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Properties

Label 1089.6.a.f.1.1
Level $1089$
Weight $6$
Character 1089.1
Self dual yes
Analytic conductor $174.658$
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] = [1089,6,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(174.657979776\)
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 \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-32.0000 q^{4} -25.0000 q^{7} +O(q^{10})\) \(q-32.0000 q^{4} -25.0000 q^{7} +1202.00 q^{13} +1024.00 q^{16} +3143.00 q^{19} -3125.00 q^{25} +800.000 q^{28} +7601.00 q^{31} -9889.00 q^{37} -3352.00 q^{43} -16182.0 q^{49} -38464.0 q^{52} -18301.0 q^{61} -32768.0 q^{64} -37939.0 q^{67} +79577.0 q^{73} -100576. q^{76} +90857.0 q^{79} -30050.0 q^{91} +177725. q^{97} +O(q^{100})\)

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −32.0000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −25.0000 −0.192839 −0.0964195 − 0.995341i \(-0.530739\pi\)
−0.0964195 + 0.995341i \(0.530739\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 1202.00 1.97263 0.986316 − 0.164866i \(-0.0527191\pi\)
0.986316 + 0.164866i \(0.0527191\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1024.00 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 3143.00 1.99738 0.998689 − 0.0511835i \(-0.0162993\pi\)
0.998689 + 0.0511835i \(0.0162993\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −3125.00 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 800.000 0.192839
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7601.00 1.42058 0.710291 − 0.703908i \(-0.248563\pi\)
0.710291 + 0.703908i \(0.248563\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −9889.00 −1.18754 −0.593770 − 0.804635i \(-0.702361\pi\)
−0.593770 + 0.804635i \(0.702361\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −3352.00 −0.276460 −0.138230 − 0.990400i \(-0.544141\pi\)
−0.138230 + 0.990400i \(0.544141\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −16182.0 −0.962813
\(50\) 0 0
\(51\) 0 0
\(52\) −38464.0 −1.97263
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 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
\(61\) −18301.0 −0.629724 −0.314862 − 0.949137i \(-0.601958\pi\)
−0.314862 + 0.949137i \(0.601958\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −32768.0 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −37939.0 −1.03252 −0.516260 − 0.856432i \(-0.672676\pi\)
−0.516260 + 0.856432i \(0.672676\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 79577.0 1.74775 0.873877 − 0.486147i \(-0.161598\pi\)
0.873877 + 0.486147i \(0.161598\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −100576. −1.99738
\(77\) 0 0
\(78\) 0 0
\(79\) 90857.0 1.63791 0.818956 − 0.573856i \(-0.194553\pi\)
0.818956 + 0.573856i \(0.194553\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −30050.0 −0.380400
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 177725. 1.91787 0.958935 − 0.283626i \(-0.0915373\pi\)
0.958935 + 0.283626i \(0.0915373\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 100000. 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −211477. −1.96413 −0.982065 − 0.188544i \(-0.939623\pi\)
−0.982065 + 0.188544i \(0.939623\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 133361. 1.07513 0.537567 − 0.843221i \(-0.319344\pi\)
0.537567 + 0.843221i \(0.319344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −25600.0 −0.192839
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) −243232. −1.42058
\(125\) 0 0
\(126\) 0 0
\(127\) −80011.0 −0.440190 −0.220095 − 0.975478i \(-0.570637\pi\)
−0.220095 + 0.975478i \(0.570637\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −78575.0 −0.385173
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 252464. 1.10831 0.554157 − 0.832412i \(-0.313041\pi\)
0.554157 + 0.832412i \(0.313041\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 316448. 1.18754
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −408724. −1.45877 −0.729387 − 0.684102i \(-0.760194\pi\)
−0.729387 + 0.684102i \(0.760194\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 471911. 1.52796 0.763978 − 0.645242i \(-0.223243\pi\)
0.763978 + 0.645242i \(0.223243\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −325873. −0.960681 −0.480341 − 0.877082i \(-0.659487\pi\)
−0.480341 + 0.877082i \(0.659487\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.07351e6 2.89128
\(170\) 0 0
\(171\) 0 0
\(172\) 107264. 0.276460
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 78125.0 0.192839
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 619001. 1.40441 0.702207 − 0.711973i \(-0.252198\pi\)
0.702207 + 0.711973i \(0.252198\pi\)
\(182\) 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
\(193\) −1.02118e6 −1.97337 −0.986683 − 0.162653i \(-0.947995\pi\)
−0.986683 + 0.162653i \(0.947995\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 517824. 0.962813
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 1.01476e6 1.81648 0.908241 − 0.418448i \(-0.137426\pi\)
0.908241 + 0.418448i \(0.137426\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1.23085e6 1.97263
\(209\) 0 0
\(210\) 0 0
\(211\) −947323. −1.46485 −0.732423 − 0.680850i \(-0.761611\pi\)
−0.732423 + 0.680850i \(0.761611\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −190025. −0.273944
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.41103e6 −1.90009 −0.950043 − 0.312120i \(-0.898961\pi\)
−0.950043 + 0.312120i \(0.898961\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 1.26982e6 1.60012 0.800060 − 0.599919i \(-0.204801\pi\)
0.800060 + 0.599919i \(0.204801\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 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
\(241\) 1.29697e6 1.43843 0.719215 − 0.694788i \(-0.244502\pi\)
0.719215 + 0.694788i \(0.244502\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 585632. 0.629724
\(245\) 0 0
\(246\) 0 0
\(247\) 3.77789e6 3.94009
\(248\) 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
\(256\) 1.04858e6 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 247225. 0.229004
\(260\) 0 0
\(261\) 0 0
\(262\) 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
\(268\) 1.21405e6 1.03252
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 2.25285e6 1.86341 0.931707 − 0.363210i \(-0.118319\pi\)
0.931707 + 0.363210i \(0.118319\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 738389. 0.578210 0.289105 − 0.957297i \(-0.406642\pi\)
0.289105 + 0.957297i \(0.406642\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −2.33458e6 −1.73277 −0.866387 − 0.499373i \(-0.833564\pi\)
−0.866387 + 0.499373i \(0.833564\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41986e6 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −2.54646e6 −1.74775
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 83800.0 0.0533123
\(302\) 0 0
\(303\) 0 0
\(304\) 3.21843e6 1.99738
\(305\) 0 0
\(306\) 0 0
\(307\) −901189. −0.545720 −0.272860 − 0.962054i \(-0.587970\pi\)
−0.272860 + 0.962054i \(0.587970\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 733898. 0.423423 0.211712 − 0.977332i \(-0.432096\pi\)
0.211712 + 0.977332i \(0.432096\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −2.90742e6 −1.63791
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.75625e6 −1.97263
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 3.69300e6 1.85272 0.926359 − 0.376642i \(-0.122921\pi\)
0.926359 + 0.376642i \(0.122921\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.63172e6 1.26231 0.631155 − 0.775657i \(-0.282581\pi\)
0.631155 + 0.775657i \(0.282581\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 824725. 0.378507
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −3.48766e6 −1.53275 −0.766373 − 0.642396i \(-0.777941\pi\)
−0.766373 + 0.642396i \(0.777941\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 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
\(361\) 7.40235e6 2.98952
\(362\) 0 0
\(363\) 0 0
\(364\) 961600. 0.380400
\(365\) 0 0
\(366\) 0 0
\(367\) −2.57736e6 −0.998874 −0.499437 − 0.866350i \(-0.666460\pi\)
−0.499437 + 0.866350i \(0.666460\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.98388e6 0.738316 0.369158 − 0.929367i \(-0.379646\pi\)
0.369158 + 0.929367i \(0.379646\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 4.80817e6 1.71942 0.859709 − 0.510784i \(-0.170645\pi\)
0.859709 + 0.510784i \(0.170645\pi\)
\(380\) 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
\(388\) −5.68720e6 −1.91787
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5.00342e6 −1.59328 −0.796638 − 0.604456i \(-0.793390\pi\)
−0.796638 + 0.604456i \(0.793390\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.20000e6 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 9.13640e6 2.80229
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 6.36489e6 1.88141 0.940703 − 0.339231i \(-0.110167\pi\)
0.940703 + 0.339231i \(0.110167\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 6.76726e6 1.96413
\(413\) 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
\(421\) −2.23610e6 −0.614874 −0.307437 − 0.951568i \(-0.599471\pi\)
−0.307437 + 0.951568i \(0.599471\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 457525. 0.121435
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 5.63432e6 1.44418 0.722091 − 0.691798i \(-0.243181\pi\)
0.722091 + 0.691798i \(0.243181\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4.26755e6 −1.07513
\(437\) 0 0
\(438\) 0 0
\(439\) 4.08241e6 1.01101 0.505505 − 0.862824i \(-0.331306\pi\)
0.505505 + 0.862824i \(0.331306\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 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
\(448\) 819200. 0.192839
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.25745e6 −1.84950 −0.924752 − 0.380569i \(-0.875728\pi\)
−0.924752 + 0.380569i \(0.875728\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 6.11690e6 1.32611 0.663054 − 0.748572i \(-0.269260\pi\)
0.663054 + 0.748572i \(0.269260\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 948475. 0.199110
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −9.82188e6 −1.99738
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −1.18866e7 −2.34258
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 9.67036e6 1.84765 0.923827 − 0.382811i \(-0.125044\pi\)
0.923827 + 0.382811i \(0.125044\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 7.78342e6 1.42058
\(497\) 0 0
\(498\) 0 0
\(499\) 1.05353e7 1.89406 0.947030 − 0.321144i \(-0.104067\pi\)
0.947030 + 0.321144i \(0.104067\pi\)
\(500\) 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
\(508\) 2.56035e6 0.440190
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −1.98942e6 −0.337035
\(512\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 2.09363e6 0.334692 0.167346 − 0.985898i \(-0.446480\pi\)
0.167346 + 0.985898i \(0.446480\pi\)
\(524\) 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
\(532\) 2.51440e6 0.385173
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −9.98573e6 −1.46685 −0.733426 − 0.679769i \(-0.762080\pi\)
−0.733426 + 0.679769i \(0.762080\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27982e7 1.82886 0.914430 − 0.404744i \(-0.132639\pi\)
0.914430 + 0.404744i \(0.132639\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.27142e6 −0.315853
\(554\) 0 0
\(555\) 0 0
\(556\) −8.07885e6 −1.10831
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −4.02910e6 −0.545355
\(560\) 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
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 9.54998e6 1.22578 0.612889 − 0.790169i \(-0.290007\pi\)
0.612889 + 0.790169i \(0.290007\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.46322e7 1.82966 0.914830 − 0.403839i \(-0.132324\pi\)
0.914830 + 0.403839i \(0.132324\pi\)
\(578\) 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
\(589\) 2.38899e7 2.83744
\(590\) 0 0
\(591\) 0 0
\(592\) −1.01263e7 −1.18754
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 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
\(601\) −1.74342e7 −1.96887 −0.984435 − 0.175749i \(-0.943765\pi\)
−0.984435 + 0.175749i \(0.943765\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.30792e7 1.45877
\(605\) 0 0
\(606\) 0 0
\(607\) 1.57649e7 1.73668 0.868339 − 0.495970i \(-0.165188\pi\)
0.868339 + 0.495970i \(0.165188\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.63902e7 1.76171 0.880853 − 0.473389i \(-0.156969\pi\)
0.880853 + 0.473389i \(0.156969\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −6.55677e6 −0.687802 −0.343901 − 0.939006i \(-0.611748\pi\)
−0.343901 + 0.939006i \(0.611748\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.76562e6 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.51012e7 −1.52796
\(629\) 0 0
\(630\) 0 0
\(631\) −1.62439e7 −1.62411 −0.812057 − 0.583579i \(-0.801652\pi\)
−0.812057 + 0.583579i \(0.801652\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.94508e7 −1.89928
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −7.28542e6 −0.694908 −0.347454 − 0.937697i \(-0.612954\pi\)
−0.347454 + 0.937697i \(0.612954\pi\)
\(644\) 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
\(652\) 1.04279e7 0.960681
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.97928e7 1.76199 0.880993 − 0.473129i \(-0.156876\pi\)
0.880993 + 0.473129i \(0.156876\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.24188e7 −1.90798 −0.953991 − 0.299836i \(-0.903068\pi\)
−0.953991 + 0.299836i \(0.903068\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −3.43524e7 −2.89128
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −4.44312e6 −0.369840
\(680\) 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
\(688\) −3.43245e6 −0.276460
\(689\) 0 0
\(690\) 0 0
\(691\) 7.01920e6 0.559233 0.279616 − 0.960112i \(-0.409793\pi\)
0.279616 + 0.960112i \(0.409793\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −2.50000e6 −0.192839
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) −3.10811e7 −2.37197
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.27014e7 −1.69604 −0.848021 − 0.529962i \(-0.822206\pi\)
−0.848021 + 0.529962i \(0.822206\pi\)
\(710\) 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
\(721\) 5.28692e6 0.378761
\(722\) 0 0
\(723\) 0 0
\(724\) −1.98080e7 −1.40441
\(725\) 0 0
\(726\) 0 0
\(727\) −2.84512e7 −1.99648 −0.998240 − 0.0592978i \(-0.981114\pi\)
−0.998240 + 0.0592978i \(0.981114\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.13028e7 −0.777006 −0.388503 − 0.921448i \(-0.627008\pi\)
−0.388503 + 0.921448i \(0.627008\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.28170e7 0.863330 0.431665 − 0.902034i \(-0.357926\pi\)
0.431665 + 0.902034i \(0.357926\pi\)
\(740\) 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
\(751\) 2.61832e7 1.69404 0.847019 − 0.531563i \(-0.178395\pi\)
0.847019 + 0.531563i \(0.178395\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −3.11435e7 −1.97527 −0.987637 − 0.156758i \(-0.949896\pi\)
−0.987637 + 0.156758i \(0.949896\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −3.33402e6 −0.207328
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −3.08846e7 −1.88333 −0.941664 − 0.336554i \(-0.890739\pi\)
−0.941664 + 0.336554i \(0.890739\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.26777e7 1.97337
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −2.37531e7 −1.42058
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.65704e7 −0.962813
\(785\) 0 0
\(786\) 0 0
\(787\) 3.31061e7 1.90534 0.952668 − 0.304012i \(-0.0983263\pi\)
0.952668 + 0.304012i \(0.0983263\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.19978e7 −1.24221
\(794\) 0 0
\(795\) 0 0
\(796\) −3.24724e7 −1.81648
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 1.43756e7 0.767491 0.383745 − 0.923439i \(-0.374634\pi\)
0.383745 + 0.923439i \(0.374634\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.05353e7 −0.552196
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 1.64220e7 0.845134 0.422567 − 0.906332i \(-0.361129\pi\)
0.422567 + 0.906332i \(0.361129\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −2.26485e7 −1.14460 −0.572299 − 0.820045i \(-0.693948\pi\)
−0.572299 + 0.820045i \(0.693948\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −3.93871e7 −1.97263
\(833\) 0 0
\(834\) 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
\(841\) −2.05111e7 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 3.03143e7 1.46485
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 3.82191e7 1.79849 0.899245 − 0.437445i \(-0.144116\pi\)
0.899245 + 0.437445i \(0.144116\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −3.88974e7 −1.79861 −0.899307 − 0.437317i \(-0.855929\pi\)
−0.899307 + 0.437317i \(0.855929\pi\)
\(860\) 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
\(868\) 6.08080e6 0.273944
\(869\) 0 0
\(870\) 0 0
\(871\) −4.56027e7 −2.03678
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.12304e7 1.81017 0.905084 − 0.425232i \(-0.139808\pi\)
0.905084 + 0.425232i \(0.139808\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 1.68468e7 0.727135 0.363567 − 0.931568i \(-0.381559\pi\)
0.363567 + 0.931568i \(0.381559\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 2.00028e6 0.0848859
\(890\) 0 0
\(891\) 0 0
\(892\) 4.51529e7 1.90009
\(893\) 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
\(907\) −4.95222e7 −1.99886 −0.999428 − 0.0338109i \(-0.989236\pi\)
−0.999428 + 0.0338109i \(0.989236\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 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
\(916\) −4.06342e7 −1.60012
\(917\) 0 0
\(918\) 0 0
\(919\) −4.21821e6 −0.164755 −0.0823777 − 0.996601i \(-0.526251\pi\)
−0.0823777 + 0.996601i \(0.526251\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 3.09031e7 1.18754
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −5.08600e7 −1.92310
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.29708e6 0.197100 0.0985501 − 0.995132i \(-0.468580\pi\)
0.0985501 + 0.995132i \(0.468580\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 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
\(949\) 9.56516e7 3.44768
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.91461e7 1.01805
\(962\) 0 0
\(963\) 0 0
\(964\) −4.15032e7 −1.43843
\(965\) 0 0
\(966\) 0 0
\(967\) −2.56836e7 −0.883262 −0.441631 − 0.897197i \(-0.645600\pi\)
−0.441631 + 0.897197i \(0.645600\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −6.31160e6 −0.213726
\(974\) 0 0
\(975\) 0 0
\(976\) −1.87402e7 −0.629724
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(988\) −1.20892e8 −3.94009
\(989\) 0 0
\(990\) 0 0
\(991\) −4.11127e7 −1.32982 −0.664908 − 0.746925i \(-0.731529\pi\)
−0.664908 + 0.746925i \(0.731529\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 6.04728e7 1.92674 0.963368 − 0.268183i \(-0.0864230\pi\)
0.963368 + 0.268183i \(0.0864230\pi\)
\(998\) 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 1089.6.a.f.1.1 ✓ 1
3.2 odd 2 CM 1089.6.a.f.1.1 ✓ 1
11.10 odd 2 1089.6.a.g.1.1 yes 1
33.32 even 2 1089.6.a.g.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1089.6.a.f.1.1 ✓ 1 1.1 even 1 trivial
1089.6.a.f.1.1 ✓ 1 3.2 odd 2 CM
1089.6.a.g.1.1 yes 1 11.10 odd 2
1089.6.a.g.1.1 yes 1 33.32 even 2