Properties

Label 1000.1.e.a.499.2
Level $1000$
Weight $1$
Character 1000.499
Self dual yes
Analytic conductor $0.499$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -40
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1000.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.499065012633\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.1000000.1

Embedding invariants

Embedding label 499.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1000.499

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.618034 q^{11} -0.618034 q^{13} -1.61803 q^{14} +1.00000 q^{16} -1.00000 q^{18} -1.61803 q^{19} -0.618034 q^{22} -0.618034 q^{23} +0.618034 q^{26} +1.61803 q^{28} -1.00000 q^{32} +1.00000 q^{36} +1.61803 q^{37} +1.61803 q^{38} -1.61803 q^{41} +0.618034 q^{44} +0.618034 q^{46} -0.618034 q^{47} +1.61803 q^{49} -0.618034 q^{52} +1.61803 q^{53} -1.61803 q^{56} -1.61803 q^{59} +1.61803 q^{63} +1.00000 q^{64} -1.00000 q^{72} -1.61803 q^{74} -1.61803 q^{76} +1.00000 q^{77} +1.00000 q^{81} +1.61803 q^{82} -0.618034 q^{88} +0.618034 q^{89} -1.00000 q^{91} -0.618034 q^{92} +0.618034 q^{94} -1.61803 q^{98} +0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + q^{7} - 2q^{8} + 2q^{9} - q^{11} + q^{13} - q^{14} + 2q^{16} - 2q^{18} - q^{19} + q^{22} + q^{23} - q^{26} + q^{28} - 2q^{32} + 2q^{36} + q^{37} + q^{38} - q^{41} - q^{44} - q^{46} + q^{47} + q^{49} + q^{52} + q^{53} - q^{56} - q^{59} + q^{63} + 2q^{64} - 2q^{72} - q^{74} - q^{76} + 2q^{77} + 2q^{81} + q^{82} + q^{88} - q^{89} - 2q^{91} + q^{92} - q^{94} - q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(8\) −1.00000 −1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(12\) 0 0
\(13\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(14\) −1.61803 −1.61803
\(15\) 0 0
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.00000 −1.00000
\(19\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.618034 −0.618034
\(23\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.618034 0.618034
\(27\) 0 0
\(28\) 1.61803 1.61803
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.00000 −1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 1.00000
\(37\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(38\) 1.61803 1.61803
\(39\) 0 0
\(40\) 0 0
\(41\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0.618034 0.618034
\(45\) 0 0
\(46\) 0.618034 0.618034
\(47\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(48\) 0 0
\(49\) 1.61803 1.61803
\(50\) 0 0
\(51\) 0 0
\(52\) −0.618034 −0.618034
\(53\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.61803 −1.61803
\(57\) 0 0
\(58\) 0 0
\(59\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 1.61803 1.61803
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −1.00000 −1.00000
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −1.61803 −1.61803
\(75\) 0 0
\(76\) −1.61803 −1.61803
\(77\) 1.00000 1.00000
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 1.61803 1.61803
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −0.618034 −0.618034
\(89\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) −0.618034 −0.618034
\(93\) 0 0
\(94\) 0.618034 0.618034
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.61803 −1.61803
\(99\) 0.618034 0.618034
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(104\) 0.618034 0.618034
\(105\) 0 0
\(106\) −1.61803 −1.61803
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.61803 1.61803
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.618034 −0.618034
\(118\) 1.61803 1.61803
\(119\) 0 0
\(120\) 0 0
\(121\) −0.618034 −0.618034
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.61803 −1.61803
\(127\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(128\) −1.00000 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(132\) 0 0
\(133\) −2.61803 −2.61803
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.381966 −0.381966
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 1.61803 1.61803
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 1.61803 1.61803
\(153\) 0 0
\(154\) −1.00000 −1.00000
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.00000 −1.00000
\(162\) −1.00000 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −1.61803 −1.61803
\(165\) 0 0
\(166\) 0 0
\(167\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(168\) 0 0
\(169\) −0.618034 −0.618034
\(170\) 0 0
\(171\) −1.61803 −1.61803
\(172\) 0 0
\(173\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.618034 0.618034
\(177\) 0 0
\(178\) −0.618034 −0.618034
\(179\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 1.00000 1.00000
\(183\) 0 0
\(184\) 0.618034 0.618034
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −0.618034 −0.618034
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.61803 1.61803
\(197\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(198\) −0.618034 −0.618034
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0.618034 0.618034
\(207\) −0.618034 −0.618034
\(208\) −0.618034 −0.618034
\(209\) −1.00000 −1.00000
\(210\) 0 0
\(211\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(212\) 1.61803 1.61803
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(224\) −1.61803 −1.61803
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0.618034 0.618034
\(235\) 0 0
\(236\) −1.61803 −1.61803
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(242\) 0.618034 0.618034
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000 1.00000
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(252\) 1.61803 1.61803
\(253\) −0.381966 −0.381966
\(254\) 2.00000 2.00000
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 2.61803 2.61803
\(260\) 0 0
\(261\) 0 0
\(262\) −0.618034 −0.618034
\(263\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.61803 2.61803
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(278\) −0.618034 −0.618034
\(279\) 0 0
\(280\) 0 0
\(281\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0.381966 0.381966
\(287\) −2.61803 −2.61803
\(288\) −1.00000 −1.00000
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.61803 −1.61803
\(297\) 0 0
\(298\) 0 0
\(299\) 0.381966 0.381966
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.61803 −1.61803
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 1.00000 1.00000
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 2.00000 2.00000
\(315\) 0 0
\(316\) 0 0
\(317\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 1.00000 1.00000
\(323\) 0 0
\(324\) 1.00000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 1.61803 1.61803
\(329\) −1.00000 −1.00000
\(330\) 0 0
\(331\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(332\) 0 0
\(333\) 1.61803 1.61803
\(334\) −1.61803 −1.61803
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0.618034 0.618034
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 1.61803 1.61803
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) −1.61803 −1.61803
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.618034 −0.618034
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.618034 0.618034
\(357\) 0 0
\(358\) 1.61803 1.61803
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.61803 1.61803
\(362\) 0 0
\(363\) 0 0
\(364\) −1.00000 −1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(368\) −0.618034 −0.618034
\(369\) −1.61803 −1.61803
\(370\) 0 0
\(371\) 2.61803 2.61803
\(372\) 0 0
\(373\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0.618034 0.618034
\(377\) 0 0
\(378\) 0 0
\(379\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.61803 −1.61803
\(393\) 0 0
\(394\) 2.00000 2.00000
\(395\) 0 0
\(396\) 0.618034 0.618034
\(397\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.00000 1.00000
\(408\) 0 0
\(409\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.618034 −0.618034
\(413\) −2.61803 −2.61803
\(414\) 0.618034 0.618034
\(415\) 0 0
\(416\) 0.618034 0.618034
\(417\) 0 0
\(418\) 1.00000 1.00000
\(419\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 1.61803 1.61803
\(423\) −0.618034 −0.618034
\(424\) −1.61803 −1.61803
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 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
\(441\) 1.61803 1.61803
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.00000 2.00000
\(447\) 0 0
\(448\) 1.61803 1.61803
\(449\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(450\) 0 0
\(451\) −1.00000 −1.00000
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −0.618034 −0.618034
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.61803 1.61803
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.61803 1.61803
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1.00000 −1.00000
\(482\) −0.618034 −0.618034
\(483\) 0 0
\(484\) −0.618034 −0.618034
\(485\) 0 0
\(486\) 0 0
\(487\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −1.00000 −1.00000
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.00000 −2.00000
\(503\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(504\) −1.61803 −1.61803
\(505\) 0 0
\(506\) 0.381966 0.381966
\(507\) 0 0
\(508\) −2.00000 −2.00000
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −0.381966 −0.381966
\(518\) −2.61803 −2.61803
\(519\) 0 0
\(520\) 0 0
\(521\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0.618034 0.618034
\(525\) 0 0
\(526\) −1.61803 −1.61803
\(527\) 0 0
\(528\) 0 0
\(529\) −0.618034 −0.618034
\(530\) 0 0
\(531\) −1.61803 −1.61803
\(532\) −2.61803 −2.61803
\(533\) 1.00000 1.00000
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.00000 1.00000
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.618034 0.618034
\(555\) 0 0
\(556\) 0.618034 0.618034
\(557\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.618034 −0.618034
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.61803 1.61803
\(568\) 0 0
\(569\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(570\) 0 0
\(571\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) −0.381966 −0.381966
\(573\) 0 0
\(574\) 2.61803 2.61803
\(575\) 0 0
\(576\) 1.00000 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −1.00000 −1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1.00000 1.00000
\(584\) 0 0
\(585\) 0 0
\(586\) 0.618034 0.618034
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.61803 1.61803
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −0.381966 −0.381966
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(608\) 1.61803 1.61803
\(609\) 0 0
\(610\) 0 0
\(611\) 0.381966 0.381966
\(612\) 0 0
\(613\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −1.00000 −1.00000
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.00000 1.00000
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −2.00000 −2.00000
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.618034 0.618034
\(635\) 0 0
\(636\) 0 0
\(637\) −1.00000 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) −1.00000 −1.00000
\(645\) 0 0
\(646\) 0 0
\(647\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(648\) −1.00000 −1.00000
\(649\) −1.00000 −1.00000
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.61803 −1.61803
\(657\) 0 0
\(658\) 1.00000 1.00000
\(659\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −2.00000 −2.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.61803 −1.61803
\(667\) 0 0
\(668\) 1.61803 1.61803
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −0.618034 −0.618034
\(677\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −1.61803 −1.61803
\(685\) 0 0
\(686\) −1.00000 −1.00000
\(687\) 0 0
\(688\) 0 0
\(689\) −1.00000 −1.00000
\(690\) 0 0
\(691\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(692\) 1.61803 1.61803
\(693\) 1.00000 1.00000
\(694\) 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
\(703\) −2.61803 −2.61803
\(704\) 0.618034 0.618034
\(705\) 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
\(712\) −0.618034 −0.618034
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.61803 −1.61803
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1.00000 −1.00000
\(722\) −1.61803 −1.61803
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(728\) 1.00000 1.00000
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(734\) 2.00000 2.00000
\(735\) 0 0
\(736\) 0.618034 0.618034
\(737\) 0 0
\(738\) 1.61803 1.61803
\(739\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.61803 −2.61803
\(743\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.61803 −1.61803
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −0.618034 −0.618034
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(758\) −0.618034 −0.618034
\(759\) 0 0
\(760\) 0 0
\(761\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −1.61803 −1.61803
\(767\) 1.00000 1.00000
\(768\) 0 0
\(769\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.61803 2.61803
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.61803 1.61803
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −2.00000 −2.00000
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.618034 −0.618034
\(793\) 0 0
\(794\) −1.61803 −1.61803
\(795\) 0 0
\(796\) 0 0
\(797\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.618034 0.618034
\(802\) 1.61803 1.61803
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) 0 0
\(811\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −1.00000 −1.00000
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.61803 1.61803
\(819\) −1.00000 −1.00000
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(824\) 0.618034 0.618034
\(825\) 0 0
\(826\) 2.61803 2.61803
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −0.618034 −0.618034
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.618034 −0.618034
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −1.00000 −1.00000
\(837\) 0 0
\(838\) −2.00000 −2.00000
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −1.61803 −1.61803
\(845\) 0 0
\(846\) 0.618034 0.618034
\(847\) −1.00000 −1.00000
\(848\) 1.61803 1.61803
\(849\) 0 0
\(850\) 0 0
\(851\) −1.00000 −1.00000
\(852\) 0 0
\(853\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) −1.00000 −1.00000
\(875\) 0 0
\(876\) 0 0
\(877\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(882\) −1.61803 −1.61803
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(888\) 0 0
\(889\) −3.23607 −3.23607
\(890\) 0 0
\(891\) 0.618034 0.618034
\(892\) −2.00000 −2.00000
\(893\) 1.00000 1.00000
\(894\) 0 0
\(895\) 0 0
\(896\) −1.61803 −1.61803
\(897\) 0 0
\(898\) 1.61803 1.61803
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 1.00000 1.00000
\(903\) 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
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.00000 1.00000
\(918\) 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
\(926\) 2.00000 2.00000
\(927\) −0.618034 −0.618034
\(928\) 0 0
\(929\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(930\) 0 0
\(931\) −2.61803 −2.61803
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.618034 0.618034
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 1.00000 1.00000
\(944\) −1.61803 −1.61803
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) −1.61803 −1.61803
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 1.00000 1.00000
\(963\) 0 0
\(964\) 0.618034 0.618034
\(965\) 0 0
\(966\) 0 0
\(967\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(968\) 0.618034 0.618034
\(969\) 0 0
\(970\) 0 0
\(971\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(972\) 0 0
\(973\) 1.00000 1.00000
\(974\) 0.618034 0.618034
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0.381966 0.381966
\(980\) 0 0
\(981\) 0 0
\(982\) 1.61803 1.61803
\(983\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.00000 1.00000
\(989\) 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
\(997\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(998\) −0.618034 −0.618034
\(999\) 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 1000.1.e.a.499.2 2
4.3 odd 2 4000.1.e.a.1999.1 2
5.2 odd 4 1000.1.g.a.251.2 4
5.3 odd 4 1000.1.g.a.251.3 yes 4
5.4 even 2 1000.1.e.b.499.1 2
8.3 odd 2 1000.1.e.b.499.1 2
8.5 even 2 4000.1.e.b.1999.2 2
20.3 even 4 4000.1.g.a.751.4 4
20.7 even 4 4000.1.g.a.751.1 4
20.19 odd 2 4000.1.e.b.1999.2 2
40.3 even 4 1000.1.g.a.251.2 4
40.13 odd 4 4000.1.g.a.751.1 4
40.19 odd 2 CM 1000.1.e.a.499.2 2
40.27 even 4 1000.1.g.a.251.3 yes 4
40.29 even 2 4000.1.e.a.1999.1 2
40.37 odd 4 4000.1.g.a.751.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.1.e.a.499.2 2 1.1 even 1 trivial
1000.1.e.a.499.2 2 40.19 odd 2 CM
1000.1.e.b.499.1 2 5.4 even 2
1000.1.e.b.499.1 2 8.3 odd 2
1000.1.g.a.251.2 4 5.2 odd 4
1000.1.g.a.251.2 4 40.3 even 4
1000.1.g.a.251.3 yes 4 5.3 odd 4
1000.1.g.a.251.3 yes 4 40.27 even 4
4000.1.e.a.1999.1 2 4.3 odd 2
4000.1.e.a.1999.1 2 40.29 even 2
4000.1.e.b.1999.2 2 8.5 even 2
4000.1.e.b.1999.2 2 20.19 odd 2
4000.1.g.a.751.1 4 20.7 even 4
4000.1.g.a.751.1 4 40.13 odd 4
4000.1.g.a.751.4 4 20.3 even 4
4000.1.g.a.751.4 4 40.37 odd 4