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Properties

Label 11.13.b.a.10.1
Level $11$
Weight $13$
Character 11.10
Self dual yes
Analytic conductor $10.054$
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] = [11,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 13 \)
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: \(10.0539319900\)
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)
 
\(f(q)\) \(=\) \(q-1358.00 q^{3} +4096.00 q^{4} -25774.0 q^{5} +1.31272e6 q^{9} +O(q^{10})\) \(q-1358.00 q^{3} +4096.00 q^{4} -25774.0 q^{5} +1.31272e6 q^{9} +1.77156e6 q^{11} -5.56237e6 q^{12} +3.50011e7 q^{15} +1.67772e7 q^{16} -1.05570e8 q^{20} -1.35543e8 q^{23} +4.20158e8 q^{25} -1.06098e9 q^{27} +1.36301e9 q^{31} -2.40578e9 q^{33} +5.37691e9 q^{36} +2.44625e9 q^{37} +7.25631e9 q^{44} -3.38341e10 q^{45} +2.10219e10 q^{47} -2.27835e10 q^{48} +1.38413e10 q^{49} +1.66259e10 q^{53} -4.56602e10 q^{55} -7.27743e10 q^{59} +1.43364e11 q^{60} +6.87195e10 q^{64} +2.66978e9 q^{67} +1.84067e11 q^{69} -1.39730e11 q^{71} -5.70575e11 q^{75} -4.32416e11 q^{80} +7.43177e11 q^{81} +9.44643e11 q^{89} -5.55184e11 q^{92} -1.85097e12 q^{93} +1.66173e12 q^{97} +2.32557e12 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −1358.00 −1.86283 −0.931413 − 0.363964i \(-0.881423\pi\)
−0.931413 + 0.363964i \(0.881423\pi\)
\(4\) 4096.00 1.00000
\(5\) −25774.0 −1.64954 −0.824768 − 0.565471i \(-0.808694\pi\)
−0.824768 + 0.565471i \(0.808694\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.31272e6 2.47012
\(10\) 0 0
\(11\) 1.77156e6 1.00000
\(12\) −5.56237e6 −1.86283
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 3.50011e7 3.07280
\(16\) 1.67772e7 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −1.05570e8 −1.64954
\(21\) 0 0
\(22\) 0 0
\(23\) −1.35543e8 −0.915608 −0.457804 − 0.889053i \(-0.651364\pi\)
−0.457804 + 0.889053i \(0.651364\pi\)
\(24\) 0 0
\(25\) 4.20158e8 1.72097
\(26\) 0 0
\(27\) −1.06098e9 −2.73858
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.36301e9 1.53578 0.767889 − 0.640583i \(-0.221307\pi\)
0.767889 + 0.640583i \(0.221307\pi\)
\(32\) 0 0
\(33\) −2.40578e9 −1.86283
\(34\) 0 0
\(35\) 0 0
\(36\) 5.37691e9 2.47012
\(37\) 2.44625e9 0.953434 0.476717 − 0.879057i \(-0.341827\pi\)
0.476717 + 0.879057i \(0.341827\pi\)
\(38\) 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 \(\pi\)
\(44\) 7.25631e9 1.00000
\(45\) −3.38341e10 −4.07455
\(46\) 0 0
\(47\) 2.10219e10 1.95022 0.975112 − 0.221711i \(-0.0711643\pi\)
0.975112 + 0.221711i \(0.0711643\pi\)
\(48\) −2.27835e10 −1.86283
\(49\) 1.38413e10 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.66259e10 0.750121 0.375060 − 0.927000i \(-0.377622\pi\)
0.375060 + 0.927000i \(0.377622\pi\)
\(54\) 0 0
\(55\) −4.56602e10 −1.64954
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.27743e10 −1.72530 −0.862652 − 0.505797i \(-0.831198\pi\)
−0.862652 + 0.505797i \(0.831198\pi\)
\(60\) 1.43364e11 3.07280
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 6.87195e10 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.66978e9 0.0295139 0.0147569 − 0.999891i \(-0.495303\pi\)
0.0147569 + 0.999891i \(0.495303\pi\)
\(68\) 0 0
\(69\) 1.84067e11 1.70562
\(70\) 0 0
\(71\) −1.39730e11 −1.09079 −0.545393 − 0.838181i \(-0.683619\pi\)
−0.545393 + 0.838181i \(0.683619\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −5.70575e11 −3.20587
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −4.32416e11 −1.64954
\(81\) 7.43177e11 2.63137
\(82\) 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
\(89\) 9.44643e11 1.90076 0.950380 − 0.311090i \(-0.100694\pi\)
0.950380 + 0.311090i \(0.100694\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −5.55184e11 −0.915608
\(93\) −1.85097e12 −2.86089
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.66173e12 1.99494 0.997468 − 0.0711226i \(-0.0226581\pi\)
0.997468 + 0.0711226i \(0.0226581\pi\)
\(98\) 0 0
\(99\) 2.32557e12 2.47012
\(100\) 1.72097e12 1.72097
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.72897e12 −1.44799 −0.723993 − 0.689807i \(-0.757695\pi\)
−0.723993 + 0.689807i \(0.757695\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −4.34578e12 −2.73858
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −3.32201e12 −1.77608
\(112\) 0 0
\(113\) −1.26584e12 −0.608006 −0.304003 − 0.952671i \(-0.598323\pi\)
−0.304003 + 0.952671i \(0.598323\pi\)
\(114\) 0 0
\(115\) 3.49348e12 1.51033
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.13843e12 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 5.58288e12 1.53578
\(125\) −4.53668e12 −1.18926
\(126\) 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 \(\pi\)
\(132\) −9.85407e12 −1.86283
\(133\) 0 0
\(134\) 0 0
\(135\) 2.73457e13 4.51738
\(136\) 0 0
\(137\) 3.79623e11 0.0574155 0.0287078 − 0.999588i \(-0.490861\pi\)
0.0287078 + 0.999588i \(0.490861\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −2.85477e13 −3.63293
\(142\) 0 0
\(143\) 0 0
\(144\) 2.20238e13 2.47012
\(145\) 0 0
\(146\) 0 0
\(147\) −1.87965e13 −1.86283
\(148\) 1.00198e13 0.953434
\(149\) 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
\(155\) −3.51302e13 −2.53332
\(156\) 0 0
\(157\) 5.57220e12 0.372073 0.186037 − 0.982543i \(-0.440436\pi\)
0.186037 + 0.982543i \(0.440436\pi\)
\(158\) 0 0
\(159\) −2.25780e13 −1.39734
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.00899e13 −0.537975 −0.268987 − 0.963144i \(-0.586689\pi\)
−0.268987 + 0.963144i \(0.586689\pi\)
\(164\) 0 0
\(165\) 6.20066e13 3.07280
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.32981e13 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
\(176\) 2.97219e13 1.00000
\(177\) 9.88275e13 3.21394
\(178\) 0 0
\(179\) −1.09331e13 −0.332374 −0.166187 − 0.986094i \(-0.553145\pi\)
−0.166187 + 0.986094i \(0.553145\pi\)
\(180\) −1.38585e14 −4.07455
\(181\) −1.16943e13 −0.332585 −0.166292 − 0.986077i \(-0.553180\pi\)
−0.166292 + 0.986077i \(0.553180\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.30496e13 −1.57272
\(186\) 0 0
\(187\) 0 0
\(188\) 8.61057e13 1.95022
\(189\) 0 0
\(190\) 0 0
\(191\) 8.01244e13 1.65031 0.825153 − 0.564909i \(-0.191089\pi\)
0.825153 + 0.564909i \(0.191089\pi\)
\(192\) −9.33210e13 −1.86283
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 5.66939e13 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.24151e14 −1.99909 −0.999545 − 0.0301463i \(-0.990403\pi\)
−0.999545 + 0.0301463i \(0.990403\pi\)
\(200\) 0 0
\(201\) −3.62556e12 −0.0549792
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.77930e14 −2.26166
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 6.80999e13 0.750121
\(213\) 1.89753e14 2.03194
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.87024e14 −1.64954
\(221\) 0 0
\(222\) 0 0
\(223\) 2.26362e14 1.84067 0.920333 − 0.391135i \(-0.127918\pi\)
0.920333 + 0.391135i \(0.127918\pi\)
\(224\) 0 0
\(225\) 5.51552e14 4.25100
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.17819e14 −0.816963 −0.408481 − 0.912767i \(-0.633942\pi\)
−0.408481 + 0.912767i \(0.633942\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −5.41818e14 −3.21697
\(236\) −2.98083e14 −1.72530
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 5.87221e14 3.07280
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −4.45386e14 −2.16321
\(244\) 0 0
\(245\) −3.56745e14 −1.64954
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.73734e14 1.89449 0.947245 − 0.320511i \(-0.103855\pi\)
0.947245 + 0.320511i \(0.103855\pi\)
\(252\) 0 0
\(253\) −2.40122e14 −0.915608
\(254\) 0 0
\(255\) 0 0
\(256\) 2.81475e14 1.00000
\(257\) 3.71685e14 1.28996 0.644979 − 0.764200i \(-0.276866\pi\)
0.644979 + 0.764200i \(0.276866\pi\)
\(258\) 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
\(265\) −4.28517e14 −1.23735
\(266\) 0 0
\(267\) −1.28282e15 −3.54079
\(268\) 1.09354e13 0.0295139
\(269\) 2.12300e14 0.560320 0.280160 − 0.959953i \(-0.409613\pi\)
0.280160 + 0.959953i \(0.409613\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.44336e14 1.72097
\(276\) 7.53939e14 1.70562
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 1.78925e15 3.79356
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −5.72334e14 −1.09079
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 5.82622e14 1.00000
\(290\) 0 0
\(291\) −2.25662e15 −3.71622
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 1.87568e15 2.84595
\(296\) 0 0
\(297\) −1.87959e15 −2.73858
\(298\) 0 0
\(299\) 0 0
\(300\) −2.33708e15 −3.20587
\(301\) 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
\(309\) 2.34794e15 2.69735
\(310\) 0 0
\(311\) −9.40419e14 −1.03934 −0.519672 − 0.854366i \(-0.673946\pi\)
−0.519672 + 0.854366i \(0.673946\pi\)
\(312\) 0 0
\(313\) 1.63301e15 1.73670 0.868348 − 0.495955i \(-0.165182\pi\)
0.868348 + 0.495955i \(0.165182\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.25764e15 −1.23937 −0.619687 − 0.784849i \(-0.712740\pi\)
−0.619687 + 0.784849i \(0.712740\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.77118e15 −1.64954
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.04405e15 2.63137
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.58676e15 −1.96693 −0.983464 − 0.181103i \(-0.942033\pi\)
−0.983464 + 0.181103i \(0.942033\pi\)
\(332\) 0 0
\(333\) 3.21125e15 2.35510
\(334\) 0 0
\(335\) −6.88108e13 −0.0486842
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 1.71901e15 1.13261
\(340\) 0 0
\(341\) 2.41465e15 1.53578
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −4.74415e15 −2.81348
\(346\) 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
\(353\) −3.75948e15 −1.94303 −0.971516 − 0.236973i \(-0.923845\pi\)
−0.971516 + 0.236973i \(0.923845\pi\)
\(354\) 0 0
\(355\) 3.60140e15 1.79929
\(356\) 3.86926e15 1.90076
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 2.21331e15 1.00000
\(362\) 0 0
\(363\) −4.26199e15 −1.86283
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4.88653e15 −1.99988 −0.999942 − 0.0108164i \(-0.996557\pi\)
−0.999942 + 0.0108164i \(0.996557\pi\)
\(368\) −2.27403e15 −0.915608
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −7.58156e15 −2.86089
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 6.16082e15 2.21539
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −2.90680e15 −0.980798 −0.490399 − 0.871498i \(-0.663149\pi\)
−0.490399 + 0.871498i \(0.663149\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.31439e15 1.05005 0.525026 − 0.851086i \(-0.324056\pi\)
0.525026 + 0.851086i \(0.324056\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 6.80643e15 1.99494
\(389\) −2.17523e14 −0.0627779 −0.0313889 − 0.999507i \(-0.509993\pi\)
−0.0313889 + 0.999507i \(0.509993\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 9.52553e15 2.47012
\(397\) 6.45140e15 1.64782 0.823912 − 0.566717i \(-0.191787\pi\)
0.823912 + 0.566717i \(0.191787\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 7.04909e15 1.72097
\(401\) −6.16878e15 −1.48366 −0.741828 − 0.670590i \(-0.766041\pi\)
−0.741828 + 0.670590i \(0.766041\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.91547e16 −4.34054
\(406\) 0 0
\(407\) 4.33368e15 0.953434
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −5.15528e14 −0.106955
\(412\) −7.08187e15 −1.44799
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.57933e15 0.661481 0.330740 − 0.943722i \(-0.392702\pi\)
0.330740 + 0.943722i \(0.392702\pi\)
\(420\) 0 0
\(421\) −1.09404e16 −1.96490 −0.982449 − 0.186530i \(-0.940276\pi\)
−0.982449 + 0.186530i \(0.940276\pi\)
\(422\) 0 0
\(423\) 2.75959e16 4.81729
\(424\) 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 \(\pi\)
\(432\) −1.78003e16 −2.73858
\(433\) −1.29840e16 −1.97006 −0.985032 − 0.172369i \(-0.944858\pi\)
−0.985032 + 0.172369i \(0.944858\pi\)
\(434\) 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
\(441\) 1.81698e16 2.47012
\(442\) 0 0
\(443\) 1.51147e16 1.99976 0.999878 − 0.0156504i \(-0.00498190\pi\)
0.999878 + 0.0156504i \(0.00498190\pi\)
\(444\) −1.36069e16 −1.77608
\(445\) −2.43472e16 −3.13537
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.83441e15 0.223882 0.111941 − 0.993715i \(-0.464293\pi\)
0.111941 + 0.993715i \(0.464293\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −5.18488e15 −0.608006
\(453\) 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
\(460\) 1.43093e16 1.51033
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.91975e16 −1.94877 −0.974383 − 0.224896i \(-0.927796\pi\)
−0.974383 + 0.224896i \(0.927796\pi\)
\(464\) 0 0
\(465\) 4.77068e16 4.71914
\(466\) 0 0
\(467\) 1.39455e16 1.34441 0.672207 − 0.740364i \(-0.265347\pi\)
0.672207 + 0.740364i \(0.265347\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −7.56704e15 −0.693108
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.18253e16 1.85289
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.28550e16 1.00000
\(485\) −4.28293e16 −3.29072
\(486\) 0 0
\(487\) −2.44388e16 −1.83192 −0.915958 − 0.401274i \(-0.868568\pi\)
−0.915958 + 0.401274i \(0.868568\pi\)
\(488\) 0 0
\(489\) 1.37021e16 1.00215
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −5.99392e16 −4.07455
\(496\) 2.28675e16 1.53578
\(497\) 0 0
\(498\) 0 0
\(499\) 2.27286e16 1.47221 0.736105 − 0.676867i \(-0.236663\pi\)
0.736105 + 0.676867i \(0.236663\pi\)
\(500\) −1.85823e16 −1.18926
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.16388e16 −1.86283
\(508\) 0 0
\(509\) 2.20843e16 1.26992 0.634962 − 0.772543i \(-0.281016\pi\)
0.634962 + 0.772543i \(0.281016\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.45625e16 2.38851
\(516\) 0 0
\(517\) 3.72416e16 1.95022
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.56085e16 −1.28043 −0.640217 − 0.768194i \(-0.721155\pi\)
−0.640217 + 0.768194i \(0.721155\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −4.03623e16 −1.86283
\(529\) −3.54275e15 −0.161662
\(530\) 0 0
\(531\) −9.55325e16 −4.26171
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.48472e16 0.619154
\(538\) 0 0
\(539\) 2.45207e16 1.00000
\(540\) 1.12008e17 4.51738
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 1.58808e16 0.619547
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 1.55494e15 0.0574155
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 8.56214e16 2.92971
\(556\) 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 \(\pi\)
\(564\) −1.16931e17 −3.63293
\(565\) 3.26258e16 1.00293
\(566\) 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
\(573\) −1.08809e17 −3.07423
\(574\) 0 0
\(575\) −5.69495e16 −1.57573
\(576\) 9.02096e16 2.47012
\(577\) −5.92795e16 −1.60639 −0.803193 − 0.595719i \(-0.796867\pi\)
−0.803193 + 0.595719i \(0.796867\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.94539e16 0.750121
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.30035e15 −0.154005 −0.0770027 − 0.997031i \(-0.524535\pi\)
−0.0770027 + 0.997031i \(0.524535\pi\)
\(588\) −7.69903e16 −1.86283
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 4.10413e16 0.953434
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.68597e17 3.72396
\(598\) 0 0
\(599\) −8.14528e16 −1.76338 −0.881689 − 0.471832i \(-0.843593\pi\)
−0.881689 + 0.471832i \(0.843593\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 3.50468e15 0.0729028
\(604\) 0 0
\(605\) −8.08899e16 −1.64954
\(606\) 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
\(617\) 9.79758e16 1.77586 0.887928 − 0.459982i \(-0.152144\pi\)
0.887928 + 0.459982i \(0.152144\pi\)
\(618\) 0 0
\(619\) 1.09249e17 1.94212 0.971058 − 0.238845i \(-0.0767689\pi\)
0.971058 + 0.238845i \(0.0767689\pi\)
\(620\) −1.43893e17 −2.53332
\(621\) 1.43808e17 2.50746
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.43507e16 0.240765
\(626\) 0 0
\(627\) 0 0
\(628\) 2.28237e16 0.372073
\(629\) 0 0
\(630\) 0 0
\(631\) −1.34389e16 −0.212906 −0.106453 − 0.994318i \(-0.533949\pi\)
−0.106453 + 0.994318i \(0.533949\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −9.24797e16 −1.39734
\(637\) 0 0
\(638\) 0 0
\(639\) −1.83427e17 −2.69437
\(640\) 0 0
\(641\) 1.16981e17 1.68643 0.843214 − 0.537579i \(-0.180661\pi\)
0.843214 + 0.537579i \(0.180661\pi\)
\(642\) 0 0
\(643\) 4.20970e16 0.595643 0.297821 − 0.954622i \(-0.403740\pi\)
0.297821 + 0.954622i \(0.403740\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.33323e17 1.81752 0.908760 − 0.417319i \(-0.137030\pi\)
0.908760 + 0.417319i \(0.137030\pi\)
\(648\) 0 0
\(649\) −1.28924e17 −1.72530
\(650\) 0 0
\(651\) 0 0
\(652\) −4.13283e16 −0.537975
\(653\) 1.10315e17 1.42283 0.711416 − 0.702771i \(-0.248054\pi\)
0.711416 + 0.702771i \(0.248054\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 2.53979e17 3.07280
\(661\) −1.66354e17 −1.99446 −0.997229 − 0.0743891i \(-0.976299\pi\)
−0.997229 + 0.0743891i \(0.976299\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.07400e17 −3.42884
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −4.45780e17 −4.71301
\(676\) 9.54290e16 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
\(683\) 1.72178e17 1.69611 0.848054 − 0.529910i \(-0.177774\pi\)
0.848054 + 0.529910i \(0.177774\pi\)
\(684\) 0 0
\(685\) −9.78441e15 −0.0947089
\(686\) 0 0
\(687\) 1.59998e17 1.52186
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.50167e17 −1.37945 −0.689727 − 0.724069i \(-0.742270\pi\)
−0.689727 + 0.724069i \(0.742270\pi\)
\(692\) 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
\(704\) 1.21741e17 1.00000
\(705\) 7.35789e17 5.99265
\(706\) 0 0
\(707\) 0 0
\(708\) 4.04797e17 3.21394
\(709\) −8.34169e16 −0.656714 −0.328357 − 0.944554i \(-0.606495\pi\)
−0.328357 + 0.944554i \(0.606495\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.84746e17 −1.40617
\(714\) 0 0
\(715\) 0 0
\(716\) −4.47821e16 −0.332374
\(717\) 0 0
\(718\) 0 0
\(719\) −2.23120e17 −1.61497 −0.807486 − 0.589887i \(-0.799172\pi\)
−0.807486 + 0.589887i \(0.799172\pi\)
\(720\) −5.67642e17 −4.07455
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −4.78998e16 −0.332585
\(725\) 0 0
\(726\) 0 0
\(727\) −1.80128e17 −1.22004 −0.610020 − 0.792386i \(-0.708839\pi\)
−0.610020 + 0.792386i \(0.708839\pi\)
\(728\) 0 0
\(729\) 2.09879e17 1.39831
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 4.84460e17 3.07280
\(736\) 0 0
\(737\) 4.72967e15 0.0295139
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −2.58251e17 −1.57272
\(741\) 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
\(751\) 1.67031e17 0.931015 0.465507 − 0.885044i \(-0.345872\pi\)
0.465507 + 0.885044i \(0.345872\pi\)
\(752\) 3.52689e17 1.95022
\(753\) −6.43331e17 −3.52910
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.61962e17 −1.92348 −0.961741 − 0.273962i \(-0.911666\pi\)
−0.961741 + 0.273962i \(0.911666\pi\)
\(758\) 0 0
\(759\) 3.26086e17 1.70562
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 3.28190e17 1.65031
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −3.82243e17 −1.86283
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −5.04748e17 −2.40297
\(772\) 0 0
\(773\) −4.13342e17 −1.93746 −0.968730 − 0.248116i \(-0.920189\pi\)
−0.968730 + 0.248116i \(0.920189\pi\)
\(774\) 0 0
\(775\) 5.72680e17 2.64303
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −2.47540e17 −1.09079
\(782\) 0 0
\(783\) 0 0
\(784\) 2.32218e17 1.00000
\(785\) −1.43618e17 −0.613748
\(786\) 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
\(795\) 5.81926e17 2.30497
\(796\) −5.08523e17 −1.99909
\(797\) 4.11800e17 1.60670 0.803352 − 0.595504i \(-0.203048\pi\)
0.803352 + 0.595504i \(0.203048\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.24005e18 4.69511
\(802\) 0 0
\(803\) 0 0
\(804\) −1.48503e16 −0.0549792
\(805\) 0 0
\(806\) 0 0
\(807\) −2.88303e17 −1.04378
\(808\) 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
\(815\) 2.60057e17 0.887408
\(816\) 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
\(823\) 5.96851e17 1.92073 0.960366 − 0.278744i \(-0.0899180\pi\)
0.960366 + 0.278744i \(0.0899180\pi\)
\(824\) 0 0
\(825\) −1.01081e18 −3.20587
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −7.28802e17 −2.26166
\(829\) 6.48356e17 1.99750 0.998749 − 0.0500031i \(-0.0159231\pi\)
0.998749 + 0.0500031i \(0.0159231\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.44613e18 −4.20585
\(838\) 0 0
\(839\) −2.20112e17 −0.631061 −0.315530 − 0.948915i \(-0.602182\pi\)
−0.315530 + 0.948915i \(0.602182\pi\)
\(840\) 0 0
\(841\) 3.53815e17 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −6.00485e17 −1.64954
\(846\) 0 0
\(847\) 0 0
\(848\) 2.78937e17 0.750121
\(849\) 0 0
\(850\) 0 0
\(851\) −3.31572e17 −0.872972
\(852\) 7.77229e17 2.03194
\(853\) 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
\(859\) 7.38884e17 1.83915 0.919575 − 0.392915i \(-0.128533\pi\)
0.919575 + 0.392915i \(0.128533\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.96255e16 −0.120127 −0.0600634 − 0.998195i \(-0.519130\pi\)
−0.0600634 + 0.998195i \(0.519130\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −7.91201e17 −1.86283
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.18138e18 4.92773
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −7.66051e17 −1.64954
\(881\) −9.04728e17 −1.93492 −0.967459 − 0.253029i \(-0.918573\pi\)
−0.967459 + 0.253029i \(0.918573\pi\)
\(882\) 0 0
\(883\) −5.47066e17 −1.15419 −0.577093 − 0.816678i \(-0.695813\pi\)
−0.577093 + 0.816678i \(0.695813\pi\)
\(884\) 0 0
\(885\) −2.54718e18 −5.30151
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.31658e18 2.63137
\(892\) 9.27180e17 1.84067
\(893\) 0 0
\(894\) 0 0
\(895\) 2.81791e17 0.548262
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 2.25916e18 4.25100
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.01408e17 0.548610
\(906\) 0 0
\(907\) −3.08845e16 −0.0554750 −0.0277375 − 0.999615i \(-0.508830\pi\)
−0.0277375 + 0.999615i \(0.508830\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.82303e17 0.493862 0.246931 − 0.969033i \(-0.420578\pi\)
0.246931 + 0.969033i \(0.420578\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −4.82587e17 −0.816963
\(917\) 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
\(925\) 1.02781e18 1.64083
\(926\) 0 0
\(927\) −2.26966e18 −3.57670
\(928\) 0 0
\(929\) 1.27806e18 1.98819 0.994097 − 0.108497i \(-0.0346038\pi\)
0.994097 + 0.108497i \(0.0346038\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.27709e18 1.93612
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −2.21763e18 −3.23516
\(940\) −2.21929e18 −3.21697
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.22095e18 −1.72530
\(945\) 0 0
\(946\) 0 0
\(947\) −1.02243e17 −0.141754 −0.0708770 − 0.997485i \(-0.522580\pi\)
−0.0708770 + 0.997485i \(0.522580\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.70788e18 2.30874
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −2.06513e18 −2.72224
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 2.40526e18 3.07280
\(961\) 1.07013e18 1.35862
\(962\) 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
\(971\) −1.62143e18 −1.93456 −0.967280 − 0.253711i \(-0.918349\pi\)
−0.967280 + 0.253711i \(0.918349\pi\)
\(972\) −1.82430e18 −2.16321
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.04827e18 1.20533 0.602663 − 0.797996i \(-0.294106\pi\)
0.602663 + 0.797996i \(0.294106\pi\)
\(978\) 0 0
\(979\) 1.67349e18 1.90076
\(980\) −1.46123e18 −1.64954
\(981\) 0 0
\(982\) 0 0
\(983\) 7.06902e17 0.783498 0.391749 − 0.920072i \(-0.371870\pi\)
0.391749 + 0.920072i \(0.371870\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.40209e18 −1.48025 −0.740124 − 0.672470i \(-0.765233\pi\)
−0.740124 + 0.672470i \(0.765233\pi\)
\(992\) 0 0
\(993\) 3.51282e18 3.66404
\(994\) 0 0
\(995\) 3.19987e18 3.29757
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −2.59542e18 −2.61105
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 11.13.b.a.10.1 ✓ 1
3.2 odd 2 99.13.c.a.10.1 1
4.3 odd 2 176.13.h.a.65.1 1
11.10 odd 2 CM 11.13.b.a.10.1 ✓ 1
33.32 even 2 99.13.c.a.10.1 1
44.43 even 2 176.13.h.a.65.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.13.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.13.b.a.10.1 ✓ 1 11.10 odd 2 CM
99.13.c.a.10.1 1 3.2 odd 2
99.13.c.a.10.1 1 33.32 even 2
176.13.h.a.65.1 1 4.3 odd 2
176.13.h.a.65.1 1 44.43 even 2