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Properties

Label 91.11.b.a.90.1
Level $91$
Weight $11$
Character 91.90
Self dual yes
Analytic conductor $57.818$
Analytic rank $0$
Dimension $1$
CM discriminant -91
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,11,Mod(90,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.90");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 91.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: \(57.8175099933\)
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 90.1
Character \(\chi\) \(=\) 91.90

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1024.00 q^{4} -6243.00 q^{5} +16807.0 q^{7} +59049.0 q^{9} +O(q^{10})\) \(q+1024.00 q^{4} -6243.00 q^{5} +16807.0 q^{7} +59049.0 q^{9} -371293. q^{13} +1.04858e6 q^{16} -2.14738e6 q^{19} -6.39283e6 q^{20} -6.46672e6 q^{23} +2.92094e7 q^{25} +1.72104e7 q^{28} -4.72167e6 q^{29} +4.21792e7 q^{31} -1.04926e8 q^{35} +6.04662e7 q^{36} +2.21229e8 q^{41} -8.43175e7 q^{43} -3.68643e8 q^{45} -4.06711e8 q^{47} +2.82475e8 q^{49} -3.80204e8 q^{52} +5.45144e8 q^{53} +1.33053e9 q^{59} +9.92437e8 q^{63} +1.07374e9 q^{64} +2.31798e9 q^{65} +3.48841e9 q^{73} -2.19891e9 q^{76} +5.01309e9 q^{79} -6.54626e9 q^{80} +3.48678e9 q^{81} -8.93027e8 q^{83} -3.85016e9 q^{89} -6.24032e9 q^{91} -6.62193e9 q^{92} +1.34061e10 q^{95} +9.19527e9 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1024.00 1.00000
\(5\) −6243.00 −1.99776 −0.998880 − 0.0473154i \(-0.984933\pi\)
−0.998880 + 0.0473154i \(0.984933\pi\)
\(6\) 0 0
\(7\) 16807.0 1.00000
\(8\) 0 0
\(9\) 59049.0 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −371293. −1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2.14738e6 −0.867241 −0.433621 − 0.901096i \(-0.642764\pi\)
−0.433621 + 0.901096i \(0.642764\pi\)
\(20\) −6.39283e6 −1.99776
\(21\) 0 0
\(22\) 0 0
\(23\) −6.46672e6 −1.00472 −0.502360 − 0.864658i \(-0.667535\pi\)
−0.502360 + 0.864658i \(0.667535\pi\)
\(24\) 0 0
\(25\) 2.92094e7 2.99105
\(26\) 0 0
\(27\) 0 0
\(28\) 1.72104e7 1.00000
\(29\) −4.72167e6 −0.230200 −0.115100 − 0.993354i \(-0.536719\pi\)
−0.115100 + 0.993354i \(0.536719\pi\)
\(30\) 0 0
\(31\) 4.21792e7 1.47330 0.736648 − 0.676276i \(-0.236407\pi\)
0.736648 + 0.676276i \(0.236407\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.04926e8 −1.99776
\(36\) 6.04662e7 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.21229e8 1.90951 0.954757 − 0.297386i \(-0.0961147\pi\)
0.954757 + 0.297386i \(0.0961147\pi\)
\(42\) 0 0
\(43\) −8.43175e7 −0.573556 −0.286778 − 0.957997i \(-0.592584\pi\)
−0.286778 + 0.957997i \(0.592584\pi\)
\(44\) 0 0
\(45\) −3.68643e8 −1.99776
\(46\) 0 0
\(47\) −4.06711e8 −1.77336 −0.886679 − 0.462386i \(-0.846993\pi\)
−0.886679 + 0.462386i \(0.846993\pi\)
\(48\) 0 0
\(49\) 2.82475e8 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −3.80204e8 −1.00000
\(53\) 5.45144e8 1.30356 0.651781 − 0.758407i \(-0.274022\pi\)
0.651781 + 0.758407i \(0.274022\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.33053e9 1.86108 0.930541 − 0.366188i \(-0.119337\pi\)
0.930541 + 0.366188i \(0.119337\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 9.92437e8 1.00000
\(64\) 1.07374e9 1.00000
\(65\) 2.31798e9 1.99776
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 3.48841e9 1.68273 0.841363 − 0.540471i \(-0.181754\pi\)
0.841363 + 0.540471i \(0.181754\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −2.19891e9 −0.867241
\(77\) 0 0
\(78\) 0 0
\(79\) 5.01309e9 1.62918 0.814592 − 0.580034i \(-0.196961\pi\)
0.814592 + 0.580034i \(0.196961\pi\)
\(80\) −6.54626e9 −1.99776
\(81\) 3.48678e9 1.00000
\(82\) 0 0
\(83\) −8.93027e8 −0.226712 −0.113356 − 0.993554i \(-0.536160\pi\)
−0.113356 + 0.993554i \(0.536160\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.85016e9 −0.689491 −0.344745 − 0.938696i \(-0.612035\pi\)
−0.344745 + 0.938696i \(0.612035\pi\)
\(90\) 0 0
\(91\) −6.24032e9 −1.00000
\(92\) −6.62193e9 −1.00472
\(93\) 0 0
\(94\) 0 0
\(95\) 1.34061e10 1.73254
\(96\) 0 0
\(97\) 9.19527e9 1.07079 0.535397 − 0.844601i \(-0.320162\pi\)
0.535397 + 0.844601i \(0.320162\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 2.99105e10 2.99105
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.89547e10 1.35144 0.675721 − 0.737157i \(-0.263832\pi\)
0.675721 + 0.737157i \(0.263832\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.76234e10 1.00000
\(113\) −2.18534e9 −0.118611 −0.0593057 − 0.998240i \(-0.518889\pi\)
−0.0593057 + 0.998240i \(0.518889\pi\)
\(114\) 0 0
\(115\) 4.03718e10 2.00719
\(116\) −4.83499e9 −0.230200
\(117\) −2.19245e10 −1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.59374e10 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 4.31915e10 1.47330
\(125\) −1.21388e11 −3.97763
\(126\) 0 0
\(127\) −5.18462e10 −1.56927 −0.784636 − 0.619957i \(-0.787150\pi\)
−0.784636 + 0.619957i \(0.787150\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −3.60909e10 −0.867241
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −1.07444e11 −1.99776
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.19174e10 1.00000
\(145\) 2.94774e10 0.459885
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 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
\(155\) −2.63325e11 −2.94329
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.08686e11 −1.00472
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 2.26539e11 1.90951
\(165\) 0 0
\(166\) 0 0
\(167\) 3.48992e10 0.268679 0.134339 − 0.990935i \(-0.457109\pi\)
0.134339 + 0.990935i \(0.457109\pi\)
\(168\) 0 0
\(169\) 1.37858e11 1.00000
\(170\) 0 0
\(171\) −1.26800e11 −0.867241
\(172\) −8.63411e10 −0.573556
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.90923e11 2.99105
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.21896e11 −1.75166 −0.875832 − 0.482616i \(-0.839687\pi\)
−0.875832 + 0.482616i \(0.839687\pi\)
\(180\) −3.77490e11 −1.99776
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −4.16472e11 −1.77336
\(189\) 0 0
\(190\) 0 0
\(191\) 1.18716e11 0.467027 0.233513 − 0.972354i \(-0.424978\pi\)
0.233513 + 0.972354i \(0.424978\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 2.89255e11 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.93572e10 −0.230200
\(204\) 0 0
\(205\) −1.38113e12 −3.81475
\(206\) 0 0
\(207\) −3.81854e11 −1.00472
\(208\) −3.89329e11 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 1.32343e11 0.316438 0.158219 − 0.987404i \(-0.449425\pi\)
0.158219 + 0.987404i \(0.449425\pi\)
\(212\) 5.58227e11 1.30356
\(213\) 0 0
\(214\) 0 0
\(215\) 5.26394e11 1.14583
\(216\) 0 0
\(217\) 7.08906e11 1.47330
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.08160e12 1.96129 0.980643 − 0.195804i \(-0.0627315\pi\)
0.980643 + 0.195804i \(0.0627315\pi\)
\(224\) 0 0
\(225\) 1.72479e12 2.99105
\(226\) 0 0
\(227\) 3.31296e11 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(228\) 0 0
\(229\) −1.37241e11 −0.217924 −0.108962 − 0.994046i \(-0.534753\pi\)
−0.108962 + 0.994046i \(0.534753\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.37253e12 −1.99868 −0.999339 − 0.0363540i \(-0.988426\pi\)
−0.999339 + 0.0363540i \(0.988426\pi\)
\(234\) 0 0
\(235\) 2.53909e12 3.54274
\(236\) 1.36247e12 1.86108
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.48675e12 −1.82875 −0.914373 − 0.404873i \(-0.867316\pi\)
−0.914373 + 0.404873i \(0.867316\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.76349e12 −1.99776
\(246\) 0 0
\(247\) 7.97305e11 0.867241
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.01626e12 1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.09951e12 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 2.37361e12 1.99776
\(261\) −2.78810e11 −0.230200
\(262\) 0 0
\(263\) −2.48588e12 −1.97561 −0.987805 − 0.155693i \(-0.950239\pi\)
−0.987805 + 0.155693i \(0.950239\pi\)
\(264\) 0 0
\(265\) −3.40333e12 −2.60420
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −9.16044e11 −0.626715 −0.313357 − 0.949635i \(-0.601454\pi\)
−0.313357 + 0.949635i \(0.601454\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.96810e12 −1.20683 −0.603417 − 0.797426i \(-0.706194\pi\)
−0.603417 + 0.797426i \(0.706194\pi\)
\(278\) 0 0
\(279\) 2.49064e12 1.47330
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.71820e12 1.90951
\(288\) 0 0
\(289\) 2.01599e12 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 3.57213e12 1.68273
\(293\) 3.19728e12 1.48062 0.740309 − 0.672267i \(-0.234679\pi\)
0.740309 + 0.672267i \(0.234679\pi\)
\(294\) 0 0
\(295\) −8.30651e12 −3.71799
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.40105e12 1.00472
\(300\) 0 0
\(301\) −1.41712e12 −0.573556
\(302\) 0 0
\(303\) 0 0
\(304\) −2.25169e12 −0.867241
\(305\) 0 0
\(306\) 0 0
\(307\) 4.06648e12 1.49117 0.745584 − 0.666411i \(-0.232170\pi\)
0.745584 + 0.666411i \(0.232170\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −6.19578e12 −1.99776
\(316\) 5.13341e12 1.62918
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −6.70337e12 −1.99776
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.57047e12 1.00000
\(325\) −1.08453e13 −2.99105
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.83559e12 −1.77336
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −9.14459e11 −0.226712
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.68394e12 −1.76781 −0.883903 − 0.467671i \(-0.845093\pi\)
−0.883903 + 0.467671i \(0.845093\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.74756e12 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.20009e12 1.23240 0.616198 − 0.787591i \(-0.288672\pi\)
0.616198 + 0.787591i \(0.288672\pi\)
\(348\) 0 0
\(349\) −4.72500e12 −0.912588 −0.456294 − 0.889829i \(-0.650823\pi\)
−0.456294 + 0.889829i \(0.650823\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.05153e13 1.91843 0.959216 − 0.282675i \(-0.0912219\pi\)
0.959216 + 0.282675i \(0.0912219\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.94256e12 −0.689491
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −1.51985e12 −0.247893
\(362\) 0 0
\(363\) 0 0
\(364\) −6.39009e12 −1.00000
\(365\) −2.17782e13 −3.36168
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −6.78085e12 −1.00472
\(369\) 1.30634e13 1.90951
\(370\) 0 0
\(371\) 9.16223e12 1.30356
\(372\) 0 0
\(373\) −1.60857e11 −0.0222790 −0.0111395 − 0.999938i \(-0.503546\pi\)
−0.0111395 + 0.999938i \(0.503546\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.75312e12 0.230200
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 1.37278e13 1.73254
\(381\) 0 0
\(382\) 0 0
\(383\) −6.89122e12 −0.836185 −0.418092 − 0.908405i \(-0.637301\pi\)
−0.418092 + 0.908405i \(0.637301\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.97887e12 −0.573556
\(388\) 9.41596e12 1.07079
\(389\) 1.49157e13 1.67454 0.837272 − 0.546786i \(-0.184149\pi\)
0.837272 + 0.546786i \(0.184149\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.12967e13 −3.25472
\(396\) 0 0
\(397\) 1.21670e13 1.23376 0.616878 − 0.787059i \(-0.288397\pi\)
0.616878 + 0.787059i \(0.288397\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.06283e13 2.99105
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.56609e13 −1.47330
\(404\) 0 0
\(405\) −2.17680e13 −1.99776
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 9.00609e12 0.786900 0.393450 − 0.919346i \(-0.371281\pi\)
0.393450 + 0.919346i \(0.371281\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.23623e13 1.86108
\(414\) 0 0
\(415\) 5.57517e12 0.452916
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −2.40159e13 −1.77336
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.94096e13 1.35144
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.38865e13 0.871335
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.66799e13 1.00000
\(442\) 0 0
\(443\) −2.23772e13 −1.31156 −0.655778 − 0.754953i \(-0.727659\pi\)
−0.655778 + 0.754953i \(0.727659\pi\)
\(444\) 0 0
\(445\) 2.40365e13 1.37744
\(446\) 0 0
\(447\) 0 0
\(448\) 1.80464e13 1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −2.23779e12 −0.118611
\(453\) 0 0
\(454\) 0 0
\(455\) 3.89583e13 1.99776
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 4.13407e13 2.00719
\(461\) −4.83915e12 −0.232415 −0.116207 − 0.993225i \(-0.537074\pi\)
−0.116207 + 0.993225i \(0.537074\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −4.95103e12 −0.230200
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −2.24507e13 −1.00000
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −6.27236e13 −2.59396
\(476\) 0 0
\(477\) 3.21902e13 1.30356
\(478\) 0 0
\(479\) 1.69934e13 0.673913 0.336956 − 0.941520i \(-0.390603\pi\)
0.336956 + 0.941520i \(0.390603\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.65599e13 1.00000
\(485\) −5.74061e13 −2.13919
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.47745e13 −0.517731 −0.258866 − 0.965913i \(-0.583349\pi\)
−0.258866 + 0.965913i \(0.583349\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 4.42281e13 1.47330
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.24301e14 −3.97763
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −5.30905e13 −1.56927
\(509\) 4.76108e13 1.39353 0.696765 − 0.717300i \(-0.254622\pi\)
0.696765 + 0.717300i \(0.254622\pi\)
\(510\) 0 0
\(511\) 5.86297e13 1.68273
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 3.92021e11 0.00946305
\(530\) 0 0
\(531\) 7.85666e13 1.86108
\(532\) −3.69571e13 −0.867241
\(533\) −8.21408e13 −1.90951
\(534\) 0 0
\(535\) −1.18334e14 −2.69986
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.35580e13 1.91049 0.955244 − 0.295820i \(-0.0955927\pi\)
0.955244 + 0.295820i \(0.0955927\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.01392e13 0.199639
\(552\) 0 0
\(553\) 8.42551e13 1.62918
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 3.13065e13 0.573556
\(560\) −1.10023e14 −1.99776
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.36431e13 0.236957
\(566\) 0 0
\(567\) 5.86024e13 1.00000
\(568\) 0 0
\(569\) −1.16675e14 −1.95622 −0.978109 − 0.208094i \(-0.933274\pi\)
−0.978109 + 0.208094i \(0.933274\pi\)
\(570\) 0 0
\(571\) −9.82253e13 −1.61824 −0.809120 − 0.587644i \(-0.800056\pi\)
−0.809120 + 0.587644i \(0.800056\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.88889e14 −3.00516
\(576\) 6.34034e13 1.00000
\(577\) −7.58002e13 −1.18520 −0.592599 − 0.805497i \(-0.701898\pi\)
−0.592599 + 0.805497i \(0.701898\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 3.01849e13 0.459885
\(581\) −1.50091e13 −0.226712
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 1.36875e14 1.99776
\(586\) 0 0
\(587\) −1.09211e14 −1.56703 −0.783515 − 0.621372i \(-0.786575\pi\)
−0.783515 + 0.621372i \(0.786575\pi\)
\(588\) 0 0
\(589\) −9.05746e13 −1.27770
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.35598e14 1.84918 0.924592 − 0.380960i \(-0.124406\pi\)
0.924592 + 0.380960i \(0.124406\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 9.88604e13 1.28200 0.641000 − 0.767541i \(-0.278520\pi\)
0.641000 + 0.767541i \(0.278520\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.61927e14 −1.99776
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.51009e14 1.77336
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −1.78094e14 −1.95973 −0.979865 − 0.199659i \(-0.936017\pi\)
−0.979865 + 0.199659i \(0.936017\pi\)
\(620\) −2.69645e14 −2.94329
\(621\) 0 0
\(622\) 0 0
\(623\) −6.47096e13 −0.689491
\(624\) 0 0
\(625\) 4.72575e14 4.95531
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.23676e14 3.13503
\(636\) 0 0
\(637\) −1.04881e14 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.07870e14 1.92089 0.960443 − 0.278475i \(-0.0898291\pi\)
0.960443 + 0.278475i \(0.0898291\pi\)
\(642\) 0 0
\(643\) −1.53964e14 −1.40076 −0.700382 − 0.713768i \(-0.746987\pi\)
−0.700382 + 0.713768i \(0.746987\pi\)
\(644\) −1.11295e14 −1.00472
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.29249e14 −1.08858 −0.544290 − 0.838897i \(-0.683201\pi\)
−0.544290 + 0.838897i \(0.683201\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.31976e14 1.90951
\(657\) 2.05987e14 1.68273
\(658\) 0 0
\(659\) 1.98003e14 1.59311 0.796553 − 0.604568i \(-0.206654\pi\)
0.796553 + 0.604568i \(0.206654\pi\)
\(660\) 0 0
\(661\) −1.31616e14 −1.04304 −0.521519 − 0.853239i \(-0.674635\pi\)
−0.521519 + 0.853239i \(0.674635\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.25316e14 1.73254
\(666\) 0 0
\(667\) 3.05338e13 0.231287
\(668\) 3.57368e13 0.268679
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −7.58589e13 −0.549454 −0.274727 − 0.961522i \(-0.588587\pi\)
−0.274727 + 0.961522i \(0.588587\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.41167e14 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 1.54545e14 1.07079
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −1.29844e14 −0.867241
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −8.84133e13 −0.573556
\(689\) −2.02408e14 −1.30356
\(690\) 0 0
\(691\) 3.78055e12 0.0239974 0.0119987 − 0.999928i \(-0.496181\pi\)
0.0119987 + 0.999928i \(0.496181\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 5.02705e14 2.99105
\(701\) 7.36283e13 0.434966 0.217483 − 0.976064i \(-0.430215\pi\)
0.217483 + 0.976064i \(0.430215\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 2.96018e14 1.62918
\(712\) 0 0
\(713\) −2.72761e14 −1.48025
\(714\) 0 0
\(715\) 0 0
\(716\) −3.29622e14 −1.75166
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −3.86550e14 −1.99776
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.37917e14 −0.688540
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.05891e14 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −3.73644e14 −1.76578 −0.882892 − 0.469575i \(-0.844407\pi\)
−0.882892 + 0.469575i \(0.844407\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.27323e13 −0.226712
\(748\) 0 0
\(749\) 3.18571e14 1.35144
\(750\) 0 0
\(751\) −2.02906e14 −0.849367 −0.424684 − 0.905342i \(-0.639615\pi\)
−0.424684 + 0.905342i \(0.639615\pi\)
\(752\) −4.26467e14 −1.77336
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.41422e14 1.37345 0.686723 − 0.726919i \(-0.259049\pi\)
0.686723 + 0.726919i \(0.259049\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.81503e14 −1.49477 −0.747386 − 0.664390i \(-0.768691\pi\)
−0.747386 + 0.664390i \(0.768691\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 1.21565e14 0.467027
\(765\) 0 0
\(766\) 0 0
\(767\) −4.94017e14 −1.86108
\(768\) 0 0
\(769\) 5.37552e14 1.99889 0.999446 − 0.0332963i \(-0.0106005\pi\)
0.999446 + 0.0332963i \(0.0106005\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.95584e14 1.43331 0.716657 − 0.697425i \(-0.245671\pi\)
0.716657 + 0.697425i \(0.245671\pi\)
\(774\) 0 0
\(775\) 1.23203e15 4.40670
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.75062e14 −1.65601
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.96197e14 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 1.91809e12 0.00635323 0.00317661 − 0.999995i \(-0.498989\pi\)
0.00317661 + 0.999995i \(0.498989\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.67290e13 −0.118611
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2.27348e14 −0.689491
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 6.78528e14 2.00719
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −6.01495e14 −1.73576 −0.867879 − 0.496775i \(-0.834517\pi\)
−0.867879 + 0.496775i \(0.834517\pi\)
\(810\) 0 0
\(811\) −4.90338e14 −1.39763 −0.698814 − 0.715304i \(-0.746288\pi\)
−0.698814 + 0.715304i \(0.746288\pi\)
\(812\) −8.12617e13 −0.230200
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.81061e14 0.497411
\(818\) 0 0
\(819\) −3.68485e14 −1.00000
\(820\) −1.41428e15 −3.81475
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −5.78672e14 −1.53262 −0.766308 − 0.642474i \(-0.777908\pi\)
−0.766308 + 0.642474i \(0.777908\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −3.91018e14 −1.00472
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.98673e14 −1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) −2.17876e14 −0.536756
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −5.07219e14 −1.22007 −0.610036 − 0.792373i \(-0.708845\pi\)
−0.610036 + 0.792373i \(0.708845\pi\)
\(840\) 0 0
\(841\) −3.98413e14 −0.947008
\(842\) 0 0
\(843\) 0 0
\(844\) 1.35519e14 0.316438
\(845\) −8.60651e14 −1.99776
\(846\) 0 0
\(847\) 4.35930e14 1.00000
\(848\) 5.71624e14 1.30356
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −9.03174e14 −1.99998 −0.999992 − 0.00407606i \(-0.998703\pi\)
−0.999992 + 0.00407606i \(0.998703\pi\)
\(854\) 0 0
\(855\) 7.91615e14 1.73254
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 5.39028e14 1.14583
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 7.25920e14 1.47330
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 5.42972e14 1.07079
\(874\) 0 0
\(875\) −2.04016e15 −3.97763
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 9.82125e14 1.82963 0.914815 − 0.403874i \(-0.132337\pi\)
0.914815 + 0.403874i \(0.132337\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −8.71378e14 −1.56927
\(890\) 0 0
\(891\) 0 0
\(892\) 1.10755e15 1.96129
\(893\) 8.73360e14 1.53793
\(894\) 0 0
\(895\) 2.00960e15 3.49941
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.99157e14 −0.339153
\(900\) 1.76618e15 2.99105
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.90233e14 0.472835 0.236418 − 0.971652i \(-0.424027\pi\)
0.236418 + 0.971652i \(0.424027\pi\)
\(908\) 3.39247e14 0.549650
\(909\) 0 0
\(910\) 0 0
\(911\) −1.19339e15 −1.90192 −0.950958 − 0.309321i \(-0.899898\pi\)
−0.950958 + 0.309321i \(0.899898\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.40535e14 −0.217924
\(917\) 0 0
\(918\) 0 0
\(919\) 1.27940e15 1.95177 0.975884 − 0.218289i \(-0.0700477\pi\)
0.975884 + 0.218289i \(0.0700477\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.14855e15 1.65986 0.829928 − 0.557870i \(-0.188381\pi\)
0.829928 + 0.557870i \(0.188381\pi\)
\(930\) 0 0
\(931\) −6.06580e14 −0.867241
\(932\) −1.40547e15 −1.99868
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.60003e15 3.54274
\(941\) −1.34045e15 −1.81678 −0.908390 − 0.418124i \(-0.862688\pi\)
−0.908390 + 0.418124i \(0.862688\pi\)
\(942\) 0 0
\(943\) −1.43063e15 −1.91853
\(944\) 1.39516e15 1.86108
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.29522e15 −1.68273
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.06959e13 0.0390495 0.0195247 − 0.999809i \(-0.493785\pi\)
0.0195247 + 0.999809i \(0.493785\pi\)
\(954\) 0 0
\(955\) −7.41143e14 −0.933007
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 9.59459e14 1.17060
\(962\) 0 0
\(963\) 1.11925e15 1.35144
\(964\) −1.52243e15 −1.82875
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.80582e15 −1.99776
\(981\) 0 0
\(982\) 0 0
\(983\) 9.67041e14 1.05360 0.526802 − 0.849988i \(-0.323391\pi\)
0.526802 + 0.849988i \(0.323391\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 8.16441e14 0.867241
\(989\) 5.45258e14 0.576263
\(990\) 0 0
\(991\) −1.90783e15 −1.99604 −0.998022 − 0.0628585i \(-0.979978\pi\)
−0.998022 + 0.0628585i \(0.979978\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 0 0
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 91.11.b.a.90.1 ✓ 1
7.6 odd 2 91.11.b.b.90.1 yes 1
13.12 even 2 91.11.b.b.90.1 yes 1
91.90 odd 2 CM 91.11.b.a.90.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.11.b.a.90.1 ✓ 1 1.1 even 1 trivial
91.11.b.a.90.1 ✓ 1 91.90 odd 2 CM
91.11.b.b.90.1 yes 1 7.6 odd 2
91.11.b.b.90.1 yes 1 13.12 even 2