Properties

Label 588.7.c.a.197.1
Level $588$
Weight $7$
Character 588.197
Self dual yes
Analytic conductor $135.272$
Analytic rank $0$
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] = [588,7,Mod(197,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.197");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 588.c (of order \(2\), degree \(1\), 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: \(135.271801168\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 197.1
Character \(\chi\) \(=\) 588.197

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} +729.000 q^{9} -4033.00 q^{13} -2269.00 q^{19} +15625.0 q^{25} -19683.0 q^{27} +23939.0 q^{31} +3023.00 q^{37} +108891. q^{39} -153973. q^{43} +61263.0 q^{57} -420838. q^{61} -585397. q^{67} -704593. q^{73} -421875. q^{75} +937691. q^{79} +531441. q^{81} -646353. q^{93} -56446.0 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −27.0000 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −4033.00 −1.83569 −0.917843 0.396945i \(-0.870070\pi\)
−0.917843 + 0.396945i \(0.870070\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2269.00 −0.330806 −0.165403 0.986226i \(-0.552893\pi\)
−0.165403 + 0.986226i \(0.552893\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) −19683.0 −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 23939.0 0.803565 0.401782 0.915735i \(-0.368391\pi\)
0.401782 + 0.915735i \(0.368391\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3023.00 0.0596806 0.0298403 0.999555i \(-0.490500\pi\)
0.0298403 + 0.999555i \(0.490500\pi\)
\(38\) 0 0
\(39\) 108891. 1.83569
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −153973. −1.93660 −0.968298 0.249796i \(-0.919636\pi\)
−0.968298 + 0.249796i \(0.919636\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 61263.0 0.330806
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −420838. −1.85407 −0.927034 0.374978i \(-0.877650\pi\)
−0.927034 + 0.374978i \(0.877650\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −585397. −1.94637 −0.973187 0.230017i \(-0.926122\pi\)
−0.973187 + 0.230017i \(0.926122\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −704593. −1.81121 −0.905607 0.424118i \(-0.860584\pi\)
−0.905607 + 0.424118i \(0.860584\pi\)
\(74\) 0 0
\(75\) −421875. −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 937691. 1.90186 0.950930 0.309407i \(-0.100130\pi\)
0.950930 + 0.309407i \(0.100130\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −646353. −0.803565
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −56446.0 −0.0618469 −0.0309235 0.999522i \(-0.509845\pi\)
−0.0309235 + 0.999522i \(0.509845\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.05815e6 0.968354 0.484177 0.874970i \(-0.339119\pi\)
0.484177 + 0.874970i \(0.339119\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.30731e6 1.78167 0.890834 0.454329i \(-0.150121\pi\)
0.890834 + 0.454329i \(0.150121\pi\)
\(110\) 0 0
\(111\) −81621.0 −0.0596806
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.94006e6 −1.83569
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.04252e6 0.508949 0.254475 0.967079i \(-0.418097\pi\)
0.254475 + 0.967079i \(0.418097\pi\)
\(128\) 0 0
\(129\) 4.15727e6 1.93660
\(130\) 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
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −5.15315e6 −1.91879 −0.959397 0.282060i \(-0.908982\pi\)
−0.959397 + 0.282060i \(0.908982\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −3.83040e6 −1.11253 −0.556267 0.831004i \(-0.687767\pi\)
−0.556267 + 0.831004i \(0.687767\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.08271e6 1.83021 0.915105 0.403216i \(-0.132108\pi\)
0.915105 + 0.403216i \(0.132108\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.89851e6 0.669285 0.334643 0.942345i \(-0.391384\pi\)
0.334643 + 0.942345i \(0.391384\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.14383e7 2.36974
\(170\) 0 0
\(171\) −1.65410e6 −0.330806
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 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
\(181\) 1.02211e7 1.72370 0.861852 0.507160i \(-0.169305\pi\)
0.861852 + 0.507160i \(0.169305\pi\)
\(182\) 0 0
\(183\) 1.13626e7 1.85407
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.31141e6 0.182417 0.0912086 0.995832i \(-0.470927\pi\)
0.0912086 + 0.995832i \(0.470927\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 1.16545e7 1.47888 0.739442 0.673220i \(-0.235089\pi\)
0.739442 + 0.673220i \(0.235089\pi\)
\(200\) 0 0
\(201\) 1.58057e7 1.94637
\(202\) 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
\(211\) 1.75972e7 1.87325 0.936624 0.350336i \(-0.113933\pi\)
0.936624 + 0.350336i \(0.113933\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.90240e7 1.81121
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.18107e7 −1.06503 −0.532516 0.846420i \(-0.678753\pi\)
−0.532516 + 0.846420i \(0.678753\pi\)
\(224\) 0 0
\(225\) 1.13906e7 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.84625e7 −1.53739 −0.768696 0.639614i \(-0.779094\pi\)
−0.768696 + 0.639614i \(0.779094\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.53177e7 −1.90186
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 2.64398e7 1.88889 0.944447 0.328663i \(-0.106598\pi\)
0.944447 + 0.328663i \(0.106598\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.15088e6 0.607256
\(248\) 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
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 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
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −3.91457e7 −1.96687 −0.983436 0.181258i \(-0.941983\pi\)
−0.983436 + 0.181258i \(0.941983\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.20769e7 1.97973 0.989863 0.142029i \(-0.0453626\pi\)
0.989863 + 0.142029i \(0.0453626\pi\)
\(278\) 0 0
\(279\) 1.74515e7 0.803565
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 4.43316e7 1.95593 0.977966 0.208762i \(-0.0669434\pi\)
0.977966 + 0.208762i \(0.0669434\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 1.52404e6 0.0618469
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −4.23216e7 −1.46267 −0.731337 0.682017i \(-0.761103\pi\)
−0.731337 + 0.682017i \(0.761103\pi\)
\(308\) 0 0
\(309\) −2.85700e7 −0.968354
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.16453e7 0.705879 0.352940 0.935646i \(-0.385182\pi\)
0.352940 + 0.935646i \(0.385182\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −6.30156e7 −1.83569
\(326\) 0 0
\(327\) −6.22974e7 −1.78167
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.60845e7 1.27078 0.635391 0.772191i \(-0.280839\pi\)
0.635391 + 0.772191i \(0.280839\pi\)
\(332\) 0 0
\(333\) 2.20377e6 0.0596806
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.45723e7 1.94844 0.974222 0.225590i \(-0.0724309\pi\)
0.974222 + 0.225590i \(0.0724309\pi\)
\(338\) 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
\(349\) 5.69263e7 1.33917 0.669586 0.742734i \(-0.266471\pi\)
0.669586 + 0.742734i \(0.266471\pi\)
\(350\) 0 0
\(351\) 7.93815e7 1.83569
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 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
\(361\) −4.18975e7 −0.890567
\(362\) 0 0
\(363\) −4.78321e7 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.02572e7 −0.207505 −0.103753 0.994603i \(-0.533085\pi\)
−0.103753 + 0.994603i \(0.533085\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.49950e7 1.05973 0.529866 0.848081i \(-0.322242\pi\)
0.529866 + 0.848081i \(0.322242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −7.16333e7 −1.31582 −0.657911 0.753096i \(-0.728560\pi\)
−0.657911 + 0.753096i \(0.728560\pi\)
\(380\) 0 0
\(381\) −2.81481e7 −0.508949
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.12246e8 −1.93660
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 7.81205e7 1.24851 0.624257 0.781219i \(-0.285402\pi\)
0.624257 + 0.781219i \(0.285402\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −9.65460e7 −1.47509
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.36835e8 1.99999 0.999994 0.00355979i \(-0.00113312\pi\)
0.999994 + 0.00355979i \(0.00113312\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.39135e8 1.91879
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.04628e8 −1.40217 −0.701086 0.713077i \(-0.747301\pi\)
−0.701086 + 0.713077i \(0.747301\pi\)
\(422\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 1.17829e8 1.45140 0.725700 0.688011i \(-0.241516\pi\)
0.725700 + 0.688011i \(0.241516\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 5.35167e7 0.632552 0.316276 0.948667i \(-0.397567\pi\)
0.316276 + 0.948667i \(0.397567\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.03421e8 1.11253
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.48677e8 −1.55774 −0.778870 0.627185i \(-0.784207\pi\)
−0.778870 + 0.627185i \(0.784207\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −5.52487e7 −0.556646 −0.278323 0.960487i \(-0.589779\pi\)
−0.278323 + 0.960487i \(0.589779\pi\)
\(464\) 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
\(471\) −1.91233e8 −1.83021
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.54531e7 −0.330806
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1.21918e7 −0.109555
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.93756e8 −1.67752 −0.838761 0.544500i \(-0.816720\pi\)
−0.838761 + 0.544500i \(0.816720\pi\)
\(488\) 0 0
\(489\) −7.82597e7 −0.669285
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.24255e8 1.80485 0.902424 0.430849i \(-0.141786\pi\)
0.902424 + 0.430849i \(0.141786\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.08834e8 −2.36974
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4.46607e7 0.330806
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 2.86103e8 1.99994 0.999971 0.00767504i \(-0.00244307\pi\)
0.999971 + 0.00767504i \(0.00244307\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 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
\(541\) −3.13113e8 −1.97746 −0.988732 0.149696i \(-0.952171\pi\)
−0.988732 + 0.149696i \(0.952171\pi\)
\(542\) 0 0
\(543\) −2.75970e8 −1.72370
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.24645e8 −1.98357 −0.991783 0.127929i \(-0.959167\pi\)
−0.991783 + 0.127929i \(0.959167\pi\)
\(548\) 0 0
\(549\) −3.06791e8 −1.85407
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 6.20973e8 3.55498
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 2.00833e8 1.07876 0.539381 0.842062i \(-0.318658\pi\)
0.539381 + 0.842062i \(0.318658\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.91790e8 −1.51895 −0.759475 0.650537i \(-0.774544\pi\)
−0.759475 + 0.650537i \(0.774544\pi\)
\(578\) 0 0
\(579\) −3.54080e7 −0.182417
\(580\) 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
\(589\) −5.43176e7 −0.265824
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.14671e8 −1.47888
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.33106e8 −0.613162 −0.306581 0.951845i \(-0.599185\pi\)
−0.306581 + 0.951845i \(0.599185\pi\)
\(602\) 0 0
\(603\) −4.26754e8 −1.94637
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.09634e8 −1.83160 −0.915799 0.401636i \(-0.868442\pi\)
−0.915799 + 0.401636i \(0.868442\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.58281e8 −0.687142 −0.343571 0.939127i \(-0.611637\pi\)
−0.343571 + 0.939127i \(0.611637\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.38163e8 1.00416 0.502079 0.864822i \(-0.332569\pi\)
0.502079 + 0.864822i \(0.332569\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −4.98900e8 −1.98575 −0.992876 0.119152i \(-0.961982\pi\)
−0.992876 + 0.119152i \(0.961982\pi\)
\(632\) 0 0
\(633\) −4.75123e8 −1.87325
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.60786e8 −0.604806 −0.302403 0.953180i \(-0.597789\pi\)
−0.302403 + 0.953180i \(0.597789\pi\)
\(644\) 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
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −5.13648e8 −1.81121
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 5.60171e8 1.93962 0.969809 0.243867i \(-0.0784160\pi\)
0.969809 + 0.243867i \(0.0784160\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.18890e8 1.06503
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −5.11652e8 −1.67853 −0.839266 0.543721i \(-0.817015\pi\)
−0.839266 + 0.543721i \(0.817015\pi\)
\(674\) 0 0
\(675\) −3.07547e8 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 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
\(685\) 0 0
\(686\) 0 0
\(687\) 4.98488e8 1.53739
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −2.52687e8 −0.765858 −0.382929 0.923778i \(-0.625085\pi\)
−0.382929 + 0.923778i \(0.625085\pi\)
\(692\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −6.85919e6 −0.0197427
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.93732e8 1.66591 0.832955 0.553341i \(-0.186647\pi\)
0.832955 + 0.553341i \(0.186647\pi\)
\(710\) 0 0
\(711\) 6.83577e8 1.90186
\(712\) 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
\(723\) −7.13876e8 −1.88889
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.78028e8 1.76459 0.882296 0.470695i \(-0.155997\pi\)
0.882296 + 0.470695i \(0.155997\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 7.59262e8 1.92788 0.963939 0.266122i \(-0.0857425\pi\)
0.963939 + 0.266122i \(0.0857425\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −7.70601e8 −1.90940 −0.954698 0.297576i \(-0.903822\pi\)
−0.954698 + 0.297576i \(0.903822\pi\)
\(740\) 0 0
\(741\) −2.47074e8 −0.607256
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −5.38460e8 −1.27126 −0.635629 0.771995i \(-0.719259\pi\)
−0.635629 + 0.771995i \(0.719259\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.52165e8 1.96443 0.982214 0.187767i \(-0.0601251\pi\)
0.982214 + 0.187767i \(0.0601251\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.75017e8 0.604756 0.302378 0.953188i \(-0.402220\pi\)
0.302378 + 0.953188i \(0.402220\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 3.74047e8 0.803565
\(776\) 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
\(787\) 9.08673e8 1.86416 0.932081 0.362251i \(-0.117992\pi\)
0.932081 + 0.362251i \(0.117992\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.69724e9 3.40348
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 4.83016e8 0.905522 0.452761 0.891632i \(-0.350439\pi\)
0.452761 + 0.891632i \(0.350439\pi\)
\(812\) 0 0
\(813\) 1.05693e9 1.96687
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.49365e8 0.640638
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −9.65555e8 −1.73212 −0.866059 0.499941i \(-0.833355\pi\)
−0.866059 + 0.499941i \(0.833355\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −2.34820e8 −0.412165 −0.206082 0.978535i \(-0.566071\pi\)
−0.206082 + 0.978535i \(0.566071\pi\)
\(830\) 0 0
\(831\) −1.13608e9 −1.97973
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.71191e8 −0.803565
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −1.19695e9 −1.95593
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.17276e9 −1.88956 −0.944782 0.327699i \(-0.893727\pi\)
−0.944782 + 0.327699i \(0.893727\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −2.87739e8 −0.453962 −0.226981 0.973899i \(-0.572886\pi\)
−0.226981 + 0.973899i \(0.572886\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.51714e8 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.36091e9 3.57293
\(872\) 0 0
\(873\) −4.11491e7 −0.0618469
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.16597e9 1.72858 0.864288 0.502997i \(-0.167769\pi\)
0.864288 + 0.502997i \(0.167769\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.33842e9 1.94406 0.972029 0.234859i \(-0.0754630\pi\)
0.972029 + 0.234859i \(0.0754630\pi\)
\(884\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 9.52721e8 1.27686 0.638431 0.769679i \(-0.279584\pi\)
0.638431 + 0.769679i \(0.279584\pi\)
\(908\) 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
\(919\) 6.39528e8 0.823974 0.411987 0.911190i \(-0.364835\pi\)
0.411987 + 0.911190i \(0.364835\pi\)
\(920\) 0 0
\(921\) 1.14268e9 1.46267
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 4.72344e7 0.0596806
\(926\) 0 0
\(927\) 7.71389e8 0.968354
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.25938e7 −0.0274644 −0.0137322 0.999906i \(-0.504371\pi\)
−0.0137322 + 0.999906i \(0.504371\pi\)
\(938\) 0 0
\(939\) −5.84423e8 −0.705879
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 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
\(949\) 2.84162e9 3.32482
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −3.14428e8 −0.354284
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.79320e9 −1.98312 −0.991559 0.129654i \(-0.958613\pi\)
−0.991559 + 0.129654i \(0.958613\pi\)
\(968\) 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
\(975\) 1.70142e9 1.83569
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.68203e9 1.78167
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.74933e9 −1.79743 −0.898715 0.438533i \(-0.855498\pi\)
−0.898715 + 0.438533i \(0.855498\pi\)
\(992\) 0 0
\(993\) −1.24428e9 −1.27078
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.97348e9 1.99134 0.995672 0.0929375i \(-0.0296257\pi\)
0.995672 + 0.0929375i \(0.0296257\pi\)
\(998\) 0 0
\(999\) −5.95017e7 −0.0596806
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 588.7.c.a.197.1 1
3.2 odd 2 CM 588.7.c.a.197.1 1
7.2 even 3 84.7.p.a.53.1 2
7.4 even 3 84.7.p.a.65.1 yes 2
7.6 odd 2 588.7.c.d.197.1 1
21.2 odd 6 84.7.p.a.53.1 2
21.11 odd 6 84.7.p.a.65.1 yes 2
21.20 even 2 588.7.c.d.197.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.7.p.a.53.1 2 7.2 even 3
84.7.p.a.53.1 2 21.2 odd 6
84.7.p.a.65.1 yes 2 7.4 even 3
84.7.p.a.65.1 yes 2 21.11 odd 6
588.7.c.a.197.1 1 1.1 even 1 trivial
588.7.c.a.197.1 1 3.2 odd 2 CM
588.7.c.d.197.1 1 7.6 odd 2
588.7.c.d.197.1 1 21.20 even 2