↑

Properties

Label 27.11.b.a.26.1
Level $27$
Weight $11$
Character 27.26
Self dual yes
Analytic conductor $17.155$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,11,Mod(26,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(17.1546458222\)
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 26.1
Character \(\chi\) \(=\) 27.26

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1024.00 q^{4} +10907.0 q^{7} +O(q^{10})\) \(q+1024.00 q^{4} +10907.0 q^{7} -141961. q^{13} +1.04858e6 q^{16} +4.92625e6 q^{19} +9.76562e6 q^{25} +1.11688e7 q^{28} +4.93267e7 q^{31} -4.08956e7 q^{37} -2.82781e8 q^{43} -1.63513e8 q^{49} -1.45368e8 q^{52} -1.35427e9 q^{61} +1.07374e9 q^{64} +2.69833e9 q^{67} +1.95768e9 q^{73} +5.04448e9 q^{76} -6.05989e9 q^{79} -1.54837e9 q^{91} -1.52964e10 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\pi\)
\(3\) 0 0
\(4\) 1024.00 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 10907.0 0.648956 0.324478 − 0.945893i \(-0.394811\pi\)
0.324478 + 0.945893i \(0.394811\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −141961. −0.382342 −0.191171 − 0.981557i \(-0.561229\pi\)
−0.191171 + 0.981557i \(0.561229\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 4.92625e6 1.98952 0.994761 − 0.102233i \(-0.0325986\pi\)
0.994761 + 0.102233i \(0.0325986\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 1.11688e7 0.648956
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 4.93267e7 1.72295 0.861476 − 0.507798i \(-0.169540\pi\)
0.861476 + 0.507798i \(0.169540\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.08956e7 −0.589750 −0.294875 − 0.955536i \(-0.595278\pi\)
−0.294875 + 0.955536i \(0.595278\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2.82781e8 −1.92357 −0.961785 − 0.273806i \(-0.911717\pi\)
−0.961785 + 0.273806i \(0.911717\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −1.63513e8 −0.578856
\(50\) 0 0
\(51\) 0 0
\(52\) −1.45368e8 −0.382342
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.35427e9 −1.60345 −0.801724 − 0.597695i \(-0.796084\pi\)
−0.801724 + 0.597695i \(0.796084\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.07374e9 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.69833e9 1.99857 0.999287 − 0.0377510i \(-0.0120194\pi\)
0.999287 + 0.0377510i \(0.0120194\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.95768e9 0.944340 0.472170 − 0.881507i \(-0.343471\pi\)
0.472170 + 0.881507i \(0.343471\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 5.04448e9 1.98952
\(77\) 0 0
\(78\) 0 0
\(79\) −6.05989e9 −1.96938 −0.984689 − 0.174321i \(-0.944227\pi\)
−0.984689 + 0.174321i \(0.944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.54837e9 −0.248123
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.52964e10 −1.78127 −0.890637 − 0.454714i \(-0.849741\pi\)
−0.890637 + 0.454714i \(0.849741\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e10 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.82044e10 −1.57032 −0.785162 − 0.619290i \(-0.787421\pi\)
−0.785162 + 0.619290i \(0.787421\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.76664e10 −1.14819 −0.574096 − 0.818788i \(-0.694646\pi\)
−0.574096 + 0.818788i \(0.694646\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.14368e10 0.648956
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.59374e10 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 5.05105e10 1.72295
\(125\) 0 0
\(126\) 0 0
\(127\) 5.26567e9 0.159380 0.0796902 − 0.996820i \(-0.474607\pi\)
0.0796902 + 0.996820i \(0.474607\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 5.37306e10 1.29111
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −6.28957e10 −1.21212 −0.606062 − 0.795418i \(-0.707252\pi\)
−0.606062 + 0.795418i \(0.707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −4.18771e10 −0.589750
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 1.30667e11 1.66449 0.832245 − 0.554409i \(-0.187055\pi\)
0.832245 + 0.554409i \(0.187055\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.78850e11 −1.87496 −0.937479 − 0.348043i \(-0.886846\pi\)
−0.937479 + 0.348043i \(0.886846\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.05959e11 −0.920874 −0.460437 − 0.887692i \(-0.652307\pi\)
−0.460437 + 0.887692i \(0.652307\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −1.17706e11 −0.853814
\(170\) 0 0
\(171\) 0 0
\(172\) −2.89568e11 −1.92357
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 1.06514e11 0.648956
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 3.39127e11 1.74570 0.872849 − 0.487991i \(-0.162270\pi\)
0.872849 + 0.487991i \(0.162270\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 5.07232e11 1.89418 0.947088 − 0.320975i \(-0.104011\pi\)
0.947088 + 0.320975i \(0.104011\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1.67437e11 −0.578856
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 4.05581e11 1.29961 0.649803 − 0.760102i \(-0.274851\pi\)
0.649803 + 0.760102i \(0.274851\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.48857e11 −0.382342
\(209\) 0 0
\(210\) 0 0
\(211\) 6.09665e10 0.145774 0.0728868 − 0.997340i \(-0.476779\pi\)
0.0728868 + 0.997340i \(0.476779\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.38006e11 1.11812
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.01050e12 −1.83236 −0.916181 − 0.400764i \(-0.868745\pi\)
−0.916181 + 0.400764i \(0.868745\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 3.52911e11 0.560387 0.280193 − 0.959944i \(-0.409601\pi\)
0.280193 + 0.959944i \(0.409601\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.43538e12 −1.76556 −0.882779 − 0.469789i \(-0.844330\pi\)
−0.882779 + 0.469789i \(0.844330\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.38677e12 −1.60345
\(245\) 0 0
\(246\) 0 0
\(247\) −6.99336e11 −0.760678
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 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
\(259\) −4.46048e11 −0.382722
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 2.76309e12 1.99857
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 6.37452e11 0.436115 0.218058 − 0.975936i \(-0.430028\pi\)
0.218058 + 0.975936i \(0.430028\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.92166e12 1.79156 0.895780 − 0.444498i \(-0.146618\pi\)
0.895780 + 0.444498i \(0.146618\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.63046e12 −2.00000 −0.999999 − 0.00144882i \(-0.999539\pi\)
−0.999999 + 0.00144882i \(0.999539\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.01599e12 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00467e12 0.944340
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −3.08429e12 −1.24831
\(302\) 0 0
\(303\) 0 0
\(304\) 5.16555e12 1.98952
\(305\) 0 0
\(306\) 0 0
\(307\) −1.58858e11 −0.0582528 −0.0291264 − 0.999576i \(-0.509273\pi\)
−0.0291264 + 0.999576i \(0.509273\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.81573e11 0.193590 0.0967949 − 0.995304i \(-0.469141\pi\)
0.0967949 + 0.995304i \(0.469141\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −6.20532e12 −1.96938
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.38634e12 −0.382342
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7.64811e12 −1.92493 −0.962463 − 0.271412i \(-0.912510\pi\)
−0.962463 + 0.271412i \(0.912510\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.76722e12 −0.406575 −0.203288 − 0.979119i \(-0.565163\pi\)
−0.203288 + 0.979119i \(0.565163\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.86439e12 −1.02461
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 1.80858e12 0.349310 0.174655 − 0.984630i \(-0.444119\pi\)
0.174655 + 0.984630i \(0.444119\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.81369e13 2.95819
\(362\) 0 0
\(363\) 0 0
\(364\) −1.58553e12 −0.248123
\(365\) 0 0
\(366\) 0 0
\(367\) −6.64279e12 −0.997747 −0.498874 − 0.866675i \(-0.666253\pi\)
−0.498874 + 0.866675i \(0.666253\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.38332e13 1.91592 0.957960 − 0.286903i \(-0.0926256\pi\)
0.957960 + 0.286903i \(0.0926256\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.15586e12 1.04298 0.521488 − 0.853259i \(-0.325377\pi\)
0.521488 + 0.853259i \(0.325377\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −1.56635e13 −1.78127
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.37947e13 1.39881 0.699407 − 0.714723i \(-0.253447\pi\)
0.699407 + 0.714723i \(0.253447\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.02400e13 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −7.00246e12 −0.658758
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.76218e13 1.53969 0.769845 − 0.638231i \(-0.220334\pi\)
0.769845 + 0.638231i \(0.220334\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.86413e13 −1.57032
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −2.67749e12 −0.202449 −0.101225 − 0.994864i \(-0.532276\pi\)
−0.101225 + 0.994864i \(0.532276\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.47710e13 −1.04057
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −2.69911e13 −1.77329 −0.886646 − 0.462448i \(-0.846971\pi\)
−0.886646 + 0.462448i \(0.846971\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.80903e13 −1.14819
\(437\) 0 0
\(438\) 0 0
\(439\) −1.66634e13 −1.02198 −0.510989 − 0.859587i \(-0.670721\pi\)
−0.510989 + 0.859587i \(0.670721\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.17113e13 0.648956
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.83187e13 1.42067 0.710334 − 0.703865i \(-0.248544\pi\)
0.710334 + 0.703865i \(0.248544\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −3.40144e13 −1.59867 −0.799333 − 0.600889i \(-0.794813\pi\)
−0.799333 + 0.600889i \(0.794813\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 2.94306e13 1.29699
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.81079e13 1.98952
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 5.80558e12 0.225486
\(482\) 0 0
\(483\) 0 0
\(484\) 2.65599e13 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −5.29238e13 −1.93199 −0.965997 − 0.258551i \(-0.916755\pi\)
−0.965997 + 0.258551i \(0.916755\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 5.17228e13 1.72295
\(497\) 0 0
\(498\) 0 0
\(499\) 8.03832e12 0.259814 0.129907 − 0.991526i \(-0.458532\pi\)
0.129907 + 0.991526i \(0.458532\pi\)
\(500\) 0 0
\(501\) 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
\(508\) 5.39205e12 0.159380
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 2.13525e13 0.612835
\(512\) 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
\(523\) −7.38765e13 −1.88798 −0.943991 − 0.329972i \(-0.892961\pi\)
−0.943991 + 0.329972i \(0.892961\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 4.14265e13 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 5.50202e13 1.29111
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8.35519e13 −1.80289 −0.901446 − 0.432891i \(-0.857493\pi\)
−0.901446 + 0.432891i \(0.857493\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −9.57115e13 −1.95446 −0.977232 − 0.212176i \(-0.931945\pi\)
−0.977232 + 0.212176i \(0.931945\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −6.60952e13 −1.27804
\(554\) 0 0
\(555\) 0 0
\(556\) −6.44052e13 −1.21212
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 4.01439e13 0.735462
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −8.67315e13 −1.42888 −0.714441 − 0.699695i \(-0.753319\pi\)
−0.714441 + 0.699695i \(0.753319\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.61903e13 1.34766 0.673828 − 0.738888i \(-0.264649\pi\)
0.673828 + 0.738888i \(0.264649\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 2.42996e14 3.42785
\(590\) 0 0
\(591\) 0 0
\(592\) −4.28821e13 −0.589750
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.20560e14 −1.53756 −0.768780 − 0.639513i \(-0.779136\pi\)
−0.768780 + 0.639513i \(0.779136\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.33803e14 1.66449
\(605\) 0 0
\(606\) 0 0
\(607\) 8.10731e13 0.983861 0.491930 − 0.870635i \(-0.336291\pi\)
0.491930 + 0.870635i \(0.336291\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.72793e14 −1.99629 −0.998144 − 0.0608915i \(-0.980606\pi\)
−0.998144 + 0.0608915i \(0.980606\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 1.71040e14 1.88211 0.941055 − 0.338255i \(-0.109837\pi\)
0.941055 + 0.338255i \(0.109837\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.53674e13 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.83143e14 −1.87496
\(629\) 0 0
\(630\) 0 0
\(631\) 1.32321e14 1.32276 0.661381 − 0.750050i \(-0.269971\pi\)
0.661381 + 0.750050i \(0.269971\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.32124e13 0.221321
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 2.07428e14 1.88718 0.943589 − 0.331118i \(-0.107426\pi\)
0.943589 + 0.331118i \(0.107426\pi\)
\(644\) 0 0
\(645\) 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
\(652\) −1.08502e14 −0.920874
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −2.51892e14 −1.99622 −0.998108 − 0.0614804i \(-0.980418\pi\)
−0.998108 + 0.0614804i \(0.980418\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.35641e13 0.170677 0.0853386 − 0.996352i \(-0.472803\pi\)
0.0853386 + 0.996352i \(0.472803\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −1.20530e14 −0.853814
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1.66838e14 −1.15597
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2.96517e14 −1.92357
\(689\) 0 0
\(690\) 0 0
\(691\) −1.36041e13 −0.0863536 −0.0431768 − 0.999067i \(-0.513748\pi\)
−0.0431768 + 0.999067i \(0.513748\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.09070e14 0.648956
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −2.01462e14 −1.17332
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.00396e14 1.11855 0.559277 − 0.828981i \(-0.311079\pi\)
0.559277 + 0.828981i \(0.311079\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1.98555e14 −1.01907
\(722\) 0 0
\(723\) 0 0
\(724\) 3.47266e14 1.74570
\(725\) 0 0
\(726\) 0 0
\(727\) 4.03308e14 1.98594 0.992968 − 0.118387i \(-0.0377723\pi\)
0.992968 + 0.118387i \(0.0377723\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −2.95452e14 −1.39626 −0.698131 − 0.715970i \(-0.745985\pi\)
−0.698131 + 0.715970i \(0.745985\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.35559e14 1.97617 0.988085 − 0.153908i \(-0.0491859\pi\)
0.988085 + 0.153908i \(0.0491859\pi\)
\(740\) 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
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.68707e14 1.12481 0.562405 − 0.826862i \(-0.309876\pi\)
0.562405 + 0.826862i \(0.309876\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.69691e14 −1.48717 −0.743583 − 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.92687e14 −0.745126
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 8.72349e13 0.324383 0.162192 − 0.986759i \(-0.448144\pi\)
0.162192 + 0.986759i \(0.448144\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.19406e14 1.89418
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 4.81706e14 1.72295
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.71455e14 −0.578856
\(785\) 0 0
\(786\) 0 0
\(787\) −5.48999e14 −1.81844 −0.909218 − 0.416321i \(-0.863319\pi\)
−0.909218 + 0.416321i \(0.863319\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.92253e14 0.613066
\(794\) 0 0
\(795\) 0 0
\(796\) 4.15315e14 1.29961
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −1.83164e14 −0.522078 −0.261039 − 0.965328i \(-0.584065\pi\)
−0.261039 + 0.965328i \(0.584065\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.39305e15 −3.82698
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −4.85462e14 −1.28575 −0.642874 − 0.765972i \(-0.722258\pi\)
−0.642874 + 0.765972i \(0.722258\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −2.70119e14 −0.689894 −0.344947 − 0.938622i \(-0.612103\pi\)
−0.344947 + 0.938622i \(0.612103\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.52429e14 −0.382342
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 4.20707e14 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 6.24297e13 0.145774
\(845\) 0 0
\(846\) 0 0
\(847\) 2.82899e14 0.648956
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −8.94131e14 −1.97996 −0.989979 − 0.141213i \(-0.954900\pi\)
−0.989979 + 0.141213i \(0.954900\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 5.77614e14 1.23501 0.617507 − 0.786565i \(-0.288143\pi\)
0.617507 + 0.786565i \(0.288143\pi\)
\(860\) 0 0
\(861\) 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
\(868\) 5.50918e14 1.11812
\(869\) 0 0
\(870\) 0 0
\(871\) −3.83057e14 −0.764139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 6.62354e14 1.27671 0.638355 − 0.769742i \(-0.279615\pi\)
0.638355 + 0.769742i \(0.279615\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.02467e15 1.90889 0.954443 − 0.298393i \(-0.0964506\pi\)
0.954443 + 0.298393i \(0.0964506\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 5.74327e13 0.103431
\(890\) 0 0
\(891\) 0 0
\(892\) −1.03475e15 −1.83236
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.22482e15 1.99543 0.997714 − 0.0675832i \(-0.0215288\pi\)
0.997714 + 0.0675832i \(0.0215288\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 3.61381e14 0.560387
\(917\) 0 0
\(918\) 0 0
\(919\) 8.33034e14 1.27082 0.635411 − 0.772174i \(-0.280831\pi\)
0.635411 + 0.772174i \(0.280831\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.99371e14 −0.589750
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −8.05504e14 −1.15165
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 9.53611e14 1.32030 0.660151 − 0.751133i \(-0.270492\pi\)
0.660151 + 0.751133i \(0.270492\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −2.77915e14 −0.361061
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.61349e15 1.96857
\(962\) 0 0
\(963\) 0 0
\(964\) −1.46983e15 −1.76556
\(965\) 0 0
\(966\) 0 0
\(967\) −1.03143e15 −1.21985 −0.609924 − 0.792460i \(-0.708800\pi\)
−0.609924 + 0.792460i \(0.708800\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −6.86003e14 −0.786614
\(974\) 0 0
\(975\) 0 0
\(976\) −1.42005e15 −1.60345
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −7.16120e14 −0.760678
\(989\) 0 0
\(990\) 0 0
\(991\) −1.53369e15 −1.60461 −0.802303 − 0.596918i \(-0.796392\pi\)
−0.802303 + 0.596918i \(0.796392\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.82451e13 0.0388240 0.0194120 − 0.999812i \(-0.493821\pi\)
0.0194120 + 0.999812i \(0.493821\pi\)
\(998\) 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 27.11.b.a.26.1 ✓ 1
3.2 odd 2 CM 27.11.b.a.26.1 ✓ 1
9.2 odd 6 81.11.d.b.53.1 2
9.4 even 3 81.11.d.b.26.1 2
9.5 odd 6 81.11.d.b.26.1 2
9.7 even 3 81.11.d.b.53.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.11.b.a.26.1 ✓ 1 1.1 even 1 trivial
27.11.b.a.26.1 ✓ 1 3.2 odd 2 CM
81.11.d.b.26.1 2 9.4 even 3
81.11.d.b.26.1 2 9.5 odd 6
81.11.d.b.53.1 2 9.2 odd 6
81.11.d.b.53.1 2 9.7 even 3