Properties

Label 208.8.a.c.1.1
Level $208$
Weight $8$
Character 208.1
Self dual yes
Analytic conductor $64.976$
Analytic rank $0$
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] = [208,8,Mod(1,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, 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("208.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 208.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: \(64.9760853007\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 208.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+39.0000 q^{3} +385.000 q^{5} +293.000 q^{7} -666.000 q^{9} +O(q^{10})\) \(q+39.0000 q^{3} +385.000 q^{5} +293.000 q^{7} -666.000 q^{9} +5402.00 q^{11} +2197.00 q^{13} +15015.0 q^{15} -21011.0 q^{17} +27326.0 q^{19} +11427.0 q^{21} +63072.0 q^{23} +70100.0 q^{25} -111267. q^{27} +122238. q^{29} +208396. q^{31} +210678. q^{33} +112805. q^{35} -442379. q^{37} +85683.0 q^{39} +58000.0 q^{41} +202025. q^{43} -256410. q^{45} -588511. q^{47} -737694. q^{49} -819429. q^{51} +1.68434e6 q^{53} +2.07977e6 q^{55} +1.06571e6 q^{57} +442630. q^{59} -1.08361e6 q^{61} -195138. q^{63} +845845. q^{65} -3.44349e6 q^{67} +2.45981e6 q^{69} -2.08470e6 q^{71} +5.93789e6 q^{73} +2.73390e6 q^{75} +1.58279e6 q^{77} +6.60926e6 q^{79} -2.88287e6 q^{81} +142740. q^{83} -8.08924e6 q^{85} +4.76728e6 q^{87} -6.98529e6 q^{89} +643721. q^{91} +8.12744e6 q^{93} +1.05205e7 q^{95} -200762. q^{97} -3.59773e6 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 39.0000 0.833950 0.416975 0.908918i \(-0.363090\pi\)
0.416975 + 0.908918i \(0.363090\pi\)
\(4\) 0 0
\(5\) 385.000 1.37742 0.688709 0.725038i \(-0.258178\pi\)
0.688709 + 0.725038i \(0.258178\pi\)
\(6\) 0 0
\(7\) 293.000 0.322868 0.161434 0.986884i \(-0.448388\pi\)
0.161434 + 0.986884i \(0.448388\pi\)
\(8\) 0 0
\(9\) −666.000 −0.304527
\(10\) 0 0
\(11\) 5402.00 1.22371 0.611857 0.790968i \(-0.290423\pi\)
0.611857 + 0.790968i \(0.290423\pi\)
\(12\) 0 0
\(13\) 2197.00 0.277350
\(14\) 0 0
\(15\) 15015.0 1.14870
\(16\) 0 0
\(17\) −21011.0 −1.03723 −0.518616 0.855008i \(-0.673552\pi\)
−0.518616 + 0.855008i \(0.673552\pi\)
\(18\) 0 0
\(19\) 27326.0 0.913984 0.456992 0.889471i \(-0.348927\pi\)
0.456992 + 0.889471i \(0.348927\pi\)
\(20\) 0 0
\(21\) 11427.0 0.269256
\(22\) 0 0
\(23\) 63072.0 1.08091 0.540455 0.841373i \(-0.318252\pi\)
0.540455 + 0.841373i \(0.318252\pi\)
\(24\) 0 0
\(25\) 70100.0 0.897280
\(26\) 0 0
\(27\) −111267. −1.08791
\(28\) 0 0
\(29\) 122238. 0.930708 0.465354 0.885125i \(-0.345927\pi\)
0.465354 + 0.885125i \(0.345927\pi\)
\(30\) 0 0
\(31\) 208396. 1.25639 0.628194 0.778057i \(-0.283795\pi\)
0.628194 + 0.778057i \(0.283795\pi\)
\(32\) 0 0
\(33\) 210678. 1.02052
\(34\) 0 0
\(35\) 112805. 0.444724
\(36\) 0 0
\(37\) −442379. −1.43578 −0.717891 0.696156i \(-0.754892\pi\)
−0.717891 + 0.696156i \(0.754892\pi\)
\(38\) 0 0
\(39\) 85683.0 0.231296
\(40\) 0 0
\(41\) 58000.0 0.131427 0.0657135 0.997839i \(-0.479068\pi\)
0.0657135 + 0.997839i \(0.479068\pi\)
\(42\) 0 0
\(43\) 202025. 0.387494 0.193747 0.981051i \(-0.437936\pi\)
0.193747 + 0.981051i \(0.437936\pi\)
\(44\) 0 0
\(45\) −256410. −0.419461
\(46\) 0 0
\(47\) −588511. −0.826822 −0.413411 0.910545i \(-0.635663\pi\)
−0.413411 + 0.910545i \(0.635663\pi\)
\(48\) 0 0
\(49\) −737694. −0.895757
\(50\) 0 0
\(51\) −819429. −0.864999
\(52\) 0 0
\(53\) 1.68434e6 1.55404 0.777022 0.629474i \(-0.216729\pi\)
0.777022 + 0.629474i \(0.216729\pi\)
\(54\) 0 0
\(55\) 2.07977e6 1.68557
\(56\) 0 0
\(57\) 1.06571e6 0.762217
\(58\) 0 0
\(59\) 442630. 0.280581 0.140291 0.990110i \(-0.455196\pi\)
0.140291 + 0.990110i \(0.455196\pi\)
\(60\) 0 0
\(61\) −1.08361e6 −0.611248 −0.305624 0.952152i \(-0.598865\pi\)
−0.305624 + 0.952152i \(0.598865\pi\)
\(62\) 0 0
\(63\) −195138. −0.0983218
\(64\) 0 0
\(65\) 845845. 0.382027
\(66\) 0 0
\(67\) −3.44349e6 −1.39874 −0.699369 0.714761i \(-0.746536\pi\)
−0.699369 + 0.714761i \(0.746536\pi\)
\(68\) 0 0
\(69\) 2.45981e6 0.901425
\(70\) 0 0
\(71\) −2.08470e6 −0.691258 −0.345629 0.938371i \(-0.612334\pi\)
−0.345629 + 0.938371i \(0.612334\pi\)
\(72\) 0 0
\(73\) 5.93789e6 1.78650 0.893248 0.449564i \(-0.148421\pi\)
0.893248 + 0.449564i \(0.148421\pi\)
\(74\) 0 0
\(75\) 2.73390e6 0.748287
\(76\) 0 0
\(77\) 1.58279e6 0.395098
\(78\) 0 0
\(79\) 6.60926e6 1.50820 0.754098 0.656762i \(-0.228074\pi\)
0.754098 + 0.656762i \(0.228074\pi\)
\(80\) 0 0
\(81\) −2.88287e6 −0.602737
\(82\) 0 0
\(83\) 142740. 0.0274014 0.0137007 0.999906i \(-0.495639\pi\)
0.0137007 + 0.999906i \(0.495639\pi\)
\(84\) 0 0
\(85\) −8.08924e6 −1.42870
\(86\) 0 0
\(87\) 4.76728e6 0.776164
\(88\) 0 0
\(89\) −6.98529e6 −1.05031 −0.525157 0.851005i \(-0.675993\pi\)
−0.525157 + 0.851005i \(0.675993\pi\)
\(90\) 0 0
\(91\) 643721. 0.0895474
\(92\) 0 0
\(93\) 8.12744e6 1.04776
\(94\) 0 0
\(95\) 1.05205e7 1.25894
\(96\) 0 0
\(97\) −200762. −0.0223347 −0.0111674 0.999938i \(-0.503555\pi\)
−0.0111674 + 0.999938i \(0.503555\pi\)
\(98\) 0 0
\(99\) −3.59773e6 −0.372654
\(100\) 0 0
\(101\) −5.42144e6 −0.523588 −0.261794 0.965124i \(-0.584314\pi\)
−0.261794 + 0.965124i \(0.584314\pi\)
\(102\) 0 0
\(103\) 1.71897e7 1.55002 0.775011 0.631948i \(-0.217745\pi\)
0.775011 + 0.631948i \(0.217745\pi\)
\(104\) 0 0
\(105\) 4.39940e6 0.370877
\(106\) 0 0
\(107\) −1.23582e7 −0.975242 −0.487621 0.873055i \(-0.662135\pi\)
−0.487621 + 0.873055i \(0.662135\pi\)
\(108\) 0 0
\(109\) 1.70569e7 1.26156 0.630778 0.775964i \(-0.282736\pi\)
0.630778 + 0.775964i \(0.282736\pi\)
\(110\) 0 0
\(111\) −1.72528e7 −1.19737
\(112\) 0 0
\(113\) 2.11250e7 1.37728 0.688639 0.725104i \(-0.258208\pi\)
0.688639 + 0.725104i \(0.258208\pi\)
\(114\) 0 0
\(115\) 2.42827e7 1.48886
\(116\) 0 0
\(117\) −1.46320e6 −0.0844605
\(118\) 0 0
\(119\) −6.15622e6 −0.334888
\(120\) 0 0
\(121\) 9.69443e6 0.497478
\(122\) 0 0
\(123\) 2.26200e6 0.109604
\(124\) 0 0
\(125\) −3.08962e6 −0.141488
\(126\) 0 0
\(127\) 3.24008e7 1.40360 0.701800 0.712374i \(-0.252380\pi\)
0.701800 + 0.712374i \(0.252380\pi\)
\(128\) 0 0
\(129\) 7.87898e6 0.323151
\(130\) 0 0
\(131\) 2.64669e7 1.02862 0.514308 0.857605i \(-0.328049\pi\)
0.514308 + 0.857605i \(0.328049\pi\)
\(132\) 0 0
\(133\) 8.00652e6 0.295096
\(134\) 0 0
\(135\) −4.28378e7 −1.49851
\(136\) 0 0
\(137\) 5.36201e7 1.78158 0.890791 0.454413i \(-0.150151\pi\)
0.890791 + 0.454413i \(0.150151\pi\)
\(138\) 0 0
\(139\) −7.58784e6 −0.239644 −0.119822 0.992795i \(-0.538232\pi\)
−0.119822 + 0.992795i \(0.538232\pi\)
\(140\) 0 0
\(141\) −2.29519e7 −0.689529
\(142\) 0 0
\(143\) 1.18682e7 0.339397
\(144\) 0 0
\(145\) 4.70616e7 1.28197
\(146\) 0 0
\(147\) −2.87701e7 −0.747016
\(148\) 0 0
\(149\) −5.70297e7 −1.41237 −0.706187 0.708026i \(-0.749586\pi\)
−0.706187 + 0.708026i \(0.749586\pi\)
\(150\) 0 0
\(151\) 2.00648e7 0.474259 0.237130 0.971478i \(-0.423793\pi\)
0.237130 + 0.971478i \(0.423793\pi\)
\(152\) 0 0
\(153\) 1.39933e7 0.315865
\(154\) 0 0
\(155\) 8.02325e7 1.73057
\(156\) 0 0
\(157\) −3.15314e7 −0.650272 −0.325136 0.945667i \(-0.605410\pi\)
−0.325136 + 0.945667i \(0.605410\pi\)
\(158\) 0 0
\(159\) 6.56891e7 1.29600
\(160\) 0 0
\(161\) 1.84801e7 0.348991
\(162\) 0 0
\(163\) 3.13938e7 0.567789 0.283895 0.958855i \(-0.408373\pi\)
0.283895 + 0.958855i \(0.408373\pi\)
\(164\) 0 0
\(165\) 8.11110e7 1.40568
\(166\) 0 0
\(167\) −9.22170e7 −1.53216 −0.766079 0.642747i \(-0.777795\pi\)
−0.766079 + 0.642747i \(0.777795\pi\)
\(168\) 0 0
\(169\) 4.82681e6 0.0769231
\(170\) 0 0
\(171\) −1.81991e7 −0.278332
\(172\) 0 0
\(173\) −6.57015e7 −0.964748 −0.482374 0.875965i \(-0.660225\pi\)
−0.482374 + 0.875965i \(0.660225\pi\)
\(174\) 0 0
\(175\) 2.05393e7 0.289703
\(176\) 0 0
\(177\) 1.72626e7 0.233991
\(178\) 0 0
\(179\) 3.20402e6 0.0417551 0.0208776 0.999782i \(-0.493354\pi\)
0.0208776 + 0.999782i \(0.493354\pi\)
\(180\) 0 0
\(181\) −4.45759e7 −0.558760 −0.279380 0.960181i \(-0.590129\pi\)
−0.279380 + 0.960181i \(0.590129\pi\)
\(182\) 0 0
\(183\) −4.22607e7 −0.509751
\(184\) 0 0
\(185\) −1.70316e8 −1.97767
\(186\) 0 0
\(187\) −1.13501e8 −1.26927
\(188\) 0 0
\(189\) −3.26012e7 −0.351251
\(190\) 0 0
\(191\) −1.86394e8 −1.93559 −0.967797 0.251733i \(-0.919000\pi\)
−0.967797 + 0.251733i \(0.919000\pi\)
\(192\) 0 0
\(193\) −1.52927e8 −1.53120 −0.765602 0.643314i \(-0.777559\pi\)
−0.765602 + 0.643314i \(0.777559\pi\)
\(194\) 0 0
\(195\) 3.29880e7 0.318592
\(196\) 0 0
\(197\) 9.51837e7 0.887015 0.443507 0.896271i \(-0.353734\pi\)
0.443507 + 0.896271i \(0.353734\pi\)
\(198\) 0 0
\(199\) −1.78585e8 −1.60642 −0.803212 0.595693i \(-0.796878\pi\)
−0.803212 + 0.595693i \(0.796878\pi\)
\(200\) 0 0
\(201\) −1.34296e8 −1.16648
\(202\) 0 0
\(203\) 3.58157e7 0.300495
\(204\) 0 0
\(205\) 2.23300e7 0.181030
\(206\) 0 0
\(207\) −4.20060e7 −0.329166
\(208\) 0 0
\(209\) 1.47615e8 1.11846
\(210\) 0 0
\(211\) 1.33235e8 0.976406 0.488203 0.872730i \(-0.337652\pi\)
0.488203 + 0.872730i \(0.337652\pi\)
\(212\) 0 0
\(213\) −8.13035e7 −0.576475
\(214\) 0 0
\(215\) 7.77796e7 0.533742
\(216\) 0 0
\(217\) 6.10600e7 0.405647
\(218\) 0 0
\(219\) 2.31578e8 1.48985
\(220\) 0 0
\(221\) −4.61612e7 −0.287676
\(222\) 0 0
\(223\) −1.19394e8 −0.720969 −0.360484 0.932765i \(-0.617389\pi\)
−0.360484 + 0.932765i \(0.617389\pi\)
\(224\) 0 0
\(225\) −4.66866e7 −0.273246
\(226\) 0 0
\(227\) −1.13656e7 −0.0644911 −0.0322456 0.999480i \(-0.510266\pi\)
−0.0322456 + 0.999480i \(0.510266\pi\)
\(228\) 0 0
\(229\) −1.46559e7 −0.0806470 −0.0403235 0.999187i \(-0.512839\pi\)
−0.0403235 + 0.999187i \(0.512839\pi\)
\(230\) 0 0
\(231\) 6.17287e7 0.329492
\(232\) 0 0
\(233\) −2.46924e8 −1.27885 −0.639423 0.768855i \(-0.720827\pi\)
−0.639423 + 0.768855i \(0.720827\pi\)
\(234\) 0 0
\(235\) −2.26577e8 −1.13888
\(236\) 0 0
\(237\) 2.57761e8 1.25776
\(238\) 0 0
\(239\) 1.61239e7 0.0763971 0.0381985 0.999270i \(-0.487838\pi\)
0.0381985 + 0.999270i \(0.487838\pi\)
\(240\) 0 0
\(241\) 1.14256e8 0.525798 0.262899 0.964823i \(-0.415321\pi\)
0.262899 + 0.964823i \(0.415321\pi\)
\(242\) 0 0
\(243\) 1.30909e8 0.585258
\(244\) 0 0
\(245\) −2.84012e8 −1.23383
\(246\) 0 0
\(247\) 6.00352e7 0.253493
\(248\) 0 0
\(249\) 5.56686e6 0.0228514
\(250\) 0 0
\(251\) −2.22704e8 −0.888935 −0.444467 0.895795i \(-0.646607\pi\)
−0.444467 + 0.895795i \(0.646607\pi\)
\(252\) 0 0
\(253\) 3.40715e8 1.32272
\(254\) 0 0
\(255\) −3.15480e8 −1.19147
\(256\) 0 0
\(257\) 2.82302e8 1.03741 0.518703 0.854955i \(-0.326415\pi\)
0.518703 + 0.854955i \(0.326415\pi\)
\(258\) 0 0
\(259\) −1.29617e8 −0.463567
\(260\) 0 0
\(261\) −8.14105e7 −0.283425
\(262\) 0 0
\(263\) 2.36490e8 0.801619 0.400809 0.916162i \(-0.368729\pi\)
0.400809 + 0.916162i \(0.368729\pi\)
\(264\) 0 0
\(265\) 6.48469e8 2.14057
\(266\) 0 0
\(267\) −2.72426e8 −0.875910
\(268\) 0 0
\(269\) −4.82172e8 −1.51032 −0.755160 0.655541i \(-0.772441\pi\)
−0.755160 + 0.655541i \(0.772441\pi\)
\(270\) 0 0
\(271\) −4.66372e8 −1.42344 −0.711721 0.702462i \(-0.752084\pi\)
−0.711721 + 0.702462i \(0.752084\pi\)
\(272\) 0 0
\(273\) 2.51051e7 0.0746781
\(274\) 0 0
\(275\) 3.78680e8 1.09801
\(276\) 0 0
\(277\) −1.88709e8 −0.533475 −0.266738 0.963769i \(-0.585946\pi\)
−0.266738 + 0.963769i \(0.585946\pi\)
\(278\) 0 0
\(279\) −1.38792e8 −0.382603
\(280\) 0 0
\(281\) −7.15402e8 −1.92344 −0.961718 0.274040i \(-0.911640\pi\)
−0.961718 + 0.274040i \(0.911640\pi\)
\(282\) 0 0
\(283\) 4.04602e8 1.06115 0.530573 0.847639i \(-0.321977\pi\)
0.530573 + 0.847639i \(0.321977\pi\)
\(284\) 0 0
\(285\) 4.10300e8 1.04989
\(286\) 0 0
\(287\) 1.69940e7 0.0424335
\(288\) 0 0
\(289\) 3.11234e7 0.0758482
\(290\) 0 0
\(291\) −7.82972e6 −0.0186260
\(292\) 0 0
\(293\) −8.11321e8 −1.88433 −0.942163 0.335156i \(-0.891211\pi\)
−0.942163 + 0.335156i \(0.891211\pi\)
\(294\) 0 0
\(295\) 1.70413e8 0.386478
\(296\) 0 0
\(297\) −6.01064e8 −1.33129
\(298\) 0 0
\(299\) 1.38569e8 0.299790
\(300\) 0 0
\(301\) 5.91933e7 0.125109
\(302\) 0 0
\(303\) −2.11436e8 −0.436646
\(304\) 0 0
\(305\) −4.17189e8 −0.841945
\(306\) 0 0
\(307\) −4.60958e8 −0.909237 −0.454618 0.890686i \(-0.650224\pi\)
−0.454618 + 0.890686i \(0.650224\pi\)
\(308\) 0 0
\(309\) 6.70398e8 1.29264
\(310\) 0 0
\(311\) 2.87718e8 0.542383 0.271192 0.962525i \(-0.412582\pi\)
0.271192 + 0.962525i \(0.412582\pi\)
\(312\) 0 0
\(313\) −9.56179e8 −1.76252 −0.881260 0.472632i \(-0.843304\pi\)
−0.881260 + 0.472632i \(0.843304\pi\)
\(314\) 0 0
\(315\) −7.51281e7 −0.135430
\(316\) 0 0
\(317\) 4.92761e8 0.868818 0.434409 0.900716i \(-0.356957\pi\)
0.434409 + 0.900716i \(0.356957\pi\)
\(318\) 0 0
\(319\) 6.60330e8 1.13892
\(320\) 0 0
\(321\) −4.81970e8 −0.813303
\(322\) 0 0
\(323\) −5.74147e8 −0.948012
\(324\) 0 0
\(325\) 1.54010e8 0.248861
\(326\) 0 0
\(327\) 6.65218e8 1.05207
\(328\) 0 0
\(329\) −1.72434e8 −0.266954
\(330\) 0 0
\(331\) 4.83358e8 0.732607 0.366304 0.930495i \(-0.380623\pi\)
0.366304 + 0.930495i \(0.380623\pi\)
\(332\) 0 0
\(333\) 2.94624e8 0.437234
\(334\) 0 0
\(335\) −1.32574e9 −1.92665
\(336\) 0 0
\(337\) 1.30823e9 1.86200 0.930998 0.365025i \(-0.118940\pi\)
0.930998 + 0.365025i \(0.118940\pi\)
\(338\) 0 0
\(339\) 8.23874e8 1.14858
\(340\) 0 0
\(341\) 1.12576e9 1.53746
\(342\) 0 0
\(343\) −4.57442e8 −0.612078
\(344\) 0 0
\(345\) 9.47026e8 1.24164
\(346\) 0 0
\(347\) 8.94842e8 1.14972 0.574861 0.818251i \(-0.305056\pi\)
0.574861 + 0.818251i \(0.305056\pi\)
\(348\) 0 0
\(349\) −5.41626e8 −0.682041 −0.341020 0.940056i \(-0.610772\pi\)
−0.341020 + 0.940056i \(0.610772\pi\)
\(350\) 0 0
\(351\) −2.44454e8 −0.301732
\(352\) 0 0
\(353\) 2.25334e8 0.272656 0.136328 0.990664i \(-0.456470\pi\)
0.136328 + 0.990664i \(0.456470\pi\)
\(354\) 0 0
\(355\) −8.02611e8 −0.952152
\(356\) 0 0
\(357\) −2.40093e8 −0.279280
\(358\) 0 0
\(359\) 4.38763e8 0.500495 0.250247 0.968182i \(-0.419488\pi\)
0.250247 + 0.968182i \(0.419488\pi\)
\(360\) 0 0
\(361\) −1.47161e8 −0.164634
\(362\) 0 0
\(363\) 3.78083e8 0.414872
\(364\) 0 0
\(365\) 2.28609e9 2.46075
\(366\) 0 0
\(367\) −8.08568e8 −0.853857 −0.426929 0.904285i \(-0.640404\pi\)
−0.426929 + 0.904285i \(0.640404\pi\)
\(368\) 0 0
\(369\) −3.86280e7 −0.0400230
\(370\) 0 0
\(371\) 4.93510e8 0.501750
\(372\) 0 0
\(373\) −1.17884e9 −1.17618 −0.588092 0.808794i \(-0.700121\pi\)
−0.588092 + 0.808794i \(0.700121\pi\)
\(374\) 0 0
\(375\) −1.20495e8 −0.117994
\(376\) 0 0
\(377\) 2.68557e8 0.258132
\(378\) 0 0
\(379\) 1.79168e9 1.69053 0.845266 0.534345i \(-0.179442\pi\)
0.845266 + 0.534345i \(0.179442\pi\)
\(380\) 0 0
\(381\) 1.26363e9 1.17053
\(382\) 0 0
\(383\) 1.19775e9 1.08936 0.544680 0.838644i \(-0.316651\pi\)
0.544680 + 0.838644i \(0.316651\pi\)
\(384\) 0 0
\(385\) 6.09373e8 0.544215
\(386\) 0 0
\(387\) −1.34549e8 −0.118002
\(388\) 0 0
\(389\) −1.43672e8 −0.123751 −0.0618754 0.998084i \(-0.519708\pi\)
−0.0618754 + 0.998084i \(0.519708\pi\)
\(390\) 0 0
\(391\) −1.32521e9 −1.12115
\(392\) 0 0
\(393\) 1.03221e9 0.857815
\(394\) 0 0
\(395\) 2.54456e9 2.07742
\(396\) 0 0
\(397\) −6.17334e8 −0.495169 −0.247584 0.968866i \(-0.579637\pi\)
−0.247584 + 0.968866i \(0.579637\pi\)
\(398\) 0 0
\(399\) 3.12254e8 0.246095
\(400\) 0 0
\(401\) 1.13305e9 0.877491 0.438746 0.898611i \(-0.355423\pi\)
0.438746 + 0.898611i \(0.355423\pi\)
\(402\) 0 0
\(403\) 4.57846e8 0.348459
\(404\) 0 0
\(405\) −1.10991e9 −0.830220
\(406\) 0 0
\(407\) −2.38973e9 −1.75699
\(408\) 0 0
\(409\) −1.04283e9 −0.753670 −0.376835 0.926280i \(-0.622988\pi\)
−0.376835 + 0.926280i \(0.622988\pi\)
\(410\) 0 0
\(411\) 2.09119e9 1.48575
\(412\) 0 0
\(413\) 1.29691e8 0.0905906
\(414\) 0 0
\(415\) 5.49549e7 0.0377431
\(416\) 0 0
\(417\) −2.95926e8 −0.199851
\(418\) 0 0
\(419\) −7.09302e8 −0.471066 −0.235533 0.971866i \(-0.575684\pi\)
−0.235533 + 0.971866i \(0.575684\pi\)
\(420\) 0 0
\(421\) −1.19877e9 −0.782974 −0.391487 0.920184i \(-0.628039\pi\)
−0.391487 + 0.920184i \(0.628039\pi\)
\(422\) 0 0
\(423\) 3.91948e8 0.251789
\(424\) 0 0
\(425\) −1.47287e9 −0.930687
\(426\) 0 0
\(427\) −3.17497e8 −0.197352
\(428\) 0 0
\(429\) 4.62860e8 0.283041
\(430\) 0 0
\(431\) −9.54153e8 −0.574047 −0.287024 0.957924i \(-0.592666\pi\)
−0.287024 + 0.957924i \(0.592666\pi\)
\(432\) 0 0
\(433\) −3.81628e8 −0.225908 −0.112954 0.993600i \(-0.536031\pi\)
−0.112954 + 0.993600i \(0.536031\pi\)
\(434\) 0 0
\(435\) 1.83540e9 1.06910
\(436\) 0 0
\(437\) 1.72351e9 0.987933
\(438\) 0 0
\(439\) −1.11683e8 −0.0630031 −0.0315015 0.999504i \(-0.510029\pi\)
−0.0315015 + 0.999504i \(0.510029\pi\)
\(440\) 0 0
\(441\) 4.91304e8 0.272782
\(442\) 0 0
\(443\) −1.45991e9 −0.797837 −0.398919 0.916986i \(-0.630614\pi\)
−0.398919 + 0.916986i \(0.630614\pi\)
\(444\) 0 0
\(445\) −2.68934e9 −1.44672
\(446\) 0 0
\(447\) −2.22416e9 −1.17785
\(448\) 0 0
\(449\) −6.34009e8 −0.330547 −0.165273 0.986248i \(-0.552851\pi\)
−0.165273 + 0.986248i \(0.552851\pi\)
\(450\) 0 0
\(451\) 3.13316e8 0.160829
\(452\) 0 0
\(453\) 7.82528e8 0.395509
\(454\) 0 0
\(455\) 2.47833e8 0.123344
\(456\) 0 0
\(457\) 6.04376e8 0.296211 0.148105 0.988972i \(-0.452683\pi\)
0.148105 + 0.988972i \(0.452683\pi\)
\(458\) 0 0
\(459\) 2.33783e9 1.12841
\(460\) 0 0
\(461\) 2.20565e9 1.04853 0.524267 0.851554i \(-0.324339\pi\)
0.524267 + 0.851554i \(0.324339\pi\)
\(462\) 0 0
\(463\) −1.04925e9 −0.491299 −0.245650 0.969359i \(-0.579001\pi\)
−0.245650 + 0.969359i \(0.579001\pi\)
\(464\) 0 0
\(465\) 3.12907e9 1.44321
\(466\) 0 0
\(467\) 2.01461e9 0.915337 0.457668 0.889123i \(-0.348685\pi\)
0.457668 + 0.889123i \(0.348685\pi\)
\(468\) 0 0
\(469\) −1.00894e9 −0.451607
\(470\) 0 0
\(471\) −1.22973e9 −0.542295
\(472\) 0 0
\(473\) 1.09134e9 0.474183
\(474\) 0 0
\(475\) 1.91555e9 0.820099
\(476\) 0 0
\(477\) −1.12177e9 −0.473248
\(478\) 0 0
\(479\) −3.67842e9 −1.52928 −0.764639 0.644458i \(-0.777083\pi\)
−0.764639 + 0.644458i \(0.777083\pi\)
\(480\) 0 0
\(481\) −9.71907e8 −0.398214
\(482\) 0 0
\(483\) 7.20724e8 0.291041
\(484\) 0 0
\(485\) −7.72934e7 −0.0307642
\(486\) 0 0
\(487\) 1.91497e8 0.0751294 0.0375647 0.999294i \(-0.488040\pi\)
0.0375647 + 0.999294i \(0.488040\pi\)
\(488\) 0 0
\(489\) 1.22436e9 0.473508
\(490\) 0 0
\(491\) 3.22321e8 0.122886 0.0614431 0.998111i \(-0.480430\pi\)
0.0614431 + 0.998111i \(0.480430\pi\)
\(492\) 0 0
\(493\) −2.56834e9 −0.965359
\(494\) 0 0
\(495\) −1.38513e9 −0.513300
\(496\) 0 0
\(497\) −6.10819e8 −0.223185
\(498\) 0 0
\(499\) −3.86695e9 −1.39321 −0.696604 0.717455i \(-0.745307\pi\)
−0.696604 + 0.717455i \(0.745307\pi\)
\(500\) 0 0
\(501\) −3.59646e9 −1.27774
\(502\) 0 0
\(503\) 3.43814e8 0.120458 0.0602290 0.998185i \(-0.480817\pi\)
0.0602290 + 0.998185i \(0.480817\pi\)
\(504\) 0 0
\(505\) −2.08725e9 −0.721199
\(506\) 0 0
\(507\) 1.88246e8 0.0641500
\(508\) 0 0
\(509\) 2.11533e9 0.710993 0.355497 0.934678i \(-0.384312\pi\)
0.355497 + 0.934678i \(0.384312\pi\)
\(510\) 0 0
\(511\) 1.73980e9 0.576802
\(512\) 0 0
\(513\) −3.04048e9 −0.994333
\(514\) 0 0
\(515\) 6.61803e9 2.13503
\(516\) 0 0
\(517\) −3.17914e9 −1.01179
\(518\) 0 0
\(519\) −2.56236e9 −0.804552
\(520\) 0 0
\(521\) −1.40622e9 −0.435634 −0.217817 0.975990i \(-0.569894\pi\)
−0.217817 + 0.975990i \(0.569894\pi\)
\(522\) 0 0
\(523\) −2.18120e9 −0.666712 −0.333356 0.942801i \(-0.608181\pi\)
−0.333356 + 0.942801i \(0.608181\pi\)
\(524\) 0 0
\(525\) 8.01033e8 0.241598
\(526\) 0 0
\(527\) −4.37861e9 −1.30316
\(528\) 0 0
\(529\) 5.73252e8 0.168365
\(530\) 0 0
\(531\) −2.94792e8 −0.0854445
\(532\) 0 0
\(533\) 1.27426e8 0.0364513
\(534\) 0 0
\(535\) −4.75791e9 −1.34332
\(536\) 0 0
\(537\) 1.24957e8 0.0348217
\(538\) 0 0
\(539\) −3.98502e9 −1.09615
\(540\) 0 0
\(541\) 2.54634e8 0.0691395 0.0345698 0.999402i \(-0.488994\pi\)
0.0345698 + 0.999402i \(0.488994\pi\)
\(542\) 0 0
\(543\) −1.73846e9 −0.465978
\(544\) 0 0
\(545\) 6.56689e9 1.73769
\(546\) 0 0
\(547\) −2.15158e9 −0.562085 −0.281043 0.959695i \(-0.590680\pi\)
−0.281043 + 0.959695i \(0.590680\pi\)
\(548\) 0 0
\(549\) 7.21683e8 0.186142
\(550\) 0 0
\(551\) 3.34028e9 0.850652
\(552\) 0 0
\(553\) 1.93651e9 0.486948
\(554\) 0 0
\(555\) −6.64232e9 −1.64928
\(556\) 0 0
\(557\) −7.71518e9 −1.89170 −0.945852 0.324599i \(-0.894771\pi\)
−0.945852 + 0.324599i \(0.894771\pi\)
\(558\) 0 0
\(559\) 4.43849e8 0.107472
\(560\) 0 0
\(561\) −4.42656e9 −1.05851
\(562\) 0 0
\(563\) 8.12996e7 0.0192003 0.00960017 0.999954i \(-0.496944\pi\)
0.00960017 + 0.999954i \(0.496944\pi\)
\(564\) 0 0
\(565\) 8.13312e9 1.89709
\(566\) 0 0
\(567\) −8.44681e8 −0.194604
\(568\) 0 0
\(569\) −5.08814e9 −1.15789 −0.578944 0.815367i \(-0.696535\pi\)
−0.578944 + 0.815367i \(0.696535\pi\)
\(570\) 0 0
\(571\) 5.61762e9 1.26277 0.631387 0.775468i \(-0.282486\pi\)
0.631387 + 0.775468i \(0.282486\pi\)
\(572\) 0 0
\(573\) −7.26935e9 −1.61419
\(574\) 0 0
\(575\) 4.42135e9 0.969878
\(576\) 0 0
\(577\) 4.12728e9 0.894435 0.447218 0.894425i \(-0.352415\pi\)
0.447218 + 0.894425i \(0.352415\pi\)
\(578\) 0 0
\(579\) −5.96415e9 −1.27695
\(580\) 0 0
\(581\) 4.18228e7 0.00884702
\(582\) 0 0
\(583\) 9.09878e9 1.90171
\(584\) 0 0
\(585\) −5.63333e8 −0.116337
\(586\) 0 0
\(587\) −1.86734e9 −0.381056 −0.190528 0.981682i \(-0.561020\pi\)
−0.190528 + 0.981682i \(0.561020\pi\)
\(588\) 0 0
\(589\) 5.69463e9 1.14832
\(590\) 0 0
\(591\) 3.71216e9 0.739726
\(592\) 0 0
\(593\) 3.31544e9 0.652905 0.326453 0.945214i \(-0.394147\pi\)
0.326453 + 0.945214i \(0.394147\pi\)
\(594\) 0 0
\(595\) −2.37015e9 −0.461281
\(596\) 0 0
\(597\) −6.96483e9 −1.33968
\(598\) 0 0
\(599\) −1.93367e9 −0.367610 −0.183805 0.982963i \(-0.558842\pi\)
−0.183805 + 0.982963i \(0.558842\pi\)
\(600\) 0 0
\(601\) −5.88820e9 −1.10643 −0.553213 0.833040i \(-0.686598\pi\)
−0.553213 + 0.833040i \(0.686598\pi\)
\(602\) 0 0
\(603\) 2.29336e9 0.425953
\(604\) 0 0
\(605\) 3.73236e9 0.685235
\(606\) 0 0
\(607\) −7.94197e9 −1.44135 −0.720673 0.693276i \(-0.756167\pi\)
−0.720673 + 0.693276i \(0.756167\pi\)
\(608\) 0 0
\(609\) 1.39681e9 0.250598
\(610\) 0 0
\(611\) −1.29296e9 −0.229319
\(612\) 0 0
\(613\) −2.36146e8 −0.0414065 −0.0207033 0.999786i \(-0.506591\pi\)
−0.0207033 + 0.999786i \(0.506591\pi\)
\(614\) 0 0
\(615\) 8.70870e8 0.150970
\(616\) 0 0
\(617\) 1.27029e9 0.217723 0.108862 0.994057i \(-0.465279\pi\)
0.108862 + 0.994057i \(0.465279\pi\)
\(618\) 0 0
\(619\) −1.63555e9 −0.277170 −0.138585 0.990351i \(-0.544255\pi\)
−0.138585 + 0.990351i \(0.544255\pi\)
\(620\) 0 0
\(621\) −7.01783e9 −1.17593
\(622\) 0 0
\(623\) −2.04669e9 −0.339112
\(624\) 0 0
\(625\) −6.66607e9 −1.09217
\(626\) 0 0
\(627\) 5.75699e9 0.932736
\(628\) 0 0
\(629\) 9.29483e9 1.48924
\(630\) 0 0
\(631\) −1.68242e9 −0.266582 −0.133291 0.991077i \(-0.542555\pi\)
−0.133291 + 0.991077i \(0.542555\pi\)
\(632\) 0 0
\(633\) 5.19618e9 0.814275
\(634\) 0 0
\(635\) 1.24743e10 1.93334
\(636\) 0 0
\(637\) −1.62071e9 −0.248438
\(638\) 0 0
\(639\) 1.38841e9 0.210507
\(640\) 0 0
\(641\) −1.70575e9 −0.255807 −0.127903 0.991787i \(-0.540825\pi\)
−0.127903 + 0.991787i \(0.540825\pi\)
\(642\) 0 0
\(643\) 1.45635e9 0.216036 0.108018 0.994149i \(-0.465550\pi\)
0.108018 + 0.994149i \(0.465550\pi\)
\(644\) 0 0
\(645\) 3.03341e9 0.445114
\(646\) 0 0
\(647\) 3.56464e9 0.517430 0.258715 0.965954i \(-0.416701\pi\)
0.258715 + 0.965954i \(0.416701\pi\)
\(648\) 0 0
\(649\) 2.39109e9 0.343352
\(650\) 0 0
\(651\) 2.38134e9 0.338289
\(652\) 0 0
\(653\) −5.86806e9 −0.824705 −0.412352 0.911024i \(-0.635293\pi\)
−0.412352 + 0.911024i \(0.635293\pi\)
\(654\) 0 0
\(655\) 1.01898e10 1.41683
\(656\) 0 0
\(657\) −3.95463e9 −0.544036
\(658\) 0 0
\(659\) 2.73239e9 0.371915 0.185958 0.982558i \(-0.440461\pi\)
0.185958 + 0.982558i \(0.440461\pi\)
\(660\) 0 0
\(661\) 8.50066e9 1.14485 0.572424 0.819958i \(-0.306003\pi\)
0.572424 + 0.819958i \(0.306003\pi\)
\(662\) 0 0
\(663\) −1.80029e9 −0.239908
\(664\) 0 0
\(665\) 3.08251e9 0.406470
\(666\) 0 0
\(667\) 7.70980e9 1.00601
\(668\) 0 0
\(669\) −4.65638e9 −0.601252
\(670\) 0 0
\(671\) −5.85365e9 −0.747994
\(672\) 0 0
\(673\) −3.85727e7 −0.00487784 −0.00243892 0.999997i \(-0.500776\pi\)
−0.00243892 + 0.999997i \(0.500776\pi\)
\(674\) 0 0
\(675\) −7.79982e9 −0.976160
\(676\) 0 0
\(677\) −7.34428e9 −0.909681 −0.454840 0.890573i \(-0.650304\pi\)
−0.454840 + 0.890573i \(0.650304\pi\)
\(678\) 0 0
\(679\) −5.88233e7 −0.00721116
\(680\) 0 0
\(681\) −4.43257e8 −0.0537824
\(682\) 0 0
\(683\) 7.49577e9 0.900210 0.450105 0.892976i \(-0.351387\pi\)
0.450105 + 0.892976i \(0.351387\pi\)
\(684\) 0 0
\(685\) 2.06438e10 2.45398
\(686\) 0 0
\(687\) −5.71580e8 −0.0672556
\(688\) 0 0
\(689\) 3.70049e9 0.431014
\(690\) 0 0
\(691\) −1.66382e10 −1.91838 −0.959188 0.282769i \(-0.908747\pi\)
−0.959188 + 0.282769i \(0.908747\pi\)
\(692\) 0 0
\(693\) −1.05414e9 −0.120318
\(694\) 0 0
\(695\) −2.92132e9 −0.330090
\(696\) 0 0
\(697\) −1.21864e9 −0.136320
\(698\) 0 0
\(699\) −9.63005e9 −1.06649
\(700\) 0 0
\(701\) 3.26804e9 0.358323 0.179161 0.983820i \(-0.442662\pi\)
0.179161 + 0.983820i \(0.442662\pi\)
\(702\) 0 0
\(703\) −1.20884e10 −1.31228
\(704\) 0 0
\(705\) −8.83649e9 −0.949769
\(706\) 0 0
\(707\) −1.58848e9 −0.169050
\(708\) 0 0
\(709\) 4.48613e9 0.472727 0.236363 0.971665i \(-0.424044\pi\)
0.236363 + 0.971665i \(0.424044\pi\)
\(710\) 0 0
\(711\) −4.40176e9 −0.459286
\(712\) 0 0
\(713\) 1.31440e10 1.35804
\(714\) 0 0
\(715\) 4.56925e9 0.467492
\(716\) 0 0
\(717\) 6.28831e8 0.0637114
\(718\) 0 0
\(719\) −5.42385e9 −0.544198 −0.272099 0.962269i \(-0.587718\pi\)
−0.272099 + 0.962269i \(0.587718\pi\)
\(720\) 0 0
\(721\) 5.03658e9 0.500452
\(722\) 0 0
\(723\) 4.45598e9 0.438490
\(724\) 0 0
\(725\) 8.56888e9 0.835105
\(726\) 0 0
\(727\) 1.50827e10 1.45582 0.727911 0.685672i \(-0.240492\pi\)
0.727911 + 0.685672i \(0.240492\pi\)
\(728\) 0 0
\(729\) 1.14103e10 1.09081
\(730\) 0 0
\(731\) −4.24475e9 −0.401921
\(732\) 0 0
\(733\) −6.75596e9 −0.633612 −0.316806 0.948490i \(-0.602610\pi\)
−0.316806 + 0.948490i \(0.602610\pi\)
\(734\) 0 0
\(735\) −1.10765e10 −1.02895
\(736\) 0 0
\(737\) −1.86017e10 −1.71166
\(738\) 0 0
\(739\) −1.08154e10 −0.985797 −0.492899 0.870087i \(-0.664063\pi\)
−0.492899 + 0.870087i \(0.664063\pi\)
\(740\) 0 0
\(741\) 2.34137e9 0.211401
\(742\) 0 0
\(743\) 3.71897e9 0.332630 0.166315 0.986073i \(-0.446813\pi\)
0.166315 + 0.986073i \(0.446813\pi\)
\(744\) 0 0
\(745\) −2.19565e10 −1.94543
\(746\) 0 0
\(747\) −9.50648e7 −0.00834445
\(748\) 0 0
\(749\) −3.62096e9 −0.314874
\(750\) 0 0
\(751\) 2.15786e10 1.85902 0.929510 0.368797i \(-0.120230\pi\)
0.929510 + 0.368797i \(0.120230\pi\)
\(752\) 0 0
\(753\) −8.68546e9 −0.741328
\(754\) 0 0
\(755\) 7.72496e9 0.653253
\(756\) 0 0
\(757\) 7.42446e9 0.622056 0.311028 0.950401i \(-0.399327\pi\)
0.311028 + 0.950401i \(0.399327\pi\)
\(758\) 0 0
\(759\) 1.32879e10 1.10309
\(760\) 0 0
\(761\) 8.57002e9 0.704913 0.352457 0.935828i \(-0.385346\pi\)
0.352457 + 0.935828i \(0.385346\pi\)
\(762\) 0 0
\(763\) 4.99766e9 0.407315
\(764\) 0 0
\(765\) 5.38743e9 0.435078
\(766\) 0 0
\(767\) 9.72458e8 0.0778193
\(768\) 0 0
\(769\) 7.81741e9 0.619899 0.309949 0.950753i \(-0.399688\pi\)
0.309949 + 0.950753i \(0.399688\pi\)
\(770\) 0 0
\(771\) 1.10098e10 0.865144
\(772\) 0 0
\(773\) 1.30864e10 1.01904 0.509522 0.860457i \(-0.329822\pi\)
0.509522 + 0.860457i \(0.329822\pi\)
\(774\) 0 0
\(775\) 1.46086e10 1.12733
\(776\) 0 0
\(777\) −5.05506e9 −0.386592
\(778\) 0 0
\(779\) 1.58491e9 0.120122
\(780\) 0 0
\(781\) −1.12616e10 −0.845903
\(782\) 0 0
\(783\) −1.36011e10 −1.01253
\(784\) 0 0
\(785\) −1.21396e10 −0.895696
\(786\) 0 0
\(787\) −5.35561e9 −0.391649 −0.195825 0.980639i \(-0.562738\pi\)
−0.195825 + 0.980639i \(0.562738\pi\)
\(788\) 0 0
\(789\) 9.22311e9 0.668510
\(790\) 0 0
\(791\) 6.18962e9 0.444679
\(792\) 0 0
\(793\) −2.38069e9 −0.169530
\(794\) 0 0
\(795\) 2.52903e10 1.78513
\(796\) 0 0
\(797\) 1.21863e10 0.852641 0.426320 0.904572i \(-0.359810\pi\)
0.426320 + 0.904572i \(0.359810\pi\)
\(798\) 0 0
\(799\) 1.23652e10 0.857606
\(800\) 0 0
\(801\) 4.65220e9 0.319849
\(802\) 0 0
\(803\) 3.20765e10 2.18616
\(804\) 0 0
\(805\) 7.11484e9 0.480706
\(806\) 0 0
\(807\) −1.88047e10 −1.25953
\(808\) 0 0
\(809\) 1.32472e10 0.879636 0.439818 0.898087i \(-0.355043\pi\)
0.439818 + 0.898087i \(0.355043\pi\)
\(810\) 0 0
\(811\) −1.45473e10 −0.957658 −0.478829 0.877908i \(-0.658939\pi\)
−0.478829 + 0.877908i \(0.658939\pi\)
\(812\) 0 0
\(813\) −1.81885e10 −1.18708
\(814\) 0 0
\(815\) 1.20866e10 0.782083
\(816\) 0 0
\(817\) 5.52054e9 0.354164
\(818\) 0 0
\(819\) −4.28718e8 −0.0272696
\(820\) 0 0
\(821\) 6.51876e9 0.411115 0.205558 0.978645i \(-0.434099\pi\)
0.205558 + 0.978645i \(0.434099\pi\)
\(822\) 0 0
\(823\) 6.77944e9 0.423930 0.211965 0.977277i \(-0.432014\pi\)
0.211965 + 0.977277i \(0.432014\pi\)
\(824\) 0 0
\(825\) 1.47685e10 0.915690
\(826\) 0 0
\(827\) 7.96808e9 0.489874 0.244937 0.969539i \(-0.421233\pi\)
0.244937 + 0.969539i \(0.421233\pi\)
\(828\) 0 0
\(829\) 3.74439e9 0.228265 0.114133 0.993466i \(-0.463591\pi\)
0.114133 + 0.993466i \(0.463591\pi\)
\(830\) 0 0
\(831\) −7.35966e9 −0.444892
\(832\) 0 0
\(833\) 1.54997e10 0.929106
\(834\) 0 0
\(835\) −3.55035e10 −2.11042
\(836\) 0 0
\(837\) −2.31876e10 −1.36684
\(838\) 0 0
\(839\) 8.06205e9 0.471280 0.235640 0.971840i \(-0.424281\pi\)
0.235640 + 0.971840i \(0.424281\pi\)
\(840\) 0 0
\(841\) −2.30775e9 −0.133783
\(842\) 0 0
\(843\) −2.79007e10 −1.60405
\(844\) 0 0
\(845\) 1.85832e9 0.105955
\(846\) 0 0
\(847\) 2.84047e9 0.160619
\(848\) 0 0
\(849\) 1.57795e10 0.884943
\(850\) 0 0
\(851\) −2.79017e10 −1.55195
\(852\) 0 0
\(853\) −3.31854e9 −0.183074 −0.0915368 0.995802i \(-0.529178\pi\)
−0.0915368 + 0.995802i \(0.529178\pi\)
\(854\) 0 0
\(855\) −7.00666e9 −0.383380
\(856\) 0 0
\(857\) −1.81939e10 −0.987398 −0.493699 0.869633i \(-0.664356\pi\)
−0.493699 + 0.869633i \(0.664356\pi\)
\(858\) 0 0
\(859\) 1.91859e10 1.03278 0.516388 0.856355i \(-0.327276\pi\)
0.516388 + 0.856355i \(0.327276\pi\)
\(860\) 0 0
\(861\) 6.62766e8 0.0353874
\(862\) 0 0
\(863\) 2.77943e10 1.47203 0.736017 0.676963i \(-0.236704\pi\)
0.736017 + 0.676963i \(0.236704\pi\)
\(864\) 0 0
\(865\) −2.52951e10 −1.32886
\(866\) 0 0
\(867\) 1.21381e9 0.0632536
\(868\) 0 0
\(869\) 3.57032e10 1.84560
\(870\) 0 0
\(871\) −7.56534e9 −0.387940
\(872\) 0 0
\(873\) 1.33707e8 0.00680152
\(874\) 0 0
\(875\) −9.05260e8 −0.0456820
\(876\) 0 0
\(877\) −3.40401e10 −1.70409 −0.852043 0.523471i \(-0.824637\pi\)
−0.852043 + 0.523471i \(0.824637\pi\)
\(878\) 0 0
\(879\) −3.16415e10 −1.57143
\(880\) 0 0
\(881\) 3.24476e10 1.59870 0.799350 0.600866i \(-0.205177\pi\)
0.799350 + 0.600866i \(0.205177\pi\)
\(882\) 0 0
\(883\) −1.15866e10 −0.566362 −0.283181 0.959066i \(-0.591390\pi\)
−0.283181 + 0.959066i \(0.591390\pi\)
\(884\) 0 0
\(885\) 6.64609e9 0.322303
\(886\) 0 0
\(887\) 2.86160e10 1.37682 0.688408 0.725323i \(-0.258310\pi\)
0.688408 + 0.725323i \(0.258310\pi\)
\(888\) 0 0
\(889\) 9.49344e9 0.453177
\(890\) 0 0
\(891\) −1.55733e10 −0.737578
\(892\) 0 0
\(893\) −1.60817e10 −0.755702
\(894\) 0 0
\(895\) 1.23355e9 0.0575142
\(896\) 0 0
\(897\) 5.40420e9 0.250010
\(898\) 0 0
\(899\) 2.54739e10 1.16933
\(900\) 0 0
\(901\) −3.53896e10 −1.61190
\(902\) 0 0
\(903\) 2.30854e9 0.104335
\(904\) 0 0
\(905\) −1.71617e10 −0.769646
\(906\) 0 0
\(907\) −2.28278e10 −1.01587 −0.507936 0.861395i \(-0.669591\pi\)
−0.507936 + 0.861395i \(0.669591\pi\)
\(908\) 0 0
\(909\) 3.61068e9 0.159446
\(910\) 0 0
\(911\) 1.76175e10 0.772024 0.386012 0.922494i \(-0.373852\pi\)
0.386012 + 0.922494i \(0.373852\pi\)
\(912\) 0 0
\(913\) 7.71081e8 0.0335315
\(914\) 0 0
\(915\) −1.62704e10 −0.702140
\(916\) 0 0
\(917\) 7.75480e9 0.332107
\(918\) 0 0
\(919\) 4.93202e9 0.209614 0.104807 0.994493i \(-0.466577\pi\)
0.104807 + 0.994493i \(0.466577\pi\)
\(920\) 0 0
\(921\) −1.79774e10 −0.758258
\(922\) 0 0
\(923\) −4.58010e9 −0.191721
\(924\) 0 0
\(925\) −3.10108e10 −1.28830
\(926\) 0 0
\(927\) −1.14483e10 −0.472023
\(928\) 0 0
\(929\) −2.68352e10 −1.09812 −0.549061 0.835783i \(-0.685014\pi\)
−0.549061 + 0.835783i \(0.685014\pi\)
\(930\) 0 0
\(931\) −2.01582e10 −0.818707
\(932\) 0 0
\(933\) 1.12210e10 0.452321
\(934\) 0 0
\(935\) −4.36980e10 −1.74832
\(936\) 0 0
\(937\) −2.08650e10 −0.828570 −0.414285 0.910147i \(-0.635968\pi\)
−0.414285 + 0.910147i \(0.635968\pi\)
\(938\) 0 0
\(939\) −3.72910e10 −1.46985
\(940\) 0 0
\(941\) −3.07099e10 −1.20147 −0.600737 0.799447i \(-0.705126\pi\)
−0.600737 + 0.799447i \(0.705126\pi\)
\(942\) 0 0
\(943\) 3.65818e9 0.142061
\(944\) 0 0
\(945\) −1.25515e10 −0.483820
\(946\) 0 0
\(947\) 1.03377e9 0.0395548 0.0197774 0.999804i \(-0.493704\pi\)
0.0197774 + 0.999804i \(0.493704\pi\)
\(948\) 0 0
\(949\) 1.30455e10 0.495485
\(950\) 0 0
\(951\) 1.92177e10 0.724552
\(952\) 0 0
\(953\) −1.78629e9 −0.0668540 −0.0334270 0.999441i \(-0.510642\pi\)
−0.0334270 + 0.999441i \(0.510642\pi\)
\(954\) 0 0
\(955\) −7.17615e10 −2.66612
\(956\) 0 0
\(957\) 2.57529e10 0.949803
\(958\) 0 0
\(959\) 1.57107e10 0.575215
\(960\) 0 0
\(961\) 1.59163e10 0.578508
\(962\) 0 0
\(963\) 8.23057e9 0.296987
\(964\) 0 0
\(965\) −5.88768e10 −2.10911
\(966\) 0 0
\(967\) 1.53418e10 0.545613 0.272806 0.962069i \(-0.412048\pi\)
0.272806 + 0.962069i \(0.412048\pi\)
\(968\) 0 0
\(969\) −2.23917e10 −0.790595
\(970\) 0 0
\(971\) 5.22182e8 0.0183043 0.00915217 0.999958i \(-0.497087\pi\)
0.00915217 + 0.999958i \(0.497087\pi\)
\(972\) 0 0
\(973\) −2.22324e9 −0.0773733
\(974\) 0 0
\(975\) 6.00638e9 0.207537
\(976\) 0 0
\(977\) 5.68857e10 1.95152 0.975758 0.218851i \(-0.0702309\pi\)
0.975758 + 0.218851i \(0.0702309\pi\)
\(978\) 0 0
\(979\) −3.77345e10 −1.28528
\(980\) 0 0
\(981\) −1.13599e10 −0.384177
\(982\) 0 0
\(983\) 1.15736e10 0.388624 0.194312 0.980940i \(-0.437753\pi\)
0.194312 + 0.980940i \(0.437753\pi\)
\(984\) 0 0
\(985\) 3.66457e10 1.22179
\(986\) 0 0
\(987\) −6.72492e9 −0.222626
\(988\) 0 0
\(989\) 1.27421e10 0.418846
\(990\) 0 0
\(991\) −1.38509e10 −0.452086 −0.226043 0.974117i \(-0.572579\pi\)
−0.226043 + 0.974117i \(0.572579\pi\)
\(992\) 0 0
\(993\) 1.88510e10 0.610958
\(994\) 0 0
\(995\) −6.87554e10 −2.21272
\(996\) 0 0
\(997\) 5.45316e9 0.174267 0.0871334 0.996197i \(-0.472229\pi\)
0.0871334 + 0.996197i \(0.472229\pi\)
\(998\) 0 0
\(999\) 4.92222e10 1.56200
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 208.8.a.c.1.1 1
4.3 odd 2 26.8.a.a.1.1 1
12.11 even 2 234.8.a.d.1.1 1
52.31 even 4 338.8.b.b.337.2 2
52.47 even 4 338.8.b.b.337.1 2
52.51 odd 2 338.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.8.a.a.1.1 1 4.3 odd 2
208.8.a.c.1.1 1 1.1 even 1 trivial
234.8.a.d.1.1 1 12.11 even 2
338.8.a.c.1.1 1 52.51 odd 2
338.8.b.b.337.1 2 52.47 even 4
338.8.b.b.337.2 2 52.31 even 4