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Properties

Label 576.5.g.b.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 $2$

Related objects

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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 4)
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)
 
\(f(q)\) \(=\) \(q-14.0000 q^{5} +O(q^{10})\) \(q-14.0000 q^{5} +238.000 q^{13} -322.000 q^{17} -429.000 q^{25} +82.0000 q^{29} -2162.00 q^{37} +3038.00 q^{41} +2401.00 q^{49} +2482.00 q^{53} +6958.00 q^{61} -3332.00 q^{65} +1442.00 q^{73} +4508.00 q^{85} +9758.00 q^{89} -1918.00 q^{97} +O(q^{100})\)

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
\(5\) −14.0000 −0.560000 −0.280000 − 0.960000i \(-0.590334\pi\)
−0.280000 + 0.960000i \(0.590334\pi\)
\(6\) 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 \(-0.251332\pi\)
0.704142 + 0.710059i \(0.251332\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −322.000 −1.11419 −0.557093 − 0.830450i \(-0.688083\pi\)
−0.557093 + 0.830450i \(0.688083\pi\)
\(18\) 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
\(25\) −429.000 −0.686400
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 82.0000 0.0975030 0.0487515 − 0.998811i \(-0.484476\pi\)
0.0487515 + 0.998811i \(0.484476\pi\)
\(30\) 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
\(41\) 3038.00 1.80726 0.903629 − 0.428316i \(-0.140893\pi\)
0.903629 + 0.428316i \(0.140893\pi\)
\(42\) 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
\(53\) 2482.00 0.883588 0.441794 − 0.897116i \(-0.354342\pi\)
0.441794 + 0.897116i \(0.354342\pi\)
\(54\) 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
\(65\) −3332.00 −0.788639
\(66\) 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 \(-0.456801\pi\)
0.135297 + 0.990805i \(0.456801\pi\)
\(74\) 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
\(85\) 4508.00 0.623945
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9758.00 1.23192 0.615958 − 0.787779i \(-0.288769\pi\)
0.615958 + 0.787779i \(0.288769\pi\)
\(90\) 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 \(-0.532500\pi\)
−0.101924 + 0.994792i \(0.532500\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18802.0 1.84315 0.921576 − 0.388197i \(-0.126902\pi\)
0.921576 + 0.388197i \(0.126902\pi\)
\(102\) 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 \(-0.628906\pi\)
−0.393990 + 0.919115i \(0.628906\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 24638.0 1.92952 0.964758 − 0.263137i \(-0.0847572\pi\)
0.964758 + 0.263137i \(0.0847572\pi\)
\(114\) 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
\(125\) 14756.0 0.944384
\(126\) 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
\(137\) −6562.00 −0.349619 −0.174810 − 0.984602i \(-0.555931\pi\)
−0.174810 + 0.984602i \(0.555931\pi\)
\(138\) 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
\(145\) −1148.00 −0.0546017
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −33998.0 −1.53137 −0.765686 − 0.643214i \(-0.777600\pi\)
−0.765686 + 0.643214i \(0.777600\pi\)
\(150\) 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
\(173\) 49042.0 1.63861 0.819306 − 0.573357i \(-0.194359\pi\)
0.819306 + 0.573357i \(0.194359\pi\)
\(174\) 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 \(-0.0669508\pi\)
0.977962 + 0.208785i \(0.0669508\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 30268.0 0.884383
\(186\) 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
\(197\) 74482.0 1.91919 0.959597 − 0.281378i \(-0.0907915\pi\)
0.959597 + 0.281378i \(0.0907915\pi\)
\(198\) 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
\(205\) −42532.0 −1.01206
\(206\) 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
\(221\) −76636.0 −1.56909
\(222\) 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 \(-0.831230\pi\)
−0.862703 + 0.505711i \(0.831230\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 64478.0 1.18768 0.593840 − 0.804583i \(-0.297611\pi\)
0.593840 + 0.804583i \(0.297611\pi\)
\(234\) 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 \(-0.331809\pi\)
0.504141 + 0.863621i \(0.331809\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −33614.0 −0.560000
\(246\) 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
\(257\) −128002. −1.93799 −0.968993 − 0.247089i \(-0.920526\pi\)
−0.968993 + 0.247089i \(0.920526\pi\)
\(258\) 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
\(265\) −34748.0 −0.494810
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −125678. −1.73682 −0.868410 − 0.495847i \(-0.834858\pi\)
−0.868410 + 0.495847i \(0.834858\pi\)
\(270\) 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 \(-0.272551\pi\)
0.655280 + 0.755386i \(0.272551\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −55522.0 −0.703157 −0.351579 − 0.936158i \(-0.614355\pi\)
−0.351579 + 0.936158i \(0.614355\pi\)
\(282\) 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
\(289\) 20163.0 0.241412
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 153202. 1.78455 0.892276 − 0.451490i \(-0.149107\pi\)
0.892276 + 0.451490i \(0.149107\pi\)
\(294\) 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
\(305\) −97412.0 −1.04716
\(306\) 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
\(317\) −178478. −1.77609 −0.888047 − 0.459752i \(-0.847938\pi\)
−0.888047 + 0.459752i \(0.847938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −102102. −0.966646
\(326\) 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 \(-0.652395\pi\)
−0.460680 + 0.887566i \(0.652395\pi\)
\(338\) 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
\(353\) 46718.0 0.374917 0.187458 − 0.982273i \(-0.439975\pi\)
0.187458 + 0.982273i \(0.439975\pi\)
\(354\) 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
\(365\) −20188.0 −0.151533
\(366\) 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
\(377\) 19516.0 0.137312
\(378\) 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
\(389\) −159758. −1.05576 −0.527878 − 0.849320i \(-0.677012\pi\)
−0.527878 + 0.849320i \(0.677012\pi\)
\(390\) 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
\(401\) −315202. −1.96020 −0.980100 − 0.198506i \(-0.936391\pi\)
−0.980100 + 0.198506i \(0.936391\pi\)
\(402\) 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 \(-0.189573\pi\)
0.827835 + 0.560972i \(0.189573\pi\)
\(410\) 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 \(-0.0438874\pi\)
0.990510 + 0.137440i \(0.0438874\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 138138. 0.764778
\(426\) 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
\(445\) −136612. −0.689872
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −89602.0 −0.444452 −0.222226 − 0.974995i \(-0.571332\pi\)
−0.222226 + 0.974995i \(0.571332\pi\)
\(450\) 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 \(-0.239651\pi\)
0.729719 + 0.683748i \(0.239651\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −152558. −0.717849 −0.358925 − 0.933367i \(-0.616856\pi\)
−0.358925 + 0.933367i \(0.616856\pi\)
\(462\) 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
\(485\) 26852.0 0.114155
\(486\) 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
\(493\) −26404.0 −0.108637
\(494\) 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
\(505\) −263228. −1.03217
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 324562. 1.25274 0.626372 − 0.779525i \(-0.284539\pi\)
0.626372 + 0.779525i \(0.284539\pi\)
\(510\) 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
\(521\) 231518. 0.852922 0.426461 − 0.904506i \(-0.359760\pi\)
0.426461 + 0.904506i \(0.359760\pi\)
\(522\) 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
\(533\) 723044. 2.54513
\(534\) 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 \(-0.434148\pi\)
0.205408 + 0.978676i \(0.434148\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 131068. 0.441269
\(546\) 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
\(557\) −511598. −1.64899 −0.824496 − 0.565868i \(-0.808541\pi\)
−0.824496 + 0.565868i \(0.808541\pi\)
\(558\) 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
\(565\) −344932. −1.08053
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 434078. 1.34074 0.670368 − 0.742029i \(-0.266136\pi\)
0.670368 + 0.742029i \(0.266136\pi\)
\(570\) 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
\(593\) −161602. −0.459555 −0.229777 − 0.973243i \(-0.573800\pi\)
−0.229777 + 0.973243i \(0.573800\pi\)
\(594\) 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
\(605\) −204974. −0.560000
\(606\) 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
\(617\) 717278. 1.88416 0.942079 − 0.335392i \(-0.108869\pi\)
0.942079 + 0.335392i \(0.108869\pi\)
\(618\) 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
\(625\) 61541.0 0.157545
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 696164. 1.75959
\(630\) 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
\(641\) −661762. −1.61059 −0.805296 − 0.592872i \(-0.797994\pi\)
−0.805296 + 0.592872i \(0.797994\pi\)
\(642\) 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
\(653\) −455918. −1.06920 −0.534602 − 0.845104i \(-0.679538\pi\)
−0.534602 + 0.845104i \(0.679538\pi\)
\(654\) 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
\(677\) 905842. 1.97640 0.988201 − 0.153165i \(-0.0489466\pi\)
0.988201 + 0.153165i \(0.0489466\pi\)
\(678\) 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
\(685\) 91868.0 0.195787
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 590716. 1.24434
\(690\) 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
\(697\) −978236. −2.01362
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 712402. 1.44974 0.724868 − 0.688887i \(-0.241901\pi\)
0.724868 + 0.688887i \(0.241901\pi\)
\(702\) 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 \(-0.238069\pi\)
0.733107 + 0.680113i \(0.238069\pi\)
\(710\) 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
\(725\) −35178.0 −0.0669260
\(726\) 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 \(-0.905858\pi\)
−0.956582 + 0.291463i \(0.905858\pi\)
\(734\) 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
\(745\) 475972. 0.857569
\(746\) 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 \(-0.575700\pi\)
−0.235584 + 0.971854i \(0.575700\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.15216e6 1.98949 0.994747 − 0.102362i \(-0.0326400\pi\)
0.994747 + 0.102362i \(0.0326400\pi\)
\(762\) 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
\(773\) −1.04296e6 −1.74545 −0.872726 − 0.488211i \(-0.837650\pi\)
−0.872726 + 0.488211i \(0.837650\pi\)
\(774\) 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
\(785\) −285572. −0.463422
\(786\) 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
\(797\) −38318.0 −0.0603235 −0.0301617 − 0.999545i \(-0.509602\pi\)
−0.0301617 + 0.999545i \(0.509602\pi\)
\(798\) 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
\(809\) −995362. −1.52084 −0.760421 − 0.649431i \(-0.775007\pi\)
−0.760421 + 0.649431i \(0.775007\pi\)
\(810\) 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
\(821\) −611918. −0.907835 −0.453917 − 0.891044i \(-0.649974\pi\)
−0.453917 + 0.891044i \(0.649974\pi\)
\(822\) 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 \(-0.548375\pi\)
−0.151389 + 0.988474i \(0.548375\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −773122. −1.11419
\(834\) 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
\(841\) −700557. −0.990493
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −393162. −0.550628
\(846\) 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
\(857\) −1.25360e6 −1.70686 −0.853430 − 0.521207i \(-0.825482\pi\)
−0.853430 + 0.521207i \(0.825482\pi\)
\(858\) 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
\(865\) −686588. −0.917622
\(866\) 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
\(881\) 1.00768e6 1.29828 0.649142 − 0.760667i \(-0.275128\pi\)
0.649142 + 0.760667i \(0.275128\pi\)
\(882\) 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
\(901\) −799204. −0.984483
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −897092. −1.09532
\(906\) 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
\(925\) 927498. 1.08400
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.65952e6 1.92287 0.961436 − 0.275027i \(-0.0886870\pi\)
0.961436 + 0.275027i \(0.0886870\pi\)
\(930\) 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
\(941\) 425362. 0.480374 0.240187 − 0.970727i \(-0.422791\pi\)
0.240187 + 0.970727i \(0.422791\pi\)
\(942\) 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
\(953\) 303518. 0.334194 0.167097 − 0.985940i \(-0.446561\pi\)
0.167097 + 0.985940i \(0.446561\pi\)
\(954\) 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
\(965\) 537572. 0.577274
\(966\) 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
\(977\) −1.66304e6 −1.74226 −0.871132 − 0.491048i \(-0.836614\pi\)
−0.871132 + 0.491048i \(0.836614\pi\)
\(978\) 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
\(985\) −1.04275e6 −1.07475
\(986\) 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.b.127.1 1
3.2 odd 2 64.5.c.a.63.1 1
4.3 odd 2 CM 576.5.g.b.127.1 1
8.3 odd 2 36.5.d.a.19.1 1
8.5 even 2 36.5.d.a.19.1 1
12.11 even 2 64.5.c.a.63.1 1
24.5 odd 2 4.5.b.a.3.1 ✓ 1
24.11 even 2 4.5.b.a.3.1 ✓ 1
48.5 odd 4 256.5.d.c.127.2 2
48.11 even 4 256.5.d.c.127.2 2
48.29 odd 4 256.5.d.c.127.1 2
48.35 even 4 256.5.d.c.127.1 2
120.29 odd 2 100.5.b.a.51.1 1
120.53 even 4 100.5.d.a.99.2 2
120.59 even 2 100.5.b.a.51.1 1
120.77 even 4 100.5.d.a.99.1 2
120.83 odd 4 100.5.d.a.99.2 2
120.107 odd 4 100.5.d.a.99.1 2
168.83 odd 2 196.5.c.a.99.1 1
168.125 even 2 196.5.c.a.99.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.5.b.a.3.1 ✓ 1 24.5 odd 2
4.5.b.a.3.1 ✓ 1 24.11 even 2
36.5.d.a.19.1 1 8.3 odd 2
36.5.d.a.19.1 1 8.5 even 2
64.5.c.a.63.1 1 3.2 odd 2
64.5.c.a.63.1 1 12.11 even 2
100.5.b.a.51.1 1 120.29 odd 2
100.5.b.a.51.1 1 120.59 even 2
100.5.d.a.99.1 2 120.77 even 4
100.5.d.a.99.1 2 120.107 odd 4
100.5.d.a.99.2 2 120.53 even 4
100.5.d.a.99.2 2 120.83 odd 4
196.5.c.a.99.1 1 168.83 odd 2
196.5.c.a.99.1 1 168.125 even 2
256.5.d.c.127.1 2 48.29 odd 4
256.5.d.c.127.1 2 48.35 even 4
256.5.d.c.127.2 2 48.5 odd 4
256.5.d.c.127.2 2 48.11 even 4
576.5.g.b.127.1 1 1.1 even 1 trivial
576.5.g.b.127.1 1 4.3 odd 2 CM