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Properties

Label 576.5.g.c.127.1
Level $576$
Weight $5$
Character 576.127
Self dual yes
Analytic conductor $59.541$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,5,Mod(127,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.127");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 576.g (of order \(2\), degree \(1\), not 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: \(59.5410987363\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 127.1
Character \(\chi\) \(=\) 576.127

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+48.0000 q^{5} +O(q^{10})\) \(q+48.0000 q^{5} -238.000 q^{13} +480.000 q^{17} +1679.00 q^{25} +1680.00 q^{29} -2162.00 q^{37} +1440.00 q^{41} +2401.00 q^{49} -5040.00 q^{53} +6958.00 q^{61} -11424.0 q^{65} -1442.00 q^{73} +23040.0 q^{85} +12480.0 q^{89} +1918.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 48.0000 1.92000 0.960000 − 0.280000i \(-0.0903345\pi\)
0.960000 + 0.280000i \(0.0903345\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −238.000 −1.40828 −0.704142 − 0.710059i \(-0.748668\pi\)
−0.704142 + 0.710059i \(0.748668\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 480.000 1.66090 0.830450 − 0.557093i \(-0.188083\pi\)
0.830450 + 0.557093i \(0.188083\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1679.00 2.68640
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1680.00 1.99762 0.998811 − 0.0487515i \(-0.0155242\pi\)
0.998811 + 0.0487515i \(0.0155242\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2162.00 −1.57925 −0.789627 − 0.613587i \(-0.789726\pi\)
−0.789627 + 0.613587i \(0.789726\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1440.00 0.856633 0.428316 − 0.903629i \(-0.359107\pi\)
0.428316 + 0.903629i \(0.359107\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5040.00 −1.79423 −0.897116 − 0.441794i \(-0.854342\pi\)
−0.897116 + 0.441794i \(0.854342\pi\)
\(54\) 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
\(61\) 6958.00 1.86993 0.934964 − 0.354743i \(-0.115432\pi\)
0.934964 + 0.354743i \(0.115432\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −11424.0 −2.70391
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1442.00 −0.270595 −0.135297 − 0.990805i \(-0.543199\pi\)
−0.135297 + 0.990805i \(0.543199\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 23040.0 3.18893
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12480.0 1.57556 0.787779 − 0.615958i \(-0.211231\pi\)
0.787779 + 0.615958i \(0.211231\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1918.00 0.203847 0.101924 − 0.994792i \(-0.467500\pi\)
0.101924 + 0.994792i \(0.467500\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7920.00 −0.776394 −0.388197 − 0.921576i \(-0.626902\pi\)
−0.388197 + 0.921576i \(0.626902\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 9362.00 0.787981 0.393990 − 0.919115i \(-0.371094\pi\)
0.393990 + 0.919115i \(0.371094\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6720.00 −0.526275 −0.263137 − 0.964758i \(-0.584757\pi\)
−0.263137 + 0.964758i \(0.584757\pi\)
\(114\) 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
\(124\) 0 0
\(125\) 50592.0 3.23789
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 36960.0 1.96920 0.984602 − 0.174810i \(-0.0559310\pi\)
0.984602 + 0.174810i \(0.0559310\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 80640.0 3.83543
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −28560.0 −1.28643 −0.643214 − 0.765686i \(-0.722400\pi\)
−0.643214 + 0.765686i \(0.722400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 20398.0 0.827539 0.413769 − 0.910382i \(-0.364212\pi\)
0.413769 + 0.910382i \(0.364212\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28083.0 0.983264
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −34320.0 −1.14671 −0.573357 − 0.819306i \(-0.694359\pi\)
−0.573357 + 0.819306i \(0.694359\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −64078.0 −1.95592 −0.977962 − 0.208785i \(-0.933049\pi\)
−0.977962 + 0.208785i \(0.933049\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −103776. −3.03217
\(186\) 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
\(193\) −38398.0 −1.03085 −0.515423 − 0.856936i \(-0.672365\pi\)
−0.515423 + 0.856936i \(0.672365\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 21840.0 0.562756 0.281378 − 0.959597i \(-0.409209\pi\)
0.281378 + 0.959597i \(0.409209\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 69120.0 1.64474
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 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
\(221\) −114240. −2.33902
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 90482.0 1.72541 0.862703 − 0.505711i \(-0.168770\pi\)
0.862703 + 0.505711i \(0.168770\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 87360.0 1.60917 0.804583 − 0.593840i \(-0.202389\pi\)
0.804583 + 0.593840i \(0.202389\pi\)
\(234\) 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
\(241\) −58562.0 −1.00828 −0.504141 − 0.863621i \(-0.668191\pi\)
−0.504141 + 0.863621i \(0.668191\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 115248. 1.92000
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(256\) 0 0
\(257\) 32640.0 0.494179 0.247089 − 0.968993i \(-0.420526\pi\)
0.247089 + 0.968993i \(0.420526\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −241920. −3.44493
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 71760.0 0.991694 0.495847 − 0.868410i \(-0.334858\pi\)
0.495847 + 0.868410i \(0.334858\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −100558. −1.31056 −0.655280 − 0.755386i \(-0.727449\pi\)
−0.655280 + 0.755386i \(0.727449\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −147840. −1.87232 −0.936158 − 0.351579i \(-0.885645\pi\)
−0.936158 + 0.351579i \(0.885645\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 146879. 1.75859
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 77520.0 0.902981 0.451490 − 0.892276i \(-0.350893\pi\)
0.451490 + 0.892276i \(0.350893\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 333984. 3.59026
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −193438. −1.97448 −0.987241 − 0.159234i \(-0.949098\pi\)
−0.987241 + 0.159234i \(0.949098\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −92400.0 −0.919504 −0.459752 − 0.888047i \(-0.652062\pi\)
−0.459752 + 0.888047i \(0.652062\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −399602. −3.78321
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 104638. 0.921361 0.460680 − 0.887566i \(-0.347605\pi\)
0.460680 + 0.887566i \(0.347605\pi\)
\(338\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −114002. −0.935969 −0.467985 − 0.883737i \(-0.655020\pi\)
−0.467985 + 0.883737i \(0.655020\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −244800. −1.96455 −0.982273 − 0.187458i \(-0.939975\pi\)
−0.982273 + 0.187458i \(0.939975\pi\)
\(354\) 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
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −69216.0 −0.519542
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −24242.0 −0.174241 −0.0871206 − 0.996198i \(-0.527767\pi\)
−0.0871206 + 0.996198i \(0.527767\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −399840. −2.81322
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 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
\(388\) 0 0
\(389\) −257040. −1.69864 −0.849320 − 0.527878i \(-0.822988\pi\)
−0.849320 + 0.527878i \(0.822988\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −107282. −0.680684 −0.340342 − 0.940302i \(-0.610543\pi\)
−0.340342 + 0.940302i \(0.610543\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −63840.0 −0.397012 −0.198506 − 0.980100i \(-0.563609\pi\)
−0.198506 + 0.980100i \(0.563609\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −276962. −1.65567 −0.827835 − 0.560972i \(-0.810427\pi\)
−0.827835 + 0.560972i \(0.810427\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −351118. −1.98102 −0.990510 − 0.137440i \(-0.956113\pi\)
−0.990510 + 0.137440i \(0.956113\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 805920. 4.46184
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −290878. −1.55144 −0.775720 − 0.631077i \(-0.782613\pi\)
−0.775720 + 0.631077i \(0.782613\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 599040. 3.02507
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −393120. −1.94999 −0.974995 − 0.222226i \(-0.928668\pi\)
−0.974995 + 0.222226i \(0.928668\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −304802. −1.45944 −0.729719 − 0.683748i \(-0.760349\pi\)
−0.729719 + 0.683748i \(0.760349\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 396720. 1.86673 0.933367 − 0.358925i \(-0.116856\pi\)
0.933367 + 0.358925i \(0.116856\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 514556. 2.22404
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 92064.0 0.391387
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 806400. 3.31785
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −380160. −1.49068
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 403920. 1.55905 0.779525 − 0.626372i \(-0.215461\pi\)
0.779525 + 0.626372i \(0.215461\pi\)
\(510\) 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
\(521\) 491040. 1.80901 0.904506 − 0.426461i \(-0.140240\pi\)
0.904506 + 0.426461i \(0.140240\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −342720. −1.20638
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −120238. −0.410816 −0.205408 − 0.978676i \(-0.565852\pi\)
−0.205408 + 0.978676i \(0.565852\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 449376. 1.51292
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −351120. −1.13174 −0.565868 − 0.824496i \(-0.691459\pi\)
−0.565868 + 0.824496i \(0.691459\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −322560. −1.01045
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −480480. −1.48406 −0.742029 − 0.670368i \(-0.766136\pi\)
−0.742029 + 0.670368i \(0.766136\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 656642. 1.97232 0.986159 − 0.165801i \(-0.0530210\pi\)
0.986159 + 0.165801i \(0.0530210\pi\)
\(578\) 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
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −684480. −1.94649 −0.973243 − 0.229777i \(-0.926200\pi\)
−0.973243 + 0.229777i \(0.926200\pi\)
\(594\) 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
\(601\) 492002. 1.36213 0.681064 − 0.732224i \(-0.261518\pi\)
0.681064 + 0.732224i \(0.261518\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 702768. 1.92000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 746638. 1.98696 0.993480 − 0.114006i \(-0.0363684\pi\)
0.993480 + 0.114006i \(0.0363684\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −255360. −0.670784 −0.335392 − 0.942079i \(-0.608869\pi\)
−0.335392 + 0.942079i \(0.608869\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.37904e6 3.53034
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.03776e6 −2.62298
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −571438. −1.40828
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −487200. −1.18574 −0.592872 − 0.805296i \(-0.702006\pi\)
−0.592872 + 0.805296i \(0.702006\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 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
\(652\) 0 0
\(653\) −720720. −1.69021 −0.845104 − 0.534602i \(-0.820462\pi\)
−0.845104 + 0.534602i \(0.820462\pi\)
\(654\) 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
\(661\) −513842. −1.17605 −0.588026 − 0.808842i \(-0.700095\pi\)
−0.588026 + 0.808842i \(0.700095\pi\)
\(662\) 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
\(673\) −312958. −0.690965 −0.345482 − 0.938425i \(-0.612285\pi\)
−0.345482 + 0.938425i \(0.612285\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 140400. 0.306330 0.153165 − 0.988201i \(-0.451053\pi\)
0.153165 + 0.988201i \(0.451053\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 1.77408e6 3.78087
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.19952e6 2.52679
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 691200. 1.42278
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −677040. −1.37777 −0.688887 − 0.724868i \(-0.741901\pi\)
−0.688887 + 0.724868i \(0.741901\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −737038. −1.46621 −0.733107 − 0.680113i \(-0.761931\pi\)
−0.733107 + 0.680113i \(0.761931\pi\)
\(710\) 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
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.82072e6 5.36641
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.02792e6 1.91316 0.956582 − 0.291463i \(-0.0941421\pi\)
0.956582 + 0.291463i \(0.0941421\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −1.37088e6 −2.46994
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 270002. 0.471167 0.235584 − 0.971854i \(-0.424300\pi\)
0.235584 + 0.971854i \(0.424300\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −118560. −0.204724 −0.102362 − 0.994747i \(-0.532640\pi\)
−0.102362 + 0.994747i \(0.532640\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −257278. −0.435061 −0.217530 − 0.976054i \(-0.569800\pi\)
−0.217530 + 0.976054i \(0.569800\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 583440. 0.976421 0.488211 − 0.872726i \(-0.337650\pi\)
0.488211 + 0.872726i \(0.337650\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 979104. 1.58887
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.65600e6 −2.63339
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.26984e6 −1.99909 −0.999545 − 0.0301617i \(-0.990398\pi\)
−0.999545 + 0.0301617i \(0.990398\pi\)
\(798\) 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
\(809\) −850080. −1.29886 −0.649431 − 0.760421i \(-0.724993\pi\)
−0.649431 + 0.760421i \(0.724993\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.20120e6 −1.78209 −0.891044 − 0.453917i \(-0.850026\pi\)
−0.891044 + 0.453917i \(0.850026\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 208082. 0.302779 0.151389 − 0.988474i \(-0.451625\pi\)
0.151389 + 0.988474i \(0.451625\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.15248e6 1.66090
\(834\) 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
\(841\) 2.11512e6 2.99049
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.34798e6 1.88787
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.28712e6 1.76897 0.884485 − 0.466569i \(-0.154510\pi\)
0.884485 + 0.466569i \(0.154510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 765600. 1.04241 0.521207 − 0.853430i \(-0.325482\pi\)
0.521207 + 0.853430i \(0.325482\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −1.64736e6 −2.20169
\(866\) 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
\(877\) −1.05384e6 −1.37018 −0.685088 − 0.728460i \(-0.740236\pi\)
−0.685088 + 0.728460i \(0.740236\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.18080e6 −1.52133 −0.760667 − 0.649142i \(-0.775128\pi\)
−0.760667 + 0.649142i \(0.775128\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2.41920e6 −2.98004
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.07574e6 −3.75537
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 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
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −3.63000e6 −4.24251
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −474720. −0.550055 −0.275027 − 0.961436i \(-0.588687\pi\)
−0.275027 + 0.961436i \(0.588687\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.57104e6 −1.78940 −0.894700 − 0.446667i \(-0.852611\pi\)
−0.894700 + 0.446667i \(0.852611\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.71912e6 −1.94145 −0.970727 − 0.240187i \(-0.922791\pi\)
−0.970727 + 0.240187i \(0.922791\pi\)
\(942\) 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
\(949\) 343196. 0.381074
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.79088e6 1.97188 0.985940 − 0.167097i \(-0.0534393\pi\)
0.985940 + 0.167097i \(0.0534393\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.84310e6 −1.97923
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −937440. −0.982097 −0.491048 − 0.871132i \(-0.663386\pi\)
−0.491048 + 0.871132i \(0.663386\pi\)
\(978\) 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
\(985\) 1.04832e6 1.08049
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.43448e6 −1.44313 −0.721564 − 0.692348i \(-0.756576\pi\)
−0.721564 + 0.692348i \(0.756576\pi\)
\(998\) 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 576.5.g.c.127.1 1
3.2 odd 2 576.5.g.a.127.1 1
4.3 odd 2 CM 576.5.g.c.127.1 1
8.3 odd 2 144.5.g.a.127.1 ✓ 1
8.5 even 2 144.5.g.a.127.1 ✓ 1
12.11 even 2 576.5.g.a.127.1 1
24.5 odd 2 144.5.g.b.127.1 yes 1
24.11 even 2 144.5.g.b.127.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.5.g.a.127.1 ✓ 1 8.3 odd 2
144.5.g.a.127.1 ✓ 1 8.5 even 2
144.5.g.b.127.1 yes 1 24.5 odd 2
144.5.g.b.127.1 yes 1 24.11 even 2
576.5.g.a.127.1 1 3.2 odd 2
576.5.g.a.127.1 1 12.11 even 2
576.5.g.c.127.1 1 1.1 even 1 trivial
576.5.g.c.127.1 1 4.3 odd 2 CM