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Properties

Label 11.45.b.a.10.1
Level $11$
Weight $45$
Character 11.10
Self dual yes
Analytic conductor $134.882$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,45,Mod(10,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 45, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 45);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 45 \)
Character orbit: \([\chi]\) \(=\) 11.b (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: \(134.881708477\)
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 10.1
Character \(\chi\) \(=\) 11.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.96014e10 q^{3} +1.75922e13 q^{4} +2.82037e15 q^{5} +2.56756e21 q^{9} +O(q^{10})\) \(q+5.96014e10 q^{3} +1.75922e13 q^{4} +2.82037e15 q^{5} +2.56756e21 q^{9} +8.14027e22 q^{11} +1.04852e24 q^{12} +1.68098e26 q^{15} +3.09485e26 q^{16} +4.96165e28 q^{20} -1.78866e30 q^{23} +2.27016e30 q^{25} +9.43364e31 q^{27} -8.86585e31 q^{31} +4.85172e33 q^{33} +4.51690e34 q^{36} -1.70311e34 q^{37} +1.43205e36 q^{44} +7.24147e36 q^{45} -1.19083e37 q^{47} +1.84457e37 q^{48} +1.52867e37 q^{49} +1.69605e38 q^{53} +2.29586e38 q^{55} -1.79835e39 q^{59} +2.95722e39 q^{60} +5.44452e39 q^{64} -2.66038e40 q^{67} -1.06606e41 q^{69} -2.21655e39 q^{71} +1.35305e41 q^{75} +8.72863e41 q^{80} +3.09412e42 q^{81} -1.53054e43 q^{89} -3.14664e43 q^{92} -5.28417e42 q^{93} -2.65640e43 q^{97} +2.09006e44 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 5.96014e10 1.89928 0.949640 − 0.313343i \(-0.101449\pi\)
0.949640 + 0.313343i \(0.101449\pi\)
\(4\) 1.75922e13 1.00000
\(5\) 2.82037e15 1.18295 0.591475 − 0.806323i \(-0.298546\pi\)
0.591475 + 0.806323i \(0.298546\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 2.56756e21 2.60726
\(10\) 0 0
\(11\) 8.14027e22 1.00000
\(12\) 1.04852e24 1.89928
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.68098e26 2.24675
\(16\) 3.09485e26 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 4.96165e28 1.18295
\(21\) 0 0
\(22\) 0 0
\(23\) −1.78866e30 −1.97022 −0.985109 − 0.171932i \(-0.944999\pi\)
−0.985109 + 0.171932i \(0.944999\pi\)
\(24\) 0 0
\(25\) 2.27016e30 0.399371
\(26\) 0 0
\(27\) 9.43364e31 3.05265
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −8.86585e31 −0.137329 −0.0686647 − 0.997640i \(-0.521874\pi\)
−0.0686647 + 0.997640i \(0.521874\pi\)
\(32\) 0 0
\(33\) 4.85172e33 1.89928
\(34\) 0 0
\(35\) 0 0
\(36\) 4.51690e34 2.60726
\(37\) −1.70311e34 −0.538029 −0.269015 − 0.963136i \(-0.586698\pi\)
−0.269015 + 0.963136i \(0.586698\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 1.43205e36 1.00000
\(45\) 7.24147e36 3.08426
\(46\) 0 0
\(47\) −1.19083e37 −1.94849 −0.974246 − 0.225488i \(-0.927602\pi\)
−0.974246 + 0.225488i \(0.927602\pi\)
\(48\) 1.84457e37 1.89928
\(49\) 1.52867e37 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.69605e38 1.97410 0.987051 − 0.160406i \(-0.0512805\pi\)
0.987051 + 0.160406i \(0.0512805\pi\)
\(54\) 0 0
\(55\) 2.29586e38 1.18295
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.79835e39 −1.97756 −0.988780 − 0.149378i \(-0.952273\pi\)
−0.988780 + 0.149378i \(0.952273\pi\)
\(60\) 2.95722e39 2.24675
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 5.44452e39 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.66038e40 −1.78360 −0.891800 − 0.452430i \(-0.850557\pi\)
−0.891800 + 0.452430i \(0.850557\pi\)
\(68\) 0 0
\(69\) −1.06606e41 −3.74199
\(70\) 0 0
\(71\) −2.21655e39 −0.0414947 −0.0207474 − 0.999785i \(-0.506605\pi\)
−0.0207474 + 0.999785i \(0.506605\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.35305e41 0.758517
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.72863e41 1.18295
\(81\) 3.09412e42 3.19056
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.53054e43 −1.98731 −0.993654 − 0.112477i \(-0.964121\pi\)
−0.993654 + 0.112477i \(0.964121\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.14664e43 −1.97022
\(93\) −5.28417e42 −0.260827
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.65640e43 −0.519176 −0.259588 − 0.965719i \(-0.583587\pi\)
−0.259588 + 0.965719i \(0.583587\pi\)
\(98\) 0 0
\(99\) 2.09006e44 2.60726
\(100\) 3.99371e43 0.399371
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.93903e44 −1.53386 −0.766929 − 0.641732i \(-0.778216\pi\)
−0.766929 + 0.641732i \(0.778216\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.65958e45 3.05265
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −1.01508e45 −1.02187
\(112\) 0 0
\(113\) 2.41371e45 1.64044 0.820218 − 0.572052i \(-0.193852\pi\)
0.820218 + 0.572052i \(0.193852\pi\)
\(114\) 0 0
\(115\) −5.04467e45 −2.33067
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 6.62641e45 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −1.55970e45 −0.137329
\(125\) −9.62926e45 −0.710514
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 8.53523e46 1.89928
\(133\) 0 0
\(134\) 0 0
\(135\) 2.66064e47 3.61113
\(136\) 0 0
\(137\) 1.64689e47 1.61738 0.808690 − 0.588235i \(-0.200177\pi\)
0.808690 + 0.588235i \(0.200177\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −7.09754e47 −3.70073
\(142\) 0 0
\(143\) 0 0
\(144\) 7.94621e47 2.60726
\(145\) 0 0
\(146\) 0 0
\(147\) 9.11109e47 1.89928
\(148\) −2.99615e47 −0.538029
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.50050e47 −0.162454
\(156\) 0 0
\(157\) 2.58585e48 1.26710 0.633549 − 0.773702i \(-0.281597\pi\)
0.633549 + 0.773702i \(0.281597\pi\)
\(158\) 0 0
\(159\) 1.01087e49 3.74937
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.52620e47 −0.0971865 −0.0485933 − 0.998819i \(-0.515474\pi\)
−0.0485933 + 0.998819i \(0.515474\pi\)
\(164\) 0 0
\(165\) 1.36837e49 2.24675
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.03159e49 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.51929e49 1.00000
\(177\) −1.07184e50 −3.75594
\(178\) 0 0
\(179\) −3.07969e49 −0.842831 −0.421415 − 0.906868i \(-0.638467\pi\)
−0.421415 + 0.906868i \(0.638467\pi\)
\(180\) 1.27393e50 3.08426
\(181\) −5.36555e49 −1.14997 −0.574984 − 0.818165i \(-0.694992\pi\)
−0.574984 + 0.818165i \(0.694992\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.80342e49 −0.636462
\(186\) 0 0
\(187\) 0 0
\(188\) −2.09494e50 −1.94849
\(189\) 0 0
\(190\) 0 0
\(191\) 3.02889e50 1.98861 0.994304 − 0.106586i \(-0.0339918\pi\)
0.994304 + 0.106586i \(0.0339918\pi\)
\(192\) 3.24501e50 1.89928
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.68926e50 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −7.46686e50 −1.98779 −0.993895 − 0.110328i \(-0.964810\pi\)
−0.993895 + 0.110328i \(0.964810\pi\)
\(200\) 0 0
\(201\) −1.58562e51 −3.38756
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −4.59248e51 −5.13688
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 2.98372e51 1.97410
\(213\) −1.32109e50 −0.0788101
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 4.03892e51 1.18295
\(221\) 0 0
\(222\) 0 0
\(223\) 8.93570e50 0.194277 0.0971385 − 0.995271i \(-0.469031\pi\)
0.0971385 + 0.995271i \(0.469031\pi\)
\(224\) 0 0
\(225\) 5.82877e51 1.04127
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 8.64396e51 1.04793 0.523965 − 0.851740i \(-0.324452\pi\)
0.523965 + 0.851740i \(0.324452\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −3.35860e52 −2.30497
\(236\) −3.16369e52 −1.97756
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 5.20239e52 2.24675
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 9.15145e52 3.00713
\(244\) 0 0
\(245\) 4.31142e52 1.18295
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.73431e52 1.24623 0.623117 − 0.782129i \(-0.285866\pi\)
0.623117 + 0.782129i \(0.285866\pi\)
\(252\) 0 0
\(253\) −1.45601e53 −1.97022
\(254\) 0 0
\(255\) 0 0
\(256\) 9.57810e52 1.00000
\(257\) 9.56261e52 0.916321 0.458161 − 0.888869i \(-0.348508\pi\)
0.458161 + 0.888869i \(0.348508\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 4.78349e53 2.33526
\(266\) 0 0
\(267\) −9.12225e53 −3.77446
\(268\) −4.68018e53 −1.78360
\(269\) 4.91579e53 1.72601 0.863005 − 0.505195i \(-0.168579\pi\)
0.863005 + 0.505195i \(0.168579\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.84797e53 0.399371
\(276\) −1.87544e54 −3.74199
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −2.27636e53 −0.358054
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −3.89939e52 −0.0414947
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.37960e54 1.00000
\(290\) 0 0
\(291\) −1.58325e54 −0.986061
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −5.07201e54 −2.33936
\(296\) 0 0
\(297\) 7.67924e54 3.05265
\(298\) 0 0
\(299\) 0 0
\(300\) 2.38031e54 0.758517
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −1.75170e55 −2.91323
\(310\) 0 0
\(311\) −1.25812e55 −1.81549 −0.907747 − 0.419519i \(-0.862199\pi\)
−0.907747 + 0.419519i \(0.862199\pi\)
\(312\) 0 0
\(313\) 1.56668e55 1.96339 0.981694 − 0.190465i \(-0.0609994\pi\)
0.981694 + 0.190465i \(0.0609994\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.08075e55 1.97204 0.986021 − 0.166619i \(-0.0532850\pi\)
0.986021 + 0.166619i \(0.0532850\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.53556e55 1.18295
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5.44324e55 3.19056
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5.07221e55 1.85772 0.928862 − 0.370425i \(-0.120788\pi\)
0.928862 + 0.370425i \(0.120788\pi\)
\(332\) 0 0
\(333\) −4.37285e55 −1.40278
\(334\) 0 0
\(335\) −7.50325e55 −2.10991
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 1.43860e56 3.11565
\(340\) 0 0
\(341\) −7.21705e54 −0.137329
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.00670e56 −4.42659
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.79027e55 −0.692642 −0.346321 − 0.938116i \(-0.612569\pi\)
−0.346321 + 0.938116i \(0.612569\pi\)
\(354\) 0 0
\(355\) −6.25148e54 −0.0490862
\(356\) −2.69256e56 −1.98731
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.84144e56 1.00000
\(362\) 0 0
\(363\) 3.94943e56 1.89928
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.82605e56 1.06789 0.533945 − 0.845519i \(-0.320709\pi\)
0.533945 + 0.845519i \(0.320709\pi\)
\(368\) −5.53562e56 −1.97022
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −9.29602e55 −0.260827
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −5.73918e56 −1.34947
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 7.94252e56 1.47875 0.739377 − 0.673292i \(-0.235120\pi\)
0.739377 + 0.673292i \(0.235120\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.12359e57 −1.66048 −0.830240 − 0.557407i \(-0.811796\pi\)
−0.830240 + 0.557407i \(0.811796\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −4.67318e56 −0.519176
\(389\) −2.18817e56 −0.229715 −0.114858 − 0.993382i \(-0.536641\pi\)
−0.114858 + 0.993382i \(0.536641\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 3.67688e57 2.60726
\(397\) −1.77672e57 −1.19187 −0.595934 − 0.803033i \(-0.703218\pi\)
−0.595934 + 0.803033i \(0.703218\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 7.02581e56 0.399371
\(401\) 3.35258e57 1.80386 0.901930 − 0.431882i \(-0.142150\pi\)
0.901930 + 0.431882i \(0.142150\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 8.72659e57 3.77428
\(406\) 0 0
\(407\) −1.38638e57 −0.538029
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 9.81569e57 3.07186
\(412\) −5.17040e57 −1.53386
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.39129e57 −1.51360 −0.756801 − 0.653645i \(-0.773239\pi\)
−0.756801 + 0.653645i \(0.773239\pi\)
\(420\) 0 0
\(421\) −1.77463e57 −0.327266 −0.163633 − 0.986521i \(-0.552321\pi\)
−0.163633 + 0.986521i \(0.552321\pi\)
\(422\) 0 0
\(423\) −3.05754e58 −5.08023
\(424\) 0 0
\(425\) 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 \(\pi\)
\(432\) 2.91957e58 3.05265
\(433\) −1.62008e58 −1.60992 −0.804958 − 0.593332i \(-0.797812\pi\)
−0.804958 + 0.593332i \(0.797812\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.92495e58 2.60726
\(442\) 0 0
\(443\) −1.82543e58 −1.09770 −0.548849 − 0.835922i \(-0.684934\pi\)
−0.548849 + 0.835922i \(0.684934\pi\)
\(444\) −1.78575e58 −1.02187
\(445\) −4.31670e58 −2.35089
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.78774e58 0.799624 0.399812 − 0.916597i \(-0.369075\pi\)
0.399812 + 0.916597i \(0.369075\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.24624e58 1.64044
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −8.87468e58 −2.33067
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −5.91850e58 −1.34719 −0.673597 − 0.739099i \(-0.735252\pi\)
−0.673597 + 0.739099i \(0.735252\pi\)
\(464\) 0 0
\(465\) −1.49033e58 −0.308545
\(466\) 0 0
\(467\) −1.05786e59 −1.99278 −0.996388 − 0.0849167i \(-0.972938\pi\)
−0.996388 + 0.0849167i \(0.972938\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.54120e59 2.40657
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.35471e59 5.14701
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.16573e59 1.00000
\(485\) −7.49203e58 −0.614160
\(486\) 0 0
\(487\) −2.23357e59 −1.67248 −0.836242 − 0.548360i \(-0.815252\pi\)
−0.836242 + 0.548360i \(0.815252\pi\)
\(488\) 0 0
\(489\) −2.69768e58 −0.184584
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 5.89476e59 3.08426
\(496\) −2.74385e58 −0.137329
\(497\) 0 0
\(498\) 0 0
\(499\) 4.92594e58 0.215912 0.107956 − 0.994156i \(-0.465569\pi\)
0.107956 + 0.994156i \(0.465569\pi\)
\(500\) −1.69400e59 −0.710514
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 6.14843e59 1.89928
\(508\) 0 0
\(509\) −7.02811e59 −1.99089 −0.995446 − 0.0953315i \(-0.969609\pi\)
−0.995446 + 0.0953315i \(0.969609\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.28917e59 −1.81448
\(516\) 0 0
\(517\) −9.69372e59 −1.94849
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.59845e59 −1.11947 −0.559734 − 0.828672i \(-0.689097\pi\)
−0.559734 + 0.828672i \(0.689097\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.50153e60 1.89928
\(529\) 2.37510e60 2.88176
\(530\) 0 0
\(531\) −4.61737e60 −5.15602
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.83554e60 −1.60077
\(538\) 0 0
\(539\) 1.24438e60 1.00000
\(540\) 4.68064e60 3.61113
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −3.19794e60 −2.18411
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 2.89724e60 1.61738
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2.86291e60 −1.20882
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) −1.24861e61 −3.70073
\(565\) 6.80756e60 1.94055
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 1.80526e61 3.77692
\(574\) 0 0
\(575\) −4.06053e60 −0.786847
\(576\) 1.39791e61 2.60726
\(577\) −1.08062e61 −1.94002 −0.970009 − 0.243069i \(-0.921846\pi\)
−0.970009 + 0.243069i \(0.921846\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.38063e61 1.97410
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.57883e61 1.94221 0.971106 − 0.238650i \(-0.0767050\pi\)
0.971106 + 0.238650i \(0.0767050\pi\)
\(588\) 1.60284e61 1.89928
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −5.27088e60 −0.538029
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.45036e61 −3.77537
\(598\) 0 0
\(599\) 5.80693e60 0.457674 0.228837 − 0.973465i \(-0.426508\pi\)
0.228837 + 0.973465i \(0.426508\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −6.83067e61 −4.65032
\(604\) 0 0
\(605\) 1.86889e61 1.18295
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.58587e61 1.88429 0.942146 − 0.335204i \(-0.108805\pi\)
0.942146 + 0.335204i \(0.108805\pi\)
\(618\) 0 0
\(619\) 3.31276e61 1.26764 0.633820 − 0.773480i \(-0.281486\pi\)
0.633820 + 0.773480i \(0.281486\pi\)
\(620\) −4.39893e60 −0.162454
\(621\) −1.68735e62 −6.01438
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −4.00625e61 −1.23987
\(626\) 0 0
\(627\) 0 0
\(628\) 4.54907e61 1.26710
\(629\) 0 0
\(630\) 0 0
\(631\) 7.90407e61 1.98246 0.991231 − 0.132143i \(-0.0421858\pi\)
0.991231 + 0.132143i \(0.0421858\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 1.77834e62 3.74937
\(637\) 0 0
\(638\) 0 0
\(639\) −5.69111e60 −0.108188
\(640\) 0 0
\(641\) −5.76438e61 −1.02300 −0.511501 − 0.859283i \(-0.670910\pi\)
−0.511501 + 0.859283i \(0.670910\pi\)
\(642\) 0 0
\(643\) −1.00605e62 −1.66716 −0.833582 − 0.552396i \(-0.813713\pi\)
−0.833582 + 0.552396i \(0.813713\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.20327e62 1.73968 0.869838 − 0.493337i \(-0.164223\pi\)
0.869838 + 0.493337i \(0.164223\pi\)
\(648\) 0 0
\(649\) −1.46391e62 −1.97756
\(650\) 0 0
\(651\) 0 0
\(652\) −7.96257e60 −0.0971865
\(653\) −1.62681e62 −1.91976 −0.959882 − 0.280403i \(-0.909532\pi\)
−0.959882 + 0.280403i \(0.909532\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 2.40725e62 2.24675
\(661\) 5.49370e61 0.495944 0.247972 − 0.968767i \(-0.420236\pi\)
0.247972 + 0.968767i \(0.420236\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 5.32580e61 0.368987
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 2.14159e62 1.21914
\(676\) 1.81479e62 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.45757e62 −1.07957 −0.539783 − 0.841804i \(-0.681494\pi\)
−0.539783 + 0.841804i \(0.681494\pi\)
\(684\) 0 0
\(685\) 4.64484e62 1.91328
\(686\) 0 0
\(687\) 5.15192e62 1.99031
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.02171e62 −0.687389 −0.343695 − 0.939082i \(-0.611678\pi\)
−0.343695 + 0.939082i \(0.611678\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.43199e62 1.00000
\(705\) −2.00177e63 −4.37778
\(706\) 0 0
\(707\) 0 0
\(708\) −1.88560e63 −3.75594
\(709\) −9.75030e62 −1.88278 −0.941392 − 0.337314i \(-0.890482\pi\)
−0.941392 + 0.337314i \(0.890482\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.58580e62 0.270569
\(714\) 0 0
\(715\) 0 0
\(716\) −5.41785e62 −0.842831
\(717\) 0 0
\(718\) 0 0
\(719\) 4.22759e62 0.599870 0.299935 − 0.953960i \(-0.403035\pi\)
0.299935 + 0.953960i \(0.403035\pi\)
\(720\) 2.24113e63 3.08426
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −9.43917e62 −1.14997
\(725\) 0 0
\(726\) 0 0
\(727\) −5.50635e62 −0.612501 −0.306251 − 0.951951i \(-0.599075\pi\)
−0.306251 + 0.951951i \(0.599075\pi\)
\(728\) 0 0
\(729\) 2.40739e63 2.52082
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 2.56967e63 2.24675
\(736\) 0 0
\(737\) −2.16562e63 −1.78360
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −8.45026e62 −0.636462
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 3.59709e63 1.95827 0.979136 − 0.203207i \(-0.0651365\pi\)
0.979136 + 0.203207i \(0.0651365\pi\)
\(752\) −3.68545e63 −1.94849
\(753\) 4.60976e63 2.36695
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.29976e63 −1.05093 −0.525464 − 0.850816i \(-0.676108\pi\)
−0.525464 + 0.850816i \(0.676108\pi\)
\(758\) 0 0
\(759\) −8.67805e63 −3.74199
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 5.32847e63 1.98861
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 5.70868e63 1.89928
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 5.69945e63 1.74035
\(772\) 0 0
\(773\) −2.67311e63 −0.771024 −0.385512 − 0.922703i \(-0.625975\pi\)
−0.385512 + 0.922703i \(0.625975\pi\)
\(774\) 0 0
\(775\) −2.01269e62 −0.0548453
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −1.80433e62 −0.0414947
\(782\) 0 0
\(783\) 0 0
\(784\) 4.73100e63 1.00000
\(785\) 7.29305e63 1.49891
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 2.85103e64 4.43532
\(796\) −1.31358e64 −1.98779
\(797\) −9.44005e63 −1.38960 −0.694802 − 0.719201i \(-0.744508\pi\)
−0.694802 + 0.719201i \(0.744508\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −3.92976e64 −5.18144
\(802\) 0 0
\(803\) 0 0
\(804\) −2.78946e64 −3.38756
\(805\) 0 0
\(806\) 0 0
\(807\) 2.92988e64 3.27818
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.27656e63 −0.114967
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.34925e64 0.980160 0.490080 − 0.871677i \(-0.336967\pi\)
0.490080 + 0.871677i \(0.336967\pi\)
\(824\) 0 0
\(825\) 1.10142e64 0.758517
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −8.07917e64 −5.13688
\(829\) −1.07777e64 −0.667309 −0.333654 − 0.942695i \(-0.608282\pi\)
−0.333654 + 0.942695i \(0.608282\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.36372e63 −0.419218
\(838\) 0 0
\(839\) 5.44561e63 0.258990 0.129495 − 0.991580i \(-0.458664\pi\)
0.129495 + 0.991580i \(0.458664\pi\)
\(840\) 0 0
\(841\) 2.21570e64 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.90947e64 1.18295
\(846\) 0 0
\(847\) 0 0
\(848\) 5.24902e64 1.97410
\(849\) 0 0
\(850\) 0 0
\(851\) 3.04628e64 1.06003
\(852\) −2.32409e63 −0.0788101
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 5.77231e64 1.63493 0.817467 − 0.575975i \(-0.195377\pi\)
0.817467 + 0.575975i \(0.195377\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.75468e64 −1.47158 −0.735790 − 0.677210i \(-0.763189\pi\)
−0.735790 + 0.677210i \(0.763189\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 8.22260e64 1.89928
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −6.82045e64 −1.35363
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 7.10535e64 1.18295
\(881\) 1.22435e65 1.98808 0.994041 − 0.109010i \(-0.0347680\pi\)
0.994041 + 0.109010i \(0.0347680\pi\)
\(882\) 0 0
\(883\) −1.00279e65 −1.54908 −0.774541 − 0.632524i \(-0.782019\pi\)
−0.774541 + 0.632524i \(0.782019\pi\)
\(884\) 0 0
\(885\) −3.02299e65 −4.44309
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.51870e65 3.19056
\(892\) 1.57198e64 0.194277
\(893\) 0 0
\(894\) 0 0
\(895\) −8.68587e64 −0.997026
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.02541e65 1.04127
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.51328e65 −1.36035
\(906\) 0 0
\(907\) 2.13077e65 1.82464 0.912320 − 0.409478i \(-0.134289\pi\)
0.912320 + 0.409478i \(0.134289\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.39463e64 0.263869 0.131934 − 0.991258i \(-0.457881\pi\)
0.131934 + 0.991258i \(0.457881\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.52066e65 1.04793
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.86634e64 −0.214873
\(926\) 0 0
\(927\) −7.54614e65 −3.99918
\(928\) 0 0
\(929\) −3.15355e65 −1.59387 −0.796936 − 0.604063i \(-0.793547\pi\)
−0.796936 + 0.604063i \(0.793547\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −7.49859e65 −3.44813
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 9.33766e65 3.72902
\(940\) −5.90851e65 −2.30497
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −5.56562e65 −1.97756
\(945\) 0 0
\(946\) 0 0
\(947\) −1.55225e65 −0.514355 −0.257177 − 0.966364i \(-0.582792\pi\)
−0.257177 + 0.966364i \(0.582792\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.24016e66 3.74546
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 8.54259e65 2.35242
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 9.15214e65 2.24675
\(961\) −4.08927e65 −0.981141
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.16906e65 −1.17868 −0.589340 − 0.807885i \(-0.700612\pi\)
−0.589340 + 0.807885i \(0.700612\pi\)
\(972\) 1.60994e66 3.00713
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 8.34342e65 1.39208 0.696039 − 0.718004i \(-0.254944\pi\)
0.696039 + 0.718004i \(0.254944\pi\)
\(978\) 0 0
\(979\) −1.24590e66 −1.98731
\(980\) 7.58473e65 1.18295
\(981\) 0 0
\(982\) 0 0
\(983\) −7.95175e65 −1.15954 −0.579769 − 0.814781i \(-0.696857\pi\)
−0.579769 + 0.814781i \(0.696857\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.41331e65 −0.172432 −0.0862159 − 0.996276i \(-0.527477\pi\)
−0.0862159 + 0.996276i \(0.527477\pi\)
\(992\) 0 0
\(993\) 3.02311e66 3.52834
\(994\) 0 0
\(995\) −2.10593e66 −2.35146
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −1.60666e66 −1.64241
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 11.45.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.45.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.45.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.45.b.a.10.1 ✓ 1 11.10 odd 2 CM