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Properties

Label 11.39.b.a.10.1
Level $11$
Weight $39$
Character 11.10
Self dual yes
Analytic conductor $100.610$
Analytic rank $0$
Dimension $1$
CM discriminant -11
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] = [11,39,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, 39, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 39);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 39 \)
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: \(100.609815937\)
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)
 
\(f(q)\) \(=\) \(q-3.06986e8 q^{3} +2.74878e11 q^{4} +3.60591e13 q^{5} -1.25661e18 q^{9} +O(q^{10})\) \(q-3.06986e8 q^{3} +2.74878e11 q^{4} +3.60591e13 q^{5} -1.25661e18 q^{9} -6.11591e19 q^{11} -8.43837e19 q^{12} -1.10696e22 q^{15} +7.55579e22 q^{16} +9.91184e24 q^{20} +9.84485e25 q^{23} +9.36459e26 q^{25} +8.00455e26 q^{27} -1.75282e28 q^{31} +1.87750e28 q^{33} -3.45415e29 q^{36} +3.26833e29 q^{37} -1.68113e31 q^{44} -4.53122e31 q^{45} +1.11021e32 q^{47} -2.31952e31 q^{48} +1.29935e32 q^{49} +1.04811e33 q^{53} -2.20534e33 q^{55} -3.57124e33 q^{59} -3.04280e33 q^{60} +2.07692e34 q^{64} +9.45027e34 q^{67} -3.02223e34 q^{69} -2.59378e35 q^{71} -2.87480e35 q^{75} +2.72455e36 q^{80} +1.45177e36 q^{81} -2.18269e37 q^{89} +2.70613e37 q^{92} +5.38091e36 q^{93} +3.29402e37 q^{97} +7.68532e37 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −3.06986e8 −0.264128 −0.132064 − 0.991241i \(-0.542160\pi\)
−0.132064 + 0.991241i \(0.542160\pi\)
\(4\) 2.74878e11 1.00000
\(5\) 3.60591e13 1.89053 0.945267 − 0.326298i \(-0.105801\pi\)
0.945267 + 0.326298i \(0.105801\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.25661e18 −0.930236
\(10\) 0 0
\(11\) −6.11591e19 −1.00000
\(12\) −8.43837e19 −0.264128
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −1.10696e22 −0.499343
\(16\) 7.55579e22 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 9.91184e24 1.89053
\(21\) 0 0
\(22\) 0 0
\(23\) 9.84485e25 1.31941 0.659706 − 0.751524i \(-0.270681\pi\)
0.659706 + 0.751524i \(0.270681\pi\)
\(24\) 0 0
\(25\) 9.36459e26 2.57412
\(26\) 0 0
\(27\) 8.00455e26 0.509830
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.75282e28 −0.808845 −0.404422 − 0.914572i \(-0.632527\pi\)
−0.404422 + 0.914572i \(0.632527\pi\)
\(32\) 0 0
\(33\) 1.87750e28 0.264128
\(34\) 0 0
\(35\) 0 0
\(36\) −3.45415e29 −0.930236
\(37\) 3.26833e29 0.522989 0.261495 − 0.965205i \(-0.415785\pi\)
0.261495 + 0.965205i \(0.415785\pi\)
\(38\) 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 \(\pi\)
\(44\) −1.68113e31 −1.00000
\(45\) −4.53122e31 −1.75864
\(46\) 0 0
\(47\) 1.11021e32 1.88602 0.943012 − 0.332760i \(-0.107980\pi\)
0.943012 + 0.332760i \(0.107980\pi\)
\(48\) −2.31952e31 −0.264128
\(49\) 1.29935e32 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.04811e33 1.81621 0.908103 − 0.418747i \(-0.137530\pi\)
0.908103 + 0.418747i \(0.137530\pi\)
\(54\) 0 0
\(55\) −2.20534e33 −1.89053
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.57124e33 −0.806549 −0.403275 − 0.915079i \(-0.632128\pi\)
−0.403275 + 0.915079i \(0.632128\pi\)
\(60\) −3.04280e33 −0.499343
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 2.07692e34 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 9.45027e34 1.90556 0.952781 − 0.303660i \(-0.0982086\pi\)
0.952781 + 0.303660i \(0.0982086\pi\)
\(68\) 0 0
\(69\) −3.02223e34 −0.348494
\(70\) 0 0
\(71\) −2.59378e35 −1.73790 −0.868951 − 0.494899i \(-0.835205\pi\)
−0.868951 + 0.494899i \(0.835205\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −2.87480e35 −0.679897
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 2.72455e36 1.89053
\(81\) 1.45177e36 0.795576
\(82\) 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
\(89\) −2.18269e37 −1.99793 −0.998967 − 0.0454385i \(-0.985531\pi\)
−0.998967 + 0.0454385i \(0.985531\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.70613e37 1.31941
\(93\) 5.38091e36 0.213639
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.29402e37 0.587575 0.293788 − 0.955871i \(-0.405084\pi\)
0.293788 + 0.955871i \(0.405084\pi\)
\(98\) 0 0
\(99\) 7.68532e37 0.930236
\(100\) 2.57412e38 2.57412
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.09034e38 −0.621805 −0.310903 − 0.950442i \(-0.600631\pi\)
−0.310903 + 0.950442i \(0.600631\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 2.20027e38 0.509830
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −1.00333e38 −0.138136
\(112\) 0 0
\(113\) 1.92888e39 1.89154 0.945770 − 0.324837i \(-0.105309\pi\)
0.945770 + 0.324837i \(0.105309\pi\)
\(114\) 0 0
\(115\) 3.54996e39 2.49439
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.74043e39 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −4.81812e39 −0.808845
\(125\) 2.06496e40 2.97592
\(126\) 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 \(\pi\)
\(132\) 5.16083e39 0.264128
\(133\) 0 0
\(134\) 0 0
\(135\) 2.88637e40 0.963851
\(136\) 0 0
\(137\) −1.34677e40 −0.340097 −0.170049 − 0.985436i \(-0.554392\pi\)
−0.170049 + 0.985436i \(0.554392\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −3.40820e40 −0.498152
\(142\) 0 0
\(143\) 0 0
\(144\) −9.49469e40 −0.930236
\(145\) 0 0
\(146\) 0 0
\(147\) −3.98882e40 −0.264128
\(148\) 8.98391e40 0.522989
\(149\) 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
\(155\) −6.32051e41 −1.52915
\(156\) 0 0
\(157\) −1.03514e42 −1.96293 −0.981466 − 0.191637i \(-0.938620\pi\)
−0.981466 + 0.191637i \(0.938620\pi\)
\(158\) 0 0
\(159\) −3.21755e41 −0.479711
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.77436e42 −1.64998 −0.824988 − 0.565150i \(-0.808818\pi\)
−0.824988 + 0.565150i \(0.808818\pi\)
\(164\) 0 0
\(165\) 6.77009e41 0.499343
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.13721e42 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
\(176\) −4.62105e42 −1.00000
\(177\) 1.09632e42 0.213032
\(178\) 0 0
\(179\) 1.22908e43 1.92918 0.964590 − 0.263754i \(-0.0849607\pi\)
0.964590 + 0.263754i \(0.0849607\pi\)
\(180\) −1.24553e43 −1.75864
\(181\) −3.99012e42 −0.507099 −0.253550 − 0.967322i \(-0.581598\pi\)
−0.253550 + 0.967322i \(0.581598\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.17853e43 0.988729
\(186\) 0 0
\(187\) 0 0
\(188\) 3.05173e43 1.88602
\(189\) 0 0
\(190\) 0 0
\(191\) 4.29035e43 1.96272 0.981362 − 0.192170i \(-0.0615525\pi\)
0.981362 + 0.192170i \(0.0615525\pi\)
\(192\) −6.37585e42 −0.264128
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.57162e43 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −9.08826e42 −0.190666 −0.0953328 − 0.995445i \(-0.530392\pi\)
−0.0953328 + 0.995445i \(0.530392\pi\)
\(200\) 0 0
\(201\) −2.90110e43 −0.503312
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.23711e44 −1.22736
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 2.88102e44 1.81621
\(213\) 7.96255e43 0.459029
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −6.06199e44 −1.89053
\(221\) 0 0
\(222\) 0 0
\(223\) −2.43730e44 −0.587647 −0.293823 − 0.955860i \(-0.594928\pi\)
−0.293823 + 0.955860i \(0.594928\pi\)
\(224\) 0 0
\(225\) −1.17676e45 −2.39454
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.37336e45 −1.99945 −0.999724 − 0.0235019i \(-0.992518\pi\)
−0.999724 + 0.0235019i \(0.992518\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 4.00333e45 3.56559
\(236\) −9.81655e44 −0.806549
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −8.36398e44 −0.499343
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −1.52697e45 −0.719964
\(244\) 0 0
\(245\) 4.68533e45 1.89053
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.82902e45 −0.975634 −0.487817 − 0.872946i \(-0.662207\pi\)
−0.487817 + 0.872946i \(0.662207\pi\)
\(252\) 0 0
\(253\) −6.02102e45 −1.31941
\(254\) 0 0
\(255\) 0 0
\(256\) 5.70899e45 1.00000
\(257\) 9.71386e45 1.58002 0.790010 − 0.613094i \(-0.210076\pi\)
0.790010 + 0.613094i \(0.210076\pi\)
\(258\) 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
\(265\) 3.77939e46 3.43360
\(266\) 0 0
\(267\) 6.70054e45 0.527711
\(268\) 2.59767e46 1.90556
\(269\) −2.90740e46 −1.98706 −0.993531 − 0.113565i \(-0.963773\pi\)
−0.993531 + 0.113565i \(0.963773\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.72730e46 −2.57412
\(276\) −8.30745e45 −0.348494
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 2.20261e46 0.752417
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −7.12974e46 −1.73790
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.71556e46 1.00000
\(290\) 0 0
\(291\) −1.01122e46 −0.155195
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −1.28776e47 −1.52481
\(296\) 0 0
\(297\) −4.89551e46 −0.509830
\(298\) 0 0
\(299\) 0 0
\(300\) −7.90218e46 −0.679897
\(301\) 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
\(309\) 3.34719e46 0.164236
\(310\) 0 0
\(311\) −3.73468e47 −1.62109 −0.810544 − 0.585678i \(-0.800828\pi\)
−0.810544 + 0.585678i \(0.800828\pi\)
\(312\) 0 0
\(313\) 4.68381e47 1.79993 0.899967 − 0.435958i \(-0.143590\pi\)
0.899967 + 0.435958i \(0.143590\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.85508e47 −0.560062 −0.280031 − 0.959991i \(-0.590345\pi\)
−0.280031 + 0.959991i \(0.590345\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7.48918e47 1.89053
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.99059e47 0.795576
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6.82647e47 −0.906702 −0.453351 − 0.891332i \(-0.649771\pi\)
−0.453351 + 0.891332i \(0.649771\pi\)
\(332\) 0 0
\(333\) −4.10702e47 −0.486503
\(334\) 0 0
\(335\) 3.40768e48 3.60253
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −5.92140e47 −0.499609
\(340\) 0 0
\(341\) 1.07201e48 0.808845
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.08979e48 −0.658839
\(346\) 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
\(353\) 4.97447e48 1.94548 0.972741 − 0.231892i \(-0.0744916\pi\)
0.972741 + 0.231892i \(0.0744916\pi\)
\(354\) 0 0
\(355\) −9.35294e48 −3.28556
\(356\) −5.99972e48 −1.99793
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 3.91414e48 1.00000
\(362\) 0 0
\(363\) −1.14826e48 −0.264128
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.08412e48 −1.69679 −0.848395 − 0.529364i \(-0.822430\pi\)
−0.848395 + 0.529364i \(0.822430\pi\)
\(368\) 7.43856e48 1.31941
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.47909e48 0.213639
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −6.33914e48 −0.786025
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −4.10426e48 −0.415996 −0.207998 − 0.978129i \(-0.566695\pi\)
−0.207998 + 0.978129i \(0.566695\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.40069e49 −1.99324 −0.996621 − 0.0821411i \(-0.973824\pi\)
−0.996621 + 0.0821411i \(0.973824\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 9.05454e48 0.587575
\(389\) 2.05010e49 1.26687 0.633434 − 0.773796i \(-0.281645\pi\)
0.633434 + 0.773796i \(0.281645\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 2.11252e49 0.930236
\(397\) 1.57630e49 0.661637 0.330818 − 0.943694i \(-0.392675\pi\)
0.330818 + 0.943694i \(0.392675\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 7.07568e49 2.57412
\(401\) −4.19145e49 −1.45419 −0.727094 − 0.686538i \(-0.759130\pi\)
−0.727094 + 0.686538i \(0.759130\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 5.23494e49 1.50406
\(406\) 0 0
\(407\) −1.99888e49 −0.522989
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 4.13440e48 0.0898293
\(412\) −2.99710e49 −0.621805
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.17595e49 −0.478418 −0.239209 − 0.970968i \(-0.576888\pi\)
−0.239209 + 0.970968i \(0.576888\pi\)
\(420\) 0 0
\(421\) 1.44980e50 1.99502 0.997512 − 0.0704996i \(-0.0224594\pi\)
0.997512 + 0.0704996i \(0.0224594\pi\)
\(422\) 0 0
\(423\) −1.39511e50 −1.75445
\(424\) 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 \(\pi\)
\(432\) 6.04807e49 0.509830
\(433\) 1.29280e50 1.04294 0.521472 − 0.853268i \(-0.325383\pi\)
0.521472 + 0.853268i \(0.325383\pi\)
\(434\) 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
\(441\) −1.63278e50 −0.930236
\(442\) 0 0
\(443\) −2.07459e50 −1.08458 −0.542290 − 0.840191i \(-0.682443\pi\)
−0.542290 + 0.840191i \(0.682443\pi\)
\(444\) −2.75793e49 −0.138136
\(445\) −7.87056e50 −3.77716
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.48976e50 −1.81779 −0.908893 − 0.417029i \(-0.863071\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 5.30207e50 1.89154
\(453\) 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
\(460\) 9.75806e50 2.49439
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −5.82601e50 −1.31623 −0.658116 − 0.752917i \(-0.728646\pi\)
−0.658116 + 0.752917i \(0.728646\pi\)
\(464\) 0 0
\(465\) 1.94031e50 0.403891
\(466\) 0 0
\(467\) 9.13913e50 1.75341 0.876706 − 0.481026i \(-0.159736\pi\)
0.876706 + 0.481026i \(0.159736\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.17773e50 0.518466
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.31707e51 −1.68950
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.02816e51 1.00000
\(485\) 1.18779e51 1.11083
\(486\) 0 0
\(487\) −1.63756e51 −1.41627 −0.708137 − 0.706075i \(-0.750464\pi\)
−0.708137 + 0.706075i \(0.750464\pi\)
\(488\) 0 0
\(489\) 5.44704e50 0.435805
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 2.77126e51 1.75864
\(496\) −1.32439e51 −0.808845
\(497\) 0 0
\(498\) 0 0
\(499\) 9.56789e50 0.521081 0.260541 − 0.965463i \(-0.416099\pi\)
0.260541 + 0.965463i \(0.416099\pi\)
\(500\) 5.67612e51 2.97592
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.56094e50 −0.264128
\(508\) 0 0
\(509\) −5.03669e51 −1.88151 −0.940756 − 0.339083i \(-0.889883\pi\)
−0.940756 + 0.339083i \(0.889883\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.93166e51 −1.17554
\(516\) 0 0
\(517\) −6.78996e51 −1.88602
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −8.23449e51 −1.97570 −0.987848 − 0.155424i \(-0.950326\pi\)
−0.987848 + 0.155424i \(0.950326\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.41860e51 0.264128
\(529\) 4.12464e51 0.740846
\(530\) 0 0
\(531\) 4.48766e51 0.750281
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −3.77311e51 −0.509551
\(538\) 0 0
\(539\) −7.94669e51 −1.00000
\(540\) 7.93398e51 0.963851
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 1.22491e51 0.133939
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −3.70198e51 −0.340097
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −3.61792e51 −0.261151
\(556\) 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 \(\pi\)
\(564\) −9.36839e51 −0.498152
\(565\) 6.95537e52 3.57602
\(566\) 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
\(573\) −1.31708e52 −0.518411
\(574\) 0 0
\(575\) 9.21929e52 3.39632
\(576\) −2.60988e52 −0.930236
\(577\) −4.26162e51 −0.146972 −0.0734859 − 0.997296i \(-0.523412\pi\)
−0.0734859 + 0.997296i \(0.523412\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −6.41014e52 −1.81621
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.89511e52 0.969160 0.484580 − 0.874747i \(-0.338972\pi\)
0.484580 + 0.874747i \(0.338972\pi\)
\(588\) −1.09644e52 −0.264128
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.46948e52 0.522989
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.78997e51 0.0503602
\(598\) 0 0
\(599\) −1.18057e53 −1.99978 −0.999890 − 0.0148000i \(-0.995289\pi\)
−0.999890 + 0.0148000i \(0.995289\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −1.18753e53 −1.77262
\(604\) 0 0
\(605\) 1.34877e53 1.89053
\(606\) 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
\(617\) −1.73246e53 −1.67204 −0.836019 − 0.548700i \(-0.815123\pi\)
−0.836019 + 0.548700i \(0.815123\pi\)
\(618\) 0 0
\(619\) −1.59165e53 −1.44453 −0.722264 − 0.691617i \(-0.756898\pi\)
−0.722264 + 0.691617i \(0.756898\pi\)
\(620\) −1.73737e53 −1.52915
\(621\) 7.88036e52 0.672675
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.03924e53 3.05197
\(626\) 0 0
\(627\) 0 0
\(628\) −2.84537e53 −1.96293
\(629\) 0 0
\(630\) 0 0
\(631\) −1.79089e53 −1.12853 −0.564263 − 0.825595i \(-0.690840\pi\)
−0.564263 + 0.825595i \(0.690840\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −8.84433e52 −0.479711
\(637\) 0 0
\(638\) 0 0
\(639\) 3.25938e53 1.61666
\(640\) 0 0
\(641\) 3.13058e53 1.46326 0.731631 − 0.681701i \(-0.238759\pi\)
0.731631 + 0.681701i \(0.238759\pi\)
\(642\) 0 0
\(643\) 1.56252e53 0.688363 0.344181 − 0.938903i \(-0.388157\pi\)
0.344181 + 0.938903i \(0.388157\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.80202e53 1.48879 0.744395 − 0.667740i \(-0.232738\pi\)
0.744395 + 0.667740i \(0.232738\pi\)
\(648\) 0 0
\(649\) 2.18414e53 0.806549
\(650\) 0 0
\(651\) 0 0
\(652\) −4.87733e53 −1.64998
\(653\) −4.75626e53 −1.56284 −0.781421 − 0.624004i \(-0.785505\pi\)
−0.781421 + 0.624004i \(0.785505\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 1.86095e53 0.499343
\(661\) 7.35556e53 1.91773 0.958866 − 0.283858i \(-0.0916145\pi\)
0.958866 + 0.283858i \(0.0916145\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 7.48217e52 0.155214
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 7.49593e53 1.31236
\(676\) 5.87472e53 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
\(683\) −1.07302e54 −1.50180 −0.750902 − 0.660413i \(-0.770381\pi\)
−0.750902 + 0.660413i \(0.770381\pi\)
\(684\) 0 0
\(685\) −4.85633e53 −0.642965
\(686\) 0 0
\(687\) 4.21602e53 0.528111
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.78024e54 −1.99709 −0.998543 − 0.0539676i \(-0.982813\pi\)
−0.998543 + 0.0539676i \(0.982813\pi\)
\(692\) 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
\(704\) −1.27022e54 −1.00000
\(705\) −1.22897e54 −0.941773
\(706\) 0 0
\(707\) 0 0
\(708\) 3.01354e53 0.213032
\(709\) −7.86493e53 −0.541273 −0.270636 − 0.962682i \(-0.587234\pi\)
−0.270636 + 0.962682i \(0.587234\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.72562e54 −1.06720
\(714\) 0 0
\(715\) 0 0
\(716\) 3.37847e54 1.92918
\(717\) 0 0
\(718\) 0 0
\(719\) 3.08612e54 1.62766 0.813830 − 0.581103i \(-0.197379\pi\)
0.813830 + 0.581103i \(0.197379\pi\)
\(720\) −3.42370e54 −1.75864
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.09680e54 −0.507099
\(725\) 0 0
\(726\) 0 0
\(727\) 3.36388e54 1.43776 0.718879 − 0.695135i \(-0.244655\pi\)
0.718879 + 0.695135i \(0.244655\pi\)
\(728\) 0 0
\(729\) −1.49236e54 −0.605413
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −1.43833e54 −0.499343
\(736\) 0 0
\(737\) −5.77970e54 −1.90556
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 3.23951e54 0.988729
\(741\) 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
\(751\) 6.35847e54 1.46620 0.733101 − 0.680120i \(-0.238072\pi\)
0.733101 + 0.680120i \(0.238072\pi\)
\(752\) 8.38853e54 1.88602
\(753\) 1.17546e54 0.257692
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 7.76821e54 1.53992 0.769962 − 0.638090i \(-0.220275\pi\)
0.769962 + 0.638090i \(0.220275\pi\)
\(758\) 0 0
\(759\) 1.84837e54 0.348494
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.17932e55 1.96272
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.75258e54 −0.264128
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −2.98202e54 −0.417328
\(772\) 0 0
\(773\) −1.44666e55 −1.92733 −0.963665 − 0.267112i \(-0.913931\pi\)
−0.963665 + 0.267112i \(0.913931\pi\)
\(774\) 0 0
\(775\) −1.64144e55 −2.08206
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 1.58633e55 1.73790
\(782\) 0 0
\(783\) 0 0
\(784\) 9.81760e54 1.00000
\(785\) −3.73261e55 −3.71099
\(786\) 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
\(795\) −1.16022e55 −0.906910
\(796\) −2.49816e54 −0.190666
\(797\) 1.16539e55 0.868487 0.434244 − 0.900795i \(-0.357016\pi\)
0.434244 + 0.900795i \(0.357016\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.74279e55 1.85855
\(802\) 0 0
\(803\) 0 0
\(804\) −7.97448e54 −0.503312
\(805\) 0 0
\(806\) 0 0
\(807\) 8.92531e54 0.524839
\(808\) 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
\(815\) −6.39818e55 −3.11934
\(816\) 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
\(823\) −1.79029e55 −0.724981 −0.362491 − 0.931987i \(-0.618074\pi\)
−0.362491 + 0.931987i \(0.618074\pi\)
\(824\) 0 0
\(825\) 1.75820e55 0.679897
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −3.40056e55 −1.22736
\(829\) −2.02483e55 −0.714253 −0.357126 − 0.934056i \(-0.616243\pi\)
−0.357126 + 0.934056i \(0.616243\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.40305e55 −0.412373
\(838\) 0 0
\(839\) 5.18684e55 1.45689 0.728443 − 0.685107i \(-0.240244\pi\)
0.728443 + 0.685107i \(0.240244\pi\)
\(840\) 0 0
\(841\) 3.72498e55 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.70658e55 1.89053
\(846\) 0 0
\(847\) 0 0
\(848\) 7.91929e55 1.81621
\(849\) 0 0
\(850\) 0 0
\(851\) 3.21762e55 0.690038
\(852\) 2.18873e55 0.459029
\(853\) 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
\(859\) 7.63519e55 1.37072 0.685362 − 0.728203i \(-0.259644\pi\)
0.685362 + 0.728203i \(0.259644\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.21374e56 −1.99489 −0.997446 − 0.0714233i \(-0.977246\pi\)
−0.997446 + 0.0714233i \(0.977246\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.75460e55 −0.264128
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −4.13931e55 −0.546584
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.66631e56 −1.89053
\(881\) −4.23140e55 −0.469831 −0.234916 − 0.972016i \(-0.575481\pi\)
−0.234916 + 0.972016i \(0.575481\pi\)
\(882\) 0 0
\(883\) 1.72605e56 1.83570 0.917848 − 0.396933i \(-0.129925\pi\)
0.917848 + 0.396933i \(0.129925\pi\)
\(884\) 0 0
\(885\) 3.95323e55 0.402745
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −8.87888e55 −0.795576
\(892\) −6.69960e55 −0.587647
\(893\) 0 0
\(894\) 0 0
\(895\) 4.43195e56 3.64718
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −3.23467e56 −2.39454
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.43880e56 −0.958689
\(906\) 0 0
\(907\) −1.07229e56 −0.685131 −0.342565 − 0.939494i \(-0.611296\pi\)
−0.342565 + 0.939494i \(0.611296\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1.71547e56 −1.00817 −0.504085 − 0.863654i \(-0.668170\pi\)
−0.504085 + 0.863654i \(0.668170\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −3.77506e56 −1.99945
\(917\) 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
\(925\) 3.06065e56 1.34624
\(926\) 0 0
\(927\) 1.37013e56 0.578426
\(928\) 0 0
\(929\) −3.76547e56 −1.52588 −0.762941 − 0.646468i \(-0.776246\pi\)
−0.762941 + 0.646468i \(0.776246\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.14650e56 0.428175
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −1.43786e56 −0.475413
\(940\) 1.10043e57 3.56559
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −2.69835e56 −0.806549
\(945\) 0 0
\(946\) 0 0
\(947\) −3.77969e56 −1.06367 −0.531835 − 0.846848i \(-0.678497\pi\)
−0.531835 + 0.846848i \(0.678497\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 5.69483e55 0.147928
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.54706e57 3.71059
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −2.29907e56 −0.499343
\(961\) −1.62380e56 −0.345770
\(962\) 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
\(971\) −8.29953e56 −1.45174 −0.725869 − 0.687833i \(-0.758562\pi\)
−0.725869 + 0.687833i \(0.758562\pi\)
\(972\) −4.19730e56 −0.719964
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.89523e56 0.606088 0.303044 − 0.952976i \(-0.401997\pi\)
0.303044 + 0.952976i \(0.401997\pi\)
\(978\) 0 0
\(979\) 1.33491e57 1.99793
\(980\) 1.28789e57 1.89053
\(981\) 0 0
\(982\) 0 0
\(983\) 4.95359e55 0.0686125 0.0343063 − 0.999411i \(-0.489078\pi\)
0.0343063 + 0.999411i \(0.489078\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.61760e57 −1.92075 −0.960376 − 0.278707i \(-0.910094\pi\)
−0.960376 + 0.278707i \(0.910094\pi\)
\(992\) 0 0
\(993\) 2.09563e56 0.239485
\(994\) 0 0
\(995\) −3.27714e56 −0.360460
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 2.61615e56 0.266635
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 11.39.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.39.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.39.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.39.b.a.10.1 ✓ 1 11.10 odd 2 CM