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Properties

Label 32.6.a.b.1.1
Level $32$
Weight $6$
Character 32.1
Self dual yes
Analytic conductor $5.132$
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] = [32,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(5.13228223402\)
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)
 
\(f(q)\) \(=\) \(q-82.0000 q^{5} -243.000 q^{9} -1194.00 q^{13} +2242.00 q^{17} +3599.00 q^{25} +2950.00 q^{29} -12242.0 q^{37} -20950.0 q^{41} +19926.0 q^{45} -16807.0 q^{49} +7294.00 q^{53} +18950.0 q^{61} +97908.0 q^{65} -88806.0 q^{73} +59049.0 q^{81} -183844. q^{85} +51050.0 q^{89} -92142.0 q^{97} +O(q^{100})\)

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
\(5\) −82.0000 −1.46686 −0.733430 − 0.679765i \(-0.762082\pi\)
−0.733430 + 0.679765i \(0.762082\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) −243.000 −1.00000
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −1194.00 −1.95950 −0.979752 − 0.200217i \(-0.935835\pi\)
−0.979752 + 0.200217i \(0.935835\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2242.00 1.88154 0.940770 − 0.339046i \(-0.110104\pi\)
0.940770 + 0.339046i \(0.110104\pi\)
\(18\) 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
\(25\) 3599.00 1.15168
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2950.00 0.651369 0.325684 − 0.945479i \(-0.394405\pi\)
0.325684 + 0.945479i \(0.394405\pi\)
\(30\) 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
\(37\) −12242.0 −1.47010 −0.735052 − 0.678011i \(-0.762842\pi\)
−0.735052 + 0.678011i \(0.762842\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −20950.0 −1.94637 −0.973183 − 0.230033i \(-0.926116\pi\)
−0.973183 + 0.230033i \(0.926116\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 19926.0 1.46686
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −16807.0 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7294.00 0.356678 0.178339 − 0.983969i \(-0.442928\pi\)
0.178339 + 0.983969i \(0.442928\pi\)
\(54\) 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
\(61\) 18950.0 0.652056 0.326028 − 0.945360i \(-0.394290\pi\)
0.326028 + 0.945360i \(0.394290\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 97908.0 2.87432
\(66\) 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
\(73\) −88806.0 −1.95045 −0.975226 − 0.221212i \(-0.928999\pi\)
−0.975226 + 0.221212i \(0.928999\pi\)
\(74\) 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
\(81\) 59049.0 1.00000
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −183844. −2.75996
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 51050.0 0.683157 0.341579 − 0.939853i \(-0.389038\pi\)
0.341579 + 0.939853i \(0.389038\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −92142.0 −0.994325 −0.497162 − 0.867657i \(-0.665625\pi\)
−0.497162 + 0.867657i \(0.665625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −98002.0 −0.955942 −0.477971 − 0.878376i \(-0.658628\pi\)
−0.477971 + 0.878376i \(0.658628\pi\)
\(102\) 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
\(109\) 246486. 1.98713 0.993564 − 0.113269i \(-0.0361321\pi\)
0.993564 + 0.113269i \(0.0361321\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 118706. 0.874534 0.437267 − 0.899332i \(-0.355947\pi\)
0.437267 + 0.899332i \(0.355947\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 290142. 1.95950
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −161051. −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −38868.0 −0.222493
\(126\) 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
\(137\) 75658.0 0.344392 0.172196 − 0.985063i \(-0.444914\pi\)
0.172196 + 0.985063i \(0.444914\pi\)
\(138\) 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
\(145\) −241900. −0.955467
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 47614.0 0.175699 0.0878494 − 0.996134i \(-0.472001\pi\)
0.0878494 + 0.996134i \(0.472001\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −544806. −1.88154
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −493658. −1.59837 −0.799185 − 0.601086i \(-0.794735\pi\)
−0.799185 + 0.601086i \(0.794735\pi\)
\(158\) 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
\(169\) 1.05434e6 2.83965
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −568906. −1.44519 −0.722596 − 0.691271i \(-0.757051\pi\)
−0.722596 + 0.691271i \(0.757051\pi\)
\(174\) 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
\(181\) 439902. 0.998067 0.499033 − 0.866583i \(-0.333689\pi\)
0.499033 + 0.866583i \(0.333689\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.00384e6 2.15644
\(186\) 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
\(193\) −497294. −0.960992 −0.480496 − 0.876997i \(-0.659543\pi\)
−0.480496 + 0.876997i \(0.659543\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −380242. −0.698063 −0.349031 − 0.937111i \(-0.613489\pi\)
−0.349031 + 0.937111i \(0.613489\pi\)
\(198\) 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
\(205\) 1.71790e6 2.85505
\(206\) 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
\(221\) −2.67695e6 −3.68688
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −874557. −1.15168
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 1.25115e6 1.57660 0.788298 − 0.615293i \(-0.210962\pi\)
0.788298 + 0.615293i \(0.210962\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.53709e6 −1.85486 −0.927429 − 0.373999i \(-0.877986\pi\)
−0.927429 + 0.373999i \(0.877986\pi\)
\(234\) 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
\(241\) −477150. −0.529191 −0.264595 − 0.964360i \(-0.585238\pi\)
−0.264595 + 0.964360i \(0.585238\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.37817e6 1.46686
\(246\) 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
\(257\) 650242. 0.614104 0.307052 − 0.951693i \(-0.400657\pi\)
0.307052 + 0.951693i \(0.400657\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −716850. −0.651369
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −598108. −0.523197
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.35141e6 1.98129 0.990646 − 0.136458i \(-0.0435720\pi\)
0.990646 + 0.136458i \(0.0435720\pi\)
\(270\) 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
\(277\) −717858. −0.562133 −0.281067 − 0.959688i \(-0.590688\pi\)
−0.281067 + 0.959688i \(0.590688\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.64305e6 1.99682 0.998412 − 0.0563421i \(-0.0179438\pi\)
0.998412 + 0.0563421i \(0.0179438\pi\)
\(282\) 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
\(289\) 3.60671e6 2.54019
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.44939e6 −1.66682 −0.833412 − 0.552653i \(-0.813616\pi\)
−0.833412 + 0.552653i \(0.813616\pi\)
\(294\) 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
\(305\) −1.55390e6 −0.956475
\(306\) 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
\(313\) −2.88909e6 −1.66687 −0.833433 − 0.552620i \(-0.813628\pi\)
−0.833433 + 0.552620i \(0.813628\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −669658. −0.374287 −0.187144 − 0.982333i \(-0.559923\pi\)
−0.187144 + 0.982333i \(0.559923\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −4.29721e6 −2.25672
\(326\) 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
\(333\) 2.97481e6 1.47010
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.28386e6 1.09545 0.547727 − 0.836657i \(-0.315493\pi\)
0.547727 + 0.836657i \(0.315493\pi\)
\(338\) 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
\(349\) −4.44505e6 −1.95350 −0.976749 − 0.214385i \(-0.931225\pi\)
−0.976749 + 0.214385i \(0.931225\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.75261e6 −1.17573 −0.587865 − 0.808959i \(-0.700031\pi\)
−0.587865 + 0.808959i \(0.700031\pi\)
\(354\) 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
\(361\) −2.47610e6 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7.28209e6 2.86104
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 5.09085e6 1.94637
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 5.15929e6 1.92008 0.960038 − 0.279871i \(-0.0902918\pi\)
0.960038 + 0.279871i \(0.0902918\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.52230e6 −1.27636
\(378\) 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
\(389\) 5.28629e6 1.77124 0.885618 − 0.464414i \(-0.153735\pi\)
0.885618 + 0.464414i \(0.153735\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 259958. 0.0827803 0.0413901 − 0.999143i \(-0.486821\pi\)
0.0413901 + 0.999143i \(0.486821\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.59200e6 0.494405 0.247202 − 0.968964i \(-0.420489\pi\)
0.247202 + 0.968964i \(0.420489\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4.84202e6 −1.46686
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.58449e6 −1.35513 −0.677567 − 0.735461i \(-0.736966\pi\)
−0.677567 + 0.735461i \(0.736966\pi\)
\(410\) 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
\(421\) −5.94885e6 −1.63579 −0.817895 − 0.575367i \(-0.804859\pi\)
−0.817895 + 0.575367i \(0.804859\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 8.06896e6 2.16693
\(426\) 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
\(433\) −7.78461e6 −1.99534 −0.997670 − 0.0682249i \(-0.978266\pi\)
−0.997670 + 0.0682249i \(0.978266\pi\)
\(434\) 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
\(441\) 4.08410e6 1.00000
\(442\) 0 0
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −4.18610e6 −1.00210
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.48961e6 −1.98734 −0.993670 − 0.112340i \(-0.964165\pi\)
−0.993670 + 0.112340i \(0.964165\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.25844e6 1.17779 0.588893 − 0.808211i \(-0.299564\pi\)
0.588893 + 0.808211i \(0.299564\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.86580e6 −1.50466 −0.752331 − 0.658785i \(-0.771071\pi\)
−0.752331 + 0.658785i \(0.771071\pi\)
\(462\) 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
\(477\) −1.77244e6 −0.356678
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 1.46169e7 2.88067
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.55564e6 1.45854
\(486\) 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
\(493\) 6.61390e6 1.22558
\(494\) 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
\(505\) 8.03616e6 1.40223
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.05090e7 −1.79791 −0.898957 − 0.438037i \(-0.855674\pi\)
−0.898957 + 0.438037i \(0.855674\pi\)
\(510\) 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
\(521\) −7.27410e6 −1.17405 −0.587023 − 0.809570i \(-0.699700\pi\)
−0.587023 + 0.809570i \(0.699700\pi\)
\(522\) 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
\(529\) −6.43634e6 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.50143e7 3.81391
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.25380e6 1.21244 0.606221 − 0.795297i \(-0.292685\pi\)
0.606221 + 0.795297i \(0.292685\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.02119e7 −2.91484
\(546\) 0 0
\(547\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −4.60485e6 −0.652056
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.46360e7 −1.99888 −0.999438 − 0.0335347i \(-0.989324\pi\)
−0.999438 + 0.0335347i \(0.989324\pi\)
\(558\) 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
\(565\) −9.73389e6 −1.28282
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.96659e6 0.513613 0.256807 − 0.966463i \(-0.417330\pi\)
0.256807 + 0.966463i \(0.417330\pi\)
\(570\) 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
\(577\) 3.30624e6 0.413423 0.206712 − 0.978402i \(-0.433724\pi\)
0.206712 + 0.978402i \(0.433724\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −2.37916e7 −2.87432
\(586\) 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
\(593\) 1.75899e6 0.205413 0.102706 − 0.994712i \(-0.467250\pi\)
0.102706 + 0.994712i \(0.467250\pi\)
\(594\) 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
\(601\) 1.51550e7 1.71148 0.855739 − 0.517408i \(-0.173103\pi\)
0.855739 + 0.517408i \(0.173103\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.32062e7 1.46686
\(606\) 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
\(613\) 1.11506e7 1.19853 0.599263 − 0.800552i \(-0.295460\pi\)
0.599263 + 0.800552i \(0.295460\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.77140e7 1.87329 0.936644 − 0.350282i \(-0.113914\pi\)
0.936644 + 0.350282i \(0.113914\pi\)
\(618\) 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
\(625\) −8.05970e6 −0.825313
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.74466e7 −2.76606
\(630\) 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
\(637\) 2.00676e7 1.95950
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.45952e7 1.40303 0.701514 − 0.712655i \(-0.252508\pi\)
0.701514 + 0.712655i \(0.252508\pi\)
\(642\) 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
\(653\) −9.92891e6 −0.911210 −0.455605 − 0.890182i \(-0.650577\pi\)
−0.455605 + 0.890182i \(0.650577\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.15799e7 1.95045
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −8.60525e6 −0.766055 −0.383027 − 0.923737i \(-0.625119\pi\)
−0.383027 + 0.923737i \(0.625119\pi\)
\(662\) 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
\(673\) 1.73990e7 1.48077 0.740383 − 0.672185i \(-0.234644\pi\)
0.740383 + 0.672185i \(0.234644\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.55624e6 −0.382063 −0.191032 − 0.981584i \(-0.561183\pi\)
−0.191032 + 0.981584i \(0.561183\pi\)
\(678\) 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
\(685\) −6.20396e6 −0.505176
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.70904e6 −0.698911
\(690\) 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
\(697\) −4.69699e7 −3.66216
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.11650e7 −1.62676 −0.813381 − 0.581731i \(-0.802376\pi\)
−0.813381 + 0.581731i \(0.802376\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.98715e6 0.297884 0.148942 − 0.988846i \(-0.452413\pi\)
0.148942 + 0.988846i \(0.452413\pi\)
\(710\) 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
\(725\) 1.06170e7 0.750169
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.71275e7 1.86488 0.932438 − 0.361331i \(-0.117678\pi\)
0.932438 + 0.361331i \(0.117678\pi\)
\(734\) 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
\(745\) −3.90435e6 −0.257726
\(746\) 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
\(757\) −3.13899e7 −1.99090 −0.995450 − 0.0952804i \(-0.969625\pi\)
−0.995450 + 0.0952804i \(0.969625\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.95198e7 1.22184 0.610919 − 0.791693i \(-0.290800\pi\)
0.610919 + 0.791693i \(0.290800\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 4.46741e7 2.75996
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −2.02848e7 −1.23695 −0.618477 − 0.785803i \(-0.712250\pi\)
−0.618477 + 0.785803i \(0.712250\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.89461e6 0.294625 0.147312 − 0.989090i \(-0.452938\pi\)
0.147312 + 0.989090i \(0.452938\pi\)
\(774\) 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
\(785\) 4.04800e7 2.34458
\(786\) 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
\(793\) −2.26263e7 −1.27770
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.25943e7 1.81759 0.908796 − 0.417241i \(-0.137003\pi\)
0.908796 + 0.417241i \(0.137003\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.24052e7 −0.683157
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.87790e7 1.54598 0.772992 − 0.634415i \(-0.218759\pi\)
0.772992 + 0.634415i \(0.218759\pi\)
\(810\) 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
\(821\) 3.21148e7 1.66283 0.831413 − 0.555655i \(-0.187533\pi\)
0.831413 + 0.555655i \(0.187533\pi\)
\(822\) 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
\(829\) −7.96819e6 −0.402692 −0.201346 − 0.979520i \(-0.564532\pi\)
−0.201346 + 0.979520i \(0.564532\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.76813e7 −1.88154
\(834\) 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
\(841\) −1.18086e7 −0.575719
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.64561e7 −4.16537
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −4.18250e7 −1.96817 −0.984087 − 0.177690i \(-0.943138\pi\)
−0.984087 + 0.177690i \(0.943138\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.32921e7 1.54842 0.774210 − 0.632929i \(-0.218148\pi\)
0.774210 + 0.632929i \(0.218148\pi\)
\(858\) 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
\(865\) 4.66503e7 2.11989
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.23905e7 0.994325
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.38381e7 −1.04658 −0.523289 − 0.852155i \(-0.675295\pi\)
−0.523289 + 0.852155i \(0.675295\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.40848e7 −1.91359 −0.956794 − 0.290765i \(-0.906090\pi\)
−0.956794 + 0.290765i \(0.906090\pi\)
\(882\) 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
\(901\) 1.63531e7 0.671103
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.60720e7 −1.46402
\(906\) 0 0
\(907\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 2.38145e7 0.955942
\(910\) 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
\(925\) −4.40590e7 −1.69309
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.76633e7 1.81194 0.905972 − 0.423337i \(-0.139141\pi\)
0.905972 + 0.423337i \(0.139141\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.10260e7 0.410271 0.205135 − 0.978734i \(-0.434237\pi\)
0.205135 + 0.978734i \(0.434237\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.85570e6 0.178763 0.0893816 − 0.995997i \(-0.471511\pi\)
0.0893816 + 0.995997i \(0.471511\pi\)
\(942\) 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
\(949\) 1.06034e8 3.82192
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.61989e7 1.64778 0.823890 − 0.566749i \(-0.191799\pi\)
0.823890 + 0.566749i \(0.191799\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.86292e7 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.07781e7 1.40964
\(966\) 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
\(977\) −4.78045e7 −1.60226 −0.801130 − 0.598491i \(-0.795767\pi\)
−0.801130 + 0.598491i \(0.795767\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −5.98961e7 −1.98713
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 3.11798e7 1.02396
\(986\) 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
\(997\) 3.62105e7 1.15371 0.576856 − 0.816846i \(-0.304280\pi\)
0.576856 + 0.816846i \(0.304280\pi\)
\(998\) 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 32.6.a.b.1.1 ✓ 1
3.2 odd 2 288.6.a.i.1.1 1
4.3 odd 2 CM 32.6.a.b.1.1 ✓ 1
5.2 odd 4 800.6.c.c.449.2 2
5.3 odd 4 800.6.c.c.449.1 2
5.4 even 2 800.6.a.d.1.1 1
8.3 odd 2 64.6.a.d.1.1 1
8.5 even 2 64.6.a.d.1.1 1
12.11 even 2 288.6.a.i.1.1 1
16.3 odd 4 256.6.b.e.129.2 2
16.5 even 4 256.6.b.e.129.1 2
16.11 odd 4 256.6.b.e.129.1 2
16.13 even 4 256.6.b.e.129.2 2
20.3 even 4 800.6.c.c.449.1 2
20.7 even 4 800.6.c.c.449.2 2
20.19 odd 2 800.6.a.d.1.1 1
24.5 odd 2 576.6.a.e.1.1 1
24.11 even 2 576.6.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.6.a.b.1.1 ✓ 1 1.1 even 1 trivial
32.6.a.b.1.1 ✓ 1 4.3 odd 2 CM
64.6.a.d.1.1 1 8.3 odd 2
64.6.a.d.1.1 1 8.5 even 2
256.6.b.e.129.1 2 16.5 even 4
256.6.b.e.129.1 2 16.11 odd 4
256.6.b.e.129.2 2 16.3 odd 4
256.6.b.e.129.2 2 16.13 even 4
288.6.a.i.1.1 1 3.2 odd 2
288.6.a.i.1.1 1 12.11 even 2
576.6.a.e.1.1 1 24.5 odd 2
576.6.a.e.1.1 1 24.11 even 2
800.6.a.d.1.1 1 5.4 even 2
800.6.a.d.1.1 1 20.19 odd 2
800.6.c.c.449.1 2 5.3 odd 4
800.6.c.c.449.1 2 20.3 even 4
800.6.c.c.449.2 2 5.2 odd 4
800.6.c.c.449.2 2 20.7 even 4