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Properties

Label 212.1.d.a.211.1
Level $212$
Weight $1$
Character 212.211
Self dual yes
Analytic conductor $0.106$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -212
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] = [212,1,Mod(211,212)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(212, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("212.211");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 212 = 2^{2} \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 212.d (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: \(0.105801782678\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.212.1
Artin image: $D_6$
Artin field: Galois closure of 6.0.179776.1

Embedding invariants

Embedding label 211.1
Character \(\chi\) \(=\) 212.211

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{19} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} -1.00000 q^{29} -2.00000 q^{31} -1.00000 q^{32} +1.00000 q^{34} -1.00000 q^{37} -1.00000 q^{38} -1.00000 q^{39} -1.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +1.00000 q^{57} +1.00000 q^{58} +2.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} -1.00000 q^{68} +1.00000 q^{69} +1.00000 q^{71} +1.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +1.00000 q^{78} +1.00000 q^{79} -1.00000 q^{81} +1.00000 q^{83} -1.00000 q^{87} +2.00000 q^{89} +1.00000 q^{92} -2.00000 q^{93} -1.00000 q^{96} -1.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/212\mathbb{Z}\right)^\times\).

\(n\) \(107\) \(161\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000
\(3\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(4\) 1.00000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.00000 −1.00000
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 1.00000
\(13\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(24\) −1.00000 −1.00000
\(25\) 1.00000 1.00000
\(26\) 1.00000 1.00000
\(27\) −1.00000 −1.00000
\(28\) 0 0
\(29\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(30\) 0 0
\(31\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) −1.00000 −1.00000
\(39\) −1.00000 −1.00000
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −1.00000
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 1.00000 1.00000
\(49\) 1.00000 1.00000
\(50\) −1.00000 −1.00000
\(51\) −1.00000 −1.00000
\(52\) −1.00000 −1.00000
\(53\) 1.00000 1.00000
\(54\) 1.00000 1.00000
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000 1.00000
\(58\) 1.00000 1.00000
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 2.00000 2.00000
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(68\) −1.00000 −1.00000
\(69\) 1.00000 1.00000
\(70\) 0 0
\(71\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 1.00000 1.00000
\(75\) 1.00000 1.00000
\(76\) 1.00000 1.00000
\(77\) 0 0
\(78\) 1.00000 1.00000
\(79\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.00000 −1.00000
\(88\) 0 0
\(89\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 1.00000
\(93\) −2.00000 −2.00000
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −1.00000
\(97\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) −1.00000 −1.00000
\(99\) 0 0
\(100\) 1.00000 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 1.00000 1.00000
\(103\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 1.00000 1.00000
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.00000 −1.00000
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) −1.00000 −1.00000
\(112\) 0 0
\(113\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) −1.00000 −1.00000
\(115\) 0 0
\(116\) −1.00000 −1.00000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −2.00000 −2.00000
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) −1.00000 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.00000 2.00000
\(135\) 0 0
\(136\) 1.00000 1.00000
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) −1.00000 −1.00000
\(139\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.00000 −1.00000
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 1.00000
\(148\) −1.00000 −1.00000
\(149\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) −1.00000 −1.00000
\(151\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) −1.00000 −1.00000
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −1.00000
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −1.00000 −1.00000
\(159\) 1.00000 1.00000
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −1.00000 −1.00000
\(167\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 1.00000 1.00000
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −2.00000 −2.00000
\(179\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1.00000 −1.00000
\(185\) 0 0
\(186\) 2.00000 2.00000
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 1.00000 1.00000
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 1.00000 1.00000
\(195\) 0 0
\(196\) 1.00000 1.00000
\(197\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.00000 −1.00000
\(201\) −2.00000 −2.00000
\(202\) 0 0
\(203\) 0 0
\(204\) −1.00000 −1.00000
\(205\) 0 0
\(206\) −1.00000 −1.00000
\(207\) 0 0
\(208\) −1.00000 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 1.00000 1.00000
\(213\) 1.00000 1.00000
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000 1.00000
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.00000 1.00000
\(222\) 1.00000 1.00000
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000 1.00000
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 1.00000 1.00000
\(229\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.00000 1.00000
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00000 1.00000
\(238\) 0 0
\(239\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(242\) −1.00000 −1.00000
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.00000 −1.00000
\(248\) 2.00000 2.00000
\(249\) 1.00000 1.00000
\(250\) 0 0
\(251\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −1.00000 −1.00000
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.00000 2.00000
\(268\) −2.00000 −2.00000
\(269\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.00000 −1.00000
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 1.00000 1.00000
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 2.00000 2.00000
\(279\) 0 0
\(280\) 0 0
\(281\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(282\) 0 0
\(283\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 1.00000 1.00000
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) −1.00000 −1.00000
\(292\) 0 0
\(293\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(294\) −1.00000 −1.00000
\(295\) 0 0
\(296\) 1.00000 1.00000
\(297\) 0 0
\(298\) 1.00000 1.00000
\(299\) −1.00000 −1.00000
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) −1.00000 −1.00000
\(303\) 0 0
\(304\) 1.00000 1.00000
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 1.00000 1.00000
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 1.00000 1.00000
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.00000 1.00000
\(317\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) −1.00000 −1.00000
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.00000 −1.00000
\(324\) −1.00000 −1.00000
\(325\) −1.00000 −1.00000
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 1.00000 1.00000
\(333\) 0 0
\(334\) −1.00000 −1.00000
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) −1.00000 −1.00000
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −1.00000 −1.00000
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 1.00000 1.00000
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 2.00000
\(357\) 0 0
\(358\) −1.00000 −1.00000
\(359\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) 1.00000 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 1.00000 1.00000
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.00000 −2.00000
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.00000 1.00000
\(378\) 0 0
\(379\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(380\) 0 0
\(381\) 1.00000 1.00000
\(382\) −1.00000 −1.00000
\(383\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(384\) −1.00000 −1.00000
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −1.00000 −1.00000
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −1.00000 −1.00000
\(392\) −1.00000 −1.00000
\(393\) 0 0
\(394\) −2.00000 −2.00000
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 2.00000 2.00000
\(403\) 2.00000 2.00000
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 1.00000 1.00000
\(409\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.00000 1.00000
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000 1.00000
\(417\) −2.00000 −2.00000
\(418\) 0 0
\(419\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −1.00000 −1.00000
\(425\) −1.00000 −1.00000
\(426\) −1.00000 −1.00000
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.00000 −1.00000
\(433\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.00000 1.00000
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.00000 −1.00000
\(443\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) −1.00000 −1.00000
\(445\) 0 0
\(446\) 0 0
\(447\) −1.00000 −1.00000
\(448\) 0 0
\(449\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.00000 −1.00000
\(453\) 1.00000 1.00000
\(454\) 0 0
\(455\) 0 0
\(456\) −1.00000 −1.00000
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 1.00000 1.00000
\(459\) 1.00000 1.00000
\(460\) 0 0
\(461\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(462\) 0 0
\(463\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) −1.00000 −1.00000
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) −1.00000 −1.00000
\(475\) 1.00000 1.00000
\(476\) 0 0
\(477\) 0 0
\(478\) 2.00000 2.00000
\(479\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(480\) 0 0
\(481\) 1.00000 1.00000
\(482\) 1.00000 1.00000
\(483\) 0 0
\(484\) 1.00000 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 1.00000 1.00000
\(494\) 1.00000 1.00000
\(495\) 0 0
\(496\) −2.00000 −2.00000
\(497\) 0 0
\(498\) −1.00000 −1.00000
\(499\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(500\) 0 0
\(501\) 1.00000 1.00000
\(502\) 2.00000 2.00000
\(503\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 1.00000 1.00000
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −1.00000
\(513\) −1.00000 −1.00000
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 2.00000 2.00000
\(527\) 2.00000 2.00000
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) −2.00000 −2.00000
\(535\) 0 0
\(536\) 2.00000 2.00000
\(537\) 1.00000 1.00000
\(538\) 1.00000 1.00000
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.00000 1.00000
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.00000 −1.00000
\(552\) −1.00000 −1.00000
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −2.00000 −2.00000
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.00000 1.00000
\(563\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.00000 −1.00000
\(567\) 0 0
\(568\) −1.00000 −1.00000
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 1.00000 1.00000
\(574\) 0 0
\(575\) 1.00000 1.00000
\(576\) 0 0
\(577\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 1.00000 1.00000
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −2.00000 −2.00000
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 1.00000 1.00000
\(589\) −2.00000 −2.00000
\(590\) 0 0
\(591\) 2.00000 2.00000
\(592\) −1.00000 −1.00000
\(593\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.00000 −1.00000
\(597\) 0 0
\(598\) 1.00000 1.00000
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) −1.00000 −1.00000
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.00000 1.00000
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −1.00000 −1.00000
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) −1.00000 −1.00000
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −1.00000 −1.00000
\(622\) 0 0
\(623\) 0 0
\(624\) −1.00000 −1.00000
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.00000 1.00000
\(630\) 0 0
\(631\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) −1.00000 −1.00000
\(633\) 0 0
\(634\) 1.00000 1.00000
\(635\) 0 0
\(636\) 1.00000 1.00000
\(637\) −1.00000 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.00000 1.00000
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.00000 1.00000
\(649\) 0 0
\(650\) 1.00000 1.00000
\(651\) 0 0
\(652\) 0 0
\(653\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 1.00000 1.00000
\(664\) −1.00000 −1.00000
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −1.00000
\(668\) 1.00000 1.00000
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(674\) 0 0
\(675\) −1.00000 −1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 1.00000 1.00000
\(679\) 0 0
\(680\) 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
\(687\) −1.00000 −1.00000
\(688\) 0 0
\(689\) −1.00000 −1.00000
\(690\) 0 0
\(691\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 1.00000 1.00000
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −1.00000 −1.00000
\(703\) −1.00000 −1.00000
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −2.00000 −2.00000
\(713\) −2.00000 −2.00000
\(714\) 0 0
\(715\) 0 0
\(716\) 1.00000 1.00000
\(717\) −2.00000 −2.00000
\(718\) −1.00000 −1.00000
\(719\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.00000 −1.00000
\(724\) 0 0
\(725\) −1.00000 −1.00000
\(726\) −1.00000 −1.00000
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(734\) 0 0
\(735\) 0 0
\(736\) −1.00000 −1.00000
\(737\) 0 0
\(738\) 0 0
\(739\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) −1.00000 −1.00000
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 2.00000 2.00000
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −2.00000 −2.00000
\(754\) −1.00000 −1.00000
\(755\) 0 0
\(756\) 0 0
\(757\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(758\) −1.00000 −1.00000
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) −1.00000 −1.00000
\(763\) 0 0
\(764\) 1.00000 1.00000
\(765\) 0 0
\(766\) 2.00000 2.00000
\(767\) 0 0
\(768\) 1.00000 1.00000
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −2.00000 −2.00000
\(776\) 1.00000 1.00000
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 1.00000 1.00000
\(783\) 1.00000 1.00000
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(788\) 2.00000 2.00000
\(789\) −2.00000 −2.00000
\(790\) 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
\(800\) −1.00000 −1.00000
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −2.00000 −2.00000
\(805\) 0 0
\(806\) −2.00000 −2.00000
\(807\) −1.00000 −1.00000
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −1.00000 −1.00000
\(817\) 0 0
\(818\) 1.00000 1.00000
\(819\) 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 \(\pi\)
\(824\) −1.00000 −1.00000
\(825\) 0 0
\(826\) 0 0
\(827\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.00000 −1.00000
\(833\) −1.00000 −1.00000
\(834\) 2.00000 2.00000
\(835\) 0 0
\(836\) 0 0
\(837\) 2.00000 2.00000
\(838\) 2.00000 2.00000
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) −1.00000 −1.00000
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 1.00000 1.00000
\(849\) 1.00000 1.00000
\(850\) 1.00000 1.00000
\(851\) −1.00000 −1.00000
\(852\) 1.00000 1.00000
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 1.00000 1.00000
\(865\) 0 0
\(866\) 1.00000 1.00000
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.00000 2.00000
\(872\) 0 0
\(873\) 0 0
\(874\) −1.00000 −1.00000
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(878\) 0 0
\(879\) 2.00000 2.00000
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(884\) 1.00000 1.00000
\(885\) 0 0
\(886\) −1.00000 −1.00000
\(887\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(888\) 1.00000 1.00000
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 1.00000 1.00000
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 −1.00000
\(898\) 1.00000 1.00000
\(899\) 2.00000 2.00000
\(900\) 0 0
\(901\) −1.00000 −1.00000
\(902\) 0 0
\(903\) 0 0
\(904\) 1.00000 1.00000
\(905\) 0 0
\(906\) −1.00000 −1.00000
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 1.00000 1.00000
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.00000 −1.00000
\(917\) 0 0
\(918\) −1.00000 −1.00000
\(919\) 1.00000 1.00000 0.500000 − 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.00000 1.00000
\(923\) −1.00000 −1.00000
\(924\) 0 0
\(925\) −1.00000 −1.00000
\(926\) −1.00000 −1.00000
\(927\) 0 0
\(928\) 1.00000 1.00000
\(929\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(930\) 0 0
\(931\) 1.00000 1.00000
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 1.00000 1.00000
\(949\) 0 0
\(950\) −1.00000 −1.00000
\(951\) −1.00000 −1.00000
\(952\) 0 0
\(953\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(954\) 0 0
\(955\) 0 0
\(956\) −2.00000 −2.00000
\(957\) 0 0
\(958\) −1.00000 −1.00000
\(959\) 0 0
\(960\) 0 0
\(961\) 3.00000 3.00000
\(962\) −1.00000 −1.00000
\(963\) 0 0
\(964\) −1.00000 −1.00000
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −1.00000 −1.00000
\(969\) −1.00000 −1.00000
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −1.00000 −1.00000
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.00000 −1.00000
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.00000 −1.00000
\(987\) 0 0
\(988\) −1.00000 −1.00000
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 2.00000 2.00000
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 1.00000 1.00000
\(997\) −1.00000 −1.00000 −0.500000 − 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) −1.00000 −1.00000
\(999\) 1.00000 1.00000
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 212.1.d.a.211.1 ✓ 1
3.2 odd 2 1908.1.b.b.847.1 1
4.3 odd 2 212.1.d.b.211.1 yes 1
8.3 odd 2 3392.1.f.c.3391.1 1
8.5 even 2 3392.1.f.a.3391.1 1
12.11 even 2 1908.1.b.a.847.1 1
53.52 even 2 212.1.d.b.211.1 yes 1
159.158 odd 2 1908.1.b.a.847.1 1
212.211 odd 2 CM 212.1.d.a.211.1 ✓ 1
424.211 odd 2 3392.1.f.a.3391.1 1
424.317 even 2 3392.1.f.c.3391.1 1
636.635 even 2 1908.1.b.b.847.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
212.1.d.a.211.1 ✓ 1 1.1 even 1 trivial
212.1.d.a.211.1 ✓ 1 212.211 odd 2 CM
212.1.d.b.211.1 yes 1 4.3 odd 2
212.1.d.b.211.1 yes 1 53.52 even 2
1908.1.b.a.847.1 1 12.11 even 2
1908.1.b.a.847.1 1 159.158 odd 2
1908.1.b.b.847.1 1 3.2 odd 2
1908.1.b.b.847.1 1 636.635 even 2
3392.1.f.a.3391.1 1 8.5 even 2
3392.1.f.a.3391.1 1 424.211 odd 2
3392.1.f.c.3391.1 1 8.3 odd 2
3392.1.f.c.3391.1 1 424.317 even 2