Properties

Label 336.8.a.j.1.1
Level $336$
Weight $8$
Character 336.1
Self dual yes
Analytic conductor $104.961$
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] = [336,8,Mod(1,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("336.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 336.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: \(104.961368563\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +100.000 q^{5} +343.000 q^{7} +729.000 q^{9} -2774.00 q^{11} -3294.00 q^{13} +2700.00 q^{15} +5900.00 q^{17} -6644.00 q^{19} +9261.00 q^{21} -1982.00 q^{23} -68125.0 q^{25} +19683.0 q^{27} -208106. q^{29} +117792. q^{31} -74898.0 q^{33} +34300.0 q^{35} -335686. q^{37} -88938.0 q^{39} -265488. q^{41} +93292.0 q^{43} +72900.0 q^{45} +657516. q^{47} +117649. q^{49} +159300. q^{51} -608718. q^{53} -277400. q^{55} -179388. q^{57} +536120. q^{59} -1.79709e6 q^{61} +250047. q^{63} -329400. q^{65} -2.12318e6 q^{67} -53514.0 q^{69} +1.19121e6 q^{71} +1.05643e6 q^{73} -1.83938e6 q^{75} -951482. q^{77} -998484. q^{79} +531441. q^{81} -3.89800e6 q^{83} +590000. q^{85} -5.61886e6 q^{87} -4.62235e6 q^{89} -1.12984e6 q^{91} +3.18038e6 q^{93} -664400. q^{95} +1.52877e7 q^{97} -2.02225e6 q^{99} +O(q^{100})\)

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
\(3\) 27.0000 0.577350
\(4\) 0 0
\(5\) 100.000 0.357771 0.178885 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 0 0
\(7\) 343.000 0.377964
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) −2774.00 −0.628394 −0.314197 0.949358i \(-0.601735\pi\)
−0.314197 + 0.949358i \(0.601735\pi\)
\(12\) 0 0
\(13\) −3294.00 −0.415836 −0.207918 0.978146i \(-0.566669\pi\)
−0.207918 + 0.978146i \(0.566669\pi\)
\(14\) 0 0
\(15\) 2700.00 0.206559
\(16\) 0 0
\(17\) 5900.00 0.291260 0.145630 0.989339i \(-0.453479\pi\)
0.145630 + 0.989339i \(0.453479\pi\)
\(18\) 0 0
\(19\) −6644.00 −0.222225 −0.111112 0.993808i \(-0.535441\pi\)
−0.111112 + 0.993808i \(0.535441\pi\)
\(20\) 0 0
\(21\) 9261.00 0.218218
\(22\) 0 0
\(23\) −1982.00 −0.0339669 −0.0169835 0.999856i \(-0.505406\pi\)
−0.0169835 + 0.999856i \(0.505406\pi\)
\(24\) 0 0
\(25\) −68125.0 −0.872000
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −208106. −1.58450 −0.792249 0.610198i \(-0.791090\pi\)
−0.792249 + 0.610198i \(0.791090\pi\)
\(30\) 0 0
\(31\) 117792. 0.710150 0.355075 0.934838i \(-0.384455\pi\)
0.355075 + 0.934838i \(0.384455\pi\)
\(32\) 0 0
\(33\) −74898.0 −0.362803
\(34\) 0 0
\(35\) 34300.0 0.135225
\(36\) 0 0
\(37\) −335686. −1.08950 −0.544750 0.838599i \(-0.683375\pi\)
−0.544750 + 0.838599i \(0.683375\pi\)
\(38\) 0 0
\(39\) −88938.0 −0.240083
\(40\) 0 0
\(41\) −265488. −0.601591 −0.300796 0.953689i \(-0.597252\pi\)
−0.300796 + 0.953689i \(0.597252\pi\)
\(42\) 0 0
\(43\) 93292.0 0.178939 0.0894695 0.995990i \(-0.471483\pi\)
0.0894695 + 0.995990i \(0.471483\pi\)
\(44\) 0 0
\(45\) 72900.0 0.119257
\(46\) 0 0
\(47\) 657516. 0.923770 0.461885 0.886940i \(-0.347173\pi\)
0.461885 + 0.886940i \(0.347173\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 159300. 0.168159
\(52\) 0 0
\(53\) −608718. −0.561630 −0.280815 0.959762i \(-0.590605\pi\)
−0.280815 + 0.959762i \(0.590605\pi\)
\(54\) 0 0
\(55\) −277400. −0.224821
\(56\) 0 0
\(57\) −179388. −0.128301
\(58\) 0 0
\(59\) 536120. 0.339844 0.169922 0.985457i \(-0.445648\pi\)
0.169922 + 0.985457i \(0.445648\pi\)
\(60\) 0 0
\(61\) −1.79709e6 −1.01371 −0.506857 0.862030i \(-0.669193\pi\)
−0.506857 + 0.862030i \(0.669193\pi\)
\(62\) 0 0
\(63\) 250047. 0.125988
\(64\) 0 0
\(65\) −329400. −0.148774
\(66\) 0 0
\(67\) −2.12318e6 −0.862431 −0.431215 0.902249i \(-0.641915\pi\)
−0.431215 + 0.902249i \(0.641915\pi\)
\(68\) 0 0
\(69\) −53514.0 −0.0196108
\(70\) 0 0
\(71\) 1.19121e6 0.394990 0.197495 0.980304i \(-0.436719\pi\)
0.197495 + 0.980304i \(0.436719\pi\)
\(72\) 0 0
\(73\) 1.05643e6 0.317842 0.158921 0.987291i \(-0.449199\pi\)
0.158921 + 0.987291i \(0.449199\pi\)
\(74\) 0 0
\(75\) −1.83938e6 −0.503449
\(76\) 0 0
\(77\) −951482. −0.237511
\(78\) 0 0
\(79\) −998484. −0.227849 −0.113924 0.993489i \(-0.536342\pi\)
−0.113924 + 0.993489i \(0.536342\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −3.89800e6 −0.748288 −0.374144 0.927371i \(-0.622063\pi\)
−0.374144 + 0.927371i \(0.622063\pi\)
\(84\) 0 0
\(85\) 590000. 0.104204
\(86\) 0 0
\(87\) −5.61886e6 −0.914810
\(88\) 0 0
\(89\) −4.62235e6 −0.695021 −0.347511 0.937676i \(-0.612973\pi\)
−0.347511 + 0.937676i \(0.612973\pi\)
\(90\) 0 0
\(91\) −1.12984e6 −0.157171
\(92\) 0 0
\(93\) 3.18038e6 0.410005
\(94\) 0 0
\(95\) −664400. −0.0795055
\(96\) 0 0
\(97\) 1.52877e7 1.70075 0.850377 0.526174i \(-0.176374\pi\)
0.850377 + 0.526174i \(0.176374\pi\)
\(98\) 0 0
\(99\) −2.02225e6 −0.209465
\(100\) 0 0
\(101\) 2.23869e6 0.216207 0.108103 0.994140i \(-0.465522\pi\)
0.108103 + 0.994140i \(0.465522\pi\)
\(102\) 0 0
\(103\) −1.24502e7 −1.12266 −0.561328 0.827593i \(-0.689709\pi\)
−0.561328 + 0.827593i \(0.689709\pi\)
\(104\) 0 0
\(105\) 926100. 0.0780720
\(106\) 0 0
\(107\) 7.58860e6 0.598851 0.299425 0.954120i \(-0.403205\pi\)
0.299425 + 0.954120i \(0.403205\pi\)
\(108\) 0 0
\(109\) 5.17097e6 0.382454 0.191227 0.981546i \(-0.438753\pi\)
0.191227 + 0.981546i \(0.438753\pi\)
\(110\) 0 0
\(111\) −9.06352e6 −0.629023
\(112\) 0 0
\(113\) −9.63868e6 −0.628410 −0.314205 0.949355i \(-0.601738\pi\)
−0.314205 + 0.949355i \(0.601738\pi\)
\(114\) 0 0
\(115\) −198200. −0.0121524
\(116\) 0 0
\(117\) −2.40133e6 −0.138612
\(118\) 0 0
\(119\) 2.02370e6 0.110086
\(120\) 0 0
\(121\) −1.17921e7 −0.605121
\(122\) 0 0
\(123\) −7.16818e6 −0.347329
\(124\) 0 0
\(125\) −1.46250e7 −0.669747
\(126\) 0 0
\(127\) −8.08309e6 −0.350158 −0.175079 0.984554i \(-0.556018\pi\)
−0.175079 + 0.984554i \(0.556018\pi\)
\(128\) 0 0
\(129\) 2.51888e6 0.103310
\(130\) 0 0
\(131\) 1.97585e7 0.767901 0.383951 0.923354i \(-0.374563\pi\)
0.383951 + 0.923354i \(0.374563\pi\)
\(132\) 0 0
\(133\) −2.27889e6 −0.0839930
\(134\) 0 0
\(135\) 1.96830e6 0.0688530
\(136\) 0 0
\(137\) −4.80896e7 −1.59783 −0.798913 0.601447i \(-0.794591\pi\)
−0.798913 + 0.601447i \(0.794591\pi\)
\(138\) 0 0
\(139\) −1.37173e7 −0.433229 −0.216615 0.976257i \(-0.569501\pi\)
−0.216615 + 0.976257i \(0.569501\pi\)
\(140\) 0 0
\(141\) 1.77529e7 0.533339
\(142\) 0 0
\(143\) 9.13756e6 0.261309
\(144\) 0 0
\(145\) −2.08106e7 −0.566887
\(146\) 0 0
\(147\) 3.17652e6 0.0824786
\(148\) 0 0
\(149\) −7.07104e7 −1.75118 −0.875591 0.483053i \(-0.839528\pi\)
−0.875591 + 0.483053i \(0.839528\pi\)
\(150\) 0 0
\(151\) 2.56970e7 0.607383 0.303691 0.952770i \(-0.401781\pi\)
0.303691 + 0.952770i \(0.401781\pi\)
\(152\) 0 0
\(153\) 4.30110e6 0.0970867
\(154\) 0 0
\(155\) 1.17792e7 0.254071
\(156\) 0 0
\(157\) 2.09180e7 0.431390 0.215695 0.976461i \(-0.430798\pi\)
0.215695 + 0.976461i \(0.430798\pi\)
\(158\) 0 0
\(159\) −1.64354e7 −0.324257
\(160\) 0 0
\(161\) −679826. −0.0128383
\(162\) 0 0
\(163\) −5.44160e7 −0.984169 −0.492084 0.870547i \(-0.663765\pi\)
−0.492084 + 0.870547i \(0.663765\pi\)
\(164\) 0 0
\(165\) −7.48980e6 −0.129801
\(166\) 0 0
\(167\) −6.41888e6 −0.106648 −0.0533238 0.998577i \(-0.516982\pi\)
−0.0533238 + 0.998577i \(0.516982\pi\)
\(168\) 0 0
\(169\) −5.18981e7 −0.827081
\(170\) 0 0
\(171\) −4.84348e6 −0.0740748
\(172\) 0 0
\(173\) −5.46532e7 −0.802517 −0.401259 0.915965i \(-0.631427\pi\)
−0.401259 + 0.915965i \(0.631427\pi\)
\(174\) 0 0
\(175\) −2.33669e7 −0.329585
\(176\) 0 0
\(177\) 1.44752e7 0.196209
\(178\) 0 0
\(179\) 3.76658e7 0.490865 0.245433 0.969414i \(-0.421070\pi\)
0.245433 + 0.969414i \(0.421070\pi\)
\(180\) 0 0
\(181\) 1.76788e7 0.221604 0.110802 0.993842i \(-0.464658\pi\)
0.110802 + 0.993842i \(0.464658\pi\)
\(182\) 0 0
\(183\) −4.85214e7 −0.585268
\(184\) 0 0
\(185\) −3.35686e7 −0.389791
\(186\) 0 0
\(187\) −1.63666e7 −0.183026
\(188\) 0 0
\(189\) 6.75127e6 0.0727393
\(190\) 0 0
\(191\) −4.09177e7 −0.424907 −0.212454 0.977171i \(-0.568145\pi\)
−0.212454 + 0.977171i \(0.568145\pi\)
\(192\) 0 0
\(193\) −1.63827e8 −1.64034 −0.820171 0.572118i \(-0.806122\pi\)
−0.820171 + 0.572118i \(0.806122\pi\)
\(194\) 0 0
\(195\) −8.89380e6 −0.0858947
\(196\) 0 0
\(197\) 8.02076e7 0.747453 0.373726 0.927539i \(-0.378080\pi\)
0.373726 + 0.927539i \(0.378080\pi\)
\(198\) 0 0
\(199\) −9.83219e7 −0.884432 −0.442216 0.896908i \(-0.645808\pi\)
−0.442216 + 0.896908i \(0.645808\pi\)
\(200\) 0 0
\(201\) −5.73258e7 −0.497925
\(202\) 0 0
\(203\) −7.13804e7 −0.598884
\(204\) 0 0
\(205\) −2.65488e7 −0.215232
\(206\) 0 0
\(207\) −1.44488e6 −0.0113223
\(208\) 0 0
\(209\) 1.84305e7 0.139645
\(210\) 0 0
\(211\) −1.36321e8 −0.999021 −0.499510 0.866308i \(-0.666487\pi\)
−0.499510 + 0.866308i \(0.666487\pi\)
\(212\) 0 0
\(213\) 3.21628e7 0.228047
\(214\) 0 0
\(215\) 9.32920e6 0.0640191
\(216\) 0 0
\(217\) 4.04027e7 0.268411
\(218\) 0 0
\(219\) 2.85236e7 0.183506
\(220\) 0 0
\(221\) −1.94346e7 −0.121116
\(222\) 0 0
\(223\) −1.26358e8 −0.763019 −0.381510 0.924365i \(-0.624596\pi\)
−0.381510 + 0.924365i \(0.624596\pi\)
\(224\) 0 0
\(225\) −4.96631e7 −0.290667
\(226\) 0 0
\(227\) −2.34007e8 −1.32782 −0.663909 0.747814i \(-0.731104\pi\)
−0.663909 + 0.747814i \(0.731104\pi\)
\(228\) 0 0
\(229\) 6.83606e7 0.376168 0.188084 0.982153i \(-0.439772\pi\)
0.188084 + 0.982153i \(0.439772\pi\)
\(230\) 0 0
\(231\) −2.56900e7 −0.137127
\(232\) 0 0
\(233\) 1.80385e8 0.934230 0.467115 0.884197i \(-0.345293\pi\)
0.467115 + 0.884197i \(0.345293\pi\)
\(234\) 0 0
\(235\) 6.57516e7 0.330498
\(236\) 0 0
\(237\) −2.69591e7 −0.131548
\(238\) 0 0
\(239\) −2.61131e8 −1.23727 −0.618637 0.785677i \(-0.712315\pi\)
−0.618637 + 0.785677i \(0.712315\pi\)
\(240\) 0 0
\(241\) 1.97756e7 0.0910061 0.0455031 0.998964i \(-0.485511\pi\)
0.0455031 + 0.998964i \(0.485511\pi\)
\(242\) 0 0
\(243\) 1.43489e7 0.0641500
\(244\) 0 0
\(245\) 1.17649e7 0.0511101
\(246\) 0 0
\(247\) 2.18853e7 0.0924089
\(248\) 0 0
\(249\) −1.05246e8 −0.432024
\(250\) 0 0
\(251\) −1.53770e8 −0.613779 −0.306890 0.951745i \(-0.599288\pi\)
−0.306890 + 0.951745i \(0.599288\pi\)
\(252\) 0 0
\(253\) 5.49807e6 0.0213446
\(254\) 0 0
\(255\) 1.59300e7 0.0601624
\(256\) 0 0
\(257\) −1.43360e8 −0.526821 −0.263411 0.964684i \(-0.584847\pi\)
−0.263411 + 0.964684i \(0.584847\pi\)
\(258\) 0 0
\(259\) −1.15140e8 −0.411792
\(260\) 0 0
\(261\) −1.51709e8 −0.528166
\(262\) 0 0
\(263\) 5.07301e8 1.71957 0.859786 0.510654i \(-0.170597\pi\)
0.859786 + 0.510654i \(0.170597\pi\)
\(264\) 0 0
\(265\) −6.08718e7 −0.200935
\(266\) 0 0
\(267\) −1.24804e8 −0.401271
\(268\) 0 0
\(269\) 5.27886e8 1.65351 0.826755 0.562562i \(-0.190184\pi\)
0.826755 + 0.562562i \(0.190184\pi\)
\(270\) 0 0
\(271\) 5.05836e8 1.54389 0.771947 0.635687i \(-0.219283\pi\)
0.771947 + 0.635687i \(0.219283\pi\)
\(272\) 0 0
\(273\) −3.05057e7 −0.0907428
\(274\) 0 0
\(275\) 1.88979e8 0.547960
\(276\) 0 0
\(277\) −5.88586e8 −1.66391 −0.831956 0.554841i \(-0.812779\pi\)
−0.831956 + 0.554841i \(0.812779\pi\)
\(278\) 0 0
\(279\) 8.58704e7 0.236717
\(280\) 0 0
\(281\) 8.65142e7 0.232603 0.116301 0.993214i \(-0.462896\pi\)
0.116301 + 0.993214i \(0.462896\pi\)
\(282\) 0 0
\(283\) 1.57301e8 0.412552 0.206276 0.978494i \(-0.433866\pi\)
0.206276 + 0.978494i \(0.433866\pi\)
\(284\) 0 0
\(285\) −1.79388e7 −0.0459025
\(286\) 0 0
\(287\) −9.10624e7 −0.227380
\(288\) 0 0
\(289\) −3.75529e8 −0.915168
\(290\) 0 0
\(291\) 4.12768e8 0.981930
\(292\) 0 0
\(293\) 6.80964e8 1.58157 0.790783 0.612097i \(-0.209674\pi\)
0.790783 + 0.612097i \(0.209674\pi\)
\(294\) 0 0
\(295\) 5.36120e7 0.121586
\(296\) 0 0
\(297\) −5.46006e7 −0.120934
\(298\) 0 0
\(299\) 6.52871e6 0.0141247
\(300\) 0 0
\(301\) 3.19992e7 0.0676326
\(302\) 0 0
\(303\) 6.04446e7 0.124827
\(304\) 0 0
\(305\) −1.79709e8 −0.362677
\(306\) 0 0
\(307\) −2.81734e8 −0.555718 −0.277859 0.960622i \(-0.589625\pi\)
−0.277859 + 0.960622i \(0.589625\pi\)
\(308\) 0 0
\(309\) −3.36156e8 −0.648166
\(310\) 0 0
\(311\) 4.19719e8 0.791220 0.395610 0.918419i \(-0.370533\pi\)
0.395610 + 0.918419i \(0.370533\pi\)
\(312\) 0 0
\(313\) −2.28684e8 −0.421532 −0.210766 0.977537i \(-0.567596\pi\)
−0.210766 + 0.977537i \(0.567596\pi\)
\(314\) 0 0
\(315\) 2.50047e7 0.0450749
\(316\) 0 0
\(317\) 2.66003e8 0.469006 0.234503 0.972115i \(-0.424654\pi\)
0.234503 + 0.972115i \(0.424654\pi\)
\(318\) 0 0
\(319\) 5.77286e8 0.995689
\(320\) 0 0
\(321\) 2.04892e8 0.345747
\(322\) 0 0
\(323\) −3.91996e7 −0.0647251
\(324\) 0 0
\(325\) 2.24404e8 0.362609
\(326\) 0 0
\(327\) 1.39616e8 0.220810
\(328\) 0 0
\(329\) 2.25528e8 0.349152
\(330\) 0 0
\(331\) 7.16107e8 1.08538 0.542688 0.839935i \(-0.317407\pi\)
0.542688 + 0.839935i \(0.317407\pi\)
\(332\) 0 0
\(333\) −2.44715e8 −0.363167
\(334\) 0 0
\(335\) −2.12318e8 −0.308553
\(336\) 0 0
\(337\) 5.70266e8 0.811657 0.405829 0.913949i \(-0.366983\pi\)
0.405829 + 0.913949i \(0.366983\pi\)
\(338\) 0 0
\(339\) −2.60244e8 −0.362813
\(340\) 0 0
\(341\) −3.26755e8 −0.446254
\(342\) 0 0
\(343\) 4.03536e7 0.0539949
\(344\) 0 0
\(345\) −5.35140e6 −0.00701618
\(346\) 0 0
\(347\) 1.64641e8 0.211536 0.105768 0.994391i \(-0.466270\pi\)
0.105768 + 0.994391i \(0.466270\pi\)
\(348\) 0 0
\(349\) 1.31564e9 1.65671 0.828357 0.560200i \(-0.189276\pi\)
0.828357 + 0.560200i \(0.189276\pi\)
\(350\) 0 0
\(351\) −6.48358e7 −0.0800276
\(352\) 0 0
\(353\) 1.56118e9 1.88904 0.944518 0.328459i \(-0.106529\pi\)
0.944518 + 0.328459i \(0.106529\pi\)
\(354\) 0 0
\(355\) 1.19121e8 0.141316
\(356\) 0 0
\(357\) 5.46399e7 0.0635581
\(358\) 0 0
\(359\) 8.60716e8 0.981814 0.490907 0.871212i \(-0.336665\pi\)
0.490907 + 0.871212i \(0.336665\pi\)
\(360\) 0 0
\(361\) −8.49729e8 −0.950616
\(362\) 0 0
\(363\) −3.18387e8 −0.349367
\(364\) 0 0
\(365\) 1.05643e8 0.113714
\(366\) 0 0
\(367\) −9.12133e8 −0.963223 −0.481612 0.876385i \(-0.659948\pi\)
−0.481612 + 0.876385i \(0.659948\pi\)
\(368\) 0 0
\(369\) −1.93541e8 −0.200530
\(370\) 0 0
\(371\) −2.08790e8 −0.212276
\(372\) 0 0
\(373\) 2.36004e8 0.235472 0.117736 0.993045i \(-0.462436\pi\)
0.117736 + 0.993045i \(0.462436\pi\)
\(374\) 0 0
\(375\) −3.94875e8 −0.386679
\(376\) 0 0
\(377\) 6.85501e8 0.658891
\(378\) 0 0
\(379\) −8.29313e8 −0.782495 −0.391247 0.920286i \(-0.627956\pi\)
−0.391247 + 0.920286i \(0.627956\pi\)
\(380\) 0 0
\(381\) −2.18243e8 −0.202164
\(382\) 0 0
\(383\) 1.21663e9 1.10653 0.553265 0.833005i \(-0.313382\pi\)
0.553265 + 0.833005i \(0.313382\pi\)
\(384\) 0 0
\(385\) −9.51482e7 −0.0849744
\(386\) 0 0
\(387\) 6.80099e7 0.0596463
\(388\) 0 0
\(389\) −1.18213e8 −0.101822 −0.0509109 0.998703i \(-0.516212\pi\)
−0.0509109 + 0.998703i \(0.516212\pi\)
\(390\) 0 0
\(391\) −1.16938e7 −0.00989321
\(392\) 0 0
\(393\) 5.33481e8 0.443348
\(394\) 0 0
\(395\) −9.98484e7 −0.0815176
\(396\) 0 0
\(397\) 3.62565e7 0.0290817 0.0145408 0.999894i \(-0.495371\pi\)
0.0145408 + 0.999894i \(0.495371\pi\)
\(398\) 0 0
\(399\) −6.15301e7 −0.0484934
\(400\) 0 0
\(401\) 5.06850e8 0.392531 0.196265 0.980551i \(-0.437119\pi\)
0.196265 + 0.980551i \(0.437119\pi\)
\(402\) 0 0
\(403\) −3.88007e8 −0.295306
\(404\) 0 0
\(405\) 5.31441e7 0.0397523
\(406\) 0 0
\(407\) 9.31193e8 0.684635
\(408\) 0 0
\(409\) 1.68121e9 1.21504 0.607519 0.794305i \(-0.292165\pi\)
0.607519 + 0.794305i \(0.292165\pi\)
\(410\) 0 0
\(411\) −1.29842e9 −0.922505
\(412\) 0 0
\(413\) 1.83889e8 0.128449
\(414\) 0 0
\(415\) −3.89800e8 −0.267716
\(416\) 0 0
\(417\) −3.70368e8 −0.250125
\(418\) 0 0
\(419\) −4.94962e8 −0.328718 −0.164359 0.986401i \(-0.552556\pi\)
−0.164359 + 0.986401i \(0.552556\pi\)
\(420\) 0 0
\(421\) −6.57487e7 −0.0429437 −0.0214719 0.999769i \(-0.506835\pi\)
−0.0214719 + 0.999769i \(0.506835\pi\)
\(422\) 0 0
\(423\) 4.79329e8 0.307923
\(424\) 0 0
\(425\) −4.01938e8 −0.253979
\(426\) 0 0
\(427\) −6.16402e8 −0.383148
\(428\) 0 0
\(429\) 2.46714e8 0.150867
\(430\) 0 0
\(431\) 1.33418e9 0.802682 0.401341 0.915929i \(-0.368544\pi\)
0.401341 + 0.915929i \(0.368544\pi\)
\(432\) 0 0
\(433\) −2.47903e7 −0.0146749 −0.00733745 0.999973i \(-0.502336\pi\)
−0.00733745 + 0.999973i \(0.502336\pi\)
\(434\) 0 0
\(435\) −5.61886e8 −0.327292
\(436\) 0 0
\(437\) 1.31684e7 0.00754828
\(438\) 0 0
\(439\) 6.89327e8 0.388866 0.194433 0.980916i \(-0.437713\pi\)
0.194433 + 0.980916i \(0.437713\pi\)
\(440\) 0 0
\(441\) 8.57661e7 0.0476190
\(442\) 0 0
\(443\) 5.74617e8 0.314026 0.157013 0.987597i \(-0.449814\pi\)
0.157013 + 0.987597i \(0.449814\pi\)
\(444\) 0 0
\(445\) −4.62235e8 −0.248658
\(446\) 0 0
\(447\) −1.90918e9 −1.01105
\(448\) 0 0
\(449\) 9.56884e7 0.0498881 0.0249440 0.999689i \(-0.492059\pi\)
0.0249440 + 0.999689i \(0.492059\pi\)
\(450\) 0 0
\(451\) 7.36464e8 0.378036
\(452\) 0 0
\(453\) 6.93818e8 0.350672
\(454\) 0 0
\(455\) −1.12984e8 −0.0562313
\(456\) 0 0
\(457\) −4.73089e7 −0.0231866 −0.0115933 0.999933i \(-0.503690\pi\)
−0.0115933 + 0.999933i \(0.503690\pi\)
\(458\) 0 0
\(459\) 1.16130e8 0.0560530
\(460\) 0 0
\(461\) −2.13567e9 −1.01527 −0.507634 0.861573i \(-0.669480\pi\)
−0.507634 + 0.861573i \(0.669480\pi\)
\(462\) 0 0
\(463\) −2.92675e8 −0.137042 −0.0685208 0.997650i \(-0.521828\pi\)
−0.0685208 + 0.997650i \(0.521828\pi\)
\(464\) 0 0
\(465\) 3.18038e8 0.146688
\(466\) 0 0
\(467\) 3.53834e9 1.60765 0.803824 0.594868i \(-0.202796\pi\)
0.803824 + 0.594868i \(0.202796\pi\)
\(468\) 0 0
\(469\) −7.28249e8 −0.325968
\(470\) 0 0
\(471\) 5.64785e8 0.249063
\(472\) 0 0
\(473\) −2.58792e8 −0.112444
\(474\) 0 0
\(475\) 4.52622e8 0.193780
\(476\) 0 0
\(477\) −4.43755e8 −0.187210
\(478\) 0 0
\(479\) 4.15698e9 1.72824 0.864120 0.503286i \(-0.167876\pi\)
0.864120 + 0.503286i \(0.167876\pi\)
\(480\) 0 0
\(481\) 1.10575e9 0.453053
\(482\) 0 0
\(483\) −1.83553e7 −0.00741219
\(484\) 0 0
\(485\) 1.52877e9 0.608480
\(486\) 0 0
\(487\) −1.24092e8 −0.0486847 −0.0243423 0.999704i \(-0.507749\pi\)
−0.0243423 + 0.999704i \(0.507749\pi\)
\(488\) 0 0
\(489\) −1.46923e9 −0.568210
\(490\) 0 0
\(491\) 3.29218e9 1.25516 0.627578 0.778554i \(-0.284046\pi\)
0.627578 + 0.778554i \(0.284046\pi\)
\(492\) 0 0
\(493\) −1.22783e9 −0.461501
\(494\) 0 0
\(495\) −2.02225e8 −0.0749404
\(496\) 0 0
\(497\) 4.08586e8 0.149292
\(498\) 0 0
\(499\) 3.60439e8 0.129861 0.0649306 0.997890i \(-0.479317\pi\)
0.0649306 + 0.997890i \(0.479317\pi\)
\(500\) 0 0
\(501\) −1.73310e8 −0.0615731
\(502\) 0 0
\(503\) 4.29760e9 1.50570 0.752849 0.658193i \(-0.228679\pi\)
0.752849 + 0.658193i \(0.228679\pi\)
\(504\) 0 0
\(505\) 2.23869e8 0.0773524
\(506\) 0 0
\(507\) −1.40125e9 −0.477515
\(508\) 0 0
\(509\) −5.42588e8 −0.182372 −0.0911860 0.995834i \(-0.529066\pi\)
−0.0911860 + 0.995834i \(0.529066\pi\)
\(510\) 0 0
\(511\) 3.62355e8 0.120133
\(512\) 0 0
\(513\) −1.30774e8 −0.0427671
\(514\) 0 0
\(515\) −1.24502e9 −0.401654
\(516\) 0 0
\(517\) −1.82395e9 −0.580492
\(518\) 0 0
\(519\) −1.47564e9 −0.463334
\(520\) 0 0
\(521\) −2.36755e8 −0.0733445 −0.0366722 0.999327i \(-0.511676\pi\)
−0.0366722 + 0.999327i \(0.511676\pi\)
\(522\) 0 0
\(523\) −6.48324e9 −1.98169 −0.990846 0.135000i \(-0.956896\pi\)
−0.990846 + 0.135000i \(0.956896\pi\)
\(524\) 0 0
\(525\) −6.30906e8 −0.190286
\(526\) 0 0
\(527\) 6.94973e8 0.206838
\(528\) 0 0
\(529\) −3.40090e9 −0.998846
\(530\) 0 0
\(531\) 3.90831e8 0.113281
\(532\) 0 0
\(533\) 8.74517e8 0.250163
\(534\) 0 0
\(535\) 7.58860e8 0.214251
\(536\) 0 0
\(537\) 1.01698e9 0.283401
\(538\) 0 0
\(539\) −3.26358e8 −0.0897706
\(540\) 0 0
\(541\) −1.50099e8 −0.0407556 −0.0203778 0.999792i \(-0.506487\pi\)
−0.0203778 + 0.999792i \(0.506487\pi\)
\(542\) 0 0
\(543\) 4.77328e8 0.127943
\(544\) 0 0
\(545\) 5.17097e8 0.136831
\(546\) 0 0
\(547\) −3.32631e9 −0.868974 −0.434487 0.900678i \(-0.643070\pi\)
−0.434487 + 0.900678i \(0.643070\pi\)
\(548\) 0 0
\(549\) −1.31008e9 −0.337905
\(550\) 0 0
\(551\) 1.38266e9 0.352114
\(552\) 0 0
\(553\) −3.42480e8 −0.0861187
\(554\) 0 0
\(555\) −9.06352e8 −0.225046
\(556\) 0 0
\(557\) 4.72512e9 1.15856 0.579282 0.815127i \(-0.303333\pi\)
0.579282 + 0.815127i \(0.303333\pi\)
\(558\) 0 0
\(559\) −3.07304e8 −0.0744092
\(560\) 0 0
\(561\) −4.41898e8 −0.105670
\(562\) 0 0
\(563\) −7.11044e9 −1.67926 −0.839628 0.543162i \(-0.817227\pi\)
−0.839628 + 0.543162i \(0.817227\pi\)
\(564\) 0 0
\(565\) −9.63868e8 −0.224827
\(566\) 0 0
\(567\) 1.82284e8 0.0419961
\(568\) 0 0
\(569\) 6.55315e9 1.49127 0.745636 0.666353i \(-0.232146\pi\)
0.745636 + 0.666353i \(0.232146\pi\)
\(570\) 0 0
\(571\) −8.29277e9 −1.86412 −0.932059 0.362306i \(-0.881990\pi\)
−0.932059 + 0.362306i \(0.881990\pi\)
\(572\) 0 0
\(573\) −1.10478e9 −0.245320
\(574\) 0 0
\(575\) 1.35024e8 0.0296192
\(576\) 0 0
\(577\) −5.88000e9 −1.27427 −0.637135 0.770752i \(-0.719881\pi\)
−0.637135 + 0.770752i \(0.719881\pi\)
\(578\) 0 0
\(579\) −4.42333e9 −0.947052
\(580\) 0 0
\(581\) −1.33702e9 −0.282826
\(582\) 0 0
\(583\) 1.68858e9 0.352925
\(584\) 0 0
\(585\) −2.40133e8 −0.0495913
\(586\) 0 0
\(587\) 1.43194e9 0.292207 0.146103 0.989269i \(-0.453327\pi\)
0.146103 + 0.989269i \(0.453327\pi\)
\(588\) 0 0
\(589\) −7.82610e8 −0.157813
\(590\) 0 0
\(591\) 2.16561e9 0.431542
\(592\) 0 0
\(593\) 3.48138e9 0.685584 0.342792 0.939411i \(-0.388627\pi\)
0.342792 + 0.939411i \(0.388627\pi\)
\(594\) 0 0
\(595\) 2.02370e8 0.0393855
\(596\) 0 0
\(597\) −2.65469e9 −0.510627
\(598\) 0 0
\(599\) −7.93597e9 −1.50871 −0.754356 0.656466i \(-0.772051\pi\)
−0.754356 + 0.656466i \(0.772051\pi\)
\(600\) 0 0
\(601\) −4.05169e9 −0.761335 −0.380668 0.924712i \(-0.624306\pi\)
−0.380668 + 0.924712i \(0.624306\pi\)
\(602\) 0 0
\(603\) −1.54780e9 −0.287477
\(604\) 0 0
\(605\) −1.17921e9 −0.216495
\(606\) 0 0
\(607\) −5.19159e9 −0.942194 −0.471097 0.882081i \(-0.656142\pi\)
−0.471097 + 0.882081i \(0.656142\pi\)
\(608\) 0 0
\(609\) −1.92727e9 −0.345766
\(610\) 0 0
\(611\) −2.16586e9 −0.384137
\(612\) 0 0
\(613\) 3.69807e9 0.648430 0.324215 0.945983i \(-0.394900\pi\)
0.324215 + 0.945983i \(0.394900\pi\)
\(614\) 0 0
\(615\) −7.16818e8 −0.124264
\(616\) 0 0
\(617\) −6.19879e9 −1.06245 −0.531226 0.847230i \(-0.678268\pi\)
−0.531226 + 0.847230i \(0.678268\pi\)
\(618\) 0 0
\(619\) 2.85137e9 0.483211 0.241605 0.970375i \(-0.422326\pi\)
0.241605 + 0.970375i \(0.422326\pi\)
\(620\) 0 0
\(621\) −3.90117e7 −0.00653694
\(622\) 0 0
\(623\) −1.58547e9 −0.262693
\(624\) 0 0
\(625\) 3.85977e9 0.632384
\(626\) 0 0
\(627\) 4.97622e8 0.0806238
\(628\) 0 0
\(629\) −1.98055e9 −0.317328
\(630\) 0 0
\(631\) 7.56414e9 1.19855 0.599276 0.800543i \(-0.295455\pi\)
0.599276 + 0.800543i \(0.295455\pi\)
\(632\) 0 0
\(633\) −3.68067e9 −0.576785
\(634\) 0 0
\(635\) −8.08309e8 −0.125276
\(636\) 0 0
\(637\) −3.87536e8 −0.0594051
\(638\) 0 0
\(639\) 8.68395e8 0.131663
\(640\) 0 0
\(641\) 1.12019e10 1.67991 0.839957 0.542653i \(-0.182580\pi\)
0.839957 + 0.542653i \(0.182580\pi\)
\(642\) 0 0
\(643\) −7.87742e9 −1.16855 −0.584273 0.811557i \(-0.698620\pi\)
−0.584273 + 0.811557i \(0.698620\pi\)
\(644\) 0 0
\(645\) 2.51888e8 0.0369615
\(646\) 0 0
\(647\) −1.24945e9 −0.181365 −0.0906825 0.995880i \(-0.528905\pi\)
−0.0906825 + 0.995880i \(0.528905\pi\)
\(648\) 0 0
\(649\) −1.48720e9 −0.213556
\(650\) 0 0
\(651\) 1.09087e9 0.154967
\(652\) 0 0
\(653\) 7.52696e9 1.05785 0.528924 0.848669i \(-0.322596\pi\)
0.528924 + 0.848669i \(0.322596\pi\)
\(654\) 0 0
\(655\) 1.97585e9 0.274733
\(656\) 0 0
\(657\) 7.70137e8 0.105947
\(658\) 0 0
\(659\) −2.29417e9 −0.312268 −0.156134 0.987736i \(-0.549903\pi\)
−0.156134 + 0.987736i \(0.549903\pi\)
\(660\) 0 0
\(661\) −5.08384e9 −0.684679 −0.342339 0.939576i \(-0.611219\pi\)
−0.342339 + 0.939576i \(0.611219\pi\)
\(662\) 0 0
\(663\) −5.24734e8 −0.0699265
\(664\) 0 0
\(665\) −2.27889e8 −0.0300502
\(666\) 0 0
\(667\) 4.12466e8 0.0538205
\(668\) 0 0
\(669\) −3.41166e9 −0.440529
\(670\) 0 0
\(671\) 4.98513e9 0.637012
\(672\) 0 0
\(673\) −5.62649e9 −0.711516 −0.355758 0.934578i \(-0.615777\pi\)
−0.355758 + 0.934578i \(0.615777\pi\)
\(674\) 0 0
\(675\) −1.34090e9 −0.167816
\(676\) 0 0
\(677\) −1.02745e10 −1.27263 −0.636314 0.771430i \(-0.719542\pi\)
−0.636314 + 0.771430i \(0.719542\pi\)
\(678\) 0 0
\(679\) 5.24368e9 0.642824
\(680\) 0 0
\(681\) −6.31819e9 −0.766616
\(682\) 0 0
\(683\) −1.55246e10 −1.86443 −0.932217 0.361901i \(-0.882128\pi\)
−0.932217 + 0.361901i \(0.882128\pi\)
\(684\) 0 0
\(685\) −4.80896e9 −0.571655
\(686\) 0 0
\(687\) 1.84574e9 0.217181
\(688\) 0 0
\(689\) 2.00512e9 0.233546
\(690\) 0 0
\(691\) −8.23449e9 −0.949432 −0.474716 0.880139i \(-0.657449\pi\)
−0.474716 + 0.880139i \(0.657449\pi\)
\(692\) 0 0
\(693\) −6.93630e8 −0.0791702
\(694\) 0 0
\(695\) −1.37173e9 −0.154997
\(696\) 0 0
\(697\) −1.56638e9 −0.175219
\(698\) 0 0
\(699\) 4.87039e9 0.539378
\(700\) 0 0
\(701\) −4.88820e9 −0.535964 −0.267982 0.963424i \(-0.586357\pi\)
−0.267982 + 0.963424i \(0.586357\pi\)
\(702\) 0 0
\(703\) 2.23030e9 0.242114
\(704\) 0 0
\(705\) 1.77529e9 0.190813
\(706\) 0 0
\(707\) 7.67870e8 0.0817184
\(708\) 0 0
\(709\) −7.95106e9 −0.837844 −0.418922 0.908022i \(-0.637592\pi\)
−0.418922 + 0.908022i \(0.637592\pi\)
\(710\) 0 0
\(711\) −7.27895e8 −0.0759495
\(712\) 0 0
\(713\) −2.33464e8 −0.0241216
\(714\) 0 0
\(715\) 9.13756e8 0.0934887
\(716\) 0 0
\(717\) −7.05053e9 −0.714340
\(718\) 0 0
\(719\) 1.30035e10 1.30469 0.652346 0.757921i \(-0.273785\pi\)
0.652346 + 0.757921i \(0.273785\pi\)
\(720\) 0 0
\(721\) −4.27043e9 −0.424324
\(722\) 0 0
\(723\) 5.33942e8 0.0525424
\(724\) 0 0
\(725\) 1.41772e10 1.38168
\(726\) 0 0
\(727\) 1.73805e10 1.67761 0.838805 0.544431i \(-0.183255\pi\)
0.838805 + 0.544431i \(0.183255\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) 5.50423e8 0.0521177
\(732\) 0 0
\(733\) 9.47114e9 0.888256 0.444128 0.895963i \(-0.353513\pi\)
0.444128 + 0.895963i \(0.353513\pi\)
\(734\) 0 0
\(735\) 3.17652e8 0.0295084
\(736\) 0 0
\(737\) 5.88969e9 0.541946
\(738\) 0 0
\(739\) −5.52914e9 −0.503967 −0.251983 0.967732i \(-0.581083\pi\)
−0.251983 + 0.967732i \(0.581083\pi\)
\(740\) 0 0
\(741\) 5.90904e8 0.0533523
\(742\) 0 0
\(743\) 1.53701e10 1.37472 0.687362 0.726315i \(-0.258769\pi\)
0.687362 + 0.726315i \(0.258769\pi\)
\(744\) 0 0
\(745\) −7.07104e9 −0.626522
\(746\) 0 0
\(747\) −2.84164e9 −0.249429
\(748\) 0 0
\(749\) 2.60289e9 0.226344
\(750\) 0 0
\(751\) 8.51950e9 0.733964 0.366982 0.930228i \(-0.380391\pi\)
0.366982 + 0.930228i \(0.380391\pi\)
\(752\) 0 0
\(753\) −4.15178e9 −0.354366
\(754\) 0 0
\(755\) 2.56970e9 0.217304
\(756\) 0 0
\(757\) −4.72648e8 −0.0396006 −0.0198003 0.999804i \(-0.506303\pi\)
−0.0198003 + 0.999804i \(0.506303\pi\)
\(758\) 0 0
\(759\) 1.48448e8 0.0123233
\(760\) 0 0
\(761\) −1.43465e10 −1.18005 −0.590025 0.807385i \(-0.700882\pi\)
−0.590025 + 0.807385i \(0.700882\pi\)
\(762\) 0 0
\(763\) 1.77364e9 0.144554
\(764\) 0 0
\(765\) 4.30110e8 0.0347348
\(766\) 0 0
\(767\) −1.76598e9 −0.141319
\(768\) 0 0
\(769\) 1.39480e10 1.10603 0.553017 0.833170i \(-0.313476\pi\)
0.553017 + 0.833170i \(0.313476\pi\)
\(770\) 0 0
\(771\) −3.87073e9 −0.304160
\(772\) 0 0
\(773\) −4.32337e9 −0.336662 −0.168331 0.985731i \(-0.553838\pi\)
−0.168331 + 0.985731i \(0.553838\pi\)
\(774\) 0 0
\(775\) −8.02458e9 −0.619250
\(776\) 0 0
\(777\) −3.10879e9 −0.237748
\(778\) 0 0
\(779\) 1.76390e9 0.133688
\(780\) 0 0
\(781\) −3.30443e9 −0.248209
\(782\) 0 0
\(783\) −4.09615e9 −0.304937
\(784\) 0 0
\(785\) 2.09180e9 0.154339
\(786\) 0 0
\(787\) −6.71201e9 −0.490842 −0.245421 0.969417i \(-0.578926\pi\)
−0.245421 + 0.969417i \(0.578926\pi\)
\(788\) 0 0
\(789\) 1.36971e10 0.992796
\(790\) 0 0
\(791\) −3.30607e9 −0.237517
\(792\) 0 0
\(793\) 5.91961e9 0.421539
\(794\) 0 0
\(795\) −1.64354e9 −0.116010
\(796\) 0 0
\(797\) 5.61333e9 0.392750 0.196375 0.980529i \(-0.437083\pi\)
0.196375 + 0.980529i \(0.437083\pi\)
\(798\) 0 0
\(799\) 3.87934e9 0.269057
\(800\) 0 0
\(801\) −3.36969e9 −0.231674
\(802\) 0 0
\(803\) −2.93054e9 −0.199730
\(804\) 0 0
\(805\) −6.79826e7 −0.00459317
\(806\) 0 0
\(807\) 1.42529e10 0.954655
\(808\) 0 0
\(809\) 4.15844e9 0.276128 0.138064 0.990423i \(-0.455912\pi\)
0.138064 + 0.990423i \(0.455912\pi\)
\(810\) 0 0
\(811\) 1.11994e10 0.737259 0.368629 0.929576i \(-0.379827\pi\)
0.368629 + 0.929576i \(0.379827\pi\)
\(812\) 0 0
\(813\) 1.36576e10 0.891367
\(814\) 0 0
\(815\) −5.44160e9 −0.352107
\(816\) 0 0
\(817\) −6.19832e8 −0.0397646
\(818\) 0 0
\(819\) −8.23655e8 −0.0523904
\(820\) 0 0
\(821\) 2.32872e10 1.46864 0.734322 0.678801i \(-0.237500\pi\)
0.734322 + 0.678801i \(0.237500\pi\)
\(822\) 0 0
\(823\) −9.31943e9 −0.582761 −0.291380 0.956607i \(-0.594114\pi\)
−0.291380 + 0.956607i \(0.594114\pi\)
\(824\) 0 0
\(825\) 5.10243e9 0.316365
\(826\) 0 0
\(827\) 1.16608e10 0.716902 0.358451 0.933548i \(-0.383305\pi\)
0.358451 + 0.933548i \(0.383305\pi\)
\(828\) 0 0
\(829\) −2.31007e8 −0.0140826 −0.00704132 0.999975i \(-0.502241\pi\)
−0.00704132 + 0.999975i \(0.502241\pi\)
\(830\) 0 0
\(831\) −1.58918e10 −0.960660
\(832\) 0 0
\(833\) 6.94129e8 0.0416086
\(834\) 0 0
\(835\) −6.41888e8 −0.0381554
\(836\) 0 0
\(837\) 2.31850e9 0.136668
\(838\) 0 0
\(839\) 2.77032e10 1.61943 0.809716 0.586822i \(-0.199621\pi\)
0.809716 + 0.586822i \(0.199621\pi\)
\(840\) 0 0
\(841\) 2.60582e10 1.51063
\(842\) 0 0
\(843\) 2.33588e9 0.134293
\(844\) 0 0
\(845\) −5.18981e9 −0.295905
\(846\) 0 0
\(847\) −4.04469e9 −0.228714
\(848\) 0 0
\(849\) 4.24712e9 0.238187
\(850\) 0 0
\(851\) 6.65330e8 0.0370070
\(852\) 0 0
\(853\) 2.87010e10 1.58335 0.791673 0.610944i \(-0.209210\pi\)
0.791673 + 0.610944i \(0.209210\pi\)
\(854\) 0 0
\(855\) −4.84348e8 −0.0265018
\(856\) 0 0
\(857\) 1.44528e9 0.0784366 0.0392183 0.999231i \(-0.487513\pi\)
0.0392183 + 0.999231i \(0.487513\pi\)
\(858\) 0 0
\(859\) −2.94906e10 −1.58748 −0.793738 0.608260i \(-0.791868\pi\)
−0.793738 + 0.608260i \(0.791868\pi\)
\(860\) 0 0
\(861\) −2.45868e9 −0.131278
\(862\) 0 0
\(863\) −8.87258e9 −0.469907 −0.234954 0.972007i \(-0.575494\pi\)
−0.234954 + 0.972007i \(0.575494\pi\)
\(864\) 0 0
\(865\) −5.46532e9 −0.287117
\(866\) 0 0
\(867\) −1.01393e10 −0.528372
\(868\) 0 0
\(869\) 2.76979e9 0.143179
\(870\) 0 0
\(871\) 6.99374e9 0.358630
\(872\) 0 0
\(873\) 1.11447e10 0.566918
\(874\) 0 0
\(875\) −5.01638e9 −0.253141
\(876\) 0 0
\(877\) 1.57308e10 0.787504 0.393752 0.919217i \(-0.371177\pi\)
0.393752 + 0.919217i \(0.371177\pi\)
\(878\) 0 0
\(879\) 1.83860e10 0.913117
\(880\) 0 0
\(881\) −9.12522e9 −0.449602 −0.224801 0.974405i \(-0.572173\pi\)
−0.224801 + 0.974405i \(0.572173\pi\)
\(882\) 0 0
\(883\) −2.24538e10 −1.09756 −0.548778 0.835968i \(-0.684907\pi\)
−0.548778 + 0.835968i \(0.684907\pi\)
\(884\) 0 0
\(885\) 1.44752e9 0.0701979
\(886\) 0 0
\(887\) 3.43569e10 1.65303 0.826516 0.562913i \(-0.190320\pi\)
0.826516 + 0.562913i \(0.190320\pi\)
\(888\) 0 0
\(889\) −2.77250e9 −0.132347
\(890\) 0 0
\(891\) −1.47422e9 −0.0698216
\(892\) 0 0
\(893\) −4.36854e9 −0.205284
\(894\) 0 0
\(895\) 3.76658e9 0.175617
\(896\) 0 0
\(897\) 1.76275e8 0.00815488
\(898\) 0 0
\(899\) −2.45132e10 −1.12523
\(900\) 0 0
\(901\) −3.59144e9 −0.163580
\(902\) 0 0
\(903\) 8.63977e8 0.0390477
\(904\) 0 0
\(905\) 1.76788e9 0.0792835
\(906\) 0 0
\(907\) 4.01062e10 1.78478 0.892392 0.451261i \(-0.149026\pi\)
0.892392 + 0.451261i \(0.149026\pi\)
\(908\) 0 0
\(909\) 1.63200e9 0.0720688
\(910\) 0 0
\(911\) 3.70230e10 1.62240 0.811200 0.584769i \(-0.198815\pi\)
0.811200 + 0.584769i \(0.198815\pi\)
\(912\) 0 0
\(913\) 1.08131e10 0.470220
\(914\) 0 0
\(915\) −4.85214e9 −0.209392
\(916\) 0 0
\(917\) 6.77718e9 0.290239
\(918\) 0 0
\(919\) 3.37645e10 1.43501 0.717507 0.696551i \(-0.245283\pi\)
0.717507 + 0.696551i \(0.245283\pi\)
\(920\) 0 0
\(921\) −7.60681e9 −0.320844
\(922\) 0 0
\(923\) −3.92386e9 −0.164251
\(924\) 0 0
\(925\) 2.28686e10 0.950044
\(926\) 0 0
\(927\) −9.07621e9 −0.374219
\(928\) 0 0
\(929\) −4.77100e10 −1.95233 −0.976167 0.217019i \(-0.930367\pi\)
−0.976167 + 0.217019i \(0.930367\pi\)
\(930\) 0 0
\(931\) −7.81660e8 −0.0317464
\(932\) 0 0
\(933\) 1.13324e10 0.456811
\(934\) 0 0
\(935\) −1.63666e9 −0.0654814
\(936\) 0 0
\(937\) 3.86353e8 0.0153425 0.00767125 0.999971i \(-0.497558\pi\)
0.00767125 + 0.999971i \(0.497558\pi\)
\(938\) 0 0
\(939\) −6.17447e9 −0.243372
\(940\) 0 0
\(941\) −2.79446e10 −1.09329 −0.546643 0.837366i \(-0.684095\pi\)
−0.546643 + 0.837366i \(0.684095\pi\)
\(942\) 0 0
\(943\) 5.26197e8 0.0204342
\(944\) 0 0
\(945\) 6.75127e8 0.0260240
\(946\) 0 0
\(947\) 5.50921e9 0.210797 0.105399 0.994430i \(-0.466388\pi\)
0.105399 + 0.994430i \(0.466388\pi\)
\(948\) 0 0
\(949\) −3.47988e9 −0.132170
\(950\) 0 0
\(951\) 7.18207e9 0.270781
\(952\) 0 0
\(953\) −7.40406e9 −0.277105 −0.138553 0.990355i \(-0.544245\pi\)
−0.138553 + 0.990355i \(0.544245\pi\)
\(954\) 0 0
\(955\) −4.09177e9 −0.152019
\(956\) 0 0
\(957\) 1.55867e10 0.574861
\(958\) 0 0
\(959\) −1.64947e10 −0.603921
\(960\) 0 0
\(961\) −1.36377e10 −0.495687
\(962\) 0 0
\(963\) 5.53209e9 0.199617
\(964\) 0 0
\(965\) −1.63827e10 −0.586867
\(966\) 0 0
\(967\) −1.97731e10 −0.703204 −0.351602 0.936149i \(-0.614363\pi\)
−0.351602 + 0.936149i \(0.614363\pi\)
\(968\) 0 0
\(969\) −1.05839e9 −0.0373691
\(970\) 0 0
\(971\) 2.81294e10 0.986037 0.493018 0.870019i \(-0.335894\pi\)
0.493018 + 0.870019i \(0.335894\pi\)
\(972\) 0 0
\(973\) −4.70504e9 −0.163745
\(974\) 0 0
\(975\) 6.05890e9 0.209352
\(976\) 0 0
\(977\) 9.63337e8 0.0330482 0.0165241 0.999863i \(-0.494740\pi\)
0.0165241 + 0.999863i \(0.494740\pi\)
\(978\) 0 0
\(979\) 1.28224e10 0.436747
\(980\) 0 0
\(981\) 3.76963e9 0.127485
\(982\) 0 0
\(983\) −4.20434e10 −1.41176 −0.705879 0.708332i \(-0.749448\pi\)
−0.705879 + 0.708332i \(0.749448\pi\)
\(984\) 0 0
\(985\) 8.02076e9 0.267417
\(986\) 0 0
\(987\) 6.08926e9 0.201583
\(988\) 0 0
\(989\) −1.84905e8 −0.00607800
\(990\) 0 0
\(991\) 2.87955e10 0.939869 0.469934 0.882701i \(-0.344278\pi\)
0.469934 + 0.882701i \(0.344278\pi\)
\(992\) 0 0
\(993\) 1.93349e10 0.626642
\(994\) 0 0
\(995\) −9.83219e9 −0.316424
\(996\) 0 0
\(997\) −3.54828e10 −1.13393 −0.566963 0.823743i \(-0.691882\pi\)
−0.566963 + 0.823743i \(0.691882\pi\)
\(998\) 0 0
\(999\) −6.60731e9 −0.209674
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 336.8.a.j.1.1 1
4.3 odd 2 84.8.a.a.1.1 1
12.11 even 2 252.8.a.a.1.1 1
28.3 even 6 588.8.i.c.373.1 2
28.11 odd 6 588.8.i.f.373.1 2
28.19 even 6 588.8.i.c.361.1 2
28.23 odd 6 588.8.i.f.361.1 2
28.27 even 2 588.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.a.1.1 1 4.3 odd 2
252.8.a.a.1.1 1 12.11 even 2
336.8.a.j.1.1 1 1.1 even 1 trivial
588.8.a.c.1.1 1 28.27 even 2
588.8.i.c.361.1 2 28.19 even 6
588.8.i.c.373.1 2 28.3 even 6
588.8.i.f.361.1 2 28.23 odd 6
588.8.i.f.373.1 2 28.11 odd 6