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Properties

Label 64.9.c.a.63.1
Level $64$
Weight $9$
Character 64.63
Self dual yes
Analytic conductor $26.072$
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] = [64,9,Mod(63,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.63");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 64.c (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: \(26.0722310439\)
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 63.1
Character \(\chi\) \(=\) 64.63

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1054.00 q^{5} +6561.00 q^{9} +478.000 q^{13} -63358.0 q^{17} +720291. q^{25} +1.40784e6 q^{29} -925922. q^{37} +3.57792e6 q^{41} +6.91529e6 q^{45} +5.76480e6 q^{49} +9.62064e6 q^{53} -2.07221e7 q^{61} +503812. q^{65} -5.47171e7 q^{73} +4.30467e7 q^{81} -6.67793e7 q^{85} -3.02659e7 q^{89} -1.73380e8 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 1054.00 1.68640 0.843200 − 0.537600i \(-0.180669\pi\)
0.843200 + 0.537600i \(0.180669\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 478.000 0.0167361 0.00836805 − 0.999965i \(-0.497336\pi\)
0.00836805 + 0.999965i \(0.497336\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −63358.0 −0.758588 −0.379294 − 0.925276i \(-0.623833\pi\)
−0.379294 + 0.925276i \(0.623833\pi\)
\(18\) 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
\(25\) 720291. 1.84394
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.40784e6 1.99049 0.995247 − 0.0973870i \(-0.0310485\pi\)
0.995247 + 0.0973870i \(0.0310485\pi\)
\(30\) 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
\(37\) −925922. −0.494046 −0.247023 − 0.969010i \(-0.579452\pi\)
−0.247023 + 0.969010i \(0.579452\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.57792e6 1.26618 0.633090 − 0.774078i \(-0.281786\pi\)
0.633090 + 0.774078i \(0.281786\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 6.91529e6 1.68640
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.62064e6 1.21927 0.609636 − 0.792682i \(-0.291316\pi\)
0.609636 + 0.792682i \(0.291316\pi\)
\(54\) 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
\(61\) −2.07221e7 −1.49663 −0.748314 − 0.663344i \(-0.769136\pi\)
−0.748314 + 0.663344i \(0.769136\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 503812. 0.0282238
\(66\) 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
\(73\) −5.47171e7 −1.92678 −0.963389 − 0.268107i \(-0.913602\pi\)
−0.963389 + 0.268107i \(0.913602\pi\)
\(74\) 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
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −6.67793e7 −1.27928
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.02659e7 −0.482385 −0.241193 − 0.970477i \(-0.577539\pi\)
−0.241193 + 0.970477i \(0.577539\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.73380e8 −1.95845 −0.979223 − 0.202786i \(-0.935000\pi\)
−0.979223 + 0.202786i \(0.935000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.45394e8 −1.39721 −0.698606 − 0.715507i \(-0.746196\pi\)
−0.698606 + 0.715507i \(0.746196\pi\)
\(102\) 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
\(109\) 1.94669e8 1.37909 0.689543 − 0.724245i \(-0.257811\pi\)
0.689543 + 0.724245i \(0.257811\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.80936e8 1.72304 0.861518 − 0.507728i \(-0.169514\pi\)
0.861518 + 0.507728i \(0.169514\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.13616e6 0.0167361
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.47468e8 1.42323
\(126\) 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
\(137\) −6.61491e8 −1.87777 −0.938883 − 0.344236i \(-0.888138\pi\)
−0.938883 + 0.344236i \(0.888138\pi\)
\(138\) 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
\(145\) 1.48386e9 3.35677
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.70095e8 −0.345102 −0.172551 − 0.985001i \(-0.555201\pi\)
−0.172551 + 0.985001i \(0.555201\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −4.15692e8 −0.758588
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.99068e8 1.31518 0.657590 − 0.753376i \(-0.271576\pi\)
0.657590 + 0.753376i \(0.271576\pi\)
\(158\) 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
\(169\) −8.15502e8 −0.999720
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.13628e8 −0.685047 −0.342524 − 0.939509i \(-0.611282\pi\)
−0.342524 + 0.939509i \(0.611282\pi\)
\(174\) 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
\(181\) −1.95942e9 −1.82564 −0.912818 − 0.408367i \(-0.866098\pi\)
−0.912818 + 0.408367i \(0.866098\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.75922e8 −0.833159
\(186\) 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
\(193\) −1.30057e9 −0.937356 −0.468678 − 0.883369i \(-0.655269\pi\)
−0.468678 + 0.883369i \(0.655269\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.53529e9 −1.68331 −0.841653 − 0.540019i \(-0.818417\pi\)
−0.841653 + 0.540019i \(0.818417\pi\)
\(198\) 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
\(205\) 3.77113e9 2.13529
\(206\) 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
\(221\) −3.02851e7 −0.0126958
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 4.72583e9 1.84394
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −2.68688e9 −0.977025 −0.488512 − 0.872557i \(-0.662460\pi\)
−0.488512 + 0.872557i \(0.662460\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.73718e9 −0.589414 −0.294707 − 0.955588i \(-0.595222\pi\)
−0.294707 + 0.955588i \(0.595222\pi\)
\(234\) 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
\(241\) −3.31730e9 −0.983368 −0.491684 − 0.870774i \(-0.663619\pi\)
−0.491684 + 0.870774i \(0.663619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 6.07610e9 1.68640
\(246\) 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
\(257\) 7.65957e9 1.75579 0.877894 − 0.478855i \(-0.158948\pi\)
0.877894 + 0.478855i \(0.158948\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 9.23683e9 1.99049
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 1.01402e10 2.05618
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.32273e9 −1.01654 −0.508271 − 0.861197i \(-0.669715\pi\)
−0.508271 + 0.861197i \(0.669715\pi\)
\(270\) 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
\(277\) 1.66277e9 0.282431 0.141216 − 0.989979i \(-0.454899\pi\)
0.141216 + 0.989979i \(0.454899\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −9.38699e9 −1.50557 −0.752785 − 0.658267i \(-0.771290\pi\)
−0.752785 + 0.658267i \(0.771290\pi\)
\(282\) 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
\(289\) −2.96152e9 −0.424545
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.73075e9 −1.18463 −0.592313 − 0.805708i \(-0.701785\pi\)
−0.592313 + 0.805708i \(0.701785\pi\)
\(294\) 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
\(305\) −2.18411e10 −2.52391
\(306\) 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
\(313\) 1.82224e10 1.89858 0.949289 − 0.314405i \(-0.101805\pi\)
0.949289 + 0.314405i \(0.101805\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.16583e10 −1.15451 −0.577257 − 0.816563i \(-0.695877\pi\)
−0.577257 + 0.816563i \(0.695877\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 3.44299e8 0.0308605
\(326\) 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
\(333\) −6.07497e9 −0.494046
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.48467e10 −1.15109 −0.575547 − 0.817768i \(-0.695211\pi\)
−0.575547 + 0.817768i \(0.695211\pi\)
\(338\) 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
\(349\) 1.66745e10 1.12396 0.561981 − 0.827150i \(-0.310040\pi\)
0.561981 + 0.827150i \(0.310040\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.88722e10 −1.85944 −0.929719 − 0.368270i \(-0.879950\pi\)
−0.929719 + 0.368270i \(0.879950\pi\)
\(354\) 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
\(361\) 1.69836e10 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.76718e10 −3.24932
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 2.34747e10 1.26618
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3.81261e10 1.96964 0.984820 − 0.173579i \(-0.0555331\pi\)
0.984820 + 0.173579i \(0.0555331\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.72947e8 0.0333131
\(378\) 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
\(389\) 2.02735e10 0.885380 0.442690 − 0.896675i \(-0.354024\pi\)
0.442690 + 0.896675i \(0.354024\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.81718e10 1.53667 0.768334 − 0.640049i \(-0.221086\pi\)
0.768334 + 0.640049i \(0.221086\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.76384e10 1.84238 0.921191 − 0.389112i \(-0.127218\pi\)
0.921191 + 0.389112i \(0.127218\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.53712e10 1.68640
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.07421e10 0.741241 0.370620 − 0.928784i \(-0.379145\pi\)
0.370620 + 0.928784i \(0.379145\pi\)
\(410\) 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
\(421\) −6.04551e10 −1.92444 −0.962221 − 0.272271i \(-0.912225\pi\)
−0.962221 + 0.272271i \(0.912225\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −4.56362e10 −1.39879
\(426\) 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
\(433\) 1.43058e10 0.406967 0.203484 − 0.979078i \(-0.434774\pi\)
0.203484 + 0.979078i \(0.434774\pi\)
\(434\) 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
\(441\) 3.78229e10 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −3.19003e10 −0.813494
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.32574e10 −1.80246 −0.901231 − 0.433339i \(-0.857335\pi\)
−0.901231 + 0.433339i \(0.857335\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.66845e9 0.129957 0.0649785 − 0.997887i \(-0.479302\pi\)
0.0649785 + 0.997887i \(0.479302\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.70564e10 1.48469 0.742346 − 0.670016i \(-0.233713\pi\)
0.742346 + 0.670016i \(0.233713\pi\)
\(462\) 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
\(477\) 6.31210e10 1.21927
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −4.42591e8 −0.00826841
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.82742e11 −3.30272
\(486\) 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
\(493\) −8.91978e10 −1.50996
\(494\) 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
\(505\) −1.53246e11 −2.35626
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.89054e10 0.430634 0.215317 − 0.976544i \(-0.430922\pi\)
0.215317 + 0.976544i \(0.430922\pi\)
\(510\) 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
\(521\) −9.37598e10 −1.27252 −0.636262 − 0.771473i \(-0.719520\pi\)
−0.636262 + 0.771473i \(0.719520\pi\)
\(522\) 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
\(529\) 7.83110e10 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.71025e9 0.0211909
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.56867e11 1.83123 0.915615 − 0.402056i \(-0.131704\pi\)
0.915615 + 0.402056i \(0.131704\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.05181e11 2.32569
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −1.35958e11 −1.49663
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.92236e10 −0.719173 −0.359587 − 0.933112i \(-0.617082\pi\)
−0.359587 + 0.933112i \(0.617082\pi\)
\(558\) 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
\(565\) 2.96107e11 2.90573
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.12187e10 −0.202427 −0.101214 − 0.994865i \(-0.532273\pi\)
−0.101214 + 0.994865i \(0.532273\pi\)
\(570\) 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
\(577\) 2.09495e11 1.89004 0.945020 − 0.327013i \(-0.106042\pi\)
0.945020 + 0.327013i \(0.106042\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 3.30551e9 0.0282238
\(586\) 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
\(593\) −2.21199e11 −1.78881 −0.894405 − 0.447259i \(-0.852400\pi\)
−0.894405 + 0.447259i \(0.852400\pi\)
\(594\) 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
\(601\) −1.88664e10 −0.144607 −0.0723036 − 0.997383i \(-0.523035\pi\)
−0.0723036 + 0.997383i \(0.523035\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.25934e11 1.68640
\(606\) 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
\(613\) −2.75064e11 −1.94801 −0.974005 − 0.226526i \(-0.927263\pi\)
−0.974005 + 0.226526i \(0.927263\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.24639e11 1.55005 0.775025 − 0.631931i \(-0.217737\pi\)
0.775025 + 0.631931i \(0.217737\pi\)
\(618\) 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
\(625\) 8.48676e10 0.556188
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.86646e10 0.374777
\(630\) 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
\(637\) 2.75557e9 0.0167361
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.00283e11 0.594009 0.297005 − 0.954876i \(-0.404012\pi\)
0.297005 + 0.954876i \(0.404012\pi\)
\(642\) 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
\(653\) 1.55788e11 0.856804 0.428402 − 0.903588i \(-0.359077\pi\)
0.428402 + 0.903588i \(0.359077\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.58999e11 −1.92678
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 1.17766e11 0.616901 0.308450 − 0.951240i \(-0.400190\pi\)
0.308450 + 0.951240i \(0.400190\pi\)
\(662\) 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
\(673\) −3.12347e11 −1.52257 −0.761284 − 0.648419i \(-0.775431\pi\)
−0.761284 + 0.648419i \(0.775431\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.00419e11 −1.90616 −0.953081 − 0.302716i \(-0.902107\pi\)
−0.953081 + 0.302716i \(0.902107\pi\)
\(678\) 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
\(685\) −6.97211e11 −3.16667
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.59866e9 0.0204059
\(690\) 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
\(697\) −2.26690e11 −0.960509
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.45667e10 −0.101736 −0.0508681 − 0.998705i \(-0.516199\pi\)
−0.0508681 + 0.998705i \(0.516199\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.78486e10 −0.149784 −0.0748920 − 0.997192i \(-0.523861\pi\)
−0.0748920 + 0.997192i \(0.523861\pi\)
\(710\) 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
\(725\) 1.01405e12 3.67036
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.82430e11 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −4.79265e11 −1.66020 −0.830098 − 0.557617i \(-0.811716\pi\)
−0.830098 + 0.557617i \(0.811716\pi\)
\(734\) 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
\(745\) −1.79280e11 −0.581979
\(746\) 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
\(757\) 5.83869e11 1.77800 0.889001 − 0.457906i \(-0.151400\pi\)
0.889001 + 0.457906i \(0.151400\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.56706e11 1.95809 0.979044 − 0.203649i \(-0.0652800\pi\)
0.979044 + 0.203649i \(0.0652800\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.38139e11 −1.27928
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.33224e11 −1.81072 −0.905361 − 0.424643i \(-0.860400\pi\)
−0.905361 + 0.424643i \(0.860400\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.73680e11 −1.04660 −0.523301 − 0.852148i \(-0.675300\pi\)
−0.523301 + 0.852148i \(0.675300\pi\)
\(774\) 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
\(785\) 8.42218e11 2.21792
\(786\) 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
\(793\) −9.90516e9 −0.0250477
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 8.05513e11 1.99636 0.998181 − 0.0602960i \(-0.0192045\pi\)
0.998181 + 0.0602960i \(0.0192045\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.98575e11 −0.482385
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.34055e11 0.312959 0.156480 − 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(810\) 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
\(821\) 5.34219e11 1.17584 0.587918 − 0.808921i \(-0.299948\pi\)
0.587918 + 0.808921i \(0.299948\pi\)
\(822\) 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
\(829\) 9.01302e11 1.90833 0.954163 − 0.299289i \(-0.0967495\pi\)
0.954163 + 0.299289i \(0.0967495\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.65246e11 −0.758588
\(834\) 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
\(841\) 1.48176e12 2.96206
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.59539e11 −1.68593
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −5.97843e11 −1.12925 −0.564626 − 0.825347i \(-0.690980\pi\)
−0.564626 + 0.825347i \(0.690980\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.92687e11 0.913373 0.456686 − 0.889628i \(-0.349036\pi\)
0.456686 + 0.889628i \(0.349036\pi\)
\(858\) 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
\(865\) −6.46764e11 −1.15526
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1.13755e12 −1.95845
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.25359e10 0.122618 0.0613090 − 0.998119i \(-0.480472\pi\)
0.0613090 + 0.998119i \(0.480472\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.89437e11 −0.314457 −0.157228 − 0.987562i \(-0.550256\pi\)
−0.157228 + 0.987562i \(0.550256\pi\)
\(882\) 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
\(901\) −6.09544e11 −0.924924
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.06523e12 −3.07875
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −9.53933e11 −1.39721
\(910\) 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
\(925\) −6.66933e11 −0.910994
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.26432e12 1.69744 0.848720 − 0.528843i \(-0.177374\pi\)
0.848720 + 0.528843i \(0.177374\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 9.26501e11 1.20195 0.600977 − 0.799267i \(-0.294778\pi\)
0.600977 + 0.799267i \(0.294778\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.38722e12 1.76924 0.884620 − 0.466312i \(-0.154418\pi\)
0.884620 + 0.466312i \(0.154418\pi\)
\(942\) 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
\(949\) −2.61548e10 −0.0322468
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.55756e12 −1.88831 −0.944157 − 0.329495i \(-0.893121\pi\)
−0.944157 + 0.329495i \(0.893121\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.52891e11 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.37080e12 −1.58076
\(966\) 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
\(977\) 9.43457e11 1.03549 0.517743 − 0.855536i \(-0.326772\pi\)
0.517743 + 0.855536i \(0.326772\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.27723e12 1.37909
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −2.67220e12 −2.83873
\(986\) 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
\(997\) −8.16308e10 −0.0826178 −0.0413089 − 0.999146i \(-0.513153\pi\)
−0.0413089 + 0.999146i \(0.513153\pi\)
\(998\) 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 64.9.c.a.63.1 1
4.3 odd 2 CM 64.9.c.a.63.1 1
8.3 odd 2 4.9.b.a.3.1 ✓ 1
8.5 even 2 4.9.b.a.3.1 ✓ 1
16.3 odd 4 256.9.d.a.127.1 2
16.5 even 4 256.9.d.a.127.2 2
16.11 odd 4 256.9.d.a.127.2 2
16.13 even 4 256.9.d.a.127.1 2
24.5 odd 2 36.9.d.a.19.1 1
24.11 even 2 36.9.d.a.19.1 1
40.3 even 4 100.9.d.a.99.1 2
40.13 odd 4 100.9.d.a.99.1 2
40.19 odd 2 100.9.b.a.51.1 1
40.27 even 4 100.9.d.a.99.2 2
40.29 even 2 100.9.b.a.51.1 1
40.37 odd 4 100.9.d.a.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4.9.b.a.3.1 ✓ 1 8.3 odd 2
4.9.b.a.3.1 ✓ 1 8.5 even 2
36.9.d.a.19.1 1 24.5 odd 2
36.9.d.a.19.1 1 24.11 even 2
64.9.c.a.63.1 1 1.1 even 1 trivial
64.9.c.a.63.1 1 4.3 odd 2 CM
100.9.b.a.51.1 1 40.19 odd 2
100.9.b.a.51.1 1 40.29 even 2
100.9.d.a.99.1 2 40.3 even 4
100.9.d.a.99.1 2 40.13 odd 4
100.9.d.a.99.2 2 40.27 even 4
100.9.d.a.99.2 2 40.37 odd 4
256.9.d.a.127.1 2 16.3 odd 4
256.9.d.a.127.1 2 16.13 even 4
256.9.d.a.127.2 2 16.5 even 4
256.9.d.a.127.2 2 16.11 odd 4