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Properties

Label 176.9.h.a.65.1
Level $176$
Weight $9$
Character 176.65
Self dual yes
Analytic conductor $71.699$
Analytic rank $0$
Dimension $1$
CM discriminant -11
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] = [176,9,Mod(65,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.65");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 176.h (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: \(71.6986353708\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.1
Character \(\chi\) \(=\) 176.65

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+113.000 q^{3} +1151.00 q^{5} +6208.00 q^{9} +O(q^{10})\) \(q+113.000 q^{3} +1151.00 q^{5} +6208.00 q^{9} -14641.0 q^{11} +130063. q^{15} +531793. q^{23} +934176. q^{25} -39889.0 q^{27} +1.54123e6 q^{31} -1.65443e6 q^{33} +716447. q^{37} +7.14541e6 q^{45} +6.08064e6 q^{47} +5.76480e6 q^{49} -1.52654e7 q^{53} -1.68518e7 q^{55} +4.10155e6 q^{59} -1.98068e7 q^{67} +6.00926e7 q^{69} -7.04309e6 q^{71} +1.05562e8 q^{75} -4.52381e7 q^{81} -8.41010e7 q^{89} +1.74159e8 q^{93} -8.11557e7 q^{97} -9.08913e7 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 113.000 1.39506 0.697531 − 0.716555i \(-0.254282\pi\)
0.697531 + 0.716555i \(0.254282\pi\)
\(4\) 0 0
\(5\) 1151.00 1.84160 0.920800 − 0.390035i \(-0.127537\pi\)
0.920800 + 0.390035i \(0.127537\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 6208.00 0.946197
\(10\) 0 0
\(11\) −14641.0 −1.00000
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 130063. 2.56915
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 531793. 1.90034 0.950170 − 0.311732i \(-0.100909\pi\)
0.950170 + 0.311732i \(0.100909\pi\)
\(24\) 0 0
\(25\) 934176. 2.39149
\(26\) 0 0
\(27\) −39889.0 −0.0750582
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.54123e6 1.66887 0.834433 − 0.551109i \(-0.185795\pi\)
0.834433 + 0.551109i \(0.185795\pi\)
\(32\) 0 0
\(33\) −1.65443e6 −1.39506
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 716447. 0.382276 0.191138 − 0.981563i \(-0.438782\pi\)
0.191138 + 0.981563i \(0.438782\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 7.14541e6 1.74252
\(46\) 0 0
\(47\) 6.08064e6 1.24611 0.623057 − 0.782177i \(-0.285890\pi\)
0.623057 + 0.782177i \(0.285890\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.52654e7 −1.93467 −0.967333 − 0.253511i \(-0.918415\pi\)
−0.967333 + 0.253511i \(0.918415\pi\)
\(54\) 0 0
\(55\) −1.68518e7 −1.84160
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.10155e6 0.338486 0.169243 − 0.985574i \(-0.445868\pi\)
0.169243 + 0.985574i \(0.445868\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −1.98068e7 −0.982911 −0.491456 − 0.870903i \(-0.663535\pi\)
−0.491456 + 0.870903i \(0.663535\pi\)
\(68\) 0 0
\(69\) 6.00926e7 2.65109
\(70\) 0 0
\(71\) −7.04309e6 −0.277159 −0.138580 − 0.990351i \(-0.544254\pi\)
−0.138580 + 0.990351i \(0.544254\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.05562e8 3.33628
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −4.52381e7 −1.05091
\(82\) 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
\(89\) −8.41010e7 −1.34042 −0.670210 − 0.742171i \(-0.733796\pi\)
−0.670210 + 0.742171i \(0.733796\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.74159e8 2.32817
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.11557e7 −0.916710 −0.458355 − 0.888769i \(-0.651561\pi\)
−0.458355 + 0.888769i \(0.651561\pi\)
\(98\) 0 0
\(99\) −9.08913e7 −0.946197
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 3.62784e6 0.0322329 0.0161164 − 0.999870i \(-0.494870\pi\)
0.0161164 + 0.999870i \(0.494870\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 8.09585e7 0.533299
\(112\) 0 0
\(113\) 1.01857e8 0.624710 0.312355 − 0.949966i \(-0.398882\pi\)
0.312355 + 0.949966i \(0.398882\pi\)
\(114\) 0 0
\(115\) 6.12094e8 3.49967
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.25627e8 2.56257
\(126\) 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
\(135\) −4.59122e7 −0.138227
\(136\) 0 0
\(137\) 3.63889e8 1.03297 0.516484 − 0.856297i \(-0.327241\pi\)
0.516484 + 0.856297i \(0.327241\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 6.87112e8 1.73841
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.51423e8 1.39506
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.77396e9 3.07338
\(156\) 0 0
\(157\) −1.20570e9 −1.98446 −0.992229 − 0.124428i \(-0.960290\pi\)
−0.992229 + 0.124428i \(0.960290\pi\)
\(158\) 0 0
\(159\) −1.72499e9 −2.69898
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −9.15081e8 −1.29631 −0.648155 − 0.761508i \(-0.724459\pi\)
−0.648155 + 0.761508i \(0.724459\pi\)
\(164\) 0 0
\(165\) −1.90425e9 −2.56915
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.63475e8 0.472208
\(178\) 0 0
\(179\) −1.21779e9 −1.18620 −0.593102 − 0.805127i \(-0.702097\pi\)
−0.593102 + 0.805127i \(0.702097\pi\)
\(180\) 0 0
\(181\) −2.13327e9 −1.98761 −0.993804 − 0.111149i \(-0.964547\pi\)
−0.993804 + 0.111149i \(0.964547\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 8.24630e8 0.704000
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.27581e8 0.246142 0.123071 − 0.992398i \(-0.460726\pi\)
0.123071 + 0.992398i \(0.460726\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.13584e9 −1.99960 −0.999798 − 0.0200992i \(-0.993602\pi\)
−0.999798 + 0.0200992i \(0.993602\pi\)
\(200\) 0 0
\(201\) −2.23816e9 −1.37122
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.30137e9 1.79810
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −7.95869e8 −0.386655
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −4.76951e9 −1.92865 −0.964326 − 0.264716i \(-0.914722\pi\)
−0.964326 + 0.264716i \(0.914722\pi\)
\(224\) 0 0
\(225\) 5.79936e9 2.26282
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 1.32380e9 0.481371 0.240686 − 0.970603i \(-0.422628\pi\)
0.240686 + 0.970603i \(0.422628\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 6.99881e9 2.29484
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −4.85020e9 −1.39102
\(244\) 0 0
\(245\) 6.63529e9 1.84160
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.39202e9 0.602658 0.301329 − 0.953520i \(-0.402570\pi\)
0.301329 + 0.953520i \(0.402570\pi\)
\(252\) 0 0
\(253\) −7.78598e9 −1.90034
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.90675e8 0.112476 0.0562382 − 0.998417i \(-0.482089\pi\)
0.0562382 + 0.998417i \(0.482089\pi\)
\(258\) 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
\(265\) −1.75705e10 −3.56288
\(266\) 0 0
\(267\) −9.50341e9 −1.86997
\(268\) 0 0
\(269\) −1.02851e10 −1.96427 −0.982135 − 0.188178i \(-0.939742\pi\)
−0.982135 + 0.188178i \(0.939742\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.36773e10 −2.39149
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 9.56797e9 1.57908
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 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
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) −9.17060e9 −1.27887
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 4.72089e9 0.623355
\(296\) 0 0
\(297\) 5.84015e8 0.0750582
\(298\) 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
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 4.09946e8 0.0449668
\(310\) 0 0
\(311\) 1.74820e10 1.86874 0.934369 − 0.356306i \(-0.115964\pi\)
0.934369 + 0.356306i \(0.115964\pi\)
\(312\) 0 0
\(313\) −3.39209e9 −0.353419 −0.176710 − 0.984263i \(-0.556545\pi\)
−0.176710 + 0.984263i \(0.556545\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.66499e10 1.64882 0.824411 − 0.565991i \(-0.191506\pi\)
0.824411 + 0.565991i \(0.191506\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.44332e10 1.20240 0.601202 − 0.799097i \(-0.294689\pi\)
0.601202 + 0.799097i \(0.294689\pi\)
\(332\) 0 0
\(333\) 4.44770e9 0.361709
\(334\) 0 0
\(335\) −2.27976e10 −1.81013
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 1.15099e10 0.871508
\(340\) 0 0
\(341\) −2.25652e10 −1.66887
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.91666e10 4.88225
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.10589e10 −0.712216 −0.356108 − 0.934445i \(-0.615896\pi\)
−0.356108 + 0.934445i \(0.615896\pi\)
\(354\) 0 0
\(355\) −8.10659e9 −0.510417
\(356\) 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
\(363\) 2.42226e10 1.39506
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.83672e10 1.01246 0.506232 − 0.862397i \(-0.331038\pi\)
0.506232 + 0.862397i \(0.331038\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 7.06959e10 3.57494
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.14149e10 −1.52258 −0.761288 − 0.648414i \(-0.775433\pi\)
−0.761288 + 0.648414i \(0.775433\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.34982e10 −1.55678 −0.778389 − 0.627782i \(-0.783963\pi\)
−0.778389 + 0.627782i \(0.783963\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.57861e10 −1.99956 −0.999781 − 0.0209279i \(-0.993338\pi\)
−0.999781 + 0.0209279i \(0.993338\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.57269e10 1.84081 0.920407 − 0.390963i \(-0.127858\pi\)
0.920407 + 0.390963i \(0.127858\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.56288e10 1.76466 0.882332 − 0.470628i \(-0.155973\pi\)
0.882332 + 0.470628i \(0.155973\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −5.20691e10 −1.93535
\(406\) 0 0
\(407\) −1.04895e10 −0.382276
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 4.11195e10 1.44105
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.81505e10 −0.588887 −0.294444 − 0.955669i \(-0.595134\pi\)
−0.294444 + 0.955669i \(0.595134\pi\)
\(420\) 0 0
\(421\) −2.43806e10 −0.776097 −0.388048 − 0.921639i \(-0.626851\pi\)
−0.388048 + 0.921639i \(0.626851\pi\)
\(422\) 0 0
\(423\) 3.77486e10 1.17907
\(424\) 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
\(433\) 6.98359e10 1.98668 0.993338 − 0.115233i \(-0.0367614\pi\)
0.993338 + 0.115233i \(0.0367614\pi\)
\(434\) 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
\(441\) 3.57879e10 0.946197
\(442\) 0 0
\(443\) 3.92076e10 1.01802 0.509009 − 0.860761i \(-0.330012\pi\)
0.509009 + 0.860761i \(0.330012\pi\)
\(444\) 0 0
\(445\) −9.68002e10 −2.46852
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.10587e10 −1.99441 −0.997205 − 0.0747142i \(-0.976196\pi\)
−0.997205 + 0.0747142i \(0.976196\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −9.08592e10 −1.97717 −0.988587 − 0.150648i \(-0.951864\pi\)
−0.988587 + 0.150648i \(0.951864\pi\)
\(464\) 0 0
\(465\) 2.00457e11 4.28756
\(466\) 0 0
\(467\) −8.08102e10 −1.69902 −0.849510 − 0.527573i \(-0.823102\pi\)
−0.849510 + 0.527573i \(0.823102\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −1.36244e11 −2.76844
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −9.47678e10 −1.83057
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9.34102e10 −1.68821
\(486\) 0 0
\(487\) 2.76502e10 0.491566 0.245783 − 0.969325i \(-0.420955\pi\)
0.245783 + 0.969325i \(0.420955\pi\)
\(488\) 0 0
\(489\) −1.03404e11 −1.80843
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.04616e11 −1.74252
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.05616e11 1.70345 0.851723 − 0.523993i \(-0.175558\pi\)
0.851723 + 0.523993i \(0.175558\pi\)
\(500\) 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
\(507\) 9.21776e10 1.39506
\(508\) 0 0
\(509\) 1.11658e11 1.66348 0.831742 − 0.555162i \(-0.187344\pi\)
0.831742 + 0.555162i \(0.187344\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.17564e9 0.0593601
\(516\) 0 0
\(517\) −8.90266e10 −1.24611
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.31834e11 −1.78927 −0.894633 − 0.446801i \(-0.852563\pi\)
−0.894633 + 0.446801i \(0.852563\pi\)
\(522\) 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
\(529\) 2.04493e11 2.61129
\(530\) 0 0
\(531\) 2.54624e10 0.320274
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.37610e11 −1.65483
\(538\) 0 0
\(539\) −8.44025e10 −1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −2.41059e11 −2.77284
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 9.31832e10 0.982123
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 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
\(565\) 1.17238e11 1.15047
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 3.70166e10 0.343383
\(574\) 0 0
\(575\) 4.96788e11 4.54464
\(576\) 0 0
\(577\) −2.17251e10 −0.196001 −0.0980006 − 0.995186i \(-0.531245\pi\)
−0.0980006 + 0.995186i \(0.531245\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.23501e11 1.93467
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.08009e11 −0.909716 −0.454858 − 0.890564i \(-0.650310\pi\)
−0.454858 + 0.890564i \(0.650310\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 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
\(597\) −3.54350e11 −2.78956
\(598\) 0 0
\(599\) −2.43785e11 −1.89365 −0.946824 − 0.321751i \(-0.895729\pi\)
−0.946824 + 0.321751i \(0.895729\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −1.22960e11 −0.930028
\(604\) 0 0
\(605\) 2.46727e11 1.84160
\(606\) 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
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.89891e10 −0.407035 −0.203517 − 0.979071i \(-0.565237\pi\)
−0.203517 + 0.979071i \(0.565237\pi\)
\(618\) 0 0
\(619\) −2.89838e11 −1.97420 −0.987102 − 0.160093i \(-0.948821\pi\)
−0.987102 + 0.160093i \(0.948821\pi\)
\(620\) 0 0
\(621\) −2.12127e10 −0.142636
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 3.55184e11 2.32774
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.38624e11 −0.874422 −0.437211 − 0.899359i \(-0.644034\pi\)
−0.437211 + 0.899359i \(0.644034\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4.37235e10 −0.262247
\(640\) 0 0
\(641\) −2.64901e11 −1.56910 −0.784552 − 0.620063i \(-0.787107\pi\)
−0.784552 + 0.620063i \(0.787107\pi\)
\(642\) 0 0
\(643\) −1.08190e11 −0.632914 −0.316457 − 0.948607i \(-0.602493\pi\)
−0.316457 + 0.948607i \(0.602493\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.21502e10 0.468804 0.234402 − 0.972140i \(-0.424687\pi\)
0.234402 + 0.972140i \(0.424687\pi\)
\(648\) 0 0
\(649\) −6.00508e10 −0.338486
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.15668e11 1.73611 0.868056 − 0.496466i \(-0.165369\pi\)
0.868056 + 0.496466i \(0.165369\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.74259e11 −0.912827 −0.456413 − 0.889768i \(-0.650866\pi\)
−0.456413 + 0.889768i \(0.650866\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −5.38954e11 −2.69059
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −3.72633e10 −0.179501
\(676\) 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
\(683\) 3.39818e11 1.56158 0.780790 − 0.624794i \(-0.214817\pi\)
0.780790 + 0.624794i \(0.214817\pi\)
\(684\) 0 0
\(685\) 4.18837e11 1.90231
\(686\) 0 0
\(687\) 1.49589e11 0.671543
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 3.98526e11 1.74801 0.874007 − 0.485914i \(-0.161513\pi\)
0.874007 + 0.485914i \(0.161513\pi\)
\(692\) 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
\(703\) 0 0
\(704\) 0 0
\(705\) 7.90866e11 3.20145
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.92858e11 −1.95046 −0.975229 − 0.221198i \(-0.929003\pi\)
−0.975229 + 0.221198i \(0.929003\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.19617e11 3.17141
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.32961e11 1.62007 0.810035 − 0.586382i \(-0.199448\pi\)
0.810035 + 0.586382i \(0.199448\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −4.81038e10 −0.172203 −0.0861017 − 0.996286i \(-0.527441\pi\)
−0.0861017 + 0.996286i \(0.527441\pi\)
\(728\) 0 0
\(729\) −2.51265e11 −0.889655
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 7.49787e11 2.56915
\(736\) 0 0
\(737\) 2.89991e11 0.982911
\(738\) 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
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 6.03334e11 1.89670 0.948349 − 0.317228i \(-0.102752\pi\)
0.948349 + 0.317228i \(0.102752\pi\)
\(752\) 0 0
\(753\) 2.70299e11 0.840745
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.45563e11 1.96587 0.982935 − 0.183953i \(-0.0588894\pi\)
0.982935 + 0.183953i \(0.0588894\pi\)
\(758\) 0 0
\(759\) −8.79816e11 −2.65109
\(760\) 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
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 5.54463e10 0.156912
\(772\) 0 0
\(773\) −2.49173e11 −0.697884 −0.348942 − 0.937144i \(-0.613459\pi\)
−0.348942 + 0.937144i \(0.613459\pi\)
\(774\) 0 0
\(775\) 1.43978e12 3.99108
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 1.03118e11 0.277159
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.38776e12 −3.65458
\(786\) 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
\(793\) 0 0
\(794\) 0 0
\(795\) −1.98547e12 −4.97044
\(796\) 0 0
\(797\) 7.35104e11 1.82186 0.910931 − 0.412560i \(-0.135365\pi\)
0.910931 + 0.412560i \(0.135365\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −5.22099e11 −1.26830
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.16222e12 −2.74028
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.05326e12 −2.38729
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 3.01900e11 0.658058 0.329029 − 0.944320i \(-0.393279\pi\)
0.329029 + 0.944320i \(0.393279\pi\)
\(824\) 0 0
\(825\) −1.54553e12 −3.33628
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −4.44761e11 −0.941692 −0.470846 − 0.882215i \(-0.656051\pi\)
−0.470846 + 0.882215i \(0.656051\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.14782e10 −0.125262
\(838\) 0 0
\(839\) −6.66474e11 −1.34504 −0.672520 − 0.740079i \(-0.734788\pi\)
−0.672520 + 0.740079i \(0.734788\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.38906e11 1.84160
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.81001e11 0.726455
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 2.74046e11 0.503328 0.251664 − 0.967815i \(-0.419022\pi\)
0.251664 + 0.967815i \(0.419022\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.92719e11 −1.06858 −0.534288 − 0.845303i \(-0.679420\pi\)
−0.534288 + 0.845303i \(0.679420\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.88261e11 1.39506
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −5.03815e11 −0.867389
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −7.70772e11 −1.27945 −0.639724 − 0.768605i \(-0.720951\pi\)
−0.639724 + 0.768605i \(0.720951\pi\)
\(882\) 0 0
\(883\) 1.11501e12 1.83415 0.917077 − 0.398710i \(-0.130542\pi\)
0.917077 + 0.398710i \(0.130542\pi\)
\(884\) 0 0
\(885\) 5.33460e11 0.869619
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6.62332e11 1.05091
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −1.40167e12 −2.18451
\(896\) 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
\(905\) −2.45539e12 −3.66038
\(906\) 0 0
\(907\) −6.54959e11 −0.967798 −0.483899 − 0.875124i \(-0.660780\pi\)
−0.483899 + 0.875124i \(0.660780\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.76790e11 −1.27298 −0.636491 − 0.771284i \(-0.719615\pi\)
−0.636491 + 0.771284i \(0.719615\pi\)
\(912\) 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
\(925\) 6.69288e11 0.914210
\(926\) 0 0
\(927\) 2.25216e10 0.0304987
\(928\) 0 0
\(929\) −8.36302e11 −1.12279 −0.561397 − 0.827547i \(-0.689736\pi\)
−0.561397 + 0.827547i \(0.689736\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.97546e12 2.60701
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) −3.83306e11 −0.493042
\(940\) 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
\(947\) 1.60673e12 1.99776 0.998882 − 0.0472733i \(-0.0150532\pi\)
0.998882 + 0.0472733i \(0.0150532\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 1.88144e12 2.30021
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 3.77045e11 0.453294
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.52251e12 1.78511
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1.75196e12 −1.97083 −0.985413 − 0.170178i \(-0.945566\pi\)
−0.985413 + 0.170178i \(0.945566\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.65543e12 −1.81691 −0.908455 − 0.417982i \(-0.862737\pi\)
−0.908455 + 0.417982i \(0.862737\pi\)
\(978\) 0 0
\(979\) 1.23132e12 1.34042
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.71492e11 −0.504964 −0.252482 − 0.967602i \(-0.581247\pi\)
−0.252482 + 0.967602i \(0.581247\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.63892e12 1.69927 0.849635 − 0.527371i \(-0.176822\pi\)
0.849635 + 0.527371i \(0.176822\pi\)
\(992\) 0 0
\(993\) 1.63095e12 1.67743
\(994\) 0 0
\(995\) −3.60936e12 −3.68246
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −2.85784e10 −0.0286930
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 176.9.h.a.65.1 1
4.3 odd 2 11.9.b.a.10.1 ✓ 1
11.10 odd 2 CM 176.9.h.a.65.1 1
12.11 even 2 99.9.c.a.10.1 1
44.43 even 2 11.9.b.a.10.1 ✓ 1
132.131 odd 2 99.9.c.a.10.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.9.b.a.10.1 ✓ 1 4.3 odd 2
11.9.b.a.10.1 ✓ 1 44.43 even 2
99.9.c.a.10.1 1 12.11 even 2
99.9.c.a.10.1 1 132.131 odd 2
176.9.h.a.65.1 1 1.1 even 1 trivial
176.9.h.a.65.1 1 11.10 odd 2 CM