Properties

Label 4000.1.g.a.751.2
Level 4000
Weight 1
Character 4000.751
Analytic conductor 1.996
Analytic rank 0
Dimension 4
Projective image \(D_{5}\)
CM discriminant -40
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1000)
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.1000000.1

Embedding invariants

Embedding label 751.2
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 4000.751
Dual form 4000.1.g.a.751.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-0.618034i q^{7} -1.00000 q^{9} +1.61803 q^{11} +1.61803i q^{13} +0.618034 q^{19} -1.61803i q^{23} +0.618034i q^{37} +0.618034 q^{41} +1.61803i q^{47} +0.618034 q^{49} -0.618034i q^{53} +0.618034 q^{59} +0.618034i q^{63} -1.00000i q^{77} +1.00000 q^{81} +1.61803 q^{89} +1.00000 q^{91} -1.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 2q^{11} - 2q^{19} - 2q^{41} - 2q^{49} - 2q^{59} + 4q^{81} + 2q^{89} + 4q^{91} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(8\) 0 0
\(9\) −1.00000 −1.00000
\(10\) 0 0
\(11\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(12\) 0 0
\(13\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 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 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(48\) 0 0
\(49\) 0.618034 0.618034
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0.618034i 0.618034i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.00000i − 1.00000i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(90\) 0 0
\(91\) 1.00000 1.00000
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −1.61803 −1.61803
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 1.61803i − 1.61803i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.61803 1.61803
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(132\) 0 0
\(133\) − 0.381966i − 0.381966i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.61803i 2.61803i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 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 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.00000 −1.00000
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(168\) 0 0
\(169\) −1.61803 −1.61803
\(170\) 0 0
\(171\) −0.618034 −0.618034
\(172\) 0 0
\(173\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(180\) 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
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 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 0
\(207\) 1.61803i 1.61803i
\(208\) 0 0
\(209\) 1.00000 1.00000
\(210\) 0 0
\(211\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(212\) 0 0
\(213\) 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
\(223\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 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
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00000i 1.00000i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) − 2.61803i − 2.61803i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0.381966 0.381966
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 0.381966i − 0.381966i
\(288\) 0 0
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.61803 2.61803
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.00000 1.00000
\(330\) 0 0
\(331\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) − 0.618034i − 0.618034i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 1.00000i − 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −0.618034 −0.618034
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(368\) 0 0
\(369\) −0.618034 −0.618034
\(370\) 0 0
\(371\) −0.381966 −0.381966
\(372\) 0 0
\(373\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(384\) 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
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.00000i 1.00000i
\(408\) 0 0
\(409\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 0.381966i − 0.381966i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) − 1.61803i − 1.61803i
\(424\) 0 0
\(425\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.00000i − 1.00000i
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −0.618034 −0.618034
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(450\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 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
\(475\) 0 0
\(476\) 0 0
\(477\) 0.618034i 0.618034i
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −1.00000 −1.00000
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(504\) 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
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.61803i 2.61803i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −1.61803 −1.61803
\(530\) 0 0
\(531\) −0.618034 −0.618034
\(532\) 0 0
\(533\) 1.00000i 1.00000i
\(534\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 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
\(557\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(558\) 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
\(567\) − 0.618034i − 0.618034i
\(568\) 0 0
\(569\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(570\) 0 0
\(571\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 1.00000i − 1.00000i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 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
\(601\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.61803 −2.61803
\(612\) 0 0
\(613\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1.00000i − 1.00000i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 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
\(637\) 1.00000i 1.00000i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(648\) 0 0
\(649\) 1.00000 1.00000
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.00000 1.00000
\(690\) 0 0
\(691\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 1.00000i 1.00000i
\(694\) 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
\(703\) 0.381966i 0.381966i
\(704\) 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
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1.00000 −1.00000
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(728\) 0 0
\(729\) −1.00000 −1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(744\) 0 0
\(745\) 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
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.00000i 1.00000i
\(768\) 0 0
\(769\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0.381966 0.381966
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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
\(797\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.61803 −1.61803
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(810\) 0 0
\(811\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.00000 −1.00000
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.00000i − 1.00000i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.00000 1.00000
\(852\) 0 0
\(853\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(864\) 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
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 0.618034i − 0.618034i −0.951057 0.309017i \(-0.900000\pi\)
0.951057 0.309017i \(-0.100000\pi\)
\(888\) 0 0
\(889\) −1.23607 −1.23607
\(890\) 0 0
\(891\) 1.61803 1.61803
\(892\) 0 0
\(893\) 1.00000i 1.00000i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 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
\(917\) − 1.00000i − 1.00000i
\(918\) 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
\(927\) 1.61803i 1.61803i
\(928\) 0 0
\(929\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(930\) 0 0
\(931\) 0.381966 0.381966
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) − 1.00000i − 1.00000i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 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
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.61803i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(972\) 0 0
\(973\) 1.00000i 1.00000i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 2.61803 2.61803
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1.61803i − 1.61803i −0.587785 0.809017i \(-0.700000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(984\) 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
\(997\) 0.618034i 0.618034i 0.951057 + 0.309017i \(0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(998\) 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 4000.1.g.a.751.2 4
4.3 odd 2 1000.1.g.a.251.4 yes 4
5.2 odd 4 4000.1.e.a.1999.2 2
5.3 odd 4 4000.1.e.b.1999.1 2
5.4 even 2 inner 4000.1.g.a.751.3 4
8.3 odd 2 inner 4000.1.g.a.751.3 4
8.5 even 2 1000.1.g.a.251.1 4
20.3 even 4 1000.1.e.b.499.2 2
20.7 even 4 1000.1.e.a.499.1 2
20.19 odd 2 1000.1.g.a.251.1 4
40.3 even 4 4000.1.e.a.1999.2 2
40.13 odd 4 1000.1.e.a.499.1 2
40.19 odd 2 CM 4000.1.g.a.751.2 4
40.27 even 4 4000.1.e.b.1999.1 2
40.29 even 2 1000.1.g.a.251.4 yes 4
40.37 odd 4 1000.1.e.b.499.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.1.e.a.499.1 2 20.7 even 4
1000.1.e.a.499.1 2 40.13 odd 4
1000.1.e.b.499.2 2 20.3 even 4
1000.1.e.b.499.2 2 40.37 odd 4
1000.1.g.a.251.1 4 8.5 even 2
1000.1.g.a.251.1 4 20.19 odd 2
1000.1.g.a.251.4 yes 4 4.3 odd 2
1000.1.g.a.251.4 yes 4 40.29 even 2
4000.1.e.a.1999.2 2 5.2 odd 4
4000.1.e.a.1999.2 2 40.3 even 4
4000.1.e.b.1999.1 2 5.3 odd 4
4000.1.e.b.1999.1 2 40.27 even 4
4000.1.g.a.751.2 4 1.1 even 1 trivial
4000.1.g.a.751.2 4 40.19 odd 2 CM
4000.1.g.a.751.3 4 5.4 even 2 inner
4000.1.g.a.751.3 4 8.3 odd 2 inner