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Properties

Label 432.8.a.b.1.1
Level $432$
Weight $8$
Character 432.1
Self dual yes
Analytic conductor $134.950$
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] = [432,8,Mod(1,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, 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("432.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.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: \(134.950331009\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 432.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-120.000 q^{5} -377.000 q^{7} +O(q^{10})\) \(q-120.000 q^{5} -377.000 q^{7} +600.000 q^{11} +5369.00 q^{13} -12168.0 q^{17} -16211.0 q^{19} +106392. q^{23} -63725.0 q^{25} -177216. q^{29} +268060. q^{31} +45240.0 q^{35} +114959. q^{37} +112128. q^{41} +115048. q^{43} +561336. q^{47} -681414. q^{49} +1.78776e6 q^{53} -72000.0 q^{55} -1.78634e6 q^{59} -1.30684e6 q^{61} -644280. q^{65} +2.01382e6 q^{67} -4.06094e6 q^{71} -3.85064e6 q^{73} -226200. q^{77} -1.03723e6 q^{79} +9.20357e6 q^{83} +1.46016e6 q^{85} +1.28930e6 q^{89} -2.02411e6 q^{91} +1.94532e6 q^{95} +8.55588e6 q^{97} +O(q^{100})\)

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
\(5\) −120.000 −0.429325 −0.214663 − 0.976688i \(-0.568865\pi\)
−0.214663 + 0.976688i \(0.568865\pi\)
\(6\) 0 0
\(7\) −377.000 −0.415430 −0.207715 − 0.978189i \(-0.566603\pi\)
−0.207715 + 0.978189i \(0.566603\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 600.000 0.135918 0.0679590 − 0.997688i \(-0.478351\pi\)
0.0679590 + 0.997688i \(0.478351\pi\)
\(12\) 0 0
\(13\) 5369.00 0.677785 0.338892 − 0.940825i \(-0.389948\pi\)
0.338892 + 0.940825i \(0.389948\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −12168.0 −0.600687 −0.300343 − 0.953831i \(-0.597101\pi\)
−0.300343 + 0.953831i \(0.597101\pi\)
\(18\) 0 0
\(19\) −16211.0 −0.542216 −0.271108 − 0.962549i \(-0.587390\pi\)
−0.271108 + 0.962549i \(0.587390\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 106392. 1.82331 0.911657 − 0.410952i \(-0.134803\pi\)
0.911657 + 0.410952i \(0.134803\pi\)
\(24\) 0 0
\(25\) −63725.0 −0.815680
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −177216. −1.34930 −0.674652 − 0.738136i \(-0.735706\pi\)
−0.674652 + 0.738136i \(0.735706\pi\)
\(30\) 0 0
\(31\) 268060. 1.61609 0.808046 − 0.589119i \(-0.200525\pi\)
0.808046 + 0.589119i \(0.200525\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 45240.0 0.178355
\(36\) 0 0
\(37\) 114959. 0.373110 0.186555 − 0.982445i \(-0.440268\pi\)
0.186555 + 0.982445i \(0.440268\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 112128. 0.254080 0.127040 − 0.991898i \(-0.459452\pi\)
0.127040 + 0.991898i \(0.459452\pi\)
\(42\) 0 0
\(43\) 115048. 0.220668 0.110334 − 0.993895i \(-0.464808\pi\)
0.110334 + 0.993895i \(0.464808\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 561336. 0.788643 0.394321 − 0.918973i \(-0.370980\pi\)
0.394321 + 0.918973i \(0.370980\pi\)
\(48\) 0 0
\(49\) −681414. −0.827418
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.78776e6 1.64947 0.824734 − 0.565521i \(-0.191325\pi\)
0.824734 + 0.565521i \(0.191325\pi\)
\(54\) 0 0
\(55\) −72000.0 −0.0583530
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.78634e6 −1.13236 −0.566178 − 0.824283i \(-0.691579\pi\)
−0.566178 + 0.824283i \(0.691579\pi\)
\(60\) 0 0
\(61\) −1.30684e6 −0.737169 −0.368584 − 0.929594i \(-0.620157\pi\)
−0.368584 + 0.929594i \(0.620157\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −644280. −0.290990
\(66\) 0 0
\(67\) 2.01382e6 0.818009 0.409005 − 0.912532i \(-0.365876\pi\)
0.409005 + 0.912532i \(0.365876\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.06094e6 −1.34655 −0.673275 − 0.739392i \(-0.735113\pi\)
−0.673275 + 0.739392i \(0.735113\pi\)
\(72\) 0 0
\(73\) −3.85064e6 −1.15852 −0.579259 − 0.815144i \(-0.696658\pi\)
−0.579259 + 0.815144i \(0.696658\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −226200. −0.0564644
\(78\) 0 0
\(79\) −1.03723e6 −0.236690 −0.118345 − 0.992973i \(-0.537759\pi\)
−0.118345 + 0.992973i \(0.537759\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.20357e6 1.76678 0.883391 − 0.468637i \(-0.155255\pi\)
0.883391 + 0.468637i \(0.155255\pi\)
\(84\) 0 0
\(85\) 1.46016e6 0.257890
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.28930e6 0.193861 0.0969305 − 0.995291i \(-0.469098\pi\)
0.0969305 + 0.995291i \(0.469098\pi\)
\(90\) 0 0
\(91\) −2.02411e6 −0.281572
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.94532e6 0.232787
\(96\) 0 0
\(97\) 8.55588e6 0.951840 0.475920 − 0.879489i \(-0.342115\pi\)
0.475920 + 0.879489i \(0.342115\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.99006e7 −1.92195 −0.960974 − 0.276639i \(-0.910779\pi\)
−0.960974 + 0.276639i \(0.910779\pi\)
\(102\) 0 0
\(103\) −6.99984e6 −0.631187 −0.315594 − 0.948894i \(-0.602204\pi\)
−0.315594 + 0.948894i \(0.602204\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.48063e6 −0.353587 −0.176793 − 0.984248i \(-0.556572\pi\)
−0.176793 + 0.984248i \(0.556572\pi\)
\(108\) 0 0
\(109\) −1.47074e6 −0.108779 −0.0543893 − 0.998520i \(-0.517321\pi\)
−0.0543893 + 0.998520i \(0.517321\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.47786e6 −0.226745 −0.113373 − 0.993553i \(-0.536165\pi\)
−0.113373 + 0.993553i \(0.536165\pi\)
\(114\) 0 0
\(115\) −1.27670e7 −0.782795
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.58734e6 0.249543
\(120\) 0 0
\(121\) −1.91272e7 −0.981526
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.70220e7 0.779517
\(126\) 0 0
\(127\) −2.10378e7 −0.911353 −0.455677 − 0.890145i \(-0.650603\pi\)
−0.455677 + 0.890145i \(0.650603\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.06906e7 −0.804125 −0.402062 − 0.915612i \(-0.631706\pi\)
−0.402062 + 0.915612i \(0.631706\pi\)
\(132\) 0 0
\(133\) 6.11155e6 0.225253
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.31521e7 1.76603 0.883016 − 0.469343i \(-0.155509\pi\)
0.883016 + 0.469343i \(0.155509\pi\)
\(138\) 0 0
\(139\) 1.95009e7 0.615890 0.307945 − 0.951404i \(-0.400359\pi\)
0.307945 + 0.951404i \(0.400359\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.22140e6 0.0921231
\(144\) 0 0
\(145\) 2.12659e7 0.579290
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.51563e6 −0.210894 −0.105447 − 0.994425i \(-0.533627\pi\)
−0.105447 + 0.994425i \(0.533627\pi\)
\(150\) 0 0
\(151\) −6.37020e7 −1.50568 −0.752841 − 0.658202i \(-0.771317\pi\)
−0.752841 + 0.658202i \(0.771317\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.21672e7 −0.693829
\(156\) 0 0
\(157\) −6.36342e7 −1.31233 −0.656163 − 0.754619i \(-0.727822\pi\)
−0.656163 + 0.754619i \(0.727822\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.01098e7 −0.757460
\(162\) 0 0
\(163\) −6.87301e7 −1.24305 −0.621527 − 0.783392i \(-0.713488\pi\)
−0.621527 + 0.783392i \(0.713488\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.29434e7 0.215051 0.107525 − 0.994202i \(-0.465707\pi\)
0.107525 + 0.994202i \(0.465707\pi\)
\(168\) 0 0
\(169\) −3.39224e7 −0.540608
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.22124e7 0.619838 0.309919 − 0.950763i \(-0.399698\pi\)
0.309919 + 0.950763i \(0.399698\pi\)
\(174\) 0 0
\(175\) 2.40243e7 0.338858
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.13640e8 1.48096 0.740482 − 0.672076i \(-0.234597\pi\)
0.740482 + 0.672076i \(0.234597\pi\)
\(180\) 0 0
\(181\) −1.27922e8 −1.60350 −0.801752 − 0.597658i \(-0.796098\pi\)
−0.801752 + 0.597658i \(0.796098\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.37951e7 −0.160185
\(186\) 0 0
\(187\) −7.30080e6 −0.0816441
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.10840e7 −0.218946 −0.109473 − 0.993990i \(-0.534916\pi\)
−0.109473 + 0.993990i \(0.534916\pi\)
\(192\) 0 0
\(193\) −1.39821e8 −1.39998 −0.699990 − 0.714152i \(-0.746812\pi\)
−0.699990 + 0.714152i \(0.746812\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.45966e8 −1.36025 −0.680127 − 0.733094i \(-0.738076\pi\)
−0.680127 + 0.733094i \(0.738076\pi\)
\(198\) 0 0
\(199\) 5.11146e7 0.459790 0.229895 − 0.973215i \(-0.426162\pi\)
0.229895 + 0.973215i \(0.426162\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.68104e7 0.560542
\(204\) 0 0
\(205\) −1.34554e7 −0.109083
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.72660e6 −0.0736969
\(210\) 0 0
\(211\) −9.80537e7 −0.718581 −0.359290 − 0.933226i \(-0.616981\pi\)
−0.359290 + 0.933226i \(0.616981\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.38058e7 −0.0947383
\(216\) 0 0
\(217\) −1.01059e8 −0.671374
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.53300e7 −0.407136
\(222\) 0 0
\(223\) −2.16758e8 −1.30891 −0.654453 − 0.756103i \(-0.727101\pi\)
−0.654453 + 0.756103i \(0.727101\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.39865e8 −0.793629 −0.396814 − 0.917899i \(-0.629884\pi\)
−0.396814 + 0.917899i \(0.629884\pi\)
\(228\) 0 0
\(229\) −9.49569e7 −0.522519 −0.261260 − 0.965269i \(-0.584138\pi\)
−0.261260 + 0.965269i \(0.584138\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.10948e8 1.09252 0.546261 − 0.837615i \(-0.316051\pi\)
0.546261 + 0.837615i \(0.316051\pi\)
\(234\) 0 0
\(235\) −6.73603e7 −0.338584
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.29152e8 −0.611939 −0.305969 − 0.952041i \(-0.598981\pi\)
−0.305969 + 0.952041i \(0.598981\pi\)
\(240\) 0 0
\(241\) −4.01868e7 −0.184937 −0.0924685 − 0.995716i \(-0.529476\pi\)
−0.0924685 + 0.995716i \(0.529476\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.17697e7 0.355231
\(246\) 0 0
\(247\) −8.70369e7 −0.367506
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.68201e8 −1.07054 −0.535270 − 0.844681i \(-0.679790\pi\)
−0.535270 + 0.844681i \(0.679790\pi\)
\(252\) 0 0
\(253\) 6.38352e7 0.247821
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.71864e8 0.631566 0.315783 − 0.948831i \(-0.397733\pi\)
0.315783 + 0.948831i \(0.397733\pi\)
\(258\) 0 0
\(259\) −4.33395e7 −0.155001
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.02033e8 0.684822 0.342411 − 0.939550i \(-0.388757\pi\)
0.342411 + 0.939550i \(0.388757\pi\)
\(264\) 0 0
\(265\) −2.14531e8 −0.708158
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.73917e7 0.305063 0.152531 − 0.988299i \(-0.451257\pi\)
0.152531 + 0.988299i \(0.451257\pi\)
\(270\) 0 0
\(271\) −4.51945e8 −1.37941 −0.689705 − 0.724091i \(-0.742260\pi\)
−0.689705 + 0.724091i \(0.742260\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.82350e7 −0.110866
\(276\) 0 0
\(277\) −2.09703e8 −0.592823 −0.296411 − 0.955060i \(-0.595790\pi\)
−0.296411 + 0.955060i \(0.595790\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.32204e8 −1.43089 −0.715444 − 0.698670i \(-0.753776\pi\)
−0.715444 + 0.698670i \(0.753776\pi\)
\(282\) 0 0
\(283\) −4.74761e8 −1.24515 −0.622576 − 0.782559i \(-0.713914\pi\)
−0.622576 + 0.782559i \(0.713914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.22723e7 −0.105553
\(288\) 0 0
\(289\) −2.62278e8 −0.639176
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.32118e8 0.539104 0.269552 − 0.962986i \(-0.413124\pi\)
0.269552 + 0.962986i \(0.413124\pi\)
\(294\) 0 0
\(295\) 2.14361e8 0.486149
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.71219e8 1.23581
\(300\) 0 0
\(301\) −4.33731e7 −0.0916722
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.56820e8 0.316485
\(306\) 0 0
\(307\) −7.53314e8 −1.48591 −0.742953 − 0.669343i \(-0.766576\pi\)
−0.742953 + 0.669343i \(0.766576\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.79778e8 0.904439 0.452219 − 0.891907i \(-0.350632\pi\)
0.452219 + 0.891907i \(0.350632\pi\)
\(312\) 0 0
\(313\) 6.55525e8 1.20833 0.604163 − 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.93361e8 −1.22251 −0.611255 − 0.791434i \(-0.709335\pi\)
−0.611255 + 0.791434i \(0.709335\pi\)
\(318\) 0 0
\(319\) −1.06330e8 −0.183395
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.97255e8 0.325702
\(324\) 0 0
\(325\) −3.42140e8 −0.552855
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.11624e8 −0.327626
\(330\) 0 0
\(331\) 1.84653e8 0.279872 0.139936 − 0.990161i \(-0.455310\pi\)
0.139936 + 0.990161i \(0.455310\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.41658e8 −0.351192
\(336\) 0 0
\(337\) 8.43883e8 1.20110 0.600548 − 0.799589i \(-0.294949\pi\)
0.600548 + 0.799589i \(0.294949\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.60836e8 0.219656
\(342\) 0 0
\(343\) 5.67369e8 0.759165
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.22250e8 0.157071 0.0785353 − 0.996911i \(-0.474976\pi\)
0.0785353 + 0.996911i \(0.474976\pi\)
\(348\) 0 0
\(349\) −1.05296e9 −1.32594 −0.662969 − 0.748646i \(-0.730704\pi\)
−0.662969 + 0.748646i \(0.730704\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.24584e8 0.755752 0.377876 − 0.925856i \(-0.376654\pi\)
0.377876 + 0.925856i \(0.376654\pi\)
\(354\) 0 0
\(355\) 4.87313e8 0.578108
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −7.95194e8 −0.907074 −0.453537 − 0.891237i \(-0.649838\pi\)
−0.453537 + 0.891237i \(0.649838\pi\)
\(360\) 0 0
\(361\) −6.31075e8 −0.706002
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.62077e8 0.497381
\(366\) 0 0
\(367\) 1.09445e9 1.15575 0.577876 − 0.816125i \(-0.303882\pi\)
0.577876 + 0.816125i \(0.303882\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.73986e8 −0.685239
\(372\) 0 0
\(373\) 8.82044e8 0.880054 0.440027 − 0.897985i \(-0.354969\pi\)
0.440027 + 0.897985i \(0.354969\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.51473e8 −0.914538
\(378\) 0 0
\(379\) 1.04251e9 0.983651 0.491826 − 0.870694i \(-0.336330\pi\)
0.491826 + 0.870694i \(0.336330\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.36847e8 −0.761115 −0.380558 − 0.924757i \(-0.624268\pi\)
−0.380558 + 0.924757i \(0.624268\pi\)
\(384\) 0 0
\(385\) 2.71440e7 0.0242416
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.47060e9 1.26669 0.633345 − 0.773870i \(-0.281681\pi\)
0.633345 + 0.773870i \(0.281681\pi\)
\(390\) 0 0
\(391\) −1.29458e9 −1.09524
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.24468e8 0.101617
\(396\) 0 0
\(397\) −1.26746e9 −1.01664 −0.508322 − 0.861167i \(-0.669734\pi\)
−0.508322 + 0.861167i \(0.669734\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.27858e8 −0.408801 −0.204401 − 0.978887i \(-0.565524\pi\)
−0.204401 + 0.978887i \(0.565524\pi\)
\(402\) 0 0
\(403\) 1.43921e9 1.09536
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.89754e7 0.0507124
\(408\) 0 0
\(409\) −2.22791e8 −0.161015 −0.0805073 − 0.996754i \(-0.525654\pi\)
−0.0805073 + 0.996754i \(0.525654\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 6.73452e8 0.470415
\(414\) 0 0
\(415\) −1.10443e9 −0.758524
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.40581e8 −0.292602 −0.146301 − 0.989240i \(-0.546737\pi\)
−0.146301 + 0.989240i \(0.546737\pi\)
\(420\) 0 0
\(421\) 2.18934e9 1.42997 0.714984 − 0.699140i \(-0.246434\pi\)
0.714984 + 0.699140i \(0.246434\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.75406e8 0.489968
\(426\) 0 0
\(427\) 4.92678e8 0.306242
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.03634e9 1.82675 0.913377 − 0.407114i \(-0.133465\pi\)
0.913377 + 0.407114i \(0.133465\pi\)
\(432\) 0 0
\(433\) 2.21735e9 1.31258 0.656290 − 0.754509i \(-0.272125\pi\)
0.656290 + 0.754509i \(0.272125\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.72472e9 −0.988630
\(438\) 0 0
\(439\) 2.50094e9 1.41084 0.705419 − 0.708791i \(-0.250759\pi\)
0.705419 + 0.708791i \(0.250759\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.40387e9 −1.31370 −0.656852 − 0.754019i \(-0.728113\pi\)
−0.656852 + 0.754019i \(0.728113\pi\)
\(444\) 0 0
\(445\) −1.54716e8 −0.0832294
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.14935e9 −1.12059 −0.560294 − 0.828294i \(-0.689312\pi\)
−0.560294 + 0.828294i \(0.689312\pi\)
\(450\) 0 0
\(451\) 6.72768e7 0.0345340
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.42894e8 0.120886
\(456\) 0 0
\(457\) 2.30159e9 1.12803 0.564015 − 0.825765i \(-0.309256\pi\)
0.564015 + 0.825765i \(0.309256\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.37928e9 −0.655689 −0.327844 − 0.944732i \(-0.606322\pi\)
−0.327844 + 0.944732i \(0.606322\pi\)
\(462\) 0 0
\(463\) 1.72054e9 0.805620 0.402810 − 0.915284i \(-0.368033\pi\)
0.402810 + 0.915284i \(0.368033\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.58750e9 0.721280 0.360640 − 0.932705i \(-0.382558\pi\)
0.360640 + 0.932705i \(0.382558\pi\)
\(468\) 0 0
\(469\) −7.59209e8 −0.339826
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.90288e7 0.0299928
\(474\) 0 0
\(475\) 1.03305e9 0.442275
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.75670e9 0.730336 0.365168 − 0.930942i \(-0.381012\pi\)
0.365168 + 0.930942i \(0.381012\pi\)
\(480\) 0 0
\(481\) 6.17215e8 0.252888
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.02671e9 −0.408649
\(486\) 0 0
\(487\) −1.52004e9 −0.596353 −0.298177 − 0.954511i \(-0.596378\pi\)
−0.298177 + 0.954511i \(0.596378\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.64824e9 1.39091 0.695453 − 0.718572i \(-0.255204\pi\)
0.695453 + 0.718572i \(0.255204\pi\)
\(492\) 0 0
\(493\) 2.15636e9 0.810509
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.53098e9 0.559398
\(498\) 0 0
\(499\) −4.83503e8 −0.174200 −0.0870998 − 0.996200i \(-0.527760\pi\)
−0.0870998 + 0.996200i \(0.527760\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.20389e9 −1.47287 −0.736434 − 0.676510i \(-0.763492\pi\)
−0.736434 + 0.676510i \(0.763492\pi\)
\(504\) 0 0
\(505\) 2.38807e9 0.825140
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.67771e9 0.900018 0.450009 − 0.893024i \(-0.351421\pi\)
0.450009 + 0.893024i \(0.351421\pi\)
\(510\) 0 0
\(511\) 1.45169e9 0.481284
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.39981e8 0.270984
\(516\) 0 0
\(517\) 3.36802e8 0.107191
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.83951e9 1.49923 0.749616 − 0.661873i \(-0.230238\pi\)
0.749616 + 0.661873i \(0.230238\pi\)
\(522\) 0 0
\(523\) −6.47019e8 −0.197770 −0.0988851 − 0.995099i \(-0.531528\pi\)
−0.0988851 + 0.995099i \(0.531528\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.26175e9 −0.970765
\(528\) 0 0
\(529\) 7.91443e9 2.32448
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.02015e8 0.172212
\(534\) 0 0
\(535\) 5.37676e8 0.151804
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4.08848e8 −0.112461
\(540\) 0 0
\(541\) −3.47237e8 −0.0942835 −0.0471418 − 0.998888i \(-0.515011\pi\)
−0.0471418 + 0.998888i \(0.515011\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.76489e8 0.0467014
\(546\) 0 0
\(547\) 2.88628e9 0.754021 0.377010 − 0.926209i \(-0.376952\pi\)
0.377010 + 0.926209i \(0.376952\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.87285e9 0.731614
\(552\) 0 0
\(553\) 3.91036e8 0.0983284
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.74890e8 0.214516 0.107258 − 0.994231i \(-0.465793\pi\)
0.107258 + 0.994231i \(0.465793\pi\)
\(558\) 0 0
\(559\) 6.17693e8 0.149565
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.07444e9 −1.90692 −0.953462 − 0.301515i \(-0.902508\pi\)
−0.953462 + 0.301515i \(0.902508\pi\)
\(564\) 0 0
\(565\) 4.17344e8 0.0973474
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.50834e9 −1.25351 −0.626755 − 0.779216i \(-0.715617\pi\)
−0.626755 + 0.779216i \(0.715617\pi\)
\(570\) 0 0
\(571\) 2.81462e9 0.632693 0.316347 − 0.948644i \(-0.397544\pi\)
0.316347 + 0.948644i \(0.397544\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.77983e9 −1.48724
\(576\) 0 0
\(577\) −4.74376e9 −1.02803 −0.514017 − 0.857780i \(-0.671843\pi\)
−0.514017 + 0.857780i \(0.671843\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.46975e9 −0.733975
\(582\) 0 0
\(583\) 1.07266e9 0.224192
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.92039e9 1.00408 0.502038 − 0.864846i \(-0.332584\pi\)
0.502038 + 0.864846i \(0.332584\pi\)
\(588\) 0 0
\(589\) −4.34552e9 −0.876271
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.96899e9 0.387750 0.193875 − 0.981026i \(-0.437894\pi\)
0.193875 + 0.981026i \(0.437894\pi\)
\(594\) 0 0
\(595\) −5.50480e8 −0.107135
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8.28842e9 1.57572 0.787858 − 0.615857i \(-0.211190\pi\)
0.787858 + 0.615857i \(0.211190\pi\)
\(600\) 0 0
\(601\) 7.19345e9 1.35169 0.675844 − 0.737045i \(-0.263779\pi\)
0.675844 + 0.737045i \(0.263779\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.29526e9 0.421394
\(606\) 0 0
\(607\) 5.35656e9 0.972132 0.486066 − 0.873922i \(-0.338431\pi\)
0.486066 + 0.873922i \(0.338431\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.01381e9 0.534530
\(612\) 0 0
\(613\) −1.01444e10 −1.77874 −0.889371 − 0.457187i \(-0.848857\pi\)
−0.889371 + 0.457187i \(0.848857\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.40428e9 0.412085 0.206043 − 0.978543i \(-0.433941\pi\)
0.206043 + 0.978543i \(0.433941\pi\)
\(618\) 0 0
\(619\) 2.75619e9 0.467081 0.233541 − 0.972347i \(-0.424969\pi\)
0.233541 + 0.972347i \(0.424969\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.86068e8 −0.0805357
\(624\) 0 0
\(625\) 2.93588e9 0.481014
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.39882e9 −0.224122
\(630\) 0 0
\(631\) −3.92975e9 −0.622676 −0.311338 − 0.950299i \(-0.600777\pi\)
−0.311338 + 0.950299i \(0.600777\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 2.52453e9 0.391267
\(636\) 0 0
\(637\) −3.65851e9 −0.560811
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.21986e10 −1.82939 −0.914696 − 0.404143i \(-0.867570\pi\)
−0.914696 + 0.404143i \(0.867570\pi\)
\(642\) 0 0
\(643\) −4.64582e8 −0.0689166 −0.0344583 − 0.999406i \(-0.510971\pi\)
−0.0344583 + 0.999406i \(0.510971\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.98889e9 1.01448 0.507239 − 0.861805i \(-0.330666\pi\)
0.507239 + 0.861805i \(0.330666\pi\)
\(648\) 0 0
\(649\) −1.07181e9 −0.153908
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.44988e9 −0.484850 −0.242425 − 0.970170i \(-0.577943\pi\)
−0.242425 + 0.970170i \(0.577943\pi\)
\(654\) 0 0
\(655\) 2.48287e9 0.345231
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.87453e9 −1.34406 −0.672029 − 0.740525i \(-0.734577\pi\)
−0.672029 + 0.740525i \(0.734577\pi\)
\(660\) 0 0
\(661\) −1.22825e10 −1.65417 −0.827087 − 0.562075i \(-0.810003\pi\)
−0.827087 + 0.562075i \(0.810003\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.33386e8 −0.0967067
\(666\) 0 0
\(667\) −1.88544e10 −2.46021
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −7.84102e8 −0.100194
\(672\) 0 0
\(673\) 9.21233e9 1.16498 0.582488 − 0.812840i \(-0.302079\pi\)
0.582488 + 0.812840i \(0.302079\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.74226e9 −0.463525 −0.231763 − 0.972772i \(-0.574449\pi\)
−0.231763 + 0.972772i \(0.574449\pi\)
\(678\) 0 0
\(679\) −3.22557e9 −0.395423
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4.17620e9 −0.501544 −0.250772 − 0.968046i \(-0.580684\pi\)
−0.250772 + 0.968046i \(0.580684\pi\)
\(684\) 0 0
\(685\) −6.37825e9 −0.758202
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.59848e9 1.11798
\(690\) 0 0
\(691\) 6.94731e9 0.801020 0.400510 − 0.916292i \(-0.368833\pi\)
0.400510 + 0.916292i \(0.368833\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.34011e9 −0.264417
\(696\) 0 0
\(697\) −1.36437e9 −0.152622
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −9.64715e9 −1.05776 −0.528879 − 0.848697i \(-0.677387\pi\)
−0.528879 + 0.848697i \(0.677387\pi\)
\(702\) 0 0
\(703\) −1.86360e9 −0.202306
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.50253e9 0.798435
\(708\) 0 0
\(709\) 7.72407e9 0.813925 0.406962 − 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.85194e10 2.94664
\(714\) 0 0
\(715\) −3.86568e8 −0.0395508
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.11572e9 0.111945 0.0559725 − 0.998432i \(-0.482174\pi\)
0.0559725 + 0.998432i \(0.482174\pi\)
\(720\) 0 0
\(721\) 2.63894e9 0.262214
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.12931e10 1.10060
\(726\) 0 0
\(727\) 7.85026e9 0.757728 0.378864 − 0.925452i \(-0.376315\pi\)
0.378864 + 0.925452i \(0.376315\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.39990e9 −0.132552
\(732\) 0 0
\(733\) −1.43652e10 −1.34725 −0.673625 − 0.739073i \(-0.735264\pi\)
−0.673625 + 0.739073i \(0.735264\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.20829e9 0.111182
\(738\) 0 0
\(739\) −5.13869e9 −0.468379 −0.234189 − 0.972191i \(-0.575244\pi\)
−0.234189 + 0.972191i \(0.575244\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.12182e10 −1.00338 −0.501688 − 0.865049i \(-0.667287\pi\)
−0.501688 + 0.865049i \(0.667287\pi\)
\(744\) 0 0
\(745\) 1.02188e9 0.0905422
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.68920e9 0.146891
\(750\) 0 0
\(751\) −2.16341e10 −1.86380 −0.931899 − 0.362717i \(-0.881849\pi\)
−0.931899 + 0.362717i \(0.881849\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.64424e9 0.646427
\(756\) 0 0
\(757\) −1.67963e10 −1.40727 −0.703635 − 0.710561i \(-0.748441\pi\)
−0.703635 + 0.710561i \(0.748441\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.41213e10 −1.98406 −0.992030 − 0.126002i \(-0.959785\pi\)
−0.992030 + 0.126002i \(0.959785\pi\)
\(762\) 0 0
\(763\) 5.54470e8 0.0451900
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9.59088e9 −0.767494
\(768\) 0 0
\(769\) 1.66703e10 1.32191 0.660954 − 0.750426i \(-0.270152\pi\)
0.660954 + 0.750426i \(0.270152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.34400e10 1.04658 0.523288 − 0.852156i \(-0.324705\pi\)
0.523288 + 0.852156i \(0.324705\pi\)
\(774\) 0 0
\(775\) −1.70821e10 −1.31821
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.81771e9 −0.137766
\(780\) 0 0
\(781\) −2.43657e9 −0.183020
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 7.63611e9 0.563415
\(786\) 0 0
\(787\) −1.88162e10 −1.37601 −0.688004 − 0.725707i \(-0.741513\pi\)
−0.688004 + 0.725707i \(0.741513\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.31115e9 0.0941968
\(792\) 0 0
\(793\) −7.01641e9 −0.499642
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.78944e9 −0.405072 −0.202536 − 0.979275i \(-0.564918\pi\)
−0.202536 + 0.979275i \(0.564918\pi\)
\(798\) 0 0
\(799\) −6.83034e9 −0.473727
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.31038e9 −0.157463
\(804\) 0 0
\(805\) 4.81317e9 0.325197
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.81286e10 −1.86779 −0.933897 − 0.357543i \(-0.883615\pi\)
−0.933897 + 0.357543i \(0.883615\pi\)
\(810\) 0 0
\(811\) −6.63925e9 −0.437065 −0.218532 − 0.975830i \(-0.570127\pi\)
−0.218532 + 0.975830i \(0.570127\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.24761e9 0.533674
\(816\) 0 0
\(817\) −1.86504e9 −0.119650
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.35199e9 −0.337532 −0.168766 − 0.985656i \(-0.553978\pi\)
−0.168766 + 0.985656i \(0.553978\pi\)
\(822\) 0 0
\(823\) −1.10559e9 −0.0691343 −0.0345672 − 0.999402i \(-0.511005\pi\)
−0.0345672 + 0.999402i \(0.511005\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.56978e9 0.588346 0.294173 − 0.955752i \(-0.404956\pi\)
0.294173 + 0.955752i \(0.404956\pi\)
\(828\) 0 0
\(829\) −2.36632e10 −1.44256 −0.721278 − 0.692646i \(-0.756445\pi\)
−0.721278 + 0.692646i \(0.756445\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.29145e9 0.497019
\(834\) 0 0
\(835\) −1.55321e9 −0.0923267
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.07240e10 1.21145 0.605726 − 0.795673i \(-0.292883\pi\)
0.605726 + 0.795673i \(0.292883\pi\)
\(840\) 0 0
\(841\) 1.41556e10 0.820622
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.07068e9 0.232097
\(846\) 0 0
\(847\) 7.21094e9 0.407756
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.22307e10 0.680297
\(852\) 0 0
\(853\) −1.96254e10 −1.08267 −0.541336 − 0.840806i \(-0.682081\pi\)
−0.541336 + 0.840806i \(0.682081\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.34086e10 −1.27040 −0.635202 − 0.772346i \(-0.719083\pi\)
−0.635202 + 0.772346i \(0.719083\pi\)
\(858\) 0 0
\(859\) −6.37986e9 −0.343428 −0.171714 − 0.985147i \(-0.554930\pi\)
−0.171714 + 0.985147i \(0.554930\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.63727e9 0.0867127 0.0433564 − 0.999060i \(-0.486195\pi\)
0.0433564 + 0.999060i \(0.486195\pi\)
\(864\) 0 0
\(865\) −5.06548e9 −0.266112
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.22339e8 −0.0321705
\(870\) 0 0
\(871\) 1.08122e10 0.554434
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −6.41729e9 −0.323835
\(876\) 0 0
\(877\) −3.31776e10 −1.66091 −0.830455 − 0.557085i \(-0.811920\pi\)
−0.830455 + 0.557085i \(0.811920\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 5.88175e9 0.289795 0.144897 − 0.989447i \(-0.453715\pi\)
0.144897 + 0.989447i \(0.453715\pi\)
\(882\) 0 0
\(883\) −5.84882e9 −0.285894 −0.142947 − 0.989730i \(-0.545658\pi\)
−0.142947 + 0.989730i \(0.545658\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.81095e9 −0.375813 −0.187906 − 0.982187i \(-0.560170\pi\)
−0.187906 + 0.982187i \(0.560170\pi\)
\(888\) 0 0
\(889\) 7.93124e9 0.378604
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.09982e9 −0.427615
\(894\) 0 0
\(895\) −1.36368e10 −0.635815
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.75045e10 −2.18060
\(900\) 0 0
\(901\) −2.17535e10 −0.990813
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.53506e10 0.688424
\(906\) 0 0
\(907\) 4.12701e10 1.83658 0.918289 − 0.395910i \(-0.129571\pi\)
0.918289 + 0.395910i \(0.129571\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.66422e9 0.292035 0.146017 − 0.989282i \(-0.453354\pi\)
0.146017 + 0.989282i \(0.453354\pi\)
\(912\) 0 0
\(913\) 5.52214e9 0.240137
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7.80035e9 0.334058
\(918\) 0 0
\(919\) −5.45018e9 −0.231636 −0.115818 − 0.993270i \(-0.536949\pi\)
−0.115818 + 0.993270i \(0.536949\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.18032e10 −0.912671
\(924\) 0 0
\(925\) −7.32576e9 −0.304338
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −7.78356e9 −0.318510 −0.159255 − 0.987237i \(-0.550909\pi\)
−0.159255 + 0.987237i \(0.550909\pi\)
\(930\) 0 0
\(931\) 1.10464e10 0.448639
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.76096e8 0.0350519
\(936\) 0 0
\(937\) 4.59669e10 1.82539 0.912697 − 0.408638i \(-0.133996\pi\)
0.912697 + 0.408638i \(0.133996\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.63152e10 0.638305 0.319152 − 0.947703i \(-0.396602\pi\)
0.319152 + 0.947703i \(0.396602\pi\)
\(942\) 0 0
\(943\) 1.19295e10 0.463268
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.13934e10 1.20119 0.600597 − 0.799552i \(-0.294929\pi\)
0.600597 + 0.799552i \(0.294929\pi\)
\(948\) 0 0
\(949\) −2.06741e10 −0.785226
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.82350e10 1.80525 0.902625 − 0.430428i \(-0.141637\pi\)
0.902625 + 0.430428i \(0.141637\pi\)
\(954\) 0 0
\(955\) 2.53008e9 0.0939989
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00383e10 −0.733663
\(960\) 0 0
\(961\) 4.43435e10 1.61175
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.67785e10 0.601047
\(966\) 0 0
\(967\) −3.22238e9 −0.114600 −0.0572999 − 0.998357i \(-0.518249\pi\)
−0.0572999 + 0.998357i \(0.518249\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.25397e10 0.790096 0.395048 − 0.918660i \(-0.370728\pi\)
0.395048 + 0.918660i \(0.370728\pi\)
\(972\) 0 0
\(973\) −7.35184e9 −0.255859
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.31576e10 −1.48056 −0.740280 − 0.672298i \(-0.765307\pi\)
−0.740280 + 0.672298i \(0.765307\pi\)
\(978\) 0 0
\(979\) 7.73582e8 0.0263492
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.28659e10 −0.767806 −0.383903 − 0.923373i \(-0.625420\pi\)
−0.383903 + 0.923373i \(0.625420\pi\)
\(984\) 0 0
\(985\) 1.75159e10 0.583991
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.22402e10 0.402347
\(990\) 0 0
\(991\) −1.13727e10 −0.371198 −0.185599 − 0.982626i \(-0.559423\pi\)
−0.185599 + 0.982626i \(0.559423\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.13376e9 −0.197399
\(996\) 0 0
\(997\) −4.31535e10 −1.37906 −0.689530 − 0.724257i \(-0.742183\pi\)
−0.689530 + 0.724257i \(0.742183\pi\)
\(998\) 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 432.8.a.b.1.1 1
3.2 odd 2 432.8.a.g.1.1 1
4.3 odd 2 54.8.a.a.1.1 ✓ 1
12.11 even 2 54.8.a.f.1.1 yes 1
36.7 odd 6 162.8.c.k.109.1 2
36.11 even 6 162.8.c.b.109.1 2
36.23 even 6 162.8.c.b.55.1 2
36.31 odd 6 162.8.c.k.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.8.a.a.1.1 ✓ 1 4.3 odd 2
54.8.a.f.1.1 yes 1 12.11 even 2
162.8.c.b.55.1 2 36.23 even 6
162.8.c.b.109.1 2 36.11 even 6
162.8.c.k.55.1 2 36.31 odd 6
162.8.c.k.109.1 2 36.7 odd 6
432.8.a.b.1.1 1 1.1 even 1 trivial
432.8.a.g.1.1 1 3.2 odd 2