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Properties

Label 32.10.a.a.1.1
Level $32$
Weight $10$
Character 32.1
Self dual yes
Analytic conductor $16.481$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,10,Mod(1,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)

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: \(16.4811467572\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 32.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2398.00 q^{5} -19683.0 q^{9} +O(q^{10})\) \(q+2398.00 q^{5} -19683.0 q^{9} +112806. q^{13} +554882. q^{17} +3.79728e6 q^{25} +7.31471e6 q^{29} -2.27189e7 q^{37} +3.53885e7 q^{41} -4.71998e7 q^{45} -4.03536e7 q^{49} -9.23630e7 q^{53} +6.60431e6 q^{61} +2.70509e8 q^{65} +4.23312e7 q^{73} +3.87420e8 q^{81} +1.33061e9 q^{85} -1.12557e9 q^{89} -1.41680e9 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 2398.00 1.71587 0.857935 − 0.513759i \(-0.171747\pi\)
0.857935 + 0.513759i \(0.171747\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −19683.0 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 112806. 1.09544 0.547718 − 0.836663i \(-0.315497\pi\)
0.547718 + 0.836663i \(0.315497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 554882. 1.61132 0.805658 − 0.592382i \(-0.201812\pi\)
0.805658 + 0.592382i \(0.201812\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 3.79728e6 1.94421
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.31471e6 1.92046 0.960232 − 0.279204i \(-0.0900705\pi\)
0.960232 + 0.279204i \(0.0900705\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.27189e7 −1.99287 −0.996436 − 0.0843579i \(-0.973116\pi\)
−0.996436 + 0.0843579i \(0.973116\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.53885e7 1.95585 0.977923 − 0.208965i \(-0.0670096\pi\)
0.977923 + 0.208965i \(0.0670096\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −4.71998e7 −1.71587
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −4.03536e7 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.23630e7 −1.60789 −0.803946 − 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 6.60431e6 0.0610721 0.0305361 − 0.999534i \(-0.490279\pi\)
0.0305361 + 0.999534i \(0.490279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.70509e8 1.87963
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 4.23312e7 0.174465 0.0872324 − 0.996188i \(-0.472198\pi\)
0.0872324 + 0.996188i \(0.472198\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 3.87420e8 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 1.33061e9 2.76481
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.12557e9 −1.90159 −0.950795 − 0.309821i \(-0.899731\pi\)
−0.950795 + 0.309821i \(0.899731\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.41680e9 −1.62493 −0.812466 − 0.583008i \(-0.801876\pi\)
−0.812466 + 0.583008i \(0.801876\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.63451e9 −1.56294 −0.781470 − 0.623943i \(-0.785530\pi\)
−0.781470 + 0.623943i \(0.785530\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 1.46065e9 0.991124 0.495562 − 0.868573i \(-0.334962\pi\)
0.495562 + 0.868573i \(0.334962\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.42015e8 −0.370418 −0.185209 − 0.982699i \(-0.559296\pi\)
−0.185209 + 0.982699i \(0.559296\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.22036e9 −1.09544
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.35795e9 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.42228e9 1.62014
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.24653e9 1.99999 0.999996 − 0.00265362i \(-0.000844674\pi\)
0.999996 + 0.00265362i \(0.000844674\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 1.75407e10 3.29526
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.51916e9 −1.41599 −0.707993 − 0.706220i \(-0.750399\pi\)
−0.707993 + 0.706220i \(0.750399\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −1.09217e10 −1.61132
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −3.29697e9 −0.433079 −0.216540 − 0.976274i \(-0.569477\pi\)
−0.216540 + 0.976274i \(0.569477\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 2.12069e9 0.199981
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.61092e9 −0.391363 −0.195682 − 0.980667i \(-0.562692\pi\)
−0.195682 + 0.980667i \(0.562692\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −8.86894e9 −0.614212 −0.307106 − 0.951675i \(-0.599361\pi\)
−0.307106 + 0.951675i \(0.599361\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.44799e10 −3.41951
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 3.85199e10 1.99838 0.999189 − 0.0402702i \(-0.0128219\pi\)
0.999189 + 0.0402702i \(0.0128219\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.53089e10 −1.19722 −0.598612 − 0.801039i \(-0.704281\pi\)
−0.598612 + 0.801039i \(0.704281\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.48616e10 3.35598
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.25940e10 1.76509
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −7.47418e10 −1.94421
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 3.07048e10 0.737814 0.368907 − 0.929466i \(-0.379732\pi\)
0.368907 + 0.929466i \(0.379732\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.24791e10 0.499662 0.249831 − 0.968289i \(-0.419625\pi\)
0.249831 + 0.968289i \(0.419625\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 7.32592e10 1.39890 0.699448 − 0.714683i \(-0.253429\pi\)
0.699448 + 0.714683i \(0.253429\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.67679e10 −1.71587
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.45073e10 1.06537 0.532684 − 0.846314i \(-0.321183\pi\)
0.532684 + 0.846314i \(0.321183\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.43975e11 −1.92046
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −2.21487e11 −2.75893
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.59382e11 −1.85590 −0.927949 − 0.372707i \(-0.878430\pi\)
−0.927949 + 0.372707i \(0.878430\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.78158e11 1.81822 0.909112 − 0.416551i \(-0.136761\pi\)
0.909112 + 0.416551i \(0.136761\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.43990e10 0.807531 0.403765 − 0.914863i \(-0.367701\pi\)
0.403765 + 0.914863i \(0.367701\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.89306e11 1.59634
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.24670e11 −0.988232 −0.494116 − 0.869396i \(-0.664508\pi\)
−0.494116 + 0.869396i \(0.664508\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.58371e10 0.104792
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 3.09315e11 1.82159 0.910795 − 0.412858i \(-0.135469\pi\)
0.910795 + 0.412858i \(0.135469\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.02638e11 −0.570874 −0.285437 − 0.958398i \(-0.592139\pi\)
−0.285437 + 0.958398i \(0.592139\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 4.28356e11 2.12975
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 4.47176e11 1.99287
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.71141e11 −1.98983 −0.994916 − 0.100713i \(-0.967888\pi\)
−0.994916 + 0.100713i \(0.967888\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −3.58390e11 −1.29313 −0.646563 − 0.762860i \(-0.723794\pi\)
−0.646563 + 0.762860i \(0.723794\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.99335e11 −1.36884 −0.684418 − 0.729089i \(-0.739944\pi\)
−0.684418 + 0.729089i \(0.739944\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.22688e11 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.01510e11 0.299359
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −6.96552e11 −1.95585
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.45925e11 −0.390336 −0.195168 − 0.980770i \(-0.562525\pi\)
−0.195168 + 0.980770i \(0.562525\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.25143e11 2.10375
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.59929e10 −0.146125 −0.0730625 − 0.997327i \(-0.523277\pi\)
−0.0730625 + 0.997327i \(0.523277\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −9.16054e11 −1.85082 −0.925409 − 0.378970i \(-0.876278\pi\)
−0.925409 + 0.378970i \(0.876278\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.50088e11 0.869257 0.434628 − 0.900610i \(-0.356880\pi\)
0.434628 + 0.900610i \(0.356880\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 9.29034e11 1.71587
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.10183e12 −1.94697 −0.973487 − 0.228744i \(-0.926538\pi\)
−0.973487 + 0.228744i \(0.926538\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 1.14632e12 1.77842 0.889212 − 0.457495i \(-0.151253\pi\)
0.889212 + 0.457495i \(0.151253\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.10704e12 3.13273
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 1.06920e12 1.46172 0.730858 − 0.682530i \(-0.239120\pi\)
0.730858 + 0.682530i \(0.239120\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 7.94280e11 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −2.69911e12 −3.26288
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.91685e11 −0.222577 −0.111288 − 0.993788i \(-0.535498\pi\)
−0.111288 + 0.993788i \(0.535498\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.29168e11 −0.245772 −0.122886 − 0.992421i \(-0.539215\pi\)
−0.122886 + 0.992421i \(0.539215\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.68844e11 −0.689716 −0.344858 − 0.938655i \(-0.612073\pi\)
−0.344858 + 0.938655i \(0.612073\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.81798e12 1.60789
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −2.56283e12 −2.18306
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.39748e12 −2.78817
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 4.05880e12 3.09447
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −3.91956e12 −2.68180
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.73961e12 −1.80908 −0.904542 − 0.426386i \(-0.859787\pi\)
−0.904542 + 0.426386i \(0.859787\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 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
\(521\) 3.30505e12 1.96521 0.982604 − 0.185716i \(-0.0594603\pi\)
0.982604 + 0.185716i \(0.0594603\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.80115e12 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.99203e12 2.14250
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.60539e12 1.30763 0.653815 − 0.756654i \(-0.273167\pi\)
0.653815 + 0.756654i \(0.273167\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3.50265e12 1.70064
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −1.29993e11 −0.0610721
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.83010e12 1.68602 0.843008 − 0.537901i \(-0.180782\pi\)
0.843008 + 0.537901i \(0.180782\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −1.53955e12 −0.635589
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.44716e12 −1.77860 −0.889298 − 0.457328i \(-0.848807\pi\)
−0.889298 + 0.457328i \(0.848807\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.94933e12 0.732142 0.366071 − 0.930587i \(-0.380703\pi\)
0.366071 + 0.930587i \(0.380703\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −5.32442e12 −1.87963
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.97248e12 1.98339 0.991696 − 0.128604i \(-0.0410496\pi\)
0.991696 + 0.128604i \(0.0410496\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 6.15165e12 1.92334 0.961671 − 0.274204i \(-0.0884144\pi\)
0.961671 + 0.274204i \(0.0884144\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.65436e12 −1.71587
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.52459e12 −1.00818 −0.504088 − 0.863653i \(-0.668171\pi\)
−0.504088 + 0.863653i \(0.668171\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7.19878e12 −1.99975 −0.999874 − 0.0158499i \(-0.994955\pi\)
−0.999874 + 0.0158499i \(0.994955\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.18807e12 0.835733
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.26063e13 −3.21114
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.55213e12 −1.09544
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.21742e12 0.284826 0.142413 − 0.989807i \(-0.454514\pi\)
0.142413 + 0.989807i \(0.454514\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.25421e12 1.99173 0.995864 − 0.0908593i \(-0.0289613\pi\)
0.995864 + 0.0908593i \(0.0289613\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −8.33205e11 −0.174465
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 5.12326e12 1.04385 0.521927 − 0.852990i \(-0.325213\pi\)
0.521927 + 0.852990i \(0.325213\pi\)
\(662\) 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
\(673\) 3.99150e12 0.750012 0.375006 − 0.927022i \(-0.377641\pi\)
0.375006 + 0.927022i \(0.377641\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.70710e12 −0.678244 −0.339122 − 0.940742i \(-0.610130\pi\)
−0.339122 + 0.940742i \(0.610130\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 1.97752e13 3.43173
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.04191e13 −1.76134
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.96364e13 3.15148
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.26633e13 −1.98068 −0.990341 − 0.138650i \(-0.955724\pi\)
−0.990341 + 0.138650i \(0.955724\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.05193e13 −1.56344 −0.781718 − 0.623633i \(-0.785656\pi\)
−0.781718 + 0.623633i \(0.785656\pi\)
\(710\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.77760e13 3.73378
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −7.62560e12 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.22963e13 1.57328 0.786638 − 0.617414i \(-0.211820\pi\)
0.786638 + 0.617414i \(0.211820\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −2.04290e13 −2.42965
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.56439e12 −0.283826 −0.141913 − 0.989879i \(-0.545325\pi\)
−0.141913 + 0.989879i \(0.545325\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.74541e12 −1.05334 −0.526671 − 0.850069i \(-0.676560\pi\)
−0.526671 + 0.850069i \(0.676560\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.61903e13 −2.76481
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −1.22666e13 −1.26490 −0.632448 − 0.774603i \(-0.717950\pi\)
−0.632448 + 0.774603i \(0.717950\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.03452e12 0.708642 0.354321 − 0.935124i \(-0.384712\pi\)
0.354321 + 0.935124i \(0.384712\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.90614e12 −0.743107
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.45006e11 0.0669006
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.01257e13 −0.888923 −0.444462 − 0.895798i \(-0.646605\pi\)
−0.444462 + 0.895798i \(0.646605\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 2.21546e13 1.90159
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.43621e13 1.99962 0.999808 − 0.0195821i \(-0.00623358\pi\)
0.999808 + 0.0195821i \(0.00623358\pi\)
\(810\) 0 0
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.06497e12 0.235441 0.117720 − 0.993047i \(-0.462441\pi\)
0.117720 + 0.993047i \(0.462441\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −2.71622e13 −1.99742 −0.998712 − 0.0507364i \(-0.983843\pi\)
−0.998712 + 0.0507364i \(0.983843\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.23915e13 −1.61132
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 3.89978e13 2.68818
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.08542e12 0.343141
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 2.94806e13 1.90663 0.953314 − 0.301981i \(-0.0976479\pi\)
0.953314 + 0.301981i \(0.0976479\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.04489e13 0.661694 0.330847 − 0.943684i \(-0.392666\pi\)
0.330847 + 0.943684i \(0.392666\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −1.10570e13 −0.671528
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.78868e13 1.62493
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.18890e12 0.524524 0.262262 − 0.964997i \(-0.415532\pi\)
0.262262 + 0.964997i \(0.415532\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.43019e13 0.799840 0.399920 − 0.916550i \(-0.369038\pi\)
0.399920 + 0.916550i \(0.369038\pi\)
\(882\) 0 0
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −5.12506e13 −2.59082
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.12677e13 −1.05391
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 3.21721e13 1.56294
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −8.62699e13 −3.87455
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.48355e13 −1.97493 −0.987464 − 0.157844i \(-0.949546\pi\)
−0.987464 + 0.157844i \(0.949546\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.19692e13 0.507267 0.253634 − 0.967300i \(-0.418374\pi\)
0.253634 + 0.967300i \(0.418374\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.75419e13 1.97662 0.988310 − 0.152461i \(-0.0487197\pi\)
0.988310 + 0.152461i \(0.0487197\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 4.77521e12 0.191115
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.54681e13 −1.39290 −0.696451 − 0.717605i \(-0.745238\pi\)
−0.696451 + 0.717605i \(0.745238\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.64396e13 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.23708e13 3.42895
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.30112e13 −0.808003 −0.404002 − 0.914758i \(-0.632381\pi\)
−0.404002 + 0.914758i \(0.632381\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −2.87500e13 −0.991124
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −6.06908e13 −2.05428
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.31597e12 −0.298607 −0.149303 − 0.988791i \(-0.547703\pi\)
−0.149303 + 0.988791i \(0.547703\pi\)
\(998\) 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 32.10.a.a.1.1 ✓ 1
3.2 odd 2 288.10.a.a.1.1 1
4.3 odd 2 CM 32.10.a.a.1.1 ✓ 1
8.3 odd 2 64.10.a.e.1.1 1
8.5 even 2 64.10.a.e.1.1 1
12.11 even 2 288.10.a.a.1.1 1
16.3 odd 4 256.10.b.f.129.1 2
16.5 even 4 256.10.b.f.129.2 2
16.11 odd 4 256.10.b.f.129.2 2
16.13 even 4 256.10.b.f.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.10.a.a.1.1 ✓ 1 1.1 even 1 trivial
32.10.a.a.1.1 ✓ 1 4.3 odd 2 CM
64.10.a.e.1.1 1 8.3 odd 2
64.10.a.e.1.1 1 8.5 even 2
256.10.b.f.129.1 2 16.3 odd 4
256.10.b.f.129.1 2 16.13 even 4
256.10.b.f.129.2 2 16.5 even 4
256.10.b.f.129.2 2 16.11 odd 4
288.10.a.a.1.1 1 3.2 odd 2
288.10.a.a.1.1 1 12.11 even 2