Properties

Label 54.8.a.a.1.1
Level $54$
Weight $8$
Character 54.1
Self dual yes
Analytic conductor $16.869$
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] = [54,8,Mod(1,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, 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("54.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 54.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: \(16.8687913761\)
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 \(\chi\) \(=\) 54.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{2} +64.0000 q^{4} -120.000 q^{5} +377.000 q^{7} -512.000 q^{8} +960.000 q^{10} -600.000 q^{11} +5369.00 q^{13} -3016.00 q^{14} +4096.00 q^{16} -12168.0 q^{17} +16211.0 q^{19} -7680.00 q^{20} +4800.00 q^{22} -106392. q^{23} -63725.0 q^{25} -42952.0 q^{26} +24128.0 q^{28} -177216. q^{29} -268060. q^{31} -32768.0 q^{32} +97344.0 q^{34} -45240.0 q^{35} +114959. q^{37} -129688. q^{38} +61440.0 q^{40} +112128. q^{41} -115048. q^{43} -38400.0 q^{44} +851136. q^{46} -561336. q^{47} -681414. q^{49} +509800. q^{50} +343616. q^{52} +1.78776e6 q^{53} +72000.0 q^{55} -193024. q^{56} +1.41773e6 q^{58} +1.78634e6 q^{59} -1.30684e6 q^{61} +2.14448e6 q^{62} +262144. q^{64} -644280. q^{65} -2.01382e6 q^{67} -778752. q^{68} +361920. q^{70} +4.06094e6 q^{71} -3.85064e6 q^{73} -919672. q^{74} +1.03750e6 q^{76} -226200. q^{77} +1.03723e6 q^{79} -491520. q^{80} -897024. q^{82} -9.20357e6 q^{83} +1.46016e6 q^{85} +920384. q^{86} +307200. q^{88} +1.28930e6 q^{89} +2.02411e6 q^{91} -6.80909e6 q^{92} +4.49069e6 q^{94} -1.94532e6 q^{95} +8.55588e6 q^{97} +5.45131e6 q^{98} +O(q^{100})\)

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
\(5\) −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 \(-0.433397\pi\)
0.207715 + 0.978189i \(0.433397\pi\)
\(8\) −512.000 −0.353553
\(9\) 0 0
\(10\) 960.000 0.303579
\(11\) −600.000 −0.135918 −0.0679590 0.997688i \(-0.521649\pi\)
−0.0679590 + 0.997688i \(0.521649\pi\)
\(12\) 0 0
\(13\) 5369.00 0.677785 0.338892 0.940825i \(-0.389948\pi\)
0.338892 + 0.940825i \(0.389948\pi\)
\(14\) −3016.00 −0.293754
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) −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 \(-0.412610\pi\)
0.271108 + 0.962549i \(0.412610\pi\)
\(20\) −7680.00 −0.214663
\(21\) 0 0
\(22\) 4800.00 0.0961085
\(23\) −106392. −1.82331 −0.911657 0.410952i \(-0.865197\pi\)
−0.911657 + 0.410952i \(0.865197\pi\)
\(24\) 0 0
\(25\) −63725.0 −0.815680
\(26\) −42952.0 −0.479266
\(27\) 0 0
\(28\) 24128.0 0.207715
\(29\) −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 \(-0.799475\pi\)
−0.808046 + 0.589119i \(0.799475\pi\)
\(32\) −32768.0 −0.176777
\(33\) 0 0
\(34\) 97344.0 0.424750
\(35\) −45240.0 −0.178355
\(36\) 0 0
\(37\) 114959. 0.373110 0.186555 0.982445i \(-0.440268\pi\)
0.186555 + 0.982445i \(0.440268\pi\)
\(38\) −129688. −0.383404
\(39\) 0 0
\(40\) 61440.0 0.151789
\(41\) 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 \(-0.535192\pi\)
−0.110334 + 0.993895i \(0.535192\pi\)
\(44\) −38400.0 −0.0679590
\(45\) 0 0
\(46\) 851136. 1.28928
\(47\) −561336. −0.788643 −0.394321 0.918973i \(-0.629020\pi\)
−0.394321 + 0.918973i \(0.629020\pi\)
\(48\) 0 0
\(49\) −681414. −0.827418
\(50\) 509800. 0.576773
\(51\) 0 0
\(52\) 343616. 0.338892
\(53\) 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
\(56\) −193024. −0.146877
\(57\) 0 0
\(58\) 1.41773e6 0.954102
\(59\) 1.78634e6 1.13236 0.566178 0.824283i \(-0.308421\pi\)
0.566178 + 0.824283i \(0.308421\pi\)
\(60\) 0 0
\(61\) −1.30684e6 −0.737169 −0.368584 0.929594i \(-0.620157\pi\)
−0.368584 + 0.929594i \(0.620157\pi\)
\(62\) 2.14448e6 1.14275
\(63\) 0 0
\(64\) 262144. 0.125000
\(65\) −644280. −0.290990
\(66\) 0 0
\(67\) −2.01382e6 −0.818009 −0.409005 0.912532i \(-0.634124\pi\)
−0.409005 + 0.912532i \(0.634124\pi\)
\(68\) −778752. −0.300343
\(69\) 0 0
\(70\) 361920. 0.126116
\(71\) 4.06094e6 1.34655 0.673275 0.739392i \(-0.264887\pi\)
0.673275 + 0.739392i \(0.264887\pi\)
\(72\) 0 0
\(73\) −3.85064e6 −1.15852 −0.579259 0.815144i \(-0.696658\pi\)
−0.579259 + 0.815144i \(0.696658\pi\)
\(74\) −919672. −0.263829
\(75\) 0 0
\(76\) 1.03750e6 0.271108
\(77\) −226200. −0.0564644
\(78\) 0 0
\(79\) 1.03723e6 0.236690 0.118345 0.992973i \(-0.462241\pi\)
0.118345 + 0.992973i \(0.462241\pi\)
\(80\) −491520. −0.107331
\(81\) 0 0
\(82\) −897024. −0.179662
\(83\) −9.20357e6 −1.76678 −0.883391 0.468637i \(-0.844745\pi\)
−0.883391 + 0.468637i \(0.844745\pi\)
\(84\) 0 0
\(85\) 1.46016e6 0.257890
\(86\) 920384. 0.156036
\(87\) 0 0
\(88\) 307200. 0.0480543
\(89\) 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
\(92\) −6.80909e6 −0.911657
\(93\) 0 0
\(94\) 4.49069e6 0.557655
\(95\) −1.94532e6 −0.232787
\(96\) 0 0
\(97\) 8.55588e6 0.951840 0.475920 0.879489i \(-0.342115\pi\)
0.475920 + 0.879489i \(0.342115\pi\)
\(98\) 5.45131e6 0.585073
\(99\) 0 0
\(100\) −4.07840e6 −0.407840
\(101\) −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 \(-0.397796\pi\)
0.315594 + 0.948894i \(0.397796\pi\)
\(104\) −2.74893e6 −0.239633
\(105\) 0 0
\(106\) −1.43021e7 −1.16635
\(107\) 4.48063e6 0.353587 0.176793 0.984248i \(-0.443428\pi\)
0.176793 + 0.984248i \(0.443428\pi\)
\(108\) 0 0
\(109\) −1.47074e6 −0.108779 −0.0543893 0.998520i \(-0.517321\pi\)
−0.0543893 + 0.998520i \(0.517321\pi\)
\(110\) −576000. −0.0412618
\(111\) 0 0
\(112\) 1.54419e6 0.103858
\(113\) −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
\(116\) −1.13418e7 −0.674652
\(117\) 0 0
\(118\) −1.42908e7 −0.800697
\(119\) −4.58734e6 −0.249543
\(120\) 0 0
\(121\) −1.91272e7 −0.981526
\(122\) 1.04547e7 0.521257
\(123\) 0 0
\(124\) −1.71558e7 −0.808046
\(125\) 1.70220e7 0.779517
\(126\) 0 0
\(127\) 2.10378e7 0.911353 0.455677 0.890145i \(-0.349397\pi\)
0.455677 + 0.890145i \(0.349397\pi\)
\(128\) −2.09715e6 −0.0883883
\(129\) 0 0
\(130\) 5.15424e6 0.205761
\(131\) 2.06906e7 0.804125 0.402062 0.915612i \(-0.368294\pi\)
0.402062 + 0.915612i \(0.368294\pi\)
\(132\) 0 0
\(133\) 6.11155e6 0.225253
\(134\) 1.61105e7 0.578420
\(135\) 0 0
\(136\) 6.23002e6 0.212375
\(137\) 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 \(-0.599641\pi\)
−0.307945 + 0.951404i \(0.599641\pi\)
\(140\) −2.89536e6 −0.0891773
\(141\) 0 0
\(142\) −3.24876e7 −0.952155
\(143\) −3.22140e6 −0.0921231
\(144\) 0 0
\(145\) 2.12659e7 0.579290
\(146\) 3.08051e7 0.819196
\(147\) 0 0
\(148\) 7.35738e6 0.186555
\(149\) −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 \(-0.228683\pi\)
0.752841 + 0.658202i \(0.228683\pi\)
\(152\) −8.30003e6 −0.191702
\(153\) 0 0
\(154\) 1.80960e6 0.0399264
\(155\) 3.21672e7 0.693829
\(156\) 0 0
\(157\) −6.36342e7 −1.31233 −0.656163 0.754619i \(-0.727822\pi\)
−0.656163 + 0.754619i \(0.727822\pi\)
\(158\) −8.29785e6 −0.167365
\(159\) 0 0
\(160\) 3.93216e6 0.0758947
\(161\) −4.01098e7 −0.757460
\(162\) 0 0
\(163\) 6.87301e7 1.24305 0.621527 0.783392i \(-0.286512\pi\)
0.621527 + 0.783392i \(0.286512\pi\)
\(164\) 7.17619e6 0.127040
\(165\) 0 0
\(166\) 7.36285e7 1.24930
\(167\) −1.29434e7 −0.215051 −0.107525 0.994202i \(-0.534293\pi\)
−0.107525 + 0.994202i \(0.534293\pi\)
\(168\) 0 0
\(169\) −3.39224e7 −0.540608
\(170\) −1.16813e7 −0.182356
\(171\) 0 0
\(172\) −7.36307e6 −0.110334
\(173\) 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
\(176\) −2.45760e6 −0.0339795
\(177\) 0 0
\(178\) −1.03144e7 −0.137080
\(179\) −1.13640e8 −1.48096 −0.740482 0.672076i \(-0.765403\pi\)
−0.740482 + 0.672076i \(0.765403\pi\)
\(180\) 0 0
\(181\) −1.27922e8 −1.60350 −0.801752 0.597658i \(-0.796098\pi\)
−0.801752 + 0.597658i \(0.796098\pi\)
\(182\) −1.61929e7 −0.199102
\(183\) 0 0
\(184\) 5.44727e7 0.644639
\(185\) −1.37951e7 −0.160185
\(186\) 0 0
\(187\) 7.30080e6 0.0816441
\(188\) −3.59255e7 −0.394321
\(189\) 0 0
\(190\) 1.55626e7 0.164605
\(191\) 2.10840e7 0.218946 0.109473 0.993990i \(-0.465084\pi\)
0.109473 + 0.993990i \(0.465084\pi\)
\(192\) 0 0
\(193\) −1.39821e8 −1.39998 −0.699990 0.714152i \(-0.746812\pi\)
−0.699990 + 0.714152i \(0.746812\pi\)
\(194\) −6.84471e7 −0.673052
\(195\) 0 0
\(196\) −4.36105e7 −0.413709
\(197\) −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 \(-0.573838\pi\)
−0.229895 + 0.973215i \(0.573838\pi\)
\(200\) 3.26272e7 0.288386
\(201\) 0 0
\(202\) 1.59205e8 1.35902
\(203\) −6.68104e7 −0.560542
\(204\) 0 0
\(205\) −1.34554e7 −0.109083
\(206\) −5.59988e7 −0.446317
\(207\) 0 0
\(208\) 2.19914e7 0.169446
\(209\) −9.72660e6 −0.0736969
\(210\) 0 0
\(211\) 9.80537e7 0.718581 0.359290 0.933226i \(-0.383019\pi\)
0.359290 + 0.933226i \(0.383019\pi\)
\(212\) 1.14417e8 0.824734
\(213\) 0 0
\(214\) −3.58451e7 −0.250024
\(215\) 1.38058e7 0.0947383
\(216\) 0 0
\(217\) −1.01059e8 −0.671374
\(218\) 1.17659e7 0.0769181
\(219\) 0 0
\(220\) 4.60800e6 0.0291765
\(221\) −6.53300e7 −0.407136
\(222\) 0 0
\(223\) 2.16758e8 1.30891 0.654453 0.756103i \(-0.272899\pi\)
0.654453 + 0.756103i \(0.272899\pi\)
\(224\) −1.23535e7 −0.0734384
\(225\) 0 0
\(226\) 2.78229e7 0.160333
\(227\) 1.39865e8 0.793629 0.396814 0.917899i \(-0.370116\pi\)
0.396814 + 0.917899i \(0.370116\pi\)
\(228\) 0 0
\(229\) −9.49569e7 −0.522519 −0.261260 0.965269i \(-0.584138\pi\)
−0.261260 + 0.965269i \(0.584138\pi\)
\(230\) −1.02136e8 −0.553519
\(231\) 0 0
\(232\) 9.07346e7 0.477051
\(233\) 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
\(236\) 1.14326e8 0.566178
\(237\) 0 0
\(238\) 3.66987e7 0.176454
\(239\) 1.29152e8 0.611939 0.305969 0.952041i \(-0.401019\pi\)
0.305969 + 0.952041i \(0.401019\pi\)
\(240\) 0 0
\(241\) −4.01868e7 −0.184937 −0.0924685 0.995716i \(-0.529476\pi\)
−0.0924685 + 0.995716i \(0.529476\pi\)
\(242\) 1.53017e8 0.694044
\(243\) 0 0
\(244\) −8.36376e7 −0.368584
\(245\) 8.17697e7 0.355231
\(246\) 0 0
\(247\) 8.70369e7 0.367506
\(248\) 1.37247e8 0.571375
\(249\) 0 0
\(250\) −1.36176e8 −0.551202
\(251\) 2.68201e8 1.07054 0.535270 0.844681i \(-0.320210\pi\)
0.535270 + 0.844681i \(0.320210\pi\)
\(252\) 0 0
\(253\) 6.38352e7 0.247821
\(254\) −1.68302e8 −0.644424
\(255\) 0 0
\(256\) 1.67772e7 0.0625000
\(257\) 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
\(260\) −4.12339e7 −0.145495
\(261\) 0 0
\(262\) −1.65525e8 −0.568602
\(263\) −2.02033e8 −0.684822 −0.342411 0.939550i \(-0.611243\pi\)
−0.342411 + 0.939550i \(0.611243\pi\)
\(264\) 0 0
\(265\) −2.14531e8 −0.708158
\(266\) −4.88924e7 −0.159278
\(267\) 0 0
\(268\) −1.28884e8 −0.409005
\(269\) 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 \(-0.257740\pi\)
0.689705 + 0.724091i \(0.257740\pi\)
\(272\) −4.98401e7 −0.150172
\(273\) 0 0
\(274\) −4.25217e8 −1.24877
\(275\) 3.82350e7 0.110866
\(276\) 0 0
\(277\) −2.09703e8 −0.592823 −0.296411 0.955060i \(-0.595790\pi\)
−0.296411 + 0.955060i \(0.595790\pi\)
\(278\) 1.56007e8 0.435500
\(279\) 0 0
\(280\) 2.31629e7 0.0630579
\(281\) −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 \(-0.286086\pi\)
0.622576 + 0.782559i \(0.286086\pi\)
\(284\) 2.59900e8 0.673275
\(285\) 0 0
\(286\) 2.57712e7 0.0651409
\(287\) 4.22723e7 0.105553
\(288\) 0 0
\(289\) −2.62278e8 −0.639176
\(290\) −1.70127e8 −0.409620
\(291\) 0 0
\(292\) −2.46441e8 −0.579259
\(293\) 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
\(296\) −5.88590e7 −0.131914
\(297\) 0 0
\(298\) 6.81251e7 0.149125
\(299\) −5.71219e8 −1.23581
\(300\) 0 0
\(301\) −4.33731e7 −0.0916722
\(302\) −5.09616e8 −1.06468
\(303\) 0 0
\(304\) 6.64003e7 0.135554
\(305\) 1.56820e8 0.316485
\(306\) 0 0
\(307\) 7.53314e8 1.48591 0.742953 0.669343i \(-0.233424\pi\)
0.742953 + 0.669343i \(0.233424\pi\)
\(308\) −1.44768e7 −0.0282322
\(309\) 0 0
\(310\) −2.57338e8 −0.490611
\(311\) −4.79778e8 −0.904439 −0.452219 0.891907i \(-0.649368\pi\)
−0.452219 + 0.891907i \(0.649368\pi\)
\(312\) 0 0
\(313\) 6.55525e8 1.20833 0.604163 0.796861i \(-0.293508\pi\)
0.604163 + 0.796861i \(0.293508\pi\)
\(314\) 5.09074e8 0.927955
\(315\) 0 0
\(316\) 6.63828e7 0.118345
\(317\) −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
\(320\) −3.14573e7 −0.0536656
\(321\) 0 0
\(322\) 3.20878e8 0.535605
\(323\) −1.97255e8 −0.325702
\(324\) 0 0
\(325\) −3.42140e8 −0.552855
\(326\) −5.49841e8 −0.878972
\(327\) 0 0
\(328\) −5.74095e7 −0.0898309
\(329\) −2.11624e8 −0.327626
\(330\) 0 0
\(331\) −1.84653e8 −0.279872 −0.139936 0.990161i \(-0.544690\pi\)
−0.139936 + 0.990161i \(0.544690\pi\)
\(332\) −5.89028e8 −0.883391
\(333\) 0 0
\(334\) 1.03547e8 0.152064
\(335\) 2.41658e8 0.351192
\(336\) 0 0
\(337\) 8.43883e8 1.20110 0.600548 0.799589i \(-0.294949\pi\)
0.600548 + 0.799589i \(0.294949\pi\)
\(338\) 2.71379e8 0.382268
\(339\) 0 0
\(340\) 9.34502e7 0.128945
\(341\) 1.60836e8 0.219656
\(342\) 0 0
\(343\) −5.67369e8 −0.759165
\(344\) 5.89046e7 0.0780179
\(345\) 0 0
\(346\) −3.37699e8 −0.438292
\(347\) −1.22250e8 −0.157071 −0.0785353 0.996911i \(-0.525024\pi\)
−0.0785353 + 0.996911i \(0.525024\pi\)
\(348\) 0 0
\(349\) −1.05296e9 −1.32594 −0.662969 0.748646i \(-0.730704\pi\)
−0.662969 + 0.748646i \(0.730704\pi\)
\(350\) 1.92195e8 0.239609
\(351\) 0 0
\(352\) 1.96608e7 0.0240271
\(353\) 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
\(356\) 8.25155e7 0.0969305
\(357\) 0 0
\(358\) 9.09118e8 1.04720
\(359\) 7.95194e8 0.907074 0.453537 0.891237i \(-0.350162\pi\)
0.453537 + 0.891237i \(0.350162\pi\)
\(360\) 0 0
\(361\) −6.31075e8 −0.706002
\(362\) 1.02337e9 1.13385
\(363\) 0 0
\(364\) 1.29543e8 0.140786
\(365\) 4.62077e8 0.497381
\(366\) 0 0
\(367\) −1.09445e9 −1.15575 −0.577876 0.816125i \(-0.696118\pi\)
−0.577876 + 0.816125i \(0.696118\pi\)
\(368\) −4.35782e8 −0.455829
\(369\) 0 0
\(370\) 1.10361e8 0.113268
\(371\) 6.73986e8 0.685239
\(372\) 0 0
\(373\) 8.82044e8 0.880054 0.440027 0.897985i \(-0.354969\pi\)
0.440027 + 0.897985i \(0.354969\pi\)
\(374\) −5.84064e7 −0.0577311
\(375\) 0 0
\(376\) 2.87404e8 0.278827
\(377\) −9.51473e8 −0.914538
\(378\) 0 0
\(379\) −1.04251e9 −0.983651 −0.491826 0.870694i \(-0.663670\pi\)
−0.491826 + 0.870694i \(0.663670\pi\)
\(380\) −1.24500e8 −0.116393
\(381\) 0 0
\(382\) −1.68672e8 −0.154818
\(383\) 8.36847e8 0.761115 0.380558 0.924757i \(-0.375732\pi\)
0.380558 + 0.924757i \(0.375732\pi\)
\(384\) 0 0
\(385\) 2.71440e7 0.0242416
\(386\) 1.11857e9 0.989936
\(387\) 0 0
\(388\) 5.47577e8 0.475920
\(389\) 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
\(392\) 3.48884e8 0.292536
\(393\) 0 0
\(394\) 1.16773e9 0.961845
\(395\) −1.24468e8 −0.101617
\(396\) 0 0
\(397\) −1.26746e9 −1.01664 −0.508322 0.861167i \(-0.669734\pi\)
−0.508322 + 0.861167i \(0.669734\pi\)
\(398\) 4.08917e8 0.325121
\(399\) 0 0
\(400\) −2.61018e8 −0.203920
\(401\) −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
\(404\) −1.27364e9 −0.960974
\(405\) 0 0
\(406\) 5.34483e8 0.396363
\(407\) −6.89754e7 −0.0507124
\(408\) 0 0
\(409\) −2.22791e8 −0.161015 −0.0805073 0.996754i \(-0.525654\pi\)
−0.0805073 + 0.996754i \(0.525654\pi\)
\(410\) 1.07643e8 0.0771333
\(411\) 0 0
\(412\) 4.47990e8 0.315594
\(413\) 6.73452e8 0.470415
\(414\) 0 0
\(415\) 1.10443e9 0.758524
\(416\) −1.75931e8 −0.119817
\(417\) 0 0
\(418\) 7.78128e7 0.0521116
\(419\) 4.40581e8 0.292602 0.146301 0.989240i \(-0.453263\pi\)
0.146301 + 0.989240i \(0.453263\pi\)
\(420\) 0 0
\(421\) 2.18934e9 1.42997 0.714984 0.699140i \(-0.246434\pi\)
0.714984 + 0.699140i \(0.246434\pi\)
\(422\) −7.84430e8 −0.508113
\(423\) 0 0
\(424\) −9.15333e8 −0.583175
\(425\) 7.75406e8 0.489968
\(426\) 0 0
\(427\) −4.92678e8 −0.306242
\(428\) 2.86760e8 0.176793
\(429\) 0 0
\(430\) −1.10446e8 −0.0669901
\(431\) −3.03634e9 −1.82675 −0.913377 0.407114i \(-0.866535\pi\)
−0.913377 + 0.407114i \(0.866535\pi\)
\(432\) 0 0
\(433\) 2.21735e9 1.31258 0.656290 0.754509i \(-0.272125\pi\)
0.656290 + 0.754509i \(0.272125\pi\)
\(434\) 8.08469e8 0.474733
\(435\) 0 0
\(436\) −9.41275e7 −0.0543893
\(437\) −1.72472e9 −0.988630
\(438\) 0 0
\(439\) −2.50094e9 −1.41084 −0.705419 0.708791i \(-0.749241\pi\)
−0.705419 + 0.708791i \(0.749241\pi\)
\(440\) −3.68640e7 −0.0206309
\(441\) 0 0
\(442\) 5.22640e8 0.287889
\(443\) 2.40387e9 1.31370 0.656852 0.754019i \(-0.271887\pi\)
0.656852 + 0.754019i \(0.271887\pi\)
\(444\) 0 0
\(445\) −1.54716e8 −0.0832294
\(446\) −1.73407e9 −0.925536
\(447\) 0 0
\(448\) 9.88283e7 0.0519288
\(449\) −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
\(452\) −2.22583e8 −0.113373
\(453\) 0 0
\(454\) −1.11892e9 −0.561180
\(455\) −2.42894e8 −0.120886
\(456\) 0 0
\(457\) 2.30159e9 1.12803 0.564015 0.825765i \(-0.309256\pi\)
0.564015 + 0.825765i \(0.309256\pi\)
\(458\) 7.59655e8 0.369477
\(459\) 0 0
\(460\) 8.17091e8 0.391397
\(461\) −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 \(-0.631967\pi\)
−0.402810 + 0.915284i \(0.631967\pi\)
\(464\) −7.25877e8 −0.337326
\(465\) 0 0
\(466\) −1.68759e9 −0.772530
\(467\) −1.58750e9 −0.721280 −0.360640 0.932705i \(-0.617442\pi\)
−0.360640 + 0.932705i \(0.617442\pi\)
\(468\) 0 0
\(469\) −7.59209e8 −0.339826
\(470\) −5.38883e8 −0.239415
\(471\) 0 0
\(472\) −9.14608e8 −0.400348
\(473\) 6.90288e7 0.0299928
\(474\) 0 0
\(475\) −1.03305e9 −0.442275
\(476\) −2.93590e8 −0.124772
\(477\) 0 0
\(478\) −1.03321e9 −0.432706
\(479\) −1.75670e9 −0.730336 −0.365168 0.930942i \(-0.618988\pi\)
−0.365168 + 0.930942i \(0.618988\pi\)
\(480\) 0 0
\(481\) 6.17215e8 0.252888
\(482\) 3.21494e8 0.130770
\(483\) 0 0
\(484\) −1.22414e9 −0.490763
\(485\) −1.02671e9 −0.408649
\(486\) 0 0
\(487\) 1.52004e9 0.596353 0.298177 0.954511i \(-0.403622\pi\)
0.298177 + 0.954511i \(0.403622\pi\)
\(488\) 6.69101e8 0.260629
\(489\) 0 0
\(490\) −6.54157e8 −0.251186
\(491\) −3.64824e9 −1.39091 −0.695453 0.718572i \(-0.744796\pi\)
−0.695453 + 0.718572i \(0.744796\pi\)
\(492\) 0 0
\(493\) 2.15636e9 0.810509
\(494\) −6.96295e8 −0.259866
\(495\) 0 0
\(496\) −1.09797e9 −0.404023
\(497\) 1.53098e9 0.559398
\(498\) 0 0
\(499\) 4.83503e8 0.174200 0.0870998 0.996200i \(-0.472240\pi\)
0.0870998 + 0.996200i \(0.472240\pi\)
\(500\) 1.08941e9 0.389758
\(501\) 0 0
\(502\) −2.14561e9 −0.756986
\(503\) 4.20389e9 1.47287 0.736434 0.676510i \(-0.236508\pi\)
0.736434 + 0.676510i \(0.236508\pi\)
\(504\) 0 0
\(505\) 2.38807e9 0.825140
\(506\) −5.10682e8 −0.175236
\(507\) 0 0
\(508\) 1.34642e9 0.455677
\(509\) 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
\(512\) −1.34218e8 −0.0441942
\(513\) 0 0
\(514\) −1.37491e9 −0.446585
\(515\) −8.39981e8 −0.270984
\(516\) 0 0
\(517\) 3.36802e8 0.107191
\(518\) −3.46716e8 −0.109602
\(519\) 0 0
\(520\) 3.29871e8 0.102880
\(521\) 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 \(-0.468472\pi\)
0.0988851 + 0.995099i \(0.468472\pi\)
\(524\) 1.32420e9 0.402062
\(525\) 0 0
\(526\) 1.61626e9 0.484242
\(527\) 3.26175e9 0.970765
\(528\) 0 0
\(529\) 7.91443e9 2.32448
\(530\) 1.71625e9 0.500743
\(531\) 0 0
\(532\) 3.91139e8 0.112626
\(533\) 6.02015e8 0.172212
\(534\) 0 0
\(535\) −5.37676e8 −0.151804
\(536\) 1.03107e9 0.289210
\(537\) 0 0
\(538\) −7.79134e8 −0.215712
\(539\) 4.08848e8 0.112461
\(540\) 0 0
\(541\) −3.47237e8 −0.0942835 −0.0471418 0.998888i \(-0.515011\pi\)
−0.0471418 + 0.998888i \(0.515011\pi\)
\(542\) −3.61556e9 −0.975390
\(543\) 0 0
\(544\) 3.98721e8 0.106187
\(545\) 1.76489e8 0.0467014
\(546\) 0 0
\(547\) −2.88628e9 −0.754021 −0.377010 0.926209i \(-0.623048\pi\)
−0.377010 + 0.926209i \(0.623048\pi\)
\(548\) 3.40173e9 0.883016
\(549\) 0 0
\(550\) −3.05880e8 −0.0783938
\(551\) −2.87285e9 −0.731614
\(552\) 0 0
\(553\) 3.91036e8 0.0983284
\(554\) 1.67762e9 0.419189
\(555\) 0 0
\(556\) −1.24806e9 −0.307945
\(557\) 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
\(560\) −1.85303e8 −0.0445887
\(561\) 0 0
\(562\) 4.25763e9 1.01179
\(563\) 8.07444e9 1.90692 0.953462 0.301515i \(-0.0974922\pi\)
0.953462 + 0.301515i \(0.0974922\pi\)
\(564\) 0 0
\(565\) 4.17344e8 0.0973474
\(566\) −3.79809e9 −0.880456
\(567\) 0 0
\(568\) −2.07920e9 −0.476078
\(569\) −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 \(-0.602456\pi\)
−0.316347 + 0.948644i \(0.602456\pi\)
\(572\) −2.06170e8 −0.0460615
\(573\) 0 0
\(574\) −3.38178e8 −0.0746369
\(575\) 6.77983e9 1.48724
\(576\) 0 0
\(577\) −4.74376e9 −1.02803 −0.514017 0.857780i \(-0.671843\pi\)
−0.514017 + 0.857780i \(0.671843\pi\)
\(578\) 2.09823e9 0.451965
\(579\) 0 0
\(580\) 1.36102e9 0.289645
\(581\) −3.46975e9 −0.733975
\(582\) 0 0
\(583\) −1.07266e9 −0.224192
\(584\) 1.97153e9 0.409598
\(585\) 0 0
\(586\) −1.85695e9 −0.381204
\(587\) −4.92039e9 −1.00408 −0.502038 0.864846i \(-0.667416\pi\)
−0.502038 + 0.864846i \(0.667416\pi\)
\(588\) 0 0
\(589\) −4.34552e9 −0.876271
\(590\) 1.71489e9 0.343759
\(591\) 0 0
\(592\) 4.70872e8 0.0932775
\(593\) 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
\(596\) −5.45000e8 −0.105447
\(597\) 0 0
\(598\) 4.56975e9 0.873853
\(599\) −8.28842e9 −1.57572 −0.787858 0.615857i \(-0.788810\pi\)
−0.787858 + 0.615857i \(0.788810\pi\)
\(600\) 0 0
\(601\) 7.19345e9 1.35169 0.675844 0.737045i \(-0.263779\pi\)
0.675844 + 0.737045i \(0.263779\pi\)
\(602\) 3.46985e8 0.0648220
\(603\) 0 0
\(604\) 4.07693e9 0.752841
\(605\) 2.29526e9 0.421394
\(606\) 0 0
\(607\) −5.35656e9 −0.972132 −0.486066 0.873922i \(-0.661569\pi\)
−0.486066 + 0.873922i \(0.661569\pi\)
\(608\) −5.31202e8 −0.0958511
\(609\) 0 0
\(610\) −1.25456e9 −0.223789
\(611\) −3.01381e9 −0.534530
\(612\) 0 0
\(613\) −1.01444e10 −1.77874 −0.889371 0.457187i \(-0.848857\pi\)
−0.889371 + 0.457187i \(0.848857\pi\)
\(614\) −6.02651e9 −1.05069
\(615\) 0 0
\(616\) 1.15814e8 0.0199632
\(617\) 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 \(-0.575031\pi\)
−0.233541 + 0.972347i \(0.575031\pi\)
\(620\) 2.05870e9 0.346914
\(621\) 0 0
\(622\) 3.83823e9 0.639535
\(623\) 4.86068e8 0.0805357
\(624\) 0 0
\(625\) 2.93588e9 0.481014
\(626\) −5.24420e9 −0.854415
\(627\) 0 0
\(628\) −4.07259e9 −0.656163
\(629\) −1.39882e9 −0.224122
\(630\) 0 0
\(631\) 3.92975e9 0.622676 0.311338 0.950299i \(-0.399223\pi\)
0.311338 + 0.950299i \(0.399223\pi\)
\(632\) −5.31062e8 −0.0836827
\(633\) 0 0
\(634\) 5.54689e9 0.864445
\(635\) −2.52453e9 −0.391267
\(636\) 0 0
\(637\) −3.65851e9 −0.560811
\(638\) −8.50637e8 −0.129680
\(639\) 0 0
\(640\) 2.51658e8 0.0379473
\(641\) −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 \(-0.489029\pi\)
0.0344583 + 0.999406i \(0.489029\pi\)
\(644\) −2.56703e9 −0.378730
\(645\) 0 0
\(646\) 1.57804e9 0.230306
\(647\) −6.98889e9 −1.01448 −0.507239 0.861805i \(-0.669334\pi\)
−0.507239 + 0.861805i \(0.669334\pi\)
\(648\) 0 0
\(649\) −1.07181e9 −0.153908
\(650\) 2.73712e9 0.390928
\(651\) 0 0
\(652\) 4.39872e9 0.621527
\(653\) −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
\(656\) 4.59276e8 0.0635200
\(657\) 0 0
\(658\) 1.69299e9 0.231667
\(659\) 9.87453e9 1.34406 0.672029 0.740525i \(-0.265423\pi\)
0.672029 + 0.740525i \(0.265423\pi\)
\(660\) 0 0
\(661\) −1.22825e10 −1.65417 −0.827087 0.562075i \(-0.810003\pi\)
−0.827087 + 0.562075i \(0.810003\pi\)
\(662\) 1.47723e9 0.197899
\(663\) 0 0
\(664\) 4.71223e9 0.624652
\(665\) −7.33386e8 −0.0967067
\(666\) 0 0
\(667\) 1.88544e10 2.46021
\(668\) −8.28379e8 −0.107525
\(669\) 0 0
\(670\) −1.93326e9 −0.248330
\(671\) 7.84102e8 0.100194
\(672\) 0 0
\(673\) 9.21233e9 1.16498 0.582488 0.812840i \(-0.302079\pi\)
0.582488 + 0.812840i \(0.302079\pi\)
\(674\) −6.75106e9 −0.849303
\(675\) 0 0
\(676\) −2.17103e9 −0.270304
\(677\) −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
\(680\) −7.47602e8 −0.0911778
\(681\) 0 0
\(682\) −1.28669e9 −0.155320
\(683\) 4.17620e9 0.501544 0.250772 0.968046i \(-0.419316\pi\)
0.250772 + 0.968046i \(0.419316\pi\)
\(684\) 0 0
\(685\) −6.37825e9 −0.758202
\(686\) 4.53895e9 0.536811
\(687\) 0 0
\(688\) −4.71237e8 −0.0551670
\(689\) 9.59848e9 1.11798
\(690\) 0 0
\(691\) −6.94731e9 −0.801020 −0.400510 0.916292i \(-0.631167\pi\)
−0.400510 + 0.916292i \(0.631167\pi\)
\(692\) 2.70159e9 0.309919
\(693\) 0 0
\(694\) 9.77998e8 0.111066
\(695\) 2.34011e9 0.264417
\(696\) 0 0
\(697\) −1.36437e9 −0.152622
\(698\) 8.42369e9 0.937580
\(699\) 0 0
\(700\) −1.53756e9 −0.169429
\(701\) −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
\(704\) −1.57286e8 −0.0169897
\(705\) 0 0
\(706\) −4.99667e9 −0.534398
\(707\) −7.50253e9 −0.798435
\(708\) 0 0
\(709\) 7.72407e9 0.813925 0.406962 0.913445i \(-0.366588\pi\)
0.406962 + 0.913445i \(0.366588\pi\)
\(710\) 3.89851e9 0.408784
\(711\) 0 0
\(712\) −6.60124e8 −0.0685402
\(713\) 2.85194e10 2.94664
\(714\) 0 0
\(715\) 3.86568e8 0.0395508
\(716\) −7.27294e9 −0.740482
\(717\) 0 0
\(718\) −6.36156e9 −0.641398
\(719\) −1.11572e9 −0.111945 −0.0559725 0.998432i \(-0.517826\pi\)
−0.0559725 + 0.998432i \(0.517826\pi\)
\(720\) 0 0
\(721\) 2.63894e9 0.262214
\(722\) 5.04860e9 0.499219
\(723\) 0 0
\(724\) −8.18700e9 −0.801752
\(725\) 1.12931e10 1.10060
\(726\) 0 0
\(727\) −7.85026e9 −0.757728 −0.378864 0.925452i \(-0.623685\pi\)
−0.378864 + 0.925452i \(0.623685\pi\)
\(728\) −1.03635e9 −0.0995508
\(729\) 0 0
\(730\) −3.69661e9 −0.351701
\(731\) 1.39990e9 0.132552
\(732\) 0 0
\(733\) −1.43652e10 −1.34725 −0.673625 0.739073i \(-0.735264\pi\)
−0.673625 + 0.739073i \(0.735264\pi\)
\(734\) 8.75560e9 0.817240
\(735\) 0 0
\(736\) 3.48625e9 0.322319
\(737\) 1.20829e9 0.111182
\(738\) 0 0
\(739\) 5.13869e9 0.468379 0.234189 0.972191i \(-0.424756\pi\)
0.234189 + 0.972191i \(0.424756\pi\)
\(740\) −8.82885e8 −0.0800927
\(741\) 0 0
\(742\) −5.39188e9 −0.484537
\(743\) 1.12182e10 1.00338 0.501688 0.865049i \(-0.332713\pi\)
0.501688 + 0.865049i \(0.332713\pi\)
\(744\) 0 0
\(745\) 1.02188e9 0.0905422
\(746\) −7.05635e9 −0.622292
\(747\) 0 0
\(748\) 4.67251e8 0.0408221
\(749\) 1.68920e9 0.146891
\(750\) 0 0
\(751\) 2.16341e10 1.86380 0.931899 0.362717i \(-0.118151\pi\)
0.931899 + 0.362717i \(0.118151\pi\)
\(752\) −2.29923e9 −0.197161
\(753\) 0 0
\(754\) 7.61178e9 0.646676
\(755\) −7.64424e9 −0.646427
\(756\) 0 0
\(757\) −1.67963e10 −1.40727 −0.703635 0.710561i \(-0.748441\pi\)
−0.703635 + 0.710561i \(0.748441\pi\)
\(758\) 8.34005e9 0.695547
\(759\) 0 0
\(760\) 9.96004e8 0.0823026
\(761\) −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
\(764\) 1.34938e9 0.109473
\(765\) 0 0
\(766\) −6.69478e9 −0.538190
\(767\) 9.59088e9 0.767494
\(768\) 0 0
\(769\) 1.66703e10 1.32191 0.660954 0.750426i \(-0.270152\pi\)
0.660954 + 0.750426i \(0.270152\pi\)
\(770\) −2.17152e8 −0.0171414
\(771\) 0 0
\(772\) −8.94855e9 −0.699990
\(773\) 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
\(776\) −4.38061e9 −0.336526
\(777\) 0 0
\(778\) −1.17648e10 −0.895685
\(779\) 1.81771e9 0.137766
\(780\) 0 0
\(781\) −2.43657e9 −0.183020
\(782\) −1.03566e10 −0.774452
\(783\) 0 0
\(784\) −2.79107e9 −0.206854
\(785\) 7.63611e9 0.563415
\(786\) 0 0
\(787\) 1.88162e10 1.37601 0.688004 0.725707i \(-0.258487\pi\)
0.688004 + 0.725707i \(0.258487\pi\)
\(788\) −9.34183e9 −0.680127
\(789\) 0 0
\(790\) 9.95742e8 0.0718542
\(791\) −1.31115e9 −0.0941968
\(792\) 0 0
\(793\) −7.01641e9 −0.499642
\(794\) 1.01397e10 0.718876
\(795\) 0 0
\(796\) −3.27134e9 −0.229895
\(797\) −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
\(800\) 2.08814e9 0.144193
\(801\) 0 0
\(802\) 4.22287e9 0.289066
\(803\) 2.31038e9 0.157463
\(804\) 0 0
\(805\) 4.81317e9 0.325197
\(806\) 1.15137e10 0.774538
\(807\) 0 0
\(808\) 1.01891e10 0.679511
\(809\) −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 \(-0.429873\pi\)
0.218532 + 0.975830i \(0.429873\pi\)
\(812\) −4.27587e9 −0.280271
\(813\) 0 0
\(814\) 5.51803e8 0.0358591
\(815\) −8.24761e9 −0.533674
\(816\) 0 0
\(817\) −1.86504e9 −0.119650
\(818\) 1.78233e9 0.113855
\(819\) 0 0
\(820\) −8.61143e8 −0.0545415
\(821\) −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 \(-0.488995\pi\)
0.0345672 + 0.999402i \(0.488995\pi\)
\(824\) −3.58392e9 −0.223158
\(825\) 0 0
\(826\) −5.38761e9 −0.332634
\(827\) −9.56978e9 −0.588346 −0.294173 0.955752i \(-0.595044\pi\)
−0.294173 + 0.955752i \(0.595044\pi\)
\(828\) 0 0
\(829\) −2.36632e10 −1.44256 −0.721278 0.692646i \(-0.756445\pi\)
−0.721278 + 0.692646i \(0.756445\pi\)
\(830\) −8.83543e9 −0.536357
\(831\) 0 0
\(832\) 1.40745e9 0.0847231
\(833\) 8.29145e9 0.497019
\(834\) 0 0
\(835\) 1.55321e9 0.0923267
\(836\) −6.22502e8 −0.0368484
\(837\) 0 0
\(838\) −3.52465e9 −0.206901
\(839\) −2.07240e10 −1.21145 −0.605726 0.795673i \(-0.707117\pi\)
−0.605726 + 0.795673i \(0.707117\pi\)
\(840\) 0 0
\(841\) 1.41556e10 0.820622
\(842\) −1.75147e10 −1.01114
\(843\) 0 0
\(844\) 6.27544e9 0.359290
\(845\) 4.07068e9 0.232097
\(846\) 0 0
\(847\) −7.21094e9 −0.407756
\(848\) 7.32266e9 0.412367
\(849\) 0 0
\(850\) −6.20325e9 −0.346460
\(851\) −1.22307e10 −0.680297
\(852\) 0 0
\(853\) −1.96254e10 −1.08267 −0.541336 0.840806i \(-0.682081\pi\)
−0.541336 + 0.840806i \(0.682081\pi\)
\(854\) 3.94142e9 0.216546
\(855\) 0 0
\(856\) −2.29408e9 −0.125012
\(857\) −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 \(-0.445070\pi\)
0.171714 + 0.985147i \(0.445070\pi\)
\(860\) 8.83569e8 0.0473692
\(861\) 0 0
\(862\) 2.42907e10 1.29171
\(863\) −1.63727e9 −0.0867127 −0.0433564 0.999060i \(-0.513805\pi\)
−0.0433564 + 0.999060i \(0.513805\pi\)
\(864\) 0 0
\(865\) −5.06548e9 −0.266112
\(866\) −1.77388e10 −0.928134
\(867\) 0 0
\(868\) −6.46775e9 −0.335687
\(869\) −6.22339e8 −0.0321705
\(870\) 0 0
\(871\) −1.08122e10 −0.554434
\(872\) 7.53020e8 0.0384591
\(873\) 0 0
\(874\) 1.37978e10 0.699067
\(875\) 6.41729e9 0.323835
\(876\) 0 0
\(877\) −3.31776e10 −1.66091 −0.830455 0.557085i \(-0.811920\pi\)
−0.830455 + 0.557085i \(0.811920\pi\)
\(878\) 2.00075e10 0.997613
\(879\) 0 0
\(880\) 2.94912e8 0.0145882
\(881\) 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 \(-0.454342\pi\)
0.142947 + 0.989730i \(0.454342\pi\)
\(884\) −4.18112e9 −0.203568
\(885\) 0 0
\(886\) −1.92310e10 −0.928930
\(887\) 7.81095e9 0.375813 0.187906 0.982187i \(-0.439830\pi\)
0.187906 + 0.982187i \(0.439830\pi\)
\(888\) 0 0
\(889\) 7.93124e9 0.378604
\(890\) 1.23773e9 0.0588520
\(891\) 0 0
\(892\) 1.38725e10 0.654453
\(893\) −9.09982e9 −0.427615
\(894\) 0 0
\(895\) 1.36368e10 0.635815
\(896\) −7.90626e8 −0.0367192
\(897\) 0 0
\(898\) 1.71948e10 0.792375
\(899\) 4.75045e10 2.18060
\(900\) 0 0
\(901\) −2.17535e10 −0.990813
\(902\) 5.38214e8 0.0244193
\(903\) 0 0
\(904\) 1.78067e9 0.0801665
\(905\) 1.53506e10 0.688424
\(906\) 0 0
\(907\) −4.12701e10 −1.83658 −0.918289 0.395910i \(-0.870429\pi\)
−0.918289 + 0.395910i \(0.870429\pi\)
\(908\) 8.95134e9 0.396814
\(909\) 0 0
\(910\) 1.94315e9 0.0854793
\(911\) −6.66422e9 −0.292035 −0.146017 0.989282i \(-0.546646\pi\)
−0.146017 + 0.989282i \(0.546646\pi\)
\(912\) 0 0
\(913\) 5.52214e9 0.240137
\(914\) −1.84127e10 −0.797637
\(915\) 0 0
\(916\) −6.07724e9 −0.261260
\(917\) 7.80035e9 0.334058
\(918\) 0 0
\(919\) 5.45018e9 0.231636 0.115818 0.993270i \(-0.463051\pi\)
0.115818 + 0.993270i \(0.463051\pi\)
\(920\) −6.53672e9 −0.276760
\(921\) 0 0
\(922\) 1.10342e10 0.463642
\(923\) 2.18032e10 0.912671
\(924\) 0 0
\(925\) −7.32576e9 −0.304338
\(926\) 1.37643e10 0.569660
\(927\) 0 0
\(928\) 5.80701e9 0.238526
\(929\) −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
\(932\) 1.35007e10 0.546261
\(933\) 0 0
\(934\) 1.27000e10 0.510022
\(935\) −8.76096e8 −0.0350519
\(936\) 0 0
\(937\) 4.59669e10 1.82539 0.912697 0.408638i \(-0.133996\pi\)
0.912697 + 0.408638i \(0.133996\pi\)
\(938\) 6.07367e9 0.240293
\(939\) 0 0
\(940\) 4.31106e9 0.169292
\(941\) 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
\(944\) 7.31687e9 0.283089
\(945\) 0 0
\(946\) −5.52230e8 −0.0212081
\(947\) −3.13934e10 −1.20119 −0.600597 0.799552i \(-0.705071\pi\)
−0.600597 + 0.799552i \(0.705071\pi\)
\(948\) 0 0
\(949\) −2.06741e10 −0.785226
\(950\) 8.26437e9 0.312735
\(951\) 0 0
\(952\) 2.34872e9 0.0882269
\(953\) 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
\(956\) 8.26572e9 0.305969
\(957\) 0 0
\(958\) 1.40536e10 0.516425
\(959\) 2.00383e10 0.733663
\(960\) 0 0
\(961\) 4.43435e10 1.61175
\(962\) −4.93772e9 −0.178819
\(963\) 0 0
\(964\) −2.57195e9 −0.0924685
\(965\) 1.67785e10 0.601047
\(966\) 0 0
\(967\) 3.22238e9 0.114600 0.0572999 0.998357i \(-0.481751\pi\)
0.0572999 + 0.998357i \(0.481751\pi\)
\(968\) 9.79311e9 0.347022
\(969\) 0 0
\(970\) 8.21365e9 0.288958
\(971\) −2.25397e10 −0.790096 −0.395048 0.918660i \(-0.629272\pi\)
−0.395048 + 0.918660i \(0.629272\pi\)
\(972\) 0 0
\(973\) −7.35184e9 −0.255859
\(974\) −1.21603e10 −0.421685
\(975\) 0 0
\(976\) −5.35280e9 −0.184292
\(977\) −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
\(980\) 5.23326e9 0.177616
\(981\) 0 0
\(982\) 2.91859e10 0.983519
\(983\) 2.28659e10 0.767806 0.383903 0.923373i \(-0.374580\pi\)
0.383903 + 0.923373i \(0.374580\pi\)
\(984\) 0 0
\(985\) 1.75159e10 0.583991
\(986\) −1.72509e10 −0.573117
\(987\) 0 0
\(988\) 5.57036e9 0.183753
\(989\) 1.22402e10 0.402347
\(990\) 0 0
\(991\) 1.13727e10 0.371198 0.185599 0.982626i \(-0.440577\pi\)
0.185599 + 0.982626i \(0.440577\pi\)
\(992\) 8.78379e9 0.285687
\(993\) 0 0
\(994\) −1.22478e10 −0.395554
\(995\) 6.13376e9 0.197399
\(996\) 0 0
\(997\) −4.31535e10 −1.37906 −0.689530 0.724257i \(-0.742183\pi\)
−0.689530 + 0.724257i \(0.742183\pi\)
\(998\) −3.86802e9 −0.123178
\(999\) 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 54.8.a.a.1.1 1
3.2 odd 2 54.8.a.f.1.1 yes 1
4.3 odd 2 432.8.a.b.1.1 1
9.2 odd 6 162.8.c.b.109.1 2
9.4 even 3 162.8.c.k.55.1 2
9.5 odd 6 162.8.c.b.55.1 2
9.7 even 3 162.8.c.k.109.1 2
12.11 even 2 432.8.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.8.a.a.1.1 1 1.1 even 1 trivial
54.8.a.f.1.1 yes 1 3.2 odd 2
162.8.c.b.55.1 2 9.5 odd 6
162.8.c.b.109.1 2 9.2 odd 6
162.8.c.k.55.1 2 9.4 even 3
162.8.c.k.109.1 2 9.7 even 3
432.8.a.b.1.1 1 4.3 odd 2
432.8.a.g.1.1 1 12.11 even 2