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Properties

Label 48.9.e.a.17.1
Level $48$
Weight $9$
Character 48.17
Self dual yes
Analytic conductor $19.554$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [48,9,Mod(17,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, 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("48.17");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.e (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: \(19.5541732829\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 17.1
Character \(\chi\) \(=\) 48.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-81.0000 q^{3} -4034.00 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q-81.0000 q^{3} -4034.00 q^{7} +6561.00 q^{9} -35806.0 q^{13} +258526. q^{19} +326754. q^{21} +390625. q^{25} -531441. q^{27} +1.80941e6 q^{31} +503522. q^{37} +2.90029e6 q^{39} -3.49219e6 q^{43} +1.05084e7 q^{49} -2.09406e7 q^{57} -2.38265e7 q^{61} -2.64671e7 q^{63} +5.42141e6 q^{67} +1.61693e7 q^{73} -3.16406e7 q^{75} +1.88870e7 q^{79} +4.30467e7 q^{81} +1.44441e8 q^{91} -1.46562e8 q^{93} +1.76908e8 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −81.0000 −1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −4034.00 −1.68013 −0.840067 − 0.542483i \(-0.817484\pi\)
−0.840067 + 0.542483i \(0.817484\pi\)
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −35806.0 −1.25367 −0.626834 − 0.779153i \(-0.715650\pi\)
−0.626834 + 0.779153i \(0.715650\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 258526. 1.98376 0.991882 − 0.127165i \(-0.0405878\pi\)
0.991882 + 0.127165i \(0.0405878\pi\)
\(20\) 0 0
\(21\) 326754. 1.68013
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) −531441. −1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.80941e6 1.95925 0.979624 − 0.200842i \(-0.0643678\pi\)
0.979624 + 0.200842i \(0.0643678\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 503522. 0.268665 0.134333 − 0.990936i \(-0.457111\pi\)
0.134333 + 0.990936i \(0.457111\pi\)
\(38\) 0 0
\(39\) 2.90029e6 1.25367
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3.49219e6 −1.02147 −0.510734 − 0.859739i \(-0.670626\pi\)
−0.510734 + 0.859739i \(0.670626\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.05084e7 1.82285
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.09406e7 −1.98376
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −2.38265e7 −1.72084 −0.860422 − 0.509583i \(-0.829800\pi\)
−0.860422 + 0.509583i \(0.829800\pi\)
\(62\) 0 0
\(63\) −2.64671e7 −1.68013
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.42141e6 0.269037 0.134519 − 0.990911i \(-0.457051\pi\)
0.134519 + 0.990911i \(0.457051\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.61693e7 0.569376 0.284688 − 0.958620i \(-0.408110\pi\)
0.284688 + 0.958620i \(0.408110\pi\)
\(74\) 0 0
\(75\) −3.16406e7 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.88870e7 0.484904 0.242452 − 0.970163i \(-0.422048\pi\)
0.242452 + 0.970163i \(0.422048\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.44441e8 2.10633
\(92\) 0 0
\(93\) −1.46562e8 −1.95925
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.76908e8 1.99830 0.999150 − 0.0412262i \(-0.0131264\pi\)
0.999150 + 0.0412262i \(0.0131264\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −4.44490e7 −0.394923 −0.197462 − 0.980311i \(-0.563270\pi\)
−0.197462 + 0.980311i \(0.563270\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.03181e8 1.43938 0.719692 − 0.694293i \(-0.244283\pi\)
0.719692 + 0.694293i \(0.244283\pi\)
\(110\) 0 0
\(111\) −4.07853e7 −0.268665
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.34923e8 −1.25367
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00562e8 1.53977 0.769883 − 0.638185i \(-0.220314\pi\)
0.769883 + 0.638185i \(0.220314\pi\)
\(128\) 0 0
\(129\) 2.82868e8 1.02147
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −1.04289e9 −3.33299
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −7.09431e8 −1.90043 −0.950213 − 0.311602i \(-0.899135\pi\)
−0.950213 + 0.311602i \(0.899135\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −8.51177e8 −1.82285
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −2.70234e8 −0.519796 −0.259898 − 0.965636i \(-0.583689\pi\)
−0.259898 + 0.965636i \(0.583689\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3.61735e7 0.0595376 0.0297688 − 0.999557i \(-0.490523\pi\)
0.0297688 + 0.999557i \(0.490523\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.17139e9 1.65940 0.829698 − 0.558212i \(-0.188512\pi\)
0.829698 + 0.558212i \(0.188512\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.66339e8 0.571682
\(170\) 0 0
\(171\) 1.69619e9 1.98376
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.57578e9 −1.68013
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.05268e9 −0.980801 −0.490400 − 0.871497i \(-0.663149\pi\)
−0.490400 + 0.871497i \(0.663149\pi\)
\(182\) 0 0
\(183\) 1.92995e9 1.72084
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.14383e9 1.68013
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 2.32670e9 1.67691 0.838457 − 0.544968i \(-0.183458\pi\)
0.838457 + 0.544968i \(0.183458\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.73472e9 −1.10616 −0.553080 − 0.833128i \(-0.686548\pi\)
−0.553080 + 0.833128i \(0.686548\pi\)
\(200\) 0 0
\(201\) −4.39134e8 −0.269037
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.83711e8 0.0926840 0.0463420 − 0.998926i \(-0.485244\pi\)
0.0463420 + 0.998926i \(0.485244\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.29914e9 −3.29180
\(218\) 0 0
\(219\) −1.30971e9 −0.569376
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.72732e9 1.91159 0.955797 − 0.294027i \(-0.0949957\pi\)
0.955797 + 0.294027i \(0.0949957\pi\)
\(224\) 0 0
\(225\) 2.56289e9 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 5.35877e9 1.94860 0.974302 − 0.225247i \(-0.0723189\pi\)
0.974302 + 0.225247i \(0.0723189\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.52985e9 −0.484904
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −5.56578e9 −1.64990 −0.824951 − 0.565204i \(-0.808797\pi\)
−0.824951 + 0.565204i \(0.808797\pi\)
\(242\) 0 0
\(243\) −3.48678e9 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.25678e9 −2.48698
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −2.03121e9 −0.451393
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 2.98709e9 0.553824 0.276912 − 0.960895i \(-0.410689\pi\)
0.276912 + 0.960895i \(0.410689\pi\)
\(272\) 0 0
\(273\) −1.16998e10 −2.10633
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 7.42807e9 1.26170 0.630852 − 0.775904i \(-0.282706\pi\)
0.630852 + 0.775904i \(0.282706\pi\)
\(278\) 0 0
\(279\) 1.18715e10 1.95925
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.03697e10 1.61667 0.808335 − 0.588723i \(-0.200369\pi\)
0.808335 + 0.588723i \(0.200369\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) −1.43296e10 −1.99830
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.40875e10 1.71620
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.41256e10 1.59020 0.795101 − 0.606478i \(-0.207418\pi\)
0.795101 + 0.606478i \(0.207418\pi\)
\(308\) 0 0
\(309\) 3.60037e9 0.394923
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.17006e10 1.21908 0.609541 − 0.792755i \(-0.291354\pi\)
0.609541 + 0.792755i \(0.291354\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.39867e10 −1.25367
\(326\) 0 0
\(327\) −1.64576e10 −1.43938
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.62495e10 1.35372 0.676858 − 0.736113i \(-0.263341\pi\)
0.676858 + 0.736113i \(0.263341\pi\)
\(332\) 0 0
\(333\) 3.30361e9 0.268665
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.57689e10 −1.99791 −0.998957 − 0.0456520i \(-0.985463\pi\)
−0.998957 + 0.0456520i \(0.985463\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.91355e10 −1.38249
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.96004e10 −1.99524 −0.997621 − 0.0689403i \(-0.978038\pi\)
−0.997621 + 0.0689403i \(0.978038\pi\)
\(350\) 0 0
\(351\) 1.90288e10 1.25367
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.98521e10 2.93532
\(362\) 0 0
\(363\) −1.73631e10 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.43056e10 −1.33981 −0.669903 − 0.742448i \(-0.733664\pi\)
−0.669903 + 0.742448i \(0.733664\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.94467e9 0.255448 0.127724 − 0.991810i \(-0.459233\pi\)
0.127724 + 0.991810i \(0.459233\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.49200e9 0.169245 0.0846227 − 0.996413i \(-0.473032\pi\)
0.0846227 + 0.996413i \(0.473032\pi\)
\(380\) 0 0
\(381\) −3.24455e10 −1.53977
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.29123e10 −1.02147
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.57611e10 −0.634491 −0.317245 − 0.948343i \(-0.602758\pi\)
−0.317245 + 0.948343i \(0.602758\pi\)
\(398\) 0 0
\(399\) 8.44744e10 3.33299
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.47876e10 −2.45624
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.26810e10 −1.88261 −0.941305 − 0.337556i \(-0.890400\pi\)
−0.941305 + 0.337556i \(0.890400\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 5.74639e10 1.90043
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.16089e10 −0.369541 −0.184771 − 0.982782i \(-0.559154\pi\)
−0.184771 + 0.982782i \(0.559154\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 9.61162e10 2.89125
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 6.51683e10 1.85389 0.926947 − 0.375192i \(-0.122423\pi\)
0.926947 + 0.375192i \(0.122423\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −6.75493e10 −1.81871 −0.909354 − 0.416024i \(-0.863423\pi\)
−0.909354 + 0.416024i \(0.863423\pi\)
\(440\) 0 0
\(441\) 6.89453e10 1.82285
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.18890e10 0.519796
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.72608e10 −0.624991 −0.312495 − 0.949919i \(-0.601165\pi\)
−0.312495 + 0.949919i \(0.601165\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 6.87853e10 1.49683 0.748413 − 0.663232i \(-0.230816\pi\)
0.748413 + 0.663232i \(0.230816\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −2.18700e10 −0.452019
\(470\) 0 0
\(471\) −2.93005e9 −0.0595376
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.00987e11 1.98376
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1.80291e10 −0.336817
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −9.09984e10 −1.61777 −0.808887 − 0.587965i \(-0.799929\pi\)
−0.808887 + 0.587965i \(0.799929\pi\)
\(488\) 0 0
\(489\) −9.48824e10 −1.65940
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.23330e11 −1.98915 −0.994574 − 0.104032i \(-0.966826\pi\)
−0.994574 + 0.104032i \(0.966826\pi\)
\(500\) 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
\(507\) −3.77735e10 −0.571682
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.52269e10 −0.956628
\(512\) 0 0
\(513\) −1.37391e11 −1.98376
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −2.44747e10 −0.327122 −0.163561 − 0.986533i \(-0.552298\pi\)
−0.163561 + 0.986533i \(0.552298\pi\)
\(524\) 0 0
\(525\) 1.27638e11 1.68013
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.83110e10 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 6.27323e10 0.732322 0.366161 − 0.930551i \(-0.380672\pi\)
0.366161 + 0.930551i \(0.380672\pi\)
\(542\) 0 0
\(543\) 8.52668e10 0.980801
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.18266e10 0.690599 0.345300 − 0.938492i \(-0.387777\pi\)
0.345300 + 0.938492i \(0.387777\pi\)
\(548\) 0 0
\(549\) −1.56326e11 −1.72084
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −7.61903e10 −0.814703
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 1.25041e11 1.28058
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.73650e11 −1.68013
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1.94939e11 1.83381 0.916907 − 0.399102i \(-0.130678\pi\)
0.916907 + 0.399102i \(0.130678\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.54299e11 −1.39206 −0.696031 − 0.718012i \(-0.745052\pi\)
−0.696031 + 0.718012i \(0.745052\pi\)
\(578\) 0 0
\(579\) −1.88462e11 −1.67691
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 4.67778e11 3.88668
\(590\) 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
\(597\) 1.40513e11 1.10616
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −6.22607e10 −0.477217 −0.238608 − 0.971116i \(-0.576691\pi\)
−0.238608 + 0.971116i \(0.576691\pi\)
\(602\) 0 0
\(603\) 3.55698e10 0.269037
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.65989e11 1.95933 0.979666 − 0.200634i \(-0.0643001\pi\)
0.979666 + 0.200634i \(0.0643001\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.57998e10 −0.111894 −0.0559472 − 0.998434i \(-0.517818\pi\)
−0.0559472 + 0.998434i \(0.517818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −2.25533e11 −1.53620 −0.768100 − 0.640330i \(-0.778797\pi\)
−0.768100 + 0.640330i \(0.778797\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 2.00069e11 1.26201 0.631004 − 0.775780i \(-0.282643\pi\)
0.631004 + 0.775780i \(0.282643\pi\)
\(632\) 0 0
\(633\) −1.48806e10 −0.0926840
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.76262e11 −2.28525
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.88544e11 −1.10298 −0.551490 − 0.834181i \(-0.685941\pi\)
−0.551490 + 0.834181i \(0.685941\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 5.91231e11 3.29180
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.06087e11 0.569376
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.56009e11 1.86490 0.932449 − 0.361302i \(-0.117668\pi\)
0.932449 + 0.361302i \(0.117668\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −3.82913e11 −1.91159
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.28934e11 −1.11597 −0.557983 − 0.829853i \(-0.688424\pi\)
−0.557983 + 0.829853i \(0.688424\pi\)
\(674\) 0 0
\(675\) −2.07594e11 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −7.13647e11 −3.35741
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4.34061e11 −1.94860
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.55801e11 −0.683374 −0.341687 − 0.939814i \(-0.610998\pi\)
−0.341687 + 0.939814i \(0.610998\pi\)
\(692\) 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
\(703\) 1.30174e11 0.532968
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −4.87686e11 −1.92999 −0.964995 − 0.262268i \(-0.915530\pi\)
−0.964995 + 0.262268i \(0.915530\pi\)
\(710\) 0 0
\(711\) 1.23918e11 0.484904
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 1.79307e11 0.663524
\(722\) 0 0
\(723\) 4.50828e11 1.64990
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.21500e11 −0.434951 −0.217475 − 0.976066i \(-0.569782\pi\)
−0.217475 + 0.976066i \(0.569782\pi\)
\(728\) 0 0
\(729\) 2.82430e11 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.77330e11 −1.99990 −0.999949 − 0.0100913i \(-0.996788\pi\)
−0.999949 + 0.0100913i \(0.996788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 4.35662e11 1.46074 0.730368 − 0.683054i \(-0.239349\pi\)
0.730368 + 0.683054i \(0.239349\pi\)
\(740\) 0 0
\(741\) 7.49799e11 2.48698
\(742\) 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
\(751\) 5.35831e11 1.68449 0.842244 − 0.539097i \(-0.181234\pi\)
0.842244 + 0.539097i \(0.181234\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −4.59764e11 −1.40008 −0.700038 − 0.714106i \(-0.746833\pi\)
−0.700038 + 0.714106i \(0.746833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −8.19631e11 −2.41836
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.70500e11 1.91732 0.958658 − 0.284562i \(-0.0918483\pi\)
0.958658 + 0.284562i \(0.0918483\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 7.06799e11 1.95925
\(776\) 0 0
\(777\) 1.64528e11 0.451393
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −6.75420e11 −1.76066 −0.880329 − 0.474364i \(-0.842678\pi\)
−0.880329 + 0.474364i \(0.842678\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 8.53133e11 2.15737
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −8.88545e10 −0.205398 −0.102699 − 0.994712i \(-0.532748\pi\)
−0.102699 + 0.994712i \(0.532748\pi\)
\(812\) 0 0
\(813\) −2.41954e11 −0.553824
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −9.02823e11 −2.02635
\(818\) 0 0
\(819\) 9.47680e11 2.10633
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 8.62186e11 1.87932 0.939662 − 0.342105i \(-0.111140\pi\)
0.939662 + 0.342105i \(0.111140\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 4.11968e11 0.872258 0.436129 − 0.899884i \(-0.356349\pi\)
0.436129 + 0.899884i \(0.356349\pi\)
\(830\) 0 0
\(831\) −6.01674e11 −1.26170
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −9.61593e11 −1.95925
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −8.64724e11 −1.68013
\(848\) 0 0
\(849\) −8.39948e11 −1.61667
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 6.14949e11 1.16156 0.580782 − 0.814059i \(-0.302747\pi\)
0.580782 + 0.814059i \(0.302747\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 8.02752e11 1.47438 0.737189 − 0.675686i \(-0.236153\pi\)
0.737189 + 0.675686i \(0.236153\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −5.65036e11 −1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −1.94119e11 −0.337284
\(872\) 0 0
\(873\) 1.16069e12 1.99830
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.10825e11 0.356389 0.178194 − 0.983995i \(-0.442974\pi\)
0.178194 + 0.983995i \(0.442974\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −5.72878e11 −0.942366 −0.471183 − 0.882035i \(-0.656173\pi\)
−0.471183 + 0.882035i \(0.656173\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1.61587e12 −2.58701
\(890\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) −1.14109e12 −1.71620
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.20490e12 −1.78042 −0.890212 − 0.455547i \(-0.849444\pi\)
−0.890212 + 0.455547i \(0.849444\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.44534e11 0.763419 0.381709 − 0.924282i \(-0.375336\pi\)
0.381709 + 0.924282i \(0.375336\pi\)
\(920\) 0 0
\(921\) −1.14417e12 −1.59020
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.96688e11 0.268665
\(926\) 0 0
\(927\) −2.91630e11 −0.394923
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 2.71668e12 3.61610
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.43879e12 −1.86655 −0.933273 − 0.359168i \(-0.883061\pi\)
−0.933273 + 0.359168i \(0.883061\pi\)
\(938\) 0 0
\(939\) −9.47753e11 −1.21908
\(940\) 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
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −5.78957e11 −0.713808
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.42106e12 2.83865
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.51552e12 1.73322 0.866611 − 0.498984i \(-0.166293\pi\)
0.866611 + 0.498984i \(0.166293\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 2.86184e12 3.19297
\(974\) 0 0
\(975\) 1.13292e12 1.25367
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.33307e12 1.43938
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.92066e12 −1.99139 −0.995693 − 0.0927105i \(-0.970447\pi\)
−0.995693 + 0.0927105i \(0.970447\pi\)
\(992\) 0 0
\(993\) −1.31621e12 −1.35372
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.81390e11 0.588420 0.294210 − 0.955741i \(-0.404944\pi\)
0.294210 + 0.955741i \(0.404944\pi\)
\(998\) 0 0
\(999\) −2.67592e11 −0.268665
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 48.9.e.a.17.1 1
3.2 odd 2 CM 48.9.e.a.17.1 1
4.3 odd 2 12.9.c.a.5.1 ✓ 1
8.3 odd 2 192.9.e.a.65.1 1
8.5 even 2 192.9.e.b.65.1 1
12.11 even 2 12.9.c.a.5.1 ✓ 1
20.3 even 4 300.9.b.b.149.2 2
20.7 even 4 300.9.b.b.149.1 2
20.19 odd 2 300.9.g.a.101.1 1
24.5 odd 2 192.9.e.b.65.1 1
24.11 even 2 192.9.e.a.65.1 1
36.7 odd 6 324.9.g.a.53.1 2
36.11 even 6 324.9.g.a.53.1 2
36.23 even 6 324.9.g.a.269.1 2
36.31 odd 6 324.9.g.a.269.1 2
60.23 odd 4 300.9.b.b.149.2 2
60.47 odd 4 300.9.b.b.149.1 2
60.59 even 2 300.9.g.a.101.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.9.c.a.5.1 ✓ 1 4.3 odd 2
12.9.c.a.5.1 ✓ 1 12.11 even 2
48.9.e.a.17.1 1 1.1 even 1 trivial
48.9.e.a.17.1 1 3.2 odd 2 CM
192.9.e.a.65.1 1 8.3 odd 2
192.9.e.a.65.1 1 24.11 even 2
192.9.e.b.65.1 1 8.5 even 2
192.9.e.b.65.1 1 24.5 odd 2
300.9.b.b.149.1 2 20.7 even 4
300.9.b.b.149.1 2 60.47 odd 4
300.9.b.b.149.2 2 20.3 even 4
300.9.b.b.149.2 2 60.23 odd 4
300.9.g.a.101.1 1 20.19 odd 2
300.9.g.a.101.1 1 60.59 even 2
324.9.g.a.53.1 2 36.7 odd 6
324.9.g.a.53.1 2 36.11 even 6
324.9.g.a.269.1 2 36.23 even 6
324.9.g.a.269.1 2 36.31 odd 6