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Properties

Label 72.11.b.a.19.1
Level $72$
Weight $11$
Character 72.19
Self dual yes
Analytic conductor $45.746$
Analytic rank $0$
Dimension $1$
CM discriminant -8
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] = [72,11,Mod(19,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.19");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), 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: \(45.7457221925\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.1
Character \(\chi\) \(=\) 72.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+32.0000 q^{2} +1024.00 q^{4} +32768.0 q^{8} +O(q^{10})\) \(q+32.0000 q^{2} +1024.00 q^{4} +32768.0 q^{8} +97426.0 q^{11} +1.04858e6 q^{16} -823682. q^{17} +3.35373e6 q^{19} +3.11763e6 q^{22} +9.76562e6 q^{25} +3.35544e7 q^{32} -2.63578e7 q^{34} +1.07319e8 q^{38} +3.77789e7 q^{41} +2.14486e8 q^{43} +9.97642e7 q^{44} +2.82475e8 q^{49} +3.12500e8 q^{50} -9.21044e8 q^{59} +1.07374e9 q^{64} +1.81371e9 q^{67} -8.43450e8 q^{68} -1.60578e9 q^{73} +3.43422e9 q^{76} +1.20893e9 q^{82} -9.60515e7 q^{83} +6.86354e9 q^{86} +3.19246e9 q^{88} +1.11160e10 q^{89} -9.87298e9 q^{97} +9.03921e9 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.0000 1.00000
\(3\) 0 0
\(4\) 1024.00 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 32768.0 1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 97426.0 0.604939 0.302469 − 0.953159i \(-0.402189\pi\)
0.302469 + 0.953159i \(0.402189\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) −823682. −0.580116 −0.290058 − 0.957009i \(-0.593675\pi\)
−0.290058 + 0.957009i \(0.593675\pi\)
\(18\) 0 0
\(19\) 3.35373e6 1.35444 0.677220 − 0.735781i \(-0.263185\pi\)
0.677220 + 0.735781i \(0.263185\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.11763e6 0.604939
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.76562e6 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 3.35544e7 1.00000
\(33\) 0 0
\(34\) −2.63578e7 −0.580116
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.07319e8 1.35444
\(39\) 0 0
\(40\) 0 0
\(41\) 3.77789e7 0.326085 0.163042 − 0.986619i \(-0.447869\pi\)
0.163042 + 0.986619i \(0.447869\pi\)
\(42\) 0 0
\(43\) 2.14486e8 1.45900 0.729501 − 0.683980i \(-0.239752\pi\)
0.729501 + 0.683980i \(0.239752\pi\)
\(44\) 9.97642e7 0.604939
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 2.82475e8 1.00000
\(50\) 3.12500e8 1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.21044e8 −1.28831 −0.644155 − 0.764895i \(-0.722791\pi\)
−0.644155 + 0.764895i \(0.722791\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.07374e9 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 1.81371e9 1.34336 0.671682 − 0.740840i \(-0.265572\pi\)
0.671682 + 0.740840i \(0.265572\pi\)
\(68\) −8.43450e8 −0.580116
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.60578e9 −0.774591 −0.387295 − 0.921956i \(-0.626591\pi\)
−0.387295 + 0.921956i \(0.626591\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 3.43422e9 1.35444
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.20893e9 0.326085
\(83\) −9.60515e7 −0.0243845 −0.0121922 − 0.999926i \(-0.503881\pi\)
−0.0121922 + 0.999926i \(0.503881\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 6.86354e9 1.45900
\(87\) 0 0
\(88\) 3.19246e9 0.604939
\(89\) 1.11160e10 1.99067 0.995335 − 0.0964794i \(-0.0307582\pi\)
0.995335 + 0.0964794i \(0.0307582\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.87298e9 −1.14971 −0.574857 − 0.818254i \(-0.694942\pi\)
−0.574857 + 0.818254i \(0.694942\pi\)
\(98\) 9.03921e9 1.00000
\(99\) 0 0
\(100\) 1.00000e10 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.74969e10 −1.96049 −0.980244 − 0.197792i \(-0.936623\pi\)
−0.980244 + 0.197792i \(0.936623\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.88420e10 −1.56543 −0.782714 − 0.622381i \(-0.786165\pi\)
−0.782714 + 0.622381i \(0.786165\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −2.94734e10 −1.28831
\(119\) 0 0
\(120\) 0 0
\(121\) −1.64456e10 −0.634049
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 3.43597e10 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −7.17614e10 −1.86009 −0.930046 − 0.367444i \(-0.880233\pi\)
−0.930046 + 0.367444i \(0.880233\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 5.80387e10 1.34336
\(135\) 0 0
\(136\) −2.69904e10 −0.580116
\(137\) −8.23091e10 −1.70547 −0.852737 − 0.522340i \(-0.825059\pi\)
−0.852737 + 0.522340i \(0.825059\pi\)
\(138\) 0 0
\(139\) −8.90354e10 −1.71589 −0.857943 − 0.513745i \(-0.828258\pi\)
−0.857943 + 0.513745i \(0.828258\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −5.13850e10 −0.774591
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.09895e11 1.35444
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.62942e11 −1.41610 −0.708050 − 0.706162i \(-0.750425\pi\)
−0.708050 + 0.706162i \(0.750425\pi\)
\(164\) 3.86856e10 0.326085
\(165\) 0 0
\(166\) −3.07365e9 −0.0243845
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.37858e11 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 2.19633e11 1.45900
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.02159e11 0.604939
\(177\) 0 0
\(178\) 3.55713e11 1.99067
\(179\) 1.68275e11 0.915703 0.457852 − 0.889029i \(-0.348619\pi\)
0.457852 + 0.889029i \(0.348619\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −8.02480e10 −0.350935
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 5.13616e11 1.91802 0.959008 − 0.283381i \(-0.0914559\pi\)
0.959008 + 0.283381i \(0.0914559\pi\)
\(194\) −3.15935e11 −1.14971
\(195\) 0 0
\(196\) 2.89255e11 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 3.20000e11 1.00000
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.26740e11 0.819353
\(210\) 0 0
\(211\) −2.59806e11 −0.621207 −0.310603 − 0.950540i \(-0.600531\pi\)
−0.310603 + 0.950540i \(0.600531\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −8.79899e11 −1.96049
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −9.22944e11 −1.56543
\(227\) −7.10947e11 −1.17953 −0.589764 − 0.807576i \(-0.700779\pi\)
−0.589764 + 0.807576i \(0.700779\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.96604e11 0.577534 0.288767 − 0.957399i \(-0.406755\pi\)
0.288767 + 0.957399i \(0.406755\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.43149e11 −1.28831
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 1.42664e12 1.75480 0.877401 − 0.479757i \(-0.159275\pi\)
0.877401 + 0.479757i \(0.159275\pi\)
\(242\) −5.26259e11 −0.634049
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.53736e11 −0.656197 −0.328098 − 0.944644i \(-0.606408\pi\)
−0.328098 + 0.944644i \(0.606408\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.09951e12 1.00000
\(257\) 2.00451e12 1.78790 0.893948 − 0.448172i \(-0.147925\pi\)
0.893948 + 0.448172i \(0.147925\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −2.29636e12 −1.86009
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.85724e12 1.34336
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −8.63693e11 −0.580116
\(273\) 0 0
\(274\) −2.63389e12 −1.70547
\(275\) 9.51426e11 0.604939
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −2.84913e12 −1.71589
\(279\) 0 0
\(280\) 0 0
\(281\) 2.86062e12 1.63278 0.816391 − 0.577499i \(-0.195971\pi\)
0.816391 + 0.577499i \(0.195971\pi\)
\(282\) 0 0
\(283\) −2.41276e12 −1.32917 −0.664586 − 0.747212i \(-0.731392\pi\)
−0.664586 + 0.747212i \(0.731392\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.33754e12 −0.663465
\(290\) 0 0
\(291\) 0 0
\(292\) −1.64432e12 −0.774591
\(293\) 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
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 3.51664e12 1.35444
\(305\) 0 0
\(306\) 0 0
\(307\) −4.18595e12 −1.53498 −0.767489 − 0.641062i \(-0.778494\pi\)
−0.767489 + 0.641062i \(0.778494\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 5.78028e12 1.92410 0.962049 − 0.272878i \(-0.0879755\pi\)
0.962049 + 0.272878i \(0.0879755\pi\)
\(314\) 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
\(323\) −2.76240e12 −0.785732
\(324\) 0 0
\(325\) 0 0
\(326\) −5.21413e12 −1.41610
\(327\) 0 0
\(328\) 1.23794e12 0.326085
\(329\) 0 0
\(330\) 0 0
\(331\) 8.38750e11 0.211102 0.105551 − 0.994414i \(-0.466339\pi\)
0.105551 + 0.994414i \(0.466339\pi\)
\(332\) −9.83568e10 −0.0243845
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 6.23741e12 1.43501 0.717504 − 0.696554i \(-0.245284\pi\)
0.717504 + 0.696554i \(0.245284\pi\)
\(338\) 4.41147e12 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 7.02826e12 1.45900
\(345\) 0 0
\(346\) 0 0
\(347\) −4.70042e11 −0.0934307 −0.0467153 − 0.998908i \(-0.514875\pi\)
−0.0467153 + 0.998908i \(0.514875\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.26907e12 0.604939
\(353\) −1.07873e13 −1.96806 −0.984031 − 0.177999i \(-0.943038\pi\)
−0.984031 + 0.177999i \(0.943038\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.13828e13 1.99067
\(357\) 0 0
\(358\) 5.38480e12 0.915703
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 5.11641e12 0.834506
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −2.56794e12 −0.350935
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −9.16504e12 −1.17203 −0.586015 − 0.810300i \(-0.699304\pi\)
−0.586015 + 0.810300i \(0.699304\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.64357e13 1.91802
\(387\) 0 0
\(388\) −1.01099e13 −1.14971
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 9.25615e12 1.00000
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.02400e13 1.00000
\(401\) 1.38849e12 0.133913 0.0669563 − 0.997756i \(-0.478671\pi\)
0.0669563 + 0.997756i \(0.478671\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.97238e13 −1.72336 −0.861678 − 0.507456i \(-0.830586\pi\)
−0.861678 + 0.507456i \(0.830586\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 1.04557e13 0.819353
\(419\) −4.11443e12 −0.318595 −0.159298 − 0.987231i \(-0.550923\pi\)
−0.159298 + 0.987231i \(0.550923\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −8.31378e12 −0.621207
\(423\) 0 0
\(424\) 0 0
\(425\) −8.04377e12 −0.580116
\(426\) 0 0
\(427\) 0 0
\(428\) −2.81568e13 −1.96049
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −1.49197e13 −0.980213 −0.490107 − 0.871662i \(-0.663042\pi\)
−0.490107 + 0.871662i \(0.663042\pi\)
\(434\) 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
\(441\) 0 0
\(442\) 0 0
\(443\) −2.66951e13 −1.56463 −0.782317 − 0.622881i \(-0.785962\pi\)
−0.782317 + 0.622881i \(0.785962\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.39104e12 −0.459816 −0.229908 − 0.973212i \(-0.573843\pi\)
−0.229908 + 0.973212i \(0.573843\pi\)
\(450\) 0 0
\(451\) 3.68065e12 0.197261
\(452\) −2.95342e13 −1.56543
\(453\) 0 0
\(454\) −2.27503e13 −1.17953
\(455\) 0 0
\(456\) 0 0
\(457\) −3.85912e13 −1.93601 −0.968004 − 0.250935i \(-0.919262\pi\)
−0.968004 + 0.250935i \(0.919262\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.26913e13 0.577534
\(467\) 8.04286e12 0.362098 0.181049 − 0.983474i \(-0.442051\pi\)
0.181049 + 0.983474i \(0.442051\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −3.01808e13 −1.28831
\(473\) 2.08965e13 0.882607
\(474\) 0 0
\(475\) 3.27512e13 1.35444
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 4.56524e13 1.75480
\(483\) 0 0
\(484\) −1.68403e13 −0.634049
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.67515e13 1.98870 0.994351 − 0.106139i \(-0.0338489\pi\)
0.994351 + 0.106139i \(0.0338489\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.18128e13 1.99791 0.998955 − 0.0457152i \(-0.0145567\pi\)
0.998955 + 0.0457152i \(0.0145567\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.09196e13 −0.656197
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 3.51844e13 1.00000
\(513\) 0 0
\(514\) 6.41442e13 1.78790
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.91921e13 0.499958 0.249979 − 0.968251i \(-0.419576\pi\)
0.249979 + 0.968251i \(0.419576\pi\)
\(522\) 0 0
\(523\) 7.26517e13 1.85668 0.928341 − 0.371731i \(-0.121236\pi\)
0.928341 + 0.371731i \(0.121236\pi\)
\(524\) −7.34836e13 −1.86009
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 4.14265e13 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 5.94316e13 1.34336
\(537\) 0 0
\(538\) 0 0
\(539\) 2.75204e13 0.604939
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −2.76382e13 −0.580116
\(545\) 0 0
\(546\) 0 0
\(547\) −2.45465e13 −0.501249 −0.250624 − 0.968084i \(-0.580636\pi\)
−0.250624 + 0.968084i \(0.580636\pi\)
\(548\) −8.42845e13 −1.70547
\(549\) 0 0
\(550\) 3.04456e13 0.604939
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −9.11722e13 −1.71589
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 9.15398e13 1.63278
\(563\) 9.58258e13 1.69411 0.847053 − 0.531509i \(-0.178375\pi\)
0.847053 + 0.531509i \(0.178375\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7.72082e13 −1.32917
\(567\) 0 0
\(568\) 0 0
\(569\) −2.71070e13 −0.454485 −0.227243 − 0.973838i \(-0.572971\pi\)
−0.227243 + 0.973838i \(0.572971\pi\)
\(570\) 0 0
\(571\) −8.65363e12 −0.142567 −0.0712833 − 0.997456i \(-0.522709\pi\)
−0.0712833 + 0.997456i \(0.522709\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.10840e12 0.0173308 0.00866541 − 0.999962i \(-0.497242\pi\)
0.00866541 + 0.999962i \(0.497242\pi\)
\(578\) −4.28013e13 −0.663465
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −5.26183e13 −0.774591
\(585\) 0 0
\(586\) 0 0
\(587\) 4.50851e13 0.646908 0.323454 − 0.946244i \(-0.395156\pi\)
0.323454 + 0.946244i \(0.395156\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.17278e14 −1.59935 −0.799677 − 0.600430i \(-0.794996\pi\)
−0.799677 + 0.600430i \(0.794996\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.44880e14 −1.84772 −0.923860 − 0.382731i \(-0.874984\pi\)
−0.923860 + 0.382731i \(0.874984\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 1.12532e14 1.35444
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −1.33950e14 −1.53498
\(615\) 0 0
\(616\) 0 0
\(617\) 1.75258e14 1.95998 0.979991 − 0.199042i \(-0.0637830\pi\)
0.979991 + 0.199042i \(0.0637830\pi\)
\(618\) 0 0
\(619\) −1.28543e14 −1.41447 −0.707236 − 0.706978i \(-0.750058\pi\)
−0.707236 + 0.706978i \(0.750058\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.53674e13 1.00000
\(626\) 1.84969e14 1.92410
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.02827e14 −1.87428 −0.937142 − 0.348947i \(-0.886539\pi\)
−0.937142 + 0.348947i \(0.886539\pi\)
\(642\) 0 0
\(643\) −2.41858e13 −0.220042 −0.110021 − 0.993929i \(-0.535092\pi\)
−0.110021 + 0.993929i \(0.535092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −8.83969e13 −0.785732
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −8.97336e13 −0.779348
\(650\) 0 0
\(651\) 0 0
\(652\) −1.66852e14 −1.41610
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.96141e13 0.326085
\(657\) 0 0
\(658\) 0 0
\(659\) −2.24854e14 −1.80915 −0.904573 − 0.426319i \(-0.859810\pi\)
−0.904573 + 0.426319i \(0.859810\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 2.68400e13 0.211102
\(663\) 0 0
\(664\) −3.14742e12 −0.0243845
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 9.95159e13 0.720804 0.360402 − 0.932797i \(-0.382640\pi\)
0.360402 + 0.932797i \(0.382640\pi\)
\(674\) 1.99597e14 1.43501
\(675\) 0 0
\(676\) 1.41167e14 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.95986e14 −1.99144 −0.995721 − 0.0924146i \(-0.970541\pi\)
−0.995721 + 0.0924146i \(0.970541\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 2.24904e14 1.45900
\(689\) 0 0
\(690\) 0 0
\(691\) 1.05674e14 0.670775 0.335388 − 0.942080i \(-0.391133\pi\)
0.335388 + 0.942080i \(0.391133\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.50413e13 −0.0934307
\(695\) 0 0
\(696\) 0 0
\(697\) −3.11178e13 −0.189167
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.04610e14 0.604939
\(705\) 0 0
\(706\) −3.45193e14 −1.96806
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.64250e14 1.99067
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.72314e14 0.915703
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.63725e14 0.834506
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1.76668e14 −0.846391
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.76702e14 0.812653
\(738\) 0 0
\(739\) −3.92476e14 −1.78070 −0.890351 − 0.455274i \(-0.849541\pi\)
−0.890351 + 0.455274i \(0.849541\pi\)
\(740\) 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
\(748\) −8.21740e13 −0.350935
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) −2.93281e14 −1.17203
\(759\) 0 0
\(760\) 0 0
\(761\) 2.42238e14 0.949116 0.474558 − 0.880224i \(-0.342608\pi\)
0.474558 + 0.880224i \(0.342608\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 3.18401e14 1.18398 0.591988 − 0.805947i \(-0.298343\pi\)
0.591988 + 0.805947i \(0.298343\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5.25943e14 1.91802
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.23518e14 −1.14971
\(777\) 0 0
\(778\) 0 0
\(779\) 1.26700e14 0.441662
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2.96197e14 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −1.95884e12 −0.00648823 −0.00324411 − 0.999995i \(-0.501033\pi\)
−0.00324411 + 0.999995i \(0.501033\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.27680e14 1.00000
\(801\) 0 0
\(802\) 4.44318e13 0.133913
\(803\) −1.56445e14 −0.468580
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 3.40807e14 0.983480 0.491740 − 0.870742i \(-0.336361\pi\)
0.491740 + 0.870742i \(0.336361\pi\)
\(810\) 0 0
\(811\) −3.66643e14 −1.04506 −0.522528 − 0.852622i \(-0.675011\pi\)
−0.522528 + 0.852622i \(0.675011\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.19326e14 1.97613
\(818\) −6.31163e14 −1.72336
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.20561e14 1.86270 0.931351 − 0.364122i \(-0.118631\pi\)
0.931351 + 0.364122i \(0.118631\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −2.32670e14 −0.580116
\(834\) 0 0
\(835\) 0 0
\(836\) 3.34582e14 0.819353
\(837\) 0 0
\(838\) −1.31662e14 −0.318595
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 4.20707e14 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −2.66041e14 −0.621207
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −2.57401e14 −0.580116
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −9.01017e14 −1.96049
\(857\) 6.26409e14 1.35505 0.677523 − 0.735502i \(-0.263053\pi\)
0.677523 + 0.735502i \(0.263053\pi\)
\(858\) 0 0
\(859\) 1.10234e14 0.235695 0.117848 − 0.993032i \(-0.462401\pi\)
0.117848 + 0.993032i \(0.462401\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −4.77430e14 −0.980213
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 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
\(881\) −1.05951e15 −1.99629 −0.998145 − 0.0608798i \(-0.980609\pi\)
−0.998145 + 0.0608798i \(0.980609\pi\)
\(882\) 0 0
\(883\) −1.01334e15 −1.88777 −0.943886 − 0.330271i \(-0.892860\pi\)
−0.943886 + 0.330271i \(0.892860\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −8.54243e14 −1.56463
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 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
\(898\) −2.68513e14 −0.459816
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 1.17781e14 0.197261
\(903\) 0 0
\(904\) −9.45095e14 −1.56543
\(905\) 0 0
\(906\) 0 0
\(907\) 1.19076e14 0.193994 0.0969968 − 0.995285i \(-0.469076\pi\)
0.0969968 + 0.995285i \(0.469076\pi\)
\(908\) −7.28010e14 −1.17953
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −9.35792e12 −0.0147511
\(914\) −1.23492e15 −1.93601
\(915\) 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
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.55409e14 −0.224594 −0.112297 − 0.993675i \(-0.535821\pi\)
−0.112297 + 0.993675i \(0.535821\pi\)
\(930\) 0 0
\(931\) 9.47345e14 1.35444
\(932\) 4.06122e14 0.577534
\(933\) 0 0
\(934\) 2.57372e14 0.362098
\(935\) 0 0
\(936\) 0 0
\(937\) −1.33321e15 −1.84587 −0.922936 − 0.384954i \(-0.874217\pi\)
−0.922936 + 0.384954i \(0.874217\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −9.65784e14 −1.28831
\(945\) 0 0
\(946\) 6.68687e14 0.882607
\(947\) 5.63746e14 0.740174 0.370087 − 0.928997i \(-0.379328\pi\)
0.370087 + 0.928997i \(0.379328\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1.04804e15 1.35444
\(951\) 0 0
\(952\) 0 0
\(953\) 5.72694e14 0.728548 0.364274 − 0.931292i \(-0.381317\pi\)
0.364274 + 0.931292i \(0.381317\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.19628e14 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 1.46088e15 1.75480
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −5.38889e14 −0.634049
\(969\) 0 0
\(970\) 0 0
\(971\) −8.49794e14 −0.984504 −0.492252 − 0.870453i \(-0.663826\pi\)
−0.492252 + 0.870453i \(0.663826\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.01921e15 1.14497 0.572483 − 0.819916i \(-0.305980\pi\)
0.572483 + 0.819916i \(0.305980\pi\)
\(978\) 0 0
\(979\) 1.08299e15 1.20423
\(980\) 0 0
\(981\) 0 0
\(982\) 1.81605e15 1.98870
\(983\) 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
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 1.97801e15 1.99791
\(999\) 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 72.11.b.a.19.1 1
3.2 odd 2 8.11.d.a.3.1 ✓ 1
4.3 odd 2 288.11.b.a.271.1 1
8.3 odd 2 CM 72.11.b.a.19.1 1
8.5 even 2 288.11.b.a.271.1 1
12.11 even 2 32.11.d.a.15.1 1
24.5 odd 2 32.11.d.a.15.1 1
24.11 even 2 8.11.d.a.3.1 ✓ 1
48.5 odd 4 256.11.c.c.255.2 2
48.11 even 4 256.11.c.c.255.1 2
48.29 odd 4 256.11.c.c.255.1 2
48.35 even 4 256.11.c.c.255.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.11.d.a.3.1 ✓ 1 3.2 odd 2
8.11.d.a.3.1 ✓ 1 24.11 even 2
32.11.d.a.15.1 1 12.11 even 2
32.11.d.a.15.1 1 24.5 odd 2
72.11.b.a.19.1 1 1.1 even 1 trivial
72.11.b.a.19.1 1 8.3 odd 2 CM
256.11.c.c.255.1 2 48.11 even 4
256.11.c.c.255.1 2 48.29 odd 4
256.11.c.c.255.2 2 48.5 odd 4
256.11.c.c.255.2 2 48.35 even 4
288.11.b.a.271.1 1 4.3 odd 2
288.11.b.a.271.1 1 8.5 even 2