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Properties

Label 11.53.b.a.10.1
Level $11$
Weight $53$
Character 11.10
Self dual yes
Analytic conductor $188.379$
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] = [11,53,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, 53, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 53);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 53 \)
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: \(188.379295253\)
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)
 
\(f(q)\) \(=\) \(q-4.50892e12 q^{3} +4.50360e15 q^{4} +2.56039e18 q^{5} +1.38693e25 q^{9} +O(q^{10})\) \(q-4.50892e12 q^{3} +4.50360e15 q^{4} +2.56039e18 q^{5} +1.38693e25 q^{9} +1.19182e27 q^{11} -2.03064e28 q^{12} -1.15446e31 q^{15} +2.02824e31 q^{16} +1.15310e34 q^{20} +4.48364e35 q^{23} +4.33515e36 q^{25} -3.34028e37 q^{27} +7.23932e38 q^{31} -5.37381e39 q^{33} +6.24616e40 q^{36} +1.06070e41 q^{37} +5.36747e42 q^{44} +3.55107e43 q^{45} +2.56855e43 q^{47} -9.14517e43 q^{48} +8.81248e43 q^{49} -1.34968e45 q^{53} +3.05152e45 q^{55} +6.93259e45 q^{59} -5.19922e46 q^{60} +9.13439e46 q^{64} -5.00329e47 q^{67} -2.02164e48 q^{69} -2.69589e48 q^{71} -1.95469e49 q^{75} +5.19309e49 q^{80} +6.10003e49 q^{81} +7.24276e50 q^{89} +2.01925e51 q^{92} -3.26415e51 q^{93} -6.69771e51 q^{97} +1.65296e52 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −4.50892e12 −1.77386 −0.886931 − 0.461903i \(-0.847167\pi\)
−0.886931 + 0.461903i \(0.847167\pi\)
\(4\) 4.50360e15 1.00000
\(5\) 2.56039e18 1.71825 0.859125 − 0.511767i \(-0.171009\pi\)
0.859125 + 0.511767i \(0.171009\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 1.38693e25 2.14658
\(10\) 0 0
\(11\) 1.19182e27 1.00000
\(12\) −2.03064e28 −1.77386
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −1.15446e31 −3.04794
\(16\) 2.02824e31 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.15310e34 1.71825
\(21\) 0 0
\(22\) 0 0
\(23\) 4.48364e35 1.76485 0.882424 − 0.470455i \(-0.155910\pi\)
0.882424 + 0.470455i \(0.155910\pi\)
\(24\) 0 0
\(25\) 4.33515e36 1.95238
\(26\) 0 0
\(27\) −3.34028e37 −2.03388
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 7.23932e38 1.21421 0.607105 − 0.794622i \(-0.292331\pi\)
0.607105 + 0.794622i \(0.292331\pi\)
\(32\) 0 0
\(33\) −5.37381e39 −1.77386
\(34\) 0 0
\(35\) 0 0
\(36\) 6.24616e40 2.14658
\(37\) 1.06070e41 1.78792 0.893960 − 0.448147i \(-0.147916\pi\)
0.893960 + 0.448147i \(0.147916\pi\)
\(38\) 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 \(\pi\)
\(44\) 5.36747e42 1.00000
\(45\) 3.55107e43 3.68837
\(46\) 0 0
\(47\) 2.56855e43 0.861278 0.430639 − 0.902524i \(-0.358288\pi\)
0.430639 + 0.902524i \(0.358288\pi\)
\(48\) −9.14517e43 −1.77386
\(49\) 8.81248e43 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.34968e45 −1.99094 −0.995471 − 0.0950618i \(-0.969695\pi\)
−0.995471 + 0.0950618i \(0.969695\pi\)
\(54\) 0 0
\(55\) 3.05152e45 1.71825
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.93259e45 0.629135 0.314567 − 0.949235i \(-0.398141\pi\)
0.314567 + 0.949235i \(0.398141\pi\)
\(60\) −5.19922e46 −3.04794
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 9.13439e46 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00329e47 −1.66460 −0.832302 − 0.554322i \(-0.812978\pi\)
−0.832302 + 0.554322i \(0.812978\pi\)
\(68\) 0 0
\(69\) −2.02164e48 −3.13060
\(70\) 0 0
\(71\) −2.69589e48 −1.98603 −0.993013 − 0.118003i \(-0.962351\pi\)
−0.993013 + 0.118003i \(0.962351\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −1.95469e49 −3.46325
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 5.19309e49 1.71825
\(81\) 6.10003e49 1.46124
\(82\) 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
\(89\) 7.24276e50 1.49887 0.749435 − 0.662078i \(-0.230325\pi\)
0.749435 + 0.662078i \(0.230325\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.01925e51 1.76485
\(93\) −3.26415e51 −2.15384
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.69771e51 −1.47864 −0.739318 − 0.673356i \(-0.764852\pi\)
−0.739318 + 0.673356i \(0.764852\pi\)
\(98\) 0 0
\(99\) 1.65296e52 2.14658
\(100\) 1.95238e52 1.95238
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.71423e52 −1.25858 −0.629288 − 0.777172i \(-0.716653\pi\)
−0.629288 + 0.777172i \(0.716653\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.50433e53 −2.03388
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −4.78261e53 −3.17152
\(112\) 0 0
\(113\) −1.37816e53 −0.574461 −0.287230 − 0.957862i \(-0.592735\pi\)
−0.287230 + 0.957862i \(0.592735\pi\)
\(114\) 0 0
\(115\) 1.14799e54 3.03245
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.42043e54 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 3.26030e54 1.21421
\(125\) 5.41448e54 1.63643
\(126\) 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 \(\pi\)
\(132\) −2.42015e55 −1.77386
\(133\) 0 0
\(134\) 0 0
\(135\) −8.55243e55 −3.49471
\(136\) 0 0
\(137\) 6.61003e55 1.84276 0.921379 − 0.388664i \(-0.127063\pi\)
0.921379 + 0.388664i \(0.127063\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −1.15814e56 −1.52779
\(142\) 0 0
\(143\) 0 0
\(144\) 2.81302e56 2.14658
\(145\) 0 0
\(146\) 0 0
\(147\) −3.97347e56 −1.77386
\(148\) 4.77697e56 1.78792
\(149\) 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
\(155\) 1.85355e57 2.08631
\(156\) 0 0
\(157\) −1.79761e57 −1.44979 −0.724896 − 0.688859i \(-0.758112\pi\)
−0.724896 + 0.688859i \(0.758112\pi\)
\(158\) 0 0
\(159\) 6.08560e57 3.53166
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.20823e57 −1.58421 −0.792104 − 0.610386i \(-0.791015\pi\)
−0.792104 + 0.610386i \(0.791015\pi\)
\(164\) 0 0
\(165\) −1.37590e58 −3.04794
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.41500e57 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
\(176\) 2.41729e58 1.00000
\(177\) −3.12585e58 −1.11600
\(178\) 0 0
\(179\) −7.35314e58 −1.96017 −0.980086 − 0.198572i \(-0.936370\pi\)
−0.980086 + 0.198572i \(0.936370\pi\)
\(180\) 1.59926e59 3.68837
\(181\) 6.63386e58 1.32472 0.662359 − 0.749187i \(-0.269555\pi\)
0.662359 + 0.749187i \(0.269555\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.71581e59 3.07209
\(186\) 0 0
\(187\) 0 0
\(188\) 1.15677e59 0.861278
\(189\) 0 0
\(190\) 0 0
\(191\) −6.72758e57 −0.0331888 −0.0165944 − 0.999862i \(-0.505282\pi\)
−0.0165944 + 0.999862i \(0.505282\pi\)
\(192\) −4.11862e59 −1.77386
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.96879e59 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.16813e60 −1.98295 −0.991477 − 0.130282i \(-0.958412\pi\)
−0.991477 + 0.130282i \(0.958412\pi\)
\(200\) 0 0
\(201\) 2.25594e60 2.95278
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.21848e60 3.78839
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −6.07843e60 −1.99094
\(213\) 1.21555e61 3.52294
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 1.37428e61 1.71825
\(221\) 0 0
\(222\) 0 0
\(223\) −3.86253e60 −0.339583 −0.169791 − 0.985480i \(-0.554309\pi\)
−0.169791 + 0.985480i \(0.554309\pi\)
\(224\) 0 0
\(225\) 6.01254e61 4.19095
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −3.17846e61 −1.40118 −0.700591 − 0.713563i \(-0.747080\pi\)
−0.700591 + 0.713563i \(0.747080\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 6.57649e61 1.47989
\(236\) 3.12216e61 0.629135
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −2.34152e62 −3.04794
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −5.92267e61 −0.558154
\(244\) 0 0
\(245\) 2.25634e62 1.71825
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.59917e62 −1.86708 −0.933539 − 0.358477i \(-0.883296\pi\)
−0.933539 + 0.358477i \(0.883296\pi\)
\(252\) 0 0
\(253\) 5.34368e62 1.76485
\(254\) 0 0
\(255\) 0 0
\(256\) 4.11376e62 1.00000
\(257\) 8.31516e62 1.82646 0.913229 − 0.407447i \(-0.133581\pi\)
0.913229 + 0.407447i \(0.133581\pi\)
\(258\) 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
\(265\) −3.45571e63 −3.42094
\(266\) 0 0
\(267\) −3.26570e63 −2.65879
\(268\) −2.25328e63 −1.66460
\(269\) −2.81152e63 −1.88531 −0.942654 − 0.333771i \(-0.891679\pi\)
−0.942654 + 0.333771i \(0.891679\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.16671e63 1.95238
\(276\) −9.10465e63 −3.13060
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 1.00404e64 2.60640
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −1.21412e64 −1.98603
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 9.62374e63 1.00000
\(290\) 0 0
\(291\) 3.01994e64 2.62290
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 1.77502e64 1.08101
\(296\) 0 0
\(297\) −3.98101e64 −2.03388
\(298\) 0 0
\(299\) 0 0
\(300\) −8.80312e64 −3.46325
\(301\) 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
\(309\) 1.22383e65 2.23254
\(310\) 0 0
\(311\) 9.05914e64 1.39739 0.698694 − 0.715421i \(-0.253765\pi\)
0.698694 + 0.715421i \(0.253765\pi\)
\(312\) 0 0
\(313\) −5.52866e64 −0.721884 −0.360942 − 0.932588i \(-0.617545\pi\)
−0.360942 + 0.932588i \(0.617545\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.53125e65 1.43716 0.718582 − 0.695442i \(-0.244791\pi\)
0.718582 + 0.695442i \(0.244791\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 2.33876e65 1.71825
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 2.74721e65 1.46124
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.72016e65 −0.524856 −0.262428 − 0.964952i \(-0.584523\pi\)
−0.262428 + 0.964952i \(0.584523\pi\)
\(332\) 0 0
\(333\) 1.47111e66 3.83792
\(334\) 0 0
\(335\) −1.28104e66 −2.86021
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 6.21401e65 1.01901
\(340\) 0 0
\(341\) 8.62794e65 1.21421
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.17618e66 −5.37914
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.49260e66 1.99989 0.999945 − 0.0104526i \(-0.00332722\pi\)
0.999945 + 0.0104526i \(0.00332722\pi\)
\(354\) 0 0
\(355\) −6.90253e66 −3.41249
\(356\) 3.26185e66 1.49887
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 3.12743e66 1.00000
\(362\) 0 0
\(363\) −6.40460e66 −1.77386
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.40596e66 0.917747 0.458874 − 0.888502i \(-0.348253\pi\)
0.458874 + 0.888502i \(0.348253\pi\)
\(368\) 9.09391e66 1.76485
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.47004e67 −2.15384
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −2.44134e67 −2.90279
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.79947e66 0.252614 0.126307 − 0.991991i \(-0.459688\pi\)
0.126307 + 0.991991i \(0.459688\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.62897e66 −0.592640 −0.296320 − 0.955089i \(-0.595759\pi\)
−0.296320 + 0.955089i \(0.595759\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −3.01638e67 −1.47864
\(389\) 5.91629e66 0.271244 0.135622 − 0.990761i \(-0.456697\pi\)
0.135622 + 0.990761i \(0.456697\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 7.44428e67 2.14658
\(397\) −6.38735e67 −1.72492 −0.862458 − 0.506129i \(-0.831076\pi\)
−0.862458 + 0.506129i \(0.831076\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 8.79274e67 1.95238
\(401\) 9.60227e67 1.99811 0.999057 − 0.0434110i \(-0.0138225\pi\)
0.999057 + 0.0434110i \(0.0138225\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.56185e68 2.51077
\(406\) 0 0
\(407\) 1.26416e68 1.78792
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −2.98041e68 −3.26880
\(412\) −1.22238e68 −1.25858
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.20960e68 0.803666 0.401833 − 0.915713i \(-0.368373\pi\)
0.401833 + 0.915713i \(0.368373\pi\)
\(420\) 0 0
\(421\) 3.31398e68 1.94543 0.972714 − 0.232007i \(-0.0745293\pi\)
0.972714 + 0.232007i \(0.0745293\pi\)
\(422\) 0 0
\(423\) 3.56239e68 1.84881
\(424\) 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 \(\pi\)
\(432\) −6.77490e68 −2.03388
\(433\) −5.17328e68 −1.46245 −0.731224 − 0.682137i \(-0.761051\pi\)
−0.731224 + 0.682137i \(0.761051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.22223e69 2.14658
\(442\) 0 0
\(443\) −5.63817e68 −0.880321 −0.440160 − 0.897919i \(-0.645078\pi\)
−0.440160 + 0.897919i \(0.645078\pi\)
\(444\) −2.15390e69 −3.17152
\(445\) 1.85443e69 2.57543
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.49017e68 −0.934354 −0.467177 − 0.884164i \(-0.654729\pi\)
−0.467177 + 0.884164i \(0.654729\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.20668e68 −0.574461
\(453\) 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
\(460\) 5.17008e69 3.03245
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −2.23918e69 −1.10913 −0.554566 − 0.832140i \(-0.687116\pi\)
−0.554566 + 0.832140i \(0.687116\pi\)
\(464\) 0 0
\(465\) −8.35749e69 −3.70083
\(466\) 0 0
\(467\) −4.50054e69 −1.78248 −0.891241 − 0.453529i \(-0.850165\pi\)
−0.891241 + 0.453529i \(0.850165\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 8.10530e69 2.57173
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.87191e70 −4.27373
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 6.39704e69 1.00000
\(485\) −1.71488e70 −2.54067
\(486\) 0 0
\(487\) 1.10737e70 1.47415 0.737073 − 0.675813i \(-0.236207\pi\)
0.737073 + 0.675813i \(0.236207\pi\)
\(488\) 0 0
\(489\) 2.34835e70 2.81017
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 4.23223e70 3.68837
\(496\) 1.46831e70 1.21421
\(497\) 0 0
\(498\) 0 0
\(499\) 1.21362e70 0.857963 0.428982 − 0.903313i \(-0.358873\pi\)
0.428982 + 0.903313i \(0.358873\pi\)
\(500\) 2.43847e70 1.63643
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.79426e70 −1.77386
\(508\) 0 0
\(509\) −3.67291e70 −1.55006 −0.775030 − 0.631925i \(-0.782265\pi\)
−0.775030 + 0.631925i \(0.782265\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.94950e70 −2.16255
\(516\) 0 0
\(517\) 3.06124e70 0.861278
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.13354e70 0.261010 0.130505 − 0.991448i \(-0.458340\pi\)
0.130505 + 0.991448i \(0.458340\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −1.08994e71 −1.77386
\(529\) 1.36488e71 2.11469
\(530\) 0 0
\(531\) 9.61499e70 1.35049
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 3.31547e71 3.47707
\(538\) 0 0
\(539\) 1.05029e71 1.00000
\(540\) −3.85167e71 −3.49471
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −2.99115e71 −2.34987
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 2.97689e71 1.84276
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.22454e72 −5.44947
\(556\) 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 \(\pi\)
\(564\) −5.21579e71 −1.52779
\(565\) −3.52863e71 −0.987066
\(566\) 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
\(573\) 3.03341e70 0.0588723
\(574\) 0 0
\(575\) 1.94373e72 3.44565
\(576\) 1.26687e72 2.14658
\(577\) −1.81317e71 −0.293675 −0.146837 − 0.989161i \(-0.546909\pi\)
−0.146837 + 0.989161i \(0.546909\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.60857e72 −1.99094
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.26288e72 1.30850 0.654248 − 0.756280i \(-0.272985\pi\)
0.654248 + 0.756280i \(0.272985\pi\)
\(588\) −1.78949e72 −1.77386
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 2.15136e72 1.78792
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.26702e72 3.51749
\(598\) 0 0
\(599\) 1.73104e72 1.05976 0.529882 − 0.848072i \(-0.322236\pi\)
0.529882 + 0.848072i \(0.322236\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −6.93920e72 −3.57321
\(604\) 0 0
\(605\) 3.63685e72 1.71825
\(606\) 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
\(617\) −3.66767e72 −1.03986 −0.519931 − 0.854208i \(-0.674042\pi\)
−0.519931 + 0.854208i \(0.674042\pi\)
\(618\) 0 0
\(619\) 3.85063e72 1.00363 0.501817 − 0.864974i \(-0.332665\pi\)
0.501817 + 0.864974i \(0.332665\pi\)
\(620\) 8.34764e72 2.08631
\(621\) −1.49766e73 −3.58949
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 4.23721e72 0.859408
\(626\) 0 0
\(627\) 0 0
\(628\) −8.09574e72 −1.44979
\(629\) 0 0
\(630\) 0 0
\(631\) 6.98082e72 1.10444 0.552221 − 0.833698i \(-0.313780\pi\)
0.552221 + 0.833698i \(0.313780\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 2.74071e73 3.53166
\(637\) 0 0
\(638\) 0 0
\(639\) −3.73900e73 −4.26317
\(640\) 0 0
\(641\) 1.77720e73 1.86822 0.934108 − 0.356990i \(-0.116197\pi\)
0.934108 + 0.356990i \(0.116197\pi\)
\(642\) 0 0
\(643\) 5.36842e72 0.520428 0.260214 − 0.965551i \(-0.416207\pi\)
0.260214 + 0.965551i \(0.416207\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.65680e73 −1.36697 −0.683484 − 0.729965i \(-0.739536\pi\)
−0.683484 + 0.729965i \(0.739536\pi\)
\(648\) 0 0
\(649\) 8.26239e72 0.629135
\(650\) 0 0
\(651\) 0 0
\(652\) −2.34558e73 −1.58421
\(653\) −2.99665e73 −1.94488 −0.972442 − 0.233143i \(-0.925099\pi\)
−0.972442 + 0.233143i \(0.925099\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) −6.19652e73 −3.04794
\(661\) 3.16690e73 1.49760 0.748800 − 0.662796i \(-0.230630\pi\)
0.748800 + 0.662796i \(0.230630\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 1.74158e73 0.602373
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.44806e74 −3.97091
\(676\) 3.78978e73 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
\(683\) 9.38657e73 1.89482 0.947411 − 0.320018i \(-0.103689\pi\)
0.947411 + 0.320018i \(0.103689\pi\)
\(684\) 0 0
\(685\) 1.69243e74 3.16632
\(686\) 0 0
\(687\) 1.43314e74 2.48550
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.09105e73 −0.311842 −0.155921 − 0.987770i \(-0.549835\pi\)
−0.155921 + 0.987770i \(0.549835\pi\)
\(692\) 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
\(704\) 1.08865e74 1.00000
\(705\) −2.96528e74 −2.62512
\(706\) 0 0
\(707\) 0 0
\(708\) −1.40776e74 −1.11600
\(709\) 2.59803e74 1.98537 0.992686 − 0.120724i \(-0.0385218\pi\)
0.992686 + 0.120724i \(0.0385218\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.24585e74 2.14289
\(714\) 0 0
\(715\) 0 0
\(716\) −3.31156e74 −1.96017
\(717\) 0 0
\(718\) 0 0
\(719\) −3.02232e74 −1.60468 −0.802342 − 0.596864i \(-0.796413\pi\)
−0.802342 + 0.596864i \(0.796413\pi\)
\(720\) 7.20243e74 3.68837
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 2.98763e74 1.32472
\(725\) 0 0
\(726\) 0 0
\(727\) −4.89591e74 −1.94957 −0.974784 − 0.223149i \(-0.928366\pi\)
−0.974784 + 0.223149i \(0.928366\pi\)
\(728\) 0 0
\(729\) −1.27080e74 −0.471151
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −1.01736e75 −3.04794
\(736\) 0 0
\(737\) −5.96301e74 −1.66460
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1.22309e75 3.07209
\(741\) 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
\(751\) −1.00979e75 −1.72820 −0.864100 − 0.503321i \(-0.832111\pi\)
−0.864100 + 0.503321i \(0.832111\pi\)
\(752\) 5.20964e74 0.861278
\(753\) 2.07373e75 3.31194
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −5.17380e74 −0.719974 −0.359987 − 0.932957i \(-0.617219\pi\)
−0.359987 + 0.932957i \(0.617219\pi\)
\(758\) 0 0
\(759\) −2.40942e75 −3.13060
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3.02983e73 −0.0331888
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.85486e75 −1.77386
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −3.74924e75 −3.23988
\(772\) 0 0
\(773\) 2.47378e75 1.99845 0.999227 − 0.0393104i \(-0.0125161\pi\)
0.999227 + 0.0393104i \(0.0125161\pi\)
\(774\) 0 0
\(775\) 3.13836e75 2.37060
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −3.21301e75 −1.98603
\(782\) 0 0
\(783\) 0 0
\(784\) 1.78738e75 1.00000
\(785\) −4.60260e75 −2.49110
\(786\) 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
\(795\) 1.55815e76 6.06827
\(796\) −5.26081e75 −1.98295
\(797\) −5.09632e75 −1.85926 −0.929630 − 0.368495i \(-0.879873\pi\)
−0.929630 + 0.368495i \(0.879873\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 1.00452e76 3.21745
\(802\) 0 0
\(803\) 0 0
\(804\) 1.01599e76 2.95278
\(805\) 0 0
\(806\) 0 0
\(807\) 1.26769e76 3.34428
\(808\) 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
\(815\) −1.33351e76 −2.72207
\(816\) 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
\(823\) −1.24335e76 −1.96879 −0.984393 − 0.175985i \(-0.943689\pi\)
−0.984393 + 0.175985i \(0.943689\pi\)
\(824\) 0 0
\(825\) −2.32963e76 −3.46325
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 2.80055e76 3.78839
\(829\) −1.02913e76 −1.34912 −0.674561 − 0.738219i \(-0.735667\pi\)
−0.674561 + 0.738219i \(0.735667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.41814e76 −2.46956
\(838\) 0 0
\(839\) −1.34767e76 −1.29352 −0.646759 − 0.762694i \(-0.723876\pi\)
−0.646759 + 0.762694i \(0.723876\pi\)
\(840\) 0 0
\(841\) 1.10840e76 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.15457e76 1.71825
\(846\) 0 0
\(847\) 0 0
\(848\) −2.73748e76 −1.99094
\(849\) 0 0
\(850\) 0 0
\(851\) 4.75580e76 3.15541
\(852\) 5.47437e76 3.52294
\(853\) 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
\(859\) −2.93407e76 −1.52633 −0.763164 − 0.646205i \(-0.776355\pi\)
−0.763164 + 0.646205i \(0.776355\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.18885e76 −1.93114 −0.965571 − 0.260140i \(-0.916231\pi\)
−0.965571 + 0.260140i \(0.916231\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.33927e76 −1.77386
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −9.28923e76 −3.17402
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 6.18922e76 1.71825
\(881\) −4.09677e76 −1.10425 −0.552126 − 0.833761i \(-0.686183\pi\)
−0.552126 + 0.833761i \(0.686183\pi\)
\(882\) 0 0
\(883\) 3.60568e76 0.916239 0.458120 − 0.888891i \(-0.348523\pi\)
0.458120 + 0.888891i \(0.348523\pi\)
\(884\) 0 0
\(885\) −8.00340e76 −1.91756
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 7.27012e76 1.46124
\(892\) −1.73953e76 −0.339583
\(893\) 0 0
\(894\) 0 0
\(895\) −1.88269e77 −3.36806
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 2.70781e77 4.19095
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.69853e77 2.27620
\(906\) 0 0
\(907\) 1.26418e77 1.59963 0.799815 − 0.600247i \(-0.204931\pi\)
0.799815 + 0.600247i \(0.204931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.50372e77 1.69703 0.848517 − 0.529168i \(-0.177496\pi\)
0.848517 + 0.529168i \(0.177496\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.43145e77 −1.40118
\(917\) 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
\(925\) 4.59830e77 3.49070
\(926\) 0 0
\(927\) −3.76444e77 −2.70164
\(928\) 0 0
\(929\) −1.54717e76 −0.104986 −0.0524929 − 0.998621i \(-0.516717\pi\)
−0.0524929 + 0.998621i \(0.516717\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4.08469e77 −2.47877
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 2.49283e77 1.28052
\(940\) 2.96179e77 1.47989
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.40610e77 0.629135
\(945\) 0 0
\(946\) 0 0
\(947\) 1.46879e77 0.605146 0.302573 − 0.953126i \(-0.402154\pi\)
0.302573 + 0.953126i \(0.402154\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −6.90430e77 −2.54933
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −1.72252e76 −0.0570266
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −1.05453e78 −3.04794
\(961\) 1.68603e77 0.474304
\(962\) 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
\(971\) −4.12466e77 −0.886520 −0.443260 − 0.896393i \(-0.646178\pi\)
−0.443260 + 0.896393i \(0.646178\pi\)
\(972\) −2.66733e77 −0.558154
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.62826e77 −0.664416 −0.332208 − 0.943206i \(-0.607794\pi\)
−0.332208 + 0.943206i \(0.607794\pi\)
\(978\) 0 0
\(979\) 8.63205e77 1.49887
\(980\) 1.01616e78 1.71825
\(981\) 0 0
\(982\) 0 0
\(983\) 8.22163e77 1.28400 0.642002 − 0.766703i \(-0.278104\pi\)
0.642002 + 0.766703i \(0.278104\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −7.14879e77 −0.904311 −0.452155 − 0.891939i \(-0.649345\pi\)
−0.452155 + 0.891939i \(0.649345\pi\)
\(992\) 0 0
\(993\) 7.75604e77 0.931022
\(994\) 0 0
\(995\) −2.99088e78 −3.40721
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −3.54304e78 −3.63642
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 11.53.b.a.10.1 ✓ 1
11.10 odd 2 CM 11.53.b.a.10.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.53.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.53.b.a.10.1 ✓ 1 11.10 odd 2 CM