Properties

Label 208.10.a.b.1.1
Level $208$
Weight $10$
Character 208.1
Self dual yes
Analytic conductor $107.127$
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] = [208,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(107.127453922\)
Analytic rank: \(1\)
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)
 
\(f(q)\) \(=\) \(q-75.0000 q^{3} -1979.00 q^{5} +10115.0 q^{7} -14058.0 q^{9} +O(q^{10})\) \(q-75.0000 q^{3} -1979.00 q^{5} +10115.0 q^{7} -14058.0 q^{9} -18850.0 q^{11} +28561.0 q^{13} +148425. q^{15} -142403. q^{17} -83302.0 q^{19} -758625. q^{21} +536544. q^{23} +1.96332e6 q^{25} +2.53058e6 q^{27} -2.60044e6 q^{29} +2.21400e6 q^{31} +1.41375e6 q^{33} -2.00176e7 q^{35} +1.80992e7 q^{37} -2.14208e6 q^{39} +2.68122e7 q^{41} +4.22535e7 q^{43} +2.78208e7 q^{45} -3.59150e7 q^{47} +6.19596e7 q^{49} +1.06802e7 q^{51} -6.65141e7 q^{53} +3.73042e7 q^{55} +6.24765e6 q^{57} +1.08164e8 q^{59} -2.07450e8 q^{61} -1.42197e8 q^{63} -5.65222e7 q^{65} -1.93016e8 q^{67} -4.02408e7 q^{69} +2.01833e8 q^{71} -1.21628e8 q^{73} -1.47249e8 q^{75} -1.90668e8 q^{77} -1.12872e8 q^{79} +8.69105e7 q^{81} -3.08254e8 q^{83} +2.81816e8 q^{85} +1.95033e8 q^{87} -6.37487e6 q^{89} +2.88895e8 q^{91} -1.66050e8 q^{93} +1.64855e8 q^{95} +8.71267e8 q^{97} +2.64993e8 q^{99} +O(q^{100})\)

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
\(3\) −75.0000 −0.534584 −0.267292 0.963616i \(-0.586129\pi\)
−0.267292 + 0.963616i \(0.586129\pi\)
\(4\) 0 0
\(5\) −1979.00 −1.41606 −0.708029 0.706184i \(-0.750415\pi\)
−0.708029 + 0.706184i \(0.750415\pi\)
\(6\) 0 0
\(7\) 10115.0 1.59230 0.796150 0.605100i \(-0.206867\pi\)
0.796150 + 0.605100i \(0.206867\pi\)
\(8\) 0 0
\(9\) −14058.0 −0.714220
\(10\) 0 0
\(11\) −18850.0 −0.388190 −0.194095 0.980983i \(-0.562177\pi\)
−0.194095 + 0.980983i \(0.562177\pi\)
\(12\) 0 0
\(13\) 28561.0 0.277350
\(14\) 0 0
\(15\) 148425. 0.757001
\(16\) 0 0
\(17\) −142403. −0.413522 −0.206761 0.978391i \(-0.566292\pi\)
−0.206761 + 0.978391i \(0.566292\pi\)
\(18\) 0 0
\(19\) −83302.0 −0.146644 −0.0733220 0.997308i \(-0.523360\pi\)
−0.0733220 + 0.997308i \(0.523360\pi\)
\(20\) 0 0
\(21\) −758625. −0.851217
\(22\) 0 0
\(23\) 536544. 0.399788 0.199894 0.979817i \(-0.435940\pi\)
0.199894 + 0.979817i \(0.435940\pi\)
\(24\) 0 0
\(25\) 1.96332e6 1.00522
\(26\) 0 0
\(27\) 2.53058e6 0.916394
\(28\) 0 0
\(29\) −2.60044e6 −0.682741 −0.341371 0.939929i \(-0.610891\pi\)
−0.341371 + 0.939929i \(0.610891\pi\)
\(30\) 0 0
\(31\) 2.21400e6 0.430577 0.215288 0.976550i \(-0.430931\pi\)
0.215288 + 0.976550i \(0.430931\pi\)
\(32\) 0 0
\(33\) 1.41375e6 0.207520
\(34\) 0 0
\(35\) −2.00176e7 −2.25479
\(36\) 0 0
\(37\) 1.80992e7 1.58764 0.793821 0.608151i \(-0.208089\pi\)
0.793821 + 0.608151i \(0.208089\pi\)
\(38\) 0 0
\(39\) −2.14208e6 −0.148267
\(40\) 0 0
\(41\) 2.68122e7 1.48186 0.740928 0.671585i \(-0.234386\pi\)
0.740928 + 0.671585i \(0.234386\pi\)
\(42\) 0 0
\(43\) 4.22535e7 1.88475 0.942376 0.334555i \(-0.108586\pi\)
0.942376 + 0.334555i \(0.108586\pi\)
\(44\) 0 0
\(45\) 2.78208e7 1.01138
\(46\) 0 0
\(47\) −3.59150e7 −1.07358 −0.536791 0.843715i \(-0.680364\pi\)
−0.536791 + 0.843715i \(0.680364\pi\)
\(48\) 0 0
\(49\) 6.19596e7 1.53542
\(50\) 0 0
\(51\) 1.06802e7 0.221062
\(52\) 0 0
\(53\) −6.65141e7 −1.15790 −0.578951 0.815362i \(-0.696538\pi\)
−0.578951 + 0.815362i \(0.696538\pi\)
\(54\) 0 0
\(55\) 3.73042e7 0.549699
\(56\) 0 0
\(57\) 6.24765e6 0.0783935
\(58\) 0 0
\(59\) 1.08164e8 1.16211 0.581057 0.813863i \(-0.302639\pi\)
0.581057 + 0.813863i \(0.302639\pi\)
\(60\) 0 0
\(61\) −2.07450e8 −1.91836 −0.959178 0.282805i \(-0.908735\pi\)
−0.959178 + 0.282805i \(0.908735\pi\)
\(62\) 0 0
\(63\) −1.42197e8 −1.13725
\(64\) 0 0
\(65\) −5.65222e7 −0.392744
\(66\) 0 0
\(67\) −1.93016e8 −1.17019 −0.585094 0.810966i \(-0.698942\pi\)
−0.585094 + 0.810966i \(0.698942\pi\)
\(68\) 0 0
\(69\) −4.02408e7 −0.213720
\(70\) 0 0
\(71\) 2.01833e8 0.942607 0.471304 0.881971i \(-0.343784\pi\)
0.471304 + 0.881971i \(0.343784\pi\)
\(72\) 0 0
\(73\) −1.21628e8 −0.501281 −0.250640 0.968080i \(-0.580641\pi\)
−0.250640 + 0.968080i \(0.580641\pi\)
\(74\) 0 0
\(75\) −1.47249e8 −0.537373
\(76\) 0 0
\(77\) −1.90668e8 −0.618115
\(78\) 0 0
\(79\) −1.12872e8 −0.326035 −0.163017 0.986623i \(-0.552123\pi\)
−0.163017 + 0.986623i \(0.552123\pi\)
\(80\) 0 0
\(81\) 8.69105e7 0.224331
\(82\) 0 0
\(83\) −3.08254e8 −0.712948 −0.356474 0.934305i \(-0.616021\pi\)
−0.356474 + 0.934305i \(0.616021\pi\)
\(84\) 0 0
\(85\) 2.81816e8 0.585571
\(86\) 0 0
\(87\) 1.95033e8 0.364982
\(88\) 0 0
\(89\) −6.37487e6 −0.0107700 −0.00538501 0.999986i \(-0.501714\pi\)
−0.00538501 + 0.999986i \(0.501714\pi\)
\(90\) 0 0
\(91\) 2.88895e8 0.441624
\(92\) 0 0
\(93\) −1.66050e8 −0.230179
\(94\) 0 0
\(95\) 1.64855e8 0.207656
\(96\) 0 0
\(97\) 8.71267e8 0.999260 0.499630 0.866239i \(-0.333469\pi\)
0.499630 + 0.866239i \(0.333469\pi\)
\(98\) 0 0
\(99\) 2.64993e8 0.277253
\(100\) 0 0
\(101\) −8.24412e8 −0.788312 −0.394156 0.919044i \(-0.628963\pi\)
−0.394156 + 0.919044i \(0.628963\pi\)
\(102\) 0 0
\(103\) 1.65896e9 1.45234 0.726168 0.687517i \(-0.241299\pi\)
0.726168 + 0.687517i \(0.241299\pi\)
\(104\) 0 0
\(105\) 1.50132e9 1.20537
\(106\) 0 0
\(107\) −1.15165e9 −0.849366 −0.424683 0.905342i \(-0.639614\pi\)
−0.424683 + 0.905342i \(0.639614\pi\)
\(108\) 0 0
\(109\) −2.78480e9 −1.88962 −0.944810 0.327620i \(-0.893753\pi\)
−0.944810 + 0.327620i \(0.893753\pi\)
\(110\) 0 0
\(111\) −1.35744e9 −0.848727
\(112\) 0 0
\(113\) 6.78547e8 0.391496 0.195748 0.980654i \(-0.437287\pi\)
0.195748 + 0.980654i \(0.437287\pi\)
\(114\) 0 0
\(115\) −1.06182e9 −0.566123
\(116\) 0 0
\(117\) −4.01511e8 −0.198089
\(118\) 0 0
\(119\) −1.44041e9 −0.658451
\(120\) 0 0
\(121\) −2.00263e9 −0.849309
\(122\) 0 0
\(123\) −2.01092e9 −0.792175
\(124\) 0 0
\(125\) −2.01680e7 −0.00738869
\(126\) 0 0
\(127\) 3.48292e9 1.18803 0.594014 0.804455i \(-0.297542\pi\)
0.594014 + 0.804455i \(0.297542\pi\)
\(128\) 0 0
\(129\) −3.16901e9 −1.00756
\(130\) 0 0
\(131\) −5.02701e9 −1.49138 −0.745691 0.666292i \(-0.767881\pi\)
−0.745691 + 0.666292i \(0.767881\pi\)
\(132\) 0 0
\(133\) −8.42600e8 −0.233501
\(134\) 0 0
\(135\) −5.00801e9 −1.29767
\(136\) 0 0
\(137\) −6.38904e9 −1.54950 −0.774752 0.632265i \(-0.782125\pi\)
−0.774752 + 0.632265i \(0.782125\pi\)
\(138\) 0 0
\(139\) −7.62665e9 −1.73287 −0.866437 0.499286i \(-0.833596\pi\)
−0.866437 + 0.499286i \(0.833596\pi\)
\(140\) 0 0
\(141\) 2.69362e9 0.573920
\(142\) 0 0
\(143\) −5.38375e8 −0.107665
\(144\) 0 0
\(145\) 5.14627e9 0.966801
\(146\) 0 0
\(147\) −4.64697e9 −0.820809
\(148\) 0 0
\(149\) −9.23455e9 −1.53489 −0.767445 0.641114i \(-0.778472\pi\)
−0.767445 + 0.641114i \(0.778472\pi\)
\(150\) 0 0
\(151\) 3.25451e9 0.509436 0.254718 0.967015i \(-0.418017\pi\)
0.254718 + 0.967015i \(0.418017\pi\)
\(152\) 0 0
\(153\) 2.00190e9 0.295346
\(154\) 0 0
\(155\) −4.38151e9 −0.609722
\(156\) 0 0
\(157\) 1.62825e9 0.213881 0.106940 0.994265i \(-0.465895\pi\)
0.106940 + 0.994265i \(0.465895\pi\)
\(158\) 0 0
\(159\) 4.98855e9 0.618996
\(160\) 0 0
\(161\) 5.42714e9 0.636583
\(162\) 0 0
\(163\) 1.13187e10 1.25590 0.627948 0.778255i \(-0.283895\pi\)
0.627948 + 0.778255i \(0.283895\pi\)
\(164\) 0 0
\(165\) −2.79781e9 −0.293860
\(166\) 0 0
\(167\) −1.72306e9 −0.171426 −0.0857131 0.996320i \(-0.527317\pi\)
−0.0857131 + 0.996320i \(0.527317\pi\)
\(168\) 0 0
\(169\) 8.15731e8 0.0769231
\(170\) 0 0
\(171\) 1.17106e9 0.104736
\(172\) 0 0
\(173\) 2.76347e9 0.234557 0.117278 0.993099i \(-0.462583\pi\)
0.117278 + 0.993099i \(0.462583\pi\)
\(174\) 0 0
\(175\) 1.98589e10 1.60061
\(176\) 0 0
\(177\) −8.11230e9 −0.621247
\(178\) 0 0
\(179\) −6.86682e9 −0.499939 −0.249969 0.968254i \(-0.580421\pi\)
−0.249969 + 0.968254i \(0.580421\pi\)
\(180\) 0 0
\(181\) −2.41534e10 −1.67273 −0.836364 0.548174i \(-0.815323\pi\)
−0.836364 + 0.548174i \(0.815323\pi\)
\(182\) 0 0
\(183\) 1.55587e10 1.02552
\(184\) 0 0
\(185\) −3.58184e10 −2.24819
\(186\) 0 0
\(187\) 2.68430e9 0.160525
\(188\) 0 0
\(189\) 2.55968e10 1.45917
\(190\) 0 0
\(191\) −3.59983e10 −1.95719 −0.978593 0.205805i \(-0.934019\pi\)
−0.978593 + 0.205805i \(0.934019\pi\)
\(192\) 0 0
\(193\) −1.70031e10 −0.882107 −0.441054 0.897481i \(-0.645395\pi\)
−0.441054 + 0.897481i \(0.645395\pi\)
\(194\) 0 0
\(195\) 4.23917e9 0.209954
\(196\) 0 0
\(197\) 3.98292e10 1.88410 0.942049 0.335477i \(-0.108897\pi\)
0.942049 + 0.335477i \(0.108897\pi\)
\(198\) 0 0
\(199\) −1.31081e9 −0.0592518 −0.0296259 0.999561i \(-0.509432\pi\)
−0.0296259 + 0.999561i \(0.509432\pi\)
\(200\) 0 0
\(201\) 1.44762e10 0.625563
\(202\) 0 0
\(203\) −2.63035e10 −1.08713
\(204\) 0 0
\(205\) −5.30614e10 −2.09839
\(206\) 0 0
\(207\) −7.54274e9 −0.285537
\(208\) 0 0
\(209\) 1.57024e9 0.0569257
\(210\) 0 0
\(211\) −2.35777e10 −0.818898 −0.409449 0.912333i \(-0.634279\pi\)
−0.409449 + 0.912333i \(0.634279\pi\)
\(212\) 0 0
\(213\) −1.51375e10 −0.503902
\(214\) 0 0
\(215\) −8.36196e10 −2.66892
\(216\) 0 0
\(217\) 2.23947e10 0.685607
\(218\) 0 0
\(219\) 9.12211e9 0.267976
\(220\) 0 0
\(221\) −4.06717e9 −0.114690
\(222\) 0 0
\(223\) −2.38326e9 −0.0645356 −0.0322678 0.999479i \(-0.510273\pi\)
−0.0322678 + 0.999479i \(0.510273\pi\)
\(224\) 0 0
\(225\) −2.76003e10 −0.717947
\(226\) 0 0
\(227\) 7.46548e9 0.186613 0.0933064 0.995637i \(-0.470256\pi\)
0.0933064 + 0.995637i \(0.470256\pi\)
\(228\) 0 0
\(229\) 2.63966e10 0.634292 0.317146 0.948377i \(-0.397276\pi\)
0.317146 + 0.948377i \(0.397276\pi\)
\(230\) 0 0
\(231\) 1.43001e10 0.330434
\(232\) 0 0
\(233\) 2.40457e10 0.534485 0.267242 0.963629i \(-0.413888\pi\)
0.267242 + 0.963629i \(0.413888\pi\)
\(234\) 0 0
\(235\) 7.10758e10 1.52025
\(236\) 0 0
\(237\) 8.46539e9 0.174293
\(238\) 0 0
\(239\) −5.96318e10 −1.18219 −0.591095 0.806602i \(-0.701304\pi\)
−0.591095 + 0.806602i \(0.701304\pi\)
\(240\) 0 0
\(241\) 1.25639e10 0.239910 0.119955 0.992779i \(-0.461725\pi\)
0.119955 + 0.992779i \(0.461725\pi\)
\(242\) 0 0
\(243\) −5.63276e10 −1.03632
\(244\) 0 0
\(245\) −1.22618e11 −2.17424
\(246\) 0 0
\(247\) −2.37919e9 −0.0406717
\(248\) 0 0
\(249\) 2.31191e10 0.381130
\(250\) 0 0
\(251\) −2.41771e10 −0.384479 −0.192240 0.981348i \(-0.561575\pi\)
−0.192240 + 0.981348i \(0.561575\pi\)
\(252\) 0 0
\(253\) −1.01139e10 −0.155194
\(254\) 0 0
\(255\) −2.11362e10 −0.313037
\(256\) 0 0
\(257\) 2.96868e10 0.424488 0.212244 0.977217i \(-0.431923\pi\)
0.212244 + 0.977217i \(0.431923\pi\)
\(258\) 0 0
\(259\) 1.83074e11 2.52800
\(260\) 0 0
\(261\) 3.65570e10 0.487628
\(262\) 0 0
\(263\) 7.59146e10 0.978418 0.489209 0.872167i \(-0.337286\pi\)
0.489209 + 0.872167i \(0.337286\pi\)
\(264\) 0 0
\(265\) 1.31631e11 1.63966
\(266\) 0 0
\(267\) 4.78115e8 0.00575747
\(268\) 0 0
\(269\) −2.57149e10 −0.299433 −0.149716 0.988729i \(-0.547836\pi\)
−0.149716 + 0.988729i \(0.547836\pi\)
\(270\) 0 0
\(271\) 8.08890e10 0.911020 0.455510 0.890231i \(-0.349457\pi\)
0.455510 + 0.890231i \(0.349457\pi\)
\(272\) 0 0
\(273\) −2.16671e10 −0.236085
\(274\) 0 0
\(275\) −3.70085e10 −0.390215
\(276\) 0 0
\(277\) −3.40035e10 −0.347028 −0.173514 0.984831i \(-0.555512\pi\)
−0.173514 + 0.984831i \(0.555512\pi\)
\(278\) 0 0
\(279\) −3.11245e10 −0.307527
\(280\) 0 0
\(281\) −7.14831e9 −0.0683951 −0.0341976 0.999415i \(-0.510888\pi\)
−0.0341976 + 0.999415i \(0.510888\pi\)
\(282\) 0 0
\(283\) 7.80508e10 0.723333 0.361667 0.932308i \(-0.382208\pi\)
0.361667 + 0.932308i \(0.382208\pi\)
\(284\) 0 0
\(285\) −1.23641e10 −0.111010
\(286\) 0 0
\(287\) 2.71206e11 2.35956
\(288\) 0 0
\(289\) −9.83093e10 −0.828999
\(290\) 0 0
\(291\) −6.53450e10 −0.534188
\(292\) 0 0
\(293\) −1.26662e11 −1.00402 −0.502009 0.864862i \(-0.667406\pi\)
−0.502009 + 0.864862i \(0.667406\pi\)
\(294\) 0 0
\(295\) −2.14057e11 −1.64562
\(296\) 0 0
\(297\) −4.77013e10 −0.355735
\(298\) 0 0
\(299\) 1.53242e10 0.110881
\(300\) 0 0
\(301\) 4.27394e11 3.00109
\(302\) 0 0
\(303\) 6.18309e10 0.421419
\(304\) 0 0
\(305\) 4.10543e11 2.71650
\(306\) 0 0
\(307\) −6.15064e10 −0.395182 −0.197591 0.980285i \(-0.563312\pi\)
−0.197591 + 0.980285i \(0.563312\pi\)
\(308\) 0 0
\(309\) −1.24422e11 −0.776395
\(310\) 0 0
\(311\) −2.16398e11 −1.31169 −0.655846 0.754894i \(-0.727688\pi\)
−0.655846 + 0.754894i \(0.727688\pi\)
\(312\) 0 0
\(313\) 2.44634e11 1.44068 0.720340 0.693621i \(-0.243986\pi\)
0.720340 + 0.693621i \(0.243986\pi\)
\(314\) 0 0
\(315\) 2.81407e11 1.61041
\(316\) 0 0
\(317\) 7.77399e10 0.432392 0.216196 0.976350i \(-0.430635\pi\)
0.216196 + 0.976350i \(0.430635\pi\)
\(318\) 0 0
\(319\) 4.90183e10 0.265033
\(320\) 0 0
\(321\) 8.63740e10 0.454057
\(322\) 0 0
\(323\) 1.18625e10 0.0606406
\(324\) 0 0
\(325\) 5.60743e10 0.278797
\(326\) 0 0
\(327\) 2.08860e11 1.01016
\(328\) 0 0
\(329\) −3.63280e11 −1.70946
\(330\) 0 0
\(331\) 1.68625e11 0.772139 0.386070 0.922470i \(-0.373832\pi\)
0.386070 + 0.922470i \(0.373832\pi\)
\(332\) 0 0
\(333\) −2.54439e11 −1.13393
\(334\) 0 0
\(335\) 3.81978e11 1.65705
\(336\) 0 0
\(337\) −7.70797e10 −0.325541 −0.162770 0.986664i \(-0.552043\pi\)
−0.162770 + 0.986664i \(0.552043\pi\)
\(338\) 0 0
\(339\) −5.08910e10 −0.209287
\(340\) 0 0
\(341\) −4.17340e10 −0.167146
\(342\) 0 0
\(343\) 2.18545e11 0.852544
\(344\) 0 0
\(345\) 7.96365e10 0.302640
\(346\) 0 0
\(347\) −3.54609e11 −1.31301 −0.656504 0.754322i \(-0.727966\pi\)
−0.656504 + 0.754322i \(0.727966\pi\)
\(348\) 0 0
\(349\) 1.70460e11 0.615048 0.307524 0.951540i \(-0.400500\pi\)
0.307524 + 0.951540i \(0.400500\pi\)
\(350\) 0 0
\(351\) 7.22758e10 0.254162
\(352\) 0 0
\(353\) 2.96506e11 1.01636 0.508180 0.861251i \(-0.330318\pi\)
0.508180 + 0.861251i \(0.330318\pi\)
\(354\) 0 0
\(355\) −3.99428e11 −1.33479
\(356\) 0 0
\(357\) 1.08030e11 0.351997
\(358\) 0 0
\(359\) −7.20144e10 −0.228820 −0.114410 0.993434i \(-0.536498\pi\)
−0.114410 + 0.993434i \(0.536498\pi\)
\(360\) 0 0
\(361\) −3.15748e11 −0.978496
\(362\) 0 0
\(363\) 1.50197e11 0.454026
\(364\) 0 0
\(365\) 2.40702e11 0.709842
\(366\) 0 0
\(367\) 1.05092e11 0.302394 0.151197 0.988504i \(-0.451687\pi\)
0.151197 + 0.988504i \(0.451687\pi\)
\(368\) 0 0
\(369\) −3.76926e11 −1.05837
\(370\) 0 0
\(371\) −6.72790e11 −1.84373
\(372\) 0 0
\(373\) 2.19888e11 0.588182 0.294091 0.955777i \(-0.404983\pi\)
0.294091 + 0.955777i \(0.404983\pi\)
\(374\) 0 0
\(375\) 1.51260e9 0.00394987
\(376\) 0 0
\(377\) −7.42712e10 −0.189358
\(378\) 0 0
\(379\) 3.14748e11 0.783586 0.391793 0.920053i \(-0.371855\pi\)
0.391793 + 0.920053i \(0.371855\pi\)
\(380\) 0 0
\(381\) −2.61219e11 −0.635100
\(382\) 0 0
\(383\) 3.41027e10 0.0809831 0.0404915 0.999180i \(-0.487108\pi\)
0.0404915 + 0.999180i \(0.487108\pi\)
\(384\) 0 0
\(385\) 3.77331e11 0.875286
\(386\) 0 0
\(387\) −5.93999e11 −1.34613
\(388\) 0 0
\(389\) 4.49612e11 0.995554 0.497777 0.867305i \(-0.334150\pi\)
0.497777 + 0.867305i \(0.334150\pi\)
\(390\) 0 0
\(391\) −7.64055e10 −0.165321
\(392\) 0 0
\(393\) 3.77026e11 0.797269
\(394\) 0 0
\(395\) 2.23374e11 0.461684
\(396\) 0 0
\(397\) 2.29976e11 0.464649 0.232324 0.972638i \(-0.425367\pi\)
0.232324 + 0.972638i \(0.425367\pi\)
\(398\) 0 0
\(399\) 6.31950e10 0.124826
\(400\) 0 0
\(401\) −6.69163e11 −1.29236 −0.646178 0.763187i \(-0.723634\pi\)
−0.646178 + 0.763187i \(0.723634\pi\)
\(402\) 0 0
\(403\) 6.32342e10 0.119421
\(404\) 0 0
\(405\) −1.71996e11 −0.317666
\(406\) 0 0
\(407\) −3.41171e11 −0.616307
\(408\) 0 0
\(409\) −6.73923e11 −1.19085 −0.595423 0.803413i \(-0.703015\pi\)
−0.595423 + 0.803413i \(0.703015\pi\)
\(410\) 0 0
\(411\) 4.79178e11 0.828340
\(412\) 0 0
\(413\) 1.09408e12 1.85043
\(414\) 0 0
\(415\) 6.10035e11 1.00957
\(416\) 0 0
\(417\) 5.71999e11 0.926366
\(418\) 0 0
\(419\) 3.45053e11 0.546919 0.273460 0.961883i \(-0.411832\pi\)
0.273460 + 0.961883i \(0.411832\pi\)
\(420\) 0 0
\(421\) −5.57817e11 −0.865411 −0.432706 0.901535i \(-0.642441\pi\)
−0.432706 + 0.901535i \(0.642441\pi\)
\(422\) 0 0
\(423\) 5.04893e11 0.766775
\(424\) 0 0
\(425\) −2.79582e11 −0.415680
\(426\) 0 0
\(427\) −2.09836e12 −3.05460
\(428\) 0 0
\(429\) 4.03781e10 0.0575557
\(430\) 0 0
\(431\) 6.39243e11 0.892315 0.446158 0.894954i \(-0.352792\pi\)
0.446158 + 0.894954i \(0.352792\pi\)
\(432\) 0 0
\(433\) −1.23759e12 −1.69193 −0.845965 0.533238i \(-0.820975\pi\)
−0.845965 + 0.533238i \(0.820975\pi\)
\(434\) 0 0
\(435\) −3.85971e11 −0.516836
\(436\) 0 0
\(437\) −4.46952e10 −0.0586265
\(438\) 0 0
\(439\) 9.14852e11 1.17560 0.587801 0.809006i \(-0.299994\pi\)
0.587801 + 0.809006i \(0.299994\pi\)
\(440\) 0 0
\(441\) −8.71028e11 −1.09663
\(442\) 0 0
\(443\) 1.18651e12 1.46370 0.731852 0.681464i \(-0.238656\pi\)
0.731852 + 0.681464i \(0.238656\pi\)
\(444\) 0 0
\(445\) 1.26159e10 0.0152510
\(446\) 0 0
\(447\) 6.92591e11 0.820527
\(448\) 0 0
\(449\) −2.70808e10 −0.0314452 −0.0157226 0.999876i \(-0.505005\pi\)
−0.0157226 + 0.999876i \(0.505005\pi\)
\(450\) 0 0
\(451\) −5.05411e11 −0.575241
\(452\) 0 0
\(453\) −2.44088e11 −0.272336
\(454\) 0 0
\(455\) −5.71722e11 −0.625365
\(456\) 0 0
\(457\) 2.02586e11 0.217263 0.108632 0.994082i \(-0.465353\pi\)
0.108632 + 0.994082i \(0.465353\pi\)
\(458\) 0 0
\(459\) −3.60361e11 −0.378949
\(460\) 0 0
\(461\) 8.96346e11 0.924319 0.462159 0.886797i \(-0.347075\pi\)
0.462159 + 0.886797i \(0.347075\pi\)
\(462\) 0 0
\(463\) 5.17740e11 0.523597 0.261799 0.965123i \(-0.415684\pi\)
0.261799 + 0.965123i \(0.415684\pi\)
\(464\) 0 0
\(465\) 3.28614e11 0.325947
\(466\) 0 0
\(467\) −9.20785e11 −0.895843 −0.447922 0.894073i \(-0.647836\pi\)
−0.447922 + 0.894073i \(0.647836\pi\)
\(468\) 0 0
\(469\) −1.95235e12 −1.86329
\(470\) 0 0
\(471\) −1.22119e11 −0.114337
\(472\) 0 0
\(473\) −7.96478e11 −0.731642
\(474\) 0 0
\(475\) −1.63548e11 −0.147409
\(476\) 0 0
\(477\) 9.35055e11 0.826998
\(478\) 0 0
\(479\) −1.73253e12 −1.50373 −0.751867 0.659315i \(-0.770846\pi\)
−0.751867 + 0.659315i \(0.770846\pi\)
\(480\) 0 0
\(481\) 5.16932e11 0.440333
\(482\) 0 0
\(483\) −4.07036e11 −0.340307
\(484\) 0 0
\(485\) −1.72424e12 −1.41501
\(486\) 0 0
\(487\) −1.49591e12 −1.20511 −0.602554 0.798078i \(-0.705850\pi\)
−0.602554 + 0.798078i \(0.705850\pi\)
\(488\) 0 0
\(489\) −8.48906e11 −0.671382
\(490\) 0 0
\(491\) 4.28954e11 0.333076 0.166538 0.986035i \(-0.446741\pi\)
0.166538 + 0.986035i \(0.446741\pi\)
\(492\) 0 0
\(493\) 3.70311e11 0.282329
\(494\) 0 0
\(495\) −5.24422e11 −0.392606
\(496\) 0 0
\(497\) 2.04155e12 1.50091
\(498\) 0 0
\(499\) 9.12174e11 0.658606 0.329303 0.944224i \(-0.393186\pi\)
0.329303 + 0.944224i \(0.393186\pi\)
\(500\) 0 0
\(501\) 1.29230e11 0.0916416
\(502\) 0 0
\(503\) 1.26835e12 0.883456 0.441728 0.897149i \(-0.354366\pi\)
0.441728 + 0.897149i \(0.354366\pi\)
\(504\) 0 0
\(505\) 1.63151e12 1.11629
\(506\) 0 0
\(507\) −6.11798e10 −0.0411218
\(508\) 0 0
\(509\) −1.54192e12 −1.01820 −0.509098 0.860708i \(-0.670021\pi\)
−0.509098 + 0.860708i \(0.670021\pi\)
\(510\) 0 0
\(511\) −1.23027e12 −0.798189
\(512\) 0 0
\(513\) −2.10802e11 −0.134384
\(514\) 0 0
\(515\) −3.28307e12 −2.05659
\(516\) 0 0
\(517\) 6.76998e11 0.416754
\(518\) 0 0
\(519\) −2.07261e11 −0.125390
\(520\) 0 0
\(521\) −1.48896e12 −0.885345 −0.442672 0.896683i \(-0.645969\pi\)
−0.442672 + 0.896683i \(0.645969\pi\)
\(522\) 0 0
\(523\) −2.55715e12 −1.49451 −0.747256 0.664536i \(-0.768629\pi\)
−0.747256 + 0.664536i \(0.768629\pi\)
\(524\) 0 0
\(525\) −1.48942e12 −0.855659
\(526\) 0 0
\(527\) −3.15281e11 −0.178053
\(528\) 0 0
\(529\) −1.51327e12 −0.840169
\(530\) 0 0
\(531\) −1.52057e12 −0.830005
\(532\) 0 0
\(533\) 7.65784e11 0.410993
\(534\) 0 0
\(535\) 2.27912e12 1.20275
\(536\) 0 0
\(537\) 5.15011e11 0.267259
\(538\) 0 0
\(539\) −1.16794e12 −0.596033
\(540\) 0 0
\(541\) −2.64921e12 −1.32962 −0.664811 0.747011i \(-0.731488\pi\)
−0.664811 + 0.747011i \(0.731488\pi\)
\(542\) 0 0
\(543\) 1.81151e12 0.894213
\(544\) 0 0
\(545\) 5.51111e12 2.67581
\(546\) 0 0
\(547\) −2.16400e12 −1.03351 −0.516754 0.856134i \(-0.672860\pi\)
−0.516754 + 0.856134i \(0.672860\pi\)
\(548\) 0 0
\(549\) 2.91633e12 1.37013
\(550\) 0 0
\(551\) 2.16622e11 0.100120
\(552\) 0 0
\(553\) −1.14170e12 −0.519145
\(554\) 0 0
\(555\) 2.68638e12 1.20185
\(556\) 0 0
\(557\) −2.96364e12 −1.30460 −0.652300 0.757961i \(-0.726196\pi\)
−0.652300 + 0.757961i \(0.726196\pi\)
\(558\) 0 0
\(559\) 1.20680e12 0.522736
\(560\) 0 0
\(561\) −2.01322e11 −0.0858141
\(562\) 0 0
\(563\) 3.46859e12 1.45501 0.727504 0.686104i \(-0.240680\pi\)
0.727504 + 0.686104i \(0.240680\pi\)
\(564\) 0 0
\(565\) −1.34284e12 −0.554380
\(566\) 0 0
\(567\) 8.79100e11 0.357202
\(568\) 0 0
\(569\) −3.83703e12 −1.53458 −0.767290 0.641300i \(-0.778395\pi\)
−0.767290 + 0.641300i \(0.778395\pi\)
\(570\) 0 0
\(571\) −3.35374e11 −0.132028 −0.0660142 0.997819i \(-0.521028\pi\)
−0.0660142 + 0.997819i \(0.521028\pi\)
\(572\) 0 0
\(573\) 2.69987e12 1.04628
\(574\) 0 0
\(575\) 1.05341e12 0.401874
\(576\) 0 0
\(577\) −1.74089e11 −0.0653852 −0.0326926 0.999465i \(-0.510408\pi\)
−0.0326926 + 0.999465i \(0.510408\pi\)
\(578\) 0 0
\(579\) 1.27524e12 0.471560
\(580\) 0 0
\(581\) −3.11799e12 −1.13523
\(582\) 0 0
\(583\) 1.25379e12 0.449486
\(584\) 0 0
\(585\) 7.94589e11 0.280505
\(586\) 0 0
\(587\) −1.87089e12 −0.650395 −0.325198 0.945646i \(-0.605431\pi\)
−0.325198 + 0.945646i \(0.605431\pi\)
\(588\) 0 0
\(589\) −1.84431e11 −0.0631415
\(590\) 0 0
\(591\) −2.98719e12 −1.00721
\(592\) 0 0
\(593\) 4.16793e12 1.38412 0.692061 0.721839i \(-0.256703\pi\)
0.692061 + 0.721839i \(0.256703\pi\)
\(594\) 0 0
\(595\) 2.85056e12 0.932405
\(596\) 0 0
\(597\) 9.83110e10 0.0316751
\(598\) 0 0
\(599\) −8.16635e10 −0.0259183 −0.0129592 0.999916i \(-0.504125\pi\)
−0.0129592 + 0.999916i \(0.504125\pi\)
\(600\) 0 0
\(601\) −4.00769e12 −1.25302 −0.626511 0.779412i \(-0.715518\pi\)
−0.626511 + 0.779412i \(0.715518\pi\)
\(602\) 0 0
\(603\) 2.71341e12 0.835772
\(604\) 0 0
\(605\) 3.96320e12 1.20267
\(606\) 0 0
\(607\) −1.45542e12 −0.435149 −0.217575 0.976044i \(-0.569815\pi\)
−0.217575 + 0.976044i \(0.569815\pi\)
\(608\) 0 0
\(609\) 1.97276e12 0.581161
\(610\) 0 0
\(611\) −1.02577e12 −0.297758
\(612\) 0 0
\(613\) −2.55645e12 −0.731248 −0.365624 0.930763i \(-0.619144\pi\)
−0.365624 + 0.930763i \(0.619144\pi\)
\(614\) 0 0
\(615\) 3.97961e12 1.12177
\(616\) 0 0
\(617\) 2.69727e11 0.0749276 0.0374638 0.999298i \(-0.488072\pi\)
0.0374638 + 0.999298i \(0.488072\pi\)
\(618\) 0 0
\(619\) −4.28111e11 −0.117206 −0.0586028 0.998281i \(-0.518665\pi\)
−0.0586028 + 0.998281i \(0.518665\pi\)
\(620\) 0 0
\(621\) 1.35776e12 0.366364
\(622\) 0 0
\(623\) −6.44818e10 −0.0171491
\(624\) 0 0
\(625\) −3.79469e12 −0.994755
\(626\) 0 0
\(627\) −1.17768e11 −0.0304316
\(628\) 0 0
\(629\) −2.57739e12 −0.656525
\(630\) 0 0
\(631\) 2.98421e11 0.0749372 0.0374686 0.999298i \(-0.488071\pi\)
0.0374686 + 0.999298i \(0.488071\pi\)
\(632\) 0 0
\(633\) 1.76833e12 0.437770
\(634\) 0 0
\(635\) −6.89269e12 −1.68231
\(636\) 0 0
\(637\) 1.76963e12 0.425848
\(638\) 0 0
\(639\) −2.83738e12 −0.673229
\(640\) 0 0
\(641\) −2.34144e12 −0.547799 −0.273899 0.961758i \(-0.588314\pi\)
−0.273899 + 0.961758i \(0.588314\pi\)
\(642\) 0 0
\(643\) −7.75186e12 −1.78837 −0.894183 0.447701i \(-0.852243\pi\)
−0.894183 + 0.447701i \(0.852243\pi\)
\(644\) 0 0
\(645\) 6.27147e12 1.42676
\(646\) 0 0
\(647\) −3.74980e12 −0.841278 −0.420639 0.907228i \(-0.638194\pi\)
−0.420639 + 0.907228i \(0.638194\pi\)
\(648\) 0 0
\(649\) −2.03889e12 −0.451121
\(650\) 0 0
\(651\) −1.67960e12 −0.366514
\(652\) 0 0
\(653\) 2.82022e12 0.606979 0.303490 0.952835i \(-0.401848\pi\)
0.303490 + 0.952835i \(0.401848\pi\)
\(654\) 0 0
\(655\) 9.94846e12 2.11188
\(656\) 0 0
\(657\) 1.70985e12 0.358025
\(658\) 0 0
\(659\) 1.16074e12 0.239746 0.119873 0.992789i \(-0.461751\pi\)
0.119873 + 0.992789i \(0.461751\pi\)
\(660\) 0 0
\(661\) −3.18976e11 −0.0649907 −0.0324954 0.999472i \(-0.510345\pi\)
−0.0324954 + 0.999472i \(0.510345\pi\)
\(662\) 0 0
\(663\) 3.05038e11 0.0613116
\(664\) 0 0
\(665\) 1.66750e12 0.330651
\(666\) 0 0
\(667\) −1.39525e12 −0.272952
\(668\) 0 0
\(669\) 1.78745e11 0.0344997
\(670\) 0 0
\(671\) 3.91043e12 0.744686
\(672\) 0 0
\(673\) 4.82897e12 0.907375 0.453687 0.891161i \(-0.350108\pi\)
0.453687 + 0.891161i \(0.350108\pi\)
\(674\) 0 0
\(675\) 4.96832e12 0.921176
\(676\) 0 0
\(677\) −7.47095e12 −1.36687 −0.683434 0.730012i \(-0.739514\pi\)
−0.683434 + 0.730012i \(0.739514\pi\)
\(678\) 0 0
\(679\) 8.81286e12 1.59112
\(680\) 0 0
\(681\) −5.59911e11 −0.0997601
\(682\) 0 0
\(683\) 4.13060e12 0.726306 0.363153 0.931730i \(-0.381700\pi\)
0.363153 + 0.931730i \(0.381700\pi\)
\(684\) 0 0
\(685\) 1.26439e13 2.19419
\(686\) 0 0
\(687\) −1.97975e12 −0.339082
\(688\) 0 0
\(689\) −1.89971e12 −0.321144
\(690\) 0 0
\(691\) −4.50776e12 −0.752159 −0.376079 0.926587i \(-0.622728\pi\)
−0.376079 + 0.926587i \(0.622728\pi\)
\(692\) 0 0
\(693\) 2.68041e12 0.441470
\(694\) 0 0
\(695\) 1.50931e13 2.45385
\(696\) 0 0
\(697\) −3.81814e12 −0.612780
\(698\) 0 0
\(699\) −1.80343e12 −0.285727
\(700\) 0 0
\(701\) −1.38907e12 −0.217267 −0.108633 0.994082i \(-0.534647\pi\)
−0.108633 + 0.994082i \(0.534647\pi\)
\(702\) 0 0
\(703\) −1.50770e12 −0.232818
\(704\) 0 0
\(705\) −5.33068e12 −0.812703
\(706\) 0 0
\(707\) −8.33893e12 −1.25523
\(708\) 0 0
\(709\) 5.59251e12 0.831187 0.415593 0.909551i \(-0.363574\pi\)
0.415593 + 0.909551i \(0.363574\pi\)
\(710\) 0 0
\(711\) 1.58675e12 0.232861
\(712\) 0 0
\(713\) 1.18791e12 0.172140
\(714\) 0 0
\(715\) 1.06544e12 0.152459
\(716\) 0 0
\(717\) 4.47238e12 0.631979
\(718\) 0 0
\(719\) 8.33742e12 1.16346 0.581730 0.813382i \(-0.302376\pi\)
0.581730 + 0.813382i \(0.302376\pi\)
\(720\) 0 0
\(721\) 1.67803e13 2.31255
\(722\) 0 0
\(723\) −9.42295e11 −0.128252
\(724\) 0 0
\(725\) −5.10549e12 −0.686304
\(726\) 0 0
\(727\) 5.13697e12 0.682028 0.341014 0.940058i \(-0.389230\pi\)
0.341014 + 0.940058i \(0.389230\pi\)
\(728\) 0 0
\(729\) 2.51391e12 0.329667
\(730\) 0 0
\(731\) −6.01702e12 −0.779387
\(732\) 0 0
\(733\) 9.72563e12 1.24437 0.622185 0.782870i \(-0.286245\pi\)
0.622185 + 0.782870i \(0.286245\pi\)
\(734\) 0 0
\(735\) 9.19636e12 1.16231
\(736\) 0 0
\(737\) 3.63834e12 0.454255
\(738\) 0 0
\(739\) 1.15533e13 1.42498 0.712488 0.701684i \(-0.247568\pi\)
0.712488 + 0.701684i \(0.247568\pi\)
\(740\) 0 0
\(741\) 1.78439e11 0.0217424
\(742\) 0 0
\(743\) 1.04495e13 1.25790 0.628948 0.777447i \(-0.283486\pi\)
0.628948 + 0.777447i \(0.283486\pi\)
\(744\) 0 0
\(745\) 1.82752e13 2.17349
\(746\) 0 0
\(747\) 4.33344e12 0.509202
\(748\) 0 0
\(749\) −1.16490e13 −1.35245
\(750\) 0 0
\(751\) −1.92230e12 −0.220516 −0.110258 0.993903i \(-0.535168\pi\)
−0.110258 + 0.993903i \(0.535168\pi\)
\(752\) 0 0
\(753\) 1.81328e12 0.205536
\(754\) 0 0
\(755\) −6.44068e12 −0.721390
\(756\) 0 0
\(757\) −1.27412e12 −0.141019 −0.0705096 0.997511i \(-0.522463\pi\)
−0.0705096 + 0.997511i \(0.522463\pi\)
\(758\) 0 0
\(759\) 7.58539e11 0.0829641
\(760\) 0 0
\(761\) 1.21619e13 1.31453 0.657267 0.753658i \(-0.271712\pi\)
0.657267 + 0.753658i \(0.271712\pi\)
\(762\) 0 0
\(763\) −2.81682e13 −3.00884
\(764\) 0 0
\(765\) −3.96176e12 −0.418227
\(766\) 0 0
\(767\) 3.08927e12 0.322312
\(768\) 0 0
\(769\) −5.98877e12 −0.617546 −0.308773 0.951136i \(-0.599918\pi\)
−0.308773 + 0.951136i \(0.599918\pi\)
\(770\) 0 0
\(771\) −2.22651e12 −0.226924
\(772\) 0 0
\(773\) −9.64838e12 −0.971956 −0.485978 0.873971i \(-0.661536\pi\)
−0.485978 + 0.873971i \(0.661536\pi\)
\(774\) 0 0
\(775\) 4.34679e12 0.432824
\(776\) 0 0
\(777\) −1.37305e13 −1.35143
\(778\) 0 0
\(779\) −2.23351e12 −0.217305
\(780\) 0 0
\(781\) −3.80456e12 −0.365911
\(782\) 0 0
\(783\) −6.58061e12 −0.625660
\(784\) 0 0
\(785\) −3.22230e12 −0.302867
\(786\) 0 0
\(787\) −1.40829e13 −1.30860 −0.654299 0.756236i \(-0.727036\pi\)
−0.654299 + 0.756236i \(0.727036\pi\)
\(788\) 0 0
\(789\) −5.69360e12 −0.523046
\(790\) 0 0
\(791\) 6.86350e12 0.623378
\(792\) 0 0
\(793\) −5.92498e12 −0.532056
\(794\) 0 0
\(795\) −9.87235e12 −0.876533
\(796\) 0 0
\(797\) −8.45863e12 −0.742570 −0.371285 0.928519i \(-0.621083\pi\)
−0.371285 + 0.928519i \(0.621083\pi\)
\(798\) 0 0
\(799\) 5.11440e12 0.443950
\(800\) 0 0
\(801\) 8.96179e10 0.00769216
\(802\) 0 0
\(803\) 2.29269e12 0.194592
\(804\) 0 0
\(805\) −1.07403e13 −0.901437
\(806\) 0 0
\(807\) 1.92862e12 0.160072
\(808\) 0 0
\(809\) −9.95988e12 −0.817496 −0.408748 0.912647i \(-0.634035\pi\)
−0.408748 + 0.912647i \(0.634035\pi\)
\(810\) 0 0
\(811\) −9.83262e12 −0.798133 −0.399067 0.916922i \(-0.630666\pi\)
−0.399067 + 0.916922i \(0.630666\pi\)
\(812\) 0 0
\(813\) −6.06668e12 −0.487016
\(814\) 0 0
\(815\) −2.23998e13 −1.77842
\(816\) 0 0
\(817\) −3.51980e12 −0.276388
\(818\) 0 0
\(819\) −4.06128e12 −0.315417
\(820\) 0 0
\(821\) 8.67049e12 0.666039 0.333019 0.942920i \(-0.391933\pi\)
0.333019 + 0.942920i \(0.391933\pi\)
\(822\) 0 0
\(823\) 1.26458e13 0.960834 0.480417 0.877040i \(-0.340485\pi\)
0.480417 + 0.877040i \(0.340485\pi\)
\(824\) 0 0
\(825\) 2.77564e12 0.208603
\(826\) 0 0
\(827\) 2.04741e13 1.52205 0.761027 0.648720i \(-0.224695\pi\)
0.761027 + 0.648720i \(0.224695\pi\)
\(828\) 0 0
\(829\) −1.00940e13 −0.742283 −0.371141 0.928576i \(-0.621033\pi\)
−0.371141 + 0.928576i \(0.621033\pi\)
\(830\) 0 0
\(831\) 2.55026e12 0.185516
\(832\) 0 0
\(833\) −8.82324e12 −0.634929
\(834\) 0 0
\(835\) 3.40994e12 0.242749
\(836\) 0 0
\(837\) 5.60270e12 0.394578
\(838\) 0 0
\(839\) −1.54505e13 −1.07650 −0.538249 0.842786i \(-0.680914\pi\)
−0.538249 + 0.842786i \(0.680914\pi\)
\(840\) 0 0
\(841\) −7.74485e12 −0.533864
\(842\) 0 0
\(843\) 5.36123e11 0.0365629
\(844\) 0 0
\(845\) −1.61433e12 −0.108927
\(846\) 0 0
\(847\) −2.02566e13 −1.35235
\(848\) 0 0
\(849\) −5.85381e12 −0.386682
\(850\) 0 0
\(851\) 9.71104e12 0.634721
\(852\) 0 0
\(853\) 2.56415e13 1.65834 0.829170 0.558997i \(-0.188814\pi\)
0.829170 + 0.558997i \(0.188814\pi\)
\(854\) 0 0
\(855\) −2.31753e12 −0.148312
\(856\) 0 0
\(857\) −1.44941e11 −0.00917862 −0.00458931 0.999989i \(-0.501461\pi\)
−0.00458931 + 0.999989i \(0.501461\pi\)
\(858\) 0 0
\(859\) 9.47728e12 0.593902 0.296951 0.954893i \(-0.404030\pi\)
0.296951 + 0.954893i \(0.404030\pi\)
\(860\) 0 0
\(861\) −2.03404e13 −1.26138
\(862\) 0 0
\(863\) 4.25726e12 0.261265 0.130633 0.991431i \(-0.458299\pi\)
0.130633 + 0.991431i \(0.458299\pi\)
\(864\) 0 0
\(865\) −5.46892e12 −0.332146
\(866\) 0 0
\(867\) 7.37319e12 0.443169
\(868\) 0 0
\(869\) 2.12764e12 0.126563
\(870\) 0 0
\(871\) −5.51272e12 −0.324552
\(872\) 0 0
\(873\) −1.22483e13 −0.713692
\(874\) 0 0
\(875\) −2.03999e11 −0.0117650
\(876\) 0 0
\(877\) 1.42035e13 0.810768 0.405384 0.914147i \(-0.367138\pi\)
0.405384 + 0.914147i \(0.367138\pi\)
\(878\) 0 0
\(879\) 9.49964e12 0.536732
\(880\) 0 0
\(881\) −1.12878e13 −0.631275 −0.315638 0.948880i \(-0.602218\pi\)
−0.315638 + 0.948880i \(0.602218\pi\)
\(882\) 0 0
\(883\) −2.25433e12 −0.124794 −0.0623972 0.998051i \(-0.519875\pi\)
−0.0623972 + 0.998051i \(0.519875\pi\)
\(884\) 0 0
\(885\) 1.60542e13 0.879721
\(886\) 0 0
\(887\) 3.13450e13 1.70024 0.850122 0.526585i \(-0.176528\pi\)
0.850122 + 0.526585i \(0.176528\pi\)
\(888\) 0 0
\(889\) 3.52297e13 1.89170
\(890\) 0 0
\(891\) −1.63826e12 −0.0870831
\(892\) 0 0
\(893\) 2.99179e12 0.157434
\(894\) 0 0
\(895\) 1.35894e13 0.707942
\(896\) 0 0
\(897\) −1.14932e12 −0.0592753
\(898\) 0 0
\(899\) −5.75739e12 −0.293973
\(900\) 0 0
\(901\) 9.47180e12 0.478819
\(902\) 0 0
\(903\) −3.20545e13 −1.60433
\(904\) 0 0
\(905\) 4.77997e13 2.36868
\(906\) 0 0
\(907\) −1.77171e13 −0.869280 −0.434640 0.900604i \(-0.643124\pi\)
−0.434640 + 0.900604i \(0.643124\pi\)
\(908\) 0 0
\(909\) 1.15896e13 0.563028
\(910\) 0 0
\(911\) −2.93419e13 −1.41142 −0.705710 0.708501i \(-0.749372\pi\)
−0.705710 + 0.708501i \(0.749372\pi\)
\(912\) 0 0
\(913\) 5.81059e12 0.276759
\(914\) 0 0
\(915\) −3.07908e13 −1.45220
\(916\) 0 0
\(917\) −5.08482e13 −2.37473
\(918\) 0 0
\(919\) 1.66013e13 0.767754 0.383877 0.923384i \(-0.374589\pi\)
0.383877 + 0.923384i \(0.374589\pi\)
\(920\) 0 0
\(921\) 4.61298e12 0.211258
\(922\) 0 0
\(923\) 5.76457e12 0.261432
\(924\) 0 0
\(925\) 3.55345e13 1.59593
\(926\) 0 0
\(927\) −2.33216e13 −1.03729
\(928\) 0 0
\(929\) −1.45567e13 −0.641197 −0.320598 0.947215i \(-0.603884\pi\)
−0.320598 + 0.947215i \(0.603884\pi\)
\(930\) 0 0
\(931\) −5.16136e12 −0.225160
\(932\) 0 0
\(933\) 1.62299e13 0.701210
\(934\) 0 0
\(935\) −5.31222e12 −0.227313
\(936\) 0 0
\(937\) −1.38682e13 −0.587751 −0.293875 0.955844i \(-0.594945\pi\)
−0.293875 + 0.955844i \(0.594945\pi\)
\(938\) 0 0
\(939\) −1.83476e13 −0.770164
\(940\) 0 0
\(941\) −2.15767e13 −0.897081 −0.448540 0.893763i \(-0.648056\pi\)
−0.448540 + 0.893763i \(0.648056\pi\)
\(942\) 0 0
\(943\) 1.43859e13 0.592428
\(944\) 0 0
\(945\) −5.06560e13 −2.06627
\(946\) 0 0
\(947\) −7.20814e12 −0.291238 −0.145619 0.989341i \(-0.546517\pi\)
−0.145619 + 0.989341i \(0.546517\pi\)
\(948\) 0 0
\(949\) −3.47382e12 −0.139030
\(950\) 0 0
\(951\) −5.83049e12 −0.231149
\(952\) 0 0
\(953\) 2.71758e13 1.06724 0.533622 0.845723i \(-0.320830\pi\)
0.533622 + 0.845723i \(0.320830\pi\)
\(954\) 0 0
\(955\) 7.12407e13 2.77149
\(956\) 0 0
\(957\) −3.67637e12 −0.141682
\(958\) 0 0
\(959\) −6.46251e13 −2.46727
\(960\) 0 0
\(961\) −2.15378e13 −0.814603
\(962\) 0 0
\(963\) 1.61899e13 0.606635
\(964\) 0 0
\(965\) 3.36492e13 1.24911
\(966\) 0 0
\(967\) −1.20281e13 −0.442364 −0.221182 0.975233i \(-0.570991\pi\)
−0.221182 + 0.975233i \(0.570991\pi\)
\(968\) 0 0
\(969\) −8.89684e11 −0.0324174
\(970\) 0 0
\(971\) 1.10321e12 0.0398265 0.0199132 0.999802i \(-0.493661\pi\)
0.0199132 + 0.999802i \(0.493661\pi\)
\(972\) 0 0
\(973\) −7.71436e13 −2.75926
\(974\) 0 0
\(975\) −4.20557e12 −0.149040
\(976\) 0 0
\(977\) −4.02945e13 −1.41488 −0.707441 0.706773i \(-0.750150\pi\)
−0.707441 + 0.706773i \(0.750150\pi\)
\(978\) 0 0
\(979\) 1.20166e11 0.00418081
\(980\) 0 0
\(981\) 3.91487e13 1.34960
\(982\) 0 0
\(983\) −5.04270e13 −1.72255 −0.861276 0.508137i \(-0.830334\pi\)
−0.861276 + 0.508137i \(0.830334\pi\)
\(984\) 0 0
\(985\) −7.88219e13 −2.66799
\(986\) 0 0
\(987\) 2.72460e13 0.913852
\(988\) 0 0
\(989\) 2.26708e13 0.753502
\(990\) 0 0
\(991\) 2.37023e13 0.780653 0.390327 0.920676i \(-0.372362\pi\)
0.390327 + 0.920676i \(0.372362\pi\)
\(992\) 0 0
\(993\) −1.26469e13 −0.412773
\(994\) 0 0
\(995\) 2.59410e12 0.0839040
\(996\) 0 0
\(997\) −3.80094e13 −1.21832 −0.609162 0.793045i \(-0.708494\pi\)
−0.609162 + 0.793045i \(0.708494\pi\)
\(998\) 0 0
\(999\) 4.58015e13 1.45491
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 208.10.a.b.1.1 1
4.3 odd 2 26.10.a.c.1.1 1
12.11 even 2 234.10.a.a.1.1 1
52.51 odd 2 338.10.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.c.1.1 1 4.3 odd 2
208.10.a.b.1.1 1 1.1 even 1 trivial
234.10.a.a.1.1 1 12.11 even 2
338.10.a.b.1.1 1 52.51 odd 2