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Properties

Label 8.59.d.a.3.1
Level $8$
Weight $59$
Character 8.3
Self dual yes
Analytic conductor $170.439$
Analytic rank $0$
Dimension $1$
CM discriminant -8
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] = [8,59,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, 59, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 59);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 59 \)
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: \(170.439115075\)
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)
 
\(f(q)\) \(=\) \(q-5.36871e8 q^{2} +5.72814e13 q^{3} +2.88230e17 q^{4} -3.07527e22 q^{6} -1.54743e26 q^{8} -1.42897e27 q^{9} +O(q^{10})\) \(q-5.36871e8 q^{2} +5.72814e13 q^{3} +2.88230e17 q^{4} -3.07527e22 q^{6} -1.54743e26 q^{8} -1.42897e27 q^{9} +2.90091e30 q^{11} +1.65102e31 q^{12} +8.30767e34 q^{16} +9.55027e35 q^{17} +7.67171e35 q^{18} -1.58405e37 q^{19} -1.55741e39 q^{22} -8.86387e39 q^{24} +3.46945e40 q^{25} -3.51656e41 q^{27} -4.46015e43 q^{32} +1.66168e44 q^{33} -5.12726e44 q^{34} -4.11872e44 q^{36} +8.50428e45 q^{38} +1.17964e47 q^{41} -4.69272e47 q^{43} +8.36130e47 q^{44} +4.75876e48 q^{48} +1.03678e49 q^{49} -1.86265e49 q^{50} +5.47053e49 q^{51} +1.88794e50 q^{54} -9.07364e50 q^{57} +1.29260e51 q^{59} +2.39452e52 q^{64} -8.92109e52 q^{66} +1.77632e53 q^{67} +2.75268e53 q^{68} +2.21122e53 q^{72} -1.88636e54 q^{73} +1.98735e54 q^{75} -4.56570e54 q^{76} -1.34128e55 q^{81} -6.33315e55 q^{82} -8.34261e55 q^{83} +2.51938e56 q^{86} -4.48894e56 q^{88} +2.53990e56 q^{89} -2.55484e57 q^{96} -7.12555e57 q^{97} -5.56617e57 q^{98} -4.14530e57 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.36871e8 −1.00000
\(3\) 5.72814e13 0.834637 0.417318 − 0.908760i \(-0.362970\pi\)
0.417318 + 0.908760i \(0.362970\pi\)
\(4\) 2.88230e17 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −3.07527e22 −0.834637
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.54743e26 −1.00000
\(9\) −1.42897e27 −0.303382
\(10\) 0 0
\(11\) 2.90091e30 1.82872 0.914358 − 0.404907i \(-0.132696\pi\)
0.914358 + 0.404907i \(0.132696\pi\)
\(12\) 1.65102e31 0.834637
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 8.30767e34 1.00000
\(17\) 9.55027e35 1.98151 0.990757 − 0.135652i \(-0.0433129\pi\)
0.990757 + 0.135652i \(0.0433129\pi\)
\(18\) 7.67171e35 0.303382
\(19\) −1.58405e37 −1.30591 −0.652955 − 0.757396i \(-0.726471\pi\)
−0.652955 + 0.757396i \(0.726471\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.55741e39 −1.82872
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −8.86387e39 −0.834637
\(25\) 3.46945e40 1.00000
\(26\) 0 0
\(27\) −3.51656e41 −1.08785
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −4.46015e43 −1.00000
\(33\) 1.66168e44 1.52631
\(34\) −5.12726e44 −1.98151
\(35\) 0 0
\(36\) −4.11872e44 −0.303382
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 8.50428e45 1.30591
\(39\) 0 0
\(40\) 0 0
\(41\) 1.17964e47 1.99995 0.999973 − 0.00740813i \(-0.00235810\pi\)
0.999973 + 0.00740813i \(0.00235810\pi\)
\(42\) 0 0
\(43\) −4.69272e47 −1.99912 −0.999561 − 0.0296410i \(-0.990564\pi\)
−0.999561 + 0.0296410i \(0.990564\pi\)
\(44\) 8.36130e47 1.82872
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 4.75876e48 0.834637
\(49\) 1.03678e49 1.00000
\(50\) −1.86265e49 −1.00000
\(51\) 5.47053e49 1.65384
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.88794e50 1.08785
\(55\) 0 0
\(56\) 0 0
\(57\) −9.07364e50 −1.08996
\(58\) 0 0
\(59\) 1.29260e51 0.571158 0.285579 − 0.958355i \(-0.407814\pi\)
0.285579 + 0.958355i \(0.407814\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 2.39452e52 1.00000
\(65\) 0 0
\(66\) −8.92109e52 −1.52631
\(67\) 1.77632e53 1.96495 0.982475 − 0.186396i \(-0.0596806\pi\)
0.982475 + 0.186396i \(0.0596806\pi\)
\(68\) 2.75268e53 1.98151
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.21122e53 0.303382
\(73\) −1.88636e54 −1.73485 −0.867423 − 0.497571i \(-0.834225\pi\)
−0.867423 + 0.497571i \(0.834225\pi\)
\(74\) 0 0
\(75\) 1.98735e54 0.834637
\(76\) −4.56570e54 −1.30591
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −1.34128e55 −0.604578
\(82\) −6.33315e55 −1.99995
\(83\) −8.34261e55 −1.85369 −0.926845 − 0.375444i \(-0.877490\pi\)
−0.926845 + 0.375444i \(0.877490\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.51938e56 1.99912
\(87\) 0 0
\(88\) −4.48894e56 −1.82872
\(89\) 2.53990e56 0.745601 0.372801 − 0.927912i \(-0.378398\pi\)
0.372801 + 0.927912i \(0.378398\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) −2.55484e57 −0.834637
\(97\) −7.12555e57 −1.72361 −0.861803 − 0.507243i \(-0.830665\pi\)
−0.861803 + 0.507243i \(0.830665\pi\)
\(98\) −5.56617e57 −1.00000
\(99\) −4.14530e57 −0.554799
\(100\) 1.00000e58 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) −2.93697e58 −1.65384
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.99854e58 0.421483 0.210742 − 0.977542i \(-0.432412\pi\)
0.210742 + 0.977542i \(0.432412\pi\)
\(108\) −1.01358e59 −1.08785
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.95247e59 0.852925 0.426462 − 0.904505i \(-0.359760\pi\)
0.426462 + 0.904505i \(0.359760\pi\)
\(114\) 4.87138e59 1.08996
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −6.93960e59 −0.571158
\(119\) 0 0
\(120\) 0 0
\(121\) 5.89889e60 2.34420
\(122\) 0 0
\(123\) 6.75715e60 1.66923
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.28555e61 −1.00000
\(129\) −2.68806e61 −1.66854
\(130\) 0 0
\(131\) 3.02685e61 1.20260 0.601302 − 0.799022i \(-0.294649\pi\)
0.601302 + 0.799022i \(0.294649\pi\)
\(132\) 4.78947e61 1.52631
\(133\) 0 0
\(134\) −9.53653e61 −1.96495
\(135\) 0 0
\(136\) −1.47783e62 −1.98151
\(137\) 1.44090e62 1.56220 0.781102 − 0.624403i \(-0.214658\pi\)
0.781102 + 0.624403i \(0.214658\pi\)
\(138\) 0 0
\(139\) −2.25167e62 −1.60353 −0.801763 − 0.597642i \(-0.796104\pi\)
−0.801763 + 0.597642i \(0.796104\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.18714e62 −0.303382
\(145\) 0 0
\(146\) 1.01273e63 1.73485
\(147\) 5.93882e62 0.834637
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −1.06695e63 −0.834637
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 2.45119e63 1.30591
\(153\) −1.36470e63 −0.601155
\(154\) 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
\(162\) 7.20092e63 0.604578
\(163\) 4.67211e63 0.328150 0.164075 − 0.986448i \(-0.447536\pi\)
0.164075 + 0.986448i \(0.447536\pi\)
\(164\) 3.40008e64 1.99995
\(165\) 0 0
\(166\) 4.47890e64 1.85369
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.06176e64 1.00000
\(170\) 0 0
\(171\) 2.26355e64 0.396189
\(172\) −1.35258e65 −1.99912
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.40998e65 1.82872
\(177\) 7.40421e64 0.476710
\(178\) −1.36360e65 −0.745601
\(179\) −1.60902e65 −0.747865 −0.373932 − 0.927456i \(-0.621991\pi\)
−0.373932 + 0.927456i \(0.621991\pi\)
\(180\) 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
\(187\) 2.77045e66 3.62362
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.37162e66 0.834637
\(193\) 3.50474e66 1.83440 0.917199 − 0.398429i \(-0.130444\pi\)
0.917199 + 0.398429i \(0.130444\pi\)
\(194\) 3.82550e66 1.72361
\(195\) 0 0
\(196\) 2.98831e66 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 2.22549e66 0.554799
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −5.36871e66 −1.00000
\(201\) 1.01750e67 1.64002
\(202\) 0 0
\(203\) 0 0
\(204\) 1.57677e67 1.65384
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.59517e67 −2.38814
\(210\) 0 0
\(211\) 3.19535e67 1.25988 0.629938 − 0.776645i \(-0.283080\pi\)
0.629938 + 0.776645i \(0.283080\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −1.60983e67 −0.421483
\(215\) 0 0
\(216\) 5.44162e67 1.08785
\(217\) 0 0
\(218\) 0 0
\(219\) −1.08053e68 −1.44797
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −4.95772e67 −0.303382
\(226\) −1.58510e68 −0.852925
\(227\) 2.84789e68 1.34826 0.674131 − 0.738612i \(-0.264518\pi\)
0.674131 + 0.738612i \(0.264518\pi\)
\(228\) −2.61530e68 −1.08996
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.66400e68 −0.369684 −0.184842 − 0.982768i \(-0.559177\pi\)
−0.184842 + 0.982768i \(0.559177\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.72567e68 0.571158
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.25973e69 −1.05144 −0.525718 − 0.850659i \(-0.676203\pi\)
−0.525718 + 0.850659i \(0.676203\pi\)
\(242\) −3.16694e69 −2.34420
\(243\) 8.88044e68 0.583247
\(244\) 0 0
\(245\) 0 0
\(246\) −3.62772e69 −1.66923
\(247\) 0 0
\(248\) 0 0
\(249\) −4.77876e69 −1.54716
\(250\) 0 0
\(251\) −4.19899e68 −0.107797 −0.0538986 − 0.998546i \(-0.517165\pi\)
−0.0538986 + 0.998546i \(0.517165\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 6.90175e69 1.00000
\(257\) −7.35905e69 −0.952272 −0.476136 − 0.879372i \(-0.657963\pi\)
−0.476136 + 0.879372i \(0.657963\pi\)
\(258\) 1.44314e70 1.66854
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.62503e70 −1.20260
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −2.57133e70 −1.52631
\(265\) 0 0
\(266\) 0 0
\(267\) 1.45489e70 0.622306
\(268\) 5.11989e70 1.96495
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 7.93405e70 1.98151
\(273\) 0 0
\(274\) −7.73578e70 −1.56220
\(275\) 1.00645e71 1.82872
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 1.20886e71 1.60353
\(279\) 0 0
\(280\) 0 0
\(281\) −2.36365e70 −0.229668 −0.114834 − 0.993385i \(-0.536634\pi\)
−0.114834 + 0.993385i \(0.536634\pi\)
\(282\) 0 0
\(283\) 2.22350e71 1.75886 0.879428 − 0.476032i \(-0.157926\pi\)
0.879428 + 0.476032i \(0.157926\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 6.37340e70 0.303382
\(289\) 6.79783e71 2.92639
\(290\) 0 0
\(291\) −4.08162e71 −1.43858
\(292\) −5.43706e71 −1.73485
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) −3.18838e71 −0.834637
\(295\) 0 0
\(296\) 0 0
\(297\) −1.02012e72 −1.98937
\(298\) 0 0
\(299\) 0 0
\(300\) 5.72814e71 0.834637
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.31597e72 −1.30591
\(305\) 0 0
\(306\) 7.32669e71 0.601155
\(307\) −1.28461e71 −0.0958869 −0.0479434 − 0.998850i \(-0.515267\pi\)
−0.0479434 + 0.998850i \(0.515267\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.88180e72 1.22710 0.613548 − 0.789658i \(-0.289742\pi\)
0.613548 + 0.789658i \(0.289742\pi\)
\(314\) 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
\(321\) 1.71761e72 0.351785
\(322\) 0 0
\(323\) −1.51281e73 −2.58768
\(324\) −3.86596e72 −0.604578
\(325\) 0 0
\(326\) −2.50832e72 −0.328150
\(327\) 0 0
\(328\) −1.82541e73 −1.99995
\(329\) 0 0
\(330\) 0 0
\(331\) 1.36824e73 1.15120 0.575601 − 0.817731i \(-0.304768\pi\)
0.575601 + 0.817731i \(0.304768\pi\)
\(332\) −2.40459e73 −1.85369
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.06240e73 1.53036 0.765182 − 0.643814i \(-0.222649\pi\)
0.765182 + 0.643814i \(0.222649\pi\)
\(338\) −2.18064e73 −1.00000
\(339\) 1.69122e73 0.711882
\(340\) 0 0
\(341\) 0 0
\(342\) −1.21523e73 −0.396189
\(343\) 0 0
\(344\) 7.26163e73 1.99912
\(345\) 0 0
\(346\) 0 0
\(347\) 9.33643e73 1.99814 0.999071 − 0.0430984i \(-0.0137229\pi\)
0.999071 + 0.0430984i \(0.0137229\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.29385e74 −1.82872
\(353\) 1.49977e74 1.95236 0.976178 − 0.216972i \(-0.0696180\pi\)
0.976178 + 0.216972i \(0.0696180\pi\)
\(354\) −3.97510e73 −0.476710
\(355\) 0 0
\(356\) 7.32076e73 0.745601
\(357\) 0 0
\(358\) 8.63835e73 0.747865
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.03788e74 0.705403
\(362\) 0 0
\(363\) 3.37897e74 1.95656
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −1.68567e74 −0.606747
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −1.48737e75 −3.62362
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.18801e75 1.96917 0.984585 − 0.174907i \(-0.0559626\pi\)
0.984585 + 0.174907i \(0.0559626\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −7.36382e74 −0.834637
\(385\) 0 0
\(386\) −1.88159e75 −1.83440
\(387\) 6.70573e74 0.606497
\(388\) −2.05380e75 −1.72361
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.60434e75 −1.00000
\(393\) 1.73382e75 1.00374
\(394\) 0 0
\(395\) 0 0
\(396\) −1.19480e75 −0.554799
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.88230e75 1.00000
\(401\) 4.72893e75 1.52607 0.763037 − 0.646354i \(-0.223707\pi\)
0.763037 + 0.646354i \(0.223707\pi\)
\(402\) −5.46266e75 −1.64002
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −8.46524e75 −1.65384
\(409\) 3.96214e75 0.721031 0.360515 − 0.932753i \(-0.382601\pi\)
0.360515 + 0.932753i \(0.382601\pi\)
\(410\) 0 0
\(411\) 8.25369e75 1.30387
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.28979e76 −1.33836
\(418\) 2.46702e76 2.38814
\(419\) −6.53440e75 −0.590200 −0.295100 − 0.955466i \(-0.595353\pi\)
−0.295100 + 0.955466i \(0.595353\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −1.71549e76 −1.25988
\(423\) 0 0
\(424\) 0 0
\(425\) 3.31342e76 1.98151
\(426\) 0 0
\(427\) 0 0
\(428\) 8.64270e75 0.421483
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −2.92144e76 −1.08785
\(433\) 4.11648e76 1.43343 0.716715 − 0.697366i \(-0.245645\pi\)
0.716715 + 0.697366i \(0.245645\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 5.80107e76 1.44797
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −1.48152e76 −0.303382
\(442\) 0 0
\(443\) −8.08388e76 −1.45182 −0.725909 − 0.687791i \(-0.758580\pi\)
−0.725909 + 0.687791i \(0.758580\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.45993e76 0.177496 0.0887480 − 0.996054i \(-0.471713\pi\)
0.0887480 + 0.996054i \(0.471713\pi\)
\(450\) 2.66166e76 0.303382
\(451\) 3.42203e77 3.65733
\(452\) 8.50992e76 0.852925
\(453\) 0 0
\(454\) −1.52895e77 −1.34826
\(455\) 0 0
\(456\) 1.40408e77 1.08996
\(457\) −2.68244e77 −1.95416 −0.977081 − 0.212867i \(-0.931720\pi\)
−0.977081 + 0.212867i \(0.931720\pi\)
\(458\) 0 0
\(459\) −3.35841e77 −2.15559
\(460\) 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
\(466\) 8.93352e76 0.369684
\(467\) −4.47623e77 −1.74070 −0.870348 − 0.492438i \(-0.836106\pi\)
−0.870348 + 0.492438i \(0.836106\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −2.00020e77 −0.571158
\(473\) −1.36131e78 −3.65582
\(474\) 0 0
\(475\) −5.49576e77 −1.30591
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 6.76312e77 1.05144
\(483\) 0 0
\(484\) 1.70024e78 2.34420
\(485\) 0 0
\(486\) −4.76765e77 −0.583247
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 2.67625e77 0.273886
\(490\) 0 0
\(491\) 2.19970e78 1.99987 0.999933 − 0.0115489i \(-0.00367623\pi\)
0.999933 + 0.0115489i \(0.00367623\pi\)
\(492\) 1.94762e78 1.66923
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 2.56558e78 1.54716
\(499\) −3.28601e78 −1.86962 −0.934808 − 0.355153i \(-0.884429\pi\)
−0.934808 + 0.355153i \(0.884429\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.25432e77 0.107797
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.32664e78 0.834637
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3.70535e78 −1.00000
\(513\) 5.57040e78 1.42063
\(514\) 3.95086e78 0.952272
\(515\) 0 0
\(516\) −7.74779e78 −1.66854
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.54601e78 −0.414540 −0.207270 − 0.978284i \(-0.566458\pi\)
−0.207270 + 0.978284i \(0.566458\pi\)
\(522\) 0 0
\(523\) −1.30124e79 −1.89587 −0.947936 − 0.318461i \(-0.896834\pi\)
−0.947936 + 0.318461i \(0.896834\pi\)
\(524\) 8.72429e78 1.20260
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.38047e79 1.52631
\(529\) 9.55464e78 1.00000
\(530\) 0 0
\(531\) −1.84708e78 −0.173279
\(532\) 0 0
\(533\) 0 0
\(534\) −7.81089e78 −0.622306
\(535\) 0 0
\(536\) −2.74872e79 −1.96495
\(537\) −9.21669e78 −0.624196
\(538\) 0 0
\(539\) 3.00760e79 1.82872
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −4.25956e79 −1.98151
\(545\) 0 0
\(546\) 0 0
\(547\) −2.03626e79 −0.807607 −0.403803 − 0.914846i \(-0.632312\pi\)
−0.403803 + 0.914846i \(0.632312\pi\)
\(548\) 4.15312e79 1.56220
\(549\) 0 0
\(550\) −5.40336e79 −1.82872
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −6.49000e79 −1.60353
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 1.58695e80 3.02441
\(562\) 1.26898e79 0.229668
\(563\) 4.77384e79 0.820586 0.410293 − 0.911954i \(-0.365426\pi\)
0.410293 + 0.911954i \(0.365426\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.19373e80 −1.75886
\(567\) 0 0
\(568\) 0 0
\(569\) −1.46518e80 −1.85198 −0.925991 − 0.377547i \(-0.876768\pi\)
−0.925991 + 0.377547i \(0.876768\pi\)
\(570\) 0 0
\(571\) 1.51260e80 1.72694 0.863472 − 0.504397i \(-0.168285\pi\)
0.863472 + 0.504397i \(0.168285\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −3.42169e79 −0.303382
\(577\) 1.19170e79 0.100478 0.0502388 − 0.998737i \(-0.484002\pi\)
0.0502388 + 0.998737i \(0.484002\pi\)
\(578\) −3.64956e80 −2.92639
\(579\) 2.00756e80 1.53106
\(580\) 0 0
\(581\) 0 0
\(582\) 2.19130e80 1.43858
\(583\) 0 0
\(584\) 2.91900e80 1.73485
\(585\) 0 0
\(586\) 0 0
\(587\) −2.37443e80 −1.21634 −0.608170 − 0.793807i \(-0.708096\pi\)
−0.608170 + 0.793807i \(0.708096\pi\)
\(588\) 1.71175e80 0.834637
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.24032e80 1.99881 0.999405 − 0.0344810i \(-0.0109778\pi\)
0.999405 + 0.0344810i \(0.0109778\pi\)
\(594\) 5.47674e80 1.98937
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −3.07527e80 −0.834637
\(601\) −6.09065e79 −0.157509 −0.0787543 − 0.996894i \(-0.525094\pi\)
−0.0787543 + 0.996894i \(0.525094\pi\)
\(602\) 0 0
\(603\) −2.53830e80 −0.596130
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 7.06508e80 1.30591
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −3.93348e80 −0.601155
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 6.89671e79 0.0958869
\(615\) 0 0
\(616\) 0 0
\(617\) −1.64953e81 −1.99109 −0.995545 − 0.0942919i \(-0.969941\pi\)
−0.995545 + 0.0942919i \(0.969941\pi\)
\(618\) 0 0
\(619\) −1.62071e81 −1.78105 −0.890526 − 0.454933i \(-0.849663\pi\)
−0.890526 + 0.454933i \(0.849663\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.20371e81 1.00000
\(626\) −1.54716e81 −1.22710
\(627\) −2.63218e81 −1.99323
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.83034e81 1.05154
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.92771e81 −1.96681 −0.983404 − 0.181430i \(-0.941927\pi\)
−0.983404 + 0.181430i \(0.941927\pi\)
\(642\) −9.22133e80 −0.351785
\(643\) −5.19708e81 −1.89514 −0.947569 − 0.319552i \(-0.896468\pi\)
−0.947569 + 0.319552i \(0.896468\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.12182e81 2.58768
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 2.07552e81 0.604578
\(649\) 3.74972e81 1.04449
\(650\) 0 0
\(651\) 0 0
\(652\) 1.34664e81 0.328150
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.80008e81 1.99995
\(657\) 2.69554e81 0.526321
\(658\) 0 0
\(659\) 3.89180e81 0.695782 0.347891 − 0.937535i \(-0.386898\pi\)
0.347891 + 0.937535i \(0.386898\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −7.34570e81 −1.15120
\(663\) 0 0
\(664\) 1.29096e82 1.85369
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.16152e81 −0.501581 −0.250790 − 0.968041i \(-0.580690\pi\)
−0.250790 + 0.968041i \(0.580690\pi\)
\(674\) −1.64411e82 −1.53036
\(675\) −1.22005e82 −1.08785
\(676\) 1.17072e82 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) −9.07966e81 −0.711882
\(679\) 0 0
\(680\) 0 0
\(681\) 1.63131e82 1.12531
\(682\) 0 0
\(683\) 2.37349e82 1.50379 0.751894 − 0.659284i \(-0.229141\pi\)
0.751894 + 0.659284i \(0.229141\pi\)
\(684\) 6.52424e81 0.396189
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −3.89856e82 −1.99912
\(689\) 0 0
\(690\) 0 0
\(691\) −2.23527e82 −1.01034 −0.505168 − 0.863021i \(-0.668570\pi\)
−0.505168 + 0.863021i \(0.668570\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −5.01246e82 −1.99814
\(695\) 0 0
\(696\) 0 0
\(697\) 1.12659e83 3.96292
\(698\) 0 0
\(699\) −9.53162e81 −0.308552
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 6.94630e82 1.82872
\(705\) 0 0
\(706\) −8.05184e82 −1.95236
\(707\) 0 0
\(708\) 2.13412e82 0.476710
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.93031e82 −0.745601
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −4.63768e82 −0.747865
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −5.57206e82 −0.705403
\(723\) −7.21591e82 −0.877568
\(724\) 0 0
\(725\) 0 0
\(726\) −1.81407e83 −1.95656
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.14044e83 1.09138
\(730\) 0 0
\(731\) −4.48167e83 −3.96128
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.15294e83 3.59333
\(738\) 9.04986e82 0.606747
\(739\) −2.66151e82 −0.171569 −0.0857845 − 0.996314i \(-0.527340\pi\)
−0.0857845 + 0.996314i \(0.527340\pi\)
\(740\) 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
\(747\) 1.19213e83 0.562375
\(748\) 7.98527e83 3.62362
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −2.40524e82 −0.0899714
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −6.37808e83 −1.96917
\(759\) 0 0
\(760\) 0 0
\(761\) 6.04467e83 1.66425 0.832123 − 0.554592i \(-0.187126\pi\)
0.832123 + 0.554592i \(0.187126\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 3.95342e83 0.834637
\(769\) −9.37826e82 −0.190659 −0.0953297 − 0.995446i \(-0.530391\pi\)
−0.0953297 + 0.995446i \(0.530391\pi\)
\(770\) 0 0
\(771\) −4.21537e83 −0.794801
\(772\) 1.01017e84 1.83440
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −3.60011e83 −0.606497
\(775\) 0 0
\(776\) 1.10263e84 1.72361
\(777\) 0 0
\(778\) 0 0
\(779\) −1.86861e84 −2.61175
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.61323e83 1.00000
\(785\) 0 0
\(786\) −9.30838e83 −1.00374
\(787\) −1.10165e84 −1.14492 −0.572460 − 0.819933i \(-0.694011\pi\)
−0.572460 + 0.819933i \(0.694011\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 6.41454e83 0.554799
\(793\) 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
\(800\) −1.54743e84 −1.00000
\(801\) −3.62943e83 −0.226202
\(802\) −2.53882e84 −1.52607
\(803\) −5.47215e84 −3.17254
\(804\) 2.93274e84 1.64002
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.23008e84 −1.97625 −0.988124 − 0.153660i \(-0.950894\pi\)
−0.988124 + 0.153660i \(0.950894\pi\)
\(810\) 0 0
\(811\) 2.17533e83 0.0946063 0.0473032 − 0.998881i \(-0.484937\pi\)
0.0473032 + 0.998881i \(0.484937\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 4.54474e84 1.65384
\(817\) 7.43348e84 2.61067
\(818\) −2.12716e84 −0.721031
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −4.43117e84 −1.30387
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 5.76512e84 1.52631
\(826\) 0 0
\(827\) 3.03908e83 0.0750036 0.0375018 − 0.999297i \(-0.488060\pi\)
0.0375018 + 0.999297i \(0.488060\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 9.90152e84 1.98151
\(834\) 6.92451e84 1.33836
\(835\) 0 0
\(836\) −1.32447e85 −2.38814
\(837\) 0 0
\(838\) 3.50813e84 0.590200
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 6.59301e84 1.00000
\(842\) 0 0
\(843\) −1.35394e84 −0.191689
\(844\) 9.20998e84 1.25988
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 1.27365e85 1.46801
\(850\) −1.77888e85 −1.98151
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.64002e84 −0.421483
\(857\) −1.18570e85 −1.04119 −0.520595 − 0.853804i \(-0.674290\pi\)
−0.520595 + 0.853804i \(0.674290\pi\)
\(858\) 0 0
\(859\) −1.31815e85 −1.08184 −0.540918 − 0.841075i \(-0.681923\pi\)
−0.540918 + 0.841075i \(0.681923\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 1.56844e85 1.08785
\(865\) 0 0
\(866\) −2.21002e85 −1.43343
\(867\) 3.89389e85 2.44248
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.01822e85 0.522910
\(874\) 0 0
\(875\) 0 0
\(876\) −3.11442e85 −1.44797
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.81930e85 −1.11132 −0.555660 − 0.831410i \(-0.687534\pi\)
−0.555660 + 0.831410i \(0.687534\pi\)
\(882\) 7.95387e84 0.303382
\(883\) 2.01707e85 0.744492 0.372246 − 0.928134i \(-0.378588\pi\)
0.372246 + 0.928134i \(0.378588\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 4.34000e85 1.45182
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −3.89092e85 −1.10560
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −7.83794e84 −0.177496
\(899\) 0 0
\(900\) −1.42897e85 −0.303382
\(901\) 0 0
\(902\) −1.83719e86 −3.65733
\(903\) 0 0
\(904\) −4.56873e85 −0.852925
\(905\) 0 0
\(906\) 0 0
\(907\) 1.14288e86 1.93817 0.969083 − 0.246733i \(-0.0793572\pi\)
0.969083 + 0.246733i \(0.0793572\pi\)
\(908\) 8.20850e85 1.34826
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −7.53809e85 −1.08996
\(913\) −2.42011e86 −3.38987
\(914\) 1.44013e86 1.95416
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 1.80303e86 2.15559
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −7.35844e84 −0.0800307
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.26459e86 −1.91661 −0.958307 − 0.285740i \(-0.907760\pi\)
−0.958307 + 0.285740i \(0.907760\pi\)
\(930\) 0 0
\(931\) −1.64231e86 −1.30591
\(932\) −4.79615e85 −0.369684
\(933\) 0 0
\(934\) 2.40316e86 1.74070
\(935\) 0 0
\(936\) 0 0
\(937\) 2.78276e86 1.83666 0.918329 − 0.395818i \(-0.129539\pi\)
0.918329 + 0.395818i \(0.129539\pi\)
\(938\) 0 0
\(939\) 1.65074e86 1.02418
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.07385e86 0.571158
\(945\) 0 0
\(946\) 7.30850e86 3.65582
\(947\) 1.27523e86 0.618642 0.309321 − 0.950958i \(-0.399898\pi\)
0.309321 + 0.950958i \(0.399898\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 2.95052e86 1.30591
\(951\) 0 0
\(952\) 0 0
\(953\) −4.10945e86 −1.65993 −0.829964 − 0.557817i \(-0.811639\pi\)
−0.829964 + 0.557817i \(0.811639\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 3.15485e86 1.00000
\(962\) 0 0
\(963\) −4.28481e85 −0.127870
\(964\) −3.63092e86 −1.05144
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −9.12810e86 −2.34420
\(969\) −8.66558e86 −2.15977
\(970\) 0 0
\(971\) 3.87212e86 0.909056 0.454528 − 0.890732i \(-0.349808\pi\)
0.454528 + 0.890732i \(0.349808\pi\)
\(972\) 2.55961e86 0.583247
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −7.73837e86 −1.51952 −0.759762 − 0.650202i \(-0.774684\pi\)
−0.759762 + 0.650202i \(0.774684\pi\)
\(978\) −1.43680e86 −0.273886
\(979\) 7.36802e86 1.36349
\(980\) 0 0
\(981\) 0 0
\(982\) −1.18096e87 −1.99987
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −1.04562e87 −1.66923
\(985\) 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
\(993\) 7.83749e86 0.960835
\(994\) 0 0
\(995\) 0 0
\(996\) −1.37739e87 −1.54716
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 1.76416e87 1.86962
\(999\) 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 8.59.d.a.3.1 ✓ 1
8.3 odd 2 CM 8.59.d.a.3.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.59.d.a.3.1 ✓ 1 1.1 even 1 trivial
8.59.d.a.3.1 ✓ 1 8.3 odd 2 CM