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Properties

Label 27.5.b.a.26.1
Level $27$
Weight $5$
Character 27.26
Self dual yes
Analytic conductor $2.791$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(2.79098900326\)
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)
 
\(f(q)\) \(=\) \(q+16.0000 q^{4} +71.0000 q^{7} +O(q^{10})\) \(q+16.0000 q^{4} +71.0000 q^{7} -337.000 q^{13} +256.000 q^{16} -601.000 q^{19} +625.000 q^{25} +1136.00 q^{28} +194.000 q^{31} -529.000 q^{37} -3214.00 q^{43} +2640.00 q^{49} -5392.00 q^{52} +7199.00 q^{61} +4096.00 q^{64} +2903.00 q^{67} -1249.00 q^{73} -9616.00 q^{76} +4679.00 q^{79} -23927.0 q^{91} +9071.00 q^{97} +O(q^{100})\)

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
\(4\) 16.0000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 71.0000 1.44898 0.724490 − 0.689286i \(-0.242075\pi\)
0.724490 + 0.689286i \(0.242075\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −337.000 −1.99408 −0.997041 − 0.0768662i \(-0.975509\pi\)
−0.997041 + 0.0768662i \(0.975509\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −601.000 −1.66482 −0.832410 − 0.554160i \(-0.813039\pi\)
−0.832410 + 0.554160i \(0.813039\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 1136.00 1.44898
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 194.000 0.201873 0.100937 − 0.994893i \(-0.467816\pi\)
0.100937 + 0.994893i \(0.467816\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −529.000 −0.386413 −0.193207 − 0.981158i \(-0.561889\pi\)
−0.193207 + 0.981158i \(0.561889\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3214.00 −1.73824 −0.869118 − 0.494604i \(-0.835313\pi\)
−0.869118 + 0.494604i \(0.835313\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2640.00 1.09954
\(50\) 0 0
\(51\) 0 0
\(52\) −5392.00 −1.99408
\(53\) 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
\(61\) 7199.00 1.93469 0.967347 − 0.253454i \(-0.0815666\pi\)
0.967347 + 0.253454i \(0.0815666\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 4096.00 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 2903.00 0.646692 0.323346 − 0.946281i \(-0.395192\pi\)
0.323346 + 0.946281i \(0.395192\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1249.00 −0.234378 −0.117189 − 0.993110i \(-0.537388\pi\)
−0.117189 + 0.993110i \(0.537388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −9616.00 −1.66482
\(77\) 0 0
\(78\) 0 0
\(79\) 4679.00 0.749720 0.374860 − 0.927082i \(-0.377691\pi\)
0.374860 + 0.927082i \(0.377691\pi\)
\(80\) 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
\(91\) −23927.0 −2.88939
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9071.00 0.964077 0.482038 − 0.876150i \(-0.339897\pi\)
0.482038 + 0.876150i \(0.339897\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 10000.0 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −19849.0 −1.87096 −0.935479 − 0.353381i \(-0.885032\pi\)
−0.935479 + 0.353381i \(0.885032\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 22034.0 1.85456 0.927279 − 0.374371i \(-0.122141\pi\)
0.927279 + 0.374371i \(0.122141\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 18176.0 1.44898
\(113\) 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
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 3104.00 0.201873
\(125\) 0 0
\(126\) 0 0
\(127\) −10942.0 −0.678405 −0.339203 − 0.940713i \(-0.610157\pi\)
−0.339203 + 0.940713i \(0.610157\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −42671.0 −2.41229
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 24359.0 1.26075 0.630376 − 0.776290i \(-0.282901\pi\)
0.630376 + 0.776290i \(0.282901\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −8464.00 −0.386413
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −42121.0 −1.84733 −0.923666 − 0.383199i \(-0.874822\pi\)
−0.923666 + 0.383199i \(0.874822\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −35374.0 −1.43511 −0.717554 − 0.696502i \(-0.754739\pi\)
−0.717554 + 0.696502i \(0.754739\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 36263.0 1.36486 0.682431 − 0.730950i \(-0.260923\pi\)
0.682431 + 0.730950i \(0.260923\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 85008.0 2.97637
\(170\) 0 0
\(171\) 0 0
\(172\) −51424.0 −1.73824
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 44375.0 1.44898
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 32447.0 0.990415 0.495208 − 0.868775i \(-0.335092\pi\)
0.495208 + 0.868775i \(0.335092\pi\)
\(182\) 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
\(193\) −54049.0 −1.45102 −0.725509 − 0.688212i \(-0.758396\pi\)
−0.725509 + 0.688212i \(0.758396\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 42240.0 1.09954
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −2473.00 −0.0624479 −0.0312240 − 0.999512i \(-0.509941\pi\)
−0.0312240 + 0.999512i \(0.509941\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −86272.0 −1.99408
\(209\) 0 0
\(210\) 0 0
\(211\) −25033.0 −0.562274 −0.281137 − 0.959668i \(-0.590712\pi\)
−0.281137 + 0.959668i \(0.590712\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 13774.0 0.292510
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 14786.0 0.297332 0.148666 − 0.988888i \(-0.452502\pi\)
0.148666 + 0.988888i \(0.452502\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −104206. −1.98711 −0.993555 − 0.113354i \(-0.963841\pi\)
−0.993555 + 0.113354i \(0.963841\pi\)
\(230\) 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
\(241\) −78913.0 −1.35867 −0.679336 − 0.733828i \(-0.737732\pi\)
−0.679336 + 0.733828i \(0.737732\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 115184. 1.93469
\(245\) 0 0
\(246\) 0 0
\(247\) 202537. 3.31979
\(248\) 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
\(256\) 65536.0 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −37559.0 −0.559905
\(260\) 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
\(268\) 46448.0 0.646692
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −57481.0 −0.782683 −0.391341 − 0.920246i \(-0.627989\pi\)
−0.391341 + 0.920246i \(0.627989\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −138574. −1.80602 −0.903009 − 0.429621i \(-0.858647\pi\)
−0.903009 + 0.429621i \(0.858647\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 49586.0 0.619136 0.309568 − 0.950877i \(-0.399816\pi\)
0.309568 + 0.950877i \(0.399816\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −19984.0 −0.234378
\(293\) 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
\(301\) −228194. −2.51867
\(302\) 0 0
\(303\) 0 0
\(304\) −153856. −1.66482
\(305\) 0 0
\(306\) 0 0
\(307\) −60334.0 −0.640155 −0.320078 − 0.947391i \(-0.603709\pi\)
−0.320078 + 0.947391i \(0.603709\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 162863. 1.66239 0.831197 − 0.555979i \(-0.187656\pi\)
0.831197 + 0.555979i \(0.187656\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 74864.0 0.749720
\(317\) 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
\(325\) −210625. −1.99408
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 217799. 1.98792 0.993962 − 0.109722i \(-0.0349962\pi\)
0.993962 + 0.109722i \(0.0349962\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 194063. 1.70877 0.854384 − 0.519643i \(-0.173935\pi\)
0.854384 + 0.519643i \(0.173935\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16969.0 0.144234
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 206639. 1.69653 0.848265 − 0.529572i \(-0.177648\pi\)
0.848265 + 0.529572i \(0.177648\pi\)
\(350\) 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
\(361\) 230880. 1.77163
\(362\) 0 0
\(363\) 0 0
\(364\) −382832. −2.88939
\(365\) 0 0
\(366\) 0 0
\(367\) −28297.0 −0.210091 −0.105046 − 0.994467i \(-0.533499\pi\)
−0.105046 + 0.994467i \(0.533499\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 54671.0 0.392952 0.196476 − 0.980509i \(-0.437050\pi\)
0.196476 + 0.980509i \(0.437050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −280393. −1.95204 −0.976020 − 0.217681i \(-0.930151\pi\)
−0.976020 + 0.217681i \(0.930151\pi\)
\(380\) 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
\(388\) 145136. 0.964077
\(389\) 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
\(397\) −184174. −1.16855 −0.584275 − 0.811556i \(-0.698621\pi\)
−0.584275 + 0.811556i \(0.698621\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 160000. 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −65378.0 −0.402552
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −314113. −1.87776 −0.938878 − 0.344249i \(-0.888133\pi\)
−0.938878 + 0.344249i \(0.888133\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −317584. −1.87096
\(413\) 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
\(421\) −123121. −0.694653 −0.347327 − 0.937744i \(-0.612910\pi\)
−0.347327 + 0.937744i \(0.612910\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 511129. 2.80333
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 368066. 1.96313 0.981567 − 0.191119i \(-0.0612116\pi\)
0.981567 + 0.191119i \(0.0612116\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 352544. 1.85456
\(437\) 0 0
\(438\) 0 0
\(439\) −376606. −1.95415 −0.977076 − 0.212892i \(-0.931712\pi\)
−0.977076 + 0.212892i \(0.931712\pi\)
\(440\) 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
\(448\) 290816. 1.44898
\(449\) 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
\(457\) 244898. 1.17261 0.586304 − 0.810091i \(-0.300582\pi\)
0.586304 + 0.810091i \(0.300582\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −271129. −1.26478 −0.632389 − 0.774651i \(-0.717925\pi\)
−0.632389 + 0.774651i \(0.717925\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 206113. 0.937043
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −375625. −1.66482
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 178273. 0.770540
\(482\) 0 0
\(483\) 0 0
\(484\) 234256. 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −352537. −1.48644 −0.743219 − 0.669048i \(-0.766702\pi\)
−0.743219 + 0.669048i \(0.766702\pi\)
\(488\) 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
\(496\) 49664.0 0.201873
\(497\) 0 0
\(498\) 0 0
\(499\) −497326. −1.99729 −0.998643 − 0.0520865i \(-0.983413\pi\)
−0.998643 + 0.0520865i \(0.983413\pi\)
\(500\) 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
\(508\) −175072. −0.678405
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −88679.0 −0.339609
\(512\) 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
\(523\) 97751.0 0.357370 0.178685 − 0.983906i \(-0.442816\pi\)
0.178685 + 0.983906i \(0.442816\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −682736. −2.41229
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −527281. −1.80156 −0.900778 − 0.434281i \(-0.857003\pi\)
−0.900778 + 0.434281i \(0.857003\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 596231. 1.99269 0.996345 − 0.0854161i \(-0.0272220\pi\)
0.996345 + 0.0854161i \(0.0272220\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 332209. 1.08633
\(554\) 0 0
\(555\) 0 0
\(556\) 389744. 1.26075
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.08312e6 3.46619
\(560\) 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
\(571\) −619321. −1.89952 −0.949759 − 0.312981i \(-0.898672\pi\)
−0.949759 + 0.312981i \(0.898672\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 401231. 1.20515 0.602577 − 0.798060i \(-0.294140\pi\)
0.602577 + 0.798060i \(0.294140\pi\)
\(578\) 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
\(589\) −116594. −0.336082
\(590\) 0 0
\(591\) 0 0
\(592\) −135424. −0.386413
\(593\) 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
\(601\) −445726. −1.23401 −0.617005 − 0.786959i \(-0.711654\pi\)
−0.617005 + 0.786959i \(0.711654\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −673936. −1.84733
\(605\) 0 0
\(606\) 0 0
\(607\) 672071. 1.82405 0.912027 − 0.410130i \(-0.134517\pi\)
0.912027 + 0.410130i \(0.134517\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 214751. 0.571497 0.285749 − 0.958305i \(-0.407758\pi\)
0.285749 + 0.958305i \(0.407758\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 134279. 0.350451 0.175225 − 0.984528i \(-0.443935\pi\)
0.175225 + 0.984528i \(0.443935\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −565984. −1.43511
\(629\) 0 0
\(630\) 0 0
\(631\) −451753. −1.13460 −0.567299 − 0.823512i \(-0.692012\pi\)
−0.567299 + 0.823512i \(0.692012\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −889680. −2.19258
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −728302. −1.76153 −0.880764 − 0.473555i \(-0.842970\pi\)
−0.880764 + 0.473555i \(0.842970\pi\)
\(644\) 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
\(652\) 580208. 1.36486
\(653\) 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
\(661\) 290399. 0.664649 0.332324 − 0.943165i \(-0.392167\pi\)
0.332324 + 0.943165i \(0.392167\pi\)
\(662\) 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
\(673\) −905329. −1.99883 −0.999416 − 0.0341703i \(-0.989121\pi\)
−0.999416 + 0.0341703i \(0.989121\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.36013e6 2.97637
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 644041. 1.39693
\(680\) 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
\(688\) −822784. −1.73824
\(689\) 0 0
\(690\) 0 0
\(691\) 782162. 1.63810 0.819050 − 0.573722i \(-0.194501\pi\)
0.819050 + 0.573722i \(0.194501\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 710000. 1.44898
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 317929. 0.643309
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 929519. 1.84912 0.924562 − 0.381033i \(-0.124432\pi\)
0.924562 + 0.381033i \(0.124432\pi\)
\(710\) 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
\(721\) −1.40928e6 −2.71098
\(722\) 0 0
\(723\) 0 0
\(724\) 519152. 0.990415
\(725\) 0 0
\(726\) 0 0
\(727\) −824734. −1.56043 −0.780216 − 0.625510i \(-0.784891\pi\)
−0.780216 + 0.625510i \(0.784891\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5422.00 −0.0100914 −0.00504570 − 0.999987i \(-0.501606\pi\)
−0.00504570 + 0.999987i \(0.501606\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 401042. 0.734346 0.367173 − 0.930153i \(-0.380326\pi\)
0.367173 + 0.930153i \(0.380326\pi\)
\(740\) 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
\(751\) −1.09596e6 −1.94319 −0.971595 − 0.236650i \(-0.923950\pi\)
−0.971595 + 0.236650i \(0.923950\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −693169. −1.20962 −0.604808 − 0.796371i \(-0.706750\pi\)
−0.604808 + 0.796371i \(0.706750\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.56441e6 2.68722
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −437953. −0.740585 −0.370292 − 0.928915i \(-0.620743\pi\)
−0.370292 + 0.928915i \(0.620743\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −864784. −1.45102
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 121250. 0.201873
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 675840. 1.09954
\(785\) 0 0
\(786\) 0 0
\(787\) −862969. −1.39330 −0.696652 − 0.717409i \(-0.745328\pi\)
−0.696652 + 0.717409i \(0.745328\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.42606e6 −3.85794
\(794\) 0 0
\(795\) 0 0
\(796\) −39568.0 −0.0624479
\(797\) 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
\(811\) 976754. 1.48506 0.742529 − 0.669814i \(-0.233626\pi\)
0.742529 + 0.669814i \(0.233626\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.93161e6 2.89385
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.27298e6 1.87942 0.939708 − 0.341978i \(-0.111097\pi\)
0.939708 + 0.341978i \(0.111097\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 1.21440e6 1.76706 0.883532 − 0.468371i \(-0.155159\pi\)
0.883532 + 0.468371i \(0.155159\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.38035e6 −1.99408
\(833\) 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
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −400528. −0.562274
\(845\) 0 0
\(846\) 0 0
\(847\) 1.03951e6 1.44898
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.22386e6 −1.68203 −0.841013 − 0.541015i \(-0.818040\pi\)
−0.841013 + 0.541015i \(0.818040\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 923639. 1.25175 0.625873 − 0.779925i \(-0.284743\pi\)
0.625873 + 0.779925i \(0.284743\pi\)
\(860\) 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
\(868\) 220384. 0.292510
\(869\) 0 0
\(870\) 0 0
\(871\) −978311. −1.28956
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.44427e6 1.87780 0.938900 − 0.344189i \(-0.111846\pi\)
0.938900 + 0.344189i \(0.111846\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.36313e6 −1.74830 −0.874149 − 0.485658i \(-0.838580\pi\)
−0.874149 + 0.485658i \(0.838580\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −776882. −0.982996
\(890\) 0 0
\(891\) 0 0
\(892\) 236576. 0.297332
\(893\) 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
\(907\) 465911. 0.566355 0.283177 − 0.959068i \(-0.408612\pi\)
0.283177 + 0.959068i \(0.408612\pi\)
\(908\) 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
\(916\) −1.66730e6 −1.98711
\(917\) 0 0
\(918\) 0 0
\(919\) −939166. −1.11202 −0.556008 − 0.831177i \(-0.687668\pi\)
−0.556008 + 0.831177i \(0.687668\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −330625. −0.386413
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.58664e6 −1.83054
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.65547e6 1.88557 0.942784 − 0.333403i \(-0.108197\pi\)
0.942784 + 0.333403i \(0.108197\pi\)
\(938\) 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
\(949\) 420913. 0.467369
\(950\) 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
\(961\) −885885. −0.959247
\(962\) 0 0
\(963\) 0 0
\(964\) −1.26261e6 −1.35867
\(965\) 0 0
\(966\) 0 0
\(967\) −1.80617e6 −1.93155 −0.965774 − 0.259386i \(-0.916480\pi\)
−0.965774 + 0.259386i \(0.916480\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 1.72949e6 1.82680
\(974\) 0 0
\(975\) 0 0
\(976\) 1.84294e6 1.93469
\(977\) 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
\(988\) 3.24059e6 3.31979
\(989\) 0 0
\(990\) 0 0
\(991\) 1.05996e6 1.07930 0.539649 − 0.841890i \(-0.318557\pi\)
0.539649 + 0.841890i \(0.318557\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.59922e6 1.60886 0.804428 − 0.594050i \(-0.202472\pi\)
0.804428 + 0.594050i \(0.202472\pi\)
\(998\) 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 27.5.b.a.26.1 ✓ 1
3.2 odd 2 CM 27.5.b.a.26.1 ✓ 1
4.3 odd 2 432.5.e.a.161.1 1
5.2 odd 4 675.5.d.c.674.2 2
5.3 odd 4 675.5.d.c.674.1 2
5.4 even 2 675.5.c.b.26.1 1
9.2 odd 6 81.5.d.a.53.1 2
9.4 even 3 81.5.d.a.26.1 2
9.5 odd 6 81.5.d.a.26.1 2
9.7 even 3 81.5.d.a.53.1 2
12.11 even 2 432.5.e.a.161.1 1
15.2 even 4 675.5.d.c.674.2 2
15.8 even 4 675.5.d.c.674.1 2
15.14 odd 2 675.5.c.b.26.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.a.26.1 ✓ 1 1.1 even 1 trivial
27.5.b.a.26.1 ✓ 1 3.2 odd 2 CM
81.5.d.a.26.1 2 9.4 even 3
81.5.d.a.26.1 2 9.5 odd 6
81.5.d.a.53.1 2 9.2 odd 6
81.5.d.a.53.1 2 9.7 even 3
432.5.e.a.161.1 1 4.3 odd 2
432.5.e.a.161.1 1 12.11 even 2
675.5.c.b.26.1 1 5.4 even 2
675.5.c.b.26.1 1 15.14 odd 2
675.5.d.c.674.1 2 5.3 odd 4
675.5.d.c.674.1 2 15.8 even 4
675.5.d.c.674.2 2 5.2 odd 4
675.5.d.c.674.2 2 15.2 even 4