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Properties

Label 800.4.a.e.1.1
Level $800$
Weight $4$
Character 800.1
Self dual yes
Analytic conductor $47.202$
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] = [800,4,Mod(1,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 800.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: \(47.2015280046\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{9} +O(q^{10})\) \(q-27.0000 q^{9} -92.0000 q^{13} +104.000 q^{17} +130.000 q^{29} +396.000 q^{37} +230.000 q^{41} -343.000 q^{49} +572.000 q^{53} -830.000 q^{61} +592.000 q^{73} +729.000 q^{81} +1670.00 q^{89} +1816.00 q^{97} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −27.0000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −92.0000 −1.96279 −0.981393 − 0.192012i \(-0.938499\pi\)
−0.981393 + 0.192012i \(0.938499\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 104.000 1.48375 0.741874 − 0.670540i \(-0.233937\pi\)
0.741874 + 0.670540i \(0.233937\pi\)
\(18\) 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
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 130.000 0.832427 0.416214 − 0.909267i \(-0.363357\pi\)
0.416214 + 0.909267i \(0.363357\pi\)
\(30\) 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
\(37\) 396.000 1.75951 0.879757 − 0.475424i \(-0.157705\pi\)
0.879757 + 0.475424i \(0.157705\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 230.000 0.876097 0.438048 − 0.898951i \(-0.355670\pi\)
0.438048 + 0.898951i \(0.355670\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 572.000 1.48246 0.741229 − 0.671253i \(-0.234243\pi\)
0.741229 + 0.671253i \(0.234243\pi\)
\(54\) 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
\(61\) −830.000 −1.74214 −0.871071 − 0.491158i \(-0.836574\pi\)
−0.871071 + 0.491158i \(0.836574\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 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
\(73\) 592.000 0.949156 0.474578 − 0.880214i \(-0.342601\pi\)
0.474578 + 0.880214i \(0.342601\pi\)
\(74\) 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
\(81\) 729.000 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1670.00 1.98898 0.994492 − 0.104809i \(-0.0334231\pi\)
0.994492 + 0.104809i \(0.0334231\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1816.00 1.90090 0.950448 − 0.310884i \(-0.100625\pi\)
0.950448 + 0.310884i \(0.100625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 598.000 0.589141 0.294570 − 0.955630i \(-0.404823\pi\)
0.294570 + 0.955630i \(0.404823\pi\)
\(102\) 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
\(109\) −1746.00 −1.53428 −0.767140 − 0.641480i \(-0.778321\pi\)
−0.767140 + 0.641480i \(0.778321\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1328.00 1.10556 0.552778 − 0.833329i \(-0.313568\pi\)
0.552778 + 0.833329i \(0.313568\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2484.00 1.96279
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 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
\(137\) 2776.00 1.73117 0.865583 − 0.500766i \(-0.166948\pi\)
0.865583 + 0.500766i \(0.166948\pi\)
\(138\) 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
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3514.00 −1.93207 −0.966034 − 0.258415i \(-0.916800\pi\)
−0.966034 + 0.258415i \(0.916800\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −2808.00 −1.48375
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3924.00 1.99471 0.997354 − 0.0726920i \(-0.0231590\pi\)
0.997354 + 0.0726920i \(0.0231590\pi\)
\(158\) 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
\(169\) 6267.00 2.85253
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2012.00 0.884217 0.442108 − 0.896962i \(-0.354231\pi\)
0.442108 + 0.896962i \(0.354231\pi\)
\(174\) 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
\(181\) 3942.00 1.61882 0.809410 − 0.587243i \(-0.199787\pi\)
0.809410 + 0.587243i \(0.199787\pi\)
\(182\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −72.0000 −0.0268532 −0.0134266 − 0.999910i \(-0.504274\pi\)
−0.0134266 + 0.999910i \(0.504274\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5404.00 −1.95441 −0.977206 − 0.212295i \(-0.931906\pi\)
−0.977206 + 0.212295i \(0.931906\pi\)
\(198\) 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
\(205\) 0 0
\(206\) 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
\(221\) −9568.00 −2.91228
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −6390.00 −1.84394 −0.921972 − 0.387257i \(-0.873423\pi\)
−0.921972 + 0.387257i \(0.873423\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7088.00 1.99292 0.996460 − 0.0840693i \(-0.0267917\pi\)
0.996460 + 0.0840693i \(0.0267917\pi\)
\(234\) 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
\(241\) 5310.00 1.41928 0.709641 − 0.704563i \(-0.248857\pi\)
0.709641 + 0.704563i \(0.248857\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 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
\(257\) −8096.00 −1.96504 −0.982519 − 0.186164i \(-0.940394\pi\)
−0.982519 + 0.186164i \(0.940394\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3510.00 −0.832427
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3406.00 0.771998 0.385999 − 0.922499i \(-0.373857\pi\)
0.385999 + 0.922499i \(0.373857\pi\)
\(270\) 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
\(277\) −1316.00 −0.285454 −0.142727 − 0.989762i \(-0.545587\pi\)
−0.142727 + 0.989762i \(0.545587\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7430.00 1.57735 0.788677 − 0.614807i \(-0.210766\pi\)
0.788677 + 0.614807i \(0.210766\pi\)
\(282\) 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
\(289\) 5903.00 1.20151
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3452.00 −0.688287 −0.344143 − 0.938917i \(-0.611831\pi\)
−0.344143 + 0.938917i \(0.611831\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 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
\(313\) −8712.00 −1.57326 −0.786632 − 0.617423i \(-0.788177\pi\)
−0.786632 + 0.617423i \(0.788177\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4676.00 −0.828487 −0.414243 − 0.910166i \(-0.635954\pi\)
−0.414243 + 0.910166i \(0.635954\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 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
\(333\) −10692.0 −1.75951
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 416.000 0.0672432 0.0336216 − 0.999435i \(-0.489296\pi\)
0.0336216 + 0.999435i \(0.489296\pi\)
\(338\) 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
\(349\) −9470.00 −1.45249 −0.726243 − 0.687438i \(-0.758735\pi\)
−0.726243 + 0.687438i \(0.758735\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12848.0 −1.93720 −0.968598 − 0.248633i \(-0.920019\pi\)
−0.968598 + 0.248633i \(0.920019\pi\)
\(354\) 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
\(361\) −6859.00 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −6210.00 −0.876097
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6372.00 0.884530 0.442265 − 0.896884i \(-0.354175\pi\)
0.442265 + 0.896884i \(0.354175\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −11960.0 −1.63388
\(378\) 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
\(389\) 374.000 0.0487469 0.0243735 − 0.999703i \(-0.492241\pi\)
0.0243735 + 0.999703i \(0.492241\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −12564.0 −1.58834 −0.794168 − 0.607699i \(-0.792093\pi\)
−0.794168 + 0.607699i \(0.792093\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2398.00 −0.298629 −0.149315 − 0.988790i \(-0.547707\pi\)
−0.149315 + 0.988790i \(0.547707\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 7146.00 0.863929 0.431964 − 0.901891i \(-0.357821\pi\)
0.431964 + 0.901891i \(0.357821\pi\)
\(410\) 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
\(421\) 10890.0 1.26068 0.630340 − 0.776319i \(-0.282916\pi\)
0.630340 + 0.776319i \(0.282916\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 17352.0 1.92583 0.962914 − 0.269807i \(-0.0869599\pi\)
0.962914 + 0.269807i \(0.0869599\pi\)
\(434\) 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
\(441\) 9261.00 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16114.0 1.69369 0.846845 − 0.531840i \(-0.178499\pi\)
0.846845 + 0.531840i \(0.178499\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10456.0 −1.07026 −0.535132 − 0.844768i \(-0.679738\pi\)
−0.535132 + 0.844768i \(0.679738\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2318.00 0.234187 0.117093 − 0.993121i \(-0.462642\pi\)
0.117093 + 0.993121i \(0.462642\pi\)
\(462\) 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
\(477\) −15444.0 −1.48246
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −36432.0 −3.45355
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 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
\(493\) 13520.0 1.23511
\(494\) 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
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14270.0 −1.24265 −0.621323 − 0.783555i \(-0.713404\pi\)
−0.621323 + 0.783555i \(0.713404\pi\)
\(510\) 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
\(521\) 23738.0 1.99612 0.998062 − 0.0622265i \(-0.0198201\pi\)
0.998062 + 0.0622265i \(0.0198201\pi\)
\(522\) 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
\(529\) −12167.0 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −21160.0 −1.71959
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5922.00 −0.470622 −0.235311 − 0.971920i \(-0.575611\pi\)
−0.235311 + 0.971920i \(0.575611\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 22410.0 1.74214
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24836.0 1.88929 0.944646 − 0.328093i \(-0.106406\pi\)
0.944646 + 0.328093i \(0.106406\pi\)
\(558\) 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
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26806.0 −1.97498 −0.987492 − 0.157669i \(-0.949602\pi\)
−0.987492 + 0.157669i \(0.949602\pi\)
\(570\) 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
\(577\) 27504.0 1.98441 0.992207 − 0.124603i \(-0.0397657\pi\)
0.992207 + 0.124603i \(0.0397657\pi\)
\(578\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −24368.0 −1.68748 −0.843738 − 0.536755i \(-0.819650\pi\)
−0.843738 + 0.536755i \(0.819650\pi\)
\(594\) 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
\(601\) 17030.0 1.15585 0.577927 − 0.816089i \(-0.303862\pi\)
0.577927 + 0.816089i \(0.303862\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 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
\(613\) 19548.0 1.28799 0.643994 − 0.765031i \(-0.277276\pi\)
0.643994 + 0.765031i \(0.277276\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26464.0 1.72674 0.863372 − 0.504569i \(-0.168348\pi\)
0.863372 + 0.504569i \(0.168348\pi\)
\(618\) 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
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 41184.0 2.61067
\(630\) 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
\(637\) 31556.0 1.96279
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28850.0 −1.77770 −0.888851 − 0.458197i \(-0.848495\pi\)
−0.888851 + 0.458197i \(0.848495\pi\)
\(642\) 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
\(653\) 1012.00 0.0606472 0.0303236 − 0.999540i \(-0.490346\pi\)
0.0303236 + 0.999540i \(0.490346\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −15984.0 −0.949156
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 25850.0 1.52110 0.760551 − 0.649278i \(-0.224929\pi\)
0.760551 + 0.649278i \(0.224929\pi\)
\(662\) 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
\(673\) 34632.0 1.98360 0.991802 − 0.127784i \(-0.0407865\pi\)
0.991802 + 0.127784i \(0.0407865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 34996.0 1.98671 0.993357 − 0.115072i \(-0.0367100\pi\)
0.993357 + 0.115072i \(0.0367100\pi\)
\(678\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −52624.0 −2.90975
\(690\) 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
\(697\) 23920.0 1.29991
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −20030.0 −1.07920 −0.539602 − 0.841920i \(-0.681425\pi\)
−0.539602 + 0.841920i \(0.681425\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 36810.0 1.94983 0.974914 − 0.222580i \(-0.0714479\pi\)
0.974914 + 0.222580i \(0.0714479\pi\)
\(710\) 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
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −19683.0 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 8732.00 0.440005 0.220003 − 0.975499i \(-0.429393\pi\)
0.220003 + 0.975499i \(0.429393\pi\)
\(734\) 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
\(745\) 0 0
\(746\) 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
\(757\) −22516.0 −1.08105 −0.540527 − 0.841327i \(-0.681775\pi\)
−0.540527 + 0.841327i \(0.681775\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 31882.0 1.51869 0.759344 − 0.650689i \(-0.225520\pi\)
0.759344 + 0.650689i \(0.225520\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −9650.00 −0.452520 −0.226260 − 0.974067i \(-0.572650\pi\)
−0.226260 + 0.974067i \(0.572650\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16852.0 −0.784119 −0.392060 − 0.919940i \(-0.628237\pi\)
−0.392060 + 0.919940i \(0.628237\pi\)
\(774\) 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
\(785\) 0 0
\(786\) 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
\(793\) 76360.0 3.41945
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16276.0 −0.723370 −0.361685 − 0.932300i \(-0.617798\pi\)
−0.361685 + 0.932300i \(0.617798\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −45090.0 −1.98898
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 23270.0 1.01129 0.505643 − 0.862743i \(-0.331255\pi\)
0.505643 + 0.862743i \(0.331255\pi\)
\(810\) 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
\(821\) 1850.00 0.0786424 0.0393212 − 0.999227i \(-0.487480\pi\)
0.0393212 + 0.999227i \(0.487480\pi\)
\(822\) 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
\(829\) 23166.0 0.970553 0.485276 − 0.874361i \(-0.338719\pi\)
0.485276 + 0.874361i \(0.338719\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −35672.0 −1.48375
\(834\) 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
\(841\) −7489.00 −0.307065
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 45468.0 1.82508 0.912541 − 0.408986i \(-0.134117\pi\)
0.912541 + 0.408986i \(0.134117\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20056.0 0.799416 0.399708 − 0.916642i \(-0.369111\pi\)
0.399708 + 0.916642i \(0.369111\pi\)
\(858\) 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
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −49032.0 −1.90090
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 29844.0 1.14910 0.574550 − 0.818470i \(-0.305177\pi\)
0.574550 + 0.818470i \(0.305177\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7150.00 0.273427 0.136714 − 0.990611i \(-0.456346\pi\)
0.136714 + 0.990611i \(0.456346\pi\)
\(882\) 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
\(901\) 59488.0 2.19959
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −16146.0 −0.589141
\(910\) 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
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 30866.0 1.09008 0.545038 − 0.838411i \(-0.316515\pi\)
0.545038 + 0.838411i \(0.316515\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −24336.0 −0.848476 −0.424238 − 0.905551i \(-0.639458\pi\)
−0.424238 + 0.905551i \(0.639458\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −31378.0 −1.08703 −0.543514 − 0.839400i \(-0.682907\pi\)
−0.543514 + 0.839400i \(0.682907\pi\)
\(942\) 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
\(949\) −54464.0 −1.86299
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −15512.0 −0.527264 −0.263632 − 0.964623i \(-0.584921\pi\)
−0.263632 + 0.964623i \(0.584921\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −29791.0 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 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
\(977\) 22936.0 0.751062 0.375531 − 0.926810i \(-0.377460\pi\)
0.375531 + 0.926810i \(0.377460\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 47142.0 1.53428
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 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
\(997\) 34164.0 1.08524 0.542620 − 0.839978i \(-0.317432\pi\)
0.542620 + 0.839978i \(0.317432\pi\)
\(998\) 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 800.4.a.e.1.1 1
4.3 odd 2 CM 800.4.a.e.1.1 1
5.2 odd 4 160.4.c.a.129.1 ✓ 2
5.3 odd 4 160.4.c.a.129.2 yes 2
5.4 even 2 800.4.a.g.1.1 1
8.3 odd 2 1600.4.a.bb.1.1 1
8.5 even 2 1600.4.a.bb.1.1 1
15.2 even 4 1440.4.f.e.289.2 2
15.8 even 4 1440.4.f.e.289.1 2
20.3 even 4 160.4.c.a.129.2 yes 2
20.7 even 4 160.4.c.a.129.1 ✓ 2
20.19 odd 2 800.4.a.g.1.1 1
40.3 even 4 320.4.c.e.129.1 2
40.13 odd 4 320.4.c.e.129.1 2
40.19 odd 2 1600.4.a.z.1.1 1
40.27 even 4 320.4.c.e.129.2 2
40.29 even 2 1600.4.a.z.1.1 1
40.37 odd 4 320.4.c.e.129.2 2
60.23 odd 4 1440.4.f.e.289.1 2
60.47 odd 4 1440.4.f.e.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.4.c.a.129.1 ✓ 2 5.2 odd 4
160.4.c.a.129.1 ✓ 2 20.7 even 4
160.4.c.a.129.2 yes 2 5.3 odd 4
160.4.c.a.129.2 yes 2 20.3 even 4
320.4.c.e.129.1 2 40.3 even 4
320.4.c.e.129.1 2 40.13 odd 4
320.4.c.e.129.2 2 40.27 even 4
320.4.c.e.129.2 2 40.37 odd 4
800.4.a.e.1.1 1 1.1 even 1 trivial
800.4.a.e.1.1 1 4.3 odd 2 CM
800.4.a.g.1.1 1 5.4 even 2
800.4.a.g.1.1 1 20.19 odd 2
1440.4.f.e.289.1 2 15.8 even 4
1440.4.f.e.289.1 2 60.23 odd 4
1440.4.f.e.289.2 2 15.2 even 4
1440.4.f.e.289.2 2 60.47 odd 4
1600.4.a.z.1.1 1 40.19 odd 2
1600.4.a.z.1.1 1 40.29 even 2
1600.4.a.bb.1.1 1 8.3 odd 2
1600.4.a.bb.1.1 1 8.5 even 2