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Properties

Label 1089.6.a.e.1.1
Level $1089$
Weight $6$
Character 1089.1
Self dual yes
Analytic conductor $174.658$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

Related objects

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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: no (minimal twist has level 121)
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)
 
\(f(q)\) \(=\) \(q-32.0000 q^{4} -57.0000 q^{5} +O(q^{10})\) \(q-32.0000 q^{4} -57.0000 q^{5} +1024.00 q^{16} +1824.00 q^{20} -981.000 q^{23} +124.000 q^{25} -7775.00 q^{31} +1267.00 q^{37} -24708.0 q^{47} -16807.0 q^{49} -34806.0 q^{53} +24825.0 q^{59} -32768.0 q^{64} -72917.0 q^{67} +66273.0 q^{71} -58368.0 q^{80} +91089.0 q^{89} +31392.0 q^{92} -163183. q^{97} +O(q^{100})\)

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
\(5\) −57.0000 −1.01965 −0.509823 − 0.860279i \(-0.670289\pi\)
−0.509823 + 0.860279i \(0.670289\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1824.00 1.01965
\(21\) 0 0
\(22\) 0 0
\(23\) −981.000 −0.386678 −0.193339 − 0.981132i \(-0.561932\pi\)
−0.193339 + 0.981132i \(0.561932\pi\)
\(24\) 0 0
\(25\) 124.000 0.0396800
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −7775.00 −1.45310 −0.726551 − 0.687112i \(-0.758878\pi\)
−0.726551 + 0.687112i \(0.758878\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1267.00 0.152150 0.0760751 − 0.997102i \(-0.475761\pi\)
0.0760751 + 0.997102i \(0.475761\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −24708.0 −1.63152 −0.815761 − 0.578389i \(-0.803682\pi\)
−0.815761 + 0.578389i \(0.803682\pi\)
\(48\) 0 0
\(49\) −16807.0 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −34806.0 −1.70202 −0.851010 − 0.525150i \(-0.824009\pi\)
−0.851010 + 0.525150i \(0.824009\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 24825.0 0.928452 0.464226 − 0.885717i \(-0.346333\pi\)
0.464226 + 0.885717i \(0.346333\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −32768.0 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −72917.0 −1.98446 −0.992229 − 0.124427i \(-0.960291\pi\)
−0.992229 + 0.124427i \(0.960291\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 66273.0 1.56024 0.780119 − 0.625631i \(-0.215159\pi\)
0.780119 + 0.625631i \(0.215159\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −58368.0 −1.01965
\(81\) 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
\(89\) 91089.0 1.21896 0.609482 − 0.792800i \(-0.291377\pi\)
0.609482 + 0.792800i \(0.291377\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 31392.0 0.386678
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −163183. −1.76094 −0.880472 − 0.474098i \(-0.842774\pi\)
−0.880472 + 0.474098i \(0.842774\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3968.00 −0.0396800
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −180244. −1.67405 −0.837024 − 0.547167i \(-0.815706\pi\)
−0.837024 + 0.547167i \(0.815706\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −192381. −1.41731 −0.708657 − 0.705553i \(-0.750699\pi\)
−0.708657 + 0.705553i \(0.750699\pi\)
\(114\) 0 0
\(115\) 55917.0 0.394275
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 248800. 1.45310
\(125\) 171057. 0.979187
\(126\) 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
\(137\) 263283. 1.19845 0.599227 − 0.800579i \(-0.295475\pi\)
0.599227 + 0.800579i \(0.295475\pi\)
\(138\) 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
\(148\) −40544.0 −0.152150
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 443175. 1.48165
\(156\) 0 0
\(157\) 280117. 0.906965 0.453482 − 0.891265i \(-0.350182\pi\)
0.453482 + 0.891265i \(0.350182\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 164144. 0.483900 0.241950 − 0.970289i \(-0.422213\pi\)
0.241950 + 0.970289i \(0.422213\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −371293. −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 840189. 1.95995 0.979974 − 0.199127i \(-0.0638106\pi\)
0.979974 + 0.199127i \(0.0638106\pi\)
\(180\) 0 0
\(181\) 279875. 0.634991 0.317496 − 0.948260i \(-0.397158\pi\)
0.317496 + 0.948260i \(0.397158\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −72219.0 −0.155139
\(186\) 0 0
\(187\) 0 0
\(188\) 790656. 1.63152
\(189\) 0 0
\(190\) 0 0
\(191\) 272325. 0.540137 0.270069 − 0.962841i \(-0.412954\pi\)
0.270069 + 0.962841i \(0.412954\pi\)
\(192\) 0 0
\(193\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 537824. 1.00000
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 799900. 1.43187 0.715934 − 0.698168i \(-0.246001\pi\)
0.715934 + 0.698168i \(0.246001\pi\)
\(200\) 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 \(0.5\pi\)
\(212\) 1.11379e6 1.70202
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −247531. −0.333325 −0.166662 − 0.986014i \(-0.553299\pi\)
−0.166662 + 0.986014i \(0.553299\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.17102e6 −1.47563 −0.737815 − 0.675003i \(-0.764142\pi\)
−0.737815 + 0.675003i \(0.764142\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 1.40836e6 1.66358
\(236\) −794400. −0.928452
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 957999. 1.01965
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.84812e6 −1.85160 −0.925799 − 0.378017i \(-0.876606\pi\)
−0.925799 + 0.378017i \(0.876606\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.04858e6 1.00000
\(257\) −339858. −0.320970 −0.160485 − 0.987038i \(-0.551306\pi\)
−0.160485 + 0.987038i \(0.551306\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 1.98394e6 1.73546
\(266\) 0 0
\(267\) 0 0
\(268\) 2.33334e6 1.98446
\(269\) 1.16085e6 0.978127 0.489064 − 0.872248i \(-0.337339\pi\)
0.489064 + 0.872248i \(0.337339\pi\)
\(270\) 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
\(277\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −2.12074e6 −1.56024
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.41986e6 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −1.41502e6 −0.946693
\(296\) 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
\(311\) 3.36505e6 1.97284 0.986418 − 0.164257i \(-0.0525226\pi\)
0.986418 + 0.164257i \(0.0525226\pi\)
\(312\) 0 0
\(313\) 1.04882e6 0.605117 0.302559 − 0.953131i \(-0.402159\pi\)
0.302559 + 0.953131i \(0.402159\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.28263e6 −1.83474 −0.917369 − 0.398037i \(-0.869691\pi\)
−0.917369 + 0.398037i \(0.869691\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.86778e6 1.01965
\(321\) 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
\(331\) 736925. 0.369703 0.184852 − 0.982766i \(-0.440820\pi\)
0.184852 + 0.982766i \(0.440820\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 4.15627e6 2.02345
\(336\) 0 0
\(337\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 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
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.37977e6 1.87074 0.935372 − 0.353665i \(-0.115065\pi\)
0.935372 + 0.353665i \(0.115065\pi\)
\(354\) 0 0
\(355\) −3.77756e6 −1.59089
\(356\) −2.91485e6 −1.21896
\(357\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) −1.31317e6 −0.508926 −0.254463 − 0.967082i \(-0.581899\pi\)
−0.254463 + 0.967082i \(0.581899\pi\)
\(368\) −1.00454e6 −0.386678
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.88862e6 0.675379 0.337690 − 0.941258i \(-0.390355\pi\)
0.337690 + 0.941258i \(0.390355\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −5.23267e6 −1.82275 −0.911373 − 0.411581i \(-0.864977\pi\)
−0.911373 + 0.411581i \(0.864977\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 5.22186e6 1.76094
\(389\) 5.54408e6 1.85761 0.928806 − 0.370566i \(-0.120836\pi\)
0.928806 + 0.370566i \(0.120836\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.56024e6 0.496839 0.248420 − 0.968653i \(-0.420089\pi\)
0.248420 + 0.968653i \(0.420089\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 126976. 0.0396800
\(401\) 5.71485e6 1.77478 0.887389 − 0.461022i \(-0.152517\pi\)
0.887389 + 0.461022i \(0.152517\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5.76781e6 1.67405
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −68664.0 −0.0191071 −0.00955353 − 0.999954i \(-0.503041\pi\)
−0.00955353 + 0.999954i \(0.503041\pi\)
\(420\) 0 0
\(421\) 6.85705e6 1.88552 0.942762 − 0.333466i \(-0.108218\pi\)
0.942762 + 0.333466i \(0.108218\pi\)
\(422\) 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
\(433\) −5.10513e6 −1.30854 −0.654270 − 0.756261i \(-0.727024\pi\)
−0.654270 + 0.756261i \(0.727024\pi\)
\(434\) 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
\(441\) 0 0
\(442\) 0 0
\(443\) −4.17713e6 −1.01127 −0.505637 − 0.862746i \(-0.668742\pi\)
−0.505637 + 0.862746i \(0.668742\pi\)
\(444\) 0 0
\(445\) −5.19207e6 −1.24291
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.63476e6 0.850864 0.425432 − 0.904990i \(-0.360122\pi\)
0.425432 + 0.904990i \(0.360122\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.15619e6 1.41731
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −1.78934e6 −0.394275
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 7.10982e6 1.54137 0.770684 − 0.637218i \(-0.219915\pi\)
0.770684 + 0.637218i \(0.219915\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.20853e6 0.680792 0.340396 − 0.940282i \(-0.389439\pi\)
0.340396 + 0.940282i \(0.389439\pi\)
\(468\) 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
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.30143e6 1.79554
\(486\) 0 0
\(487\) 4.40023e6 0.840724 0.420362 − 0.907357i \(-0.361903\pi\)
0.420362 + 0.907357i \(0.361903\pi\)
\(488\) 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
\(496\) −7.96160e6 −1.45310
\(497\) 0 0
\(498\) 0 0
\(499\) −7.47980e6 −1.34474 −0.672370 − 0.740215i \(-0.734724\pi\)
−0.672370 + 0.740215i \(0.734724\pi\)
\(500\) −5.47382e6 −0.979187
\(501\) 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
\(509\) 1.09082e7 1.86621 0.933103 − 0.359609i \(-0.117090\pi\)
0.933103 + 0.359609i \(0.117090\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.02739e7 1.70694
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.23256e7 1.98936 0.994679 − 0.103024i \(-0.0328519\pi\)
0.994679 + 0.103024i \(0.0328519\pi\)
\(522\) 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
\(529\) −5.47398e6 −0.850480
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −8.42506e6 −1.19845
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 1.09657e7 1.44516
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −121644. −0.0153434
\(576\) 0 0
\(577\) −8.05397e6 −1.00709 −0.503547 − 0.863968i \(-0.667972\pi\)
−0.503547 + 0.863968i \(0.667972\pi\)
\(578\) 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
\(587\) 1.29130e7 1.54679 0.773396 − 0.633923i \(-0.218556\pi\)
0.773396 + 0.633923i \(0.218556\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.29741e6 0.152150
\(593\) 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
\(599\) −1.46844e7 −1.67220 −0.836100 − 0.548577i \(-0.815170\pi\)
−0.836100 + 0.548577i \(0.815170\pi\)
\(600\) 0 0
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 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
\(613\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.79256e7 −1.89566 −0.947829 − 0.318781i \(-0.896727\pi\)
−0.947829 + 0.318781i \(0.896727\pi\)
\(618\) 0 0
\(619\) −1.91271e6 −0.200642 −0.100321 − 0.994955i \(-0.531987\pi\)
−0.100321 + 0.994955i \(0.531987\pi\)
\(620\) −1.41816e7 −1.48165
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.01377e7 −1.03811
\(626\) 0 0
\(627\) 0 0
\(628\) −8.96374e6 −0.906965
\(629\) 0 0
\(630\) 0 0
\(631\) 1.28703e7 1.28681 0.643405 − 0.765526i \(-0.277521\pi\)
0.643405 + 0.765526i \(0.277521\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.50793e7 −1.44956 −0.724779 − 0.688982i \(-0.758058\pi\)
−0.724779 + 0.688982i \(0.758058\pi\)
\(642\) 0 0
\(643\) −2.09678e7 −1.99998 −0.999988 − 0.00489569i \(-0.998442\pi\)
−0.999988 + 0.00489569i \(0.998442\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.18662e6 0.674938 0.337469 − 0.941337i \(-0.390429\pi\)
0.337469 + 0.941337i \(0.390429\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −5.25261e6 −0.483900
\(653\) −2.06540e7 −1.89549 −0.947746 − 0.319026i \(-0.896644\pi\)
−0.947746 + 0.319026i \(0.896644\pi\)
\(654\) 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
\(661\) 2.15101e7 1.91487 0.957433 − 0.288657i \(-0.0932086\pi\)
0.957433 + 0.288657i \(0.0932086\pi\)
\(662\) 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
\(673\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.18814e7 1.00000
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.67323e7 −1.37248 −0.686238 − 0.727377i \(-0.740739\pi\)
−0.686238 + 0.727377i \(0.740739\pi\)
\(684\) 0 0
\(685\) −1.50071e7 −1.22200
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 2.50313e7 1.99429 0.997146 − 0.0754971i \(-0.0240544\pi\)
0.997146 + 0.0754971i \(0.0240544\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.59156e7 1.93618 0.968091 − 0.250598i \(-0.0806271\pi\)
0.968091 + 0.250598i \(0.0806271\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.62728e6 0.561883
\(714\) 0 0
\(715\) 0 0
\(716\) −2.68860e7 −1.95995
\(717\) 0 0
\(718\) 0 0
\(719\) 30039.0 0.00216702 0.00108351 − 0.999999i \(-0.499655\pi\)
0.00108351 + 0.999999i \(0.499655\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −8.95600e6 −0.634991
\(725\) 0 0
\(726\) 0 0
\(727\) −1.70878e7 −1.19908 −0.599542 − 0.800343i \(-0.704651\pi\)
−0.599542 + 0.800343i \(0.704651\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 2.31101e6 0.155139
\(741\) 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
\(751\) −2.56094e7 −1.65691 −0.828455 − 0.560055i \(-0.810780\pi\)
−0.828455 + 0.560055i \(0.810780\pi\)
\(752\) −2.53010e7 −1.63152
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.95760e7 1.24161 0.620805 − 0.783965i \(-0.286806\pi\)
0.620805 + 0.783965i \(0.286806\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.71440e6 −0.540137
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.18995e7 −0.716275 −0.358137 − 0.933669i \(-0.616588\pi\)
−0.358137 + 0.933669i \(0.616588\pi\)
\(774\) 0 0
\(775\) −964100. −0.0576591
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.72104e7 −1.00000
\(785\) −1.59667e7 −0.924784
\(786\) 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
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −2.55968e7 −1.43187
\(797\) 9.42078e6 0.525341 0.262670 − 0.964886i \(-0.415397\pi\)
0.262670 + 0.964886i \(0.415397\pi\)
\(798\) 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
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9.35621e6 −0.493408
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 3.57048e7 1.83750 0.918749 − 0.394843i \(-0.129201\pi\)
0.918749 + 0.394843i \(0.129201\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1.90687e7 0.963684 0.481842 − 0.876258i \(-0.339968\pi\)
0.481842 + 0.876258i \(0.339968\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.93590e7 −1.93036 −0.965182 − 0.261577i \(-0.915757\pi\)
−0.965182 + 0.261577i \(0.915757\pi\)
\(840\) 0 0
\(841\) −2.05111e7 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.11637e7 1.01965
\(846\) 0 0
\(847\) 0 0
\(848\) −3.56413e7 −1.70202
\(849\) 0 0
\(850\) 0 0
\(851\) −1.24293e6 −0.0588331
\(852\) 0 0
\(853\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −1.50484e7 −0.695835 −0.347918 − 0.937525i \(-0.613111\pi\)
−0.347918 + 0.937525i \(0.613111\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.79604e6 −0.310620 −0.155310 − 0.987866i \(-0.549638\pi\)
−0.155310 + 0.987866i \(0.549638\pi\)
\(864\) 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
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7.12278e6 −0.309179 −0.154589 − 0.987979i \(-0.549405\pi\)
−0.154589 + 0.987979i \(0.549405\pi\)
\(882\) 0 0
\(883\) 6.29994e6 0.271916 0.135958 − 0.990715i \(-0.456589\pi\)
0.135958 + 0.990715i \(0.456589\pi\)
\(884\) 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
\(892\) 7.92099e6 0.333325
\(893\) 0 0
\(894\) 0 0
\(895\) −4.78908e7 −1.99845
\(896\) 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
\(905\) −1.59529e7 −0.647467
\(906\) 0 0
\(907\) 3.06168e7 1.23578 0.617891 − 0.786264i \(-0.287987\pi\)
0.617891 + 0.786264i \(0.287987\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.26963e7 1.70449 0.852245 − 0.523143i \(-0.175241\pi\)
0.852245 + 0.523143i \(0.175241\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 3.74728e7 1.47563
\(917\) 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
\(925\) 157108. 0.00603732
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.83412e7 −1.07740 −0.538702 − 0.842497i \(-0.681085\pi\)
−0.538702 + 0.842497i \(0.681085\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −4.50674e7 −1.66358
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 2.54208e7 0.928452
\(945\) 0 0
\(946\) 0 0
\(947\) 1.96062e7 0.710427 0.355213 − 0.934785i \(-0.384408\pi\)
0.355213 + 0.934785i \(0.384408\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −1.55225e7 −0.550749
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 3.18215e7 1.11151
\(962\) 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
\(971\) −4.57445e7 −1.55701 −0.778504 − 0.627639i \(-0.784021\pi\)
−0.778504 + 0.627639i \(0.784021\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.70589e7 1.57727 0.788633 − 0.614864i \(-0.210789\pi\)
0.788633 + 0.614864i \(0.210789\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3.06560e7 −1.01965
\(981\) 0 0
\(982\) 0 0
\(983\) 6.05507e7 1.99864 0.999322 − 0.0368136i \(-0.0117208\pi\)
0.999322 + 0.0368136i \(0.0117208\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −6.17681e7 −1.99793 −0.998965 − 0.0454936i \(-0.985514\pi\)
−0.998965 + 0.0454936i \(0.985514\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.55943e7 −1.46000
\(996\) 0 0
\(997\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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.e.1.1 1
3.2 odd 2 121.6.a.a.1.1 ✓ 1
11.10 odd 2 CM 1089.6.a.e.1.1 1
33.32 even 2 121.6.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
121.6.a.a.1.1 ✓ 1 3.2 odd 2
121.6.a.a.1.1 ✓ 1 33.32 even 2
1089.6.a.e.1.1 1 1.1 even 1 trivial
1089.6.a.e.1.1 1 11.10 odd 2 CM