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Properties

Label 1008.2.b.i.559.3
Level $1008$
Weight $2$
Character 1008.559
Analytic conductor $8.049$
Analytic rank $0$
Dimension $4$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1008,2,Mod(559,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.559"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 24x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 559.3
Root \(-2.01563i\) of defining polynomial
Character \(\chi\) \(=\) 1008.559
Dual form 1008.2.b.i.559.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.44949i q^{5} -2.64575 q^{7} +6.48074i q^{11} -7.34847i q^{17} -5.29150 q^{19} -6.48074i q^{23} -1.00000 q^{25} -10.5830 q^{31} -6.48074i q^{35} -8.00000 q^{37} +12.2474i q^{41} +7.00000 q^{49} -15.8745 q^{55} +6.48074i q^{71} -17.1464i q^{77} +18.0000 q^{85} -2.44949i q^{89} -12.9615i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{25} - 32 q^{37} + 28 q^{49} + 72 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.44949i 1.09545i 0.836660 + 0.547723i \(0.184505\pi\)
−0.836660 + 0.547723i \(0.815495\pi\)
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 6.48074i 1.95402i 0.213201 + 0.977008i \(0.431611\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.34847i − 1.78227i −0.453743 − 0.891133i \(-0.649911\pi\)
0.453743 − 0.891133i \(-0.350089\pi\)
\(18\) 0 0
\(19\) −5.29150 −1.21395 −0.606977 − 0.794719i \(-0.707618\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.48074i − 1.35133i −0.737210 − 0.675664i \(-0.763857\pi\)
0.737210 − 0.675664i \(-0.236143\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −10.5830 −1.90076 −0.950382 − 0.311086i \(-0.899307\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 6.48074i − 1.09545i
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 − 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.2474i 1.91273i 0.292174 + 0.956365i \(0.405621\pi\)
−0.292174 + 0.956365i \(0.594379\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −15.8745 −2.14052
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.48074i 0.769122i 0.923099 + 0.384561i \(0.125647\pi\)
−0.923099 + 0.384561i \(0.874353\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 17.1464i − 1.95402i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.44949i − 0.259645i −0.991537 − 0.129823i \(-0.958559\pi\)
0.991537 − 0.129823i \(-0.0414408\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 12.9615i − 1.32982i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.34847i 0.731200i 0.930772 + 0.365600i \(0.119136\pi\)
−0.930772 + 0.365600i \(0.880864\pi\)
\(102\) 0 0
\(103\) −10.5830 −1.04277 −0.521387 − 0.853320i \(-0.674585\pi\)
−0.521387 + 0.853320i \(0.674585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.4422i 1.87955i 0.341793 + 0.939775i \(0.388966\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 15.8745 1.48031
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 19.4422i 1.78227i
\(120\) 0 0
\(121\) −31.0000 −2.81818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.79796i 0.876356i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 14.0000 1.21395
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 21.1660 1.79528 0.897639 − 0.440732i \(-0.145281\pi\)
0.897639 + 0.440732i \(0.145281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 25.9230i − 2.08218i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.1464i 1.35133i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 2.44949i − 0.186231i −0.995655 − 0.0931156i \(-0.970317\pi\)
0.995655 − 0.0931156i \(-0.0296826\pi\)
\(174\) 0 0
\(175\) 2.64575 0.200000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 19.4422i − 1.45318i −0.687071 − 0.726590i \(-0.741104\pi\)
0.687071 − 0.726590i \(-0.258896\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 19.5959i − 1.44072i
\(186\) 0 0
\(187\) 47.6235 3.48258
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 6.48074i − 0.468930i −0.972125 − 0.234465i \(-0.924666\pi\)
0.972125 − 0.234465i \(-0.0753338\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −5.29150 −0.375105 −0.187552 − 0.982255i \(-0.560055\pi\)
−0.187552 + 0.982255i \(0.560055\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −30.0000 −2.09529
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 34.2929i − 2.37209i
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 28.0000 1.90076
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 26.4575 1.77173 0.885863 − 0.463947i \(-0.153567\pi\)
0.885863 + 0.463947i \(0.153567\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.4422i 1.25761i 0.777562 + 0.628806i \(0.216456\pi\)
−0.777562 + 0.628806i \(0.783544\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 17.1464i 1.09545i
\(246\) 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
\(253\) 42.0000 2.64052
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 31.8434i 1.98633i 0.116699 + 0.993167i \(0.462769\pi\)
−0.116699 + 0.993167i \(0.537231\pi\)
\(258\) 0 0
\(259\) 21.1660 1.31519
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 32.4037i 1.99810i 0.0436021 + 0.999049i \(0.486117\pi\)
−0.0436021 + 0.999049i \(0.513883\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.9444i 1.64283i 0.570332 + 0.821414i \(0.306814\pi\)
−0.570332 + 0.821414i \(0.693186\pi\)
\(270\) 0 0
\(271\) −10.5830 −0.642872 −0.321436 − 0.946931i \(-0.604165\pi\)
−0.321436 + 0.946931i \(0.604165\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 6.48074i − 0.390803i
\(276\) 0 0
\(277\) −32.0000 −1.92269 −0.961347 − 0.275340i \(-0.911209\pi\)
−0.961347 + 0.275340i \(0.911209\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 26.4575 1.57274 0.786368 − 0.617758i \(-0.211959\pi\)
0.786368 + 0.617758i \(0.211959\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 32.4037i − 1.91273i
\(288\) 0 0
\(289\) −37.0000 −2.17647
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 22.0454i − 1.28791i −0.765065 − 0.643953i \(-0.777293\pi\)
0.765065 − 0.643953i \(-0.222707\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) −5.29150 −0.302002 −0.151001 − 0.988534i \(-0.548250\pi\)
−0.151001 + 0.988534i \(0.548250\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 38.8844i 2.16359i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 − 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 68.5857i − 3.71412i
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.4037i 1.73952i 0.493473 + 0.869761i \(0.335727\pi\)
−0.493473 + 0.869761i \(0.664273\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.9444i 1.43411i 0.697019 + 0.717053i \(0.254509\pi\)
−0.697019 + 0.717053i \(0.745491\pi\)
\(354\) 0 0
\(355\) −15.8745 −0.842531
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 19.4422i − 1.02612i −0.858352 − 0.513061i \(-0.828512\pi\)
0.858352 − 0.513061i \(-0.171488\pi\)
\(360\) 0 0
\(361\) 9.00000 0.473684
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 26.4575 1.38107 0.690535 − 0.723299i \(-0.257375\pi\)
0.690535 + 0.723299i \(0.257375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −34.0000 −1.76045 −0.880227 − 0.474554i \(-0.842610\pi\)
−0.880227 + 0.474554i \(0.842610\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 42.0000 2.14052
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −47.6235 −2.40843
\(392\) 0 0
\(393\) 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
\(400\) 0 0
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 51.8459i − 2.56991i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 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
\(421\) 40.0000 1.94948 0.974740 − 0.223341i \(-0.0716964\pi\)
0.974740 + 0.223341i \(0.0716964\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7.34847i 0.356453i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 6.48074i − 0.312166i −0.987744 − 0.156083i \(-0.950113\pi\)
0.987744 − 0.156083i \(-0.0498868\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 34.2929i 1.64045i
\(438\) 0 0
\(439\) −5.29150 −0.252550 −0.126275 − 0.991995i \(-0.540302\pi\)
−0.126275 + 0.991995i \(0.540302\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 32.4037i − 1.53955i −0.638317 − 0.769773i \(-0.720369\pi\)
0.638317 − 0.769773i \(-0.279631\pi\)
\(444\) 0 0
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −79.3725 −3.73751
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.2474i 0.570421i 0.958465 + 0.285210i \(0.0920634\pi\)
−0.958465 + 0.285210i \(0.907937\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 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
\(475\) 5.29150 0.242791
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 6.48074i − 0.292472i −0.989250 − 0.146236i \(-0.953284\pi\)
0.989250 − 0.146236i \(-0.0467158\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 17.1464i − 0.769122i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 36.7423i 1.62858i 0.580461 + 0.814288i \(0.302872\pi\)
−0.580461 + 0.814288i \(0.697128\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 25.9230i − 1.14230i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 41.6413i − 1.82434i −0.409812 − 0.912170i \(-0.634406\pi\)
0.409812 − 0.912170i \(-0.365594\pi\)
\(522\) 0 0
\(523\) −42.3320 −1.85105 −0.925525 − 0.378686i \(-0.876376\pi\)
−0.925525 + 0.378686i \(0.876376\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 77.7689i 3.38767i
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −47.6235 −2.05894
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 45.3652i 1.95402i
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 24.4949i − 1.04925i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 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
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.48074i 0.270266i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 56.0000 2.30744
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.44949i 0.100588i 0.998734 + 0.0502942i \(0.0160159\pi\)
−0.998734 + 0.0502942i \(0.983984\pi\)
\(594\) 0 0
\(595\) −47.6235 −1.95237
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 45.3652i − 1.85357i −0.375591 − 0.926786i \(-0.622560\pi\)
0.375591 − 0.926786i \(-0.377440\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 75.9342i − 3.08716i
\(606\) 0 0
\(607\) 26.4575 1.07388 0.536939 − 0.843621i \(-0.319581\pi\)
0.536939 + 0.843621i \(0.319581\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −46.0000 −1.85792 −0.928961 − 0.370177i \(-0.879297\pi\)
−0.928961 + 0.370177i \(0.879297\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 21.1660 0.850734 0.425367 − 0.905021i \(-0.360145\pi\)
0.425367 + 0.905021i \(0.360145\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.48074i 0.259645i
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 58.7878i 2.34402i
\(630\) 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
\(641\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −37.0405 −1.46074 −0.730368 − 0.683054i \(-0.760651\pi\)
−0.730368 + 0.683054i \(0.760651\pi\)
\(644\) 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
\(653\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 45.3652i 1.76718i 0.468264 + 0.883588i \(0.344879\pi\)
−0.468264 + 0.883588i \(0.655121\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 34.2929i 1.32982i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 − 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.8434i 1.22384i 0.790920 + 0.611920i \(0.209603\pi\)
−0.790920 + 0.611920i \(0.790397\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 32.4037i − 1.23989i −0.784644 − 0.619947i \(-0.787154\pi\)
0.784644 − 0.619947i \(-0.212846\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −42.3320 −1.61039 −0.805193 − 0.593013i \(-0.797938\pi\)
−0.805193 + 0.593013i \(0.797938\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 51.8459i 1.96663i
\(696\) 0 0
\(697\) 90.0000 3.40899
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 42.3320 1.59658
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 19.4422i − 0.731200i
\(708\) 0 0
\(709\) 50.0000 1.87779 0.938895 − 0.344204i \(-0.111851\pi\)
0.938895 + 0.344204i \(0.111851\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 68.5857i 2.56856i
\(714\) 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
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 52.9150 1.96251 0.981255 − 0.192715i \(-0.0617292\pi\)
0.981255 + 0.192715i \(0.0617292\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 32.4037i − 1.18878i −0.804178 − 0.594388i \(-0.797394\pi\)
0.804178 − 0.594388i \(-0.202606\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 51.4393i − 1.87955i
\(750\) 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
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 36.7423i − 1.33191i −0.745992 − 0.665955i \(-0.768024\pi\)
0.745992 − 0.665955i \(-0.231976\pi\)
\(762\) 0 0
\(763\) 26.4575 0.957826
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 26.9444i − 0.969122i −0.874757 − 0.484561i \(-0.838979\pi\)
0.874757 − 0.484561i \(-0.161021\pi\)
\(774\) 0 0
\(775\) 10.5830 0.380153
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 64.8074i − 2.32197i
\(780\) 0 0
\(781\) −42.0000 −1.50288
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 21.1660 0.754487 0.377243 − 0.926114i \(-0.376872\pi\)
0.377243 + 0.926114i \(0.376872\pi\)
\(788\) 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
\(797\) − 56.3383i − 1.99560i −0.0662682 − 0.997802i \(-0.521109\pi\)
0.0662682 − 0.997802i \(-0.478891\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −42.0000 −1.48031
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −42.3320 −1.48648 −0.743239 − 0.669026i \(-0.766712\pi\)
−0.743239 + 0.669026i \(0.766712\pi\)
\(812\) 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
\(821\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 45.3652i − 1.57750i −0.614713 − 0.788751i \(-0.710728\pi\)
0.614713 − 0.788751i \(-0.289272\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 51.4393i − 1.78227i
\(834\) 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
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 31.8434i 1.09545i
\(846\) 0 0
\(847\) 82.0183 2.81818
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 51.8459i 1.77726i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.6413i 1.42244i 0.702969 + 0.711220i \(0.251857\pi\)
−0.702969 + 0.711220i \(0.748143\pi\)
\(858\) 0 0
\(859\) 58.2065 1.98598 0.992991 − 0.118194i \(-0.0377103\pi\)
0.992991 + 0.118194i \(0.0377103\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 58.3267i − 1.98546i −0.120351 − 0.992731i \(-0.538402\pi\)
0.120351 − 0.992731i \(-0.461598\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 25.9230i − 0.876356i
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 − 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 56.3383i 1.89808i 0.315149 + 0.949042i \(0.397945\pi\)
−0.315149 + 0.949042i \(0.602055\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 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
\(895\) 47.6235 1.59188
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 58.3267i 1.93245i 0.257702 + 0.966224i \(0.417035\pi\)
−0.257702 + 0.966224i \(0.582965\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 36.7423i − 1.20548i −0.797939 − 0.602739i \(-0.794076\pi\)
0.797939 − 0.602739i \(-0.205924\pi\)
\(930\) 0 0
\(931\) −37.0405 −1.21395
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 116.653i 3.81497i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 61.2372i 1.99628i 0.0609873 + 0.998139i \(0.480575\pi\)
−0.0609873 + 0.998139i \(0.519425\pi\)
\(942\) 0 0
\(943\) 79.3725 2.58473
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.4037i 1.05298i 0.850182 + 0.526489i \(0.176492\pi\)
−0.850182 + 0.526489i \(0.823508\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 15.8745 0.513687
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 81.0000 2.61290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.79796i 0.315407i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −56.0000 −1.79528
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 15.8745 0.507351
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 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
\(995\) − 12.9615i − 0.410907i
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 1008.2.b.i.559.3 yes 4
3.2 odd 2 inner 1008.2.b.i.559.1 ✓ 4
4.3 odd 2 inner 1008.2.b.i.559.4 yes 4
7.6 odd 2 inner 1008.2.b.i.559.2 yes 4
8.3 odd 2 4032.2.b.l.3583.2 4
8.5 even 2 4032.2.b.l.3583.1 4
12.11 even 2 inner 1008.2.b.i.559.2 yes 4
21.20 even 2 inner 1008.2.b.i.559.4 yes 4
24.5 odd 2 4032.2.b.l.3583.3 4
24.11 even 2 4032.2.b.l.3583.4 4
28.27 even 2 inner 1008.2.b.i.559.1 ✓ 4
56.13 odd 2 4032.2.b.l.3583.4 4
56.27 even 2 4032.2.b.l.3583.3 4
84.83 odd 2 CM 1008.2.b.i.559.3 yes 4
168.83 odd 2 4032.2.b.l.3583.1 4
168.125 even 2 4032.2.b.l.3583.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1008.2.b.i.559.1 ✓ 4 3.2 odd 2 inner
1008.2.b.i.559.1 ✓ 4 28.27 even 2 inner
1008.2.b.i.559.2 yes 4 7.6 odd 2 inner
1008.2.b.i.559.2 yes 4 12.11 even 2 inner
1008.2.b.i.559.3 yes 4 1.1 even 1 trivial
1008.2.b.i.559.3 yes 4 84.83 odd 2 CM
1008.2.b.i.559.4 yes 4 4.3 odd 2 inner
1008.2.b.i.559.4 yes 4 21.20 even 2 inner
4032.2.b.l.3583.1 4 8.5 even 2
4032.2.b.l.3583.1 4 168.83 odd 2
4032.2.b.l.3583.2 4 8.3 odd 2
4032.2.b.l.3583.2 4 168.125 even 2
4032.2.b.l.3583.3 4 24.5 odd 2
4032.2.b.l.3583.3 4 56.27 even 2
4032.2.b.l.3583.4 4 24.11 even 2
4032.2.b.l.3583.4 4 56.13 odd 2