↑

Properties

Label 40.15.e.a.19.1
Level $40$
Weight $15$
Character 40.19
Self dual yes
Analytic conductor $49.732$
Analytic rank $0$
Dimension $1$
CM discriminant -40
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,15,Mod(19,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.19");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 40.e (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: \(49.7315872608\)
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 \(\chi\) \(=\) 40.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-128.000 q^{2} +16384.0 q^{4} +78125.0 q^{5} -67866.0 q^{7} -2.09715e6 q^{8} +4.78297e6 q^{9} -1.00000e7 q^{10} +1.60524e7 q^{11} +1.25277e8 q^{13} +8.68685e6 q^{14} +2.68435e8 q^{16} -6.12220e8 q^{18} +6.44127e8 q^{19} +1.28000e9 q^{20} -2.05470e9 q^{22} -5.95134e9 q^{23} +6.10352e9 q^{25} -1.60354e10 q^{26} -1.11192e9 q^{28} -3.43597e10 q^{32} -5.30203e9 q^{35} +7.83642e10 q^{36} -1.00483e11 q^{37} -8.24483e10 q^{38} -1.63840e11 q^{40} +2.27781e11 q^{41} +2.63002e11 q^{44} +3.73669e11 q^{45} +7.61771e11 q^{46} -9.85930e11 q^{47} -6.73617e11 q^{49} -7.81250e11 q^{50} +2.05253e12 q^{52} +1.80159e12 q^{53} +1.25409e12 q^{55} +1.42325e11 q^{56} +4.68767e12 q^{59} -3.24601e11 q^{63} +4.39805e12 q^{64} +9.78723e12 q^{65} +6.78660e11 q^{70} -1.00306e13 q^{72} +1.28618e13 q^{74} +1.05534e13 q^{76} -1.08941e12 q^{77} +2.09715e13 q^{80} +2.28768e13 q^{81} -2.91560e13 q^{82} -3.36643e13 q^{88} -5.76010e13 q^{89} -4.78297e13 q^{90} -8.50202e12 q^{91} -9.75067e13 q^{92} +1.26199e14 q^{94} +5.03224e13 q^{95} +8.62230e13 q^{98} +7.67780e13 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −128.000 −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 16384.0 1.00000
\(5\) 78125.0 1.00000
\(6\) 0 0
\(7\) −67866.0 −0.0824074 −0.0412037 − 0.999151i \(-0.513119\pi\)
−0.0412037 + 0.999151i \(0.513119\pi\)
\(8\) −2.09715e6 −1.00000
\(9\) 4.78297e6 1.00000
\(10\) −1.00000e7 −1.00000
\(11\) 1.60524e7 0.823741 0.411871 − 0.911242i \(-0.364876\pi\)
0.411871 + 0.911242i \(0.364876\pi\)
\(12\) 0 0
\(13\) 1.25277e8 1.99649 0.998243 − 0.0592462i \(-0.0188697\pi\)
0.998243 + 0.0592462i \(0.0188697\pi\)
\(14\) 8.68685e6 0.0824074
\(15\) 0 0
\(16\) 2.68435e8 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −6.12220e8 −1.00000
\(19\) 6.44127e8 0.720604 0.360302 − 0.932836i \(-0.382674\pi\)
0.360302 + 0.932836i \(0.382674\pi\)
\(20\) 1.28000e9 1.00000
\(21\) 0 0
\(22\) −2.05470e9 −0.823741
\(23\) −5.95134e9 −1.74791 −0.873957 − 0.486004i \(-0.838454\pi\)
−0.873957 + 0.486004i \(0.838454\pi\)
\(24\) 0 0
\(25\) 6.10352e9 1.00000
\(26\) −1.60354e10 −1.99649
\(27\) 0 0
\(28\) −1.11192e9 −0.0824074
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −3.43597e10 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −5.30203e9 −0.0824074
\(36\) 7.83642e10 1.00000
\(37\) −1.00483e11 −1.05847 −0.529235 − 0.848475i \(-0.677521\pi\)
−0.529235 + 0.848475i \(0.677521\pi\)
\(38\) −8.24483e10 −0.720604
\(39\) 0 0
\(40\) −1.63840e11 −1.00000
\(41\) 2.27781e11 1.16958 0.584792 − 0.811183i \(-0.301176\pi\)
0.584792 + 0.811183i \(0.301176\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 2.63002e11 0.823741
\(45\) 3.73669e11 1.00000
\(46\) 7.61771e11 1.74791
\(47\) −9.85930e11 −1.94608 −0.973041 − 0.230633i \(-0.925920\pi\)
−0.973041 + 0.230633i \(0.925920\pi\)
\(48\) 0 0
\(49\) −6.73617e11 −0.993209
\(50\) −7.81250e11 −1.00000
\(51\) 0 0
\(52\) 2.05253e12 1.99649
\(53\) 1.80159e12 1.53364 0.766821 − 0.641861i \(-0.221838\pi\)
0.766821 + 0.641861i \(0.221838\pi\)
\(54\) 0 0
\(55\) 1.25409e12 0.823741
\(56\) 1.42325e11 0.0824074
\(57\) 0 0
\(58\) 0 0
\(59\) 4.68767e12 1.88362 0.941809 − 0.336148i \(-0.109124\pi\)
0.941809 + 0.336148i \(0.109124\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −3.24601e11 −0.0824074
\(64\) 4.39805e12 1.00000
\(65\) 9.78723e12 1.99649
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 6.78660e11 0.0824074
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.00306e13 −1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 1.28618e13 1.05847
\(75\) 0 0
\(76\) 1.05534e13 0.720604
\(77\) −1.08941e12 −0.0678823
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 2.09715e13 1.00000
\(81\) 2.28768e13 1.00000
\(82\) −2.91560e13 −1.16958
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −3.36643e13 −0.823741
\(89\) −5.76010e13 −1.30227 −0.651134 − 0.758963i \(-0.725706\pi\)
−0.651134 + 0.758963i \(0.725706\pi\)
\(90\) −4.78297e13 −1.00000
\(91\) −8.50202e12 −0.164525
\(92\) −9.75067e13 −1.74791
\(93\) 0 0
\(94\) 1.26199e14 1.94608
\(95\) 5.03224e13 0.720604
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 8.62230e13 0.993209
\(99\) 7.67780e13 0.823741
\(100\) 1.00000e14 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.45853e14 1.99901 0.999504 − 0.0315014i \(-0.0100289\pi\)
0.999504 + 0.0315014i \(0.0100289\pi\)
\(104\) −2.62724e14 −1.99649
\(105\) 0 0
\(106\) −2.30603e14 −1.53364
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −1.60524e14 −0.823741
\(111\) 0 0
\(112\) −1.82176e13 −0.0824074
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −4.64948e14 −1.74791
\(116\) 0 0
\(117\) 5.99194e14 1.99649
\(118\) −6.00022e14 −1.88362
\(119\) 0 0
\(120\) 0 0
\(121\) −1.22071e14 −0.321451
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.76837e14 1.00000
\(126\) 4.15489e13 0.0824074
\(127\) −2.02035e14 −0.379140 −0.189570 − 0.981867i \(-0.560709\pi\)
−0.189570 + 0.981867i \(0.560709\pi\)
\(128\) −5.62950e14 −1.00000
\(129\) 0 0
\(130\) −1.25277e15 −1.99649
\(131\) −9.43716e14 −1.42542 −0.712709 − 0.701460i \(-0.752532\pi\)
−0.712709 + 0.701460i \(0.752532\pi\)
\(132\) 0 0
\(133\) −4.37143e13 −0.0593830
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.73492e15 1.73052 0.865259 − 0.501324i \(-0.167154\pi\)
0.865259 + 0.501324i \(0.167154\pi\)
\(140\) −8.68685e13 −0.0824074
\(141\) 0 0
\(142\) 0 0
\(143\) 2.01099e15 1.64459
\(144\) 1.28392e15 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.64631e15 −1.05847
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −1.35083e15 −0.720604
\(153\) 0 0
\(154\) 1.39445e14 0.0678823
\(155\) 0 0
\(156\) 0 0
\(157\) 4.52640e15 1.92511 0.962555 − 0.271088i \(-0.0873835\pi\)
0.962555 + 0.271088i \(0.0873835\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.68435e15 −1.00000
\(161\) 4.03894e14 0.144041
\(162\) −2.92823e15 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 3.73197e15 1.16958
\(165\) 0 0
\(166\) 0 0
\(167\) 5.50909e15 1.52077 0.760386 − 0.649471i \(-0.225010\pi\)
0.760386 + 0.649471i \(0.225010\pi\)
\(168\) 0 0
\(169\) 1.17568e16 2.98596
\(170\) 0 0
\(171\) 3.08084e15 0.720604
\(172\) 0 0
\(173\) −3.27758e15 −0.706692 −0.353346 − 0.935493i \(-0.614956\pi\)
−0.353346 + 0.935493i \(0.614956\pi\)
\(174\) 0 0
\(175\) −4.14221e14 −0.0824074
\(176\) 4.30903e15 0.823741
\(177\) 0 0
\(178\) 7.37293e15 1.30227
\(179\) −1.15436e16 −1.96051 −0.980254 − 0.197743i \(-0.936639\pi\)
−0.980254 + 0.197743i \(0.936639\pi\)
\(180\) 6.12220e15 1.00000
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 1.08826e15 0.164525
\(183\) 0 0
\(184\) 1.24809e16 1.74791
\(185\) −7.85021e15 −1.05847
\(186\) 0 0
\(187\) 0 0
\(188\) −1.61535e16 −1.94608
\(189\) 0 0
\(190\) −6.44127e15 −0.720604
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.10365e16 −0.993209
\(197\) −2.17426e16 −1.88820 −0.944099 − 0.329661i \(-0.893066\pi\)
−0.944099 + 0.329661i \(0.893066\pi\)
\(198\) −9.82759e15 −0.823741
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.28000e16 −1.00000
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.77954e16 1.16958
\(206\) −3.14691e16 −1.99901
\(207\) −2.84651e16 −1.74791
\(208\) 3.36287e16 1.99649
\(209\) 1.03398e16 0.593591
\(210\) 0 0
\(211\) −3.19672e16 −1.71683 −0.858415 − 0.512955i \(-0.828551\pi\)
−0.858415 + 0.512955i \(0.828551\pi\)
\(212\) 2.95172e16 1.53364
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 2.05470e16 0.823741
\(221\) 0 0
\(222\) 0 0
\(223\) −4.11927e16 −1.50206 −0.751028 − 0.660270i \(-0.770442\pi\)
−0.751028 + 0.660270i \(0.770442\pi\)
\(224\) 2.33186e15 0.0824074
\(225\) 2.91929e16 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 5.95134e16 1.74791
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) −7.66968e16 −1.99649
\(235\) −7.70258e16 −1.94608
\(236\) 7.68028e16 1.88362
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 6.79936e16 1.43995 0.719977 − 0.693998i \(-0.244152\pi\)
0.719977 + 0.693998i \(0.244152\pi\)
\(242\) 1.56251e16 0.321451
\(243\) 0 0
\(244\) 0 0
\(245\) −5.26263e16 −0.993209
\(246\) 0 0
\(247\) 8.06940e16 1.43868
\(248\) 0 0
\(249\) 0 0
\(250\) −6.10352e16 −1.00000
\(251\) −7.95684e16 −1.26772 −0.633862 − 0.773446i \(-0.718531\pi\)
−0.633862 + 0.773446i \(0.718531\pi\)
\(252\) −5.31826e15 −0.0824074
\(253\) −9.55332e16 −1.43983
\(254\) 2.58604e16 0.379140
\(255\) 0 0
\(256\) 7.20576e16 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 6.81935e15 0.0872258
\(260\) 1.60354e17 1.99649
\(261\) 0 0
\(262\) 1.20796e17 1.42542
\(263\) −1.47100e17 −1.69014 −0.845069 − 0.534657i \(-0.820441\pi\)
−0.845069 + 0.534657i \(0.820441\pi\)
\(264\) 0 0
\(265\) 1.40749e17 1.53364
\(266\) 5.59543e15 0.0593830
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.79760e16 0.823741
\(276\) 0 0
\(277\) 3.56086e16 0.284575 0.142287 − 0.989825i \(-0.454554\pi\)
0.142287 + 0.989825i \(0.454554\pi\)
\(278\) −2.22070e17 −1.73052
\(279\) 0 0
\(280\) 1.11192e16 0.0824074
\(281\) 2.28956e17 1.65504 0.827520 − 0.561436i \(-0.189751\pi\)
0.827520 + 0.561436i \(0.189751\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.57406e17 −1.64459
\(287\) −1.54586e16 −0.0963823
\(288\) −1.64342e17 −1.00000
\(289\) 1.68378e17 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 9.64995e16 0.520537 0.260269 − 0.965536i \(-0.416189\pi\)
0.260269 + 0.965536i \(0.416189\pi\)
\(294\) 0 0
\(295\) 3.66224e17 1.88362
\(296\) 2.10727e17 1.05847
\(297\) 0 0
\(298\) 0 0
\(299\) −7.45563e17 −3.48969
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 1.72907e17 0.720604
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −1.78489e16 −0.0678823
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −5.79379e17 −1.92511
\(315\) −2.53595e16 −0.0824074
\(316\) 0 0
\(317\) 5.03679e17 1.56581 0.782905 − 0.622141i \(-0.213737\pi\)
0.782905 + 0.622141i \(0.213737\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 3.43597e17 1.00000
\(321\) 0 0
\(322\) −5.16984e16 −0.144041
\(323\) 0 0
\(324\) 3.74813e17 1.00000
\(325\) 7.64628e17 1.99649
\(326\) 0 0
\(327\) 0 0
\(328\) −4.77692e17 −1.16958
\(329\) 6.69111e16 0.160371
\(330\) 0 0
\(331\) −4.98299e17 −1.14471 −0.572354 − 0.820007i \(-0.693969\pi\)
−0.572354 + 0.820007i \(0.693969\pi\)
\(332\) 0 0
\(333\) −4.80605e17 −1.05847
\(334\) −7.05163e17 −1.52077
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −1.50488e18 −2.98596
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) −3.94348e17 −0.720604
\(343\) 9.17440e16 0.164255
\(344\) 0 0
\(345\) 0 0
\(346\) 4.19530e17 0.706692
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 5.30203e16 0.0824074
\(351\) 0 0
\(352\) −5.51556e17 −0.823741
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −9.43735e17 −1.30227
\(357\) 0 0
\(358\) 1.47758e18 1.96051
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −7.83642e17 −1.00000
\(361\) −3.84107e17 −0.480731
\(362\) 0 0
\(363\) 0 0
\(364\) −1.39297e17 −0.164525
\(365\) 0 0
\(366\) 0 0
\(367\) 1.37119e18 1.52910 0.764550 − 0.644565i \(-0.222961\pi\)
0.764550 + 0.644565i \(0.222961\pi\)
\(368\) −1.59755e18 −1.74791
\(369\) 1.08947e18 1.16958
\(370\) 1.00483e18 1.05847
\(371\) −1.22266e17 −0.126383
\(372\) 0 0
\(373\) −1.58391e18 −1.57677 −0.788386 − 0.615180i \(-0.789083\pi\)
−0.788386 + 0.615180i \(0.789083\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 2.06764e18 1.94608
\(377\) 0 0
\(378\) 0 0
\(379\) 1.49469e18 1.33069 0.665345 − 0.746536i \(-0.268285\pi\)
0.665345 + 0.746536i \(0.268285\pi\)
\(380\) 8.24483e17 0.720604
\(381\) 0 0
\(382\) 0 0
\(383\) −4.46539e17 −0.369375 −0.184688 − 0.982797i \(-0.559127\pi\)
−0.184688 + 0.982797i \(0.559127\pi\)
\(384\) 0 0
\(385\) −8.51102e16 −0.0678823
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.41268e18 0.993209
\(393\) 0 0
\(394\) 2.78305e18 1.88820
\(395\) 0 0
\(396\) 1.25793e18 0.823741
\(397\) 1.14115e16 0.00734190 0.00367095 − 0.999993i \(-0.498831\pi\)
0.00367095 + 0.999993i \(0.498831\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.63840e18 1.00000
\(401\) −7.07198e17 −0.424161 −0.212080 − 0.977252i \(-0.568024\pi\)
−0.212080 + 0.977252i \(0.568024\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.78725e18 1.00000
\(406\) 0 0
\(407\) −1.61299e18 −0.871906
\(408\) 0 0
\(409\) −8.94471e17 −0.467200 −0.233600 − 0.972333i \(-0.575051\pi\)
−0.233600 + 0.972333i \(0.575051\pi\)
\(410\) −2.27781e18 −1.16958
\(411\) 0 0
\(412\) 4.02805e18 1.99901
\(413\) −3.18133e17 −0.155224
\(414\) 3.64353e18 1.74791
\(415\) 0 0
\(416\) −4.30447e18 −1.99649
\(417\) 0 0
\(418\) −1.32349e18 −0.593591
\(419\) 4.43581e18 1.95648 0.978239 − 0.207482i \(-0.0665270\pi\)
0.978239 + 0.207482i \(0.0665270\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 4.09180e18 1.71683
\(423\) −4.71567e18 −1.94608
\(424\) −3.77820e18 −1.53364
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.83342e18 −1.25955
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) −2.63002e18 −0.823741
\(441\) −3.22189e18 −0.993209
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −4.50008e18 −1.30227
\(446\) 5.27267e18 1.50206
\(447\) 0 0
\(448\) −2.98478e17 −0.0824074
\(449\) −6.56311e18 −1.78396 −0.891981 − 0.452074i \(-0.850684\pi\)
−0.891981 + 0.452074i \(0.850684\pi\)
\(450\) −3.73669e18 −1.00000
\(451\) 3.65643e18 0.963434
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.64220e17 −0.164525
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −7.61771e18 −1.74791
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.85442e18 −0.406575 −0.203287 − 0.979119i \(-0.565163\pi\)
−0.203287 + 0.979119i \(0.565163\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 9.81719e18 1.99649
\(469\) 0 0
\(470\) 9.85930e18 1.94608
\(471\) 0 0
\(472\) −9.83076e18 −1.88362
\(473\) 0 0
\(474\) 0 0
\(475\) 3.93144e18 0.720604
\(476\) 0 0
\(477\) 8.61693e18 1.53364
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1.25881e19 −2.11322
\(482\) −8.70317e18 −1.43995
\(483\) 0 0
\(484\) −2.00001e18 −0.321451
\(485\) 0 0
\(486\) 0 0
\(487\) −5.60133e17 −0.0862161 −0.0431081 − 0.999070i \(-0.513726\pi\)
−0.0431081 + 0.999070i \(0.513726\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 6.73617e18 0.993209
\(491\) −1.11016e19 −1.61366 −0.806832 − 0.590780i \(-0.798820\pi\)
−0.806832 + 0.590780i \(0.798820\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −1.03288e19 −1.43868
\(495\) 5.99828e18 0.823741
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 4.77469e18 0.619785 0.309893 − 0.950772i \(-0.399707\pi\)
0.309893 + 0.950772i \(0.399707\pi\)
\(500\) 7.81250e18 1.00000
\(501\) 0 0
\(502\) 1.01848e19 1.26772
\(503\) 1.61818e19 1.98633 0.993166 − 0.116712i \(-0.0372355\pi\)
0.993166 + 0.116712i \(0.0372355\pi\)
\(504\) 6.80738e17 0.0824074
\(505\) 0 0
\(506\) 1.22282e19 1.43983
\(507\) 0 0
\(508\) −3.31013e18 −0.379140
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −9.22337e18 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 1.92072e19 1.99901
\(516\) 0 0
\(517\) −1.58265e19 −1.60307
\(518\) −8.72877e17 −0.0872258
\(519\) 0 0
\(520\) −2.05253e19 −1.99649
\(521\) −7.51842e18 −0.721544 −0.360772 − 0.932654i \(-0.617487\pi\)
−0.360772 + 0.932654i \(0.617487\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −1.54618e19 −1.42542
\(525\) 0 0
\(526\) 1.88288e19 1.69014
\(527\) 0 0
\(528\) 0 0
\(529\) 2.38256e19 2.05520
\(530\) −1.80159e19 −1.53364
\(531\) 2.24210e19 1.88362
\(532\) −7.16216e17 −0.0593830
\(533\) 2.85357e19 2.33506
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.08132e19 −0.818147
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −1.25409e19 −0.823741
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −4.55790e18 −0.284575
\(555\) 0 0
\(556\) 2.84250e19 1.73052
\(557\) 2.08524e19 1.25363 0.626816 − 0.779167i \(-0.284358\pi\)
0.626816 + 0.779167i \(0.284358\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1.42325e18 −0.0824074
\(561\) 0 0
\(562\) −2.93064e19 −1.65504
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.55256e18 −0.0824074
\(568\) 0 0
\(569\) −3.15378e19 −1.63322 −0.816610 − 0.577189i \(-0.804149\pi\)
−0.816610 + 0.577189i \(0.804149\pi\)
\(570\) 0 0
\(571\) −2.53139e19 −1.27911 −0.639553 − 0.768747i \(-0.720880\pi\)
−0.639553 + 0.768747i \(0.720880\pi\)
\(572\) 3.29480e19 1.64459
\(573\) 0 0
\(574\) 1.97870e18 0.0963823
\(575\) −3.63241e19 −1.74791
\(576\) 2.10357e19 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −2.15524e19 −1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.89197e19 1.26332
\(584\) 0 0
\(585\) 4.68120e19 1.99649
\(586\) −1.23519e19 −0.520537
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −4.68767e19 −1.88362
\(591\) 0 0
\(592\) −2.69731e19 −1.05847
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 9.54321e19 3.48969
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1.47841e19 0.522003 0.261002 − 0.965338i \(-0.415947\pi\)
0.261002 + 0.965338i \(0.415947\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.53679e18 −0.321451
\(606\) 0 0
\(607\) 4.21288e19 1.38758 0.693790 − 0.720178i \(-0.255940\pi\)
0.693790 + 0.720178i \(0.255940\pi\)
\(608\) −2.21320e19 −0.720604
\(609\) 0 0
\(610\) 0 0
\(611\) −1.23514e20 −3.88533
\(612\) 0 0
\(613\) −2.08455e18 −0.0640897 −0.0320448 − 0.999486i \(-0.510202\pi\)
−0.0320448 + 0.999486i \(0.510202\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 2.28466e18 0.0678823
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 4.82069e19 1.38444 0.692221 − 0.721686i \(-0.256632\pi\)
0.692221 + 0.721686i \(0.256632\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.90915e18 0.107316
\(624\) 0 0
\(625\) 3.72529e19 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 7.41605e19 1.92511
\(629\) 0 0
\(630\) 3.24601e18 0.0824074
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.44710e19 −1.56581
\(635\) −1.57839e19 −0.379140
\(636\) 0 0
\(637\) −8.43885e19 −1.98293
\(638\) 0 0
\(639\) 0 0
\(640\) −4.39805e19 −1.00000
\(641\) −7.56679e19 −1.70179 −0.850895 − 0.525337i \(-0.823939\pi\)
−0.850895 + 0.525337i \(0.823939\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 6.61739e18 0.144041
\(645\) 0 0
\(646\) 0 0
\(647\) −9.27967e19 −1.95525 −0.977625 − 0.210354i \(-0.932539\pi\)
−0.977625 + 0.210354i \(0.932539\pi\)
\(648\) −4.79761e19 −1.00000
\(649\) 7.52483e19 1.55161
\(650\) −9.78723e19 −1.99649
\(651\) 0 0
\(652\) 0 0
\(653\) −3.41655e19 −0.674831 −0.337416 − 0.941356i \(-0.609553\pi\)
−0.337416 + 0.941356i \(0.609553\pi\)
\(654\) 0 0
\(655\) −7.37278e19 −1.42542
\(656\) 6.11446e19 1.16958
\(657\) 0 0
\(658\) −8.56462e18 −0.160371
\(659\) −4.39969e19 −0.815125 −0.407563 − 0.913177i \(-0.633621\pi\)
−0.407563 + 0.913177i \(0.633621\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 6.37823e19 1.14471
\(663\) 0 0
\(664\) 0 0
\(665\) −3.41518e18 −0.0593830
\(666\) 6.15175e19 1.05847
\(667\) 0 0
\(668\) 9.02609e19 1.52077
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.92624e20 2.98596
\(677\) −8.03858e18 −0.123327 −0.0616636 − 0.998097i \(-0.519641\pi\)
−0.0616636 + 0.998097i \(0.519641\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 5.04765e19 0.720604
\(685\) 0 0
\(686\) −1.17432e19 −0.164255
\(687\) 0 0
\(688\) 0 0
\(689\) 2.25697e20 3.06190
\(690\) 0 0
\(691\) −1.39170e20 −1.85012 −0.925058 − 0.379825i \(-0.875984\pi\)
−0.925058 + 0.379825i \(0.875984\pi\)
\(692\) −5.36998e19 −0.706692
\(693\) −5.21062e18 −0.0678823
\(694\) 0 0
\(695\) 1.35541e20 1.73052
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −6.78660e18 −0.0824074
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −6.47236e19 −0.762738
\(704\) 7.05991e19 0.823741
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.20798e20 1.30227
\(713\) 0 0
\(714\) 0 0
\(715\) 1.57108e20 1.64459
\(716\) −1.89130e20 −1.96051
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.00306e20 1.00000
\(721\) −1.66850e19 −0.164733
\(722\) 4.91657e19 0.480731
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −8.52592e19 −0.794329 −0.397164 − 0.917748i \(-0.630006\pi\)
−0.397164 + 0.917748i \(0.630006\pi\)
\(728\) 1.78300e19 0.164525
\(729\) 1.09419e20 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.20448e20 1.93900 0.969499 − 0.245094i \(-0.0788188\pi\)
0.969499 + 0.245094i \(0.0788188\pi\)
\(734\) −1.75513e20 −1.52910
\(735\) 0 0
\(736\) 2.04486e20 1.74791
\(737\) 0 0
\(738\) −1.39452e20 −1.16958
\(739\) −1.47880e19 −0.122856 −0.0614281 − 0.998112i \(-0.519565\pi\)
−0.0614281 + 0.998112i \(0.519565\pi\)
\(740\) −1.28618e20 −1.05847
\(741\) 0 0
\(742\) 1.56501e19 0.126383
\(743\) −1.47711e20 −1.18165 −0.590827 − 0.806798i \(-0.701198\pi\)
−0.590827 + 0.806798i \(0.701198\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.02741e20 1.57677
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −2.64659e20 −1.94608
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.65157e20 1.15938 0.579690 − 0.814837i \(-0.303174\pi\)
0.579690 + 0.814837i \(0.303174\pi\)
\(758\) −1.91320e20 −1.33069
\(759\) 0 0
\(760\) −1.05534e20 −0.720604
\(761\) −2.53391e20 −1.71434 −0.857172 − 0.515030i \(-0.827781\pi\)
−0.857172 + 0.515030i \(0.827781\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 5.71570e19 0.369375
\(767\) 5.87255e20 3.76062
\(768\) 0 0
\(769\) −3.17054e20 −1.99365 −0.996823 − 0.0796425i \(-0.974622\pi\)
−0.996823 + 0.0796425i \(0.974622\pi\)
\(770\) 1.08941e19 0.0678823
\(771\) 0 0
\(772\) 0 0
\(773\) −2.91108e20 −1.76522 −0.882609 − 0.470109i \(-0.844215\pi\)
−0.882609 + 0.470109i \(0.844215\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.46720e20 0.842806
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.80823e20 −0.993209
\(785\) 3.53625e20 1.92511
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −3.56231e20 −1.88820
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −1.61015e20 −0.823741
\(793\) 0 0
\(794\) −1.46067e18 −0.00734190
\(795\) 0 0
\(796\) 0 0
\(797\) −4.65475e19 −0.227871 −0.113935 − 0.993488i \(-0.536346\pi\)
−0.113935 + 0.993488i \(0.536346\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.09715e20 −1.00000
\(801\) −2.75504e20 −1.30227
\(802\) 9.05213e19 0.424161
\(803\) 0 0
\(804\) 0 0
\(805\) 3.15542e19 0.144041
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.24141e20 −1.87012 −0.935062 − 0.354485i \(-0.884656\pi\)
−0.935062 + 0.354485i \(0.884656\pi\)
\(810\) −2.28768e20 −1.00000
\(811\) −4.06829e20 −1.76306 −0.881528 − 0.472132i \(-0.843485\pi\)
−0.881528 + 0.472132i \(0.843485\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 2.06462e20 0.871906
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.14492e20 0.467200
\(819\) −4.06649e19 −0.164525
\(820\) 2.91560e20 1.16958
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.14449e20 0.447521 0.223760 − 0.974644i \(-0.428167\pi\)
0.223760 + 0.974644i \(0.428167\pi\)
\(824\) −5.15590e20 −1.99901
\(825\) 0 0
\(826\) 4.07211e19 0.155224
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −4.66372e20 −1.74791
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.50972e20 1.99649
\(833\) 0 0
\(834\) 0 0
\(835\) 4.30397e20 1.52077
\(836\) 1.69407e20 0.593591
\(837\) 0 0
\(838\) −5.67784e20 −1.95648
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.97558e20 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −5.23751e20 −1.71683
\(845\) 9.18504e20 2.98596
\(846\) 6.03606e20 1.94608
\(847\) 8.28446e18 0.0264899
\(848\) 4.83610e20 1.53364
\(849\) 0 0
\(850\) 0 0
\(851\) 5.98006e20 1.85012
\(852\) 0 0
\(853\) 1.38829e20 0.422511 0.211255 − 0.977431i \(-0.432245\pi\)
0.211255 + 0.977431i \(0.432245\pi\)
\(854\) 0 0
\(855\) 2.40691e20 0.720604
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 6.71279e20 1.94514 0.972571 − 0.232608i \(-0.0747259\pi\)
0.972571 + 0.232608i \(0.0747259\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.15393e20 −0.884661 −0.442330 − 0.896852i \(-0.645848\pi\)
−0.442330 + 0.896852i \(0.645848\pi\)
\(864\) 0 0
\(865\) −2.56061e20 −0.706692
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 4.90678e20 1.25955
\(875\) −3.23610e19 −0.0824074
\(876\) 0 0
\(877\) −7.18319e20 −1.80020 −0.900099 − 0.435685i \(-0.856506\pi\)
−0.900099 + 0.435685i \(0.856506\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 3.36643e20 0.823741
\(881\) −8.16331e20 −1.98169 −0.990843 − 0.135016i \(-0.956892\pi\)
−0.990843 + 0.135016i \(0.956892\pi\)
\(882\) 4.12402e20 0.993209
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.28303e20 −0.991484 −0.495742 − 0.868470i \(-0.665104\pi\)
−0.495742 + 0.868470i \(0.665104\pi\)
\(888\) 0 0
\(889\) 1.37113e19 0.0312439
\(890\) 5.76010e20 1.30227
\(891\) 3.67227e20 0.823741
\(892\) −6.74901e20 −1.50206
\(893\) −6.35064e20 −1.40235
\(894\) 0 0
\(895\) −9.01841e20 −1.96051
\(896\) 3.82052e19 0.0824074
\(897\) 0 0
\(898\) 8.40078e20 1.78396
\(899\) 0 0
\(900\) 4.78297e20 1.00000
\(901\) 0 0
\(902\) −4.68024e20 −0.963434
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 8.50202e19 0.164525
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.40462e19 0.117465
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 9.75067e20 1.74791
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −6.13297e20 −1.05847
\(926\) 2.37365e20 0.406575
\(927\) 1.17591e21 1.99901
\(928\) 0 0
\(929\) 4.97319e20 0.832770 0.416385 − 0.909188i \(-0.363297\pi\)
0.416385 + 0.909188i \(0.363297\pi\)
\(930\) 0 0
\(931\) −4.33895e20 −0.715710
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1.25660e21 −1.99649
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −1.26199e21 −1.94608
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −1.35560e21 −2.04433
\(944\) 1.25834e21 1.88362
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −5.03224e20 −0.720604
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) −1.10297e21 −1.53364
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.56944e20 1.00000
\(962\) 1.61128e21 2.11322
\(963\) 0 0
\(964\) 1.11401e21 1.43995
\(965\) 0 0
\(966\) 0 0
\(967\) −7.15287e20 −0.904680 −0.452340 − 0.891845i \(-0.649411\pi\)
−0.452340 + 0.891845i \(0.649411\pi\)
\(968\) 2.56001e20 0.321451
\(969\) 0 0
\(970\) 0 0
\(971\) 2.39362e20 0.294117 0.147059 − 0.989128i \(-0.453019\pi\)
0.147059 + 0.989128i \(0.453019\pi\)
\(972\) 0 0
\(973\) −1.17742e20 −0.142607
\(974\) 7.16970e19 0.0862161
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) −9.24634e20 −1.07273
\(980\) −8.62230e20 −0.993209
\(981\) 0 0
\(982\) 1.42100e21 1.61366
\(983\) 1.67670e21 1.89052 0.945261 − 0.326316i \(-0.105807\pi\)
0.945261 + 0.326316i \(0.105807\pi\)
\(984\) 0 0
\(985\) −1.69864e21 −1.88820
\(986\) 0 0
\(987\) 0 0
\(988\) 1.32209e21 1.43868
\(989\) 0 0
\(990\) −7.67780e20 −0.823741
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.95441e21 −1.99594 −0.997972 − 0.0636503i \(-0.979726\pi\)
−0.997972 + 0.0636503i \(0.979726\pi\)
\(998\) −6.11160e20 −0.619785
\(999\) 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 40.15.e.a.19.1 ✓ 1
4.3 odd 2 160.15.e.b.79.1 1
5.4 even 2 40.15.e.b.19.1 yes 1
8.3 odd 2 40.15.e.b.19.1 yes 1
8.5 even 2 160.15.e.a.79.1 1
20.19 odd 2 160.15.e.a.79.1 1
40.19 odd 2 CM 40.15.e.a.19.1 ✓ 1
40.29 even 2 160.15.e.b.79.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.15.e.a.19.1 ✓ 1 1.1 even 1 trivial
40.15.e.a.19.1 ✓ 1 40.19 odd 2 CM
40.15.e.b.19.1 yes 1 5.4 even 2
40.15.e.b.19.1 yes 1 8.3 odd 2
160.15.e.a.79.1 1 8.5 even 2
160.15.e.a.79.1 1 20.19 odd 2
160.15.e.b.79.1 1 4.3 odd 2
160.15.e.b.79.1 1 40.29 even 2