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Properties

Label 11.23.b.a.10.1
Level $11$
Weight $23$
Character 11.10
Self dual yes
Analytic conductor $33.738$
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] = [11,23,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, 23, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 23);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 23 \)
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: \(33.7378178326\)
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)
 
\(f(q)\) \(=\) \(q-349805. q^{3} +4.19430e6 q^{4} +8.71134e7 q^{5} +9.09825e10 q^{9} +O(q^{10})\) \(q-349805. q^{3} +4.19430e6 q^{4} +8.71134e7 q^{5} +9.09825e10 q^{9} -2.85312e11 q^{11} -1.46719e12 q^{12} -3.04727e13 q^{15} +1.75922e13 q^{16} +3.65380e14 q^{20} +1.64432e14 q^{23} +5.20456e15 q^{25} -2.08489e16 q^{27} +3.46774e16 q^{31} +9.98034e16 q^{33} +3.81608e17 q^{36} -2.15124e17 q^{37} -1.19668e18 q^{44} +7.92579e18 q^{45} +5.61067e17 q^{47} -6.15383e18 q^{48} +3.90982e18 q^{49} +1.84780e19 q^{53} -2.48545e19 q^{55} +4.51732e18 q^{59} -1.27812e20 q^{60} +7.37870e19 q^{64} -5.68135e19 q^{67} -5.75191e19 q^{69} +3.23448e20 q^{71} -1.82058e21 q^{75} +1.53252e21 q^{80} +4.43791e21 q^{81} +3.12640e20 q^{89} +6.89677e20 q^{92} -1.21303e22 q^{93} +8.70444e21 q^{97} -2.59584e22 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −349805. −1.97466 −0.987330 − 0.158682i \(-0.949276\pi\)
−0.987330 + 0.158682i \(0.949276\pi\)
\(4\) 4.19430e6 1.00000
\(5\) 8.71134e7 1.78408 0.892041 − 0.451954i \(-0.149273\pi\)
0.892041 + 0.451954i \(0.149273\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.09825e10 2.89928
\(10\) 0 0
\(11\) −2.85312e11 −1.00000
\(12\) −1.46719e12 −1.97466
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −3.04727e13 −3.52296
\(16\) 1.75922e13 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 3.65380e14 1.78408
\(21\) 0 0
\(22\) 0 0
\(23\) 1.64432e14 0.172576 0.0862879 − 0.996270i \(-0.472500\pi\)
0.0862879 + 0.996270i \(0.472500\pi\)
\(24\) 0 0
\(25\) 5.20456e15 2.18295
\(26\) 0 0
\(27\) −2.08489e16 −3.75043
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 3.46774e16 1.36480 0.682398 − 0.730980i \(-0.260937\pi\)
0.682398 + 0.730980i \(0.260937\pi\)
\(32\) 0 0
\(33\) 9.98034e16 1.97466
\(34\) 0 0
\(35\) 0 0
\(36\) 3.81608e17 2.89928
\(37\) −2.15124e17 −1.20912 −0.604560 − 0.796560i \(-0.706651\pi\)
−0.604560 + 0.796560i \(0.706651\pi\)
\(38\) 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 \(\pi\)
\(44\) −1.19668e18 −1.00000
\(45\) 7.92579e18 5.17255
\(46\) 0 0
\(47\) 5.61067e17 0.226954 0.113477 − 0.993541i \(-0.463801\pi\)
0.113477 + 0.993541i \(0.463801\pi\)
\(48\) −6.15383e18 −1.97466
\(49\) 3.90982e18 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.84780e19 1.99352 0.996758 − 0.0804641i \(-0.0256403\pi\)
0.996758 + 0.0804641i \(0.0256403\pi\)
\(54\) 0 0
\(55\) −2.48545e19 −1.78408
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.51732e18 0.149799 0.0748995 − 0.997191i \(-0.476136\pi\)
0.0748995 + 0.997191i \(0.476136\pi\)
\(60\) −1.27812e20 −3.52296
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.37870e19 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.68135e19 −0.465188 −0.232594 − 0.972574i \(-0.574721\pi\)
−0.232594 + 0.972574i \(0.574721\pi\)
\(68\) 0 0
\(69\) −5.75191e19 −0.340778
\(70\) 0 0
\(71\) 3.23448e20 1.39947 0.699733 − 0.714404i \(-0.253302\pi\)
0.699733 + 0.714404i \(0.253302\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −1.82058e21 −4.31058
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.53252e21 1.78408
\(81\) 4.43791e21 4.50654
\(82\) 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
\(89\) 3.12640e20 0.112656 0.0563281 − 0.998412i \(-0.482061\pi\)
0.0563281 + 0.998412i \(0.482061\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.89677e20 0.172576
\(93\) −1.21303e22 −2.69501
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.70444e21 1.21689 0.608446 − 0.793596i \(-0.291793\pi\)
0.608446 + 0.793596i \(0.291793\pi\)
\(98\) 0 0
\(99\) −2.59584e22 −2.89928
\(100\) 2.18295e22 2.18295
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 9.45079e21 0.682745 0.341373 − 0.939928i \(-0.389108\pi\)
0.341373 + 0.939928i \(0.389108\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −8.74465e22 −3.75043
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 7.52514e22 2.38760
\(112\) 0 0
\(113\) −7.31879e22 −1.90799 −0.953996 − 0.299819i \(-0.903074\pi\)
−0.953996 + 0.299819i \(0.903074\pi\)
\(114\) 0 0
\(115\) 1.43242e22 0.307889
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.14027e22 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.45448e23 1.36480
\(125\) 2.45692e23 2.11048
\(126\) 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 \(\pi\)
\(132\) 4.18606e23 1.97466
\(133\) 0 0
\(134\) 0 0
\(135\) −1.81622e24 −6.69108
\(136\) 0 0
\(137\) 6.06909e23 1.90194 0.950970 − 0.309282i \(-0.100089\pi\)
0.950970 + 0.309282i \(0.100089\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −1.96264e23 −0.448157
\(142\) 0 0
\(143\) 0 0
\(144\) 1.60058e24 2.89928
\(145\) 0 0
\(146\) 0 0
\(147\) −1.36767e24 −1.97466
\(148\) −9.02294e23 −1.20912
\(149\) 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
\(155\) 3.02087e24 2.43491
\(156\) 0 0
\(157\) −2.58213e24 −1.80751 −0.903756 − 0.428048i \(-0.859201\pi\)
−0.903756 + 0.428048i \(0.859201\pi\)
\(158\) 0 0
\(159\) −6.46368e24 −3.93651
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.97688e24 1.37943 0.689713 − 0.724083i \(-0.257737\pi\)
0.689713 + 0.724083i \(0.257737\pi\)
\(164\) 0 0
\(165\) 8.69422e24 3.52296
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 3.21184e24 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
\(176\) −5.01926e24 −1.00000
\(177\) −1.58018e24 −0.295802
\(178\) 0 0
\(179\) −6.50252e24 −1.07572 −0.537859 − 0.843035i \(-0.680767\pi\)
−0.537859 + 0.843035i \(0.680767\pi\)
\(180\) 3.32432e25 5.17255
\(181\) −6.29770e24 −0.921972 −0.460986 − 0.887407i \(-0.652504\pi\)
−0.460986 + 0.887407i \(0.652504\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.87402e25 −2.15717
\(186\) 0 0
\(187\) 0 0
\(188\) 2.35328e24 0.226954
\(189\) 0 0
\(190\) 0 0
\(191\) −2.46478e25 −1.99715 −0.998575 − 0.0533689i \(-0.983004\pi\)
−0.998575 + 0.0533689i \(0.983004\pi\)
\(192\) −2.58111e25 −1.97466
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.63990e25 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −2.14158e24 −0.110497 −0.0552485 − 0.998473i \(-0.517595\pi\)
−0.0552485 + 0.998473i \(0.517595\pi\)
\(200\) 0 0
\(201\) 1.98736e25 0.918588
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.49604e25 0.500345
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 7.75022e25 1.99352
\(213\) −1.13144e26 −2.76347
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.04247e26 −1.78408
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00461e26 −1.48131 −0.740655 − 0.671886i \(-0.765484\pi\)
−0.740655 + 0.671886i \(0.765484\pi\)
\(224\) 0 0
\(225\) 4.73524e26 6.32898
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.58560e26 1.74583 0.872916 − 0.487870i \(-0.162226\pi\)
0.872916 + 0.487870i \(0.162226\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 4.88764e25 0.404905
\(236\) 1.89470e25 0.149799
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −5.36081e26 −3.52296
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −8.98145e26 −5.14846
\(244\) 0 0
\(245\) 3.40598e26 1.78408
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.48850e26 1.80173 0.900865 − 0.434098i \(-0.142933\pi\)
0.900865 + 0.434098i \(0.142933\pi\)
\(252\) 0 0
\(253\) −4.69143e25 −0.172576
\(254\) 0 0
\(255\) 0 0
\(256\) 3.09485e26 1.00000
\(257\) −5.51673e26 −1.70772 −0.853862 − 0.520500i \(-0.825746\pi\)
−0.853862 + 0.520500i \(0.825746\pi\)
\(258\) 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
\(265\) 1.60968e27 3.55660
\(266\) 0 0
\(267\) −1.09363e26 −0.222458
\(268\) −2.38293e26 −0.465188
\(269\) −1.03014e27 −1.93029 −0.965144 − 0.261720i \(-0.915710\pi\)
−0.965144 + 0.261720i \(0.915710\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.48492e27 −2.18295
\(276\) −2.41253e26 −0.340778
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 3.15504e27 3.95693
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.35664e27 1.39947
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.17456e27 1.00000
\(290\) 0 0
\(291\) −3.04486e27 −2.40295
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 3.93519e26 0.267254
\(296\) 0 0
\(297\) 5.94843e27 3.75043
\(298\) 0 0
\(299\) 0 0
\(300\) −7.63607e27 −4.31058
\(301\) 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
\(309\) −3.30593e27 −1.34819
\(310\) 0 0
\(311\) −1.13076e27 −0.429542 −0.214771 − 0.976664i \(-0.568901\pi\)
−0.214771 + 0.976664i \(0.568901\pi\)
\(312\) 0 0
\(313\) −5.62369e27 −1.99083 −0.995413 − 0.0956711i \(-0.969500\pi\)
−0.995413 + 0.0956711i \(0.969500\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.47379e27 −1.99300 −0.996499 − 0.0836021i \(-0.973358\pi\)
−0.996499 + 0.0836021i \(0.973358\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 6.42783e27 1.78408
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.86140e28 4.50654
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.02630e28 1.96411 0.982055 − 0.188597i \(-0.0603940\pi\)
0.982055 + 0.188597i \(0.0603940\pi\)
\(332\) 0 0
\(333\) −1.95725e28 −3.50558
\(334\) 0 0
\(335\) −4.94922e27 −0.829934
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 2.56015e28 3.76764
\(340\) 0 0
\(341\) −9.89387e27 −1.36480
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.01068e27 −0.607977
\(346\) 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
\(353\) −1.21260e28 −1.14340 −0.571699 − 0.820464i \(-0.693715\pi\)
−0.571699 + 0.820464i \(0.693715\pi\)
\(354\) 0 0
\(355\) 2.81766e28 2.49676
\(356\) 1.31131e27 0.112656
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.35700e28 1.00000
\(362\) 0 0
\(363\) −2.84751e28 −1.97466
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.84936e28 1.75154 0.875770 − 0.482729i \(-0.160354\pi\)
0.875770 + 0.482729i \(0.160354\pi\)
\(368\) 2.89272e27 0.172576
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −5.08783e28 −2.69501
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −8.59444e28 −4.16748
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.32258e28 1.86514 0.932571 − 0.360987i \(-0.117560\pi\)
0.932571 + 0.360987i \(0.117560\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.51572e28 −0.582684 −0.291342 − 0.956619i \(-0.594102\pi\)
−0.291342 + 0.956619i \(0.594102\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 3.65091e28 1.21689
\(389\) 4.10646e28 1.33052 0.665260 − 0.746612i \(-0.268321\pi\)
0.665260 + 0.746612i \(0.268321\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.08877e29 −2.89928
\(397\) −3.47086e28 −0.898961 −0.449481 − 0.893290i \(-0.648391\pi\)
−0.449481 + 0.893290i \(0.648391\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 9.15596e28 2.18295
\(401\) −8.40815e28 −1.95035 −0.975174 − 0.221438i \(-0.928925\pi\)
−0.975174 + 0.221438i \(0.928925\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.86602e29 8.04005
\(406\) 0 0
\(407\) 6.13773e28 1.20912
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −2.12300e29 −3.75569
\(412\) 3.96395e28 0.682745
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.87360e28 −0.697422 −0.348711 − 0.937230i \(-0.613381\pi\)
−0.348711 + 0.937230i \(0.613381\pi\)
\(420\) 0 0
\(421\) −9.52394e28 −1.29334 −0.646671 − 0.762769i \(-0.723839\pi\)
−0.646671 + 0.762769i \(0.723839\pi\)
\(422\) 0 0
\(423\) 5.10472e28 0.658004
\(424\) 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 \(\pi\)
\(432\) −3.66777e29 −3.75043
\(433\) 6.26536e28 0.624568 0.312284 − 0.949989i \(-0.398906\pi\)
0.312284 + 0.949989i \(0.398906\pi\)
\(434\) 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
\(441\) 3.55725e29 2.89928
\(442\) 0 0
\(443\) −1.22495e29 −0.949896 −0.474948 − 0.880014i \(-0.657533\pi\)
−0.474948 + 0.880014i \(0.657533\pi\)
\(444\) 3.15627e29 2.38760
\(445\) 2.72352e28 0.200988
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.50184e29 −1.67321 −0.836604 − 0.547808i \(-0.815462\pi\)
−0.836604 + 0.547808i \(0.815462\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −3.06972e29 −1.90799
\(453\) 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
\(460\) 6.00801e28 0.307889
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.69349e29 −0.807964 −0.403982 − 0.914767i \(-0.632374\pi\)
−0.403982 + 0.914767i \(0.632374\pi\)
\(464\) 0 0
\(465\) −1.05671e30 −4.80812
\(466\) 0 0
\(467\) −1.95826e28 −0.0849935 −0.0424968 − 0.999097i \(-0.513531\pi\)
−0.0424968 + 0.999097i \(0.513531\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 9.03240e29 3.56922
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.68117e30 5.77976
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 3.41428e29 1.00000
\(485\) 7.58273e29 2.17103
\(486\) 0 0
\(487\) 2.09139e29 0.572290 0.286145 − 0.958186i \(-0.407626\pi\)
0.286145 + 0.958186i \(0.407626\pi\)
\(488\) 0 0
\(489\) −1.04133e30 −2.72389
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.26132e30 −5.17255
\(496\) 6.10051e29 1.36480
\(497\) 0 0
\(498\) 0 0
\(499\) −7.11021e29 −1.48859 −0.744297 − 0.667849i \(-0.767215\pi\)
−0.744297 + 0.667849i \(0.767215\pi\)
\(500\) 1.03051e30 2.11048
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.12352e30 −1.97466
\(508\) 0 0
\(509\) −5.67057e28 −0.0954402 −0.0477201 − 0.998861i \(-0.515196\pi\)
−0.0477201 + 0.998861i \(0.515196\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.23291e29 1.21807
\(516\) 0 0
\(517\) −1.60079e29 −0.226954
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.20423e29 −0.938367 −0.469183 − 0.883101i \(-0.655452\pi\)
−0.469183 + 0.883101i \(0.655452\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 1.75576e30 1.97466
\(529\) −8.80809e29 −0.970218
\(530\) 0 0
\(531\) 4.10997e29 0.434309
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.27461e30 2.12418
\(538\) 0 0
\(539\) −1.11552e30 −1.00000
\(540\) −7.61776e30 −6.69108
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 2.20297e30 1.82058
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 2.54556e30 1.90194
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.55540e30 4.25967
\(556\) 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 \(\pi\)
\(564\) −8.23191e29 −0.448157
\(565\) −6.37565e30 −3.40402
\(566\) 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
\(573\) 8.62191e30 3.94369
\(574\) 0 0
\(575\) 8.55795e29 0.376724
\(576\) 6.71332e30 2.89928
\(577\) −5.78022e29 −0.244912 −0.122456 − 0.992474i \(-0.539077\pi\)
−0.122456 + 0.992474i \(0.539077\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −5.27198e30 −1.99352
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.66095e30 −1.98550 −0.992750 − 0.120197i \(-0.961648\pi\)
−0.992750 + 0.120197i \(0.961648\pi\)
\(588\) −5.73644e30 −1.97466
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −3.78450e30 −1.20912
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 7.49135e29 0.218194
\(598\) 0 0
\(599\) −5.58415e30 −1.56770 −0.783849 − 0.620952i \(-0.786746\pi\)
−0.783849 + 0.620952i \(0.786746\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −5.16903e30 −1.34871
\(604\) 0 0
\(605\) 7.09127e30 1.78408
\(606\) 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
\(617\) 9.72283e30 1.97086 0.985430 − 0.170080i \(-0.0544026\pi\)
0.985430 + 0.170080i \(0.0544026\pi\)
\(618\) 0 0
\(619\) −9.24090e30 −1.80766 −0.903831 − 0.427890i \(-0.859257\pi\)
−0.903831 + 0.427890i \(0.859257\pi\)
\(620\) 1.26704e31 2.43491
\(621\) −3.42822e30 −0.647233
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 8.99445e30 1.58232
\(626\) 0 0
\(627\) 0 0
\(628\) −1.08302e31 −1.80751
\(629\) 0 0
\(630\) 0 0
\(631\) 1.26008e31 1.99561 0.997805 − 0.0662167i \(-0.0210929\pi\)
0.997805 + 0.0662167i \(0.0210929\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −2.71107e31 −3.93651
\(637\) 0 0
\(638\) 0 0
\(639\) 2.94281e31 4.05744
\(640\) 0 0
\(641\) −7.41968e30 −0.988432 −0.494216 − 0.869339i \(-0.664545\pi\)
−0.494216 + 0.869339i \(0.664545\pi\)
\(642\) 0 0
\(643\) 4.48163e30 0.576920 0.288460 − 0.957492i \(-0.406857\pi\)
0.288460 + 0.957492i \(0.406857\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.60829e31 −1.93382 −0.966912 − 0.255109i \(-0.917889\pi\)
−0.966912 + 0.255109i \(0.917889\pi\)
\(648\) 0 0
\(649\) −1.28884e30 −0.149799
\(650\) 0 0
\(651\) 0 0
\(652\) 1.24860e31 1.37943
\(653\) 2.60752e30 0.283259 0.141629 − 0.989920i \(-0.454766\pi\)
0.141629 + 0.989920i \(0.454766\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 3.64662e31 3.52296
\(661\) −1.66277e31 −1.57986 −0.789928 − 0.613200i \(-0.789882\pi\)
−0.789928 + 0.613200i \(0.789882\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.51419e31 2.92508
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.08509e32 −8.18700
\(676\) 1.34714e31 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
\(683\) 1.44752e31 0.959392 0.479696 − 0.877435i \(-0.340747\pi\)
0.479696 + 0.877435i \(0.340747\pi\)
\(684\) 0 0
\(685\) 5.28699e31 3.39322
\(686\) 0 0
\(687\) −5.54650e31 −3.44742
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.96484e31 −1.14569 −0.572846 − 0.819663i \(-0.694161\pi\)
−0.572846 + 0.819663i \(0.694161\pi\)
\(692\) 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
\(704\) −2.10523e31 −1.00000
\(705\) −1.70972e31 −0.799549
\(706\) 0 0
\(707\) 0 0
\(708\) −6.62776e30 −0.295802
\(709\) 7.79115e30 0.342368 0.171184 − 0.985239i \(-0.445241\pi\)
0.171184 + 0.985239i \(0.445241\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.70207e30 0.235531
\(714\) 0 0
\(715\) 0 0
\(716\) −2.72736e31 −1.07572
\(717\) 0 0
\(718\) 0 0
\(719\) 4.28049e31 1.61241 0.806206 − 0.591636i \(-0.201518\pi\)
0.806206 + 0.591636i \(0.201518\pi\)
\(720\) 1.39432e32 5.17255
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −2.64145e31 −0.921972
\(725\) 0 0
\(726\) 0 0
\(727\) −3.53179e31 −1.17792 −0.588961 − 0.808162i \(-0.700463\pi\)
−0.588961 + 0.808162i \(0.700463\pi\)
\(728\) 0 0
\(729\) 1.74909e32 5.65991
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −1.19143e32 −3.52296
\(736\) 0 0
\(737\) 1.62095e31 0.465188
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −7.86019e31 −2.15717
\(741\) 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
\(751\) 8.52692e31 1.98954 0.994770 − 0.102138i \(-0.0325682\pi\)
0.994770 + 0.102138i \(0.0325682\pi\)
\(752\) 9.87039e30 0.226954
\(753\) −1.57010e32 −3.55780
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 4.55726e31 0.974203 0.487102 − 0.873345i \(-0.338054\pi\)
0.487102 + 0.873345i \(0.338054\pi\)
\(758\) 0 0
\(759\) 1.64109e31 0.340778
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.03380e32 −1.99715
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.08259e32 −1.97466
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.92978e32 3.37217
\(772\) 0 0
\(773\) 6.52749e31 1.10859 0.554296 − 0.832320i \(-0.312988\pi\)
0.554296 + 0.832320i \(0.312988\pi\)
\(774\) 0 0
\(775\) 1.80481e32 2.97928
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −9.22834e31 −1.39947
\(782\) 0 0
\(783\) 0 0
\(784\) 6.87823e31 1.00000
\(785\) −2.24938e32 −3.22475
\(786\) 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
\(795\) −5.63073e32 −7.02306
\(796\) −8.98243e30 −0.110497
\(797\) −6.43943e31 −0.781278 −0.390639 − 0.920544i \(-0.627746\pi\)
−0.390639 + 0.920544i \(0.627746\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.84448e31 0.326622
\(802\) 0 0
\(803\) 0 0
\(804\) 8.33561e31 0.918588
\(805\) 0 0
\(806\) 0 0
\(807\) 3.60349e32 3.81166
\(808\) 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
\(815\) 2.59326e32 2.46101
\(816\) 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
\(823\) 2.02543e32 1.72631 0.863157 − 0.504936i \(-0.168484\pi\)
0.863157 + 0.504936i \(0.168484\pi\)
\(824\) 0 0
\(825\) 5.19433e32 4.31058
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 6.27485e31 0.500345
\(829\) −1.46712e32 −1.15442 −0.577211 − 0.816595i \(-0.695859\pi\)
−0.577211 + 0.816595i \(0.695859\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −7.22985e32 −5.11858
\(838\) 0 0
\(839\) −2.17941e32 −1.50299 −0.751497 − 0.659737i \(-0.770668\pi\)
−0.751497 + 0.659737i \(0.770668\pi\)
\(840\) 0 0
\(841\) 1.48852e32 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.79794e32 1.78408
\(846\) 0 0
\(847\) 0 0
\(848\) 3.25068e32 1.99352
\(849\) 0 0
\(850\) 0 0
\(851\) −3.53732e31 −0.208665
\(852\) −4.74559e32 −2.76347
\(853\) 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
\(859\) −3.58239e32 −1.90655 −0.953275 − 0.302103i \(-0.902311\pi\)
−0.953275 + 0.302103i \(0.902311\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.43750e32 0.726925 0.363462 − 0.931609i \(-0.381594\pi\)
0.363462 + 0.931609i \(0.381594\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.10868e32 −1.97466
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 7.91951e32 3.52811
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −4.37244e32 −1.78408
\(881\) 4.95584e32 1.99702 0.998509 − 0.0545862i \(-0.0173840\pi\)
0.998509 + 0.0545862i \(0.0173840\pi\)
\(882\) 0 0
\(883\) 1.70851e32 0.671505 0.335752 − 0.941950i \(-0.391009\pi\)
0.335752 + 0.941950i \(0.391009\pi\)
\(884\) 0 0
\(885\) −1.37655e32 −0.527735
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −1.26619e33 −4.50654
\(892\) −4.21366e32 −1.48131
\(893\) 0 0
\(894\) 0 0
\(895\) −5.66457e32 −1.91917
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.98610e33 6.32898
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.48614e32 −1.64487
\(906\) 0 0
\(907\) 6.68306e32 1.95567 0.977834 − 0.209380i \(-0.0671445\pi\)
0.977834 + 0.209380i \(0.0671445\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.39669e32 −1.50462 −0.752308 − 0.658812i \(-0.771059\pi\)
−0.752308 + 0.658812i \(0.771059\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 6.65048e32 1.74583
\(917\) 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
\(925\) −1.11962e33 −2.63945
\(926\) 0 0
\(927\) 8.59857e32 1.97947
\(928\) 0 0
\(929\) −2.83468e32 −0.637281 −0.318641 − 0.947876i \(-0.603226\pi\)
−0.318641 + 0.947876i \(0.603226\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 3.95545e32 0.848200
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 1.96719e33 3.93120
\(940\) 2.05003e32 0.404905
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 7.94695e31 0.149799
\(945\) 0 0
\(946\) 0 0
\(947\) −6.69587e32 −1.21887 −0.609435 − 0.792836i \(-0.708604\pi\)
−0.609435 + 0.792836i \(0.708604\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.26457e33 3.93549
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −2.14715e33 −3.56308
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −2.24849e33 −3.52296
\(961\) 5.56932e32 0.862671
\(962\) 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
\(971\) −6.55643e32 −0.906267 −0.453133 − 0.891443i \(-0.649694\pi\)
−0.453133 + 0.891443i \(0.649694\pi\)
\(972\) −3.76709e33 −5.14846
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.42585e33 1.84176 0.920880 − 0.389847i \(-0.127472\pi\)
0.920880 + 0.389847i \(0.127472\pi\)
\(978\) 0 0
\(979\) −8.91999e31 −0.112656
\(980\) 1.42857e33 1.78408
\(981\) 0 0
\(982\) 0 0
\(983\) 7.59185e32 0.916767 0.458384 − 0.888754i \(-0.348429\pi\)
0.458384 + 0.888754i \(0.348429\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.22390e33 −1.35188 −0.675938 − 0.736959i \(-0.736261\pi\)
−0.675938 + 0.736959i \(0.736261\pi\)
\(992\) 0 0
\(993\) −3.59004e33 −3.87845
\(994\) 0 0
\(995\) −1.86560e32 −0.197136
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 4.48509e33 4.53472
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 11.23.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.23.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.23.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.23.b.a.10.1 ✓ 1 11.10 odd 2 CM