Properties

Label 49.16.a.a.1.1
Level $49$
Weight $16$
Character 49.1
Self dual yes
Analytic conductor $69.920$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,16,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(69.9198174990\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+216.000 q^{2} +3348.00 q^{3} +13888.0 q^{4} -52110.0 q^{5} +723168. q^{6} -4.07808e6 q^{8} -3.13980e6 q^{9} +O(q^{10})\) \(q+216.000 q^{2} +3348.00 q^{3} +13888.0 q^{4} -52110.0 q^{5} +723168. q^{6} -4.07808e6 q^{8} -3.13980e6 q^{9} -1.12558e7 q^{10} +2.05869e7 q^{11} +4.64970e7 q^{12} +1.90073e8 q^{13} -1.74464e8 q^{15} -1.33595e9 q^{16} -1.64653e9 q^{17} -6.78197e8 q^{18} -1.56326e9 q^{19} -7.23704e8 q^{20} +4.44676e9 q^{22} +9.45112e9 q^{23} -1.36534e10 q^{24} -2.78021e10 q^{25} +4.10558e10 q^{26} -5.85522e10 q^{27} -3.69026e10 q^{29} -3.76843e10 q^{30} -7.15885e10 q^{31} -1.54934e11 q^{32} +6.89248e10 q^{33} -3.55650e11 q^{34} -4.36056e10 q^{36} -1.03365e12 q^{37} -3.37664e11 q^{38} +6.36366e11 q^{39} +2.12509e11 q^{40} -1.64197e12 q^{41} -4.92403e11 q^{43} +2.85910e11 q^{44} +1.63615e11 q^{45} +2.04144e12 q^{46} +3.41068e12 q^{47} -4.47275e12 q^{48} -6.00526e12 q^{50} -5.51258e12 q^{51} +2.63974e12 q^{52} +6.79715e12 q^{53} -1.26473e13 q^{54} -1.07278e12 q^{55} -5.23379e12 q^{57} -7.97095e12 q^{58} -9.85886e12 q^{59} -2.42296e12 q^{60} -4.93184e12 q^{61} -1.54631e13 q^{62} +1.03106e13 q^{64} -9.90472e12 q^{65} +1.48878e13 q^{66} -2.88378e13 q^{67} -2.28670e13 q^{68} +3.16423e13 q^{69} +1.25050e14 q^{71} +1.28044e13 q^{72} +8.21715e13 q^{73} -2.23269e14 q^{74} -9.30815e13 q^{75} -2.17105e13 q^{76} +1.37455e14 q^{78} -2.54131e13 q^{79} +6.96162e13 q^{80} -1.50980e14 q^{81} -3.54666e14 q^{82} +2.81737e14 q^{83} +8.58006e13 q^{85} -1.06359e14 q^{86} -1.23550e14 q^{87} -8.39548e13 q^{88} -7.15619e14 q^{89} +3.53409e13 q^{90} +1.31257e14 q^{92} -2.39678e14 q^{93} +7.36708e14 q^{94} +8.14613e13 q^{95} -5.18719e14 q^{96} -6.12786e14 q^{97} -6.46387e13 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 216.000 1.19324 0.596621 0.802523i \(-0.296509\pi\)
0.596621 + 0.802523i \(0.296509\pi\)
\(3\) 3348.00 0.883845 0.441922 0.897053i \(-0.354297\pi\)
0.441922 + 0.897053i \(0.354297\pi\)
\(4\) 13888.0 0.423828
\(5\) −52110.0 −0.298295 −0.149148 0.988815i \(-0.547653\pi\)
−0.149148 + 0.988815i \(0.547653\pi\)
\(6\) 723168. 1.05464
\(7\) 0 0
\(8\) −4.07808e6 −0.687513
\(9\) −3.13980e6 −0.218818
\(10\) −1.12558e7 −0.355938
\(11\) 2.05869e7 0.318526 0.159263 0.987236i \(-0.449088\pi\)
0.159263 + 0.987236i \(0.449088\pi\)
\(12\) 4.64970e7 0.374598
\(13\) 1.90073e8 0.840129 0.420065 0.907494i \(-0.362007\pi\)
0.420065 + 0.907494i \(0.362007\pi\)
\(14\) 0 0
\(15\) −1.74464e8 −0.263647
\(16\) −1.33595e9 −1.24420
\(17\) −1.64653e9 −0.973200 −0.486600 0.873625i \(-0.661763\pi\)
−0.486600 + 0.873625i \(0.661763\pi\)
\(18\) −6.78197e8 −0.261103
\(19\) −1.56326e9 −0.401216 −0.200608 0.979672i \(-0.564292\pi\)
−0.200608 + 0.979672i \(0.564292\pi\)
\(20\) −7.23704e8 −0.126426
\(21\) 0 0
\(22\) 4.44676e9 0.380079
\(23\) 9.45112e9 0.578794 0.289397 0.957209i \(-0.406545\pi\)
0.289397 + 0.957209i \(0.406545\pi\)
\(24\) −1.36534e10 −0.607655
\(25\) −2.78021e10 −0.911020
\(26\) 4.10558e10 1.00248
\(27\) −5.85522e10 −1.07725
\(28\) 0 0
\(29\) −3.69026e10 −0.397257 −0.198629 0.980075i \(-0.563649\pi\)
−0.198629 + 0.980075i \(0.563649\pi\)
\(30\) −3.76843e10 −0.314594
\(31\) −7.15885e10 −0.467337 −0.233669 0.972316i \(-0.575073\pi\)
−0.233669 + 0.972316i \(0.575073\pi\)
\(32\) −1.54934e11 −0.797117
\(33\) 6.89248e10 0.281528
\(34\) −3.55650e11 −1.16126
\(35\) 0 0
\(36\) −4.36056e10 −0.0927413
\(37\) −1.03365e12 −1.79003 −0.895017 0.446031i \(-0.852837\pi\)
−0.895017 + 0.446031i \(0.852837\pi\)
\(38\) −3.37664e11 −0.478748
\(39\) 6.36366e11 0.742544
\(40\) 2.12509e11 0.205082
\(41\) −1.64197e12 −1.31670 −0.658351 0.752711i \(-0.728746\pi\)
−0.658351 + 0.752711i \(0.728746\pi\)
\(42\) 0 0
\(43\) −4.92403e11 −0.276253 −0.138127 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) 2.85910e11 0.135000
\(45\) 1.63615e11 0.0652724
\(46\) 2.04144e12 0.690642
\(47\) 3.41068e12 0.981991 0.490996 0.871162i \(-0.336633\pi\)
0.490996 + 0.871162i \(0.336633\pi\)
\(48\) −4.47275e12 −1.09968
\(49\) 0 0
\(50\) −6.00526e12 −1.08707
\(51\) −5.51258e12 −0.860158
\(52\) 2.63974e12 0.356070
\(53\) 6.79715e12 0.794800 0.397400 0.917645i \(-0.369913\pi\)
0.397400 + 0.917645i \(0.369913\pi\)
\(54\) −1.26473e13 −1.28542
\(55\) −1.07278e12 −0.0950147
\(56\) 0 0
\(57\) −5.23379e12 −0.354613
\(58\) −7.97095e12 −0.474024
\(59\) −9.85886e12 −0.515747 −0.257873 0.966179i \(-0.583022\pi\)
−0.257873 + 0.966179i \(0.583022\pi\)
\(60\) −2.42296e12 −0.111741
\(61\) −4.93184e12 −0.200926 −0.100463 0.994941i \(-0.532032\pi\)
−0.100463 + 0.994941i \(0.532032\pi\)
\(62\) −1.54631e13 −0.557647
\(63\) 0 0
\(64\) 1.03106e13 0.293044
\(65\) −9.90472e12 −0.250606
\(66\) 1.48878e13 0.335931
\(67\) −2.88378e13 −0.581302 −0.290651 0.956829i \(-0.593872\pi\)
−0.290651 + 0.956829i \(0.593872\pi\)
\(68\) −2.28670e13 −0.412470
\(69\) 3.16423e13 0.511564
\(70\) 0 0
\(71\) 1.25050e14 1.63172 0.815862 0.578247i \(-0.196263\pi\)
0.815862 + 0.578247i \(0.196263\pi\)
\(72\) 1.28044e13 0.150440
\(73\) 8.21715e13 0.870562 0.435281 0.900295i \(-0.356649\pi\)
0.435281 + 0.900295i \(0.356649\pi\)
\(74\) −2.23269e14 −2.13595
\(75\) −9.30815e13 −0.805200
\(76\) −2.17105e13 −0.170047
\(77\) 0 0
\(78\) 1.37455e14 0.886035
\(79\) −2.54131e13 −0.148886 −0.0744430 0.997225i \(-0.523718\pi\)
−0.0744430 + 0.997225i \(0.523718\pi\)
\(80\) 6.96162e13 0.371138
\(81\) −1.50980e14 −0.733300
\(82\) −3.54666e14 −1.57114
\(83\) 2.81737e14 1.13961 0.569807 0.821779i \(-0.307018\pi\)
0.569807 + 0.821779i \(0.307018\pi\)
\(84\) 0 0
\(85\) 8.58006e13 0.290301
\(86\) −1.06359e14 −0.329637
\(87\) −1.23550e14 −0.351114
\(88\) −8.39548e13 −0.218991
\(89\) −7.15619e14 −1.71497 −0.857485 0.514509i \(-0.827974\pi\)
−0.857485 + 0.514509i \(0.827974\pi\)
\(90\) 3.53409e13 0.0778858
\(91\) 0 0
\(92\) 1.31257e14 0.245309
\(93\) −2.39678e14 −0.413054
\(94\) 7.36708e14 1.17175
\(95\) 8.14613e13 0.119681
\(96\) −5.18719e14 −0.704528
\(97\) −6.12786e14 −0.770054 −0.385027 0.922905i \(-0.625808\pi\)
−0.385027 + 0.922905i \(0.625808\pi\)
\(98\) 0 0
\(99\) −6.46387e13 −0.0696993
\(100\) −3.86116e14 −0.386116
\(101\) 8.17642e14 0.758844 0.379422 0.925224i \(-0.376123\pi\)
0.379422 + 0.925224i \(0.376123\pi\)
\(102\) −1.19072e15 −1.02638
\(103\) −7.41115e14 −0.593753 −0.296877 0.954916i \(-0.595945\pi\)
−0.296877 + 0.954916i \(0.595945\pi\)
\(104\) −7.75134e14 −0.577600
\(105\) 0 0
\(106\) 1.46818e15 0.948389
\(107\) −2.51430e15 −1.51370 −0.756849 0.653590i \(-0.773262\pi\)
−0.756849 + 0.653590i \(0.773262\pi\)
\(108\) −8.13173e14 −0.456567
\(109\) 1.26835e15 0.664572 0.332286 0.943179i \(-0.392180\pi\)
0.332286 + 0.943179i \(0.392180\pi\)
\(110\) −2.31721e14 −0.113376
\(111\) −3.46067e15 −1.58211
\(112\) 0 0
\(113\) −2.05416e15 −0.821385 −0.410692 0.911774i \(-0.634713\pi\)
−0.410692 + 0.911774i \(0.634713\pi\)
\(114\) −1.13050e15 −0.423139
\(115\) −4.92498e14 −0.172652
\(116\) −5.12503e14 −0.168369
\(117\) −5.96793e14 −0.183836
\(118\) −2.12951e15 −0.615411
\(119\) 0 0
\(120\) 7.11479e14 0.181260
\(121\) −3.75343e15 −0.898541
\(122\) −1.06528e15 −0.239753
\(123\) −5.49733e15 −1.16376
\(124\) −9.94221e14 −0.198071
\(125\) 3.03904e15 0.570048
\(126\) 0 0
\(127\) 2.99068e15 0.498014 0.249007 0.968502i \(-0.419896\pi\)
0.249007 + 0.968502i \(0.419896\pi\)
\(128\) 7.30396e15 1.14679
\(129\) −1.64857e15 −0.244165
\(130\) −2.13942e15 −0.299034
\(131\) 1.62623e15 0.214608 0.107304 0.994226i \(-0.465778\pi\)
0.107304 + 0.994226i \(0.465778\pi\)
\(132\) 9.57227e14 0.119319
\(133\) 0 0
\(134\) −6.22897e15 −0.693634
\(135\) 3.05116e15 0.321337
\(136\) 6.71467e15 0.669088
\(137\) 1.05922e16 0.999038 0.499519 0.866303i \(-0.333510\pi\)
0.499519 + 0.866303i \(0.333510\pi\)
\(138\) 6.83474e15 0.610421
\(139\) 1.86709e16 1.57963 0.789813 0.613347i \(-0.210177\pi\)
0.789813 + 0.613347i \(0.210177\pi\)
\(140\) 0 0
\(141\) 1.14190e16 0.867928
\(142\) 2.70108e16 1.94704
\(143\) 3.91301e15 0.267603
\(144\) 4.19461e15 0.272253
\(145\) 1.92299e15 0.118500
\(146\) 1.77490e16 1.03879
\(147\) 0 0
\(148\) −1.43554e16 −0.758667
\(149\) −1.25560e16 −0.630889 −0.315444 0.948944i \(-0.602154\pi\)
−0.315444 + 0.948944i \(0.602154\pi\)
\(150\) −2.01056e16 −0.960799
\(151\) 2.87588e16 1.30751 0.653753 0.756708i \(-0.273194\pi\)
0.653753 + 0.756708i \(0.273194\pi\)
\(152\) 6.37509e15 0.275841
\(153\) 5.16977e15 0.212954
\(154\) 0 0
\(155\) 3.73048e15 0.139404
\(156\) 8.83784e15 0.314711
\(157\) 1.45276e16 0.493114 0.246557 0.969128i \(-0.420701\pi\)
0.246557 + 0.969128i \(0.420701\pi\)
\(158\) −5.48922e15 −0.177657
\(159\) 2.27569e16 0.702480
\(160\) 8.07362e15 0.237776
\(161\) 0 0
\(162\) −3.26117e16 −0.875005
\(163\) 1.67741e16 0.429767 0.214884 0.976640i \(-0.431063\pi\)
0.214884 + 0.976640i \(0.431063\pi\)
\(164\) −2.28037e16 −0.558055
\(165\) −3.59167e15 −0.0839783
\(166\) 6.08551e16 1.35984
\(167\) −6.41999e16 −1.37139 −0.685695 0.727889i \(-0.740502\pi\)
−0.685695 + 0.727889i \(0.740502\pi\)
\(168\) 0 0
\(169\) −1.50580e16 −0.294183
\(170\) 1.85329e16 0.346399
\(171\) 4.90832e15 0.0877934
\(172\) −6.83849e15 −0.117084
\(173\) 7.59860e16 1.24563 0.622814 0.782370i \(-0.285990\pi\)
0.622814 + 0.782370i \(0.285990\pi\)
\(174\) −2.66868e16 −0.418964
\(175\) 0 0
\(176\) −2.75029e16 −0.396309
\(177\) −3.30075e16 −0.455840
\(178\) −1.54574e17 −2.04638
\(179\) 9.33749e16 1.18531 0.592655 0.805456i \(-0.298080\pi\)
0.592655 + 0.805456i \(0.298080\pi\)
\(180\) 2.27229e15 0.0276643
\(181\) −7.43177e16 −0.867966 −0.433983 0.900921i \(-0.642892\pi\)
−0.433983 + 0.900921i \(0.642892\pi\)
\(182\) 0 0
\(183\) −1.65118e16 −0.177587
\(184\) −3.85424e16 −0.397929
\(185\) 5.38636e16 0.533958
\(186\) −5.17705e16 −0.492873
\(187\) −3.38968e16 −0.309990
\(188\) 4.73676e16 0.416196
\(189\) 0 0
\(190\) 1.75956e16 0.142808
\(191\) −9.86224e16 −0.769529 −0.384765 0.923015i \(-0.625717\pi\)
−0.384765 + 0.923015i \(0.625717\pi\)
\(192\) 3.45197e16 0.259005
\(193\) −8.91178e15 −0.0643109 −0.0321554 0.999483i \(-0.510237\pi\)
−0.0321554 + 0.999483i \(0.510237\pi\)
\(194\) −1.32362e17 −0.918861
\(195\) −3.31610e16 −0.221497
\(196\) 0 0
\(197\) 3.54176e16 0.219140 0.109570 0.993979i \(-0.465053\pi\)
0.109570 + 0.993979i \(0.465053\pi\)
\(198\) −1.39620e16 −0.0831682
\(199\) 2.86461e17 1.64311 0.821556 0.570127i \(-0.193106\pi\)
0.821556 + 0.570127i \(0.193106\pi\)
\(200\) 1.13379e17 0.626338
\(201\) −9.65490e16 −0.513780
\(202\) 1.76611e17 0.905485
\(203\) 0 0
\(204\) −7.65587e16 −0.364559
\(205\) 8.55633e16 0.392766
\(206\) −1.60081e17 −0.708492
\(207\) −2.96746e16 −0.126651
\(208\) −2.53928e17 −1.04529
\(209\) −3.21825e16 −0.127798
\(210\) 0 0
\(211\) 3.75834e17 1.38956 0.694780 0.719222i \(-0.255502\pi\)
0.694780 + 0.719222i \(0.255502\pi\)
\(212\) 9.43988e16 0.336859
\(213\) 4.18668e17 1.44219
\(214\) −5.43089e17 −1.80621
\(215\) 2.56591e16 0.0824050
\(216\) 2.38781e17 0.740621
\(217\) 0 0
\(218\) 2.73964e17 0.792995
\(219\) 2.75110e17 0.769441
\(220\) −1.48988e16 −0.0402699
\(221\) −3.12961e17 −0.817614
\(222\) −7.47504e17 −1.88784
\(223\) 2.53078e16 0.0617970 0.0308985 0.999523i \(-0.490163\pi\)
0.0308985 + 0.999523i \(0.490163\pi\)
\(224\) 0 0
\(225\) 8.72932e16 0.199348
\(226\) −4.43699e17 −0.980111
\(227\) −3.03692e17 −0.648992 −0.324496 0.945887i \(-0.605195\pi\)
−0.324496 + 0.945887i \(0.605195\pi\)
\(228\) −7.26868e16 −0.150295
\(229\) −1.07992e17 −0.216085 −0.108042 0.994146i \(-0.534458\pi\)
−0.108042 + 0.994146i \(0.534458\pi\)
\(230\) −1.06379e17 −0.206015
\(231\) 0 0
\(232\) 1.50492e17 0.273119
\(233\) −7.90506e17 −1.38911 −0.694554 0.719441i \(-0.744398\pi\)
−0.694554 + 0.719441i \(0.744398\pi\)
\(234\) −1.28907e17 −0.219360
\(235\) −1.77731e17 −0.292923
\(236\) −1.36920e17 −0.218588
\(237\) −8.50830e16 −0.131592
\(238\) 0 0
\(239\) 3.52956e17 0.512551 0.256275 0.966604i \(-0.417505\pi\)
0.256275 + 0.966604i \(0.417505\pi\)
\(240\) 2.33075e17 0.328028
\(241\) −6.85690e16 −0.0935405 −0.0467703 0.998906i \(-0.514893\pi\)
−0.0467703 + 0.998906i \(0.514893\pi\)
\(242\) −8.10741e17 −1.07218
\(243\) 3.34679e17 0.429123
\(244\) −6.84934e16 −0.0851580
\(245\) 0 0
\(246\) −1.18742e18 −1.38865
\(247\) −2.97134e17 −0.337073
\(248\) 2.91944e17 0.321300
\(249\) 9.43255e17 1.00724
\(250\) 6.56433e17 0.680205
\(251\) −1.58806e18 −1.59703 −0.798515 0.601975i \(-0.794381\pi\)
−0.798515 + 0.601975i \(0.794381\pi\)
\(252\) 0 0
\(253\) 1.94569e17 0.184361
\(254\) 6.45986e17 0.594251
\(255\) 2.87260e17 0.256581
\(256\) 1.23980e18 1.07535
\(257\) 8.28562e17 0.697954 0.348977 0.937131i \(-0.386529\pi\)
0.348977 + 0.937131i \(0.386529\pi\)
\(258\) −3.56090e17 −0.291348
\(259\) 0 0
\(260\) −1.37557e17 −0.106214
\(261\) 1.15867e17 0.0869271
\(262\) 3.51265e17 0.256080
\(263\) 1.40445e18 0.995038 0.497519 0.867453i \(-0.334244\pi\)
0.497519 + 0.867453i \(0.334244\pi\)
\(264\) −2.81081e17 −0.193554
\(265\) −3.54200e17 −0.237085
\(266\) 0 0
\(267\) −2.39589e18 −1.51577
\(268\) −4.00500e17 −0.246372
\(269\) −1.43582e18 −0.858930 −0.429465 0.903083i \(-0.641298\pi\)
−0.429465 + 0.903083i \(0.641298\pi\)
\(270\) 6.59050e17 0.383433
\(271\) −5.09160e17 −0.288127 −0.144064 0.989568i \(-0.546017\pi\)
−0.144064 + 0.989568i \(0.546017\pi\)
\(272\) 2.19967e18 1.21085
\(273\) 0 0
\(274\) 2.28792e18 1.19209
\(275\) −5.72358e17 −0.290184
\(276\) 4.39449e17 0.216815
\(277\) 5.68946e17 0.273195 0.136598 0.990627i \(-0.456383\pi\)
0.136598 + 0.990627i \(0.456383\pi\)
\(278\) 4.03292e18 1.88488
\(279\) 2.24774e17 0.102262
\(280\) 0 0
\(281\) −4.06184e18 −1.75156 −0.875780 0.482710i \(-0.839652\pi\)
−0.875780 + 0.482710i \(0.839652\pi\)
\(282\) 2.46650e18 1.03565
\(283\) −2.78506e18 −1.13877 −0.569385 0.822071i \(-0.692819\pi\)
−0.569385 + 0.822071i \(0.692819\pi\)
\(284\) 1.73670e18 0.691570
\(285\) 2.72733e17 0.105779
\(286\) 8.45211e17 0.319315
\(287\) 0 0
\(288\) 4.86463e17 0.174424
\(289\) −1.51369e17 −0.0528813
\(290\) 4.15366e17 0.141399
\(291\) −2.05161e18 −0.680608
\(292\) 1.14120e18 0.368969
\(293\) 3.63803e18 1.14646 0.573230 0.819395i \(-0.305690\pi\)
0.573230 + 0.819395i \(0.305690\pi\)
\(294\) 0 0
\(295\) 5.13745e17 0.153845
\(296\) 4.21532e18 1.23067
\(297\) −1.20541e18 −0.343131
\(298\) −2.71209e18 −0.752803
\(299\) 1.79641e18 0.486262
\(300\) −1.29272e18 −0.341267
\(301\) 0 0
\(302\) 6.21190e18 1.56017
\(303\) 2.73746e18 0.670701
\(304\) 2.08843e18 0.499192
\(305\) 2.56998e17 0.0599351
\(306\) 1.11667e18 0.254106
\(307\) 9.75296e17 0.216570 0.108285 0.994120i \(-0.465464\pi\)
0.108285 + 0.994120i \(0.465464\pi\)
\(308\) 0 0
\(309\) −2.48125e18 −0.524786
\(310\) 8.05783e17 0.166343
\(311\) −3.36692e17 −0.0678468 −0.0339234 0.999424i \(-0.510800\pi\)
−0.0339234 + 0.999424i \(0.510800\pi\)
\(312\) −2.59515e18 −0.510508
\(313\) −3.65551e18 −0.702046 −0.351023 0.936367i \(-0.614166\pi\)
−0.351023 + 0.936367i \(0.614166\pi\)
\(314\) 3.13797e18 0.588405
\(315\) 0 0
\(316\) −3.52937e17 −0.0631021
\(317\) −7.97380e17 −0.139226 −0.0696131 0.997574i \(-0.522176\pi\)
−0.0696131 + 0.997574i \(0.522176\pi\)
\(318\) 4.91548e18 0.838229
\(319\) −7.59708e17 −0.126537
\(320\) −5.37283e17 −0.0874135
\(321\) −8.41788e18 −1.33787
\(322\) 0 0
\(323\) 2.57395e18 0.390464
\(324\) −2.09681e18 −0.310793
\(325\) −5.28444e18 −0.765375
\(326\) 3.62321e18 0.512816
\(327\) 4.24645e18 0.587378
\(328\) 6.69610e18 0.905249
\(329\) 0 0
\(330\) −7.75801e17 −0.100206
\(331\) −1.01585e19 −1.28269 −0.641343 0.767255i \(-0.721622\pi\)
−0.641343 + 0.767255i \(0.721622\pi\)
\(332\) 3.91276e18 0.483000
\(333\) 3.24546e18 0.391692
\(334\) −1.38672e19 −1.63640
\(335\) 1.50274e18 0.173399
\(336\) 0 0
\(337\) −4.81465e18 −0.531301 −0.265651 0.964069i \(-0.585587\pi\)
−0.265651 + 0.964069i \(0.585587\pi\)
\(338\) −3.25253e18 −0.351032
\(339\) −6.87734e18 −0.725977
\(340\) 1.19160e18 0.123038
\(341\) −1.47378e18 −0.148859
\(342\) 1.06020e18 0.104759
\(343\) 0 0
\(344\) 2.00806e18 0.189928
\(345\) −1.64888e18 −0.152597
\(346\) 1.64130e19 1.48634
\(347\) 4.50275e18 0.399031 0.199516 0.979895i \(-0.436063\pi\)
0.199516 + 0.979895i \(0.436063\pi\)
\(348\) −1.71586e18 −0.148812
\(349\) −2.24323e19 −1.90407 −0.952036 0.305986i \(-0.901014\pi\)
−0.952036 + 0.305986i \(0.901014\pi\)
\(350\) 0 0
\(351\) −1.11292e19 −0.905026
\(352\) −3.18961e18 −0.253902
\(353\) −8.02510e18 −0.625374 −0.312687 0.949856i \(-0.601229\pi\)
−0.312687 + 0.949856i \(0.601229\pi\)
\(354\) −7.12961e18 −0.543928
\(355\) −6.51636e18 −0.486735
\(356\) −9.93851e18 −0.726853
\(357\) 0 0
\(358\) 2.01690e19 1.41436
\(359\) 1.61507e18 0.110913 0.0554567 0.998461i \(-0.482339\pi\)
0.0554567 + 0.998461i \(0.482339\pi\)
\(360\) −6.67236e17 −0.0448756
\(361\) −1.27374e19 −0.839026
\(362\) −1.60526e19 −1.03569
\(363\) −1.25665e19 −0.794171
\(364\) 0 0
\(365\) −4.28195e18 −0.259684
\(366\) −3.56655e18 −0.211905
\(367\) 9.97799e18 0.580828 0.290414 0.956901i \(-0.406207\pi\)
0.290414 + 0.956901i \(0.406207\pi\)
\(368\) −1.26262e19 −0.720135
\(369\) 5.15547e18 0.288118
\(370\) 1.16345e19 0.637142
\(371\) 0 0
\(372\) −3.32865e18 −0.175064
\(373\) −2.36866e19 −1.22092 −0.610459 0.792048i \(-0.709015\pi\)
−0.610459 + 0.792048i \(0.709015\pi\)
\(374\) −7.32171e18 −0.369893
\(375\) 1.01747e19 0.503834
\(376\) −1.39090e19 −0.675132
\(377\) −7.01419e18 −0.333747
\(378\) 0 0
\(379\) 1.86851e19 0.854480 0.427240 0.904138i \(-0.359486\pi\)
0.427240 + 0.904138i \(0.359486\pi\)
\(380\) 1.13133e18 0.0507241
\(381\) 1.00128e19 0.440167
\(382\) −2.13024e19 −0.918235
\(383\) 3.02521e19 1.27869 0.639343 0.768921i \(-0.279206\pi\)
0.639343 + 0.768921i \(0.279206\pi\)
\(384\) 2.44537e19 1.01358
\(385\) 0 0
\(386\) −1.92494e18 −0.0767385
\(387\) 1.54605e18 0.0604493
\(388\) −8.51037e18 −0.326370
\(389\) −1.00714e18 −0.0378852 −0.0189426 0.999821i \(-0.506030\pi\)
−0.0189426 + 0.999821i \(0.506030\pi\)
\(390\) −7.16278e18 −0.264300
\(391\) −1.55615e19 −0.563283
\(392\) 0 0
\(393\) 5.44461e18 0.189680
\(394\) 7.65020e18 0.261487
\(395\) 1.32428e18 0.0444120
\(396\) −8.97702e17 −0.0295405
\(397\) −3.56324e19 −1.15058 −0.575290 0.817950i \(-0.695111\pi\)
−0.575290 + 0.817950i \(0.695111\pi\)
\(398\) 6.18756e19 1.96063
\(399\) 0 0
\(400\) 3.71422e19 1.13349
\(401\) 3.94327e19 1.18106 0.590532 0.807014i \(-0.298918\pi\)
0.590532 + 0.807014i \(0.298918\pi\)
\(402\) −2.08546e19 −0.613065
\(403\) −1.36071e19 −0.392624
\(404\) 1.13554e19 0.321620
\(405\) 7.86757e18 0.218740
\(406\) 0 0
\(407\) −2.12796e19 −0.570172
\(408\) 2.24807e19 0.591370
\(409\) 5.27823e19 1.36321 0.681607 0.731719i \(-0.261282\pi\)
0.681607 + 0.731719i \(0.261282\pi\)
\(410\) 1.84817e19 0.468665
\(411\) 3.54627e19 0.882995
\(412\) −1.02926e19 −0.251649
\(413\) 0 0
\(414\) −6.40972e18 −0.151125
\(415\) −1.46813e19 −0.339941
\(416\) −2.94488e19 −0.669681
\(417\) 6.25102e19 1.39615
\(418\) −6.95143e18 −0.152494
\(419\) −8.62630e18 −0.185874 −0.0929372 0.995672i \(-0.529626\pi\)
−0.0929372 + 0.995672i \(0.529626\pi\)
\(420\) 0 0
\(421\) −4.29249e19 −0.892469 −0.446235 0.894916i \(-0.647235\pi\)
−0.446235 + 0.894916i \(0.647235\pi\)
\(422\) 8.11801e19 1.65808
\(423\) −1.07089e19 −0.214878
\(424\) −2.77193e19 −0.546435
\(425\) 4.57770e19 0.886605
\(426\) 9.04322e19 1.72088
\(427\) 0 0
\(428\) −3.49186e19 −0.641547
\(429\) 1.31008e19 0.236519
\(430\) 5.54237e18 0.0983291
\(431\) 5.04764e19 0.880053 0.440026 0.897985i \(-0.354969\pi\)
0.440026 + 0.897985i \(0.354969\pi\)
\(432\) 7.82227e19 1.34031
\(433\) −5.05734e19 −0.851653 −0.425827 0.904805i \(-0.640017\pi\)
−0.425827 + 0.904805i \(0.640017\pi\)
\(434\) 0 0
\(435\) 6.43818e18 0.104735
\(436\) 1.76149e19 0.281664
\(437\) −1.47745e19 −0.232222
\(438\) 5.94238e19 0.918130
\(439\) −2.47946e19 −0.376594 −0.188297 0.982112i \(-0.560297\pi\)
−0.188297 + 0.982112i \(0.560297\pi\)
\(440\) 4.37489e18 0.0653238
\(441\) 0 0
\(442\) −6.75996e19 −0.975612
\(443\) −1.30654e20 −1.85394 −0.926970 0.375135i \(-0.877596\pi\)
−0.926970 + 0.375135i \(0.877596\pi\)
\(444\) −4.80617e19 −0.670544
\(445\) 3.72909e19 0.511567
\(446\) 5.46648e18 0.0737389
\(447\) −4.20373e19 −0.557608
\(448\) 0 0
\(449\) −7.78280e19 −0.998363 −0.499181 0.866498i \(-0.666366\pi\)
−0.499181 + 0.866498i \(0.666366\pi\)
\(450\) 1.88553e19 0.237870
\(451\) −3.38031e19 −0.419404
\(452\) −2.85282e19 −0.348126
\(453\) 9.62844e19 1.15563
\(454\) −6.55975e19 −0.774405
\(455\) 0 0
\(456\) 2.13438e19 0.243801
\(457\) −1.18451e20 −1.33096 −0.665482 0.746414i \(-0.731774\pi\)
−0.665482 + 0.746414i \(0.731774\pi\)
\(458\) −2.33262e19 −0.257841
\(459\) 9.64078e19 1.04838
\(460\) −6.83981e18 −0.0731746
\(461\) −1.38643e20 −1.45929 −0.729644 0.683827i \(-0.760314\pi\)
−0.729644 + 0.683827i \(0.760314\pi\)
\(462\) 0 0
\(463\) 1.75645e20 1.78969 0.894846 0.446375i \(-0.147285\pi\)
0.894846 + 0.446375i \(0.147285\pi\)
\(464\) 4.92999e19 0.494266
\(465\) 1.24896e19 0.123212
\(466\) −1.70749e20 −1.65754
\(467\) 1.36631e20 1.30519 0.652593 0.757708i \(-0.273681\pi\)
0.652593 + 0.757708i \(0.273681\pi\)
\(468\) −8.28826e18 −0.0779147
\(469\) 0 0
\(470\) −3.83899e19 −0.349528
\(471\) 4.86385e19 0.435837
\(472\) 4.02052e19 0.354583
\(473\) −1.01370e19 −0.0879938
\(474\) −1.83779e19 −0.157021
\(475\) 4.34619e19 0.365516
\(476\) 0 0
\(477\) −2.13417e19 −0.173917
\(478\) 7.62386e19 0.611597
\(479\) −6.41058e19 −0.506269 −0.253134 0.967431i \(-0.581461\pi\)
−0.253134 + 0.967431i \(0.581461\pi\)
\(480\) 2.70305e19 0.210157
\(481\) −1.96470e20 −1.50386
\(482\) −1.48109e19 −0.111617
\(483\) 0 0
\(484\) −5.21276e19 −0.380827
\(485\) 3.19323e19 0.229703
\(486\) 7.22907e19 0.512047
\(487\) −2.41343e19 −0.168332 −0.0841662 0.996452i \(-0.526823\pi\)
−0.0841662 + 0.996452i \(0.526823\pi\)
\(488\) 2.01124e19 0.138139
\(489\) 5.61598e19 0.379847
\(490\) 0 0
\(491\) −2.80908e19 −0.184269 −0.0921346 0.995747i \(-0.529369\pi\)
−0.0921346 + 0.995747i \(0.529369\pi\)
\(492\) −7.63469e19 −0.493234
\(493\) 6.07611e19 0.386611
\(494\) −6.41808e19 −0.402210
\(495\) 3.36832e18 0.0207910
\(496\) 9.56384e19 0.581460
\(497\) 0 0
\(498\) 2.03743e20 1.20188
\(499\) −1.71994e20 −0.999443 −0.499722 0.866186i \(-0.666564\pi\)
−0.499722 + 0.866186i \(0.666564\pi\)
\(500\) 4.22062e19 0.241602
\(501\) −2.14941e20 −1.21210
\(502\) −3.43020e20 −1.90564
\(503\) 1.83497e20 1.00431 0.502155 0.864778i \(-0.332541\pi\)
0.502155 + 0.864778i \(0.332541\pi\)
\(504\) 0 0
\(505\) −4.26073e19 −0.226359
\(506\) 4.20268e19 0.219987
\(507\) −5.04142e19 −0.260012
\(508\) 4.15345e19 0.211072
\(509\) −2.67204e20 −1.33801 −0.669004 0.743258i \(-0.733279\pi\)
−0.669004 + 0.743258i \(0.733279\pi\)
\(510\) 6.20482e19 0.306163
\(511\) 0 0
\(512\) 2.84604e19 0.136369
\(513\) 9.15321e19 0.432209
\(514\) 1.78969e20 0.832828
\(515\) 3.86195e19 0.177114
\(516\) −2.28953e19 −0.103484
\(517\) 7.02153e19 0.312790
\(518\) 0 0
\(519\) 2.54401e20 1.10094
\(520\) 4.03922e19 0.172295
\(521\) 2.01468e20 0.847076 0.423538 0.905878i \(-0.360788\pi\)
0.423538 + 0.905878i \(0.360788\pi\)
\(522\) 2.50272e19 0.103725
\(523\) −3.58989e20 −1.46662 −0.733311 0.679894i \(-0.762026\pi\)
−0.733311 + 0.679894i \(0.762026\pi\)
\(524\) 2.25850e19 0.0909570
\(525\) 0 0
\(526\) 3.03362e20 1.18732
\(527\) 1.17872e20 0.454813
\(528\) −9.20799e19 −0.350276
\(529\) −1.77312e20 −0.664997
\(530\) −7.65071e19 −0.282900
\(531\) 3.09549e19 0.112855
\(532\) 0 0
\(533\) −3.12095e20 −1.10620
\(534\) −5.17512e20 −1.80868
\(535\) 1.31020e20 0.451528
\(536\) 1.17603e20 0.399652
\(537\) 3.12619e20 1.04763
\(538\) −3.10137e20 −1.02491
\(539\) 0 0
\(540\) 4.23744e19 0.136192
\(541\) 2.02328e20 0.641323 0.320662 0.947194i \(-0.396095\pi\)
0.320662 + 0.947194i \(0.396095\pi\)
\(542\) −1.09979e20 −0.343806
\(543\) −2.48816e20 −0.767147
\(544\) 2.55103e20 0.775755
\(545\) −6.60939e19 −0.198238
\(546\) 0 0
\(547\) 7.40963e19 0.216218 0.108109 0.994139i \(-0.465520\pi\)
0.108109 + 0.994139i \(0.465520\pi\)
\(548\) 1.47104e20 0.423420
\(549\) 1.54850e19 0.0439662
\(550\) −1.23629e20 −0.346259
\(551\) 5.76882e19 0.159386
\(552\) −1.29040e20 −0.351707
\(553\) 0 0
\(554\) 1.22892e20 0.325988
\(555\) 1.80335e20 0.471936
\(556\) 2.59302e20 0.669490
\(557\) 2.09626e18 0.00533987 0.00266994 0.999996i \(-0.499150\pi\)
0.00266994 + 0.999996i \(0.499150\pi\)
\(558\) 4.85511e19 0.122023
\(559\) −9.35927e19 −0.232088
\(560\) 0 0
\(561\) −1.13487e20 −0.273983
\(562\) −8.77357e20 −2.09004
\(563\) −6.87353e20 −1.61572 −0.807861 0.589373i \(-0.799375\pi\)
−0.807861 + 0.589373i \(0.799375\pi\)
\(564\) 1.58587e20 0.367852
\(565\) 1.07042e20 0.245015
\(566\) −6.01573e20 −1.35883
\(567\) 0 0
\(568\) −5.09964e20 −1.12183
\(569\) −9.05218e19 −0.196522 −0.0982610 0.995161i \(-0.531328\pi\)
−0.0982610 + 0.995161i \(0.531328\pi\)
\(570\) 5.89102e19 0.126220
\(571\) 2.05774e20 0.435130 0.217565 0.976046i \(-0.430189\pi\)
0.217565 + 0.976046i \(0.430189\pi\)
\(572\) 5.43439e19 0.113418
\(573\) −3.30188e20 −0.680145
\(574\) 0 0
\(575\) −2.62761e20 −0.527293
\(576\) −3.23731e19 −0.0641233
\(577\) −5.70778e20 −1.11596 −0.557980 0.829854i \(-0.688424\pi\)
−0.557980 + 0.829854i \(0.688424\pi\)
\(578\) −3.26956e19 −0.0631002
\(579\) −2.98366e19 −0.0568408
\(580\) 2.67065e19 0.0502236
\(581\) 0 0
\(582\) −4.43147e20 −0.812130
\(583\) 1.39932e20 0.253164
\(584\) −3.35102e20 −0.598522
\(585\) 3.10989e19 0.0548373
\(586\) 7.85814e20 1.36800
\(587\) 9.30363e20 1.59907 0.799534 0.600621i \(-0.205080\pi\)
0.799534 + 0.600621i \(0.205080\pi\)
\(588\) 0 0
\(589\) 1.11911e20 0.187503
\(590\) 1.10969e20 0.183574
\(591\) 1.18578e20 0.193686
\(592\) 1.38090e21 2.22716
\(593\) −3.54225e20 −0.564116 −0.282058 0.959397i \(-0.591017\pi\)
−0.282058 + 0.959397i \(0.591017\pi\)
\(594\) −2.60368e20 −0.409438
\(595\) 0 0
\(596\) −1.74377e20 −0.267388
\(597\) 9.59071e20 1.45226
\(598\) 3.88024e20 0.580229
\(599\) −3.30045e20 −0.487385 −0.243693 0.969853i \(-0.578359\pi\)
−0.243693 + 0.969853i \(0.578359\pi\)
\(600\) 3.79594e20 0.553586
\(601\) 3.35884e20 0.483761 0.241880 0.970306i \(-0.422236\pi\)
0.241880 + 0.970306i \(0.422236\pi\)
\(602\) 0 0
\(603\) 9.05451e19 0.127199
\(604\) 3.99402e20 0.554158
\(605\) 1.95591e20 0.268030
\(606\) 5.91292e20 0.800309
\(607\) 1.33438e21 1.78387 0.891934 0.452165i \(-0.149348\pi\)
0.891934 + 0.452165i \(0.149348\pi\)
\(608\) 2.42202e20 0.319816
\(609\) 0 0
\(610\) 5.55116e19 0.0715172
\(611\) 6.48280e20 0.825000
\(612\) 7.17978e19 0.0902559
\(613\) 5.68844e18 0.00706381 0.00353191 0.999994i \(-0.498876\pi\)
0.00353191 + 0.999994i \(0.498876\pi\)
\(614\) 2.10664e20 0.258421
\(615\) 2.86466e20 0.347144
\(616\) 0 0
\(617\) 3.98915e20 0.471783 0.235891 0.971779i \(-0.424199\pi\)
0.235891 + 0.971779i \(0.424199\pi\)
\(618\) −5.35950e20 −0.626197
\(619\) 5.40017e20 0.623343 0.311672 0.950190i \(-0.399111\pi\)
0.311672 + 0.950190i \(0.399111\pi\)
\(620\) 5.18088e19 0.0590835
\(621\) −5.53384e20 −0.623504
\(622\) −7.27254e19 −0.0809577
\(623\) 0 0
\(624\) −8.50151e20 −0.923871
\(625\) 6.90089e20 0.740978
\(626\) −7.89591e20 −0.837711
\(627\) −1.07747e20 −0.112953
\(628\) 2.01760e20 0.208996
\(629\) 1.70194e21 1.74206
\(630\) 0 0
\(631\) 9.59111e20 0.958625 0.479312 0.877644i \(-0.340886\pi\)
0.479312 + 0.877644i \(0.340886\pi\)
\(632\) 1.03637e20 0.102361
\(633\) 1.25829e21 1.22816
\(634\) −1.72234e20 −0.166131
\(635\) −1.55844e20 −0.148555
\(636\) 3.16047e20 0.297731
\(637\) 0 0
\(638\) −1.64097e20 −0.150989
\(639\) −3.92633e20 −0.357051
\(640\) −3.80609e20 −0.342082
\(641\) −9.25925e20 −0.822509 −0.411255 0.911521i \(-0.634909\pi\)
−0.411255 + 0.911521i \(0.634909\pi\)
\(642\) −1.81826e21 −1.59641
\(643\) 7.65928e20 0.664669 0.332335 0.943162i \(-0.392164\pi\)
0.332335 + 0.943162i \(0.392164\pi\)
\(644\) 0 0
\(645\) 8.59068e19 0.0728332
\(646\) 5.55972e20 0.465918
\(647\) −1.36075e21 −1.12719 −0.563596 0.826051i \(-0.690582\pi\)
−0.563596 + 0.826051i \(0.690582\pi\)
\(648\) 6.15709e20 0.504153
\(649\) −2.02963e20 −0.164279
\(650\) −1.14144e21 −0.913278
\(651\) 0 0
\(652\) 2.32959e20 0.182147
\(653\) −2.71809e20 −0.210094 −0.105047 0.994467i \(-0.533499\pi\)
−0.105047 + 0.994467i \(0.533499\pi\)
\(654\) 9.17233e20 0.700885
\(655\) −8.47427e19 −0.0640166
\(656\) 2.19359e21 1.63824
\(657\) −2.58002e20 −0.190495
\(658\) 0 0
\(659\) 6.74316e20 0.486657 0.243329 0.969944i \(-0.421761\pi\)
0.243329 + 0.969944i \(0.421761\pi\)
\(660\) −4.98811e19 −0.0355923
\(661\) −1.26727e21 −0.894042 −0.447021 0.894524i \(-0.647515\pi\)
−0.447021 + 0.894524i \(0.647515\pi\)
\(662\) −2.19424e21 −1.53056
\(663\) −1.04779e21 −0.722644
\(664\) −1.14894e21 −0.783499
\(665\) 0 0
\(666\) 7.01020e20 0.467384
\(667\) −3.48770e20 −0.229930
\(668\) −8.91609e20 −0.581234
\(669\) 8.47305e19 0.0546190
\(670\) 3.24592e20 0.206908
\(671\) −1.01531e20 −0.0640000
\(672\) 0 0
\(673\) −1.13945e21 −0.702394 −0.351197 0.936302i \(-0.614225\pi\)
−0.351197 + 0.936302i \(0.614225\pi\)
\(674\) −1.03997e21 −0.633971
\(675\) 1.62788e21 0.981393
\(676\) −2.09126e20 −0.124683
\(677\) −1.74431e21 −1.02851 −0.514256 0.857637i \(-0.671932\pi\)
−0.514256 + 0.857637i \(0.671932\pi\)
\(678\) −1.48550e21 −0.866266
\(679\) 0 0
\(680\) −3.49902e20 −0.199586
\(681\) −1.01676e21 −0.573608
\(682\) −3.18337e20 −0.177625
\(683\) −1.43739e21 −0.793267 −0.396634 0.917977i \(-0.629822\pi\)
−0.396634 + 0.917977i \(0.629822\pi\)
\(684\) 6.81667e19 0.0372093
\(685\) −5.51960e20 −0.298008
\(686\) 0 0
\(687\) −3.61556e20 −0.190985
\(688\) 6.57825e20 0.343714
\(689\) 1.29196e21 0.667735
\(690\) −3.56159e20 −0.182085
\(691\) 1.77548e21 0.897903 0.448951 0.893556i \(-0.351798\pi\)
0.448951 + 0.893556i \(0.351798\pi\)
\(692\) 1.05529e21 0.527932
\(693\) 0 0
\(694\) 9.72594e20 0.476141
\(695\) −9.72941e20 −0.471195
\(696\) 5.03846e20 0.241395
\(697\) 2.70356e21 1.28141
\(698\) −4.84538e21 −2.27202
\(699\) −2.64661e21 −1.22776
\(700\) 0 0
\(701\) 1.43100e21 0.649764 0.324882 0.945755i \(-0.394675\pi\)
0.324882 + 0.945755i \(0.394675\pi\)
\(702\) −2.40391e21 −1.07992
\(703\) 1.61586e21 0.718191
\(704\) 2.12262e20 0.0933420
\(705\) −5.95043e20 −0.258899
\(706\) −1.73342e21 −0.746223
\(707\) 0 0
\(708\) −4.58408e20 −0.193198
\(709\) −2.41840e21 −1.00851 −0.504257 0.863554i \(-0.668234\pi\)
−0.504257 + 0.863554i \(0.668234\pi\)
\(710\) −1.40753e21 −0.580793
\(711\) 7.97921e19 0.0325790
\(712\) 2.91835e21 1.17906
\(713\) −6.76591e20 −0.270492
\(714\) 0 0
\(715\) −2.03907e20 −0.0798246
\(716\) 1.29679e21 0.502368
\(717\) 1.18170e21 0.453015
\(718\) 3.48856e20 0.132347
\(719\) −4.74444e21 −1.78122 −0.890611 0.454766i \(-0.849723\pi\)
−0.890611 + 0.454766i \(0.849723\pi\)
\(720\) −2.18581e20 −0.0812118
\(721\) 0 0
\(722\) −2.75127e21 −1.00116
\(723\) −2.29569e20 −0.0826753
\(724\) −1.03212e21 −0.367868
\(725\) 1.02597e21 0.361909
\(726\) −2.71436e21 −0.947639
\(727\) 3.59265e21 1.24138 0.620692 0.784054i \(-0.286852\pi\)
0.620692 + 0.784054i \(0.286852\pi\)
\(728\) 0 0
\(729\) 3.28690e21 1.11258
\(730\) −9.24902e20 −0.309866
\(731\) 8.10755e20 0.268850
\(732\) −2.29316e20 −0.0752664
\(733\) 2.76824e21 0.899339 0.449669 0.893195i \(-0.351542\pi\)
0.449669 + 0.893195i \(0.351542\pi\)
\(734\) 2.15525e21 0.693069
\(735\) 0 0
\(736\) −1.46430e21 −0.461367
\(737\) −5.93680e20 −0.185160
\(738\) 1.11358e21 0.343795
\(739\) 3.55824e21 1.08743 0.543716 0.839269i \(-0.317017\pi\)
0.543716 + 0.839269i \(0.317017\pi\)
\(740\) 7.48058e20 0.226307
\(741\) −9.94803e20 −0.297921
\(742\) 0 0
\(743\) −1.94092e21 −0.569628 −0.284814 0.958583i \(-0.591932\pi\)
−0.284814 + 0.958583i \(0.591932\pi\)
\(744\) 9.77427e20 0.283980
\(745\) 6.54291e20 0.188191
\(746\) −5.11630e21 −1.45685
\(747\) −8.84598e20 −0.249368
\(748\) −4.70759e20 −0.131382
\(749\) 0 0
\(750\) 2.19774e21 0.601196
\(751\) 4.75565e21 1.28798 0.643992 0.765032i \(-0.277277\pi\)
0.643992 + 0.765032i \(0.277277\pi\)
\(752\) −4.55650e21 −1.22179
\(753\) −5.31681e21 −1.41153
\(754\) −1.51507e21 −0.398241
\(755\) −1.49862e21 −0.390022
\(756\) 0 0
\(757\) 3.62137e21 0.923960 0.461980 0.886890i \(-0.347139\pi\)
0.461980 + 0.886890i \(0.347139\pi\)
\(758\) 4.03599e21 1.01960
\(759\) 6.51416e20 0.162947
\(760\) −3.32206e20 −0.0822821
\(761\) 3.86361e21 0.947564 0.473782 0.880642i \(-0.342888\pi\)
0.473782 + 0.880642i \(0.342888\pi\)
\(762\) 2.16276e21 0.525226
\(763\) 0 0
\(764\) −1.36967e21 −0.326148
\(765\) −2.69397e20 −0.0635231
\(766\) 6.53445e21 1.52578
\(767\) −1.87391e21 −0.433294
\(768\) 4.15085e21 0.950446
\(769\) 5.39327e21 1.22294 0.611469 0.791268i \(-0.290579\pi\)
0.611469 + 0.791268i \(0.290579\pi\)
\(770\) 0 0
\(771\) 2.77403e21 0.616883
\(772\) −1.23767e20 −0.0272568
\(773\) −6.57037e21 −1.43299 −0.716496 0.697591i \(-0.754255\pi\)
−0.716496 + 0.697591i \(0.754255\pi\)
\(774\) 3.33947e20 0.0721306
\(775\) 1.99031e21 0.425754
\(776\) 2.49899e21 0.529422
\(777\) 0 0
\(778\) −2.17543e20 −0.0452062
\(779\) 2.56683e21 0.528282
\(780\) −4.60540e20 −0.0938767
\(781\) 2.57439e21 0.519746
\(782\) −3.36129e21 −0.672133
\(783\) 2.16073e21 0.427944
\(784\) 0 0
\(785\) −7.57035e20 −0.147094
\(786\) 1.17604e21 0.226335
\(787\) 3.72074e20 0.0709281 0.0354641 0.999371i \(-0.488709\pi\)
0.0354641 + 0.999371i \(0.488709\pi\)
\(788\) 4.91879e20 0.0928778
\(789\) 4.70211e21 0.879459
\(790\) 2.86044e20 0.0529943
\(791\) 0 0
\(792\) 2.63602e20 0.0479192
\(793\) −9.37412e20 −0.168804
\(794\) −7.69660e21 −1.37292
\(795\) −1.18586e21 −0.209546
\(796\) 3.97837e21 0.696397
\(797\) −2.61511e21 −0.453474 −0.226737 0.973956i \(-0.572806\pi\)
−0.226737 + 0.973956i \(0.572806\pi\)
\(798\) 0 0
\(799\) −5.61579e21 −0.955674
\(800\) 4.30750e21 0.726190
\(801\) 2.24690e21 0.375267
\(802\) 8.51746e21 1.40930
\(803\) 1.69165e21 0.277296
\(804\) −1.34087e21 −0.217755
\(805\) 0 0
\(806\) −2.93913e21 −0.468495
\(807\) −4.80713e21 −0.759161
\(808\) −3.33441e21 −0.521715
\(809\) 5.34899e21 0.829198 0.414599 0.910004i \(-0.363922\pi\)
0.414599 + 0.910004i \(0.363922\pi\)
\(810\) 1.69939e21 0.261010
\(811\) −8.46492e21 −1.28815 −0.644075 0.764962i \(-0.722758\pi\)
−0.644075 + 0.764962i \(0.722758\pi\)
\(812\) 0 0
\(813\) −1.70467e21 −0.254660
\(814\) −4.59640e21 −0.680354
\(815\) −8.74100e20 −0.128197
\(816\) 7.36451e21 1.07021
\(817\) 7.69753e20 0.110837
\(818\) 1.14010e22 1.62664
\(819\) 0 0
\(820\) 1.18830e21 0.166465
\(821\) 7.99397e21 1.10966 0.554829 0.831965i \(-0.312784\pi\)
0.554829 + 0.831965i \(0.312784\pi\)
\(822\) 7.65994e21 1.05363
\(823\) 1.96841e21 0.268297 0.134148 0.990961i \(-0.457170\pi\)
0.134148 + 0.990961i \(0.457170\pi\)
\(824\) 3.02232e21 0.408213
\(825\) −1.91626e21 −0.256477
\(826\) 0 0
\(827\) −1.43539e22 −1.88659 −0.943296 0.331954i \(-0.892292\pi\)
−0.943296 + 0.331954i \(0.892292\pi\)
\(828\) −4.12121e20 −0.0536782
\(829\) 8.83327e21 1.14015 0.570076 0.821592i \(-0.306914\pi\)
0.570076 + 0.821592i \(0.306914\pi\)
\(830\) −3.17116e21 −0.405632
\(831\) 1.90483e21 0.241462
\(832\) 1.95976e21 0.246195
\(833\) 0 0
\(834\) 1.35022e22 1.66594
\(835\) 3.34546e21 0.409079
\(836\) −4.46951e20 −0.0541643
\(837\) 4.19166e21 0.503437
\(838\) −1.86328e21 −0.221793
\(839\) −1.26696e22 −1.49469 −0.747343 0.664439i \(-0.768671\pi\)
−0.747343 + 0.664439i \(0.768671\pi\)
\(840\) 0 0
\(841\) −7.26739e21 −0.842187
\(842\) −9.27177e21 −1.06493
\(843\) −1.35990e22 −1.54811
\(844\) 5.21958e21 0.588935
\(845\) 7.84673e20 0.0877533
\(846\) −2.31312e21 −0.256401
\(847\) 0 0
\(848\) −9.08064e21 −0.988889
\(849\) −9.32438e21 −1.00650
\(850\) 9.88783e21 1.05793
\(851\) −9.76917e21 −1.03606
\(852\) 5.81446e21 0.611241
\(853\) −6.00532e21 −0.625776 −0.312888 0.949790i \(-0.601296\pi\)
−0.312888 + 0.949790i \(0.601296\pi\)
\(854\) 0 0
\(855\) −2.55773e20 −0.0261883
\(856\) 1.02535e22 1.04069
\(857\) −1.47589e22 −1.48491 −0.742453 0.669898i \(-0.766338\pi\)
−0.742453 + 0.669898i \(0.766338\pi\)
\(858\) 2.82976e21 0.282225
\(859\) −9.64956e20 −0.0954023 −0.0477012 0.998862i \(-0.515190\pi\)
−0.0477012 + 0.998862i \(0.515190\pi\)
\(860\) 3.56354e20 0.0349255
\(861\) 0 0
\(862\) 1.09029e22 1.05012
\(863\) 3.44391e21 0.328829 0.164415 0.986391i \(-0.447426\pi\)
0.164415 + 0.986391i \(0.447426\pi\)
\(864\) 9.07173e21 0.858691
\(865\) −3.95963e21 −0.371564
\(866\) −1.09239e22 −1.01623
\(867\) −5.06782e20 −0.0467389
\(868\) 0 0
\(869\) −5.23175e20 −0.0474241
\(870\) 1.39065e21 0.124975
\(871\) −5.48130e21 −0.488368
\(872\) −5.17245e21 −0.456901
\(873\) 1.92403e21 0.168502
\(874\) −3.19130e21 −0.277097
\(875\) 0 0
\(876\) 3.82073e21 0.326111
\(877\) −1.09850e22 −0.929617 −0.464808 0.885411i \(-0.653877\pi\)
−0.464808 + 0.885411i \(0.653877\pi\)
\(878\) −5.35564e21 −0.449368
\(879\) 1.21801e22 1.01329
\(880\) 1.43318e21 0.118217
\(881\) 7.98462e21 0.653033 0.326516 0.945192i \(-0.394125\pi\)
0.326516 + 0.945192i \(0.394125\pi\)
\(882\) 0 0
\(883\) 5.45236e21 0.438409 0.219204 0.975679i \(-0.429654\pi\)
0.219204 + 0.975679i \(0.429654\pi\)
\(884\) −4.34640e21 −0.346528
\(885\) 1.72002e21 0.135975
\(886\) −2.82213e22 −2.21220
\(887\) −1.67127e22 −1.29903 −0.649517 0.760347i \(-0.725029\pi\)
−0.649517 + 0.760347i \(0.725029\pi\)
\(888\) 1.41129e22 1.08772
\(889\) 0 0
\(890\) 8.05483e21 0.610424
\(891\) −3.10820e21 −0.233575
\(892\) 3.51475e20 0.0261913
\(893\) −5.33178e21 −0.393991
\(894\) −9.08007e21 −0.665361
\(895\) −4.86576e21 −0.353572
\(896\) 0 0
\(897\) 6.01436e21 0.429780
\(898\) −1.68108e22 −1.19129
\(899\) 2.64180e21 0.185653
\(900\) 1.21233e21 0.0844892
\(901\) −1.11917e22 −0.773500
\(902\) −7.30146e21 −0.500450
\(903\) 0 0
\(904\) 8.37704e21 0.564712
\(905\) 3.87269e21 0.258910
\(906\) 2.07974e22 1.37895
\(907\) −1.53384e22 −1.00862 −0.504309 0.863523i \(-0.668253\pi\)
−0.504309 + 0.863523i \(0.668253\pi\)
\(908\) −4.21767e21 −0.275061
\(909\) −2.56723e21 −0.166049
\(910\) 0 0
\(911\) −1.49134e22 −0.948829 −0.474415 0.880302i \(-0.657340\pi\)
−0.474415 + 0.880302i \(0.657340\pi\)
\(912\) 6.99206e21 0.441209
\(913\) 5.80007e21 0.362997
\(914\) −2.55854e22 −1.58816
\(915\) 8.60430e20 0.0529734
\(916\) −1.49979e21 −0.0915827
\(917\) 0 0
\(918\) 2.08241e22 1.25097
\(919\) −5.86667e21 −0.349563 −0.174782 0.984607i \(-0.555922\pi\)
−0.174782 + 0.984607i \(0.555922\pi\)
\(920\) 2.00844e21 0.118700
\(921\) 3.26529e21 0.191414
\(922\) −2.99469e22 −1.74129
\(923\) 2.37687e22 1.37086
\(924\) 0 0
\(925\) 2.87377e22 1.63076
\(926\) 3.79393e22 2.13554
\(927\) 2.32695e21 0.129924
\(928\) 5.71747e21 0.316660
\(929\) 1.67946e22 0.922684 0.461342 0.887222i \(-0.347368\pi\)
0.461342 + 0.887222i \(0.347368\pi\)
\(930\) 2.69776e21 0.147022
\(931\) 0 0
\(932\) −1.09786e22 −0.588743
\(933\) −1.12724e21 −0.0599661
\(934\) 2.95123e22 1.55740
\(935\) 1.76636e21 0.0924683
\(936\) 2.43377e21 0.126389
\(937\) −5.04466e21 −0.259887 −0.129944 0.991521i \(-0.541480\pi\)
−0.129944 + 0.991521i \(0.541480\pi\)
\(938\) 0 0
\(939\) −1.22387e22 −0.620500
\(940\) −2.46833e21 −0.124149
\(941\) −1.65425e22 −0.825430 −0.412715 0.910860i \(-0.635419\pi\)
−0.412715 + 0.910860i \(0.635419\pi\)
\(942\) 1.05059e22 0.520059
\(943\) −1.55185e22 −0.762100
\(944\) 1.31709e22 0.641691
\(945\) 0 0
\(946\) −2.18960e21 −0.104998
\(947\) −9.81583e21 −0.466984 −0.233492 0.972359i \(-0.575015\pi\)
−0.233492 + 0.972359i \(0.575015\pi\)
\(948\) −1.18163e21 −0.0557725
\(949\) 1.56186e22 0.731384
\(950\) 9.38776e21 0.436149
\(951\) −2.66963e21 −0.123054
\(952\) 0 0
\(953\) −5.97914e21 −0.271295 −0.135648 0.990757i \(-0.543311\pi\)
−0.135648 + 0.990757i \(0.543311\pi\)
\(954\) −4.60981e21 −0.207525
\(955\) 5.13921e21 0.229547
\(956\) 4.90186e21 0.217233
\(957\) −2.54350e21 −0.111839
\(958\) −1.38469e22 −0.604101
\(959\) 0 0
\(960\) −1.79882e21 −0.0772600
\(961\) −1.83404e22 −0.781596
\(962\) −4.24375e22 −1.79447
\(963\) 7.89441e21 0.331225
\(964\) −9.52286e20 −0.0396451
\(965\) 4.64393e20 0.0191836
\(966\) 0 0
\(967\) −1.44757e22 −0.588764 −0.294382 0.955688i \(-0.595114\pi\)
−0.294382 + 0.955688i \(0.595114\pi\)
\(968\) 1.53068e22 0.617759
\(969\) 8.61757e21 0.345109
\(970\) 6.89737e21 0.274092
\(971\) −1.77921e21 −0.0701590 −0.0350795 0.999385i \(-0.511168\pi\)
−0.0350795 + 0.999385i \(0.511168\pi\)
\(972\) 4.64802e21 0.181874
\(973\) 0 0
\(974\) −5.21301e21 −0.200861
\(975\) −1.76923e22 −0.676472
\(976\) 6.58868e21 0.249991
\(977\) 1.04088e22 0.391913 0.195957 0.980613i \(-0.437219\pi\)
0.195957 + 0.980613i \(0.437219\pi\)
\(978\) 1.21305e22 0.453250
\(979\) −1.47323e22 −0.546262
\(980\) 0 0
\(981\) −3.98238e21 −0.145420
\(982\) −6.06761e21 −0.219878
\(983\) −3.26461e22 −1.17403 −0.587017 0.809575i \(-0.699698\pi\)
−0.587017 + 0.809575i \(0.699698\pi\)
\(984\) 2.24185e22 0.800100
\(985\) −1.84561e21 −0.0653684
\(986\) 1.31244e22 0.461320
\(987\) 0 0
\(988\) −4.12659e21 −0.142861
\(989\) −4.65376e21 −0.159894
\(990\) 7.27557e20 0.0248087
\(991\) −7.47327e21 −0.252906 −0.126453 0.991973i \(-0.540359\pi\)
−0.126453 + 0.991973i \(0.540359\pi\)
\(992\) 1.10915e22 0.372523
\(993\) −3.40107e22 −1.13369
\(994\) 0 0
\(995\) −1.49275e22 −0.490132
\(996\) 1.30999e22 0.426897
\(997\) 3.10809e22 1.00526 0.502632 0.864500i \(-0.332365\pi\)
0.502632 + 0.864500i \(0.332365\pi\)
\(998\) −3.71506e22 −1.19258
\(999\) 6.05226e22 1.92831
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 49.16.a.a.1.1 1
7.2 even 3 49.16.c.b.18.1 2
7.3 odd 6 49.16.c.c.30.1 2
7.4 even 3 49.16.c.b.30.1 2
7.5 odd 6 49.16.c.c.18.1 2
7.6 odd 2 1.16.a.a.1.1 1
21.20 even 2 9.16.a.a.1.1 1
28.27 even 2 16.16.a.d.1.1 1
35.13 even 4 25.16.b.a.24.1 2
35.27 even 4 25.16.b.a.24.2 2
35.34 odd 2 25.16.a.a.1.1 1
56.13 odd 2 64.16.a.i.1.1 1
56.27 even 2 64.16.a.c.1.1 1
77.76 even 2 121.16.a.a.1.1 1
84.83 odd 2 144.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 1 7.6 odd 2
9.16.a.a.1.1 1 21.20 even 2
16.16.a.d.1.1 1 28.27 even 2
25.16.a.a.1.1 1 35.34 odd 2
25.16.b.a.24.1 2 35.13 even 4
25.16.b.a.24.2 2 35.27 even 4
49.16.a.a.1.1 1 1.1 even 1 trivial
49.16.c.b.18.1 2 7.2 even 3
49.16.c.b.30.1 2 7.4 even 3
49.16.c.c.18.1 2 7.5 odd 6
49.16.c.c.30.1 2 7.3 odd 6
64.16.a.c.1.1 1 56.27 even 2
64.16.a.i.1.1 1 56.13 odd 2
121.16.a.a.1.1 1 77.76 even 2
144.16.a.f.1.1 1 84.83 odd 2