Properties

Label 131.1.b.a.130.1
Level 131
Weight 1
Character 131.130
Self dual yes
Analytic conductor 0.065
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -131
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 131 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 131.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0653775166549\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.17161.1
Artin image $D_5$
Artin field Galois closure of 5.1.17161.1

Embedding invariants

Embedding label 130.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 131.130

$q$-expansion

\(f(q)\) \(=\) \(q-1.61803 q^{3} +1.00000 q^{4} +0.618034 q^{5} +0.618034 q^{7} +1.61803 q^{9} +O(q^{10})\) \(q-1.61803 q^{3} +1.00000 q^{4} +0.618034 q^{5} +0.618034 q^{7} +1.61803 q^{9} -1.61803 q^{11} -1.61803 q^{12} -1.61803 q^{13} -1.00000 q^{15} +1.00000 q^{16} +0.618034 q^{20} -1.00000 q^{21} -0.618034 q^{25} -1.00000 q^{27} +0.618034 q^{28} +2.61803 q^{33} +0.381966 q^{35} +1.61803 q^{36} +2.61803 q^{39} -1.61803 q^{41} +0.618034 q^{43} -1.61803 q^{44} +1.00000 q^{45} -1.61803 q^{48} -0.618034 q^{49} -1.61803 q^{52} +2.00000 q^{53} -1.00000 q^{55} +0.618034 q^{59} -1.00000 q^{60} +0.618034 q^{61} +1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +1.00000 q^{75} -1.00000 q^{77} +0.618034 q^{80} -1.00000 q^{84} +2.00000 q^{89} -1.00000 q^{91} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + 2q^{4} - q^{5} - q^{7} + q^{9} + O(q^{10}) \) \( 2q - q^{3} + 2q^{4} - q^{5} - q^{7} + q^{9} - q^{11} - q^{12} - q^{13} - 2q^{15} + 2q^{16} - q^{20} - 2q^{21} + q^{25} - 2q^{27} - q^{28} + 3q^{33} + 3q^{35} + q^{36} + 3q^{39} - q^{41} - q^{43} - q^{44} + 2q^{45} - q^{48} + q^{49} - q^{52} + 4q^{53} - 2q^{55} - q^{59} - 2q^{60} - q^{61} + 2q^{63} + 2q^{64} - 2q^{65} + 2q^{75} - 2q^{77} - q^{80} - 2q^{84} + 4q^{89} - 2q^{91} - 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/131\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(4\) 1.00000 1.00000
\(5\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(6\) 0 0
\(7\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(8\) 0 0
\(9\) 1.61803 1.61803
\(10\) 0 0
\(11\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(12\) −1.61803 −1.61803
\(13\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(14\) 0 0
\(15\) −1.00000 −1.00000
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0.618034 0.618034
\(21\) −1.00000 −1.00000
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.618034 −0.618034
\(26\) 0 0
\(27\) −1.00000 −1.00000
\(28\) 0.618034 0.618034
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 2.61803 2.61803
\(34\) 0 0
\(35\) 0.381966 0.381966
\(36\) 1.61803 1.61803
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 2.61803 2.61803
\(40\) 0 0
\(41\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(42\) 0 0
\(43\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(44\) −1.61803 −1.61803
\(45\) 1.00000 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.61803 −1.61803
\(49\) −0.618034 −0.618034
\(50\) 0 0
\(51\) 0 0
\(52\) −1.61803 −1.61803
\(53\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(54\) 0 0
\(55\) −1.00000 −1.00000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) −1.00000 −1.00000
\(61\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(62\) 0 0
\(63\) 1.00000 1.00000
\(64\) 1.00000 1.00000
\(65\) −1.00000 −1.00000
\(66\) 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 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) −1.00000 −1.00000
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0.618034 0.618034
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −1.00000 −1.00000
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(90\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −2.61803 −2.61803
\(100\) −0.618034 −0.618034
\(101\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −0.618034 −0.618034
\(106\) 0 0
\(107\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(108\) −1.00000 −1.00000
\(109\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.618034 0.618034
\(113\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.61803 −2.61803
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.61803 1.61803
\(122\) 0 0
\(123\) 2.61803 2.61803
\(124\) 0 0
\(125\) −1.00000 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) −1.00000 −1.00000
\(130\) 0 0
\(131\) 1.00000 1.00000
\(132\) 2.61803 2.61803
\(133\) 0 0
\(134\) 0 0
\(135\) −0.618034 −0.618034
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0.381966 0.381966
\(141\) 0 0
\(142\) 0 0
\(143\) 2.61803 2.61803
\(144\) 1.61803 1.61803
\(145\) 0 0
\(146\) 0 0
\(147\) 1.00000 1.00000
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 2.61803 2.61803
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) −3.23607 −3.23607
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −1.61803 −1.61803
\(165\) 1.61803 1.61803
\(166\) 0 0
\(167\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(168\) 0 0
\(169\) 1.61803 1.61803
\(170\) 0 0
\(171\) 0 0
\(172\) 0.618034 0.618034
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −0.381966 −0.381966
\(176\) −1.61803 −1.61803
\(177\) −1.00000 −1.00000
\(178\) 0 0
\(179\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(180\) 1.00000 1.00000
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −1.00000 −1.00000
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.618034 −0.618034
\(190\) 0 0
\(191\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(192\) −1.61803 −1.61803
\(193\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(194\) 0 0
\(195\) 1.61803 1.61803
\(196\) −0.618034 −0.618034
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 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
\(205\) −1.00000 −1.00000
\(206\) 0 0
\(207\) 0 0
\(208\) −1.61803 −1.61803
\(209\) 0 0
\(210\) 0 0
\(211\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(212\) 2.00000 2.00000
\(213\) 0 0
\(214\) 0 0
\(215\) 0.381966 0.381966
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.00000 −1.00000
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −1.00000 −1.00000
\(226\) 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
\(231\) 1.61803 1.61803
\(232\) 0 0
\(233\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0.618034 0.618034
\(237\) 0 0
\(238\) 0 0
\(239\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(240\) −1.00000 −1.00000
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 1.00000 1.00000
\(244\) 0.618034 0.618034
\(245\) −0.381966 −0.381966
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.00000 1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.00000 −1.00000
\(261\) 0 0
\(262\) 0 0
\(263\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(264\) 0 0
\(265\) 1.23607 1.23607
\(266\) 0 0
\(267\) −3.23607 −3.23607
\(268\) 0 0
\(269\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(270\) 0 0
\(271\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(272\) 0 0
\(273\) 1.61803 1.61803
\(274\) 0 0
\(275\) 1.00000 1.00000
\(276\) 0 0
\(277\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.00000 −1.00000
\(288\) 0 0
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0.381966 0.381966
\(296\) 0 0
\(297\) 1.61803 1.61803
\(298\) 0 0
\(299\) 0 0
\(300\) 1.00000 1.00000
\(301\) 0.381966 0.381966
\(302\) 0 0
\(303\) −1.00000 −1.00000
\(304\) 0 0
\(305\) 0.381966 0.381966
\(306\) 0 0
\(307\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(308\) −1.00000 −1.00000
\(309\) 0 0
\(310\) 0 0
\(311\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0.618034 0.618034
\(316\) 0 0
\(317\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.618034 0.618034
\(321\) −1.00000 −1.00000
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.00000 1.00000
\(326\) 0 0
\(327\) 2.61803 2.61803
\(328\) 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
\(336\) −1.00000 −1.00000
\(337\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(338\) 0 0
\(339\) 2.61803 2.61803
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 1.61803 1.61803
\(352\) 0 0
\(353\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.00000 2.00000
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) −2.61803 −2.61803
\(364\) −1.00000 −1.00000
\(365\) 0 0
\(366\) 0 0
\(367\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(368\) 0 0
\(369\) −2.61803 −2.61803
\(370\) 0 0
\(371\) 1.23607 1.23607
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 1.61803 1.61803
\(376\) 0 0
\(377\) 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 \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(384\) 0 0
\(385\) −0.618034 −0.618034
\(386\) 0 0
\(387\) 1.00000 1.00000
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.61803 −1.61803
\(394\) 0 0
\(395\) 0 0
\(396\) −2.61803 −2.61803
\(397\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.618034 −0.618034
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0.618034 0.618034
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.381966 0.381966
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) −0.618034 −0.618034
\(421\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.381966 0.381966
\(428\) 0.618034 0.618034
\(429\) −4.23607 −4.23607
\(430\) 0 0
\(431\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(432\) −1.00000 −1.00000
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.61803 −1.61803
\(437\) 0 0
\(438\) 0 0
\(439\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(440\) 0 0
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 1.23607 1.23607
\(446\) 0 0
\(447\) 0 0
\(448\) 0.618034 0.618034
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 2.61803 2.61803
\(452\) −1.61803 −1.61803
\(453\) 2.61803 2.61803
\(454\) 0 0
\(455\) −0.618034 −0.618034
\(456\) 0 0
\(457\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(468\) −2.61803 −2.61803
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.00000 −1.00000
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.23607 3.23607
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.61803 1.61803
\(485\) 0 0
\(486\) 0 0
\(487\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 2.61803 2.61803
\(493\) 0 0
\(494\) 0 0
\(495\) −1.61803 −1.61803
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −1.00000 −1.00000
\(501\) −3.23607 −3.23607
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0.381966 0.381966
\(506\) 0 0
\(507\) −2.61803 −2.61803
\(508\) 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
\(516\) −1.00000 −1.00000
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 1.00000 1.00000
\(525\) 0.618034 0.618034
\(526\) 0 0
\(527\) 0 0
\(528\) 2.61803 2.61803
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 1.00000 1.00000
\(532\) 0 0
\(533\) 2.61803 2.61803
\(534\) 0 0
\(535\) 0.381966 0.381966
\(536\) 0 0
\(537\) −1.00000 −1.00000
\(538\) 0 0
\(539\) 1.00000 1.00000
\(540\) −0.618034 −0.618034
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.00000 −1.00000
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 1.00000 1.00000
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(558\) 0 0
\(559\) −1.00000 −1.00000
\(560\) 0.381966 0.381966
\(561\) 0 0
\(562\) 0 0
\(563\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(564\) 0 0
\(565\) −1.00000 −1.00000
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 2.61803 2.61803
\(573\) 2.61803 2.61803
\(574\) 0 0
\(575\) 0 0
\(576\) 1.61803 1.61803
\(577\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) 0 0
\(579\) −1.00000 −1.00000
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −3.23607 −3.23607
\(584\) 0 0
\(585\) −1.61803 −1.61803
\(586\) 0 0
\(587\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(588\) 1.00000 1.00000
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(600\) 0 0
\(601\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.61803 −1.61803
\(605\) 1.00000 1.00000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) 1.61803 1.61803
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.23607 1.23607
\(624\) 2.61803 2.61803
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(632\) 0 0
\(633\) 2.61803 2.61803
\(634\) 0 0
\(635\) 0 0
\(636\) −3.23607 −3.23607
\(637\) 1.00000 1.00000
\(638\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) −0.618034 −0.618034
\(646\) 0 0
\(647\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(648\) 0 0
\(649\) −1.00000 −1.00000
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(654\) 0 0
\(655\) 0.618034 0.618034
\(656\) −1.61803 −1.61803
\(657\) 0 0
\(658\) 0 0
\(659\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(660\) 1.61803 1.61803
\(661\) 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
\(668\) 2.00000 2.00000
\(669\) 0 0
\(670\) 0 0
\(671\) −1.00000 −1.00000
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0.618034 0.618034
\(676\) 1.61803 1.61803
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0.618034 0.618034
\(689\) −3.23607 −3.23607
\(690\) 0 0
\(691\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(692\) 0 0
\(693\) −1.61803 −1.61803
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1.00000 −1.00000
\(700\) −0.381966 −0.381966
\(701\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.61803 −1.61803
\(705\) 0 0
\(706\) 0 0
\(707\) 0.381966 0.381966
\(708\) −1.00000 −1.00000
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 1.61803 1.61803
\(716\) 0.618034 0.618034
\(717\) −1.00000 −1.00000
\(718\) 0 0
\(719\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(720\) 1.00000 1.00000
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −1.61803 −1.61803
\(730\) 0 0
\(731\) 0 0
\(732\) −1.00000 −1.00000
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0.618034 0.618034
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.381966 0.381966
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.00000 −1.00000
\(756\) −0.618034 −0.618034
\(757\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.00000 −1.00000
\(764\) −1.61803 −1.61803
\(765\) 0 0
\(766\) 0 0
\(767\) −1.00000 −1.00000
\(768\) −1.61803 −1.61803
\(769\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.618034 0.618034
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 1.61803 1.61803
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.618034 −0.618034
\(785\) 0 0
\(786\) 0 0
\(787\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(788\) 0 0
\(789\) 2.61803 2.61803
\(790\) 0 0
\(791\) −1.00000 −1.00000
\(792\) 0 0
\(793\) −1.00000 −1.00000
\(794\) 0 0
\(795\) −2.00000 −2.00000
\(796\) 0 0
\(797\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 3.23607 3.23607
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.61803 2.61803
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) 2.61803 2.61803
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.61803 −1.61803
\(820\) −1.00000 −1.00000
\(821\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −1.61803 −1.61803
\(826\) 0 0
\(827\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(828\) 0 0
\(829\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(830\) 0 0
\(831\) 2.61803 2.61803
\(832\) −1.61803 −1.61803
\(833\) 0 0
\(834\) 0 0
\(835\) 1.23607 1.23607
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −1.61803 −1.61803
\(845\) 1.00000 1.00000
\(846\) 0 0
\(847\) 1.00000 1.00000
\(848\) 2.00000 2.00000
\(849\) 2.61803 2.61803
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0.381966 0.381966
\(861\) 1.61803 1.61803
\(862\) 0 0
\(863\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −1.61803 −1.61803
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.618034 −0.618034
\(876\) 0 0
\(877\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −1.00000 −1.00000
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) −0.618034 −0.618034
\(886\) 0 0
\(887\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0.381966 0.381966
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −1.00000 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −0.618034 −0.618034
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(908\) 0 0
\(909\) 1.00000 1.00000
\(910\) 0 0
\(911\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −0.618034 −0.618034
\(916\) 0 0
\(917\) 0.618034 0.618034
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −3.23607 −3.23607
\(922\) 0 0
\(923\) 0 0
\(924\) 1.61803 1.61803
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.618034 0.618034
\(933\) −1.00000 −1.00000
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.618034 0.618034
\(945\) −0.381966 −0.381966
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.61803 2.61803
\(952\) 0 0
\(953\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(954\) 0 0
\(955\) −1.00000 −1.00000
\(956\) 0.618034 0.618034
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) −1.00000 −1.00000
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 1.00000 1.00000
\(964\) 0 0
\(965\) 0.381966 0.381966
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 1.00000 1.00000
\(973\) 0 0
\(974\) 0 0
\(975\) −1.61803 −1.61803
\(976\) 0.618034 0.618034
\(977\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(978\) 0 0
\(979\) −3.23607 −3.23607
\(980\) −0.381966 −0.381966
\(981\) −2.61803 −2.61803
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(998\) 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 131.1.b.a.130.1 2
3.2 odd 2 1179.1.c.a.523.1 2
4.3 odd 2 2096.1.h.b.785.2 2
5.2 odd 4 3275.1.d.a.3274.4 4
5.3 odd 4 3275.1.d.a.3274.1 4
5.4 even 2 3275.1.c.d.2226.2 2
131.130 odd 2 CM 131.1.b.a.130.1 2
393.392 even 2 1179.1.c.a.523.1 2
524.523 even 2 2096.1.h.b.785.2 2
655.392 even 4 3275.1.d.a.3274.4 4
655.523 even 4 3275.1.d.a.3274.1 4
655.654 odd 2 3275.1.c.d.2226.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
131.1.b.a.130.1 2 1.1 even 1 trivial
131.1.b.a.130.1 2 131.130 odd 2 CM
1179.1.c.a.523.1 2 3.2 odd 2
1179.1.c.a.523.1 2 393.392 even 2
2096.1.h.b.785.2 2 4.3 odd 2
2096.1.h.b.785.2 2 524.523 even 2
3275.1.c.d.2226.2 2 5.4 even 2
3275.1.c.d.2226.2 2 655.654 odd 2
3275.1.d.a.3274.1 4 5.3 odd 4
3275.1.d.a.3274.1 4 655.523 even 4
3275.1.d.a.3274.4 4 5.2 odd 4
3275.1.d.a.3274.4 4 655.392 even 4