Properties

Label 103.1.b.a.102.2
Level 103
Weight 1
Character 103.102
Self dual yes
Analytic conductor 0.051
Analytic rank 0
Dimension 2
Projective image \(D_{5}\)
CM discriminant -103
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 103.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.0514036963012\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.10609.1
Artin image $D_5$
Artin field Galois closure of 5.1.10609.1

Embedding invariants

Embedding label 102.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 103.102

$q$-expansion

\(f(q)\) \(=\) \(q+0.618034 q^{2} -0.618034 q^{4} -1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.618034 q^{2} -0.618034 q^{4} -1.61803 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.618034 q^{13} -1.00000 q^{14} -1.61803 q^{17} +0.618034 q^{18} +0.618034 q^{19} +0.618034 q^{23} +1.00000 q^{25} +0.381966 q^{26} +1.00000 q^{28} -1.61803 q^{29} +1.00000 q^{32} -1.00000 q^{34} -0.618034 q^{36} +0.381966 q^{38} +0.618034 q^{41} +0.381966 q^{46} +1.61803 q^{49} +0.618034 q^{50} -0.381966 q^{52} +1.61803 q^{56} -1.00000 q^{58} -1.61803 q^{59} -1.61803 q^{61} -1.61803 q^{63} +0.618034 q^{64} +1.00000 q^{68} -1.00000 q^{72} -0.381966 q^{76} +0.618034 q^{79} +1.00000 q^{81} +0.381966 q^{82} -1.61803 q^{83} -1.00000 q^{91} -0.381966 q^{92} -1.61803 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{4} - q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{4} - q^{7} - 2q^{8} + 2q^{9} - q^{13} - 2q^{14} - q^{17} - q^{18} - q^{19} - q^{23} + 2q^{25} + 3q^{26} + 2q^{28} - q^{29} + 2q^{32} - 2q^{34} + q^{36} + 3q^{38} - q^{41} + 3q^{46} + q^{49} - q^{50} - 3q^{52} + q^{56} - 2q^{58} - q^{59} - q^{61} - q^{63} - q^{64} + 2q^{68} - 2q^{72} - 3q^{76} - q^{79} + 2q^{81} + 3q^{82} - q^{83} - 2q^{91} - 3q^{92} - q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/103\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −0.618034 −0.618034
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(8\) −1.00000 −1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(14\) −1.00000 −1.00000
\(15\) 0 0
\(16\) 0 0
\(17\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(18\) 0.618034 0.618034
\(19\) 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
\(23\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(24\) 0 0
\(25\) 1.00000 1.00000
\(26\) 0.381966 0.381966
\(27\) 0 0
\(28\) 1.00000 1.00000
\(29\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1.00000 1.00000
\(33\) 0 0
\(34\) −1.00000 −1.00000
\(35\) 0 0
\(36\) −0.618034 −0.618034
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0.381966 0.381966
\(39\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0.381966 0.381966
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.61803 1.61803
\(50\) 0.618034 0.618034
\(51\) 0 0
\(52\) −0.381966 −0.381966
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.61803 1.61803
\(57\) 0 0
\(58\) −1.00000 −1.00000
\(59\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(60\) 0 0
\(61\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(62\) 0 0
\(63\) −1.61803 −1.61803
\(64\) 0.618034 0.618034
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 1.00000 1.00000
\(69\) 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 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.381966 −0.381966
\(77\) 0 0
\(78\) 0 0
\(79\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 0.381966 0.381966
\(83\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.00000 −1.00000
\(92\) −0.381966 −0.381966
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(98\) 1.00000 1.00000
\(99\) 0 0
\(100\) −0.618034 −0.618034
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 1.00000 1.00000
\(104\) −0.618034 −0.618034
\(105\) 0 0
\(106\) 0 0
\(107\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.00000 1.00000
\(117\) 0.618034 0.618034
\(118\) −1.00000 −1.00000
\(119\) 2.61803 2.61803
\(120\) 0 0
\(121\) 1.00000 1.00000
\(122\) −1.00000 −1.00000
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) −1.00000 −1.00000
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −0.618034 −0.618034
\(129\) 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
\(133\) −1.00000 −1.00000
\(134\) 0 0
\(135\) 0 0
\(136\) 1.61803 1.61803
\(137\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(138\) 0 0
\(139\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(140\) 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
\(149\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) −0.618034 −0.618034
\(153\) −1.61803 −1.61803
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0.381966 0.381966
\(159\) 0 0
\(160\) 0 0
\(161\) −1.00000 −1.00000
\(162\) 0.618034 0.618034
\(163\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(164\) −0.381966 −0.381966
\(165\) 0 0
\(166\) −1.00000 −1.00000
\(167\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(168\) 0 0
\(169\) −0.618034 −0.618034
\(170\) 0 0
\(171\) 0.618034 0.618034
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.61803 −1.61803
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −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 \(\pi\)
\(182\) −0.618034 −0.618034
\(183\) 0 0
\(184\) −0.618034 −0.618034
\(185\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) −1.00000 −1.00000
\(195\) 0 0
\(196\) −1.00000 −1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.00000 −1.00000
\(201\) 0 0
\(202\) 0 0
\(203\) 2.61803 2.61803
\(204\) 0 0
\(205\) 0 0
\(206\) 0.618034 0.618034
\(207\) 0.618034 0.618034
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.23607 1.23607
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.00000 −1.00000
\(222\) 0 0
\(223\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(224\) −1.61803 −1.61803
\(225\) 1.00000 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.61803 1.61803
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0.381966 0.381966
\(235\) 0 0
\(236\) 1.00000 1.00000
\(237\) 0 0
\(238\) 1.61803 1.61803
\(239\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0.618034 0.618034
\(243\) 0 0
\(244\) 1.00000 1.00000
\(245\) 0 0
\(246\) 0 0
\(247\) 0.381966 0.381966
\(248\) 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
\(260\) 0 0
\(261\) −1.61803 −1.61803
\(262\) 0.381966 0.381966
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.618034 −0.618034
\(267\) 0 0
\(268\) 0 0
\(269\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0.381966 0.381966
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 1.23607 1.23607
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.00000 −1.00000
\(288\) 1.00000 1.00000
\(289\) 1.61803 1.61803
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0.381966 0.381966
\(299\) 0.381966 0.381966
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −1.00000 −1.00000
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(312\) 0 0
\(313\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.381966 −0.381966
\(317\) −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 0
\(321\) 0 0
\(322\) −0.618034 −0.618034
\(323\) −1.00000 −1.00000
\(324\) −0.618034 −0.618034
\(325\) 0.618034 0.618034
\(326\) −1.00000 −1.00000
\(327\) 0 0
\(328\) −0.618034 −0.618034
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 1.00000 1.00000
\(333\) 0 0
\(334\) 1.23607 1.23607
\(335\) 0 0
\(336\) 0 0
\(337\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(338\) −0.381966 −0.381966
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0.381966 0.381966
\(343\) −1.00000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −1.00000 −1.00000
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −1.00000 −1.00000
\(359\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(360\) 0 0
\(361\) −0.618034 −0.618034
\(362\) 0 0
\(363\) 0 0
\(364\) 0.618034 0.618034
\(365\) 0 0
\(366\) 0 0
\(367\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(368\) 0 0
\(369\) 0.618034 0.618034
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.00000 −1.00000
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 1.00000 1.00000
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) −1.00000 −1.00000
\(392\) −1.61803 −1.61803
\(393\) 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
\(401\) −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
\(406\) 1.61803 1.61803
\(407\) 0 0
\(408\) 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
\(414\) 0.381966 0.381966
\(415\) 0 0
\(416\) 0.618034 0.618034
\(417\) 0 0
\(418\) 0 0
\(419\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(420\) 0 0
\(421\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.61803 −1.61803
\(426\) 0 0
\(427\) 2.61803 2.61803
\(428\) −1.23607 −1.23607
\(429\) 0 0
\(430\) 0 0
\(431\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0.381966 0.381966
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 1.61803 1.61803
\(442\) −0.618034 −0.618034
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.381966 0.381966
\(447\) 0 0
\(448\) −1.00000 −1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0.618034 0.618034
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) −1.00000 −1.00000
\(459\) 0 0
\(460\) 0 0
\(461\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(462\) 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 \(0.400000\pi\)
\(468\) −0.381966 −0.381966
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 1.61803 1.61803
\(473\) 0 0
\(474\) 0 0
\(475\) 0.618034 0.618034
\(476\) −1.61803 −1.61803
\(477\) 0 0
\(478\) −1.00000 −1.00000
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.618034 −0.618034
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 1.61803 1.61803
\(489\) 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
\(493\) 2.61803 2.61803
\(494\) 0.236068 0.236068
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(504\) 1.61803 1.61803
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) −1.00000 −1.00000
\(523\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(524\) −0.381966 −0.381966
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.618034 −0.618034
\(530\) 0 0
\(531\) −1.61803 −1.61803
\(532\) 0.618034 0.618034
\(533\) 0.381966 0.381966
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.381966 0.381966
\(539\) 0 0
\(540\) 0 0
\(541\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −1.61803 −1.61803
\(545\) 0 0
\(546\) 0 0
\(547\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(548\) −0.381966 −0.381966
\(549\) −1.61803 −1.61803
\(550\) 0 0
\(551\) −1.00000 −1.00000
\(552\) 0 0
\(553\) −1.00000 −1.00000
\(554\) 0 0
\(555\) 0 0
\(556\) −1.23607 −1.23607
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.618034 −0.618034
\(575\) 0.618034 0.618034
\(576\) 0.618034 0.618034
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 1.00000 1.00000
\(579\) 0 0
\(580\) 0 0
\(581\) 2.61803 2.61803
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(588\) 0 0
\(589\) 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
\(596\) −0.381966 −0.381966
\(597\) 0 0
\(598\) 0.236068 0.236068
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(608\) 0.618034 0.618034
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 1.00000 1.00000
\(613\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 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
\(622\) −1.00000 −1.00000
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) 0.381966 0.381966
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(632\) −0.618034 −0.618034
\(633\) 0 0
\(634\) −1.00000 −1.00000
\(635\) 0 0
\(636\) 0 0
\(637\) 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
\(643\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(644\) 0.618034 0.618034
\(645\) 0 0
\(646\) −0.618034 −0.618034
\(647\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(648\) −1.00000 −1.00000
\(649\) 0 0
\(650\) 0.381966 0.381966
\(651\) 0 0
\(652\) 1.00000 1.00000
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1.61803 1.61803
\(665\) 0 0
\(666\) 0 0
\(667\) −1.00000 −1.00000
\(668\) −1.23607 −1.23607
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(674\) 0.381966 0.381966
\(675\) 0 0
\(676\) 0.381966 0.381966
\(677\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(678\) 0 0
\(679\) 2.61803 2.61803
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −0.381966 −0.381966
\(685\) 0 0
\(686\) −0.618034 −0.618034
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0.381966 0.381966
\(695\) 0 0
\(696\) 0 0
\(697\) −1.00000 −1.00000
\(698\) 0 0
\(699\) 0 0
\(700\) 1.00000 1.00000
\(701\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(710\) 0 0
\(711\) 0.618034 0.618034
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.00000 1.00000
\(717\) 0 0
\(718\) 1.23607 1.23607
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1.61803 −1.61803
\(722\) −0.381966 −0.381966
\(723\) 0 0
\(724\) 0 0
\(725\) −1.61803 −1.61803
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 1.00000 1.00000
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0.381966 0.381966
\(735\) 0 0
\(736\) 0.618034 0.618034
\(737\) 0 0
\(738\) 0.381966 0.381966
\(739\) −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
\(746\) −1.00000 −1.00000
\(747\) −1.61803 −1.61803
\(748\) 0 0
\(749\) −3.23607 −3.23607
\(750\) 0 0
\(751\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −0.618034 −0.618034
\(755\) 0 0
\(756\) 0 0
\(757\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.00000 −1.00000
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.61803 1.61803
\(777\) 0 0
\(778\) 0 0
\(779\) 0.381966 0.381966
\(780\) 0 0
\(781\) 0 0
\(782\) −0.618034 −0.618034
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.00000 −1.00000
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 1.00000
\(801\) 0 0
\(802\) −1.00000 −1.00000
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −1.61803 −1.61803
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.00000 −1.00000
\(819\) −1.00000 −1.00000
\(820\) 0 0
\(821\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) −1.00000 −1.00000
\(825\) 0 0
\(826\) 1.61803 1.61803
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −0.381966 −0.381966
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.381966 0.381966
\(833\) −2.61803 −2.61803
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0.381966 0.381966
\(839\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(840\) 0 0
\(841\) 1.61803 1.61803
\(842\) 1.23607 1.23607
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −1.61803 −1.61803
\(848\) 0 0
\(849\) 0 0
\(850\) −1.00000 −1.00000
\(851\) 0 0
\(852\) 0 0
\(853\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(854\) 1.61803 1.61803
\(855\) 0 0
\(856\) −2.00000 −2.00000
\(857\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.00000 −1.00000
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 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
\(873\) −1.61803 −1.61803
\(874\) 0.236068 0.236068
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.00000 1.00000
\(883\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(884\) 0.618034 0.618034
\(885\) 0 0
\(886\) 0 0
\(887\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −0.381966 −0.381966
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.00000 1.00000
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.618034 −0.618034
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(908\) 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
\(916\) 1.00000 1.00000
\(917\) −1.00000 −1.00000
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.23607 1.23607
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.00000 1.00000
\(928\) −1.61803 −1.61803
\(929\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(930\) 0 0
\(931\) 1.00000 1.00000
\(932\) 0 0
\(933\) 0 0
\(934\) 0.381966 0.381966
\(935\) 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
\(941\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(942\) 0 0
\(943\) 0.381966 0.381966
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0.381966 0.381966
\(951\) 0 0
\(952\) −2.61803 −2.61803
\(953\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.00000 1.00000
\(957\) 0 0
\(958\) 0 0
\(959\) −1.00000 −1.00000
\(960\) 0 0
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 2.00000 2.00000
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −1.00000 −1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −3.23607 −3.23607
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.00000 −1.00000
\(983\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.61803 1.61803
\(987\) 0 0
\(988\) −0.236068 −0.236068
\(989\) 0 0
\(990\) 0 0
\(991\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 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 103.1.b.a.102.2 2
3.2 odd 2 927.1.d.b.514.1 2
4.3 odd 2 1648.1.c.a.1441.2 2
5.2 odd 4 2575.1.c.b.2574.3 4
5.3 odd 4 2575.1.c.b.2574.2 4
5.4 even 2 2575.1.d.d.926.1 2
103.102 odd 2 CM 103.1.b.a.102.2 2
309.308 even 2 927.1.d.b.514.1 2
412.411 even 2 1648.1.c.a.1441.2 2
515.102 even 4 2575.1.c.b.2574.3 4
515.308 even 4 2575.1.c.b.2574.2 4
515.514 odd 2 2575.1.d.d.926.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
103.1.b.a.102.2 2 1.1 even 1 trivial
103.1.b.a.102.2 2 103.102 odd 2 CM
927.1.d.b.514.1 2 3.2 odd 2
927.1.d.b.514.1 2 309.308 even 2
1648.1.c.a.1441.2 2 4.3 odd 2
1648.1.c.a.1441.2 2 412.411 even 2
2575.1.c.b.2574.2 4 5.3 odd 4
2575.1.c.b.2574.2 4 515.308 even 4
2575.1.c.b.2574.3 4 5.2 odd 4
2575.1.c.b.2574.3 4 515.102 even 4
2575.1.d.d.926.1 2 5.4 even 2
2575.1.d.d.926.1 2 515.514 odd 2