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Properties

Label 11.25.b.a.10.1
Level $11$
Weight $25$
Character 11.10
Self dual yes
Analytic conductor $40.146$
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] = [11,25,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, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 25 \)
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: \(40.1463867484\)
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)
 
\(f(q)\) \(=\) \(q+781282. q^{3} +1.67772e7 q^{4} +1.76018e8 q^{5} +3.27972e11 q^{9} +O(q^{10})\) \(q+781282. q^{3} +1.67772e7 q^{4} +1.76018e8 q^{5} +3.27972e11 q^{9} +3.13843e12 q^{11} +1.31077e13 q^{12} +1.37520e14 q^{15} +2.81475e14 q^{16} +2.95309e15 q^{20} -2.54574e16 q^{23} -2.86224e16 q^{25} +3.55815e16 q^{27} +2.82468e17 q^{31} +2.45200e18 q^{33} +5.50246e18 q^{36} -7.18177e18 q^{37} +5.26541e19 q^{44} +5.77289e19 q^{45} +2.09537e20 q^{47} +2.19911e20 q^{48} +1.91581e20 q^{49} -7.06096e20 q^{53} +5.52419e20 q^{55} +1.73770e21 q^{59} +2.30720e21 q^{60} +4.72237e21 q^{64} -1.63583e22 q^{67} -1.98894e22 q^{69} -1.32949e22 q^{71} -2.23621e22 q^{75} +4.95446e22 q^{80} -6.48298e22 q^{81} +3.98369e23 q^{89} -4.27104e23 q^{92} +2.20687e23 q^{93} +1.37365e24 q^{97} +1.02932e24 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) 781282. 1.47012 0.735060 − 0.678002i \(-0.237154\pi\)
0.735060 + 0.678002i \(0.237154\pi\)
\(4\) 1.67772e7 1.00000
\(5\) 1.76018e8 0.720969 0.360485 − 0.932765i \(-0.382611\pi\)
0.360485 + 0.932765i \(0.382611\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 3.27972e11 1.16125
\(10\) 0 0
\(11\) 3.13843e12 1.00000
\(12\) 1.31077e13 1.47012
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 1.37520e14 1.05991
\(16\) 2.81475e14 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 2.95309e15 0.720969
\(21\) 0 0
\(22\) 0 0
\(23\) −2.54574e16 −1.16166 −0.580831 − 0.814024i \(-0.697272\pi\)
−0.580831 + 0.814024i \(0.697272\pi\)
\(24\) 0 0
\(25\) −2.86224e16 −0.480204
\(26\) 0 0
\(27\) 3.55815e16 0.237061
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 2.82468e17 0.358615 0.179308 − 0.983793i \(-0.442614\pi\)
0.179308 + 0.983793i \(0.442614\pi\)
\(32\) 0 0
\(33\) 2.45200e18 1.47012
\(34\) 0 0
\(35\) 0 0
\(36\) 5.50246e18 1.16125
\(37\) −7.18177e18 −1.09096 −0.545482 − 0.838122i \(-0.683653\pi\)
−0.545482 + 0.838122i \(0.683653\pi\)
\(38\) 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 \(\pi\)
\(44\) 5.26541e19 1.00000
\(45\) 5.77289e19 0.837227
\(46\) 0 0
\(47\) 2.09537e20 1.80338 0.901688 − 0.432387i \(-0.142329\pi\)
0.901688 + 0.432387i \(0.142329\pi\)
\(48\) 2.19911e20 1.47012
\(49\) 1.91581e20 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.06096e20 −1.43732 −0.718659 − 0.695362i \(-0.755244\pi\)
−0.718659 + 0.695362i \(0.755244\pi\)
\(54\) 0 0
\(55\) 5.52419e20 0.720969
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.73770e21 0.976676 0.488338 − 0.872655i \(-0.337603\pi\)
0.488338 + 0.872655i \(0.337603\pi\)
\(60\) 2.30720e21 1.05991
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4.72237e21 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −1.63583e22 −1.99913 −0.999564 − 0.0295107i \(-0.990605\pi\)
−0.999564 + 0.0295107i \(0.990605\pi\)
\(68\) 0 0
\(69\) −1.98894e22 −1.70778
\(70\) 0 0
\(71\) −1.32949e22 −0.810188 −0.405094 − 0.914275i \(-0.632761\pi\)
−0.405094 + 0.914275i \(0.632761\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −2.23621e22 −0.705957
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 4.95446e22 0.720969
\(81\) −6.48298e22 −0.812745
\(82\) 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
\(89\) 3.98369e23 1.61289 0.806446 − 0.591308i \(-0.201388\pi\)
0.806446 + 0.591308i \(0.201388\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.27104e23 −1.16166
\(93\) 2.20687e23 0.527207
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.37365e24 1.97977 0.989883 − 0.141885i \(-0.0453163\pi\)
0.989883 + 0.141885i \(0.0453163\pi\)
\(98\) 0 0
\(99\) 1.02932e24 1.16125
\(100\) −4.80204e23 −0.480204
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.37821e23 0.0966651 0.0483326 − 0.998831i \(-0.484609\pi\)
0.0483326 + 0.998831i \(0.484609\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 5.96959e23 0.237061
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −5.61098e24 −1.60385
\(112\) 0 0
\(113\) −7.06670e24 −1.63033 −0.815164 − 0.579230i \(-0.803353\pi\)
−0.815164 + 0.579230i \(0.803353\pi\)
\(114\) 0 0
\(115\) −4.48095e24 −0.837522
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 9.84973e24 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 4.73902e24 0.358615
\(125\) −1.55295e25 −1.06718
\(126\) 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 \(\pi\)
\(132\) 4.11377e25 1.47012
\(133\) 0 0
\(134\) 0 0
\(135\) 6.26298e24 0.170913
\(136\) 0 0
\(137\) −8.72892e25 −1.99670 −0.998352 − 0.0573918i \(-0.981722\pi\)
−0.998352 + 0.0573918i \(0.981722\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 1.63707e26 2.65118
\(142\) 0 0
\(143\) 0 0
\(144\) 9.23159e25 1.16125
\(145\) 0 0
\(146\) 0 0
\(147\) 1.49679e26 1.47012
\(148\) −1.20490e26 −1.09096
\(149\) 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
\(155\) 4.97194e25 0.258550
\(156\) 0 0
\(157\) −4.17516e26 −1.86156 −0.930781 − 0.365578i \(-0.880871\pi\)
−0.930781 + 0.365578i \(0.880871\pi\)
\(158\) 0 0
\(159\) −5.51660e26 −2.11303
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.01721e26 −1.71058 −0.855292 − 0.518147i \(-0.826622\pi\)
−0.855292 + 0.518147i \(0.826622\pi\)
\(164\) 0 0
\(165\) 4.31595e26 1.05991
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 5.42801e26 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
\(176\) 8.83389e26 1.00000
\(177\) 1.35763e27 1.43583
\(178\) 0 0
\(179\) −2.04451e27 −1.88953 −0.944764 − 0.327752i \(-0.893709\pi\)
−0.944764 + 0.327752i \(0.893709\pi\)
\(180\) 9.68531e26 0.837227
\(181\) −2.33595e27 −1.88939 −0.944694 − 0.327954i \(-0.893641\pi\)
−0.944694 + 0.327954i \(0.893641\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.26412e27 −0.786552
\(186\) 0 0
\(187\) 0 0
\(188\) 3.51545e27 1.80338
\(189\) 0 0
\(190\) 0 0
\(191\) 1.70548e27 0.723512 0.361756 − 0.932273i \(-0.382177\pi\)
0.361756 + 0.932273i \(0.382177\pi\)
\(192\) 3.68950e27 1.47012
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.21420e27 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 7.69975e27 1.99636 0.998182 − 0.0602652i \(-0.0191946\pi\)
0.998182 + 0.0602652i \(0.0191946\pi\)
\(200\) 0 0
\(201\) −1.27805e28 −2.93896
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.34931e27 −1.34898
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −1.18463e28 −1.43732
\(213\) −1.03871e28 −1.19107
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 9.26806e27 0.720969
\(221\) 0 0
\(222\) 0 0
\(223\) 2.09925e28 1.38805 0.694026 − 0.719950i \(-0.255835\pi\)
0.694026 + 0.719950i \(0.255835\pi\)
\(224\) 0 0
\(225\) −9.38734e27 −0.557638
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −2.77151e28 −1.33257 −0.666286 − 0.745696i \(-0.732117\pi\)
−0.666286 + 0.745696i \(0.732117\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 3.68822e28 1.30018
\(236\) 2.91538e28 0.976676
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 3.87083e28 1.05991
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −6.06996e28 −1.43189
\(244\) 0 0
\(245\) 3.37217e28 0.720969
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.93649e28 1.58909 0.794545 − 0.607205i \(-0.207709\pi\)
0.794545 + 0.607205i \(0.207709\pi\)
\(252\) 0 0
\(253\) −7.98962e28 −1.16166
\(254\) 0 0
\(255\) 0 0
\(256\) 7.92282e28 1.00000
\(257\) −2.78962e28 −0.336006 −0.168003 − 0.985786i \(-0.553732\pi\)
−0.168003 + 0.985786i \(0.553732\pi\)
\(258\) 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
\(265\) −1.24285e29 −1.03626
\(266\) 0 0
\(267\) 3.11238e29 2.37114
\(268\) −2.74447e29 −1.99913
\(269\) −2.42045e29 −1.68604 −0.843021 − 0.537881i \(-0.819225\pi\)
−0.843021 + 0.537881i \(0.819225\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −8.98293e28 −0.480204
\(276\) −3.33689e29 −1.70778
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 9.26416e28 0.416443
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −2.23052e29 −0.810188
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3.39449e29 1.00000
\(290\) 0 0
\(291\) 1.07320e30 2.91049
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 3.05866e29 0.704153
\(296\) 0 0
\(297\) 1.11670e29 0.237061
\(298\) 0 0
\(299\) 0 0
\(300\) −3.75174e29 −0.705957
\(301\) 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
\(309\) 1.07677e29 0.142109
\(310\) 0 0
\(311\) −7.53011e29 −0.919765 −0.459883 − 0.887980i \(-0.652108\pi\)
−0.459883 + 0.887980i \(0.652108\pi\)
\(312\) 0 0
\(313\) 8.98410e29 1.01611 0.508057 − 0.861323i \(-0.330364\pi\)
0.508057 + 0.861323i \(0.330364\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.77732e29 −0.463952 −0.231976 − 0.972722i \(-0.574519\pi\)
−0.231976 + 0.972722i \(0.574519\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.31221e29 0.720969
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.08766e30 −0.812745
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.23221e30 1.86881 0.934403 − 0.356218i \(-0.115934\pi\)
0.934403 + 0.356218i \(0.115934\pi\)
\(332\) 0 0
\(333\) −2.35542e30 −1.26689
\(334\) 0 0
\(335\) −2.87935e30 −1.44131
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −5.52108e30 −2.39678
\(340\) 0 0
\(341\) 8.86505e29 0.358615
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3.50089e30 −1.23126
\(346\) 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
\(353\) 6.64640e30 1.77538 0.887688 − 0.460446i \(-0.152311\pi\)
0.887688 + 0.460446i \(0.152311\pi\)
\(354\) 0 0
\(355\) −2.34014e30 −0.584120
\(356\) 6.68352e30 1.61289
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.89876e30 1.00000
\(362\) 0 0
\(363\) 7.69542e30 1.47012
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.19377e31 1.99953 0.999766 − 0.0216315i \(-0.00688605\pi\)
0.999766 + 0.0216315i \(0.00688605\pi\)
\(368\) −7.16561e30 −1.16166
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 3.70251e30 0.527207
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −1.21329e31 −1.56888
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.11764e30 −1.03803 −0.519017 − 0.854764i \(-0.673702\pi\)
−0.519017 + 0.854764i \(0.673702\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.94060e30 −0.897390 −0.448695 − 0.893685i \(-0.648111\pi\)
−0.448695 + 0.893685i \(0.648111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 2.30460e31 1.97977
\(389\) −2.39645e31 −1.99606 −0.998029 − 0.0627470i \(-0.980014\pi\)
−0.998029 + 0.0627470i \(0.980014\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 1.72691e31 1.16125
\(397\) 1.09645e31 0.715327 0.357663 − 0.933851i \(-0.383574\pi\)
0.357663 + 0.933851i \(0.383574\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −8.05648e30 −0.480204
\(401\) 3.47886e30 0.201236 0.100618 − 0.994925i \(-0.467918\pi\)
0.100618 + 0.994925i \(0.467918\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.14112e31 −0.585964
\(406\) 0 0
\(407\) −2.25395e31 −1.09096
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −6.81975e31 −2.93539
\(412\) 2.31226e30 0.0966651
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.57480e31 −1.56244 −0.781221 − 0.624254i \(-0.785403\pi\)
−0.781221 + 0.624254i \(0.785403\pi\)
\(420\) 0 0
\(421\) 5.76888e31 1.86083 0.930413 − 0.366512i \(-0.119448\pi\)
0.930413 + 0.366512i \(0.119448\pi\)
\(422\) 0 0
\(423\) 6.87223e31 2.09418
\(424\) 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 \(\pi\)
\(432\) 1.00153e31 0.237061
\(433\) 8.17108e31 1.88116 0.940578 − 0.339578i \(-0.110284\pi\)
0.940578 + 0.339578i \(0.110284\pi\)
\(434\) 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
\(441\) 6.28333e31 1.16125
\(442\) 0 0
\(443\) 1.14199e32 1.99902 0.999510 − 0.0312971i \(-0.00996379\pi\)
0.999510 + 0.0312971i \(0.00996379\pi\)
\(444\) −9.41367e31 −1.60385
\(445\) 7.01200e31 1.16284
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.30907e32 −1.94988 −0.974939 − 0.222474i \(-0.928587\pi\)
−0.974939 + 0.222474i \(0.928587\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.18559e32 −1.63033
\(453\) 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
\(460\) −7.51779e31 −0.837522
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 1.74456e32 1.79769 0.898844 − 0.438269i \(-0.144408\pi\)
0.898844 + 0.438269i \(0.144408\pi\)
\(464\) 0 0
\(465\) 3.88449e31 0.380100
\(466\) 0 0
\(467\) −2.07182e31 −0.192553 −0.0962763 − 0.995355i \(-0.530693\pi\)
−0.0962763 + 0.995355i \(0.530693\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −3.26198e32 −2.73672
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2.31580e32 −1.66909
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.65251e32 1.00000
\(485\) 2.41786e32 1.42735
\(486\) 0 0
\(487\) 2.41313e32 1.35592 0.677959 − 0.735100i \(-0.262865\pi\)
0.677959 + 0.735100i \(0.262865\pi\)
\(488\) 0 0
\(489\) −4.70114e32 −2.51476
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.81178e32 0.837227
\(496\) 7.95076e31 0.358615
\(497\) 0 0
\(498\) 0 0
\(499\) 3.98997e31 0.167403 0.0837015 − 0.996491i \(-0.473326\pi\)
0.0837015 + 0.996491i \(0.473326\pi\)
\(500\) −2.60542e32 −1.06718
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.24080e32 1.47012
\(508\) 0 0
\(509\) −1.17126e32 −0.387293 −0.193647 − 0.981071i \(-0.562032\pi\)
−0.193647 + 0.981071i \(0.562032\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.42590e31 0.0696926
\(516\) 0 0
\(517\) 6.57617e32 1.80338
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.44194e32 −0.360490 −0.180245 − 0.983622i \(-0.557689\pi\)
−0.180245 + 0.983622i \(0.557689\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 6.90176e32 1.47012
\(529\) 1.67827e32 0.349458
\(530\) 0 0
\(531\) 5.69917e32 1.13417
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.59734e33 −2.77783
\(538\) 0 0
\(539\) 6.01264e32 1.00000
\(540\) 1.05075e32 0.170913
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −1.82504e33 −2.77763
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −1.46447e33 −1.99670
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −9.87633e32 −1.15633
\(556\) 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 \(\pi\)
\(564\) 2.74656e33 2.65118
\(565\) −1.24386e33 −1.17542
\(566\) 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
\(573\) 1.33246e33 1.06365
\(574\) 0 0
\(575\) 7.28650e32 0.557834
\(576\) 1.54880e33 1.16125
\(577\) 7.90483e32 0.580474 0.290237 − 0.956955i \(-0.406266\pi\)
0.290237 + 0.956955i \(0.406266\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.21603e33 −1.43732
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.30755e33 −1.97628 −0.988141 − 0.153548i \(-0.950930\pi\)
−0.988141 + 0.153548i \(0.950930\pi\)
\(588\) 2.51120e33 1.47012
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −2.02149e33 −1.09096
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.01568e33 2.93490
\(598\) 0 0
\(599\) 2.36728e33 1.10950 0.554750 − 0.832017i \(-0.312814\pi\)
0.554750 + 0.832017i \(0.312814\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −5.36507e33 −2.32149
\(604\) 0 0
\(605\) 1.73373e33 0.720969
\(606\) 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
\(617\) 3.51158e33 1.15367 0.576834 − 0.816862i \(-0.304288\pi\)
0.576834 + 0.816862i \(0.304288\pi\)
\(618\) 0 0
\(619\) 5.60667e33 1.77181 0.885906 − 0.463865i \(-0.153538\pi\)
0.885906 + 0.463865i \(0.153538\pi\)
\(620\) 8.34153e32 0.258550
\(621\) −9.05812e32 −0.275384
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.02745e33 −0.289201
\(626\) 0 0
\(627\) 0 0
\(628\) −7.00476e33 −1.86156
\(629\) 0 0
\(630\) 0 0
\(631\) −7.78800e33 −1.95467 −0.977336 − 0.211696i \(-0.932101\pi\)
−0.977336 + 0.211696i \(0.932101\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) −9.25532e33 −2.11303
\(637\) 0 0
\(638\) 0 0
\(639\) −4.36036e33 −0.940832
\(640\) 0 0
\(641\) 4.06123e33 0.844037 0.422019 − 0.906587i \(-0.361322\pi\)
0.422019 + 0.906587i \(0.361322\pi\)
\(642\) 0 0
\(643\) −8.21775e33 −1.64521 −0.822605 − 0.568613i \(-0.807480\pi\)
−0.822605 + 0.568613i \(0.807480\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.01330e33 1.30338 0.651690 − 0.758485i \(-0.274060\pi\)
0.651690 + 0.758485i \(0.274060\pi\)
\(648\) 0 0
\(649\) 5.45364e33 0.976676
\(650\) 0 0
\(651\) 0 0
\(652\) −1.00952e34 −1.71058
\(653\) 1.46986e32 0.0244522 0.0122261 − 0.999925i \(-0.496108\pi\)
0.0122261 + 0.999925i \(0.496108\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 7.24097e33 1.05991
\(661\) 1.37599e34 1.97787 0.988933 − 0.148366i \(-0.0474014\pi\)
0.988933 + 0.148366i \(0.0474014\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.64011e34 2.04060
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.01843e33 −0.113837
\(676\) 9.10669e33 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
\(683\) 9.03523e33 0.876780 0.438390 − 0.898785i \(-0.355549\pi\)
0.438390 + 0.898785i \(0.355549\pi\)
\(684\) 0 0
\(685\) −1.53645e34 −1.43956
\(686\) 0 0
\(687\) −2.16533e34 −1.95904
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.15075e33 −0.0971060 −0.0485530 − 0.998821i \(-0.515461\pi\)
−0.0485530 + 0.998821i \(0.515461\pi\)
\(692\) 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
\(704\) 1.48208e34 1.00000
\(705\) 2.88154e34 1.91142
\(706\) 0 0
\(707\) 0 0
\(708\) 2.27773e34 1.43583
\(709\) −2.53106e34 −1.56873 −0.784363 − 0.620302i \(-0.787010\pi\)
−0.784363 + 0.620302i \(0.787010\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.19089e33 −0.416590
\(714\) 0 0
\(715\) 0 0
\(716\) −3.43012e34 −1.88953
\(717\) 0 0
\(718\) 0 0
\(719\) 1.16077e34 0.608133 0.304067 − 0.952651i \(-0.401655\pi\)
0.304067 + 0.952651i \(0.401655\pi\)
\(720\) 1.62492e34 0.837227
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −3.91908e34 −1.88939
\(725\) 0 0
\(726\) 0 0
\(727\) −1.11497e34 −0.511503 −0.255752 − 0.966742i \(-0.582323\pi\)
−0.255752 + 0.966742i \(0.582323\pi\)
\(728\) 0 0
\(729\) −2.91137e34 −1.29231
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 2.63462e34 1.05991
\(736\) 0 0
\(737\) −5.13394e34 −1.99913
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −2.12084e34 −0.786552
\(741\) 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
\(751\) −3.64746e34 −1.13321 −0.566606 − 0.823989i \(-0.691744\pi\)
−0.566606 + 0.823989i \(0.691744\pi\)
\(752\) 5.89794e34 1.80338
\(753\) 7.76320e34 2.33615
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.01926e34 1.69978 0.849890 − 0.526960i \(-0.176668\pi\)
0.849890 + 0.526960i \(0.176668\pi\)
\(758\) 0 0
\(759\) −6.24214e34 −1.70778
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 2.86132e34 0.723512
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 6.18995e34 1.47012
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −2.17948e34 −0.493970
\(772\) 0 0
\(773\) 7.98220e34 1.75375 0.876877 − 0.480715i \(-0.159623\pi\)
0.876877 + 0.480715i \(0.159623\pi\)
\(774\) 0 0
\(775\) −8.08490e33 −0.172208
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −4.17252e34 −0.810188
\(782\) 0 0
\(783\) 0 0
\(784\) 5.39253e34 1.00000
\(785\) −7.34903e34 −1.34213
\(786\) 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
\(795\) −9.71020e34 −1.52343
\(796\) 1.29180e35 1.99636
\(797\) 3.81987e34 0.581499 0.290750 − 0.956799i \(-0.406095\pi\)
0.290750 + 0.956799i \(0.406095\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.30654e35 1.87297
\(802\) 0 0
\(803\) 0 0
\(804\) −2.14420e35 −2.93896
\(805\) 0 0
\(806\) 0 0
\(807\) −1.89105e35 −2.47868
\(808\) 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
\(815\) −1.05914e35 −1.23328
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.63110e35 1.68921 0.844604 − 0.535391i \(-0.179836\pi\)
0.844604 + 0.535391i \(0.179836\pi\)
\(824\) 0 0
\(825\) −7.01820e34 −0.705957
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −1.40078e35 −1.34898
\(829\) 2.09656e35 1.99000 0.994999 − 0.0998812i \(-0.0318463\pi\)
0.994999 + 0.0998812i \(0.0318463\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.00506e34 0.0850135
\(838\) 0 0
\(839\) −1.94868e35 −1.60176 −0.800881 − 0.598823i \(-0.795635\pi\)
−0.800881 + 0.598823i \(0.795635\pi\)
\(840\) 0 0
\(841\) 1.25185e35 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.55426e34 0.720969
\(846\) 0 0
\(847\) 0 0
\(848\) −1.98748e35 −1.43732
\(849\) 0 0
\(850\) 0 0
\(851\) 1.82829e35 1.26733
\(852\) −1.74266e35 −1.19107
\(853\) 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
\(859\) 2.23138e35 1.38247 0.691235 − 0.722630i \(-0.257067\pi\)
0.691235 + 0.722630i \(0.257067\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3.38856e35 −1.98557 −0.992785 − 0.119910i \(-0.961739\pi\)
−0.992785 + 0.119910i \(0.961739\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.65205e35 1.47012
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 4.50517e35 2.29901
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1.55492e35 0.720969
\(881\) 3.81271e35 1.74390 0.871952 − 0.489591i \(-0.162854\pi\)
0.871952 + 0.489591i \(0.162854\pi\)
\(882\) 0 0
\(883\) −1.50041e35 −0.667854 −0.333927 − 0.942599i \(-0.608374\pi\)
−0.333927 + 0.942599i \(0.608374\pi\)
\(884\) 0 0
\(885\) 2.38968e35 1.03519
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.03464e35 −0.812745
\(892\) 3.52196e35 1.38805
\(893\) 0 0
\(894\) 0 0
\(895\) −3.59871e35 −1.36229
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.57493e35 −0.557638
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.11169e35 −1.36219
\(906\) 0 0
\(907\) −6.18940e35 −1.99692 −0.998461 − 0.0554537i \(-0.982339\pi\)
−0.998461 + 0.0554537i \(0.982339\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.73812e35 −1.75610 −0.878050 − 0.478569i \(-0.841156\pi\)
−0.878050 + 0.478569i \(0.841156\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −4.64982e35 −1.33257
\(917\) 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
\(925\) 2.05559e35 0.523885
\(926\) 0 0
\(927\) 4.52015e34 0.112253
\(928\) 0 0
\(929\) 8.06996e35 1.95291 0.976457 − 0.215713i \(-0.0692077\pi\)
0.976457 + 0.215713i \(0.0692077\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −5.88314e35 −1.35216
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 7.01911e35 1.49381
\(940\) 6.18781e35 1.30018
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 4.89119e35 0.976676
\(945\) 0 0
\(946\) 0 0
\(947\) −1.03002e36 −1.97991 −0.989953 − 0.141397i \(-0.954840\pi\)
−0.989953 + 0.141397i \(0.954840\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −3.73243e35 −0.682065
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 3.00195e35 0.521630
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 6.49418e35 1.05991
\(961\) −5.40625e35 −0.871395
\(962\) 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
\(971\) 1.22408e36 1.74252 0.871262 − 0.490819i \(-0.163302\pi\)
0.871262 + 0.490819i \(0.163302\pi\)
\(972\) −1.01837e36 −1.43189
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −4.13877e35 −0.547188 −0.273594 − 0.961845i \(-0.588212\pi\)
−0.273594 + 0.961845i \(0.588212\pi\)
\(978\) 0 0
\(979\) 1.25025e36 1.61289
\(980\) 5.65756e35 0.720969
\(981\) 0 0
\(982\) 0 0
\(983\) −1.12836e36 −1.38613 −0.693065 − 0.720875i \(-0.743740\pi\)
−0.693065 + 0.720875i \(0.743740\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.71486e35 0.191137 0.0955683 − 0.995423i \(-0.469533\pi\)
0.0955683 + 0.995423i \(0.469533\pi\)
\(992\) 0 0
\(993\) 2.52527e36 2.74737
\(994\) 0 0
\(995\) 1.35529e36 1.43932
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −2.55538e35 −0.258625
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 11.25.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.25.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.25.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.25.b.a.10.1 ✓ 1 11.10 odd 2 CM