
\(822\) 0 0
\(823\) 1.44369e10 0.902767 0.451384 0.892330i \(-0.350931\pi\)
0.451384 + 0.892330i \(0.350931\pi\)
\(824\) 1.05928e9 0.0659580
\(825\) 0 0
\(826\) 9.04510e9 0.558449
\(827\) 5.56666e9 0.342236 0.171118 0.985251i \(-0.445262\pi\)
0.171118 + 0.985251i \(0.445262\pi\)
\(828\) 0 0
\(829\) −1.50766e10 −0.919101 −0.459550 0.888152i \(-0.651989\pi\)
−0.459550 + 0.888152i \(0.651989\pi\)
\(830\) 3.49958e9 0.212443
\(831\) 0 0
\(832\) 1.32094e9 0.0795157
\(833\) 1.80331e9 0.108097
\(834\) 0 0
\(835\) 9.04130e9 0.537438
\(836\) 2.91852e8 0.0172759
\(837\) 0 0
\(838\) 8.33029e9 0.488997
\(839\) 1.20004e10 0.701502 0.350751 0.936469i \(-0.385926\pi\)
0.350751 + 0.936469i \(0.385926\pi\)
\(840\) 0 0
\(841\) −1.04164e10 −0.603852
\(842\) −1.56676e10 −0.904503
\(843\) 0 0
\(844\) −1.40378e8 −0.00803710
\(845\) 6.16390e9 0.351445
\(846\) 0 0
\(847\) 5.25404e9 0.297099
\(848\) 3.11380e9 0.175350
\(849\) 0 0
\(850\) 1.29856e9 0.0725264
\(851\) 2.44618e10 1.36061
\(852\) 0 0
\(853\) −9.01661e8 −0.0497418 −0.0248709 0.999691i \(-0.507917\pi\)
−0.0248709 + 0.999691i \(0.507917\pi\)
\(854\) −9.12293e9 −0.501224
\(855\) 0 0
\(856\) −4.68710e9 −0.255415
\(857\) −2.81436e10 −1.52738 −0.763690 0.645583i \(-0.776614\pi\)
−0.763690 + 0.645583i \(0.776614\pi\)
\(858\) 0 0
\(859\) −9.51543e9 −0.512215 −0.256107 0.966648i \(-0.582440\pi\)
−0.256107 + 0.966648i \(0.582440\pi\)
\(860\) −3.55534e9 −0.190606
\(861\) 0 0
\(862\) 2.27405e10 1.20927
\(863\) −1.24682e10 −0.660336 −0.330168 0.943922i \(-0.607105\pi\)
−0.330168 + 0.943922i \(0.607105\pi\)
\(864\) 0 0
\(865\) −2.04014e10 −1.07178
\(866\) −6.73033e8 −0.0352147
\(867\) 0 0
\(868\) 5.68609e9 0.295117
\(869\) 2.12550e10 1.09873
\(870\) 0 0
\(871\) 7.42339e9 0.380662
\(872\) 4.44746e9 0.227146
\(873\) 0 0
\(874\) −9.12038e8 −0.0462086
\(875\) −1.08149e10 −0.545749
\(876\) 0 0
\(877\) −6.71074e9 −0.335948 −0.167974 0.985791i \(-0.553722\pi\)
−0.167974 + 0.985791i \(0.553722\pi\)
\(878\) −1.49637e10 −0.746120
\(879\) 0 0
\(880\) −2.04374e9 −0.101097
\(881\) −3.59350e10 −1.77053 −0.885263 0.465091i \(-0.846022\pi\)
−0.885263 + 0.465091i \(0.846022\pi\)
\(882\) 0 0
\(883\) −1.40241e9 −0.0685509 −0.0342754 0.999412i \(-0.510912\pi\)
−0.0342754 + 0.999412i \(0.510912\pi\)
\(884\) −1.02844e9 −0.0500721
\(885\) 0 0
\(886\) −8.10661e9 −0.391581
\(887\) −8.19690e8 −0.0394382 −0.0197191 0.999806i \(-0.506277\pi\)
−0.0197191 + 0.999806i \(0.506277\pi\)
\(888\) 0 0
\(889\) 2.53319e9 0.120924
\(890\) 8.93759e9 0.424967
\(891\) 0 0
\(892\) 1.27643e9 0.0602174
\(893\) −5.77860e8 −0.0271545
\(894\) 0 0
\(895\) −1.62904e10 −0.759541
\(896\) −1.06535e9 −0.0494784
\(897\) 0 0
\(898\) −1.32604e9 −0.0611070
\(899\) 1.44574e10 0.663640
\(900\) 0 0
\(901\) −2.42430e9 −0.110420
\(902\) −7.45399e9 −0.338194
\(903\) 0 0
\(904\) −1.27885e10 −0.575745
\(905\) −1.51776e10 −0.680665
\(906\) 0 0
\(907\) 2.84506e9 0.126609 0.0633047 0.997994i \(-0.479836\pi\)
0.0633047 + 0.997994i \(0.479836\pi\)
\(908\) −8.06304e9 −0.357436
\(909\) 0 0
\(910\) 3.37895e9 0.148640
\(911\) −4.32996e10 −1.89745 −0.948723 0.316108i \(-0.897624\pi\)
−0.948723 + 0.316108i \(0.897624\pi\)
\(912\) 0 0
\(913\) −8.01722e9 −0.348639
\(914\) −1.10464e10 −0.478529
\(915\) 0 0
\(916\) 1.50429e9 0.0646693
\(917\) 1.13490e10 0.486030
\(918\) 0 0
\(919\) −3.26317e10 −1.38687 −0.693435 0.720520i \(-0.743903\pi\)
−0.693435 + 0.720520i \(0.743903\pi\)
\(920\) 6.38669e9 0.270407
\(921\) 0 0
\(922\) 2.22793e10 0.936146
\(923\) −2.52297e10 −1.05610
\(924\) 0 0
\(925\) 1.64697e10 0.684209
\(926\) 2.23135e10 0.923483
\(927\) 0 0
\(928\) −2.70877e9 −0.111264
\(929\) 4.69546e10 1.92142 0.960712 0.277546i \(-0.0895211\pi\)
0.960712 + 0.277546i \(0.0895211\pi\)
\(930\) 0 0
\(931\) −8.52742e8 −0.0346333
\(932\) −8.17644e9 −0.330833
\(933\) 0 0
\(934\) 3.17806e10 1.27629
\(935\) 1.59118e9 0.0636619
\(936\) 0 0
\(937\) −2.80951e10 −1.11569 −0.557843 0.829947i \(-0.688371\pi\)
−0.557843 + 0.829947i \(0.688371\pi\)
\(938\) −5.98704e9 −0.236866
\(939\) 0 0
\(940\) 4.04655e9 0.158905
\(941\) −2.72197e10 −1.06493 −0.532464 0.846453i \(-0.678734\pi\)
−0.532464 + 0.846453i \(0.678734\pi\)
\(942\) 0 0
\(943\) 2.32937e10 0.904582
\(944\) −9.11632e9 −0.352710
\(945\) 0 0
\(946\) 8.14496e9 0.312802
\(947\) −2.30441e10 −0.881729 −0.440864 0.897574i \(-0.645328\pi\)
−0.440864 + 0.897574i \(0.645328\pi\)
\(948\) 0 0
\(949\) −2.97215e10 −1.12886
\(950\) −6.14058e8 −0.0232368
\(951\) 0 0
\(952\) 8.29446e8 0.0311572
\(953\) 2.22131e10 0.831352 0.415676 0.909513i \(-0.363545\pi\)
0.415676 + 0.909513i \(0.363545\pi\)
\(954\) 0 0
\(955\) −2.17765e10 −0.809051
\(956\) −5.98322e9 −0.221479
\(957\) 0 0
\(958\) −1.87574e10 −0.689275
\(959\) −6.53444e9 −0.239245
\(960\) 0 0
\(961\) 3.07460e9 0.111752
\(962\) −1.30437e10 −0.472377
\(963\) 0 0
\(964\) −8.62593e9 −0.310125
\(965\) −2.44047e10 −0.874235
\(966\) 0 0
\(967\) 2.70771e10 0.962962 0.481481 0.876457i \(-0.340099\pi\)
0.481481 + 0.876457i \(0.340099\pi\)
\(968\) −5.29541e9 −0.187644
\(969\) 0 0
\(970\) −2.13528e10 −0.751198
\(971\) −2.57664e10 −0.903204 −0.451602 0.892220i \(-0.649147\pi\)
−0.451602 + 0.892220i \(0.649147\pi\)
\(972\) 0 0
\(973\) −2.24486e10 −0.781259
\(974\) 2.46705e10 0.855502
\(975\) 0 0
\(976\) 9.19476e9 0.316567
\(977\) −9.51873e9 −0.326549 −0.163274 0.986581i \(-0.552206\pi\)
−0.163274 + 0.986581i \(0.552206\pi\)
\(978\) 0 0
\(979\) −2.04752e10 −0.697411
\(980\) 5.97146e9 0.202670
\(981\) 0 0
\(982\) −2.97481e10 −1.00246
\(983\) −5.51917e9 −0.185326 −0.0926631 0.995698i \(-0.529538\pi\)
−0.0926631 + 0.995698i \(0.529538\pi\)
\(984\) 0 0
\(985\) 4.44776e9 0.148291
\(986\) 2.10895e9 0.0700643
\(987\) 0 0
\(988\) 4.86324e8 0.0160427
\(989\) −2.54530e10 −0.836666
\(990\) 0 0
\(991\) 1.65440e10 0.539987 0.269994 0.962862i \(-0.412978\pi\)
0.269994 + 0.962862i \(0.412978\pi\)
\(992\) −5.73086e9 −0.186393
\(993\) 0 0
\(994\) 2.03480e10 0.657158
\(995\) −2.24120e10 −0.721274
\(996\) 0 0
\(997\) 4.10055e10 1.31042 0.655208 0.755448i \(-0.272581\pi\)
0.655208 + 0.755448i \(0.272581\pi\)
\(998\) 3.49141e9 0.111184
\(999\) 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 162.8.a.b.1.1 yes 1
3.2 odd 2 162.8.a.a.1.1 1
9.2 odd 6 162.8.c.h.109.1 2
9.4 even 3 162.8.c.e.55.1 2
9.5 odd 6 162.8.c.h.55.1 2
9.7 even 3 162.8.c.e.109.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.8.a.a.1.1 1 3.2 odd 2
162.8.a.b.1.1 yes 1 1.1 even 1 trivial
162.8.c.e.55.1 2 9.4 even 3
162.8.c.e.109.1 2 9.7 even 3
162.8.c.h.55.1 2 9.5 odd 6
162.8.c.h.109.1 2 9.2 odd 6