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Properties

Label 11.9.b.a.10.1
Level $11$
Weight $9$
Character 11.10
Self dual yes
Analytic conductor $4.481$
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] = [11,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(4.48116471067\)
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)
 
\(f(q)\) \(=\) \(q-113.000 q^{3} +256.000 q^{4} +1151.00 q^{5} +6208.00 q^{9} +O(q^{10})\) \(q-113.000 q^{3} +256.000 q^{4} +1151.00 q^{5} +6208.00 q^{9} +14641.0 q^{11} -28928.0 q^{12} -130063. q^{15} +65536.0 q^{16} +294656. q^{20} -531793. q^{23} +934176. q^{25} +39889.0 q^{27} -1.54123e6 q^{31} -1.65443e6 q^{33} +1.58925e6 q^{36} +716447. q^{37} +3.74810e6 q^{44} +7.14541e6 q^{45} -6.08064e6 q^{47} -7.40557e6 q^{48} +5.76480e6 q^{49} -1.52654e7 q^{53} +1.68518e7 q^{55} -4.10155e6 q^{59} -3.32961e7 q^{60} +1.67772e7 q^{64} +1.98068e7 q^{67} +6.00926e7 q^{69} +7.04309e6 q^{71} -1.05562e8 q^{75} +7.54319e7 q^{80} -4.52381e7 q^{81} -8.41010e7 q^{89} -1.36139e8 q^{92} +1.74159e8 q^{93} -8.11557e7 q^{97} +9.08913e7 q^{99} +O(q^{100})\)

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 \(\pi\)
\(3\) −113.000 −1.39506 −0.697531 − 0.716555i \(-0.745718\pi\)
−0.697531 + 0.716555i \(0.745718\pi\)
\(4\) 256.000 1.00000
\(5\) 1151.00 1.84160 0.920800 − 0.390035i \(-0.127537\pi\)
0.920800 + 0.390035i \(0.127537\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 6208.00 0.946197
\(10\) 0 0
\(11\) 14641.0 1.00000
\(12\) −28928.0 −1.39506
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −130063. −2.56915
\(16\) 65536.0 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 294656. 1.84160
\(21\) 0 0
\(22\) 0 0
\(23\) −531793. −1.90034 −0.950170 − 0.311732i \(-0.899091\pi\)
−0.950170 + 0.311732i \(0.899091\pi\)
\(24\) 0 0
\(25\) 934176. 2.39149
\(26\) 0 0
\(27\) 39889.0 0.0750582
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.54123e6 −1.66887 −0.834433 − 0.551109i \(-0.814205\pi\)
−0.834433 + 0.551109i \(0.814205\pi\)
\(32\) 0 0
\(33\) −1.65443e6 −1.39506
\(34\) 0 0
\(35\) 0 0
\(36\) 1.58925e6 0.946197
\(37\) 716447. 0.382276 0.191138 − 0.981563i \(-0.438782\pi\)
0.191138 + 0.981563i \(0.438782\pi\)
\(38\) 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 \(\pi\)
\(44\) 3.74810e6 1.00000
\(45\) 7.14541e6 1.74252
\(46\) 0 0
\(47\) −6.08064e6 −1.24611 −0.623057 − 0.782177i \(-0.714110\pi\)
−0.623057 + 0.782177i \(0.714110\pi\)
\(48\) −7.40557e6 −1.39506
\(49\) 5.76480e6 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.52654e7 −1.93467 −0.967333 − 0.253511i \(-0.918415\pi\)
−0.967333 + 0.253511i \(0.918415\pi\)
\(54\) 0 0
\(55\) 1.68518e7 1.84160
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.10155e6 −0.338486 −0.169243 − 0.985574i \(-0.554132\pi\)
−0.169243 + 0.985574i \(0.554132\pi\)
\(60\) −3.32961e7 −2.56915
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.67772e7 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.98068e7 0.982911 0.491456 − 0.870903i \(-0.336465\pi\)
0.491456 + 0.870903i \(0.336465\pi\)
\(68\) 0 0
\(69\) 6.00926e7 2.65109
\(70\) 0 0
\(71\) 7.04309e6 0.277159 0.138580 − 0.990351i \(-0.455746\pi\)
0.138580 + 0.990351i \(0.455746\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −1.05562e8 −3.33628
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 7.54319e7 1.84160
\(81\) −4.52381e7 −1.05091
\(82\) 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
\(89\) −8.41010e7 −1.34042 −0.670210 − 0.742171i \(-0.733796\pi\)
−0.670210 + 0.742171i \(0.733796\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.36139e8 −1.90034
\(93\) 1.74159e8 2.32817
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.11557e7 −0.916710 −0.458355 − 0.888769i \(-0.651561\pi\)
−0.458355 + 0.888769i \(0.651561\pi\)
\(98\) 0 0
\(99\) 9.08913e7 0.946197
\(100\) 2.39149e8 2.39149
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −3.62784e6 −0.0322329 −0.0161164 − 0.999870i \(-0.505130\pi\)
−0.0161164 + 0.999870i \(0.505130\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 1.02116e7 0.0750582
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −8.09585e7 −0.533299
\(112\) 0 0
\(113\) 1.01857e8 0.624710 0.312355 − 0.949966i \(-0.398882\pi\)
0.312355 + 0.949966i \(0.398882\pi\)
\(114\) 0 0
\(115\) −6.12094e8 −3.49967
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −3.94556e8 −1.66887
\(125\) 6.25627e8 2.56257
\(126\) 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 \(\pi\)
\(132\) −4.23535e8 −1.39506
\(133\) 0 0
\(134\) 0 0
\(135\) 4.59122e7 0.138227
\(136\) 0 0
\(137\) 3.63889e8 1.03297 0.516484 − 0.856297i \(-0.327241\pi\)
0.516484 + 0.856297i \(0.327241\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 6.87112e8 1.73841
\(142\) 0 0
\(143\) 0 0
\(144\) 4.06847e8 0.946197
\(145\) 0 0
\(146\) 0 0
\(147\) −6.51423e8 −1.39506
\(148\) 1.83410e8 0.382276
\(149\) 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
\(155\) −1.77396e9 −3.07338
\(156\) 0 0
\(157\) −1.20570e9 −1.98446 −0.992229 − 0.124428i \(-0.960290\pi\)
−0.992229 + 0.124428i \(0.960290\pi\)
\(158\) 0 0
\(159\) 1.72499e9 2.69898
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.15081e8 1.29631 0.648155 − 0.761508i \(-0.275541\pi\)
0.648155 + 0.761508i \(0.275541\pi\)
\(164\) 0 0
\(165\) −1.90425e9 −2.56915
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.15731e8 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
\(176\) 9.59513e8 1.00000
\(177\) 4.63475e8 0.472208
\(178\) 0 0
\(179\) 1.21779e9 1.18620 0.593102 − 0.805127i \(-0.297903\pi\)
0.593102 + 0.805127i \(0.297903\pi\)
\(180\) 1.82922e9 1.74252
\(181\) −2.13327e9 −1.98761 −0.993804 − 0.111149i \(-0.964547\pi\)
−0.993804 + 0.111149i \(0.964547\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.24630e8 0.704000
\(186\) 0 0
\(187\) 0 0
\(188\) −1.55664e9 −1.24611
\(189\) 0 0
\(190\) 0 0
\(191\) −3.27581e8 −0.246142 −0.123071 − 0.992398i \(-0.539274\pi\)
−0.123071 + 0.992398i \(0.539274\pi\)
\(192\) −1.89583e9 −1.39506
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.47579e9 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 3.13584e9 1.99960 0.999798 − 0.0200992i \(-0.00639822\pi\)
0.999798 + 0.0200992i \(0.00639822\pi\)
\(200\) 0 0
\(201\) −2.23816e9 −1.37122
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.30137e9 −1.79810
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −3.90795e9 −1.93467
\(213\) −7.95869e8 −0.386655
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 4.31406e9 1.84160
\(221\) 0 0
\(222\) 0 0
\(223\) 4.76951e9 1.92865 0.964326 − 0.264716i \(-0.0852784\pi\)
0.964326 + 0.264716i \(0.0852784\pi\)
\(224\) 0 0
\(225\) 5.79936e9 2.26282
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.32380e9 0.481371 0.240686 − 0.970603i \(-0.422628\pi\)
0.240686 + 0.970603i \(0.422628\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −6.99881e9 −2.29484
\(236\) −1.05000e9 −0.338486
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −8.52381e9 −2.56915
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 4.85020e9 1.39102
\(244\) 0 0
\(245\) 6.63529e9 1.84160
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.39202e9 −0.602658 −0.301329 − 0.953520i \(-0.597430\pi\)
−0.301329 + 0.953520i \(0.597430\pi\)
\(252\) 0 0
\(253\) −7.78598e9 −1.90034
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 4.90675e8 0.112476 0.0562382 − 0.998417i \(-0.482089\pi\)
0.0562382 + 0.998417i \(0.482089\pi\)
\(258\) 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
\(265\) −1.75705e10 −3.56288
\(266\) 0 0
\(267\) 9.50341e9 1.86997
\(268\) 5.07053e9 0.982911
\(269\) −1.02851e10 −1.96427 −0.982135 − 0.188178i \(-0.939742\pi\)
−0.982135 + 0.188178i \(0.939742\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.36773e10 2.39149
\(276\) 1.53837e10 2.65109
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −9.56797e9 −1.57908
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.80303e9 0.277159
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 9.17060e9 1.27887
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −4.72089e9 −0.623355
\(296\) 0 0
\(297\) 5.84015e8 0.0750582
\(298\) 0 0
\(299\) 0 0
\(300\) −2.70238e10 −3.33628
\(301\) 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
\(309\) 4.09946e8 0.0449668
\(310\) 0 0
\(311\) −1.74820e10 −1.86874 −0.934369 − 0.356306i \(-0.884036\pi\)
−0.934369 + 0.356306i \(0.884036\pi\)
\(312\) 0 0
\(313\) −3.39209e9 −0.353419 −0.176710 − 0.984263i \(-0.556545\pi\)
−0.176710 + 0.984263i \(0.556545\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.66499e10 1.64882 0.824411 − 0.565991i \(-0.191506\pi\)
0.824411 + 0.565991i \(0.191506\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.93106e10 1.84160
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.15810e10 −1.05091
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.44332e10 −1.20240 −0.601202 − 0.799097i \(-0.705311\pi\)
−0.601202 + 0.799097i \(0.705311\pi\)
\(332\) 0 0
\(333\) 4.44770e9 0.361709
\(334\) 0 0
\(335\) 2.27976e10 1.81013
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −1.15099e10 −0.871508
\(340\) 0 0
\(341\) −2.25652e10 −1.66887
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.91666e10 4.88225
\(346\) 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
\(353\) −1.10589e10 −0.712216 −0.356108 − 0.934445i \(-0.615896\pi\)
−0.356108 + 0.934445i \(0.615896\pi\)
\(354\) 0 0
\(355\) 8.10659e9 0.510417
\(356\) −2.15299e10 −1.34042
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.69836e10 1.00000
\(362\) 0 0
\(363\) −2.42226e10 −1.39506
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.83672e10 −1.01246 −0.506232 − 0.862397i \(-0.668962\pi\)
−0.506232 + 0.862397i \(0.668962\pi\)
\(368\) −3.48516e10 −1.90034
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 4.45848e10 2.32817
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −7.06959e10 −3.57494
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.14149e10 1.52258 0.761288 − 0.648414i \(-0.224567\pi\)
0.761288 + 0.648414i \(0.224567\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.34982e10 1.55678 0.778389 − 0.627782i \(-0.216037\pi\)
0.778389 + 0.627782i \(0.216037\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −2.07759e10 −0.916710
\(389\) −4.57861e10 −1.99956 −0.999781 − 0.0209279i \(-0.993338\pi\)
−0.999781 + 0.0209279i \(0.993338\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 2.32682e10 0.946197
\(397\) 4.57269e10 1.84081 0.920407 − 0.390963i \(-0.127858\pi\)
0.920407 + 0.390963i \(0.127858\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.12222e10 2.39149
\(401\) 4.56288e10 1.76466 0.882332 − 0.470628i \(-0.155973\pi\)
0.882332 + 0.470628i \(0.155973\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −5.20691e10 −1.93535
\(406\) 0 0
\(407\) 1.04895e10 0.382276
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −4.11195e10 −1.44105
\(412\) −9.28727e8 −0.0322329
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.81505e10 0.588887 0.294444 − 0.955669i \(-0.404866\pi\)
0.294444 + 0.955669i \(0.404866\pi\)
\(420\) 0 0
\(421\) −2.43806e10 −0.776097 −0.388048 − 0.921639i \(-0.626851\pi\)
−0.388048 + 0.921639i \(0.626851\pi\)
\(422\) 0 0
\(423\) −3.77486e10 −1.17907
\(424\) 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 \(\pi\)
\(432\) 2.61417e9 0.0750582
\(433\) 6.98359e10 1.98668 0.993338 − 0.115233i \(-0.0367614\pi\)
0.993338 + 0.115233i \(0.0367614\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.57879e10 0.946197
\(442\) 0 0
\(443\) −3.92076e10 −1.01802 −0.509009 − 0.860761i \(-0.669988\pi\)
−0.509009 + 0.860761i \(0.669988\pi\)
\(444\) −2.07254e10 −0.533299
\(445\) −9.68002e10 −2.46852
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.10587e10 −1.99441 −0.997205 − 0.0747142i \(-0.976196\pi\)
−0.997205 + 0.0747142i \(0.976196\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 2.60755e10 0.624710
\(453\) 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
\(460\) −1.56696e11 −3.49967
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 9.08592e10 1.97717 0.988587 − 0.150648i \(-0.0481361\pi\)
0.988587 + 0.150648i \(0.0481361\pi\)
\(464\) 0 0
\(465\) 2.00457e11 4.28756
\(466\) 0 0
\(467\) 8.08102e10 1.69902 0.849510 − 0.527573i \(-0.176898\pi\)
0.849510 + 0.527573i \(0.176898\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 1.36244e11 2.76844
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −9.47678e10 −1.83057
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.48759e10 1.00000
\(485\) −9.34102e10 −1.68821
\(486\) 0 0
\(487\) −2.76502e10 −0.491566 −0.245783 − 0.969325i \(-0.579045\pi\)
−0.245783 + 0.969325i \(0.579045\pi\)
\(488\) 0 0
\(489\) −1.03404e11 −1.80843
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 1.04616e11 1.74252
\(496\) −1.01006e11 −1.66887
\(497\) 0 0
\(498\) 0 0
\(499\) −1.05616e11 −1.70345 −0.851723 − 0.523993i \(-0.824442\pi\)
−0.851723 + 0.523993i \(0.824442\pi\)
\(500\) 1.60161e11 2.56257
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.21776e10 −1.39506
\(508\) 0 0
\(509\) 1.11658e11 1.66348 0.831742 − 0.555162i \(-0.187344\pi\)
0.831742 + 0.555162i \(0.187344\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.17564e9 −0.0593601
\(516\) 0 0
\(517\) −8.90266e10 −1.24611
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.31834e11 −1.78927 −0.894633 − 0.446801i \(-0.852563\pi\)
−0.894633 + 0.446801i \(0.852563\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −1.08425e11 −1.39506
\(529\) 2.04493e11 2.61129
\(530\) 0 0
\(531\) −2.54624e10 −0.320274
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.37610e11 −1.65483
\(538\) 0 0
\(539\) 8.44025e10 1.00000
\(540\) 1.17535e10 0.138227
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 2.41059e11 2.77284
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 9.31556e10 1.03297
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −9.31832e10 −0.982123
\(556\) 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 \(\pi\)
\(564\) 1.75901e11 1.73841
\(565\) 1.17238e11 1.15047
\(566\) 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
\(573\) 3.70166e10 0.343383
\(574\) 0 0
\(575\) −4.96788e11 −4.54464
\(576\) 1.04153e11 0.946197
\(577\) −2.17251e10 −0.196001 −0.0980006 − 0.995186i \(-0.531245\pi\)
−0.0980006 + 0.995186i \(0.531245\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −2.23501e11 −1.93467
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.08009e11 0.909716 0.454858 − 0.890564i \(-0.349690\pi\)
0.454858 + 0.890564i \(0.349690\pi\)
\(588\) −1.66764e11 −1.39506
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 4.69531e10 0.382276
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.54350e11 −2.78956
\(598\) 0 0
\(599\) 2.43785e11 1.89365 0.946824 − 0.321751i \(-0.104271\pi\)
0.946824 + 0.321751i \(0.104271\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 1.22960e11 0.930028
\(604\) 0 0
\(605\) 2.46727e11 1.84160
\(606\) 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
\(617\) −5.89891e10 −0.407035 −0.203517 − 0.979071i \(-0.565237\pi\)
−0.203517 + 0.979071i \(0.565237\pi\)
\(618\) 0 0
\(619\) 2.89838e11 1.97420 0.987102 − 0.160093i \(-0.0511793\pi\)
0.987102 + 0.160093i \(0.0511793\pi\)
\(620\) −4.54134e11 −3.07338
\(621\) −2.12127e10 −0.142636
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.55184e11 2.32774
\(626\) 0 0
\(627\) 0 0
\(628\) −3.08660e11 −1.98446
\(629\) 0 0
\(630\) 0 0
\(631\) 1.38624e11 0.874422 0.437211 − 0.899359i \(-0.355966\pi\)
0.437211 + 0.899359i \(0.355966\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 4.41599e11 2.69898
\(637\) 0 0
\(638\) 0 0
\(639\) 4.37235e10 0.262247
\(640\) 0 0
\(641\) −2.64901e11 −1.56910 −0.784552 − 0.620063i \(-0.787107\pi\)
−0.784552 + 0.620063i \(0.787107\pi\)
\(642\) 0 0
\(643\) 1.08190e11 0.632914 0.316457 − 0.948607i \(-0.397507\pi\)
0.316457 + 0.948607i \(0.397507\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.21502e10 −0.468804 −0.234402 − 0.972140i \(-0.575313\pi\)
−0.234402 + 0.972140i \(0.575313\pi\)
\(648\) 0 0
\(649\) −6.00508e10 −0.338486
\(650\) 0 0
\(651\) 0 0
\(652\) 2.34261e11 1.29631
\(653\) 3.15668e11 1.73611 0.868056 − 0.496466i \(-0.165369\pi\)
0.868056 + 0.496466i \(0.165369\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) −4.87489e11 −2.56915
\(661\) −1.74259e11 −0.912827 −0.456413 − 0.889768i \(-0.650866\pi\)
−0.456413 + 0.889768i \(0.650866\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.38954e11 −2.69059
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 3.72633e10 0.179501
\(676\) 2.08827e11 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
\(683\) −3.39818e11 −1.56158 −0.780790 − 0.624794i \(-0.785183\pi\)
−0.780790 + 0.624794i \(0.785183\pi\)
\(684\) 0 0
\(685\) 4.18837e11 1.90231
\(686\) 0 0
\(687\) −1.49589e11 −0.671543
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −3.98526e11 −1.74801 −0.874007 − 0.485914i \(-0.838487\pi\)
−0.874007 + 0.485914i \(0.838487\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.45635e11 1.00000
\(705\) 7.90866e11 3.20145
\(706\) 0 0
\(707\) 0 0
\(708\) 1.18650e11 0.472208
\(709\) −4.92858e11 −1.95046 −0.975229 − 0.221198i \(-0.929003\pi\)
−0.975229 + 0.221198i \(0.929003\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.19617e11 3.17141
\(714\) 0 0
\(715\) 0 0
\(716\) 3.11754e11 1.18620
\(717\) 0 0
\(718\) 0 0
\(719\) −4.32961e11 −1.62007 −0.810035 − 0.586382i \(-0.800552\pi\)
−0.810035 + 0.586382i \(0.800552\pi\)
\(720\) 4.68281e11 1.74252
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −5.46116e11 −1.98761
\(725\) 0 0
\(726\) 0 0
\(727\) 4.81038e10 0.172203 0.0861017 − 0.996286i \(-0.472559\pi\)
0.0861017 + 0.996286i \(0.472559\pi\)
\(728\) 0 0
\(729\) −2.51265e11 −0.889655
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) −7.49787e11 −2.56915
\(736\) 0 0
\(737\) 2.89991e11 0.982911
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 2.11105e11 0.704000
\(741\) 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
\(751\) −6.03334e11 −1.89670 −0.948349 − 0.317228i \(-0.897248\pi\)
−0.948349 + 0.317228i \(0.897248\pi\)
\(752\) −3.98501e11 −1.24611
\(753\) 2.70299e11 0.840745
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.45563e11 1.96587 0.982935 − 0.183953i \(-0.0588894\pi\)
0.982935 + 0.183953i \(0.0588894\pi\)
\(758\) 0 0
\(759\) 8.79816e11 2.65109
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.38607e10 −0.246142
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −4.85331e11 −1.39506
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −5.54463e10 −0.156912
\(772\) 0 0
\(773\) −2.49173e11 −0.697884 −0.348942 − 0.937144i \(-0.613459\pi\)
−0.348942 + 0.937144i \(0.613459\pi\)
\(774\) 0 0
\(775\) −1.43978e12 −3.99108
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 1.03118e11 0.277159
\(782\) 0 0
\(783\) 0 0
\(784\) 3.77802e11 1.00000
\(785\) −1.38776e12 −3.65458
\(786\) 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
\(795\) 1.98547e12 4.97044
\(796\) 8.02776e11 1.99960
\(797\) 7.35104e11 1.82186 0.910931 − 0.412560i \(-0.135365\pi\)
0.910931 + 0.412560i \(0.135365\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −5.22099e11 −1.26830
\(802\) 0 0
\(803\) 0 0
\(804\) −5.72970e11 −1.37122
\(805\) 0 0
\(806\) 0 0
\(807\) 1.16222e12 2.74028
\(808\) 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
\(815\) 1.05326e12 2.38729
\(816\) 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
\(823\) −3.01900e11 −0.658058 −0.329029 − 0.944320i \(-0.606721\pi\)
−0.329029 + 0.944320i \(0.606721\pi\)
\(824\) 0 0
\(825\) −1.54553e12 −3.33628
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −8.45151e11 −1.79810
\(829\) −4.44761e11 −0.941692 −0.470846 − 0.882215i \(-0.656051\pi\)
−0.470846 + 0.882215i \(0.656051\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.14782e10 −0.125262
\(838\) 0 0
\(839\) 6.66474e11 1.34504 0.672520 − 0.740079i \(-0.265212\pi\)
0.672520 + 0.740079i \(0.265212\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.38906e11 1.84160
\(846\) 0 0
\(847\) 0 0
\(848\) −1.00044e12 −1.93467
\(849\) 0 0
\(850\) 0 0
\(851\) −3.81001e11 −0.726455
\(852\) −2.03742e11 −0.386655
\(853\) 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
\(859\) −2.74046e11 −0.503328 −0.251664 − 0.967815i \(-0.580978\pi\)
−0.251664 + 0.967815i \(0.580978\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 5.92719e11 1.06858 0.534288 − 0.845303i \(-0.320580\pi\)
0.534288 + 0.845303i \(0.320580\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −7.88261e11 −1.39506
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.03815e11 −0.867389
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 1.10440e12 1.84160
\(881\) −7.70772e11 −1.27945 −0.639724 − 0.768605i \(-0.720951\pi\)
−0.639724 + 0.768605i \(0.720951\pi\)
\(882\) 0 0
\(883\) −1.11501e12 −1.83415 −0.917077 − 0.398710i \(-0.869458\pi\)
−0.917077 + 0.398710i \(0.869458\pi\)
\(884\) 0 0
\(885\) 5.33460e11 0.869619
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −6.62332e11 −1.05091
\(892\) 1.22099e12 1.92865
\(893\) 0 0
\(894\) 0 0
\(895\) 1.40167e12 2.18451
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.48464e12 2.26282
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.45539e12 −3.66038
\(906\) 0 0
\(907\) 6.54959e11 0.967798 0.483899 − 0.875124i \(-0.339220\pi\)
0.483899 + 0.875124i \(0.339220\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 8.76790e11 1.27298 0.636491 − 0.771284i \(-0.280385\pi\)
0.636491 + 0.771284i \(0.280385\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 3.38893e11 0.481371
\(917\) 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
\(925\) 6.69288e11 0.914210
\(926\) 0 0
\(927\) −2.25216e10 −0.0304987
\(928\) 0 0
\(929\) −8.36302e11 −1.12279 −0.561397 − 0.827547i \(-0.689736\pi\)
−0.561397 + 0.827547i \(0.689736\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.97546e12 2.60701
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 3.83306e11 0.493042
\(940\) −1.79170e12 −2.29484
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −2.68799e11 −0.338486
\(945\) 0 0
\(946\) 0 0
\(947\) −1.60673e12 −1.99776 −0.998882 − 0.0472733i \(-0.984947\pi\)
−0.998882 + 0.0472733i \(0.984947\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.88144e12 −2.30021
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −3.77045e11 −0.453294
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −2.18210e12 −2.56915
\(961\) 1.52251e12 1.78511
\(962\) 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
\(971\) 1.75196e12 1.97083 0.985413 − 0.170178i \(-0.0544341\pi\)
0.985413 + 0.170178i \(0.0544341\pi\)
\(972\) 1.24165e12 1.39102
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.65543e12 −1.81691 −0.908455 − 0.417982i \(-0.862737\pi\)
−0.908455 + 0.417982i \(0.862737\pi\)
\(978\) 0 0
\(979\) −1.23132e12 −1.34042
\(980\) 1.69863e12 1.84160
\(981\) 0 0
\(982\) 0 0
\(983\) 4.71492e11 0.504964 0.252482 − 0.967602i \(-0.418753\pi\)
0.252482 + 0.967602i \(0.418753\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.63892e12 −1.69927 −0.849635 − 0.527371i \(-0.823178\pi\)
−0.849635 + 0.527371i \(0.823178\pi\)
\(992\) 0 0
\(993\) 1.63095e12 1.67743
\(994\) 0 0
\(995\) 3.60936e12 3.68246
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 2.85784e10 0.0286930
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 11.9.b.a.10.1 ✓ 1
3.2 odd 2 99.9.c.a.10.1 1
4.3 odd 2 176.9.h.a.65.1 1
11.10 odd 2 CM 11.9.b.a.10.1 ✓ 1
33.32 even 2 99.9.c.a.10.1 1
44.43 even 2 176.9.h.a.65.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.9.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.9.b.a.10.1 ✓ 1 11.10 odd 2 CM
99.9.c.a.10.1 1 3.2 odd 2
99.9.c.a.10.1 1 33.32 even 2
176.9.h.a.65.1 1 4.3 odd 2
176.9.h.a.65.1 1 44.43 even 2