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Properties

Label 256.6.a.b.1.1
Level $256$
Weight $6$
Character 256.1
Self dual yes
Analytic conductor $41.058$
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] = [256,6,Mod(1,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 256.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: \(41.0582578721\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 256.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-76.0000 q^{5} -243.000 q^{9} +244.000 q^{13} -2242.00 q^{17} +2651.00 q^{25} +8564.00 q^{29} -11292.0 q^{37} +20950.0 q^{41} +18468.0 q^{45} -16807.0 q^{49} +40244.0 q^{53} +54948.0 q^{61} -18544.0 q^{65} -88806.0 q^{73} +59049.0 q^{81} +170392. q^{85} +51050.0 q^{89} +92142.0 q^{97} +O(q^{100})\)

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
\(5\) −76.0000 −1.35953 −0.679765 − 0.733430i \(-0.737918\pi\)
−0.679765 + 0.733430i \(0.737918\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 244.000 0.400434 0.200217 − 0.979752i \(-0.435835\pi\)
0.200217 + 0.979752i \(0.435835\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2242.00 −1.88154 −0.940770 − 0.339046i \(-0.889896\pi\)
−0.940770 + 0.339046i \(0.889896\pi\)
\(18\) 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
\(25\) 2651.00 0.848320
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8564.00 1.89096 0.945479 − 0.325684i \(-0.105595\pi\)
0.945479 + 0.325684i \(0.105595\pi\)
\(30\) 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
\(37\) −11292.0 −1.35602 −0.678011 − 0.735052i \(-0.737158\pi\)
−0.678011 + 0.735052i \(0.737158\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 20950.0 1.94637 0.973183 − 0.230033i \(-0.0738836\pi\)
0.973183 + 0.230033i \(0.0738836\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 18468.0 1.35953
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −16807.0 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 40244.0 1.96794 0.983969 − 0.178339i \(-0.0570723\pi\)
0.983969 + 0.178339i \(0.0570723\pi\)
\(54\) 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
\(61\) 54948.0 1.89072 0.945360 − 0.326028i \(-0.105710\pi\)
0.945360 + 0.326028i \(0.105710\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −18544.0 −0.544402
\(66\) 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
\(73\) −88806.0 −1.95045 −0.975226 − 0.221212i \(-0.928999\pi\)
−0.975226 + 0.221212i \(0.928999\pi\)
\(74\) 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
\(81\) 59049.0 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 170392. 2.55801
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 51050.0 0.683157 0.341579 − 0.939853i \(-0.389038\pi\)
0.341579 + 0.939853i \(0.389038\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 92142.0 0.994325 0.497162 − 0.867657i \(-0.334375\pi\)
0.497162 + 0.867657i \(0.334375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 180100. 1.75675 0.878376 − 0.477971i \(-0.158628\pi\)
0.878376 + 0.477971i \(0.158628\pi\)
\(102\) 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
\(109\) 28100.0 0.226537 0.113269 − 0.993564i \(-0.463868\pi\)
0.113269 + 0.993564i \(0.463868\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 118706. 0.874534 0.437267 − 0.899332i \(-0.355947\pi\)
0.437267 + 0.899332i \(0.355947\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −59292.0 −0.400434
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 36024.0 0.206213
\(126\) 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
\(137\) −75658.0 −0.344392 −0.172196 − 0.985063i \(-0.555086\pi\)
−0.172196 + 0.985063i \(0.555086\pi\)
\(138\) 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
\(145\) −650864. −2.57081
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −539900. −1.99227 −0.996134 − 0.0878494i \(-0.972001\pi\)
−0.996134 + 0.0878494i \(0.972001\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 544806. 1.88154
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −371292. −1.20217 −0.601086 − 0.799185i \(-0.705265\pi\)
−0.601086 + 0.799185i \(0.705265\pi\)
\(158\) 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
\(169\) −311757. −0.839652
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 544244. 1.38254 0.691271 − 0.722596i \(-0.257051\pi\)
0.691271 + 0.722596i \(0.257051\pi\)
\(174\) 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
\(181\) −763900. −1.73317 −0.866583 − 0.499033i \(-0.833689\pi\)
−0.866583 + 0.499033i \(0.833689\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 858192. 1.84355
\(186\) 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
\(193\) 497294. 0.960992 0.480496 − 0.876997i \(-0.340457\pi\)
0.480496 + 0.876997i \(0.340457\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.02091e6 −1.87422 −0.937111 − 0.349031i \(-0.886511\pi\)
−0.937111 + 0.349031i \(0.886511\pi\)
\(198\) 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
\(205\) −1.59220e6 −2.64614
\(206\) 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
\(221\) −547048. −0.753433
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −644193. −0.848320
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 976564. 1.23059 0.615293 − 0.788298i \(-0.289038\pi\)
0.615293 + 0.788298i \(0.289038\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.53709e6 −1.85486 −0.927429 − 0.373999i \(-0.877986\pi\)
−0.927429 + 0.373999i \(0.877986\pi\)
\(234\) 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
\(241\) 477150. 0.529191 0.264595 − 0.964360i \(-0.414762\pi\)
0.264595 + 0.964360i \(0.414762\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.27733e6 1.35953
\(246\) 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
\(257\) 650242. 0.614104 0.307052 − 0.951693i \(-0.400657\pi\)
0.307052 + 0.951693i \(0.400657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.08105e6 −1.89096
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −3.05854e6 −2.67547
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −323900. −0.272917 −0.136458 − 0.990646i \(-0.543572\pi\)
−0.136458 + 0.990646i \(0.543572\pi\)
\(270\) 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
\(277\) 2.45109e6 1.91938 0.959688 − 0.281067i \(-0.0906882\pi\)
0.959688 + 0.281067i \(0.0906882\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.64305e6 1.99682 0.998412 − 0.0563421i \(-0.0179438\pi\)
0.998412 + 0.0563421i \(0.0179438\pi\)
\(282\) 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
\(289\) 3.60671e6 2.54019
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.62424e6 1.10531 0.552653 − 0.833412i \(-0.313616\pi\)
0.552653 + 0.833412i \(0.313616\pi\)
\(294\) 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
\(305\) −4.17605e6 −2.57049
\(306\) 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
\(313\) 2.88909e6 1.66687 0.833433 − 0.552620i \(-0.186372\pi\)
0.833433 + 0.552620i \(0.186372\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.51509e6 1.96467 0.982333 − 0.187144i \(-0.0599230\pi\)
0.982333 + 0.187144i \(0.0599230\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 646844. 0.339697
\(326\) 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
\(333\) 2.74396e6 1.35602
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.28386e6 1.09545 0.547727 − 0.836657i \(-0.315493\pi\)
0.547727 + 0.836657i \(0.315493\pi\)
\(338\) 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
\(349\) 975636. 0.428770 0.214385 − 0.976749i \(-0.431225\pi\)
0.214385 + 0.976749i \(0.431225\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.75261e6 −1.17573 −0.587865 − 0.808959i \(-0.700031\pi\)
−0.587865 + 0.808959i \(0.700031\pi\)
\(354\) 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
\(361\) −2.47610e6 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.74926e6 2.65170
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −5.09085e6 −1.94637
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.50404e6 −0.559743 −0.279871 − 0.960038i \(-0.590292\pi\)
−0.279871 + 0.960038i \(0.590292\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.08962e6 0.757204
\(378\) 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
\(389\) 2.77210e6 0.928827 0.464414 − 0.885618i \(-0.346265\pi\)
0.464414 + 0.885618i \(0.346265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −6.27529e6 −1.99829 −0.999143 − 0.0413901i \(-0.986821\pi\)
−0.999143 + 0.0413901i \(0.986821\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.59200e6 −0.494405 −0.247202 − 0.968964i \(-0.579511\pi\)
−0.247202 + 0.968964i \(0.579511\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4.48772e6 −1.35953
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.58449e6 1.35513 0.677567 − 0.735461i \(-0.263034\pi\)
0.677567 + 0.735461i \(0.263034\pi\)
\(410\) 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
\(421\) 4.18485e6 1.15073 0.575367 − 0.817895i \(-0.304859\pi\)
0.575367 + 0.817895i \(0.304859\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.94354e6 −1.59615
\(426\) 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
\(433\) 7.78461e6 1.99534 0.997670 − 0.0682249i \(-0.0217335\pi\)
0.997670 + 0.0682249i \(0.0217335\pi\)
\(434\) 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
\(441\) 4.08410e6 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −3.87980e6 −0.928772
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.48961e6 1.98734 0.993670 − 0.112340i \(-0.0358346\pi\)
0.993670 + 0.112340i \(0.0358346\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −5.25844e6 −1.17779 −0.588893 − 0.808211i \(-0.700436\pi\)
−0.588893 + 0.808211i \(0.700436\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.01210e6 1.31757 0.658785 − 0.752331i \(-0.271071\pi\)
0.658785 + 0.752331i \(0.271071\pi\)
\(462\) 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
\(477\) −9.77929e6 −1.96794
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −2.75525e6 −0.542998
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.00279e6 −1.35181
\(486\) 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
\(493\) −1.92005e7 −3.55791
\(494\) 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
\(505\) −1.36876e7 −2.38835
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.12076e6 −0.876073 −0.438037 − 0.898957i \(-0.644326\pi\)
−0.438037 + 0.898957i \(0.644326\pi\)
\(510\) 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
\(521\) 7.27410e6 1.17405 0.587023 − 0.809570i \(-0.300300\pi\)
0.587023 + 0.809570i \(0.300300\pi\)
\(522\) 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
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.11180e6 0.779392
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.08281e7 1.59059 0.795297 − 0.606221i \(-0.207315\pi\)
0.795297 + 0.606221i \(0.207315\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.13560e6 −0.307984
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −1.33524e7 −1.89072
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 491092. 0.0670695 0.0335347 − 0.999438i \(-0.489324\pi\)
0.0335347 + 0.999438i \(0.489324\pi\)
\(558\) 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
\(565\) −9.02166e6 −1.18895
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.96659e6 −0.513613 −0.256807 − 0.966463i \(-0.582670\pi\)
−0.256807 + 0.966463i \(0.582670\pi\)
\(570\) 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
\(577\) 3.30624e6 0.413423 0.206712 − 0.978402i \(-0.433724\pi\)
0.206712 + 0.978402i \(0.433724\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.50619e6 0.544402
\(586\) 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
\(593\) 1.75899e6 0.205413 0.102706 − 0.994712i \(-0.467250\pi\)
0.102706 + 0.994712i \(0.467250\pi\)
\(594\) 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
\(601\) 1.51550e7 1.71148 0.855739 − 0.517408i \(-0.173103\pi\)
0.855739 + 0.517408i \(0.173103\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.22399e7 1.35953
\(606\) 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
\(613\) −1.48960e7 −1.60110 −0.800552 − 0.599263i \(-0.795460\pi\)
−0.800552 + 0.599263i \(0.795460\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.77140e7 1.87329 0.936644 − 0.350282i \(-0.113914\pi\)
0.936644 + 0.350282i \(0.113914\pi\)
\(618\) 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
\(625\) −1.10222e7 −1.12867
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.53167e7 2.55141
\(630\) 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
\(637\) −4.10091e6 −0.400434
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.45952e7 −1.40303 −0.701514 − 0.712655i \(-0.747492\pi\)
−0.701514 + 0.712655i \(0.747492\pi\)
\(642\) 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
\(653\) 1.93996e7 1.78036 0.890182 − 0.455605i \(-0.150577\pi\)
0.890182 + 0.455605i \(0.150577\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.15799e7 1.95045
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −2.07531e7 −1.84747 −0.923737 − 0.383027i \(-0.874881\pi\)
−0.923737 + 0.383027i \(0.874881\pi\)
\(662\) 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
\(673\) −1.73990e7 −1.48077 −0.740383 − 0.672185i \(-0.765356\pi\)
−0.740383 + 0.672185i \(0.765356\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.34115e7 1.96317 0.981584 − 0.191032i \(-0.0611833\pi\)
0.981584 + 0.191032i \(0.0611833\pi\)
\(678\) 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
\(685\) 5.75001e6 0.468211
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.81954e6 0.788030
\(690\) 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
\(697\) −4.69699e7 −3.66216
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.51373e7 1.16346 0.581731 − 0.813381i \(-0.302376\pi\)
0.581731 + 0.813381i \(0.302376\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.64712e7 1.97769 0.988846 − 0.148942i \(-0.0475868\pi\)
0.988846 + 0.148942i \(0.0475868\pi\)
\(710\) 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
\(725\) 2.27032e7 1.60414
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.05122e7 0.722662 0.361331 − 0.932438i \(-0.382322\pi\)
0.361331 + 0.932438i \(0.382322\pi\)
\(734\) 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
\(745\) 4.10324e7 2.70855
\(746\) 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
\(757\) −3.00451e6 −0.190561 −0.0952804 − 0.995450i \(-0.530375\pi\)
−0.0952804 + 0.995450i \(0.530375\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.95198e7 −1.22184 −0.610919 − 0.791693i \(-0.709200\pi\)
−0.610919 + 0.791693i \(0.709200\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.14053e7 −2.55801
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.02848e7 1.23695 0.618477 − 0.785803i \(-0.287750\pi\)
0.618477 + 0.785803i \(0.287750\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.28636e7 1.97818 0.989090 − 0.147312i \(-0.0470622\pi\)
0.989090 + 0.147312i \(0.0470622\pi\)
\(774\) 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
\(785\) 2.82182e7 1.63439
\(786\) 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
\(793\) 1.34073e7 0.757110
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.49645e7 −0.834481 −0.417241 − 0.908796i \(-0.637003\pi\)
−0.417241 + 0.908796i \(0.637003\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.24052e7 −0.683157
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.87790e7 −1.54598 −0.772992 − 0.634415i \(-0.781241\pi\)
−0.772992 + 0.634415i \(0.781241\pi\)
\(810\) 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
\(821\) −2.14631e7 −1.11131 −0.555655 − 0.831413i \(-0.687533\pi\)
−0.555655 + 0.831413i \(0.687533\pi\)
\(822\) 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
\(829\) 3.87641e7 1.95904 0.979520 − 0.201346i \(-0.0645316\pi\)
0.979520 + 0.201346i \(0.0645316\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.76813e7 1.88154
\(834\) 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
\(841\) 5.28309e7 2.57572
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.36935e7 1.14153
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −7.55204e6 −0.355379 −0.177690 − 0.984087i \(-0.556862\pi\)
−0.177690 + 0.984087i \(0.556862\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.32921e7 −1.54842 −0.774210 − 0.632929i \(-0.781852\pi\)
−0.774210 + 0.632929i \(0.781852\pi\)
\(858\) 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
\(865\) −4.13625e7 −1.87961
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −2.23905e7 −0.994325
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3.88193e7 −1.70431 −0.852155 − 0.523289i \(-0.824705\pi\)
−0.852155 + 0.523289i \(0.824705\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.40848e7 −1.91359 −0.956794 − 0.290765i \(-0.906090\pi\)
−0.956794 + 0.290765i \(0.906090\pi\)
\(882\) 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
\(901\) −9.02270e7 −3.70275
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.80564e7 2.35629
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −4.37643e7 −1.75675
\(910\) 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
\(925\) −2.99351e7 −1.15034
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −4.76633e7 −1.81194 −0.905972 − 0.423337i \(-0.860859\pi\)
−0.905972 + 0.423337i \(0.860859\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.10260e7 0.410271 0.205135 − 0.978734i \(-0.434237\pi\)
0.205135 + 0.978734i \(0.434237\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.41081e7 1.99199 0.995997 − 0.0893816i \(-0.0284891\pi\)
0.995997 + 0.0893816i \(0.0284891\pi\)
\(942\) 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
\(949\) −2.16687e7 −0.781028
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4.61989e7 −1.64778 −0.823890 − 0.566749i \(-0.808201\pi\)
−0.823890 + 0.566749i \(0.808201\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.86292e7 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3.77943e7 −1.30650
\(966\) 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
\(977\) 4.78045e7 1.60226 0.801130 − 0.598491i \(-0.204233\pi\)
0.801130 + 0.598491i \(0.204233\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −6.82830e6 −0.226537
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 7.75890e7 2.54806
\(986\) 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
\(997\) −5.12753e7 −1.63369 −0.816846 − 0.576856i \(-0.804280\pi\)
−0.816846 + 0.576856i \(0.804280\pi\)
\(998\) 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 256.6.a.b.1.1 1
4.3 odd 2 CM 256.6.a.b.1.1 1
8.3 odd 2 256.6.a.c.1.1 1
8.5 even 2 256.6.a.c.1.1 1
16.3 odd 4 128.6.b.b.65.2 yes 2
16.5 even 4 128.6.b.b.65.1 ✓ 2
16.11 odd 4 128.6.b.b.65.1 ✓ 2
16.13 even 4 128.6.b.b.65.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.6.b.b.65.1 ✓ 2 16.5 even 4
128.6.b.b.65.1 ✓ 2 16.11 odd 4
128.6.b.b.65.2 yes 2 16.3 odd 4
128.6.b.b.65.2 yes 2 16.13 even 4
256.6.a.b.1.1 1 1.1 even 1 trivial
256.6.a.b.1.1 1 4.3 odd 2 CM
256.6.a.c.1.1 1 8.3 odd 2
256.6.a.c.1.1 1 8.5 even 2