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Properties

Label 32.8.a.a.1.1
Level $32$
Weight $8$
Character 32.1
Self dual yes
Analytic conductor $9.996$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [32,8,Mod(1,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 32.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: \(9.99632081549\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 32.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-58.0000 q^{5} -2187.00 q^{9} +O(q^{10})\) \(q-58.0000 q^{5} -2187.00 q^{9} -8898.00 q^{13} -40094.0 q^{17} -74761.0 q^{25} +233230. q^{29} +563974. q^{37} +9530.00 q^{41} +126846. q^{45} -823543. q^{49} -798602. q^{53} -3.50533e6 q^{61} +516084. q^{65} +3.91742e6 q^{73} +4.78297e6 q^{81} +2.32545e6 q^{85} +9.24617e6 q^{89} -1.75674e7 q^{97} +O(q^{100})\)

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
\(5\) −58.0000 −0.207507 −0.103754 − 0.994603i \(-0.533085\pi\)
−0.103754 + 0.994603i \(0.533085\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −2187.00 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −8898.00 −1.12329 −0.561643 − 0.827379i \(-0.689831\pi\)
−0.561643 + 0.827379i \(0.689831\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −40094.0 −1.97928 −0.989642 − 0.143557i \(-0.954146\pi\)
−0.989642 + 0.143557i \(0.954146\pi\)
\(18\) 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
\(25\) −74761.0 −0.956941
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 233230. 1.77579 0.887895 − 0.460047i \(-0.152167\pi\)
0.887895 + 0.460047i \(0.152167\pi\)
\(30\) 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
\(37\) 563974. 1.83043 0.915215 − 0.402966i \(-0.132021\pi\)
0.915215 + 0.402966i \(0.132021\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9530.00 0.0215948 0.0107974 − 0.999942i \(-0.496563\pi\)
0.0107974 + 0.999942i \(0.496563\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 126846. 0.207507
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −823543. −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −798602. −0.736826 −0.368413 − 0.929662i \(-0.620099\pi\)
−0.368413 + 0.929662i \(0.620099\pi\)
\(54\) 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
\(61\) −3.50533e6 −1.97731 −0.988654 − 0.150208i \(-0.952006\pi\)
−0.988654 + 0.150208i \(0.952006\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 516084. 0.233090
\(66\) 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
\(73\) 3.91742e6 1.17861 0.589305 − 0.807911i \(-0.299402\pi\)
0.589305 + 0.807911i \(0.299402\pi\)
\(74\) 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
\(81\) 4.78297e6 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 2.32545e6 0.410716
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.24617e6 1.39026 0.695131 − 0.718883i \(-0.255346\pi\)
0.695131 + 0.718883i \(0.255346\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.75674e7 −1.95437 −0.977185 − 0.212392i \(-0.931875\pi\)
−0.977185 + 0.212392i \(0.931875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.33042e7 1.28488 0.642442 − 0.766334i \(-0.277921\pi\)
0.642442 + 0.766334i \(0.277921\pi\)
\(102\) 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
\(109\) −2.41162e7 −1.78368 −0.891839 − 0.452352i \(-0.850585\pi\)
−0.891839 + 0.452352i \(0.850585\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.91247e7 −1.89883 −0.949417 − 0.314017i \(-0.898325\pi\)
−0.949417 + 0.314017i \(0.898325\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.94599e7 1.12329
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.94872e7 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.86739e6 0.406079
\(126\) 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
\(137\) 4.60312e7 1.52943 0.764716 − 0.644367i \(-0.222879\pi\)
0.764716 + 0.644367i \(0.222879\pi\)
\(138\) 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
\(145\) −1.35273e7 −0.368489
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.31577e7 1.81179 0.905895 − 0.423502i \(-0.139199\pi\)
0.905895 + 0.423502i \(0.139199\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 8.76856e7 1.97928
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −9.09715e7 −1.87610 −0.938051 − 0.346497i \(-0.887371\pi\)
−0.938051 + 0.346497i \(0.887371\pi\)
\(158\) 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
\(169\) 1.64259e7 0.261773
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.55688e7 −0.962800 −0.481400 − 0.876501i \(-0.659871\pi\)
−0.481400 + 0.876501i \(0.659871\pi\)
\(174\) 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
\(181\) −1.45860e8 −1.82836 −0.914180 − 0.405309i \(-0.867164\pi\)
−0.914180 + 0.405309i \(0.867164\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.27105e7 −0.379827
\(186\) 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
\(193\) −1.05243e8 −1.05377 −0.526883 − 0.849938i \(-0.676639\pi\)
−0.526883 + 0.849938i \(0.676639\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.02733e8 0.957363 0.478681 − 0.877989i \(-0.341115\pi\)
0.478681 + 0.877989i \(0.341115\pi\)
\(198\) 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
\(205\) −552740. −0.00448108
\(206\) 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
\(221\) 3.56756e8 2.22330
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 1.63502e8 0.956941
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 2.17910e8 1.19909 0.599547 − 0.800339i \(-0.295347\pi\)
0.599547 + 0.800339i \(0.295347\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.90325e8 −1.50362 −0.751811 − 0.659379i \(-0.770819\pi\)
−0.751811 + 0.659379i \(0.770819\pi\)
\(234\) 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
\(241\) 1.08961e8 0.501433 0.250717 − 0.968061i \(-0.419334\pi\)
0.250717 + 0.968061i \(0.419334\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.77655e7 0.207507
\(246\) 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
\(257\) −2.30304e8 −0.846323 −0.423161 − 0.906054i \(-0.639080\pi\)
−0.423161 + 0.906054i \(0.639080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −5.10074e8 −1.77579
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 4.63189e7 0.152897
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.80336e7 0.244427 0.122213 − 0.992504i \(-0.461001\pi\)
0.122213 + 0.992504i \(0.461001\pi\)
\(270\) 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
\(277\) −5.35122e8 −1.51277 −0.756386 − 0.654126i \(-0.773037\pi\)
−0.756386 + 0.654126i \(0.773037\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.34409e8 −1.70568 −0.852839 − 0.522174i \(-0.825121\pi\)
−0.852839 + 0.522174i \(0.825121\pi\)
\(282\) 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
\(289\) 1.19719e9 2.91757
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.87629e8 −1.36479 −0.682396 − 0.730983i \(-0.739062\pi\)
−0.682396 + 0.730983i \(0.739062\pi\)
\(294\) 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
\(305\) 2.03309e8 0.410306
\(306\) 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
\(313\) 5.25457e8 0.968573 0.484286 − 0.874910i \(-0.339079\pi\)
0.484286 + 0.874910i \(0.339079\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.13287e9 1.99744 0.998720 − 0.0505827i \(-0.0161079\pi\)
0.998720 + 0.0505827i \(0.0161079\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 6.65223e8 1.07492
\(326\) 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
\(333\) −1.23341e9 −1.83043
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.05044e8 0.861157 0.430578 − 0.902553i \(-0.358310\pi\)
0.430578 + 0.902553i \(0.358310\pi\)
\(338\) 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
\(349\) 1.50468e9 1.89477 0.947385 − 0.320097i \(-0.103715\pi\)
0.947385 + 0.320097i \(0.103715\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.64963e9 −1.99607 −0.998035 − 0.0626535i \(-0.980044\pi\)
−0.998035 + 0.0626535i \(0.980044\pi\)
\(354\) 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
\(361\) −8.93872e8 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.27210e8 −0.244570
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) −2.08421e7 −0.0215948
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.03979e9 −1.03744 −0.518720 − 0.854944i \(-0.673591\pi\)
−0.518720 + 0.854944i \(0.673591\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.07528e9 −1.99472
\(378\) 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
\(389\) 1.94162e9 1.67240 0.836202 − 0.548422i \(-0.184771\pi\)
0.836202 + 0.548422i \(0.184771\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.34628e9 −1.07986 −0.539932 − 0.841709i \(-0.681550\pi\)
−0.539932 + 0.841709i \(0.681550\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −8.84817e8 −0.685248 −0.342624 − 0.939473i \(-0.611316\pi\)
−0.342624 + 0.939473i \(0.611316\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −2.77412e8 −0.207507
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 2.38968e9 1.72706 0.863531 − 0.504296i \(-0.168248\pi\)
0.863531 + 0.504296i \(0.168248\pi\)
\(410\) 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
\(421\) 1.58512e9 1.03532 0.517661 − 0.855586i \(-0.326803\pi\)
0.517661 + 0.855586i \(0.326803\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.99747e9 1.89406
\(426\) 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
\(433\) −1.34596e9 −0.796757 −0.398378 − 0.917221i \(-0.630427\pi\)
−0.398378 + 0.917221i \(0.630427\pi\)
\(434\) 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
\(441\) 1.80109e9 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −5.36278e8 −0.288489
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.71111e9 1.41346 0.706732 − 0.707481i \(-0.250168\pi\)
0.706732 + 0.707481i \(0.250168\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.02241e9 0.501095 0.250548 − 0.968104i \(-0.419389\pi\)
0.250548 + 0.968104i \(0.419389\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.07657e9 −1.93795 −0.968974 − 0.247164i \(-0.920501\pi\)
−0.968974 + 0.247164i \(0.920501\pi\)
\(462\) 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
\(477\) 1.74654e9 0.736826
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −5.01824e9 −2.05610
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.01891e9 0.405545
\(486\) 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
\(493\) −9.35112e9 −3.51479
\(494\) 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
\(505\) −7.71643e8 −0.266623
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.95022e9 1.99996 0.999979 − 0.00646007i \(-0.00205632\pi\)
0.999979 + 0.00646007i \(0.00205632\pi\)
\(510\) 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
\(521\) −2.38452e9 −0.738701 −0.369350 − 0.929290i \(-0.620420\pi\)
−0.369350 + 0.929290i \(0.620420\pi\)
\(522\) 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
\(529\) −3.40483e9 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.47979e7 −0.0242572
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 7.36235e9 1.99906 0.999530 − 0.0306534i \(-0.00975880\pi\)
0.999530 + 0.0306534i \(0.00975880\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.39874e9 0.370126
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 7.66616e9 1.97731
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.15369e9 −0.528067 −0.264034 − 0.964513i \(-0.585053\pi\)
−0.264034 + 0.964513i \(0.585053\pi\)
\(558\) 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
\(565\) 1.68923e9 0.394022
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 6.84617e9 1.55796 0.778978 − 0.627052i \(-0.215738\pi\)
0.778978 + 0.627052i \(0.215738\pi\)
\(570\) 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
\(577\) −2.65224e9 −0.574775 −0.287387 − 0.957814i \(-0.592787\pi\)
−0.287387 + 0.957814i \(0.592787\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −1.12868e9 −0.233090
\(586\) 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
\(593\) 7.08713e9 1.39566 0.697829 − 0.716265i \(-0.254150\pi\)
0.697829 + 0.716265i \(0.254150\pi\)
\(594\) 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
\(601\) −1.05496e10 −1.98233 −0.991165 − 0.132637i \(-0.957656\pi\)
−0.991165 + 0.132637i \(0.957656\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.13026e9 0.207507
\(606\) 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
\(613\) −9.50665e9 −1.66692 −0.833462 − 0.552577i \(-0.813645\pi\)
−0.833462 + 0.552577i \(0.813645\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.16779e9 −0.371551 −0.185776 − 0.982592i \(-0.559480\pi\)
−0.185776 + 0.982592i \(0.559480\pi\)
\(618\) 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
\(625\) 5.32639e9 0.872676
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.26120e10 −3.62294
\(630\) 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
\(637\) 7.32789e9 1.12329
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.92310e9 0.888272 0.444136 − 0.895959i \(-0.353511\pi\)
0.444136 + 0.895959i \(0.353511\pi\)
\(642\) 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
\(653\) −7.96894e9 −1.11997 −0.559983 − 0.828504i \(-0.689192\pi\)
−0.559983 + 0.828504i \(0.689192\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −8.56739e9 −1.17861
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 1.15742e9 0.155879 0.0779394 − 0.996958i \(-0.475166\pi\)
0.0779394 + 0.996958i \(0.475166\pi\)
\(662\) 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
\(673\) 1.54182e10 1.94976 0.974879 − 0.222734i \(-0.0714982\pi\)
0.974879 + 0.222734i \(0.0714982\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.29286e9 0.531724 0.265862 − 0.964011i \(-0.414343\pi\)
0.265862 + 0.964011i \(0.414343\pi\)
\(678\) 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
\(685\) −2.66981e9 −0.317368
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.10596e9 0.827667
\(690\) 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
\(697\) −3.82096e8 −0.0427423
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.77141e10 1.94226 0.971129 − 0.238556i \(-0.0766739\pi\)
0.971129 + 0.238556i \(0.0766739\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.64383e10 1.73219 0.866096 − 0.499877i \(-0.166621\pi\)
0.866096 + 0.499877i \(0.166621\pi\)
\(710\) 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
\(725\) −1.74365e10 −1.69933
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.85321e10 1.73804 0.869022 − 0.494773i \(-0.164749\pi\)
0.869022 + 0.494773i \(0.164749\pi\)
\(734\) 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
\(745\) −4.24315e9 −0.375959
\(746\) 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
\(757\) 1.72709e10 1.44703 0.723516 − 0.690307i \(-0.242525\pi\)
0.723516 + 0.690307i \(0.242525\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.99861e10 −1.64392 −0.821961 − 0.569543i \(-0.807120\pi\)
−0.821961 + 0.569543i \(0.807120\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5.08576e9 −0.410716
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.52206e10 −1.99992 −0.999960 − 0.00895271i \(-0.997150\pi\)
−0.999960 + 0.00895271i \(0.997150\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.55364e10 −1.98853 −0.994263 − 0.106964i \(-0.965887\pi\)
−0.994263 + 0.106964i \(0.965887\pi\)
\(774\) 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
\(785\) 5.27635e9 0.389305
\(786\) 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
\(793\) 3.11904e10 2.22108
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.53016e9 −0.666801 −0.333400 − 0.942785i \(-0.608196\pi\)
−0.333400 + 0.942785i \(0.608196\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −2.02214e10 −1.39026
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.84568e10 −1.88959 −0.944793 − 0.327668i \(-0.893737\pi\)
−0.944793 + 0.327668i \(0.893737\pi\)
\(810\) 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
\(821\) 2.88014e10 1.81641 0.908203 − 0.418530i \(-0.137454\pi\)
0.908203 + 0.418530i \(0.137454\pi\)
\(822\) 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
\(829\) −2.59446e10 −1.58163 −0.790817 − 0.612052i \(-0.790344\pi\)
−0.790817 + 0.612052i \(0.790344\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.30191e10 1.97928
\(834\) 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
\(841\) 3.71464e10 2.15343
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.52701e8 −0.0543198
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.32103e9 −0.128044 −0.0640220 − 0.997948i \(-0.520393\pi\)
−0.0640220 + 0.997948i \(0.520393\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.11516e10 1.14792 0.573960 − 0.818884i \(-0.305407\pi\)
0.573960 + 0.818884i \(0.305407\pi\)
\(858\) 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
\(865\) 3.80299e9 0.199788
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 3.84199e10 1.95437
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −5.68052e9 −0.284374 −0.142187 − 0.989840i \(-0.545413\pi\)
−0.142187 + 0.989840i \(0.545413\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.69850e10 −1.32956 −0.664778 − 0.747041i \(-0.731474\pi\)
−0.664778 + 0.747041i \(0.731474\pi\)
\(882\) 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
\(901\) 3.20191e10 1.45839
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.45989e9 0.379398
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −2.90963e10 −1.28488
\(910\) 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
\(925\) −4.21633e10 −1.75161
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.43415e10 −0.586866 −0.293433 − 0.955980i \(-0.594798\pi\)
−0.293433 + 0.955980i \(0.594798\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.03482e10 −1.99938 −0.999690 − 0.0249138i \(-0.992069\pi\)
−0.999690 + 0.0249138i \(0.992069\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.49808e10 1.36857 0.684283 − 0.729217i \(-0.260115\pi\)
0.684283 + 0.729217i \(0.260115\pi\)
\(942\) 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
\(949\) −3.48572e10 −1.32392
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.27653e9 −0.197480 −0.0987400 − 0.995113i \(-0.531481\pi\)
−0.0987400 + 0.995113i \(0.531481\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.75126e10 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6.10411e9 0.218664
\(966\) 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
\(977\) −3.63185e10 −1.24594 −0.622970 − 0.782246i \(-0.714074\pi\)
−0.622970 + 0.782246i \(0.714074\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.27422e10 1.78368
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −5.95849e9 −0.198660
\(986\) 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
\(997\) 1.44203e10 0.460832 0.230416 − 0.973092i \(-0.425991\pi\)
0.230416 + 0.973092i \(0.425991\pi\)
\(998\) 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 32.8.a.a.1.1 ✓ 1
3.2 odd 2 288.8.a.b.1.1 1
4.3 odd 2 CM 32.8.a.a.1.1 ✓ 1
8.3 odd 2 64.8.a.d.1.1 1
8.5 even 2 64.8.a.d.1.1 1
12.11 even 2 288.8.a.b.1.1 1
16.3 odd 4 256.8.b.d.129.2 2
16.5 even 4 256.8.b.d.129.1 2
16.11 odd 4 256.8.b.d.129.1 2
16.13 even 4 256.8.b.d.129.2 2
24.5 odd 2 576.8.a.n.1.1 1
24.11 even 2 576.8.a.n.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.8.a.a.1.1 ✓ 1 1.1 even 1 trivial
32.8.a.a.1.1 ✓ 1 4.3 odd 2 CM
64.8.a.d.1.1 1 8.3 odd 2
64.8.a.d.1.1 1 8.5 even 2
256.8.b.d.129.1 2 16.5 even 4
256.8.b.d.129.1 2 16.11 odd 4
256.8.b.d.129.2 2 16.3 odd 4
256.8.b.d.129.2 2 16.13 even 4
288.8.a.b.1.1 1 3.2 odd 2
288.8.a.b.1.1 1 12.11 even 2
576.8.a.n.1.1 1 24.5 odd 2
576.8.a.n.1.1 1 24.11 even 2