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Properties

Label 11.37.b.a.10.1
Level $11$
Weight $37$
Character 11.10
Self dual yes
Analytic conductor $90.300$
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] = [11,37,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, 37, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 37);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 37 \)
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: \(90.3003845253\)
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)
 
\(f(q)\) \(=\) \(q-3.39284e8 q^{3} +6.87195e10 q^{4} +1.75580e12 q^{5} -3.49809e16 q^{9} +O(q^{10})\) \(q-3.39284e8 q^{3} +6.87195e10 q^{4} +1.75580e12 q^{5} -3.49809e16 q^{9} +5.55992e18 q^{11} -2.33154e19 q^{12} -5.95714e20 q^{15} +4.72237e21 q^{16} +1.20657e23 q^{20} +6.42094e24 q^{23} -1.14691e25 q^{25} +6.27932e25 q^{27} -6.88585e26 q^{31} -1.88639e27 q^{33} -2.40387e27 q^{36} -3.36719e28 q^{37} +3.82075e29 q^{44} -6.14194e28 q^{45} +1.96230e30 q^{47} -1.60222e30 q^{48} +2.65173e30 q^{49} -1.99072e31 q^{53} +9.76209e30 q^{55} +3.02001e30 q^{59} -4.09372e31 q^{60} +3.24519e32 q^{64} -6.55191e31 q^{67} -2.17852e33 q^{69} +4.15062e33 q^{71} +3.89128e33 q^{75} +8.29152e33 q^{80} -1.60543e34 q^{81} +1.42998e35 q^{89} +4.41243e35 q^{92} +2.33626e35 q^{93} +1.12965e36 q^{97} -1.94491e35 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −3.39284e8 −0.875752 −0.437876 − 0.899035i \(-0.644269\pi\)
−0.437876 + 0.899035i \(0.644269\pi\)
\(4\) 6.87195e10 1.00000
\(5\) 1.75580e12 0.460272 0.230136 − 0.973159i \(-0.426083\pi\)
0.230136 + 0.973159i \(0.426083\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −3.49809e16 −0.233059
\(10\) 0 0
\(11\) 5.55992e18 1.00000
\(12\) −2.33154e19 −0.875752
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −5.95714e20 −0.403084
\(16\) 4.72237e21 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.20657e23 0.460272
\(21\) 0 0
\(22\) 0 0
\(23\) 6.42094e24 1.97924 0.989618 − 0.143726i \(-0.0459083\pi\)
0.989618 + 0.143726i \(0.0459083\pi\)
\(24\) 0 0
\(25\) −1.14691e25 −0.788150
\(26\) 0 0
\(27\) 6.27932e25 1.07985
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −6.88585e26 −0.985025 −0.492512 − 0.870305i \(-0.663921\pi\)
−0.492512 + 0.870305i \(0.663921\pi\)
\(32\) 0 0
\(33\) −1.88639e27 −0.875752
\(34\) 0 0
\(35\) 0 0
\(36\) −2.40387e27 −0.233059
\(37\) −3.36719e28 −1.99360 −0.996798 − 0.0799629i \(-0.974520\pi\)
−0.996798 + 0.0799629i \(0.974520\pi\)
\(38\) 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 \(\pi\)
\(44\) 3.82075e29 1.00000
\(45\) −6.14194e28 −0.107271
\(46\) 0 0
\(47\) 1.96230e30 1.56676 0.783382 − 0.621540i \(-0.213493\pi\)
0.783382 + 0.621540i \(0.213493\pi\)
\(48\) −1.60222e30 −0.875752
\(49\) 2.65173e30 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.99072e31 −1.82828 −0.914142 − 0.405395i \(-0.867134\pi\)
−0.914142 + 0.405395i \(0.867134\pi\)
\(54\) 0 0
\(55\) 9.76209e30 0.460272
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.02001e30 0.0402413 0.0201206 − 0.999798i \(-0.493595\pi\)
0.0201206 + 0.999798i \(0.493595\pi\)
\(60\) −4.09372e31 −0.403084
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 3.24519e32 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.55191e31 −0.0885159 −0.0442579 − 0.999020i \(-0.514092\pi\)
−0.0442579 + 0.999020i \(0.514092\pi\)
\(68\) 0 0
\(69\) −2.17852e33 −1.73332
\(70\) 0 0
\(71\) 4.15062e33 1.97453 0.987263 − 0.159097i \(-0.0508583\pi\)
0.987263 + 0.159097i \(0.0508583\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 3.89128e33 0.690224
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.29152e33 0.460272
\(81\) −1.60543e34 −0.712624
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.42998e35 1.16496 0.582480 − 0.812845i \(-0.302082\pi\)
0.582480 + 0.812845i \(0.302082\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.41243e35 1.97924
\(93\) 2.33626e35 0.862637
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.12965e36 1.95457 0.977285 − 0.211929i \(-0.0679744\pi\)
0.977285 + 0.211929i \(0.0679744\pi\)
\(98\) 0 0
\(99\) −1.94491e35 −0.233059
\(100\) −7.88150e35 −0.788150
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 2.22681e36 1.30802 0.654008 − 0.756487i \(-0.273086\pi\)
0.654008 + 0.756487i \(0.273086\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 4.31512e36 1.07985
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 1.14244e37 1.74589
\(112\) 0 0
\(113\) 1.44321e37 1.59926 0.799628 − 0.600495i \(-0.205030\pi\)
0.799628 + 0.600495i \(0.205030\pi\)
\(114\) 0 0
\(115\) 1.12739e37 0.910986
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.09127e37 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −4.73192e37 −0.985025
\(125\) −4.56876e37 −0.823035
\(126\) 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 \(\pi\)
\(132\) −1.29632e38 −0.875752
\(133\) 0 0
\(134\) 0 0
\(135\) 1.10252e38 0.497026
\(136\) 0 0
\(137\) −4.97328e37 −0.172057 −0.0860286 − 0.996293i \(-0.527418\pi\)
−0.0860286 + 0.996293i \(0.527418\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −6.65777e38 −1.37210
\(142\) 0 0
\(143\) 0 0
\(144\) −1.65193e38 −0.233059
\(145\) 0 0
\(146\) 0 0
\(147\) −8.99690e38 −0.875752
\(148\) −2.31392e39 −1.99360
\(149\) 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
\(155\) −1.20902e39 −0.453379
\(156\) 0 0
\(157\) −3.57623e39 −1.06471 −0.532355 − 0.846521i \(-0.678693\pi\)
−0.532355 + 0.846521i \(0.678693\pi\)
\(158\) 0 0
\(159\) 6.75418e39 1.60112
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.62058e39 1.45822 0.729112 − 0.684394i \(-0.239933\pi\)
0.729112 + 0.684394i \(0.239933\pi\)
\(164\) 0 0
\(165\) −3.31212e39 −0.403084
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.26462e40 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
\(176\) 2.62560e40 1.00000
\(177\) −1.02464e39 −0.0352414
\(178\) 0 0
\(179\) 3.41828e40 0.960403 0.480202 − 0.877158i \(-0.340564\pi\)
0.480202 + 0.877158i \(0.340564\pi\)
\(180\) −4.22071e39 −0.107271
\(181\) 4.17756e40 0.960966 0.480483 − 0.877004i \(-0.340461\pi\)
0.480483 + 0.877004i \(0.340461\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.91211e40 −0.917596
\(186\) 0 0
\(187\) 0 0
\(188\) 1.34848e41 1.56676
\(189\) 0 0
\(190\) 0 0
\(191\) −5.22206e40 −0.456290 −0.228145 − 0.973627i \(-0.573266\pi\)
−0.228145 + 0.973627i \(0.573266\pi\)
\(192\) −1.10104e41 −0.875752
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.82226e41 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −4.77097e41 −1.99182 −0.995912 − 0.0903294i \(-0.971208\pi\)
−0.995912 + 0.0903294i \(0.971208\pi\)
\(200\) 0 0
\(201\) 2.22296e40 0.0775179
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.24610e41 −0.461279
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −1.36801e42 −1.82828
\(213\) −1.40824e42 −1.72919
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 6.70845e41 0.460272
\(221\) 0 0
\(222\) 0 0
\(223\) 1.32848e42 0.714276 0.357138 − 0.934052i \(-0.383753\pi\)
0.357138 + 0.934052i \(0.383753\pi\)
\(224\) 0 0
\(225\) 4.01200e41 0.183686
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 5.71579e42 1.90562 0.952812 − 0.303560i \(-0.0981754\pi\)
0.952812 + 0.303560i \(0.0981754\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 3.44540e42 0.721137
\(236\) 2.07533e41 0.0402413
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −2.81318e42 −0.403084
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −3.97796e42 −0.455772
\(244\) 0 0
\(245\) 4.65590e42 0.460272
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.74502e43 1.11602 0.558012 − 0.829833i \(-0.311564\pi\)
0.558012 + 0.829833i \(0.311564\pi\)
\(252\) 0 0
\(253\) 3.56999e43 1.97924
\(254\) 0 0
\(255\) 0 0
\(256\) 2.23007e43 1.00000
\(257\) −4.12269e43 −1.72339 −0.861697 − 0.507424i \(-0.830598\pi\)
−0.861697 + 0.507424i \(0.830598\pi\)
\(258\) 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
\(265\) −3.49529e43 −0.841507
\(266\) 0 0
\(267\) −4.85171e43 −1.02022
\(268\) −4.50244e42 −0.0885159
\(269\) −8.18634e43 −1.50504 −0.752521 − 0.658568i \(-0.771162\pi\)
−0.752521 + 0.658568i \(0.771162\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.37672e43 −0.788150
\(276\) −1.49707e44 −1.73332
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 2.40874e43 0.229569
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 2.85229e44 1.97453
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.97770e44 1.00000
\(290\) 0 0
\(291\) −3.83271e44 −1.71172
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 5.30252e42 0.0185219
\(296\) 0 0
\(297\) 3.49125e44 1.07985
\(298\) 0 0
\(299\) 0 0
\(300\) 2.67407e44 0.690224
\(301\) 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
\(309\) −7.55522e44 −1.14550
\(310\) 0 0
\(311\) 1.47807e45 1.99530 0.997648 − 0.0685501i \(-0.0218373\pi\)
0.997648 + 0.0685501i \(0.0218373\pi\)
\(312\) 0 0
\(313\) 2.32660e43 0.0279849 0.0139925 − 0.999902i \(-0.495546\pi\)
0.0139925 + 0.999902i \(0.495546\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.89581e45 1.81438 0.907192 − 0.420717i \(-0.138221\pi\)
0.907192 + 0.420717i \(0.138221\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.69789e44 0.460272
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.10324e45 −0.712624
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.88701e45 −1.70888 −0.854440 − 0.519551i \(-0.826099\pi\)
−0.854440 + 0.519551i \(0.826099\pi\)
\(332\) 0 0
\(333\) 1.17788e45 0.464626
\(334\) 0 0
\(335\) −1.15038e44 −0.0407414
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −4.89659e45 −1.40055
\(340\) 0 0
\(341\) −3.82848e45 −0.985025
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.82504e45 −0.797797
\(346\) 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
\(353\) −1.09128e46 −1.50658 −0.753290 − 0.657689i \(-0.771534\pi\)
−0.753290 + 0.657689i \(0.771534\pi\)
\(354\) 0 0
\(355\) 7.28765e45 0.908818
\(356\) 9.82677e45 1.16496
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.08425e46 1.00000
\(362\) 0 0
\(363\) −1.04882e46 −0.875752
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.91602e46 −1.99895 −0.999474 − 0.0324440i \(-0.989671\pi\)
−0.999474 + 0.0324440i \(0.989671\pi\)
\(368\) 3.03220e46 1.97924
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 1.60547e46 0.862637
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 1.55011e46 0.720774
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 5.20352e46 1.99890 0.999450 − 0.0331471i \(-0.0105530\pi\)
0.999450 + 0.0331471i \(0.0105530\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.26535e46 −1.99236 −0.996179 − 0.0873299i \(-0.972167\pi\)
−0.996179 + 0.0873299i \(0.972167\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 7.76287e46 1.95457
\(389\) 7.82438e45 0.188086 0.0940431 − 0.995568i \(-0.470021\pi\)
0.0940431 + 0.995568i \(0.470021\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −1.33653e46 −0.233059
\(397\) −2.81505e46 −0.469092 −0.234546 − 0.972105i \(-0.575360\pi\)
−0.234546 + 0.972105i \(0.575360\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −5.41613e46 −0.788150
\(401\) 8.51826e46 1.18509 0.592546 − 0.805537i \(-0.298123\pi\)
0.592546 + 0.805537i \(0.298123\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.81881e46 −0.328001
\(406\) 0 0
\(407\) −1.87213e47 −1.99360
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 1.68736e46 0.150679
\(412\) 1.53025e47 1.30802
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.68549e47 −1.69501 −0.847504 − 0.530790i \(-0.821895\pi\)
−0.847504 + 0.530790i \(0.821895\pi\)
\(420\) 0 0
\(421\) −2.91967e47 −1.69144 −0.845719 − 0.533629i \(-0.820828\pi\)
−0.845719 + 0.533629i \(0.820828\pi\)
\(422\) 0 0
\(423\) −6.86431e46 −0.365149
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 2.96532e47 1.07985
\(433\) −4.96953e47 −1.73593 −0.867967 − 0.496622i \(-0.834573\pi\)
−0.867967 + 0.496622i \(0.834573\pi\)
\(434\) 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
\(441\) −9.27601e46 −0.233059
\(442\) 0 0
\(443\) 8.62617e47 1.99780 0.998898 − 0.0469360i \(-0.0149457\pi\)
0.998898 + 0.0469360i \(0.0149457\pi\)
\(444\) 7.85076e47 1.74589
\(445\) 2.51076e47 0.536198
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.63292e47 −0.660423 −0.330212 − 0.943907i \(-0.607120\pi\)
−0.330212 + 0.943907i \(0.607120\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 9.91768e47 1.59926
\(453\) 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
\(460\) 7.74734e47 0.910986
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.48611e48 −1.55451 −0.777253 − 0.629188i \(-0.783388\pi\)
−0.777253 + 0.629188i \(0.783388\pi\)
\(464\) 0 0
\(465\) 4.10200e47 0.397047
\(466\) 0 0
\(467\) −1.78943e48 −1.60328 −0.801642 − 0.597805i \(-0.796040\pi\)
−0.801642 + 0.597805i \(0.796040\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.21336e48 0.932422
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.96371e47 0.426098
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.12430e48 1.00000
\(485\) 1.98343e48 0.899633
\(486\) 0 0
\(487\) −1.54802e48 −0.652010 −0.326005 − 0.945368i \(-0.605703\pi\)
−0.326005 + 0.945368i \(0.605703\pi\)
\(488\) 0 0
\(489\) −3.26411e48 −1.27704
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −3.41487e47 −0.107271
\(496\) −3.25175e48 −0.985025
\(497\) 0 0
\(498\) 0 0
\(499\) −4.51039e48 −1.22576 −0.612879 − 0.790177i \(-0.709989\pi\)
−0.612879 + 0.790177i \(0.709989\pi\)
\(500\) −3.13963e48 −0.823035
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4.29066e48 −0.875752
\(508\) 0 0
\(509\) −9.26544e48 −1.76176 −0.880878 − 0.473343i \(-0.843047\pi\)
−0.880878 + 0.473343i \(0.843047\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.90983e48 0.602043
\(516\) 0 0
\(517\) 1.09102e49 1.56676
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.39358e49 1.74202 0.871009 − 0.491268i \(-0.163466\pi\)
0.871009 + 0.491268i \(0.163466\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −8.90823e48 −0.875752
\(529\) 3.07039e49 2.91737
\(530\) 0 0
\(531\) −1.05643e47 −0.00937861
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.15977e49 −0.841075
\(538\) 0 0
\(539\) 1.47434e49 1.00000
\(540\) 7.57647e48 0.497026
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −1.41738e49 −0.841567
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −3.41761e48 −0.172057
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 2.00588e49 0.803586
\(556\) 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 \(\pi\)
\(564\) −4.57518e49 −1.37210
\(565\) 2.53399e49 0.736092
\(566\) 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
\(573\) 1.77176e49 0.399597
\(574\) 0 0
\(575\) −7.36423e49 −1.55993
\(576\) −1.13520e49 −0.233059
\(577\) 3.38667e49 0.673920 0.336960 − 0.941519i \(-0.390601\pi\)
0.336960 + 0.941519i \(0.390601\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.10682e50 −1.82828
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.13831e49 0.458364 0.229182 − 0.973384i \(-0.426395\pi\)
0.229182 + 0.973384i \(0.426395\pi\)
\(588\) −6.18262e49 −0.875752
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.59011e50 −1.99360
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.61871e50 1.74434
\(598\) 0 0
\(599\) −1.90301e49 −0.193089 −0.0965444 − 0.995329i \(-0.530779\pi\)
−0.0965444 + 0.995329i \(0.530779\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 2.29192e48 0.0206295
\(604\) 0 0
\(605\) 5.42764e49 0.460272
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.58271e49 0.272891 0.136446 − 0.990648i \(-0.456432\pi\)
0.136446 + 0.990648i \(0.456432\pi\)
\(618\) 0 0
\(619\) 2.66820e50 1.49895 0.749474 − 0.662034i \(-0.230307\pi\)
0.749474 + 0.662034i \(0.230307\pi\)
\(620\) −8.30829e49 −0.453379
\(621\) 4.03191e50 2.13728
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 8.66791e49 0.409330
\(626\) 0 0
\(627\) 0 0
\(628\) −2.45757e50 −1.06471
\(629\) 0 0
\(630\) 0 0
\(631\) 1.58207e50 0.629067 0.314533 − 0.949246i \(-0.398152\pi\)
0.314533 + 0.949246i \(0.398152\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 4.64144e50 1.60112
\(637\) 0 0
\(638\) 0 0
\(639\) −1.45193e50 −0.460182
\(640\) 0 0
\(641\) −8.77876e49 −0.263020 −0.131510 − 0.991315i \(-0.541983\pi\)
−0.131510 + 0.991315i \(0.541983\pi\)
\(642\) 0 0
\(643\) −5.56216e50 −1.57560 −0.787800 − 0.615931i \(-0.788780\pi\)
−0.787800 + 0.615931i \(0.788780\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.17643e50 0.551400 0.275700 − 0.961244i \(-0.411090\pi\)
0.275700 + 0.961244i \(0.411090\pi\)
\(648\) 0 0
\(649\) 1.67910e49 0.0402413
\(650\) 0 0
\(651\) 0 0
\(652\) 6.61121e50 1.45822
\(653\) −6.46904e50 −1.38804 −0.694021 − 0.719955i \(-0.744162\pi\)
−0.694021 + 0.719955i \(0.744162\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) −2.27607e50 −0.403084
\(661\) −1.13170e51 −1.95031 −0.975156 − 0.221521i \(-0.928898\pi\)
−0.975156 + 0.221521i \(0.928898\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −4.50730e50 −0.625528
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −7.20181e50 −0.851087
\(676\) 8.69042e50 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.18628e50 −0.208994 −0.104497 − 0.994525i \(-0.533323\pi\)
−0.104497 + 0.994525i \(0.533323\pi\)
\(684\) 0 0
\(685\) −8.73208e49 −0.0791931
\(686\) 0 0
\(687\) −1.93928e51 −1.66885
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 1.95236e51 1.51341 0.756704 − 0.653758i \(-0.226809\pi\)
0.756704 + 0.653758i \(0.226809\pi\)
\(692\) 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
\(704\) 1.80430e51 1.00000
\(705\) −1.16897e51 −0.631537
\(706\) 0 0
\(707\) 0 0
\(708\) −7.04128e49 −0.0352414
\(709\) 3.45722e51 1.68692 0.843460 − 0.537192i \(-0.180515\pi\)
0.843460 + 0.537192i \(0.180515\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.42136e51 −1.94960
\(714\) 0 0
\(715\) 0 0
\(716\) 2.34903e51 0.960403
\(717\) 0 0
\(718\) 0 0
\(719\) 1.66887e51 0.632854 0.316427 − 0.948617i \(-0.397517\pi\)
0.316427 + 0.948617i \(0.397517\pi\)
\(720\) −2.90045e50 −0.107271
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 2.87079e51 0.960966
\(725\) 0 0
\(726\) 0 0
\(727\) 5.93475e51 1.84409 0.922047 − 0.387078i \(-0.126516\pi\)
0.922047 + 0.387078i \(0.126516\pi\)
\(728\) 0 0
\(729\) 3.75932e51 1.11177
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −1.57967e51 −0.403084
\(736\) 0 0
\(737\) −3.64281e50 −0.0885159
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −4.06277e51 −0.917596
\(741\) 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
\(751\) −1.14686e52 −1.98605 −0.993026 − 0.117898i \(-0.962384\pi\)
−0.993026 + 0.117898i \(0.962384\pi\)
\(752\) 9.26669e51 1.56676
\(753\) −5.92058e51 −0.977360
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.96965e51 −1.34601 −0.673007 − 0.739636i \(-0.734998\pi\)
−0.673007 + 0.739636i \(0.734998\pi\)
\(758\) 0 0
\(759\) −1.21124e52 −1.73332
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3.58857e51 −0.456290
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −7.56629e51 −0.875752
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.39876e52 1.50926
\(772\) 0 0
\(773\) −1.41806e52 −1.46037 −0.730184 − 0.683250i \(-0.760566\pi\)
−0.730184 + 0.683250i \(0.760566\pi\)
\(774\) 0 0
\(775\) 7.89745e51 0.776347
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 2.30771e52 1.97453
\(782\) 0 0
\(783\) 0 0
\(784\) 1.25224e52 1.00000
\(785\) −6.27913e51 −0.490056
\(786\) 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
\(795\) 1.18590e52 0.736951
\(796\) −3.27858e52 −1.99182
\(797\) −1.13209e52 −0.672407 −0.336204 − 0.941789i \(-0.609143\pi\)
−0.336204 + 0.941789i \(0.609143\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −5.00222e51 −0.271505
\(802\) 0 0
\(803\) 0 0
\(804\) 1.52760e51 0.0775179
\(805\) 0 0
\(806\) 0 0
\(807\) 2.77749e52 1.31804
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.68918e52 0.671180
\(816\) 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
\(823\) 3.97204e52 1.32378 0.661892 − 0.749599i \(-0.269754\pi\)
0.661892 + 0.749599i \(0.269754\pi\)
\(824\) 0 0
\(825\) 2.16352e52 0.690224
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −1.54351e52 −0.461279
\(829\) 6.76243e52 1.97752 0.988760 − 0.149509i \(-0.0477694\pi\)
0.988760 + 0.149509i \(0.0477694\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.32385e52 −1.06368
\(838\) 0 0
\(839\) 6.96713e52 1.64187 0.820935 − 0.571021i \(-0.193453\pi\)
0.820935 + 0.571021i \(0.193453\pi\)
\(840\) 0 0
\(841\) 4.42923e52 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.22042e52 0.460272
\(846\) 0 0
\(847\) 0 0
\(848\) −9.40089e52 −1.82828
\(849\) 0 0
\(850\) 0 0
\(851\) −2.16205e53 −3.94579
\(852\) −9.67735e52 −1.72919
\(853\) 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
\(859\) 4.56133e52 0.703420 0.351710 − 0.936109i \(-0.385600\pi\)
0.351710 + 0.936109i \(0.385600\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.52849e52 0.358647 0.179323 − 0.983790i \(-0.442609\pi\)
0.179323 + 0.983790i \(0.442609\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.71003e52 −0.875752
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −3.95161e52 −0.455531
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 4.61001e52 0.460272
\(881\) −1.47145e53 −1.43939 −0.719697 − 0.694288i \(-0.755719\pi\)
−0.719697 + 0.694288i \(0.755719\pi\)
\(882\) 0 0
\(883\) 2.04987e53 1.92501 0.962507 − 0.271256i \(-0.0874391\pi\)
0.962507 + 0.271256i \(0.0874391\pi\)
\(884\) 0 0
\(885\) −1.79906e51 −0.0162206
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −8.92605e52 −0.712624
\(892\) 9.12921e52 0.714276
\(893\) 0 0
\(894\) 0 0
\(895\) 6.00181e52 0.442046
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 2.75702e52 0.183686
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 7.33494e52 0.442305
\(906\) 0 0
\(907\) 2.86883e52 0.166254 0.0831272 − 0.996539i \(-0.473509\pi\)
0.0831272 + 0.996539i \(0.473509\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −2.54233e53 −1.36113 −0.680567 − 0.732686i \(-0.738266\pi\)
−0.680567 + 0.732686i \(0.738266\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 3.92786e53 1.90562
\(917\) 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
\(925\) 3.86187e53 1.57125
\(926\) 0 0
\(927\) −7.78960e52 −0.304845
\(928\) 0 0
\(929\) 5.03263e53 1.89458 0.947288 − 0.320383i \(-0.103812\pi\)
0.947288 + 0.320383i \(0.103812\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −5.01485e53 −1.74738
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −7.89379e51 −0.0245078
\(940\) 2.36766e53 0.721137
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.42616e52 0.0402413
\(945\) 0 0
\(946\) 0 0
\(947\) 1.58503e53 0.422413 0.211207 − 0.977441i \(-0.432261\pi\)
0.211207 + 0.977441i \(0.432261\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.43220e53 −1.58895
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −9.16887e52 −0.210017
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −1.93320e53 −0.403084
\(961\) −1.45264e52 −0.0297260
\(962\) 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
\(971\) −8.45743e53 −1.43646 −0.718228 − 0.695808i \(-0.755046\pi\)
−0.718228 + 0.695808i \(0.755046\pi\)
\(972\) −2.73363e53 −0.455772
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.22673e54 −1.86487 −0.932433 − 0.361342i \(-0.882319\pi\)
−0.932433 + 0.361342i \(0.882319\pi\)
\(978\) 0 0
\(979\) 7.95059e53 1.16496
\(980\) 3.19951e53 0.460272
\(981\) 0 0
\(982\) 0 0
\(983\) −1.37308e54 −1.86953 −0.934764 − 0.355268i \(-0.884390\pi\)
−0.934764 + 0.355268i \(0.884390\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.01750e54 1.19732 0.598660 − 0.801004i \(-0.295700\pi\)
0.598660 + 0.801004i \(0.295700\pi\)
\(992\) 0 0
\(993\) 1.31880e54 1.49655
\(994\) 0 0
\(995\) −8.37685e53 −0.916780
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −2.11437e54 −2.15279
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 11.37.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.37.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.37.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.37.b.a.10.1 ✓ 1 11.10 odd 2 CM