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Properties

Label 588.7.c.b.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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} +729.000 q^{9} +3527.00 q^{13} +12851.0 q^{19} +15625.0 q^{25} -19683.0 q^{27} -59221.0 q^{31} +86183.0 q^{37} -95229.0 q^{39} +42587.0 q^{43} -346977. q^{57} -420838. q^{61} +412523. q^{67} +66527.0 q^{73} -421875. q^{75} -733069. q^{79} +531441. q^{81} +1.59897e6 q^{93} -56446.0 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(13\) 3527.00 1.60537 0.802685 − 0.596403i \(-0.203404\pi\)
0.802685 + 0.596403i \(0.203404\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 12851.0 1.87360 0.936798 − 0.349870i \(-0.113774\pi\)
0.936798 + 0.349870i \(0.113774\pi\)
\(20\) 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
\(31\) −59221.0 −1.98788 −0.993941 − 0.109914i \(-0.964943\pi\)
−0.993941 + 0.109914i \(0.964943\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 86183.0 1.70144 0.850720 − 0.525620i \(-0.176167\pi\)
0.850720 + 0.525620i \(0.176167\pi\)
\(38\) 0 0
\(39\) −95229.0 −1.60537
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 42587.0 0.535638 0.267819 − 0.963469i \(-0.413697\pi\)
0.267819 + 0.963469i \(0.413697\pi\)
\(44\) 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
\(57\) −346977. −1.87360
\(58\) 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
\(67\) 412523. 1.37159 0.685794 − 0.727796i \(-0.259455\pi\)
0.685794 + 0.727796i \(0.259455\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 66527.0 0.171013 0.0855065 − 0.996338i \(-0.472749\pi\)
0.0855065 + 0.996338i \(0.472749\pi\)
\(74\) 0 0
\(75\) −421875. −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −733069. −1.48684 −0.743419 − 0.668826i \(-0.766797\pi\)
−0.743419 + 0.668826i \(0.766797\pi\)
\(80\) 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
\(93\) 1.59897e6 1.98788
\(94\) 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
\(103\) −2.18509e6 −1.99967 −0.999835 − 0.0181752i \(-0.994214\pi\)
−0.999835 + 0.0181752i \(0.994214\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −134569. −0.103912 −0.0519560 − 0.998649i \(-0.516546\pi\)
−0.0519560 + 0.998649i \(0.516546\pi\)
\(110\) 0 0
\(111\) −2.32694e6 −1.70144
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.57118e6 1.60537
\(118\) 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
\(127\) 2.90984e6 1.42056 0.710278 − 0.703921i \(-0.248569\pi\)
0.710278 + 0.703921i \(0.248569\pi\)
\(128\) 0 0
\(129\) −1.14985e6 −0.535638
\(130\) 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
\(139\) 3.88861e6 1.44794 0.723969 − 0.689832i \(-0.242316\pi\)
0.723969 + 0.689832i \(0.242316\pi\)
\(140\) 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
\(169\) 7.61292e6 1.57722
\(170\) 0 0
\(171\) 9.36838e6 1.87360
\(172\) 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
\(181\) −1.03194e7 −1.74028 −0.870139 − 0.492806i \(-0.835971\pi\)
−0.870139 + 0.492806i \(0.835971\pi\)
\(182\) 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
\(193\) 1.17442e7 1.63362 0.816811 − 0.576905i \(-0.195740\pi\)
0.816811 + 0.576905i \(0.195740\pi\)
\(194\) 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
\(201\) −1.11381e7 −1.37159
\(202\) 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
\(219\) −1.79623e6 −0.171013
\(220\) 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
\(229\) 2.25354e7 1.87654 0.938270 − 0.345903i \(-0.112428\pi\)
0.938270 + 0.345903i \(0.112428\pi\)
\(230\) 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
\(237\) 1.97929e7 1.48684
\(238\) 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
\(247\) 4.53255e7 3.00782
\(248\) 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
\(277\) −1.58100e7 −0.743861 −0.371931 − 0.928261i \(-0.621304\pi\)
−0.371931 + 0.928261i \(0.621304\pi\)
\(278\) 0 0
\(279\) −4.31721e7 −1.98788
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.03612e7 −1.33955 −0.669776 − 0.742563i \(-0.733610\pi\)
−0.669776 + 0.742563i \(0.733610\pi\)
\(284\) 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
\(307\) −1.30191e7 −0.449951 −0.224975 − 0.974364i \(-0.572230\pi\)
−0.224975 + 0.974364i \(0.572230\pi\)
\(308\) 0 0
\(309\) 5.89975e7 1.99967
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −6.05168e7 −1.97353 −0.986763 − 0.162168i \(-0.948151\pi\)
−0.986763 + 0.162168i \(0.948151\pi\)
\(314\) 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
\(325\) 5.51094e7 1.60537
\(326\) 0 0
\(327\) 3.63336e6 0.103912
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 2.54608e7 0.702083 0.351041 − 0.936360i \(-0.385828\pi\)
0.351041 + 0.936360i \(0.385828\pi\)
\(332\) 0 0
\(333\) 6.28274e7 1.70144
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.23317e7 −0.583489 −0.291745 − 0.956496i \(-0.594236\pi\)
−0.291745 + 0.956496i \(0.594236\pi\)
\(338\) 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
\(351\) −6.94219e7 −1.60537
\(352\) 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
\(361\) 1.18102e8 2.51036
\(362\) 0 0
\(363\) −4.78321e7 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.02833e7 1.82646 0.913228 − 0.407449i \(-0.133582\pi\)
0.913228 + 0.407449i \(0.133582\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.03727e8 −1.99879 −0.999393 − 0.0348373i \(-0.988909\pi\)
−0.999393 + 0.0348373i \(0.988909\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.06828e8 1.96231 0.981155 − 0.193220i \(-0.0618930\pi\)
0.981155 + 0.193220i \(0.0618930\pi\)
\(380\) 0 0
\(381\) −7.85658e7 −1.42056
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.10459e7 0.535638
\(388\) 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
\(397\) 4.56049e7 0.728854 0.364427 − 0.931232i \(-0.381265\pi\)
0.364427 + 0.931232i \(0.381265\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −2.08872e8 −3.19129
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.79956e7 −0.993828 −0.496914 − 0.867800i \(-0.665534\pi\)
−0.496914 + 0.867800i \(0.665534\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −1.04992e8 −1.44794
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −3.98462e7 −0.534000 −0.267000 − 0.963697i \(-0.586032\pi\)
−0.267000 + 0.963697i \(0.586032\pi\)
\(422\) 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
\(433\) 3.78287e7 0.465970 0.232985 − 0.972480i \(-0.425151\pi\)
0.232985 + 0.972480i \(0.425151\pi\)
\(434\) 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
\(457\) 1.78021e8 1.86519 0.932593 − 0.360929i \(-0.117540\pi\)
0.932593 + 0.360929i \(0.117540\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.37494e8 −1.38529 −0.692645 − 0.721279i \(-0.743555\pi\)
−0.692645 + 0.721279i \(0.743555\pi\)
\(464\) 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
\(475\) 2.00797e8 1.87360
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 3.03967e8 2.73144
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.20514e7 −0.104340 −0.0521700 − 0.998638i \(-0.516614\pi\)
−0.0521700 + 0.998638i \(0.516614\pi\)
\(488\) 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
\(499\) −2.04851e8 −1.64868 −0.824338 − 0.566097i \(-0.808453\pi\)
−0.824338 + 0.566097i \(0.808453\pi\)
\(500\) 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
\(507\) −2.05549e8 −1.57722
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.52946e8 −1.87360
\(514\) 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
\(523\) −1.44953e8 −1.01326 −0.506632 − 0.862162i \(-0.669110\pi\)
−0.506632 + 0.862162i \(0.669110\pi\)
\(524\) 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
\(541\) 1.15502e8 0.729452 0.364726 − 0.931115i \(-0.381163\pi\)
0.364726 + 0.931115i \(0.381163\pi\)
\(542\) 0 0
\(543\) 2.78624e8 1.74028
\(544\) 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
\(559\) 1.50204e8 0.859898
\(560\) 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
\(571\) −3.71943e8 −1.99787 −0.998937 − 0.0460873i \(-0.985325\pi\)
−0.998937 + 0.0460873i \(0.985325\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.62346e8 1.88624 0.943119 − 0.332456i \(-0.107877\pi\)
0.943119 + 0.332456i \(0.107877\pi\)
\(578\) 0 0
\(579\) −3.17094e8 −1.63362
\(580\) 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
\(589\) −7.61049e8 −3.72449
\(590\) 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
\(601\) −2.91337e8 −1.34206 −0.671031 − 0.741429i \(-0.734148\pi\)
−0.671031 + 0.741429i \(0.734148\pi\)
\(602\) 0 0
\(603\) 3.00729e8 1.37159
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4.92350e7 0.220145 0.110072 − 0.993924i \(-0.464892\pi\)
0.110072 + 0.993924i \(0.464892\pi\)
\(608\) 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
\(619\) −4.74352e8 −1.99999 −0.999997 − 0.00240251i \(-0.999235\pi\)
−0.999997 + 0.00240251i \(0.999235\pi\)
\(620\) 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
\(643\) 5.19296e8 1.95336 0.976680 − 0.214701i \(-0.0688778\pi\)
0.976680 + 0.214701i \(0.0688778\pi\)
\(644\) 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
\(657\) 4.84982e7 0.171013
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −4.02074e8 −1.39220 −0.696099 − 0.717946i \(-0.745083\pi\)
−0.696099 + 0.717946i \(0.745083\pi\)
\(662\) 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
\(673\) 5.42892e8 1.78102 0.890509 − 0.454965i \(-0.150348\pi\)
0.890509 + 0.454965i \(0.150348\pi\)
\(674\) 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
\(687\) −6.08454e8 −1.87654
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 6.54256e8 1.98296 0.991480 − 0.130263i \(-0.0415821\pi\)
0.991480 + 0.130263i \(0.0415821\pi\)
\(692\) 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
\(703\) 1.10754e9 3.18781
\(704\) 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
\(711\) −5.34407e8 −1.48684
\(712\) 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
\(727\) −2.57552e7 −0.0670287 −0.0335144 − 0.999438i \(-0.510670\pi\)
−0.0335144 + 0.999438i \(0.510670\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.98099e8 −0.503002 −0.251501 − 0.967857i \(-0.580924\pi\)
−0.251501 + 0.967857i \(0.580924\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.93314e8 1.47012 0.735058 − 0.678005i \(-0.237155\pi\)
0.735058 + 0.678005i \(0.237155\pi\)
\(740\) 0 0
\(741\) −1.22379e9 −3.00782
\(742\) 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
\(751\) 8.35593e8 1.97276 0.986382 − 0.164473i \(-0.0525923\pi\)
0.986382 + 0.164473i \(0.0525923\pi\)
\(752\) 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
\(769\) 6.13281e8 1.34859 0.674296 − 0.738461i \(-0.264447\pi\)
0.674296 + 0.738461i \(0.264447\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −9.25328e8 −1.98788
\(776\) 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
\(793\) −1.48430e9 −2.97647
\(794\) 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
\(817\) 5.47286e8 1.00357
\(818\) 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
\(829\) 1.08302e9 1.90095 0.950477 − 0.310795i \(-0.100595\pi\)
0.950477 + 0.310795i \(0.100595\pi\)
\(830\) 0 0
\(831\) 4.26869e8 0.743861
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.16565e9 1.98788
\(838\) 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
\(849\) 8.19753e8 1.33955
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 2.34104e8 0.377191 0.188596 − 0.982055i \(-0.439606\pi\)
0.188596 + 0.982055i \(0.439606\pi\)
\(854\) 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
\(871\) 1.45497e9 2.20191
\(872\) 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
\(883\) −3.89149e8 −0.565241 −0.282620 − 0.959232i \(-0.591204\pi\)
−0.282620 + 0.959232i \(0.591204\pi\)
\(884\) 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
\(907\) −1.47106e9 −1.97155 −0.985777 − 0.168057i \(-0.946251\pi\)
−0.985777 + 0.168057i \(0.946251\pi\)
\(908\) 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
\(919\) 9.05179e8 1.16624 0.583120 − 0.812386i \(-0.301832\pi\)
0.583120 + 0.812386i \(0.301832\pi\)
\(920\) 0 0
\(921\) 3.51515e8 0.449951
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.34661e9 1.70144
\(926\) 0 0
\(927\) −1.59293e9 −1.99967
\(928\) 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
\(937\) −1.41345e9 −1.71816 −0.859078 − 0.511845i \(-0.828962\pi\)
−0.859078 + 0.511845i \(0.828962\pi\)
\(938\) 0 0
\(939\) 1.63395e9 1.97353
\(940\) 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
\(949\) 2.34641e8 0.274539
\(950\) 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
\(961\) 2.61962e9 2.95168
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.09966e9 1.21613 0.608063 − 0.793889i \(-0.291947\pi\)
0.608063 + 0.793889i \(0.291947\pi\)
\(968\) 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
\(975\) −1.48795e9 −1.60537
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −9.81008e7 −0.103912
\(982\) 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
\(991\) 1.61391e9 1.65828 0.829138 − 0.559043i \(-0.188832\pi\)
0.829138 + 0.559043i \(0.188832\pi\)
\(992\) 0 0
\(993\) −6.87442e8 −0.702083
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −8.27210e8 −0.834699 −0.417350 − 0.908746i \(-0.637041\pi\)
−0.417350 + 0.908746i \(0.637041\pi\)
\(998\) 0 0
\(999\) −1.69634e9 −1.70144
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.b.197.1 1
3.2 odd 2 CM 588.7.c.b.197.1 1
7.2 even 3 84.7.p.b.53.1 ✓ 2
7.4 even 3 84.7.p.b.65.1 yes 2
7.6 odd 2 588.7.c.c.197.1 1
21.2 odd 6 84.7.p.b.53.1 ✓ 2
21.11 odd 6 84.7.p.b.65.1 yes 2
21.20 even 2 588.7.c.c.197.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.7.p.b.53.1 ✓ 2 7.2 even 3
84.7.p.b.53.1 ✓ 2 21.2 odd 6
84.7.p.b.65.1 yes 2 7.4 even 3
84.7.p.b.65.1 yes 2 21.11 odd 6
588.7.c.b.197.1 1 1.1 even 1 trivial
588.7.c.b.197.1 1 3.2 odd 2 CM
588.7.c.c.197.1 1 7.6 odd 2
588.7.c.c.197.1 1 21.20 even 2