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Properties

Label 11.33.b.a.10.1
Level $11$
Weight $33$
Character 11.10
Self dual yes
Analytic conductor $71.353$
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] = [11,33,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, 33, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 33 \)
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: \(71.3533206565\)
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)
 
\(f(q)\) \(=\) \(q-8.59688e7 q^{3} +4.29497e9 q^{4} -9.72809e9 q^{5} +5.53762e15 q^{9} +O(q^{10})\) \(q-8.59688e7 q^{3} +4.29497e9 q^{4} -9.72809e9 q^{5} +5.53762e15 q^{9} +4.59497e16 q^{11} -3.69233e17 q^{12} +8.36313e17 q^{15} +1.84467e19 q^{16} -4.17818e19 q^{20} +3.65663e21 q^{23} -2.31884e22 q^{25} -3.16761e23 q^{27} -1.00646e24 q^{31} -3.95024e24 q^{33} +2.37839e25 q^{36} +1.77267e25 q^{37} +1.97353e26 q^{44} -5.38705e25 q^{45} -1.02057e27 q^{47} -1.58585e27 q^{48} +1.10443e27 q^{49} +4.02280e27 q^{53} -4.47003e26 q^{55} +3.35210e28 q^{59} +3.59194e27 q^{60} +7.92282e28 q^{64} -1.53527e29 q^{67} -3.14356e29 q^{69} +7.08325e29 q^{71} +1.99348e30 q^{75} -1.79452e29 q^{80} +1.69702e31 q^{81} -3.03531e31 q^{89} +1.57051e31 q^{92} +8.65241e31 q^{93} -4.02478e31 q^{97} +2.54452e32 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −8.59688e7 −1.99711 −0.998553 − 0.0537833i \(-0.982872\pi\)
−0.998553 + 0.0537833i \(0.982872\pi\)
\(4\) 4.29497e9 1.00000
\(5\) −9.72809e9 −0.0637540 −0.0318770 − 0.999492i \(-0.510148\pi\)
−0.0318770 + 0.999492i \(0.510148\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 5.53762e15 2.98843
\(10\) 0 0
\(11\) 4.59497e16 1.00000
\(12\) −3.69233e17 −1.99711
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 8.36313e17 0.127323
\(16\) 1.84467e19 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −4.17818e19 −0.0637540
\(21\) 0 0
\(22\) 0 0
\(23\) 3.65663e21 0.596260 0.298130 − 0.954525i \(-0.403637\pi\)
0.298130 + 0.954525i \(0.403637\pi\)
\(24\) 0 0
\(25\) −2.31884e22 −0.995935
\(26\) 0 0
\(27\) −3.16761e23 −3.97110
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.00646e24 −1.38360 −0.691798 − 0.722091i \(-0.743181\pi\)
−0.691798 + 0.722091i \(0.743181\pi\)
\(32\) 0 0
\(33\) −3.95024e24 −1.99711
\(34\) 0 0
\(35\) 0 0
\(36\) 2.37839e25 2.98843
\(37\) 1.77267e25 1.43682 0.718408 − 0.695622i \(-0.244871\pi\)
0.718408 + 0.695622i \(0.244871\pi\)
\(38\) 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 \(\pi\)
\(44\) 1.97353e26 1.00000
\(45\) −5.38705e25 −0.190524
\(46\) 0 0
\(47\) −1.02057e27 −1.80001 −0.900006 − 0.435878i \(-0.856438\pi\)
−0.900006 + 0.435878i \(0.856438\pi\)
\(48\) −1.58585e27 −1.99711
\(49\) 1.10443e27 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.02280e27 1.03780 0.518901 − 0.854835i \(-0.326341\pi\)
0.518901 + 0.854835i \(0.326341\pi\)
\(54\) 0 0
\(55\) −4.47003e26 −0.0637540
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.35210e28 1.55484 0.777418 − 0.628984i \(-0.216529\pi\)
0.777418 + 0.628984i \(0.216529\pi\)
\(60\) 3.59194e27 0.127323
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.92282e28 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.53527e29 −0.931082 −0.465541 − 0.885026i \(-0.654140\pi\)
−0.465541 + 0.885026i \(0.654140\pi\)
\(68\) 0 0
\(69\) −3.14356e29 −1.19079
\(70\) 0 0
\(71\) 7.08325e29 1.69863 0.849316 − 0.527885i \(-0.177015\pi\)
0.849316 + 0.527885i \(0.177015\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.99348e30 1.98899
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −1.79452e29 −0.0637540
\(81\) 1.69702e31 4.94228
\(82\) 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
\(89\) −3.03531e31 −1.95868 −0.979340 − 0.202220i \(-0.935184\pi\)
−0.979340 + 0.202220i \(0.935184\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.57051e31 0.596260
\(93\) 8.65241e31 2.76319
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.02478e31 −0.655230 −0.327615 − 0.944811i \(-0.606245\pi\)
−0.327615 + 0.944811i \(0.606245\pi\)
\(98\) 0 0
\(99\) 2.54452e32 2.98843
\(100\) −9.95935e31 −0.995935
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 3.20275e32 1.99585 0.997923 − 0.0644239i \(-0.0205210\pi\)
0.997923 + 0.0644239i \(0.0205210\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.36048e33 −3.97110
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −1.52395e33 −2.86947
\(112\) 0 0
\(113\) 4.17860e32 0.591256 0.295628 − 0.955303i \(-0.404471\pi\)
0.295628 + 0.955303i \(0.404471\pi\)
\(114\) 0 0
\(115\) −3.55721e31 −0.0380140
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.11138e33 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −4.32271e33 −1.38360
\(125\) 4.52079e32 0.127249
\(126\) 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 \(\pi\)
\(132\) −1.69662e34 −1.99711
\(133\) 0 0
\(134\) 0 0
\(135\) 3.08148e33 0.253174
\(136\) 0 0
\(137\) −1.73955e34 −1.12955 −0.564777 − 0.825243i \(-0.691038\pi\)
−0.564777 + 0.825243i \(0.691038\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 8.77369e34 3.59481
\(142\) 0 0
\(143\) 0 0
\(144\) 1.02151e35 2.98843
\(145\) 0 0
\(146\) 0 0
\(147\) −9.49464e34 −1.99711
\(148\) 7.61357e34 1.43682
\(149\) 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
\(155\) 9.79093e33 0.0882098
\(156\) 0 0
\(157\) 2.39303e35 1.75612 0.878059 − 0.478552i \(-0.158838\pi\)
0.878059 + 0.478552i \(0.158838\pi\)
\(158\) 0 0
\(159\) −3.45835e35 −2.07260
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.71268e35 −1.89787 −0.948935 − 0.315472i \(-0.897837\pi\)
−0.948935 + 0.315472i \(0.897837\pi\)
\(164\) 0 0
\(165\) 3.84283e34 0.127323
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.42779e35 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
\(176\) 8.47623e35 1.00000
\(177\) −2.88176e36 −3.10517
\(178\) 0 0
\(179\) −1.83115e36 −1.64845 −0.824223 − 0.566265i \(-0.808388\pi\)
−0.824223 + 0.566265i \(0.808388\pi\)
\(180\) −2.31372e35 −0.190524
\(181\) 2.39486e36 1.80478 0.902389 − 0.430923i \(-0.141812\pi\)
0.902389 + 0.430923i \(0.141812\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.72447e35 −0.0916028
\(186\) 0 0
\(187\) 0 0
\(188\) −4.38330e36 −1.80001
\(189\) 0 0
\(190\) 0 0
\(191\) 5.52553e36 1.76133 0.880664 − 0.473741i \(-0.157097\pi\)
0.880664 + 0.473741i \(0.157097\pi\)
\(192\) −6.81115e36 −1.99711
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 4.74348e36 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.20580e37 1.99354 0.996769 − 0.0803157i \(-0.0255929\pi\)
0.996769 + 0.0803157i \(0.0255929\pi\)
\(200\) 0 0
\(201\) 1.31985e37 1.85947
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.02490e37 1.78188
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.72778e37 1.03780
\(213\) −6.08939e37 −3.39235
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.91986e36 −0.0637540
\(221\) 0 0
\(222\) 0 0
\(223\) 3.58062e37 0.957371 0.478685 − 0.877987i \(-0.341113\pi\)
0.478685 + 0.877987i \(0.341113\pi\)
\(224\) 0 0
\(225\) −1.28409e38 −2.97628
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 6.44499e37 1.12682 0.563410 − 0.826178i \(-0.309489\pi\)
0.563410 + 0.826178i \(0.309489\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 9.92816e36 0.114758
\(236\) 1.43972e38 1.55484
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 1.54272e37 0.127323
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −8.71947e38 −5.89915
\(244\) 0 0
\(245\) −1.07440e37 −0.0637540
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.68551e38 0.679126 0.339563 − 0.940583i \(-0.389721\pi\)
0.339563 + 0.940583i \(0.389721\pi\)
\(252\) 0 0
\(253\) 1.68021e38 0.596260
\(254\) 0 0
\(255\) 0 0
\(256\) 3.40282e38 1.00000
\(257\) 7.06099e38 1.94956 0.974778 − 0.223176i \(-0.0716425\pi\)
0.974778 + 0.223176i \(0.0716425\pi\)
\(258\) 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
\(265\) −3.91341e37 −0.0661640
\(266\) 0 0
\(267\) 2.60942e39 3.91169
\(268\) −6.59393e38 −0.931082
\(269\) 1.09256e39 1.45349 0.726743 − 0.686910i \(-0.241033\pi\)
0.726743 + 0.686910i \(0.241033\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.06550e39 −0.995935
\(276\) −1.35015e39 −1.19079
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −5.57339e39 −4.13478
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 3.04223e39 1.69863
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.36791e39 1.00000
\(290\) 0 0
\(291\) 3.46005e39 1.30856
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −3.26095e38 −0.0991271
\(296\) 0 0
\(297\) −1.45551e40 −3.97110
\(298\) 0 0
\(299\) 0 0
\(300\) 8.56194e39 1.98899
\(301\) 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
\(309\) −2.75336e40 −3.98591
\(310\) 0 0
\(311\) 1.73563e39 0.226616 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(312\) 0 0
\(313\) 1.28648e40 1.51598 0.757990 − 0.652266i \(-0.226181\pi\)
0.757990 + 0.652266i \(0.226181\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.54263e40 −1.48359 −0.741795 − 0.670626i \(-0.766025\pi\)
−0.741795 + 0.670626i \(0.766025\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.70739e38 −0.0637540
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 7.28866e40 4.94228
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.51449e40 −1.69283 −0.846417 − 0.532520i \(-0.821245\pi\)
−0.846417 + 0.532520i \(0.821245\pi\)
\(332\) 0 0
\(333\) 9.81639e40 4.29382
\(334\) 0 0
\(335\) 1.49352e39 0.0593602
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −3.59229e40 −1.18080
\(340\) 0 0
\(341\) −4.62465e40 −1.38360
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 3.05809e39 0.0759179
\(346\) 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
\(353\) 1.32708e40 0.228297 0.114149 − 0.993464i \(-0.463586\pi\)
0.114149 + 0.993464i \(0.463586\pi\)
\(354\) 0 0
\(355\) −6.89065e39 −0.108295
\(356\) −1.30366e41 −1.95868
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 8.31984e40 1.00000
\(362\) 0 0
\(363\) −1.81513e41 −1.99711
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.13672e41 −1.04954 −0.524768 − 0.851245i \(-0.675848\pi\)
−0.524768 + 0.851245i \(0.675848\pi\)
\(368\) 6.74530e40 0.596260
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 3.71618e41 2.76319
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −3.88647e40 −0.254129
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.44104e41 −1.89872 −0.949362 − 0.314183i \(-0.898269\pi\)
−0.949362 + 0.314183i \(0.898269\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.90296e41 −1.82060 −0.910299 − 0.413952i \(-0.864148\pi\)
−0.910299 + 0.413952i \(0.864148\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −1.72863e41 −0.655230
\(389\) 5.47898e41 1.99300 0.996498 − 0.0836198i \(-0.0266481\pi\)
0.996498 + 0.0836198i \(0.0266481\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 1.09286e42 2.98843
\(397\) −2.73421e40 −0.0718098 −0.0359049 − 0.999355i \(-0.511431\pi\)
−0.0359049 + 0.999355i \(0.511431\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −4.27751e41 −0.995935
\(401\) −3.39241e41 −0.758923 −0.379462 − 0.925207i \(-0.623891\pi\)
−0.379462 + 0.925207i \(0.623891\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.65088e41 −0.315090
\(406\) 0 0
\(407\) 8.14538e41 1.43682
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 1.49547e42 2.25584
\(412\) 1.37557e42 1.99585
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.61596e41 0.733109 0.366554 − 0.930397i \(-0.380537\pi\)
0.366554 + 0.930397i \(0.380537\pi\)
\(420\) 0 0
\(421\) −4.52941e40 −0.0465080 −0.0232540 − 0.999730i \(-0.507403\pi\)
−0.0232540 + 0.999730i \(0.507403\pi\)
\(422\) 0 0
\(423\) −5.65151e42 −5.37921
\(424\) 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 \(\pi\)
\(432\) −5.84320e42 −3.97110
\(433\) 2.73368e42 1.79036 0.895182 − 0.445702i \(-0.147046\pi\)
0.895182 + 0.445702i \(0.147046\pi\)
\(434\) 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
\(441\) 6.11590e42 2.98843
\(442\) 0 0
\(443\) −2.35727e42 −1.07140 −0.535699 − 0.844409i \(-0.679952\pi\)
−0.535699 + 0.844409i \(0.679952\pi\)
\(444\) −6.54530e42 −2.86947
\(445\) 2.95278e41 0.124874
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.21487e42 1.91118 0.955592 − 0.294694i \(-0.0952178\pi\)
0.955592 + 0.294694i \(0.0952178\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 1.79470e42 0.591256
\(453\) 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
\(460\) −1.52781e41 −0.0380140
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 7.33659e42 1.64512 0.822561 − 0.568676i \(-0.192544\pi\)
0.822561 + 0.568676i \(0.192544\pi\)
\(464\) 0 0
\(465\) −8.41715e41 −0.176164
\(466\) 0 0
\(467\) −6.21189e42 −1.21382 −0.606910 − 0.794771i \(-0.707591\pi\)
−0.606910 + 0.794771i \(0.707591\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −2.05726e43 −3.50715
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.22767e43 3.10140
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 9.06830e42 1.00000
\(485\) 3.91534e41 0.0417736
\(486\) 0 0
\(487\) 1.09301e43 1.09184 0.545920 − 0.837837i \(-0.316180\pi\)
0.545920 + 0.837837i \(0.316180\pi\)
\(488\) 0 0
\(489\) 4.05144e43 3.79025
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.47533e42 −0.190524
\(496\) −1.85659e43 −1.38360
\(497\) 0 0
\(498\) 0 0
\(499\) −1.75396e43 −1.18689 −0.593444 − 0.804875i \(-0.702232\pi\)
−0.593444 + 0.804875i \(0.702232\pi\)
\(500\) 1.94166e42 0.127249
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.80652e43 −1.99711
\(508\) 0 0
\(509\) −2.86514e43 −1.41144 −0.705718 − 0.708493i \(-0.749375\pi\)
−0.705718 + 0.708493i \(0.749375\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.11566e42 −0.127243
\(516\) 0 0
\(517\) −4.68947e43 −1.80001
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.63998e43 −0.556460 −0.278230 − 0.960514i \(-0.589748\pi\)
−0.278230 + 0.960514i \(0.589748\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −7.28692e43 −1.99711
\(529\) −2.42380e43 −0.644474
\(530\) 0 0
\(531\) 1.85627e44 4.64652
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.57422e44 3.29212
\(538\) 0 0
\(539\) 5.07482e43 1.00000
\(540\) 1.32348e43 0.253174
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −2.05884e44 −3.60433
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −7.47130e43 −1.12955
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.48251e43 0.182940
\(556\) 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 \(\pi\)
\(564\) 3.76827e44 3.59481
\(565\) −4.06498e42 −0.0376950
\(566\) 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
\(573\) −4.75024e44 −3.51756
\(574\) 0 0
\(575\) −8.47915e43 −0.593837
\(576\) 4.38735e44 2.98843
\(577\) 2.78914e44 1.84781 0.923905 − 0.382622i \(-0.124979\pi\)
0.923905 + 0.382622i \(0.124979\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.84846e44 1.03780
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.24278e44 −0.625439 −0.312719 − 0.949846i \(-0.601240\pi\)
−0.312719 + 0.949846i \(0.601240\pi\)
\(588\) −4.07792e44 −1.99711
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 3.27000e44 1.43682
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.03661e45 −3.98131
\(598\) 0 0
\(599\) 1.41487e44 0.515097 0.257548 − 0.966265i \(-0.417085\pi\)
0.257548 + 0.966265i \(0.417085\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −8.50174e44 −2.78247
\(604\) 0 0
\(605\) −2.05397e43 −0.0637540
\(606\) 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
\(617\) 6.02022e44 1.36474 0.682370 − 0.731007i \(-0.260949\pi\)
0.682370 + 0.731007i \(0.260949\pi\)
\(618\) 0 0
\(619\) 7.43513e44 1.60044 0.800218 − 0.599709i \(-0.204717\pi\)
0.800218 + 0.599709i \(0.204717\pi\)
\(620\) 4.20517e43 0.0882098
\(621\) −1.15828e45 −2.36781
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.35500e44 0.987823
\(626\) 0 0
\(627\) 0 0
\(628\) 1.02780e45 1.75612
\(629\) 0 0
\(630\) 0 0
\(631\) −2.99285e44 −0.473823 −0.236911 − 0.971531i \(-0.576135\pi\)
−0.236911 + 0.971531i \(0.576135\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −1.48535e45 −2.07260
\(637\) 0 0
\(638\) 0 0
\(639\) 3.92244e45 5.07624
\(640\) 0 0
\(641\) −1.45119e45 −1.78647 −0.893236 − 0.449587i \(-0.851571\pi\)
−0.893236 + 0.449587i \(0.851571\pi\)
\(642\) 0 0
\(643\) 4.76563e44 0.558142 0.279071 − 0.960270i \(-0.409973\pi\)
0.279071 + 0.960270i \(0.409973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.10244e45 1.16919 0.584596 − 0.811325i \(-0.301253\pi\)
0.584596 + 0.811325i \(0.301253\pi\)
\(648\) 0 0
\(649\) 1.54028e45 1.55484
\(650\) 0 0
\(651\) 0 0
\(652\) −2.02408e45 −1.89787
\(653\) −1.06197e45 −0.971633 −0.485816 − 0.874061i \(-0.661478\pi\)
−0.485816 + 0.874061i \(0.661478\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 1.65048e44 0.127323
\(661\) −8.48245e44 −0.638702 −0.319351 − 0.947637i \(-0.603465\pi\)
−0.319351 + 0.947637i \(0.603465\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.07822e45 −1.91197
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 7.34518e45 3.95496
\(676\) 1.90172e45 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
\(683\) −4.05374e45 −1.80769 −0.903846 − 0.427858i \(-0.859268\pi\)
−0.903846 + 0.427858i \(0.859268\pi\)
\(684\) 0 0
\(685\) 1.69225e44 0.0720136
\(686\) 0 0
\(687\) −5.54068e45 −2.25038
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.39326e45 −0.885810 −0.442905 − 0.896569i \(-0.646052\pi\)
−0.442905 + 0.896569i \(0.646052\pi\)
\(692\) 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
\(704\) 3.64051e45 1.00000
\(705\) −8.53512e44 −0.229184
\(706\) 0 0
\(707\) 0 0
\(708\) −1.23771e46 −3.10517
\(709\) 5.11845e45 1.25545 0.627724 − 0.778436i \(-0.283987\pi\)
0.627724 + 0.778436i \(0.283987\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.68025e45 −0.824983
\(714\) 0 0
\(715\) 0 0
\(716\) −7.86472e45 −1.64845
\(717\) 0 0
\(718\) 0 0
\(719\) −8.21194e45 −1.60984 −0.804922 − 0.593381i \(-0.797793\pi\)
−0.804922 + 0.593381i \(0.797793\pi\)
\(720\) −9.93735e44 −0.190524
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 1.02859e46 1.80478
\(725\) 0 0
\(726\) 0 0
\(727\) 1.14612e46 1.88226 0.941132 − 0.338040i \(-0.109764\pi\)
0.941132 + 0.338040i \(0.109764\pi\)
\(728\) 0 0
\(729\) 4.35141e46 6.83895
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 9.23647e44 0.127323
\(736\) 0 0
\(737\) −7.05452e45 −0.931082
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −7.40655e44 −0.0916028
\(741\) 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
\(751\) 5.65062e45 0.551895 0.275948 − 0.961173i \(-0.411008\pi\)
0.275948 + 0.961173i \(0.411008\pi\)
\(752\) −1.88261e46 −1.80001
\(753\) −1.44901e46 −1.35629
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.71746e46 1.47690 0.738451 − 0.674307i \(-0.235558\pi\)
0.738451 + 0.674307i \(0.235558\pi\)
\(758\) 0 0
\(759\) −1.44446e46 −1.19079
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.37320e46 1.76133
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −2.92537e46 −1.99711
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −6.07025e46 −3.89347
\(772\) 0 0
\(773\) 4.69710e45 0.289040 0.144520 − 0.989502i \(-0.453836\pi\)
0.144520 + 0.989502i \(0.453836\pi\)
\(774\) 0 0
\(775\) 2.33382e46 1.37797
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 3.25474e46 1.69863
\(782\) 0 0
\(783\) 0 0
\(784\) 2.03731e46 1.00000
\(785\) −2.32796e45 −0.111960
\(786\) 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
\(795\) 3.36432e45 0.132136
\(796\) 5.17886e46 1.99354
\(797\) −6.88523e45 −0.259768 −0.129884 − 0.991529i \(-0.541460\pi\)
−0.129884 + 0.991529i \(0.541460\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.68084e47 −5.85338
\(802\) 0 0
\(803\) 0 0
\(804\) 5.66873e46 1.85947
\(805\) 0 0
\(806\) 0 0
\(807\) −9.39265e46 −2.90276
\(808\) 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
\(815\) 4.58454e45 0.120997
\(816\) 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
\(823\) 2.01723e46 0.455363 0.227682 − 0.973736i \(-0.426885\pi\)
0.227682 + 0.973736i \(0.426885\pi\)
\(824\) 0 0
\(825\) 9.16000e46 1.98899
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 8.69690e46 1.78188
\(829\) −3.78542e46 −0.760749 −0.380375 − 0.924832i \(-0.624205\pi\)
−0.380375 + 0.924832i \(0.624205\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.18807e47 5.49440
\(838\) 0 0
\(839\) −1.18369e47 −1.96357 −0.981785 − 0.189994i \(-0.939153\pi\)
−0.981785 + 0.189994i \(0.939153\pi\)
\(840\) 0 0
\(841\) 6.26233e46 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4.30740e45 −0.0637540
\(846\) 0 0
\(847\) 0 0
\(848\) 7.42075e46 1.03780
\(849\) 0 0
\(850\) 0 0
\(851\) 6.48201e46 0.856716
\(852\) −2.61537e47 −3.39235
\(853\) 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
\(859\) 9.23467e46 1.05083 0.525413 − 0.850847i \(-0.323911\pi\)
0.525413 + 0.850847i \(0.323911\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.19613e47 −1.26359 −0.631793 − 0.775137i \(-0.717681\pi\)
−0.631793 + 0.775137i \(0.717681\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −2.03567e47 −1.99711
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.22877e47 −1.95811
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −8.24575e45 −0.0637540
\(881\) −2.46061e47 −1.86822 −0.934111 − 0.356984i \(-0.883805\pi\)
−0.934111 + 0.356984i \(0.883805\pi\)
\(882\) 0 0
\(883\) −1.90076e46 −0.139174 −0.0695868 − 0.997576i \(-0.522168\pi\)
−0.0695868 + 0.997576i \(0.522168\pi\)
\(884\) 0 0
\(885\) 2.80340e46 0.197967
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 7.79777e47 4.94228
\(892\) 1.53786e47 0.957371
\(893\) 0 0
\(894\) 0 0
\(895\) 1.78136e46 0.105095
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −5.51511e47 −2.97628
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.32975e46 −0.115062
\(906\) 0 0
\(907\) −1.82332e47 −0.869251 −0.434626 − 0.900611i \(-0.643119\pi\)
−0.434626 + 0.900611i \(0.643119\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −4.17700e47 −1.85597 −0.927983 − 0.372621i \(-0.878459\pi\)
−0.927983 + 0.372621i \(0.878459\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 2.76810e47 1.12682
\(917\) 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
\(925\) −4.11055e47 −1.43098
\(926\) 0 0
\(927\) 1.77356e48 5.96444
\(928\) 0 0
\(929\) −4.47334e47 −1.45339 −0.726693 − 0.686963i \(-0.758944\pi\)
−0.726693 + 0.686963i \(0.758944\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1.49210e47 −0.452575
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −1.10597e48 −3.02757
\(940\) 4.26411e46 0.114758
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 6.18353e47 1.55484
\(945\) 0 0
\(946\) 0 0
\(947\) 8.21888e47 1.96432 0.982162 − 0.188038i \(-0.0602127\pi\)
0.982162 + 0.188038i \(0.0602127\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.32618e48 2.96289
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −5.37529e46 −0.112292
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 6.62595e46 0.127323
\(961\) 4.83816e47 0.914336
\(962\) 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
\(971\) 9.67953e47 1.55005 0.775026 − 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(972\) −3.74498e48 −5.89915
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.11550e47 −0.306973 −0.153486 − 0.988151i \(-0.549050\pi\)
−0.153486 + 0.988151i \(0.549050\pi\)
\(978\) 0 0
\(979\) −1.39472e48 −1.95868
\(980\) −4.61450e46 −0.0637540
\(981\) 0 0
\(982\) 0 0
\(983\) 7.94328e47 1.04507 0.522533 − 0.852619i \(-0.324987\pi\)
0.522533 + 0.852619i \(0.324987\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.04904e48 −1.21231 −0.606156 − 0.795346i \(-0.707289\pi\)
−0.606156 + 0.795346i \(0.707289\pi\)
\(992\) 0 0
\(993\) 3.02137e48 3.38077
\(994\) 0 0
\(995\) −1.17301e47 −0.127096
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −5.61513e48 −5.70574
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 11.33.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.33.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.33.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.33.b.a.10.1 ✓ 1 11.10 odd 2 CM