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Properties

Label 1323.4.a.h.1.1
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $1$
CM discriminant -3
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] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.00000 q^{4} +19.0000 q^{13} +64.0000 q^{16} -56.0000 q^{19} -125.000 q^{25} +19.0000 q^{31} +323.000 q^{37} +449.000 q^{43} -152.000 q^{52} +901.000 q^{61} -512.000 q^{64} -127.000 q^{67} -1190.00 q^{73} +448.000 q^{76} -1387.00 q^{79} -1853.00 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −8.00000 −1.00000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 19.0000 0.405358 0.202679 − 0.979245i \(-0.435035\pi\)
0.202679 + 0.979245i \(0.435035\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 64.0000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −56.0000 −0.676173 −0.338086 − 0.941115i \(-0.609780\pi\)
−0.338086 + 0.941115i \(0.609780\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −125.000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 19.0000 0.110081 0.0550403 − 0.998484i \(-0.482471\pi\)
0.0550403 + 0.998484i \(0.482471\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 323.000 1.43516 0.717579 − 0.696477i \(-0.245250\pi\)
0.717579 + 0.696477i \(0.245250\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 449.000 1.59237 0.796184 − 0.605054i \(-0.206849\pi\)
0.796184 + 0.605054i \(0.206849\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) −152.000 −0.405358
\(53\) 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
\(61\) 901.000 1.89117 0.945584 − 0.325379i \(-0.105492\pi\)
0.945584 + 0.325379i \(0.105492\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −127.000 −0.231575 −0.115787 − 0.993274i \(-0.536939\pi\)
−0.115787 + 0.993274i \(0.536939\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1190.00 −1.90793 −0.953966 − 0.299916i \(-0.903041\pi\)
−0.953966 + 0.299916i \(0.903041\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 448.000 0.676173
\(77\) 0 0
\(78\) 0 0
\(79\) −1387.00 −1.97531 −0.987656 − 0.156637i \(-0.949935\pi\)
−0.987656 + 0.156637i \(0.949935\pi\)
\(80\) 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
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1853.00 −1.93963 −0.969813 − 0.243851i \(-0.921589\pi\)
−0.969813 + 0.243851i \(0.921589\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1000.00 1.00000
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 1801.00 1.72289 0.861446 − 0.507850i \(-0.169560\pi\)
0.861446 + 0.507850i \(0.169560\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −1567.00 −1.37699 −0.688493 − 0.725243i \(-0.741727\pi\)
−0.688493 + 0.725243i \(0.741727\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 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
\(121\) −1331.00 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −152.000 −0.110081
\(125\) 0 0
\(126\) 0 0
\(127\) −2647.00 −1.84947 −0.924737 − 0.380606i \(-0.875715\pi\)
−0.924737 + 0.380606i \(0.875715\pi\)
\(128\) 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
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −467.000 −0.284967 −0.142484 − 0.989797i \(-0.545509\pi\)
−0.142484 + 0.989797i \(0.545509\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −2584.00 −1.43516
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −3709.00 −1.99890 −0.999451 − 0.0331378i \(-0.989450\pi\)
−0.999451 + 0.0331378i \(0.989450\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3850.00 1.95709 0.978546 − 0.206028i \(-0.0660539\pi\)
0.978546 + 0.206028i \(0.0660539\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −379.000 −0.182120 −0.0910600 − 0.995845i \(-0.529026\pi\)
−0.0910600 + 0.995845i \(0.529026\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −1836.00 −0.835685
\(170\) 0 0
\(171\) 0 0
\(172\) −3592.00 −1.59237
\(173\) 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
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −3458.00 −1.42006 −0.710031 − 0.704171i \(-0.751319\pi\)
−0.710031 + 0.704171i \(0.751319\pi\)
\(182\) 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
\(193\) −3961.00 −1.47730 −0.738650 − 0.674089i \(-0.764536\pi\)
−0.738650 + 0.674089i \(0.764536\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −4373.00 −1.55776 −0.778879 − 0.627175i \(-0.784211\pi\)
−0.778879 + 0.627175i \(0.784211\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 1216.00 0.405358
\(209\) 0 0
\(210\) 0 0
\(211\) −2071.00 −0.675704 −0.337852 − 0.941199i \(-0.609700\pi\)
−0.337852 + 0.941199i \(0.609700\pi\)
\(212\) 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
\(221\) 0 0
\(222\) 0 0
\(223\) 3220.00 0.966938 0.483469 − 0.875362i \(-0.339377\pi\)
0.483469 + 0.875362i \(0.339377\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 6823.00 1.96889 0.984447 − 0.175684i \(-0.0562138\pi\)
0.984447 + 0.175684i \(0.0562138\pi\)
\(230\) 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
\(241\) −4769.00 −1.27468 −0.637341 − 0.770582i \(-0.719966\pi\)
−0.637341 + 0.770582i \(0.719966\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −7208.00 −1.89117
\(245\) 0 0
\(246\) 0 0
\(247\) −1064.00 −0.274092
\(248\) 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
\(256\) 4096.00 1.00000
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 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
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1016.00 0.231575
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8101.00 1.81587 0.907935 − 0.419111i \(-0.137658\pi\)
0.907935 + 0.419111i \(0.137658\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5167.00 −1.12078 −0.560388 − 0.828230i \(-0.689348\pi\)
−0.560388 + 0.828230i \(0.689348\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −3869.00 −0.812679 −0.406340 − 0.913722i \(-0.633195\pi\)
−0.406340 + 0.913722i \(0.633195\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4913.00 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 9520.00 1.90793
\(293\) 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
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −3584.00 −0.676173
\(305\) 0 0
\(306\) 0 0
\(307\) 6697.00 1.24501 0.622505 − 0.782616i \(-0.286115\pi\)
0.622505 + 0.782616i \(0.286115\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −10010.0 −1.80766 −0.903832 − 0.427888i \(-0.859258\pi\)
−0.903832 + 0.427888i \(0.859258\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 11096.0 1.97531
\(317\) 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
\(325\) −2375.00 −0.405358
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 992.000 0.164729 0.0823644 − 0.996602i \(-0.473753\pi\)
0.0823644 + 0.996602i \(0.473753\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4930.00 −0.796897 −0.398448 − 0.917191i \(-0.630451\pi\)
−0.398448 + 0.917191i \(0.630451\pi\)
\(338\) 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
\(349\) −1367.00 −0.209667 −0.104834 − 0.994490i \(-0.533431\pi\)
−0.104834 + 0.994490i \(0.533431\pi\)
\(350\) 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
\(361\) −3723.00 −0.542790
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −4340.00 −0.617292 −0.308646 − 0.951177i \(-0.599876\pi\)
−0.308646 + 0.951177i \(0.599876\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12350.0 1.71437 0.857183 − 0.515011i \(-0.172212\pi\)
0.857183 + 0.515011i \(0.172212\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −6103.00 −0.827151 −0.413575 − 0.910470i \(-0.635720\pi\)
−0.413575 + 0.910470i \(0.635720\pi\)
\(380\) 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
\(388\) 14824.0 1.93963
\(389\) 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
\(397\) −13067.0 −1.65192 −0.825962 − 0.563726i \(-0.809368\pi\)
−0.825962 + 0.563726i \(0.809368\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −8000.00 −1.00000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 361.000 0.0446221
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 16543.0 2.00000 0.999998 − 0.00177990i \(-0.000566559\pi\)
0.999998 + 0.00177990i \(0.000566559\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −14408.0 −1.72289
\(413\) 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
\(421\) 17138.0 1.98398 0.991989 − 0.126322i \(-0.0403172\pi\)
0.991989 + 0.126322i \(0.0403172\pi\)
\(422\) 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
\(433\) 14149.0 1.57034 0.785170 − 0.619280i \(-0.212575\pi\)
0.785170 + 0.619280i \(0.212575\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12536.0 1.37699
\(437\) 0 0
\(438\) 0 0
\(439\) −14924.0 −1.62251 −0.811257 − 0.584690i \(-0.801216\pi\)
−0.811257 + 0.584690i \(0.801216\pi\)
\(440\) 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
\(448\) 0 0
\(449\) 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
\(457\) −19207.0 −1.96601 −0.983004 − 0.183585i \(-0.941230\pi\)
−0.983004 + 0.183585i \(0.941230\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −19780.0 −1.98543 −0.992716 − 0.120482i \(-0.961556\pi\)
−0.992716 + 0.120482i \(0.961556\pi\)
\(464\) 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
\(475\) 7000.00 0.676173
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 6137.00 0.581753
\(482\) 0 0
\(483\) 0 0
\(484\) 10648.0 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 20900.0 1.94470 0.972351 − 0.233526i \(-0.0750265\pi\)
0.972351 + 0.233526i \(0.0750265\pi\)
\(488\) 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
\(496\) 1216.00 0.110081
\(497\) 0 0
\(498\) 0 0
\(499\) −6607.00 −0.592725 −0.296363 − 0.955075i \(-0.595774\pi\)
−0.296363 + 0.955075i \(0.595774\pi\)
\(500\) 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
\(508\) 21176.0 1.84947
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 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
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −23921.0 −1.99999 −0.999993 − 0.00383755i \(-0.998778\pi\)
−0.999993 + 0.00383755i \(0.998778\pi\)
\(524\) 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
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22678.0 −1.80222 −0.901112 − 0.433586i \(-0.857248\pi\)
−0.901112 + 0.433586i \(0.857248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21293.0 1.66439 0.832196 − 0.554481i \(-0.187083\pi\)
0.832196 + 0.554481i \(0.187083\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 3736.00 0.284967
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 8531.00 0.645479
\(560\) 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
\(571\) 23312.0 1.70854 0.854270 − 0.519829i \(-0.174004\pi\)
0.854270 + 0.519829i \(0.174004\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9613.00 0.693578 0.346789 − 0.937943i \(-0.387272\pi\)
0.346789 + 0.937943i \(0.387272\pi\)
\(578\) 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
\(589\) −1064.00 −0.0744335
\(590\) 0 0
\(591\) 0 0
\(592\) 20672.0 1.43516
\(593\) 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
\(601\) −11951.0 −0.811134 −0.405567 − 0.914065i \(-0.632926\pi\)
−0.405567 + 0.914065i \(0.632926\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 29672.0 1.99890
\(605\) 0 0
\(606\) 0 0
\(607\) 28420.0 1.90038 0.950191 − 0.311667i \(-0.100887\pi\)
0.950191 + 0.311667i \(0.100887\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 12851.0 0.846732 0.423366 − 0.905959i \(-0.360848\pi\)
0.423366 + 0.905959i \(0.360848\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −26693.0 −1.73325 −0.866625 − 0.498959i \(-0.833716\pi\)
−0.866625 + 0.498959i \(0.833716\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 15625.0 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −30800.0 −1.95709
\(629\) 0 0
\(630\) 0 0
\(631\) 26459.0 1.66928 0.834640 − 0.550795i \(-0.185675\pi\)
0.834640 + 0.550795i \(0.185675\pi\)
\(632\) 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
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 32419.0 1.98831 0.994153 − 0.107982i \(-0.0344389\pi\)
0.994153 + 0.107982i \(0.0344389\pi\)
\(644\) 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
\(652\) 3032.00 0.182120
\(653\) 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
\(661\) 20482.0 1.20523 0.602615 − 0.798032i \(-0.294125\pi\)
0.602615 + 0.798032i \(0.294125\pi\)
\(662\) 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
\(673\) 24050.0 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 14688.0 0.835685
\(677\) 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
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 28736.0 1.59237
\(689\) 0 0
\(690\) 0 0
\(691\) −36251.0 −1.99574 −0.997868 − 0.0652705i \(-0.979209\pi\)
−0.997868 + 0.0652705i \(0.979209\pi\)
\(692\) 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
\(703\) −18088.0 −0.970415
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −8623.00 −0.456761 −0.228381 − 0.973572i \(-0.573343\pi\)
−0.228381 + 0.973572i \(0.573343\pi\)
\(710\) 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
\(724\) 27664.0 1.42006
\(725\) 0 0
\(726\) 0 0
\(727\) 27253.0 1.39031 0.695157 − 0.718858i \(-0.255335\pi\)
0.695157 + 0.718858i \(0.255335\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −24281.0 −1.22352 −0.611759 − 0.791044i \(-0.709538\pi\)
−0.611759 + 0.791044i \(0.709538\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −37423.0 −1.86282 −0.931412 − 0.363966i \(-0.881422\pi\)
−0.931412 + 0.363966i \(0.881422\pi\)
\(740\) 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
\(751\) −23452.0 −1.13951 −0.569757 − 0.821813i \(-0.692963\pi\)
−0.569757 + 0.821813i \(0.692963\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 24137.0 1.15888 0.579441 − 0.815014i \(-0.303271\pi\)
0.579441 + 0.815014i \(0.303271\pi\)
\(758\) 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
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 4606.00 0.215990 0.107995 − 0.994151i \(-0.465557\pi\)
0.107995 + 0.994151i \(0.465557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 31688.0 1.47730
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) −2375.00 −0.110081
\(776\) 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
\(787\) 14653.0 0.663689 0.331844 − 0.943334i \(-0.392329\pi\)
0.331844 + 0.943334i \(0.392329\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 17119.0 0.766600
\(794\) 0 0
\(795\) 0 0
\(796\) 34984.0 1.55776
\(797\) 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
\(811\) −39368.0 −1.70456 −0.852280 − 0.523087i \(-0.824780\pi\)
−0.852280 + 0.523087i \(0.824780\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −25144.0 −1.07672
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 45611.0 1.93183 0.965917 − 0.258852i \(-0.0833441\pi\)
0.965917 + 0.258852i \(0.0833441\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −17066.0 −0.714990 −0.357495 − 0.933915i \(-0.616369\pi\)
−0.357495 + 0.933915i \(0.616369\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −9728.00 −0.405358
\(833\) 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
\(841\) −24389.0 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 16568.0 0.675704
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46690.0 1.87413 0.937066 − 0.349151i \(-0.113530\pi\)
0.937066 + 0.349151i \(0.113530\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 49807.0 1.97834 0.989169 − 0.146779i \(-0.0468906\pi\)
0.989169 + 0.146779i \(0.0468906\pi\)
\(860\) 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
\(871\) −2413.00 −0.0938707
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −36793.0 −1.41666 −0.708330 − 0.705881i \(-0.750551\pi\)
−0.708330 + 0.705881i \(0.750551\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −20680.0 −0.788151 −0.394076 − 0.919078i \(-0.628935\pi\)
−0.394076 + 0.919078i \(0.628935\pi\)
\(884\) 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
\(892\) −25760.0 −0.966938
\(893\) 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
\(907\) −49447.0 −1.81021 −0.905105 − 0.425188i \(-0.860208\pi\)
−0.905105 + 0.425188i \(0.860208\pi\)
\(908\) 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
\(916\) −54584.0 −1.96889
\(917\) 0 0
\(918\) 0 0
\(919\) −49573.0 −1.77939 −0.889697 − 0.456552i \(-0.849084\pi\)
−0.889697 + 0.456552i \(0.849084\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −40375.0 −1.43516
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −40283.0 −1.40447 −0.702235 − 0.711945i \(-0.747814\pi\)
−0.702235 + 0.711945i \(0.747814\pi\)
\(938\) 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
\(949\) −22610.0 −0.773395
\(950\) 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
\(961\) −29430.0 −0.987882
\(962\) 0 0
\(963\) 0 0
\(964\) 38152.0 1.27468
\(965\) 0 0
\(966\) 0 0
\(967\) −3907.00 −0.129928 −0.0649641 − 0.997888i \(-0.520693\pi\)
−0.0649641 + 0.997888i \(0.520693\pi\)
\(968\) 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
\(976\) 57664.0 1.89117
\(977\) 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
\(988\) 8512.00 0.274092
\(989\) 0 0
\(990\) 0 0
\(991\) −14041.0 −0.450078 −0.225039 − 0.974350i \(-0.572251\pi\)
−0.225039 + 0.974350i \(0.572251\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −33983.0 −1.07949 −0.539745 − 0.841828i \(-0.681480\pi\)
−0.539745 + 0.841828i \(0.681480\pi\)
\(998\) 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 1323.4.a.h.1.1 1
3.2 odd 2 CM 1323.4.a.h.1.1 1
7.3 odd 6 189.4.e.b.163.1 yes 2
7.5 odd 6 189.4.e.b.109.1 ✓ 2
7.6 odd 2 1323.4.a.g.1.1 1
21.5 even 6 189.4.e.b.109.1 ✓ 2
21.17 even 6 189.4.e.b.163.1 yes 2
21.20 even 2 1323.4.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.b.109.1 ✓ 2 7.5 odd 6
189.4.e.b.109.1 ✓ 2 21.5 even 6
189.4.e.b.163.1 yes 2 7.3 odd 6
189.4.e.b.163.1 yes 2 21.17 even 6
1323.4.a.g.1.1 1 7.6 odd 2
1323.4.a.g.1.1 1 21.20 even 2
1323.4.a.h.1.1 1 1.1 even 1 trivial
1323.4.a.h.1.1 1 3.2 odd 2 CM