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Properties

Label 20.9.d.a.19.1
Level $20$
Weight $9$
Character 20.19
Self dual yes
Analytic conductor $8.148$
Analytic rank $0$
Dimension $1$
CM discriminant -20
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,9,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

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: \(8.14757220122\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.1
Character \(\chi\) \(=\) 20.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-16.0000 q^{2} +158.000 q^{3} +256.000 q^{4} +625.000 q^{5} -2528.00 q^{6} -1922.00 q^{7} -4096.00 q^{8} +18403.0 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} +158.000 q^{3} +256.000 q^{4} +625.000 q^{5} -2528.00 q^{6} -1922.00 q^{7} -4096.00 q^{8} +18403.0 q^{9} -10000.0 q^{10} +40448.0 q^{12} +30752.0 q^{14} +98750.0 q^{15} +65536.0 q^{16} -294448. q^{18} +160000. q^{20} -303676. q^{21} -211202. q^{23} -647168. q^{24} +390625. q^{25} +1.87104e6 q^{27} -492032. q^{28} +20642.0 q^{29} -1.58000e6 q^{30} -1.04858e6 q^{32} -1.20125e6 q^{35} +4.71117e6 q^{36} -2.56000e6 q^{40} -5.41920e6 q^{41} +4.85882e6 q^{42} +2.51952e6 q^{43} +1.15019e7 q^{45} +3.37923e6 q^{46} -9.61824e6 q^{47} +1.03547e7 q^{48} -2.07072e6 q^{49} -6.25000e6 q^{50} -2.99366e7 q^{54} +7.87251e6 q^{56} -330272. q^{58} +2.52800e7 q^{60} -1.10616e7 q^{61} -3.53706e7 q^{63} +1.67772e7 q^{64} +2.02498e7 q^{67} -3.33699e7 q^{69} +1.92200e7 q^{70} -7.53787e7 q^{72} +6.17188e7 q^{75} +4.09600e7 q^{80} +1.74882e8 q^{81} +8.67072e7 q^{82} +3.08846e7 q^{83} -7.77411e7 q^{84} -4.03123e7 q^{86} +3.26144e6 q^{87} -1.06805e8 q^{89} -1.84030e8 q^{90} -5.40677e7 q^{92} +1.53892e8 q^{94} -1.65675e8 q^{96} +3.31315e7 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −1.00000
\(3\) 158.000 1.95062 0.975309 − 0.220846i \(-0.0708819\pi\)
0.975309 + 0.220846i \(0.0708819\pi\)
\(4\) 256.000 1.00000
\(5\) 625.000 1.00000
\(6\) −2528.00 −1.95062
\(7\) −1922.00 −0.800500 −0.400250 − 0.916406i \(-0.631077\pi\)
−0.400250 + 0.916406i \(0.631077\pi\)
\(8\) −4096.00 −1.00000
\(9\) 18403.0 2.80491
\(10\) −10000.0 −1.00000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 40448.0 1.95062
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 30752.0 0.800500
\(15\) 98750.0 1.95062
\(16\) 65536.0 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −294448. −2.80491
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 160000. 1.00000
\(21\) −303676. −1.56147
\(22\) 0 0
\(23\) −211202. −0.754721 −0.377361 − 0.926066i \(-0.623168\pi\)
−0.377361 + 0.926066i \(0.623168\pi\)
\(24\) −647168. −1.95062
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) 1.87104e6 3.52068
\(28\) −492032. −0.800500
\(29\) 20642.0 0.0291850 0.0145925 − 0.999894i \(-0.495355\pi\)
0.0145925 + 0.999894i \(0.495355\pi\)
\(30\) −1.58000e6 −1.95062
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.04858e6 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −1.20125e6 −0.800500
\(36\) 4.71117e6 2.80491
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −2.56000e6 −1.00000
\(41\) −5.41920e6 −1.91778 −0.958892 − 0.283772i \(-0.908414\pi\)
−0.958892 + 0.283772i \(0.908414\pi\)
\(42\) 4.85882e6 1.56147
\(43\) 2.51952e6 0.736960 0.368480 − 0.929636i \(-0.379878\pi\)
0.368480 + 0.929636i \(0.379878\pi\)
\(44\) 0 0
\(45\) 1.15019e7 2.80491
\(46\) 3.37923e6 0.754721
\(47\) −9.61824e6 −1.97108 −0.985540 − 0.169443i \(-0.945803\pi\)
−0.985540 + 0.169443i \(0.945803\pi\)
\(48\) 1.03547e7 1.95062
\(49\) −2.07072e6 −0.359200
\(50\) −6.25000e6 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −2.99366e7 −3.52068
\(55\) 0 0
\(56\) 7.87251e6 0.800500
\(57\) 0 0
\(58\) −330272. −0.0291850
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 2.52800e7 1.95062
\(61\) −1.10616e7 −0.798911 −0.399456 − 0.916753i \(-0.630801\pi\)
−0.399456 + 0.916753i \(0.630801\pi\)
\(62\) 0 0
\(63\) −3.53706e7 −2.24533
\(64\) 1.67772e7 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.02498e7 1.00489 0.502447 − 0.864608i \(-0.332433\pi\)
0.502447 + 0.864608i \(0.332433\pi\)
\(68\) 0 0
\(69\) −3.33699e7 −1.47217
\(70\) 1.92200e7 0.800500
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −7.53787e7 −2.80491
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 6.17188e7 1.95062
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 4.09600e7 1.00000
\(81\) 1.74882e8 4.06260
\(82\) 8.67072e7 1.91778
\(83\) 3.08846e7 0.650774 0.325387 − 0.945581i \(-0.394505\pi\)
0.325387 + 0.945581i \(0.394505\pi\)
\(84\) −7.77411e7 −1.56147
\(85\) 0 0
\(86\) −4.03123e7 −0.736960
\(87\) 3.26144e6 0.0569288
\(88\) 0 0
\(89\) −1.06805e8 −1.70228 −0.851139 − 0.524940i \(-0.824088\pi\)
−0.851139 + 0.524940i \(0.824088\pi\)
\(90\) −1.84030e8 −2.80491
\(91\) 0 0
\(92\) −5.40677e7 −0.754721
\(93\) 0 0
\(94\) 1.53892e8 1.97108
\(95\) 0 0
\(96\) −1.65675e8 −1.95062
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 3.31315e7 0.359200
\(99\) 0 0
\(100\) 1.00000e8 1.00000
\(101\) −1.77672e8 −1.70740 −0.853699 − 0.520767i \(-0.825646\pi\)
−0.853699 + 0.520767i \(0.825646\pi\)
\(102\) 0 0
\(103\) −1.46694e8 −1.30336 −0.651678 − 0.758496i \(-0.725934\pi\)
−0.651678 + 0.758496i \(0.725934\pi\)
\(104\) 0 0
\(105\) −1.89798e8 −1.56147
\(106\) 0 0
\(107\) 2.05579e8 1.56835 0.784175 − 0.620540i \(-0.213086\pi\)
0.784175 + 0.620540i \(0.213086\pi\)
\(108\) 4.78985e8 3.52068
\(109\) 2.15777e8 1.52862 0.764309 − 0.644851i \(-0.223080\pi\)
0.764309 + 0.644851i \(0.223080\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.25960e8 −0.800500
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −1.32001e8 −0.754721
\(116\) 5.28435e6 0.0291850
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) −4.04480e8 −1.95062
\(121\) 2.14359e8 1.00000
\(122\) 1.76986e8 0.798911
\(123\) −8.56233e8 −3.74086
\(124\) 0 0
\(125\) 2.44141e8 1.00000
\(126\) 5.65929e8 2.24533
\(127\) −2.90506e7 −0.111671 −0.0558354 − 0.998440i \(-0.517782\pi\)
−0.0558354 + 0.998440i \(0.517782\pi\)
\(128\) −2.68435e8 −1.00000
\(129\) 3.98084e8 1.43753
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −3.23996e8 −1.00489
\(135\) 1.16940e9 3.52068
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 5.33919e8 1.47217
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −3.07520e8 −0.800500
\(141\) −1.51968e9 −3.84482
\(142\) 0 0
\(143\) 0 0
\(144\) 1.20606e9 2.80491
\(145\) 1.29012e7 0.0291850
\(146\) 0 0
\(147\) −3.27173e8 −0.700662
\(148\) 0 0
\(149\) 9.54571e7 0.193670 0.0968352 − 0.995300i \(-0.469128\pi\)
0.0968352 + 0.995300i \(0.469128\pi\)
\(150\) −9.87500e8 −1.95062
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −6.55360e8 −1.00000
\(161\) 4.05930e8 0.604154
\(162\) −2.79811e9 −4.06260
\(163\) 7.23181e8 1.02446 0.512232 − 0.858847i \(-0.328819\pi\)
0.512232 + 0.858847i \(0.328819\pi\)
\(164\) −1.38731e9 −1.91778
\(165\) 0 0
\(166\) −4.94154e8 −0.650774
\(167\) 1.54147e9 1.98184 0.990921 − 0.134442i \(-0.0429243\pi\)
0.990921 + 0.134442i \(0.0429243\pi\)
\(168\) 1.24386e9 1.56147
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 6.44997e8 0.736960
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) −5.21830e7 −0.0569288
\(175\) −7.50781e8 −0.800500
\(176\) 0 0
\(177\) 0 0
\(178\) 1.70888e9 1.70228
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 2.94448e9 2.80491
\(181\) 1.77745e9 1.65609 0.828045 − 0.560661i \(-0.189453\pi\)
0.828045 + 0.560661i \(0.189453\pi\)
\(182\) 0 0
\(183\) −1.74773e9 −1.55837
\(184\) 8.65083e8 0.754721
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −2.46227e9 −1.97108
\(189\) −3.59613e9 −2.81831
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 2.65080e9 1.95062
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −5.30104e8 −0.359200
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.60000e9 −1.00000
\(201\) 3.19946e9 1.96017
\(202\) 2.84276e9 1.70740
\(203\) −3.96739e7 −0.0233626
\(204\) 0 0
\(205\) −3.38700e9 −1.91778
\(206\) 2.34710e9 1.30336
\(207\) −3.88675e9 −2.11692
\(208\) 0 0
\(209\) 0 0
\(210\) 3.03676e9 1.56147
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −3.28926e9 −1.56835
\(215\) 1.57470e9 0.736960
\(216\) −7.66376e9 −3.52068
\(217\) 0 0
\(218\) −3.45243e9 −1.52862
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −3.26930e9 −1.32201 −0.661006 − 0.750381i \(-0.729870\pi\)
−0.661006 + 0.750381i \(0.729870\pi\)
\(224\) 2.01536e9 0.800500
\(225\) 7.18867e9 2.80491
\(226\) 0 0
\(227\) 4.74983e9 1.78885 0.894426 − 0.447216i \(-0.147584\pi\)
0.894426 + 0.447216i \(0.147584\pi\)
\(228\) 0 0
\(229\) −4.18692e9 −1.52249 −0.761243 − 0.648467i \(-0.775410\pi\)
−0.761243 + 0.648467i \(0.775410\pi\)
\(230\) 2.11202e9 0.754721
\(231\) 0 0
\(232\) −8.45496e7 −0.0291850
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −6.01140e9 −1.97108
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 6.47168e9 1.95062
\(241\) −6.12390e9 −1.81535 −0.907675 − 0.419675i \(-0.862144\pi\)
−0.907675 + 0.419675i \(0.862144\pi\)
\(242\) −3.42974e9 −1.00000
\(243\) 1.53554e10 4.40389
\(244\) −2.83177e9 −0.798911
\(245\) −1.29420e9 −0.359200
\(246\) 1.36997e10 3.74086
\(247\) 0 0
\(248\) 0 0
\(249\) 4.87977e9 1.26941
\(250\) −3.90625e9 −1.00000
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −9.05486e9 −2.24533
\(253\) 0 0
\(254\) 4.64809e8 0.111671
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) −6.36934e9 −1.43753
\(259\) 0 0
\(260\) 0 0
\(261\) 3.79875e8 0.0818612
\(262\) 0 0
\(263\) 6.24149e9 1.30456 0.652282 − 0.757977i \(-0.273812\pi\)
0.652282 + 0.757977i \(0.273812\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −1.68752e10 −3.32049
\(268\) 5.18394e9 1.00489
\(269\) 1.00564e10 1.92058 0.960288 − 0.279010i \(-0.0900064\pi\)
0.960288 + 0.279010i \(0.0900064\pi\)
\(270\) −1.87104e10 −3.52068
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −8.54270e9 −1.47217
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 4.92032e9 0.800500
\(281\) −1.21874e10 −1.95472 −0.977361 − 0.211581i \(-0.932139\pi\)
−0.977361 + 0.211581i \(0.932139\pi\)
\(282\) 2.43149e10 3.84482
\(283\) 9.18975e9 1.43271 0.716355 − 0.697736i \(-0.245809\pi\)
0.716355 + 0.697736i \(0.245809\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.04157e10 1.53519
\(288\) −1.92969e10 −2.80491
\(289\) 6.97576e9 1.00000
\(290\) −2.06420e8 −0.0291850
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 5.23477e9 0.700662
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −1.52731e9 −0.193670
\(299\) 0 0
\(300\) 1.58000e10 1.95062
\(301\) −4.84251e9 −0.589936
\(302\) 0 0
\(303\) −2.80723e10 −3.33048
\(304\) 0 0
\(305\) −6.91350e9 −0.798911
\(306\) 0 0
\(307\) −1.00291e10 −1.12904 −0.564521 − 0.825418i \(-0.690939\pi\)
−0.564521 + 0.825418i \(0.690939\pi\)
\(308\) 0 0
\(309\) −2.31776e10 −2.54235
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −2.21066e10 −2.24533
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.04858e10 1.00000
\(321\) 3.24814e10 3.05925
\(322\) −6.49488e9 −0.604154
\(323\) 0 0
\(324\) 4.47697e10 4.06260
\(325\) 0 0
\(326\) −1.15709e10 −1.02446
\(327\) 3.40927e10 2.98175
\(328\) 2.21970e10 1.91778
\(329\) 1.84863e10 1.57785
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 7.90647e9 0.650774
\(333\) 0 0
\(334\) −2.46635e10 −1.98184
\(335\) 1.26561e10 1.00489
\(336\) −1.99017e10 −1.56147
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −1.30517e10 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.50599e10 1.08804
\(344\) −1.03199e10 −0.736960
\(345\) −2.08562e10 −1.47217
\(346\) 0 0
\(347\) −2.26968e10 −1.56548 −0.782739 − 0.622350i \(-0.786178\pi\)
−0.782739 + 0.622350i \(0.786178\pi\)
\(348\) 8.34928e8 0.0569288
\(349\) 2.94354e10 1.98412 0.992061 − 0.125761i \(-0.0401374\pi\)
0.992061 + 0.125761i \(0.0401374\pi\)
\(350\) 1.20125e10 0.800500
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.73420e10 −1.70228
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −4.71117e10 −2.80491
\(361\) 1.69836e10 1.00000
\(362\) −2.84393e10 −1.65609
\(363\) 3.38687e10 1.95062
\(364\) 0 0
\(365\) 0 0
\(366\) 2.79637e10 1.55837
\(367\) −2.86398e10 −1.57872 −0.789360 − 0.613930i \(-0.789588\pi\)
−0.789360 + 0.613930i \(0.789588\pi\)
\(368\) −1.38413e10 −0.754721
\(369\) −9.97295e10 −5.37921
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 3.85742e10 1.95062
\(376\) 3.93963e10 1.97108
\(377\) 0 0
\(378\) 5.75381e10 2.81831
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −4.58999e9 −0.217827
\(382\) 0 0
\(383\) −4.19031e10 −1.94738 −0.973691 − 0.227872i \(-0.926823\pi\)
−0.973691 + 0.227872i \(0.926823\pi\)
\(384\) −4.24128e10 −1.95062
\(385\) 0 0
\(386\) 0 0
\(387\) 4.63667e10 2.06710
\(388\) 0 0
\(389\) 2.27630e10 0.994104 0.497052 − 0.867721i \(-0.334416\pi\)
0.497052 + 0.867721i \(0.334416\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8.48166e9 0.359200
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.56000e10 1.00000
\(401\) −4.30430e10 −1.66466 −0.832328 − 0.554283i \(-0.812993\pi\)
−0.832328 + 0.554283i \(0.812993\pi\)
\(402\) −5.11914e10 −1.96017
\(403\) 0 0
\(404\) −4.54842e10 −1.70740
\(405\) 1.09301e11 4.06260
\(406\) 6.34783e8 0.0233626
\(407\) 0 0
\(408\) 0 0
\(409\) 3.92940e10 1.40421 0.702107 − 0.712072i \(-0.252243\pi\)
0.702107 + 0.712072i \(0.252243\pi\)
\(410\) 5.41920e10 1.91778
\(411\) 0 0
\(412\) −3.75536e10 −1.30336
\(413\) 0 0
\(414\) 6.21880e10 2.11692
\(415\) 1.93029e10 0.650774
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −4.85882e10 −1.56147
\(421\) 7.28990e7 0.00232056 0.00116028 − 0.999999i \(-0.499631\pi\)
0.00116028 + 0.999999i \(0.499631\pi\)
\(422\) 0 0
\(423\) −1.77005e11 −5.52870
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.12604e10 0.639528
\(428\) 5.26282e10 1.56835
\(429\) 0 0
\(430\) −2.51952e10 −0.736960
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 1.22620e11 3.52068
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 2.03840e9 0.0569288
\(436\) 5.52389e10 1.52862
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −3.81074e10 −1.00752
\(442\) 0 0
\(443\) 1.88895e10 0.490461 0.245230 − 0.969465i \(-0.421136\pi\)
0.245230 + 0.969465i \(0.421136\pi\)
\(444\) 0 0
\(445\) −6.67530e10 −1.70228
\(446\) 5.23088e10 1.32201
\(447\) 1.50822e10 0.377777
\(448\) −3.22458e10 −0.800500
\(449\) −2.13599e10 −0.525549 −0.262774 − 0.964857i \(-0.584638\pi\)
−0.262774 + 0.964857i \(0.584638\pi\)
\(450\) −1.15019e11 −2.80491
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −7.59972e10 −1.78885
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 6.69908e10 1.52249
\(459\) 0 0
\(460\) −3.37923e10 −0.754721
\(461\) −9.71865e9 −0.215180 −0.107590 − 0.994195i \(-0.534313\pi\)
−0.107590 + 0.994195i \(0.534313\pi\)
\(462\) 0 0
\(463\) 6.78962e10 1.47748 0.738739 − 0.673991i \(-0.235421\pi\)
0.738739 + 0.673991i \(0.235421\pi\)
\(464\) 1.35279e9 0.0291850
\(465\) 0 0
\(466\) 0 0
\(467\) −8.19487e10 −1.72296 −0.861479 − 0.507794i \(-0.830461\pi\)
−0.861479 + 0.507794i \(0.830461\pi\)
\(468\) 0 0
\(469\) −3.89200e10 −0.804418
\(470\) 9.61824e10 1.97108
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −1.03547e11 −1.95062
\(481\) 0 0
\(482\) 9.79825e10 1.81535
\(483\) 6.41370e10 1.17847
\(484\) 5.48759e10 1.00000
\(485\) 0 0
\(486\) −2.45687e11 −4.40389
\(487\) 5.48589e10 0.975284 0.487642 − 0.873044i \(-0.337857\pi\)
0.487642 + 0.873044i \(0.337857\pi\)
\(488\) 4.53083e10 0.798911
\(489\) 1.14263e11 1.99834
\(490\) 2.07072e10 0.359200
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) −2.19196e11 −3.74086
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −7.80764e10 −1.26941
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 6.25000e10 1.00000
\(501\) 2.43552e11 3.86582
\(502\) 0 0
\(503\) 1.71128e10 0.267331 0.133665 − 0.991027i \(-0.457325\pi\)
0.133665 + 0.991027i \(0.457325\pi\)
\(504\) 1.44878e11 2.24533
\(505\) −1.11045e11 −1.70740
\(506\) 0 0
\(507\) 1.28885e11 1.95062
\(508\) −7.43694e9 −0.111671
\(509\) 6.48146e10 0.965610 0.482805 − 0.875728i \(-0.339618\pi\)
0.482805 + 0.875728i \(0.339618\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6.87195e10 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −9.16836e10 −1.30336
\(516\) 1.01909e11 1.43753
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.46894e11 −1.99367 −0.996834 − 0.0795049i \(-0.974666\pi\)
−0.996834 + 0.0795049i \(0.974666\pi\)
\(522\) −6.07800e9 −0.0818612
\(523\) −1.20932e11 −1.61635 −0.808175 − 0.588942i \(-0.799545\pi\)
−0.808175 + 0.588942i \(0.799545\pi\)
\(524\) 0 0
\(525\) −1.18623e11 −1.56147
\(526\) −9.98638e10 −1.30456
\(527\) 0 0
\(528\) 0 0
\(529\) −3.37047e10 −0.430396
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 2.70003e11 3.32049
\(535\) 1.28487e11 1.56835
\(536\) −8.29430e10 −1.00489
\(537\) 0 0
\(538\) −1.60902e11 −1.92058
\(539\) 0 0
\(540\) 2.99366e11 3.52068
\(541\) 3.50805e10 0.409522 0.204761 − 0.978812i \(-0.434358\pi\)
0.204761 + 0.978812i \(0.434358\pi\)
\(542\) 0 0
\(543\) 2.80838e11 3.23040
\(544\) 0 0
\(545\) 1.34861e11 1.52862
\(546\) 0 0
\(547\) −1.53459e11 −1.71413 −0.857065 − 0.515208i \(-0.827715\pi\)
−0.857065 + 0.515208i \(0.827715\pi\)
\(548\) 0 0
\(549\) −2.03567e11 −2.24087
\(550\) 0 0
\(551\) 0 0
\(552\) 1.36683e11 1.47217
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −7.87251e10 −0.800500
\(561\) 0 0
\(562\) 1.94998e11 1.95472
\(563\) −1.95254e11 −1.94341 −0.971707 − 0.236190i \(-0.924101\pi\)
−0.971707 + 0.236190i \(0.924101\pi\)
\(564\) −3.89039e11 −3.84482
\(565\) 0 0
\(566\) −1.47036e11 −1.43271
\(567\) −3.36122e11 −3.25211
\(568\) 0 0
\(569\) 1.77936e11 1.69752 0.848760 − 0.528778i \(-0.177350\pi\)
0.848760 + 0.528778i \(0.177350\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.66651e11 −1.53519
\(575\) −8.25008e10 −0.754721
\(576\) 3.08751e11 2.80491
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −1.11612e11 −1.00000
\(579\) 0 0
\(580\) 3.30272e9 0.0291850
\(581\) −5.93603e10 −0.520944
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.78340e10 0.150209 0.0751046 − 0.997176i \(-0.476071\pi\)
0.0751046 + 0.997176i \(0.476071\pi\)
\(588\) −8.37564e10 −0.700662
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.44370e10 0.193670
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −2.52800e11 −1.95062
\(601\) 3.90189e10 0.299073 0.149536 − 0.988756i \(-0.452222\pi\)
0.149536 + 0.988756i \(0.452222\pi\)
\(602\) 7.74802e10 0.589936
\(603\) 3.72656e11 2.81864
\(604\) 0 0
\(605\) 1.33974e11 1.00000
\(606\) 4.49156e11 3.33048
\(607\) 2.34492e11 1.72732 0.863662 − 0.504071i \(-0.168165\pi\)
0.863662 + 0.504071i \(0.168165\pi\)
\(608\) 0 0
\(609\) −6.26848e9 −0.0455715
\(610\) 1.10616e11 0.798911
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.60466e11 1.12904
\(615\) −5.35146e11 −3.74086
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 3.70842e11 2.54235
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −3.95167e11 −2.65714
\(622\) 0 0
\(623\) 2.05279e11 1.36267
\(624\) 0 0
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 3.53706e11 2.24533
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.81566e10 −0.111671
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.67772e11 −1.00000
\(641\) −1.13651e11 −0.673193 −0.336596 − 0.941649i \(-0.609276\pi\)
−0.336596 + 0.941649i \(0.609276\pi\)
\(642\) −5.19703e11 −3.05925
\(643\) −9.85936e10 −0.576773 −0.288387 − 0.957514i \(-0.593119\pi\)
−0.288387 + 0.957514i \(0.593119\pi\)
\(644\) 1.03918e11 0.604154
\(645\) 2.48802e11 1.43753
\(646\) 0 0
\(647\) 3.44581e11 1.96641 0.983206 − 0.182498i \(-0.0584182\pi\)
0.983206 + 0.182498i \(0.0584182\pi\)
\(648\) −7.16315e11 −4.06260
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.85134e11 1.02446
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −5.45484e11 −2.98175
\(655\) 0 0
\(656\) −3.55153e11 −1.91778
\(657\) 0 0
\(658\) −2.95780e11 −1.57785
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 2.34485e11 1.22831 0.614156 − 0.789185i \(-0.289497\pi\)
0.614156 + 0.789185i \(0.289497\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −1.26503e11 −0.650774
\(665\) 0 0
\(666\) 0 0
\(667\) −4.35963e9 −0.0220265
\(668\) 3.94616e11 1.98184
\(669\) −5.16549e11 −2.57874
\(670\) −2.02498e11 −1.00489
\(671\) 0 0
\(672\) 3.18427e11 1.56147
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 7.30873e11 3.52068
\(676\) 2.08827e11 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 7.50473e11 3.48937
\(682\) 0 0
\(683\) 4.60451e10 0.211593 0.105796 − 0.994388i \(-0.466261\pi\)
0.105796 + 0.994388i \(0.466261\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.40958e11 −1.08804
\(687\) −6.61534e11 −2.96979
\(688\) 1.65119e11 0.736960
\(689\) 0 0
\(690\) 3.33699e11 1.47217
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 3.63149e11 1.56548
\(695\) 0 0
\(696\) −1.33588e10 −0.0569288
\(697\) 0 0
\(698\) −4.70966e11 −1.98412
\(699\) 0 0
\(700\) −1.92200e11 −0.800500
\(701\) −4.54322e11 −1.88145 −0.940723 − 0.339177i \(-0.889852\pi\)
−0.940723 + 0.339177i \(0.889852\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −9.49801e11 −3.84482
\(706\) 0 0
\(707\) 3.41487e11 1.36677
\(708\) 0 0
\(709\) −2.36889e11 −0.937474 −0.468737 − 0.883338i \(-0.655291\pi\)
−0.468737 + 0.883338i \(0.655291\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.37472e11 1.70228
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 7.53787e11 2.80491
\(721\) 2.81945e11 1.04334
\(722\) −2.71737e11 −1.00000
\(723\) −9.67577e11 −3.54105
\(724\) 4.55028e11 1.65609
\(725\) 8.06328e9 0.0291850
\(726\) −5.41899e11 −1.95062
\(727\) −4.51416e11 −1.61599 −0.807995 − 0.589189i \(-0.799447\pi\)
−0.807995 + 0.589189i \(0.799447\pi\)
\(728\) 0 0
\(729\) 1.27876e12 4.52771
\(730\) 0 0
\(731\) 0 0
\(732\) −4.47420e11 −1.55837
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 4.58236e11 1.57872
\(735\) −2.04483e11 −0.700662
\(736\) 2.21461e11 0.754721
\(737\) 0 0
\(738\) 1.59567e12 5.37921
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.38698e11 1.11137 0.555683 − 0.831394i \(-0.312457\pi\)
0.555683 + 0.831394i \(0.312457\pi\)
\(744\) 0 0
\(745\) 5.96607e10 0.193670
\(746\) 0 0
\(747\) 5.68370e11 1.82536
\(748\) 0 0
\(749\) −3.95122e11 −1.25546
\(750\) −6.17188e11 −1.95062
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −6.30341e11 −1.97108
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −9.20610e11 −2.81831
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.38529e11 1.60572 0.802862 − 0.596165i \(-0.203310\pi\)
0.802862 + 0.596165i \(0.203310\pi\)
\(762\) 7.34398e10 0.217827
\(763\) −4.14723e11 −1.22366
\(764\) 0 0
\(765\) 0 0
\(766\) 6.70450e11 1.94738
\(767\) 0 0
\(768\) 6.78605e11 1.95062
\(769\) −3.17192e11 −0.907021 −0.453511 − 0.891251i \(-0.649829\pi\)
−0.453511 + 0.891251i \(0.649829\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −7.41867e11 −2.06710
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −3.64209e11 −0.994104
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.86219e10 0.102751
\(784\) −1.35707e11 −0.359200
\(785\) 0 0
\(786\) 0 0
\(787\) −7.34080e11 −1.91357 −0.956785 − 0.290795i \(-0.906080\pi\)
−0.956785 + 0.290795i \(0.906080\pi\)
\(788\) 0 0
\(789\) 9.86155e11 2.54470
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −4.09600e11 −1.00000
\(801\) −1.96553e12 −4.77474
\(802\) 6.88687e11 1.66466
\(803\) 0 0
\(804\) 8.19062e11 1.96017
\(805\) 2.53706e11 0.604154
\(806\) 0 0
\(807\) 1.58890e12 3.74631
\(808\) 7.27746e11 1.70740
\(809\) −7.15434e11 −1.67023 −0.835113 − 0.550078i \(-0.814598\pi\)
−0.835113 + 0.550078i \(0.814598\pi\)
\(810\) −1.74882e12 −4.06260
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.01565e10 −0.0233626
\(813\) 0 0
\(814\) 0 0
\(815\) 4.51988e11 1.02446
\(816\) 0 0
\(817\) 0 0
\(818\) −6.28704e11 −1.40421
\(819\) 0 0
\(820\) −8.67072e11 −1.91778
\(821\) −8.08574e10 −0.177970 −0.0889850 − 0.996033i \(-0.528362\pi\)
−0.0889850 + 0.996033i \(0.528362\pi\)
\(822\) 0 0
\(823\) 5.64525e11 1.23051 0.615254 − 0.788329i \(-0.289054\pi\)
0.615254 + 0.788329i \(0.289054\pi\)
\(824\) 6.00858e11 1.30336
\(825\) 0 0
\(826\) 0 0
\(827\) −8.99261e10 −0.192249 −0.0961244 − 0.995369i \(-0.530645\pi\)
−0.0961244 + 0.995369i \(0.530645\pi\)
\(828\) −9.95008e11 −2.11692
\(829\) −2.88497e11 −0.610834 −0.305417 − 0.952219i \(-0.598796\pi\)
−0.305417 + 0.952219i \(0.598796\pi\)
\(830\) −3.08846e11 −0.650774
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 9.63419e11 1.98184
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 7.77411e11 1.56147
\(841\) −4.99820e11 −0.999148
\(842\) −1.16638e9 −0.00232056
\(843\) −1.92560e12 −3.81291
\(844\) 0 0
\(845\) 5.09832e11 1.00000
\(846\) 2.83207e12 5.52870
\(847\) −4.11998e11 −0.800500
\(848\) 0 0
\(849\) 1.45198e12 2.79467
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) −3.40166e11 −0.639528
\(855\) 0 0
\(856\) −8.42050e11 −1.56835
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 4.03123e11 0.736960
\(861\) 1.64568e12 2.99456
\(862\) 0 0
\(863\) −2.99508e11 −0.539964 −0.269982 − 0.962865i \(-0.587018\pi\)
−0.269982 + 0.962865i \(0.587018\pi\)
\(864\) −1.96192e12 −3.52068
\(865\) 0 0
\(866\) 0 0
\(867\) 1.10217e12 1.95062
\(868\) 0 0
\(869\) 0 0
\(870\) −3.26144e10 −0.0569288
\(871\) 0 0
\(872\) −8.83822e11 −1.52862
\(873\) 0 0
\(874\) 0 0
\(875\) −4.69238e11 −0.800500
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.93442e10 −0.115108 −0.0575541 − 0.998342i \(-0.518330\pi\)
−0.0575541 + 0.998342i \(0.518330\pi\)
\(882\) 6.09718e11 1.00752
\(883\) 1.21110e12 1.99221 0.996107 − 0.0881518i \(-0.0280961\pi\)
0.996107 + 0.0881518i \(0.0280961\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −3.02231e11 −0.490461
\(887\) 8.22963e11 1.32949 0.664746 − 0.747069i \(-0.268540\pi\)
0.664746 + 0.747069i \(0.268540\pi\)
\(888\) 0 0
\(889\) 5.58352e10 0.0893925
\(890\) 1.06805e12 1.70228
\(891\) 0 0
\(892\) −8.36941e11 −1.32201
\(893\) 0 0
\(894\) −2.41316e11 −0.377777
\(895\) 0 0
\(896\) 5.15933e11 0.800500
\(897\) 0 0
\(898\) 3.41758e11 0.525549
\(899\) 0 0
\(900\) 1.84030e12 2.80491
\(901\) 0 0
\(902\) 0 0
\(903\) −7.65117e11 −1.15074
\(904\) 0 0
\(905\) 1.11091e12 1.65609
\(906\) 0 0
\(907\) −1.14390e12 −1.69028 −0.845142 − 0.534542i \(-0.820484\pi\)
−0.845142 + 0.534542i \(0.820484\pi\)
\(908\) 1.21596e12 1.78885
\(909\) −3.26971e12 −4.78909
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −1.09233e12 −1.55837
\(916\) −1.07185e12 −1.52249
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 5.40677e11 0.754721
\(921\) −1.58460e12 −2.20233
\(922\) 1.55498e11 0.215180
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −1.08634e12 −1.47748
\(927\) −2.69961e12 −3.65579
\(928\) −2.16447e10 −0.0291850
\(929\) 4.99092e11 0.670066 0.335033 − 0.942206i \(-0.391253\pi\)
0.335033 + 0.942206i \(0.391253\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 1.31118e12 1.72296
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 6.22721e11 0.804418
\(939\) 0 0
\(940\) −1.53892e12 −1.97108
\(941\) 1.51903e12 1.93735 0.968674 − 0.248336i \(-0.0798836\pi\)
0.968674 + 0.248336i \(0.0798836\pi\)
\(942\) 0 0
\(943\) 1.14455e12 1.44739
\(944\) 0 0
\(945\) −2.24758e12 −2.81831
\(946\) 0 0
\(947\) −5.25398e11 −0.653264 −0.326632 − 0.945152i \(-0.605914\pi\)
−0.326632 + 0.945152i \(0.605914\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 1.65675e12 1.95062
\(961\) 8.52891e11 1.00000
\(962\) 0 0
\(963\) 3.78327e12 4.39908
\(964\) −1.56772e12 −1.81535
\(965\) 0 0
\(966\) −1.02619e12 −1.17847
\(967\) −1.52964e12 −1.74938 −0.874689 − 0.484684i \(-0.838935\pi\)
−0.874689 + 0.484684i \(0.838935\pi\)
\(968\) −8.78014e11 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 3.93099e12 4.40389
\(973\) 0 0
\(974\) −8.77742e11 −0.975284
\(975\) 0 0
\(976\) −7.24933e11 −0.798911
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) −1.82820e12 −1.99834
\(979\) 0 0
\(980\) −3.31315e11 −0.359200
\(981\) 3.97094e12 4.28763
\(982\) 0 0
\(983\) −1.56219e12 −1.67309 −0.836544 − 0.547900i \(-0.815427\pi\)
−0.836544 + 0.547900i \(0.815427\pi\)
\(984\) 3.50713e12 3.74086
\(985\) 0 0
\(986\) 0 0
\(987\) 2.92083e12 3.07778
\(988\) 0 0
\(989\) −5.32127e11 −0.556199
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 1.24922e12 1.26941
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 20.9.d.a.19.1 ✓ 1
4.3 odd 2 20.9.d.b.19.1 yes 1
5.2 odd 4 100.9.b.b.51.1 2
5.3 odd 4 100.9.b.b.51.2 2
5.4 even 2 20.9.d.b.19.1 yes 1
8.3 odd 2 320.9.h.b.319.1 1
8.5 even 2 320.9.h.a.319.1 1
20.3 even 4 100.9.b.b.51.1 2
20.7 even 4 100.9.b.b.51.2 2
20.19 odd 2 CM 20.9.d.a.19.1 ✓ 1
40.19 odd 2 320.9.h.a.319.1 1
40.29 even 2 320.9.h.b.319.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.9.d.a.19.1 ✓ 1 1.1 even 1 trivial
20.9.d.a.19.1 ✓ 1 20.19 odd 2 CM
20.9.d.b.19.1 yes 1 4.3 odd 2
20.9.d.b.19.1 yes 1 5.4 even 2
100.9.b.b.51.1 2 5.2 odd 4
100.9.b.b.51.1 2 20.3 even 4
100.9.b.b.51.2 2 5.3 odd 4
100.9.b.b.51.2 2 20.7 even 4
320.9.h.a.319.1 1 8.5 even 2
320.9.h.a.319.1 1 40.19 odd 2
320.9.h.b.319.1 1 8.3 odd 2
320.9.h.b.319.1 1 40.29 even 2