↑

Properties

Label 8.15.d.a.3.1
Level $8$
Weight $15$
Character 8.3
Self dual yes
Analytic conductor $9.946$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,15,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8.3");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(9.94631745215\)
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 3.1
Character \(\chi\) \(=\) 8.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-128.000 q^{2} +3022.00 q^{3} +16384.0 q^{4} -386816. q^{6} -2.09715e6 q^{8} +4.34952e6 q^{9} +O(q^{10})\) \(q-128.000 q^{2} +3022.00 q^{3} +16384.0 q^{4} -386816. q^{6} -2.09715e6 q^{8} +4.34952e6 q^{9} +3.87123e7 q^{11} +4.95124e7 q^{12} +2.68435e8 q^{16} -3.28636e8 q^{17} -5.56738e8 q^{18} +1.77897e9 q^{19} -4.95517e9 q^{22} -6.33759e9 q^{24} +6.10352e9 q^{25} -1.30990e9 q^{27} -3.43597e10 q^{32} +1.16988e11 q^{33} +4.20654e10 q^{34} +7.12625e10 q^{36} -2.27709e11 q^{38} -3.33394e11 q^{41} -4.95013e11 q^{43} +6.34262e11 q^{44} +8.11212e11 q^{48} +6.78223e11 q^{49} -7.81250e11 q^{50} -9.93139e11 q^{51} +1.67667e11 q^{54} +5.37606e12 q^{57} -3.91449e12 q^{59} +4.39805e12 q^{64} -1.49745e13 q^{66} +2.71110e12 q^{67} -5.38438e12 q^{68} -9.12159e12 q^{72} +1.57940e12 q^{73} +1.84448e13 q^{75} +2.91467e13 q^{76} -2.47621e13 q^{81} +4.26744e13 q^{82} -3.11463e13 q^{83} +6.33616e13 q^{86} -8.11855e13 q^{88} -3.84337e13 q^{89} -1.03835e14 q^{96} -6.28150e13 q^{97} -8.68126e13 q^{98} +1.68380e14 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −128.000 −1.00000
\(3\) 3022.00 1.38180 0.690901 − 0.722950i \(-0.257214\pi\)
0.690901 + 0.722950i \(0.257214\pi\)
\(4\) 16384.0 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −386816. −1.38180
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −2.09715e6 −1.00000
\(9\) 4.34952e6 0.909376
\(10\) 0 0
\(11\) 3.87123e7 1.98655 0.993275 − 0.115776i \(-0.0369354\pi\)
0.993275 + 0.115776i \(0.0369354\pi\)
\(12\) 4.95124e7 1.38180
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 2.68435e8 1.00000
\(17\) −3.28636e8 −0.800890 −0.400445 − 0.916321i \(-0.631144\pi\)
−0.400445 + 0.916321i \(0.631144\pi\)
\(18\) −5.56738e8 −0.909376
\(19\) 1.77897e9 1.99019 0.995095 − 0.0989284i \(-0.0315415\pi\)
0.995095 + 0.0989284i \(0.0315415\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.95517e9 −1.98655
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −6.33759e9 −1.38180
\(25\) 6.10352e9 1.00000
\(26\) 0 0
\(27\) −1.30990e9 −0.125225
\(28\) 0 0
\(29\) 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
\(33\) 1.16988e11 2.74502
\(34\) 4.20654e10 0.800890
\(35\) 0 0
\(36\) 7.12625e10 0.909376
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −2.27709e11 −1.99019
\(39\) 0 0
\(40\) 0 0
\(41\) −3.33394e11 −1.71187 −0.855934 − 0.517085i \(-0.827017\pi\)
−0.855934 + 0.517085i \(0.827017\pi\)
\(42\) 0 0
\(43\) −4.95013e11 −1.82111 −0.910557 − 0.413384i \(-0.864347\pi\)
−0.910557 + 0.413384i \(0.864347\pi\)
\(44\) 6.34262e11 1.98655
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 8.11212e11 1.38180
\(49\) 6.78223e11 1.00000
\(50\) −7.81250e11 −1.00000
\(51\) −9.93139e11 −1.10667
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.67667e11 0.125225
\(55\) 0 0
\(56\) 0 0
\(57\) 5.37606e12 2.75005
\(58\) 0 0
\(59\) −3.91449e12 −1.57294 −0.786469 − 0.617630i \(-0.788093\pi\)
−0.786469 + 0.617630i \(0.788093\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.39805e12 1.00000
\(65\) 0 0
\(66\) −1.49745e13 −2.74502
\(67\) 2.71110e12 0.447324 0.223662 − 0.974667i \(-0.428199\pi\)
0.223662 + 0.974667i \(0.428199\pi\)
\(68\) −5.38438e12 −0.800890
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −9.12159e12 −0.909376
\(73\) 1.57940e12 0.142966 0.0714830 − 0.997442i \(-0.477227\pi\)
0.0714830 + 0.997442i \(0.477227\pi\)
\(74\) 0 0
\(75\) 1.84448e13 1.38180
\(76\) 2.91467e13 1.99019
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −2.47621e13 −1.08241
\(82\) 4.26744e13 1.71187
\(83\) −3.11463e13 −1.14778 −0.573891 − 0.818932i \(-0.694567\pi\)
−0.573891 + 0.818932i \(0.694567\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.33616e13 1.82111
\(87\) 0 0
\(88\) −8.11855e13 −1.98655
\(89\) −3.84337e13 −0.868924 −0.434462 − 0.900690i \(-0.643061\pi\)
−0.434462 + 0.900690i \(0.643061\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −1.03835e14 −1.38180
\(97\) −6.28150e13 −0.777430 −0.388715 − 0.921358i \(-0.627081\pi\)
−0.388715 + 0.921358i \(0.627081\pi\)
\(98\) −8.68126e13 −1.00000
\(99\) 1.68380e14 1.80652
\(100\) 1.00000e14 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 1.27122e14 1.10667
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.79454e14 1.11755 0.558774 − 0.829320i \(-0.311272\pi\)
0.558774 + 0.829320i \(0.311272\pi\)
\(108\) −2.14614e13 −0.125225
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.33670e11 −0.00396866 −0.00198433 − 0.999998i \(-0.500632\pi\)
−0.00198433 + 0.999998i \(0.500632\pi\)
\(114\) −6.88136e14 −2.75005
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 5.01055e14 1.57294
\(119\) 0 0
\(120\) 0 0
\(121\) 1.11889e15 2.94638
\(122\) 0 0
\(123\) −1.00752e15 −2.36546
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −5.62950e14 −1.00000
\(129\) −1.49593e15 −2.51642
\(130\) 0 0
\(131\) −1.31730e15 −1.98970 −0.994848 − 0.101381i \(-0.967674\pi\)
−0.994848 + 0.101381i \(0.967674\pi\)
\(132\) 1.91674e15 2.74502
\(133\) 0 0
\(134\) −3.47021e14 −0.447324
\(135\) 0 0
\(136\) 6.89200e14 0.800890
\(137\) 1.60085e15 1.76729 0.883644 − 0.468159i \(-0.155083\pi\)
0.883644 + 0.468159i \(0.155083\pi\)
\(138\) 0 0
\(139\) 8.56226e14 0.854053 0.427027 − 0.904239i \(-0.359561\pi\)
0.427027 + 0.904239i \(0.359561\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.16756e15 0.909376
\(145\) 0 0
\(146\) −2.02164e14 −0.142966
\(147\) 2.04959e15 1.38180
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −2.36094e15 −1.38180
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −3.73078e15 −1.99019
\(153\) −1.42941e15 −0.728310
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 3.16955e15 1.08241
\(163\) −2.78599e15 −0.911309 −0.455655 − 0.890157i \(-0.650595\pi\)
−0.455655 + 0.890157i \(0.650595\pi\)
\(164\) −5.46232e15 −1.71187
\(165\) 0 0
\(166\) 3.98672e15 1.14778
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.93738e15 1.00000
\(170\) 0 0
\(171\) 7.73767e15 1.80983
\(172\) −8.11029e15 −1.82111
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.03917e16 1.98655
\(177\) −1.18296e16 −2.17349
\(178\) 4.91951e15 0.868924
\(179\) 7.27403e15 1.23539 0.617695 − 0.786418i \(-0.288067\pi\)
0.617695 + 0.786418i \(0.288067\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.27222e16 −1.59101
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.32909e16 1.38180
\(193\) −1.94418e16 −1.94911 −0.974554 − 0.224153i \(-0.928038\pi\)
−0.974554 + 0.224153i \(0.928038\pi\)
\(194\) 8.04032e15 0.777430
\(195\) 0 0
\(196\) 1.11120e16 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −2.15526e16 −1.80652
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.28000e16 −1.00000
\(201\) 8.19296e15 0.618113
\(202\) 0 0
\(203\) 0 0
\(204\) −1.62716e16 −1.10667
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.88681e16 3.95361
\(210\) 0 0
\(211\) −2.70870e16 −1.45474 −0.727368 − 0.686248i \(-0.759256\pi\)
−0.727368 + 0.686248i \(0.759256\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −2.29701e16 −1.11755
\(215\) 0 0
\(216\) 2.74706e15 0.125225
\(217\) 0 0
\(218\) 0 0
\(219\) 4.77295e15 0.197551
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.65473e16 0.909376
\(226\) 1.19510e14 0.00396866
\(227\) 1.56540e16 0.504014 0.252007 − 0.967725i \(-0.418909\pi\)
0.252007 + 0.967725i \(0.418909\pi\)
\(228\) 8.80813e16 2.75005
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.42439e16 −1.72322 −0.861610 − 0.507571i \(-0.830543\pi\)
−0.861610 + 0.507571i \(0.830543\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.41351e16 −1.57294
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −2.26270e16 −0.479190 −0.239595 − 0.970873i \(-0.577015\pi\)
−0.239595 + 0.970873i \(0.577015\pi\)
\(242\) −1.43218e17 −2.94638
\(243\) −6.85659e16 −1.37045
\(244\) 0 0
\(245\) 0 0
\(246\) 1.28962e17 2.36546
\(247\) 0 0
\(248\) 0 0
\(249\) −9.41240e16 −1.58601
\(250\) 0 0
\(251\) 1.11663e17 1.77907 0.889534 − 0.456869i \(-0.151029\pi\)
0.889534 + 0.456869i \(0.151029\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 7.20576e16 1.00000
\(257\) 4.88939e16 0.660272 0.330136 − 0.943933i \(-0.392905\pi\)
0.330136 + 0.943933i \(0.392905\pi\)
\(258\) 1.91479e17 2.51642
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 1.68615e17 1.98970
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −2.45343e17 −2.74502
\(265\) 0 0
\(266\) 0 0
\(267\) −1.16147e17 −1.20068
\(268\) 4.44187e16 0.447324
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −8.82176e16 −0.800890
\(273\) 0 0
\(274\) −2.04909e17 −1.76729
\(275\) 2.36281e17 1.98655
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −1.09597e17 −0.854053
\(279\) 0 0
\(280\) 0 0
\(281\) 2.22626e16 0.160928 0.0804638 − 0.996758i \(-0.474360\pi\)
0.0804638 + 0.996758i \(0.474360\pi\)
\(282\) 0 0
\(283\) 2.47411e17 1.70182 0.850910 − 0.525311i \(-0.176051\pi\)
0.850910 + 0.525311i \(0.176051\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −1.49448e17 −0.909376
\(289\) −6.03761e16 −0.358575
\(290\) 0 0
\(291\) −1.89827e17 −1.07425
\(292\) 2.58769e16 0.142966
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −2.62348e17 −1.38180
\(295\) 0 0
\(296\) 0 0
\(297\) −5.07091e16 −0.248766
\(298\) 0 0
\(299\) 0 0
\(300\) 3.02200e17 1.38180
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 4.77540e17 1.99019
\(305\) 0 0
\(306\) 1.82964e17 0.728310
\(307\) −4.93680e17 −1.92078 −0.960388 − 0.278665i \(-0.910108\pi\)
−0.960388 + 0.278665i \(0.910108\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.79697e17 1.29011 0.645055 − 0.764136i \(-0.276834\pi\)
0.645055 + 0.764136i \(0.276834\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 5.42309e17 1.54423
\(322\) 0 0
\(323\) −5.84635e17 −1.59392
\(324\) −4.05702e17 −1.08241
\(325\) 0 0
\(326\) 3.56606e17 0.911309
\(327\) 0 0
\(328\) 6.99177e17 1.71187
\(329\) 0 0
\(330\) 0 0
\(331\) −1.28422e17 −0.295014 −0.147507 − 0.989061i \(-0.547125\pi\)
−0.147507 + 0.989061i \(0.547125\pi\)
\(332\) −5.10300e17 −1.14778
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −6.83482e17 −1.38458 −0.692290 − 0.721620i \(-0.743398\pi\)
−0.692290 + 0.721620i \(0.743398\pi\)
\(338\) −5.03984e17 −1.00000
\(339\) −2.82155e15 −0.00548390
\(340\) 0 0
\(341\) 0 0
\(342\) −9.90422e17 −1.80983
\(343\) 0 0
\(344\) 1.03812e18 1.82111
\(345\) 0 0
\(346\) 0 0
\(347\) 7.74680e17 1.27884 0.639420 − 0.768858i \(-0.279174\pi\)
0.639420 + 0.768858i \(0.279174\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.33014e18 −1.98655
\(353\) −1.26968e18 −1.85897 −0.929484 − 0.368862i \(-0.879747\pi\)
−0.929484 + 0.368862i \(0.879747\pi\)
\(354\) 1.51419e18 2.17349
\(355\) 0 0
\(356\) −6.29697e17 −0.868924
\(357\) 0 0
\(358\) −9.31076e17 −1.23539
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 2.36574e18 2.96085
\(362\) 0 0
\(363\) 3.38128e18 4.07132
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.45010e18 −1.55673
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 1.62845e18 1.59101
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.24161e18 −1.99566 −0.997830 − 0.0658458i \(-0.979025\pi\)
−0.997830 + 0.0658458i \(0.979025\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.70123e18 −1.38180
\(385\) 0 0
\(386\) 2.48855e18 1.94911
\(387\) −2.15306e18 −1.65608
\(388\) −1.02916e18 −0.777430
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.42234e18 −1.00000
\(393\) −3.98089e18 −2.74936
\(394\) 0 0
\(395\) 0 0
\(396\) 2.75873e18 1.80652
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.63840e18 1.00000
\(401\) 1.69870e18 1.01884 0.509419 − 0.860519i \(-0.329860\pi\)
0.509419 + 0.860519i \(0.329860\pi\)
\(402\) −1.04870e18 −0.618113
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 2.08276e18 1.10667
\(409\) 7.50795e17 0.392156 0.196078 − 0.980588i \(-0.437180\pi\)
0.196078 + 0.980588i \(0.437180\pi\)
\(410\) 0 0
\(411\) 4.83778e18 2.44204
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.58752e18 1.18013
\(418\) −8.81511e18 −3.95361
\(419\) −4.51606e18 −1.99187 −0.995936 − 0.0900663i \(-0.971292\pi\)
−0.995936 + 0.0900663i \(0.971292\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 3.46714e18 1.45474
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00584e18 −0.800890
\(426\) 0 0
\(427\) 0 0
\(428\) 2.94017e18 1.11755
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −3.51623e17 −0.125225
\(433\) 5.25040e18 1.83983 0.919913 − 0.392122i \(-0.128259\pi\)
0.919913 + 0.392122i \(0.128259\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) −6.10938e17 −0.197551
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 2.94994e18 0.909376
\(442\) 0 0
\(443\) 3.94208e18 1.17733 0.588667 − 0.808376i \(-0.299653\pi\)
0.588667 + 0.808376i \(0.299653\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.19930e18 −0.597805 −0.298902 − 0.954284i \(-0.596620\pi\)
−0.298902 + 0.954284i \(0.596620\pi\)
\(450\) −3.39806e18 −0.909376
\(451\) −1.29064e19 −3.40071
\(452\) −1.52972e16 −0.00396866
\(453\) 0 0
\(454\) −2.00371e18 −0.504014
\(455\) 0 0
\(456\) −1.12744e19 −2.75005
\(457\) 8.01731e18 1.92582 0.962908 − 0.269828i \(-0.0869669\pi\)
0.962908 + 0.269828i \(0.0869669\pi\)
\(458\) 0 0
\(459\) 4.30480e17 0.100291
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 8.22322e18 1.72322
\(467\) 2.44265e18 0.504248 0.252124 − 0.967695i \(-0.418871\pi\)
0.252124 + 0.967695i \(0.418871\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 8.20929e18 1.57294
\(473\) −1.91631e19 −3.61773
\(474\) 0 0
\(475\) 1.08580e19 1.99019
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.89626e18 0.479190
\(483\) 0 0
\(484\) 1.83319e19 2.94638
\(485\) 0 0
\(486\) 8.77643e18 1.37045
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −8.41925e18 −1.25925
\(490\) 0 0
\(491\) −2.26387e18 −0.329065 −0.164532 − 0.986372i \(-0.552612\pi\)
−0.164532 + 0.986372i \(0.552612\pi\)
\(492\) −1.65071e19 −2.36546
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 1.20479e19 1.58601
\(499\) 3.81392e18 0.495071 0.247535 − 0.968879i \(-0.420379\pi\)
0.247535 + 0.968879i \(0.420379\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.42928e19 −1.77907
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.18988e19 1.38180
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −9.22337e18 −1.00000
\(513\) −2.33027e18 −0.249221
\(514\) −6.25842e18 −0.660272
\(515\) 0 0
\(516\) −2.45093e19 −2.51642
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.65089e18 0.542317 0.271158 − 0.962535i \(-0.412593\pi\)
0.271158 + 0.962535i \(0.412593\pi\)
\(522\) 0 0
\(523\) −8.51815e18 −0.795854 −0.397927 − 0.917417i \(-0.630270\pi\)
−0.397927 + 0.917417i \(0.630270\pi\)
\(524\) −2.15827e19 −1.98970
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 3.14038e19 2.74502
\(529\) 1.15928e19 1.00000
\(530\) 0 0
\(531\) −1.70262e19 −1.43039
\(532\) 0 0
\(533\) 0 0
\(534\) 1.48668e19 1.20068
\(535\) 0 0
\(536\) −5.68560e18 −0.447324
\(537\) 2.19821e19 1.70706
\(538\) 0 0
\(539\) 2.62555e19 1.98655
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.12919e19 0.800890
\(545\) 0 0
\(546\) 0 0
\(547\) −2.43861e19 −1.66430 −0.832150 − 0.554551i \(-0.812890\pi\)
−0.832150 + 0.554551i \(0.812890\pi\)
\(548\) 2.62284e19 1.76729
\(549\) 0 0
\(550\) −3.02439e19 −1.98655
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.40284e19 0.854053
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −3.84466e19 −2.19846
\(562\) −2.84961e18 −0.160928
\(563\) 3.54216e19 1.97565 0.987823 − 0.155580i \(-0.0497246\pi\)
0.987823 + 0.155580i \(0.0497246\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −3.16686e19 −1.70182
\(567\) 0 0
\(568\) 0 0
\(569\) 3.10898e19 1.61002 0.805011 − 0.593260i \(-0.202159\pi\)
0.805011 + 0.593260i \(0.202159\pi\)
\(570\) 0 0
\(571\) −3.62362e19 −1.83101 −0.915503 − 0.402312i \(-0.868207\pi\)
−0.915503 + 0.402312i \(0.868207\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.91294e19 0.909376
\(577\) −5.16622e17 −0.0242628 −0.0121314 − 0.999926i \(-0.503862\pi\)
−0.0121314 + 0.999926i \(0.503862\pi\)
\(578\) 7.72814e18 0.358575
\(579\) −5.87532e19 −2.69328
\(580\) 0 0
\(581\) 0 0
\(582\) 2.42979e19 1.07425
\(583\) 0 0
\(584\) −3.31225e18 −0.142966
\(585\) 0 0
\(586\) 0 0
\(587\) 7.99248e18 0.332824 0.166412 − 0.986056i \(-0.446782\pi\)
0.166412 + 0.986056i \(0.446782\pi\)
\(588\) 3.35805e19 1.38180
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.85851e19 −1.10855 −0.554277 − 0.832332i \(-0.687005\pi\)
−0.554277 + 0.832332i \(0.687005\pi\)
\(594\) 6.49077e18 0.248766
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −3.86816e19 −1.38180
\(601\) 1.32271e19 0.467030 0.233515 − 0.972353i \(-0.424977\pi\)
0.233515 + 0.972353i \(0.424977\pi\)
\(602\) 0 0
\(603\) 1.17920e19 0.406786
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −6.11251e19 −1.99019
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −2.34194e19 −0.728310
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 6.31910e19 1.92078
\(615\) 0 0
\(616\) 0 0
\(617\) 6.40050e19 1.88026 0.940129 − 0.340817i \(-0.110704\pi\)
0.940129 + 0.340817i \(0.110704\pi\)
\(618\) 0 0
\(619\) −1.08767e19 −0.312365 −0.156182 − 0.987728i \(-0.549919\pi\)
−0.156182 + 0.987728i \(0.549919\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.72529e19 1.00000
\(626\) −4.86012e19 −1.29011
\(627\) 2.08119e20 5.46311
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −8.18570e19 −2.01016
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.81552e19 −0.858119 −0.429059 − 0.903276i \(-0.641155\pi\)
−0.429059 + 0.903276i \(0.641155\pi\)
\(642\) −6.94156e19 −1.54423
\(643\) −6.40914e19 −1.41034 −0.705168 − 0.709040i \(-0.749128\pi\)
−0.705168 + 0.709040i \(0.749128\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 7.48333e19 1.59392
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 5.19299e19 1.08241
\(649\) −1.51539e20 −3.12472
\(650\) 0 0
\(651\) 0 0
\(652\) −4.56456e19 −0.911309
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −8.94947e19 −1.71187
\(657\) 6.86964e18 0.130010
\(658\) 0 0
\(659\) −3.21522e19 −0.595679 −0.297839 − 0.954616i \(-0.596266\pi\)
−0.297839 + 0.954616i \(0.596266\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 1.64380e19 0.295014
\(663\) 0 0
\(664\) 6.53184e19 1.14778
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.13872e20 1.82100 0.910502 − 0.413504i \(-0.135695\pi\)
0.910502 + 0.413504i \(0.135695\pi\)
\(674\) 8.74857e19 1.38458
\(675\) −7.99498e18 −0.125225
\(676\) 6.45100e19 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 3.61158e17 0.00548390
\(679\) 0 0
\(680\) 0 0
\(681\) 4.73062e19 0.696448
\(682\) 0 0
\(683\) −1.21775e20 −1.75636 −0.878180 − 0.478331i \(-0.841242\pi\)
−0.878180 + 0.478331i \(0.841242\pi\)
\(684\) 1.26774e20 1.80983
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −1.32879e20 −1.82111
\(689\) 0 0
\(690\) 0 0
\(691\) 1.05570e20 1.40345 0.701723 − 0.712450i \(-0.252414\pi\)
0.701723 + 0.712450i \(0.252414\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −9.91591e19 −1.27884
\(695\) 0 0
\(696\) 0 0
\(697\) 1.09565e20 1.37102
\(698\) 0 0
\(699\) −1.94145e20 −2.38115
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.70258e20 1.98655
\(705\) 0 0
\(706\) 1.62519e20 1.85897
\(707\) 0 0
\(708\) −1.93816e20 −2.17349
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 8.06013e19 0.868924
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.19178e20 1.23539
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −3.02815e20 −2.96085
\(723\) −6.83788e19 −0.662145
\(724\) 0 0
\(725\) 0 0
\(726\) −4.32804e20 −4.07132
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −8.87697e19 −0.811283
\(730\) 0 0
\(731\) 1.62679e20 1.45851
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.04953e20 0.888633
\(738\) 1.85613e20 1.55673
\(739\) 2.40601e20 1.99888 0.999441 − 0.0334271i \(-0.0106422\pi\)
0.999441 + 0.0334271i \(0.0106422\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.35471e20 −1.04376
\(748\) −2.08441e20 −1.59101
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 3.37445e20 2.45832
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 2.86927e20 1.99566
\(759\) 0 0
\(760\) 0 0
\(761\) −2.86406e20 −1.93771 −0.968855 − 0.247628i \(-0.920349\pi\)
−0.968855 + 0.247628i \(0.920349\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 2.17758e20 1.38180
\(769\) −2.67301e20 −1.68080 −0.840401 − 0.541965i \(-0.817680\pi\)
−0.840401 + 0.541965i \(0.817680\pi\)
\(770\) 0 0
\(771\) 1.47757e20 0.912364
\(772\) −3.18535e20 −1.94911
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 2.75592e20 1.65608
\(775\) 0 0
\(776\) 1.31733e20 0.777430
\(777\) 0 0
\(778\) 0 0
\(779\) −5.93098e20 −3.40694
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.82059e20 1.00000
\(785\) 0 0
\(786\) 5.09554e20 2.74936
\(787\) 3.56201e20 1.90490 0.952450 − 0.304694i \(-0.0985543\pi\)
0.952450 + 0.304694i \(0.0985543\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −3.53117e20 −1.80652
\(793\) 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
\(800\) −2.09715e20 −1.00000
\(801\) −1.67168e20 −0.790178
\(802\) −2.17433e20 −1.01884
\(803\) 6.11422e19 0.284009
\(804\) 1.34233e20 0.618113
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.14025e19 −0.182552 −0.0912760 − 0.995826i \(-0.529095\pi\)
−0.0912760 + 0.995826i \(0.529095\pi\)
\(810\) 0 0
\(811\) 3.21183e20 1.39189 0.695947 − 0.718093i \(-0.254985\pi\)
0.695947 + 0.718093i \(0.254985\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −2.66594e20 −1.10667
\(817\) −8.80614e20 −3.62436
\(818\) −9.61018e19 −0.392156
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −6.19235e20 −2.44204
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 7.14041e20 2.74502
\(826\) 0 0
\(827\) 1.09065e20 0.412238 0.206119 − 0.978527i \(-0.433917\pi\)
0.206119 + 0.978527i \(0.433917\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.22889e20 −0.800890
\(834\) −3.31202e20 −1.18013
\(835\) 0 0
\(836\) 1.12833e21 3.95361
\(837\) 0 0
\(838\) 5.78055e20 1.99187
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.97558e20 1.00000
\(842\) 0 0
\(843\) 6.72774e19 0.222370
\(844\) −4.43794e20 −1.45474
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 7.47676e20 2.35158
\(850\) 2.56747e20 0.800890
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −3.76342e20 −1.11755
\(857\) 5.06915e20 1.49303 0.746516 − 0.665367i \(-0.231725\pi\)
0.746516 + 0.665367i \(0.231725\pi\)
\(858\) 0 0
\(859\) −3.08240e20 −0.893175 −0.446587 − 0.894740i \(-0.647361\pi\)
−0.446587 + 0.894740i \(0.647361\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 4.50078e19 0.125225
\(865\) 0 0
\(866\) −6.72051e20 −1.83983
\(867\) −1.82456e20 −0.495479
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.73215e20 −0.706976
\(874\) 0 0
\(875\) 0 0
\(876\) 7.82001e19 0.197551
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.20410e20 0.777812 0.388906 − 0.921277i \(-0.372853\pi\)
0.388906 + 0.921277i \(0.372853\pi\)
\(882\) −3.77593e20 −0.909376
\(883\) −7.45828e20 −1.78202 −0.891012 − 0.453979i \(-0.850004\pi\)
−0.891012 + 0.453979i \(0.850004\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −5.04587e20 −1.17733
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −9.58597e20 −2.15027
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 2.81510e20 0.597805
\(899\) 0 0
\(900\) 4.34952e20 0.909376
\(901\) 0 0
\(902\) 1.65202e21 3.40071
\(903\) 0 0
\(904\) 1.95805e18 0.00396866
\(905\) 0 0
\(906\) 0 0
\(907\) 6.98908e20 1.38410 0.692052 − 0.721848i \(-0.256707\pi\)
0.692052 + 0.721848i \(0.256707\pi\)
\(908\) 2.56474e20 0.504014
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.44312e21 2.75005
\(913\) −1.20574e21 −2.28013
\(914\) −1.02622e21 −1.92582
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −5.51014e19 −0.100291
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.49190e21 −2.65413
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.06394e21 1.78159 0.890794 − 0.454408i \(-0.150149\pi\)
0.890794 + 0.454408i \(0.150149\pi\)
\(930\) 0 0
\(931\) 1.20654e21 1.99019
\(932\) −1.05257e21 −1.72322
\(933\) 0 0
\(934\) −3.12660e20 −0.504248
\(935\) 0 0
\(936\) 0 0
\(937\) 4.81305e20 0.759002 0.379501 − 0.925191i \(-0.376096\pi\)
0.379501 + 0.925191i \(0.376096\pi\)
\(938\) 0 0
\(939\) 1.14745e21 1.78268
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.05079e21 −1.57294
\(945\) 0 0
\(946\) 2.45287e21 3.61773
\(947\) −9.06817e20 −1.32761 −0.663804 − 0.747907i \(-0.731059\pi\)
−0.663804 + 0.747907i \(0.731059\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −1.38982e21 −1.99019
\(951\) 0 0
\(952\) 0 0
\(953\) 7.11937e20 0.997220 0.498610 − 0.866826i \(-0.333844\pi\)
0.498610 + 0.866826i \(0.333844\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.56944e20 1.00000
\(962\) 0 0
\(963\) 7.80537e20 1.01627
\(964\) −3.70721e20 −0.479190
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −2.34648e21 −2.94638
\(969\) −1.76677e21 −2.20249
\(970\) 0 0
\(971\) 8.31378e20 1.02156 0.510780 − 0.859712i \(-0.329357\pi\)
0.510780 + 0.859712i \(0.329357\pi\)
\(972\) −1.12338e21 −1.37045
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.79186e20 −0.446263 −0.223131 − 0.974788i \(-0.571628\pi\)
−0.223131 + 0.974788i \(0.571628\pi\)
\(978\) 1.07766e21 1.25925
\(979\) −1.48785e21 −1.72616
\(980\) 0 0
\(981\) 0 0
\(982\) 2.89776e20 0.329065
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 2.11291e21 2.36546
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −3.88091e20 −0.407651
\(994\) 0 0
\(995\) 0 0
\(996\) −1.54213e21 −1.58601
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) −4.88181e20 −0.495071
\(999\) 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 8.15.d.a.3.1 ✓ 1
3.2 odd 2 72.15.b.a.19.1 1
4.3 odd 2 32.15.d.a.15.1 1
8.3 odd 2 CM 8.15.d.a.3.1 ✓ 1
8.5 even 2 32.15.d.a.15.1 1
24.11 even 2 72.15.b.a.19.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.15.d.a.3.1 ✓ 1 1.1 even 1 trivial
8.15.d.a.3.1 ✓ 1 8.3 odd 2 CM
32.15.d.a.15.1 1 4.3 odd 2
32.15.d.a.15.1 1 8.5 even 2
72.15.b.a.19.1 1 3.2 odd 2
72.15.b.a.19.1 1 24.11 even 2