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Properties

Label 64.12.a.d.1.1
Level $64$
Weight $12$
Character 64.1
Self dual yes
Analytic conductor $49.174$
Analytic rank $1$
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] = [64,12,Mod(1,64)]
 
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([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 64.a (trivial)

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: \(49.1739635558\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 64.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+12938.0 q^{5} -177147. q^{9} +O(q^{10})\) \(q+12938.0 q^{5} -177147. q^{9} -2.63182e6 q^{13} -5.05929e6 q^{17} +1.18564e8 q^{25} -1.11071e8 q^{29} -8.18241e8 q^{37} +6.20973e8 q^{41} -2.29193e9 q^{45} -1.97733e9 q^{49} +6.06940e9 q^{53} -1.28980e10 q^{61} -3.40505e10 q^{65} -2.55324e10 q^{73} +3.13811e10 q^{81} -6.54571e10 q^{85} +1.45554e10 q^{89} -1.88921e10 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 12938.0 1.85154 0.925768 − 0.378092i \(-0.123420\pi\)
0.925768 + 0.378092i \(0.123420\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −177147. −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.63182e6 −1.96593 −0.982965 − 0.183793i \(-0.941163\pi\)
−0.982965 + 0.183793i \(0.941163\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.05929e6 −0.864213 −0.432107 − 0.901823i \(-0.642230\pi\)
−0.432107 + 0.901823i \(0.642230\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.18564e8 2.42818
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.11071e8 −1.00557 −0.502786 − 0.864411i \(-0.667692\pi\)
−0.502786 + 0.864411i \(0.667692\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.18241e8 −1.93987 −0.969933 − 0.243370i \(-0.921747\pi\)
−0.969933 + 0.243370i \(0.921747\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.20973e8 0.837069 0.418535 − 0.908201i \(-0.362544\pi\)
0.418535 + 0.908201i \(0.362544\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −2.29193e9 −1.85154
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −1.97733e9 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.06940e9 1.99356 0.996778 − 0.0802101i \(-0.0255591\pi\)
0.996778 + 0.0802101i \(0.0255591\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.28980e10 −1.95528 −0.977641 − 0.210279i \(-0.932563\pi\)
−0.977641 + 0.210279i \(0.932563\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.40505e10 −3.63999
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −2.55324e10 −1.44150 −0.720751 − 0.693194i \(-0.756203\pi\)
−0.720751 + 0.693194i \(0.756203\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 3.13811e10 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.54571e10 −1.60012
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.45554e10 0.276299 0.138150 − 0.990411i \(-0.455885\pi\)
0.138150 + 0.990411i \(0.455885\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.88921e10 −0.223376 −0.111688 − 0.993743i \(-0.535626\pi\)
−0.111688 + 0.993743i \(0.535626\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.87917e11 −1.77910 −0.889548 − 0.456841i \(-0.848981\pi\)
−0.889548 + 0.456841i \(0.848981\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) −2.06860e10 −0.128775 −0.0643875 − 0.997925i \(-0.520509\pi\)
−0.0643875 + 0.997925i \(0.520509\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.91154e11 1.99718 0.998588 − 0.0531233i \(-0.0169176\pi\)
0.998588 + 0.0531233i \(0.0169176\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.66219e11 1.96593
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.85312e11 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.02239e11 2.64434
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.67811e11 1.53625 0.768125 − 0.640300i \(-0.221190\pi\)
0.768125 + 0.640300i \(0.221190\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.43704e12 −1.86185
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.55219e11 0.842459 0.421229 − 0.906954i \(-0.361599\pi\)
0.421229 + 0.906954i \(0.361599\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 8.96239e11 0.864213
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.68187e12 −1.40716 −0.703581 − 0.710615i \(-0.748417\pi\)
−0.703581 + 0.710615i \(0.748417\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 5.13433e12 2.86488
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.40806e11 −0.216269 −0.108134 − 0.994136i \(-0.534488\pi\)
−0.108134 + 0.994136i \(0.534488\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −5.11554e12 −1.95731 −0.978655 − 0.205510i \(-0.934115\pi\)
−0.978655 + 0.205510i \(0.934115\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.05864e13 −3.59173
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −3.39858e12 −0.913550 −0.456775 − 0.889582i \(-0.650996\pi\)
−0.456775 + 0.889582i \(0.650996\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.88352e12 −1.41278 −0.706388 − 0.707825i \(-0.749677\pi\)
−0.706388 + 0.707825i \(0.749677\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 8.03415e12 1.54986
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 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
\(221\) 1.33152e13 1.69898
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −2.10032e13 −2.42818
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −2.14409e12 −0.224982 −0.112491 − 0.993653i \(-0.535883\pi\)
−0.112491 + 0.993653i \(0.535883\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.04821e13 1.95396 0.976981 − 0.213328i \(-0.0684304\pi\)
0.976981 + 0.213328i \(0.0684304\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.42938e13 1.92487 0.962434 − 0.271515i \(-0.0875245\pi\)
0.962434 + 0.271515i \(0.0875245\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.55827e13 −1.85154
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.27874e13 −1.26783 −0.633917 − 0.773401i \(-0.718554\pi\)
−0.633917 + 0.773401i \(0.718554\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.96760e13 1.00557
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 7.85259e13 3.69114
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.76413e13 1.19652 0.598261 − 0.801301i \(-0.295858\pi\)
0.598261 + 0.801301i \(0.295858\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −8.28895e10 −0.00305394 −0.00152697 − 0.999999i \(-0.500486\pi\)
−0.00152697 + 0.999999i \(0.500486\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.11012e13 0.377995 0.188997 − 0.981978i \(-0.439476\pi\)
0.188997 + 0.981978i \(0.439476\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −8.67544e12 −0.253136
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.06148e13 0.557707 0.278854 − 0.960334i \(-0.410046\pi\)
0.278854 + 0.960334i \(0.410046\pi\)
\(294\) 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
\(305\) −1.66875e14 −3.62028
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −3.60128e13 −0.677584 −0.338792 − 0.940861i \(-0.610018\pi\)
−0.338792 + 0.940861i \(0.610018\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.84456e13 1.72731 0.863655 − 0.504083i \(-0.168169\pi\)
0.863655 + 0.504083i \(0.168169\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −3.12039e14 −4.77364
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 1.44949e14 1.93987
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.61851e13 1.20543 0.602717 − 0.797955i \(-0.294085\pi\)
0.602717 + 0.797955i \(0.294085\pi\)
\(338\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −3.10450e13 −0.320961 −0.160480 − 0.987039i \(-0.551304\pi\)
−0.160480 + 0.987039i \(0.551304\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.58582e13 0.251095 0.125547 − 0.992088i \(-0.459931\pi\)
0.125547 + 0.992088i \(0.459931\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −1.16490e14 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.30338e14 −2.66899
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −1.10003e14 −0.837069
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.24923e14 −1.61300 −0.806502 − 0.591231i \(-0.798642\pi\)
−0.806502 + 0.591231i \(0.798642\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.92320e14 1.97688
\(378\) 0 0
\(379\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.18754e14 1.81440 0.907198 − 0.420703i \(-0.138217\pi\)
0.907198 + 0.420703i \(0.138217\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.83249e14 1.95044 0.975222 − 0.221229i \(-0.0710070\pi\)
0.975222 + 0.221229i \(0.0710070\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.16891e14 −1.04459 −0.522297 − 0.852764i \(-0.674925\pi\)
−0.522297 + 0.852764i \(0.674925\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.06008e14 1.85154
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.61884e14 1.99551 0.997757 − 0.0669430i \(-0.0213246\pi\)
0.997757 + 0.0669430i \(0.0213246\pi\)
\(410\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 2.14462e14 0.790313 0.395156 − 0.918614i \(-0.370690\pi\)
0.395156 + 0.918614i \(0.370690\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.99849e14 −2.09847
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 5.62421e14 1.77573 0.887867 − 0.460099i \(-0.152186\pi\)
0.887867 + 0.460099i \(0.152186\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 3.50278e14 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 1.88318e14 0.511578
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.12000e14 −1.06547 −0.532736 − 0.846281i \(-0.678836\pi\)
−0.532736 + 0.846281i \(0.678836\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.08323e14 −0.958220 −0.479110 − 0.877755i \(-0.659040\pi\)
−0.479110 + 0.877755i \(0.659040\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.17221e14 −1.15697 −0.578483 − 0.815694i \(-0.696355\pi\)
−0.578483 + 0.815694i \(0.696355\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 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
\(477\) −1.07518e15 −1.99356
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 2.15347e15 3.81364
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.44426e14 −0.413588
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 5.61943e14 0.869029
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −2.43128e15 −3.29406
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.73371e14 −1.26279 −0.631394 − 0.775462i \(-0.717517\pi\)
−0.631394 + 0.775462i \(0.717517\pi\)
\(510\) 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
\(521\) −1.19696e15 −1.36607 −0.683035 − 0.730386i \(-0.739340\pi\)
−0.683035 + 0.730386i \(0.739340\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9.52810e14 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1.63429e15 −1.64562
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 3.77943e14 0.350623 0.175312 − 0.984513i \(-0.443907\pi\)
0.175312 + 0.984513i \(0.443907\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.67636e14 −0.238432
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 2.28485e15 1.95528
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.93211e15 −1.52696 −0.763482 − 0.645829i \(-0.776512\pi\)
−0.763482 + 0.645829i \(0.776512\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 5.06075e15 3.69784
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.61937e14 −0.113823 −0.0569114 − 0.998379i \(-0.518125\pi\)
−0.0569114 + 0.998379i \(0.518125\pi\)
\(570\) 0 0
\(571\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.35873e15 −0.884433 −0.442216 − 0.896908i \(-0.645808\pi\)
−0.442216 + 0.896908i \(0.645808\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 6.03195e15 3.63999
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.06223e15 1.71489 0.857445 − 0.514576i \(-0.172050\pi\)
0.857445 + 0.514576i \(0.172050\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −2.96859e15 −1.54433 −0.772166 − 0.635420i \(-0.780827\pi\)
−0.772166 + 0.635420i \(0.780827\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.69136e15 −1.85154
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.81903e15 −1.78205 −0.891025 − 0.453954i \(-0.850013\pi\)
−0.891025 + 0.453954i \(0.850013\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.86490e14 −0.309076 −0.154538 − 0.987987i \(-0.549389\pi\)
−0.154538 + 0.987987i \(0.549389\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 5.88393e15 2.46790
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.13972e15 1.67646
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.20397e15 1.96593
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.50868e14 −0.347057 −0.173529 − 0.984829i \(-0.555517\pi\)
−0.173529 + 0.984829i \(0.555517\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.43212e15 0.801610 0.400805 − 0.916163i \(-0.368731\pi\)
0.400805 + 0.916163i \(0.368731\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.52298e15 1.44150
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −5.52950e15 −1.70442 −0.852212 − 0.523197i \(-0.824739\pi\)
−0.852212 + 0.523197i \(0.824739\pi\)
\(662\) 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
\(673\) −3.90989e15 −1.09165 −0.545823 − 0.837900i \(-0.683783\pi\)
−0.545823 + 0.837900i \(0.683783\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.03705e15 −0.820756 −0.410378 − 0.911916i \(-0.634603\pi\)
−0.410378 + 0.911916i \(0.634603\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 1.12277e16 2.84442
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.59736e16 −3.91919
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.14168e15 −0.723406
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −7.78284e15 −1.73656 −0.868279 − 0.496077i \(-0.834773\pi\)
−0.868279 + 0.496077i \(0.834773\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.81423e15 0.589937 0.294968 − 0.955507i \(-0.404691\pi\)
0.294968 + 0.955507i \(0.404691\pi\)
\(710\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.31690e16 −2.44172
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −5.55906e15 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.87247e15 −1.37417 −0.687083 − 0.726579i \(-0.741109\pi\)
−0.687083 + 0.726579i \(0.741109\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 9.77103e15 1.55984
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.84542e15 1.00086 0.500429 − 0.865777i \(-0.333175\pi\)
0.500429 + 0.865777i \(0.333175\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.23345e16 1.75189 0.875943 − 0.482414i \(-0.160240\pi\)
0.875943 + 0.482414i \(0.160240\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.15955e16 1.60012
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 3.11402e15 0.417567 0.208783 − 0.977962i \(-0.433050\pi\)
0.208783 + 0.977962i \(0.433050\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.25153e16 −1.63100 −0.815498 − 0.578760i \(-0.803537\pi\)
−0.815498 + 0.578760i \(0.803537\pi\)
\(774\) 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
\(785\) −2.17600e16 −2.60541
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 3.39453e16 3.84395
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.72932e16 −1.90482 −0.952412 − 0.304813i \(-0.901406\pi\)
−0.952412 + 0.304813i \(0.901406\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2.57845e15 −0.276299
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.83572e16 −1.86248 −0.931238 − 0.364412i \(-0.881270\pi\)
−0.931238 + 0.364412i \(0.881270\pi\)
\(810\) 0 0
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.67836e16 1.57035 0.785177 − 0.619271i \(-0.212572\pi\)
0.785177 + 0.619271i \(0.212572\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 1.63399e16 1.44944 0.724719 − 0.689044i \(-0.241969\pi\)
0.724719 + 0.689044i \(0.241969\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.00039e16 0.864213
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 1.36344e14 0.0111753
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.64279e16 5.30443
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.37759e16 −1.04448 −0.522239 − 0.852799i \(-0.674903\pi\)
−0.522239 + 0.852799i \(0.674903\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.70588e15 0.126053 0.0630266 − 0.998012i \(-0.479925\pi\)
0.0630266 + 0.998012i \(0.479925\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −5.70315e15 −0.400430
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 3.34668e15 0.223376
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.91632e16 −1.24730 −0.623649 − 0.781705i \(-0.714350\pi\)
−0.623649 + 0.781705i \(0.714350\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 3.14995e16 1.99957 0.999785 − 0.0207387i \(-0.00660181\pi\)
0.999785 + 0.0207387i \(0.00660181\pi\)
\(882\) 0 0
\(883\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 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
\(901\) −3.07069e16 −1.72286
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.61849e16 −3.62403
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 3.32890e16 1.77910
\(910\) 0 0
\(911\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −9.70137e16 −4.71036
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.09734e14 0.0383933 0.0191967 − 0.999816i \(-0.493889\pi\)
0.0191967 + 0.999816i \(0.493889\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 3.90574e16 1.76659 0.883294 − 0.468819i \(-0.155320\pi\)
0.883294 + 0.468819i \(0.155320\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.94822e16 −1.74445 −0.872226 − 0.489103i \(-0.837324\pi\)
−0.872226 + 0.489103i \(0.837324\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 6.71966e16 2.83389
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.84180e16 1.99524 0.997622 − 0.0689287i \(-0.0219581\pi\)
0.997622 + 0.0689287i \(0.0219581\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.54085e16 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.39708e16 −1.69147
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −8.82404e15 −0.317137 −0.158569 − 0.987348i \(-0.550688\pi\)
−0.158569 + 0.987348i \(0.550688\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 3.66447e15 0.128775
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −7.61210e16 −2.61581
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.15685e16 1.01492 0.507458 − 0.861676i \(-0.330585\pi\)
0.507458 + 0.861676i \(0.330585\pi\)
\(998\) 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 64.12.a.d.1.1 1
4.3 odd 2 CM 64.12.a.d.1.1 1
8.3 odd 2 32.12.a.a.1.1 ✓ 1
8.5 even 2 32.12.a.a.1.1 ✓ 1
16.3 odd 4 256.12.b.d.129.1 2
16.5 even 4 256.12.b.d.129.2 2
16.11 odd 4 256.12.b.d.129.2 2
16.13 even 4 256.12.b.d.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.12.a.a.1.1 ✓ 1 8.3 odd 2
32.12.a.a.1.1 ✓ 1 8.5 even 2
64.12.a.d.1.1 1 1.1 even 1 trivial
64.12.a.d.1.1 1 4.3 odd 2 CM
256.12.b.d.129.1 2 16.3 odd 4
256.12.b.d.129.1 2 16.13 even 4
256.12.b.d.129.2 2 16.5 even 4
256.12.b.d.129.2 2 16.11 odd 4