Properties

Label 1560.1.y.f.389.1
Level $1560$
Weight $1$
Character 1560.389
Analytic conductor $0.779$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -39, -120, 520
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1560.y (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.778541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-30}, \sqrt{-39})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.3701505600.1

Embedding invariants

Embedding label 389.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1560.389
Dual form 1560.1.y.f.389.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.00000i q^{11} -1.00000 q^{12} -1.00000 q^{13} +1.00000i q^{15} +1.00000 q^{16} -1.00000i q^{18} -1.00000i q^{20} +2.00000 q^{22} +1.00000i q^{24} -1.00000 q^{25} +1.00000i q^{26} +1.00000 q^{27} +1.00000 q^{30} -1.00000i q^{32} +2.00000i q^{33} -1.00000 q^{36} -1.00000 q^{39} -1.00000 q^{40} +2.00000 q^{43} -2.00000i q^{44} +1.00000i q^{45} -2.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000i q^{50} +1.00000 q^{52} -1.00000i q^{54} -2.00000 q^{55} -2.00000i q^{59} -1.00000i q^{60} -1.00000 q^{64} -1.00000i q^{65} +2.00000 q^{66} +1.00000i q^{72} -1.00000 q^{75} +1.00000i q^{78} +2.00000 q^{79} +1.00000i q^{80} +1.00000 q^{81} -2.00000i q^{86} -2.00000 q^{88} +1.00000 q^{90} -2.00000 q^{94} -1.00000i q^{96} +1.00000i q^{98} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 2 q^{10} - 2 q^{12} - 2 q^{13} + 2 q^{16} + 4 q^{22} - 2 q^{25} + 2 q^{27} + 2 q^{30} - 2 q^{36} - 2 q^{39} - 2 q^{40} + 4 q^{43} + 2 q^{48} - 2 q^{49} + 2 q^{52} - 4 q^{55} - 2 q^{64} + 4 q^{66} - 2 q^{75} + 4 q^{79} + 2 q^{81} - 4 q^{88} + 2 q^{90} - 4 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).

\(n\) \(391\) \(521\) \(781\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 1.00000i
\(3\) 1.00000 1.00000
\(4\) −1.00000 −1.00000
\(5\) 1.00000i 1.00000i
\(6\) − 1.00000i − 1.00000i
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000i 1.00000i
\(9\) 1.00000 1.00000
\(10\) 1.00000 1.00000
\(11\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −1.00000
\(13\) −1.00000 −1.00000
\(14\) 0 0
\(15\) 1.00000i 1.00000i
\(16\) 1.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) − 1.00000i − 1.00000i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) − 1.00000i − 1.00000i
\(21\) 0 0
\(22\) 2.00000 2.00000
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000i 1.00000i
\(25\) −1.00000 −1.00000
\(26\) 1.00000i 1.00000i
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 1.00000 1.00000
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) − 1.00000i − 1.00000i
\(33\) 2.00000i 2.00000i
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −1.00000 −1.00000
\(40\) −1.00000 −1.00000
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(44\) − 2.00000i − 2.00000i
\(45\) 1.00000i 1.00000i
\(46\) 0 0
\(47\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(48\) 1.00000 1.00000
\(49\) −1.00000 −1.00000
\(50\) 1.00000i 1.00000i
\(51\) 0 0
\(52\) 1.00000 1.00000
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) − 1.00000i − 1.00000i
\(55\) −2.00000 −2.00000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(60\) − 1.00000i − 1.00000i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) − 1.00000i − 1.00000i
\(66\) 2.00000 2.00000
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000i 1.00000i
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) −1.00000 −1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 1.00000i 1.00000i
\(79\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(80\) 1.00000i 1.00000i
\(81\) 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
\(86\) − 2.00000i − 2.00000i
\(87\) 0 0
\(88\) −2.00000 −2.00000
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 1.00000 1.00000
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −2.00000 −2.00000
\(95\) 0 0
\(96\) − 1.00000i − 1.00000i
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 1.00000i 1.00000i
\(99\) 2.00000i 2.00000i
\(100\) 1.00000 1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) − 1.00000i − 1.00000i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.00000 −1.00000
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 2.00000i 2.00000i
\(111\) 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
\(117\) −1.00000 −1.00000
\(118\) −2.00000 −2.00000
\(119\) 0 0
\(120\) −1.00000 −1.00000
\(121\) −3.00000 −3.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1.00000i − 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 2.00000 2.00000
\(130\) −1.00000 −1.00000
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) − 2.00000i − 2.00000i
\(133\) 0 0
\(134\) 0 0
\(135\) 1.00000i 1.00000i
\(136\) 0 0
\(137\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) − 2.00000i − 2.00000i
\(142\) 0 0
\(143\) − 2.00000i − 2.00000i
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −1.00000 −1.00000
\(148\) 0 0
\(149\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 1.00000i 1.00000i
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.00000 1.00000
\(157\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(158\) − 2.00000i − 2.00000i
\(159\) 0 0
\(160\) 1.00000 1.00000
\(161\) 0 0
\(162\) − 1.00000i − 1.00000i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −2.00000 −2.00000
\(166\) 0 0
\(167\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −2.00000 −2.00000
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.00000i 2.00000i
\(177\) − 2.00000i − 2.00000i
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) − 1.00000i − 1.00000i
\(181\) 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
\(188\) 2.00000i 2.00000i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.00000 −1.00000
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) − 1.00000i − 1.00000i
\(196\) 1.00000 1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 2.00000 2.00000
\(199\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(200\) − 1.00000i − 1.00000i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −1.00000 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.00000i 2.00000i
\(216\) 1.00000i 1.00000i
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 2.00000 2.00000
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −1.00000 −1.00000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 1.00000i 1.00000i
\(235\) 2.00000 2.00000
\(236\) 2.00000i 2.00000i
\(237\) 2.00000 2.00000
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 1.00000i 1.00000i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 3.00000i 3.00000i
\(243\) 1.00000 1.00000
\(244\) 0 0
\(245\) − 1.00000i − 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −1.00000 −1.00000
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) − 2.00000i − 2.00000i
\(259\) 0 0
\(260\) 1.00000i 1.00000i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −2.00000 −2.00000
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 1.00000 1.00000
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −2.00000 −2.00000
\(275\) − 2.00000i − 2.00000i
\(276\) 0 0
\(277\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −2.00000 −2.00000
\(283\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −2.00000 −2.00000
\(287\) 0 0
\(288\) − 1.00000i − 1.00000i
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 1.00000i 1.00000i
\(295\) 2.00000 2.00000
\(296\) 0 0
\(297\) 2.00000i 2.00000i
\(298\) 2.00000 2.00000
\(299\) 0 0
\(300\) 1.00000 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) − 1.00000i − 1.00000i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 2.00000i 2.00000i
\(315\) 0 0
\(316\) −2.00000 −2.00000
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 1.00000i − 1.00000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 1.00000 1.00000
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 2.00000i 2.00000i
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 2.00000 2.00000
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) − 1.00000i − 1.00000i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 2.00000i 2.00000i
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −1.00000 −1.00000
\(352\) 2.00000 2.00000
\(353\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) −2.00000 −2.00000
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −1.00000 −1.00000
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) −3.00000 −3.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(374\) 0 0
\(375\) − 1.00000i − 1.00000i
\(376\) 2.00000 2.00000
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(384\) 1.00000i 1.00000i
\(385\) 0 0
\(386\) 0 0
\(387\) 2.00000 2.00000
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) −1.00000 −1.00000
\(391\) 0 0
\(392\) − 1.00000i − 1.00000i
\(393\) 0 0
\(394\) 0 0
\(395\) 2.00000i 2.00000i
\(396\) − 2.00000i − 2.00000i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) − 2.00000i − 2.00000i
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000i 1.00000i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) − 2.00000i − 2.00000i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.00000i 1.00000i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) − 2.00000i − 2.00000i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 2.00000i − 2.00000i
\(430\) 2.00000 2.00000
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 1.00000
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(440\) − 2.00000i − 2.00000i
\(441\) −1.00000 −1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.00000i 2.00000i
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 1.00000i 1.00000i
\(451\) 0 0
\(452\) 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
\(461\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 1.00000 1.00000
\(469\) 0 0
\(470\) − 2.00000i − 2.00000i
\(471\) −2.00000 −2.00000
\(472\) 2.00000 2.00000
\(473\) 4.00000i 4.00000i
\(474\) − 2.00000i − 2.00000i
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 1.00000 1.00000
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 3.00000 3.00000
\(485\) 0 0
\(486\) − 1.00000i − 1.00000i
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −1.00000 −1.00000
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −2.00000 −2.00000
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 1.00000i 1.00000i
\(501\) 2.00000i 2.00000i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.00000 1.00000
\(508\) 0 0
\(509\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) −2.00000 −2.00000
\(517\) 4.00000 4.00000
\(518\) 0 0
\(519\) 0 0
\(520\) 1.00000 1.00000
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 2.00000i 2.00000i
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) − 2.00000i − 2.00000i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.00000i − 2.00000i
\(540\) − 1.00000i − 1.00000i
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(548\) 2.00000i 2.00000i
\(549\) 0 0
\(550\) −2.00000 −2.00000
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000i 2.00000i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −2.00000 −2.00000
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 2.00000i 2.00000i
\(565\) 0 0
\(566\) 2.00000i 2.00000i
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 2.00000i 2.00000i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −1.00000 −1.00000
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 1.00000i 1.00000i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) − 1.00000i − 1.00000i
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.00000 1.00000
\(589\) 0 0
\(590\) − 2.00000i − 2.00000i
\(591\) 0 0
\(592\) 0 0
\(593\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(594\) 2.00000 2.00000
\(595\) 0 0
\(596\) − 2.00000i − 2.00000i
\(597\) 2.00000 2.00000
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) − 1.00000i − 1.00000i
\(601\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.00000i − 3.00000i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.00000i 2.00000i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.00000 −1.00000
\(625\) 1.00000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 2.00000 2.00000
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 2.00000i 2.00000i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.00000 1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) −1.00000 −1.00000
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 2.00000i 2.00000i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 1.00000i 1.00000i
\(649\) 4.00000 4.00000
\(650\) − 1.00000i − 1.00000i
\(651\) 0 0
\(652\) 0 0
\(653\) 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 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 2.00000 2.00000
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) − 2.00000i − 2.00000i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −1.00000 −1.00000
\(676\) −1.00000 −1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 2.00000 2.00000
\(686\) 0 0
\(687\) 0 0
\(688\) 2.00000 2.00000
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 1.00000i 1.00000i
\(703\) 0 0
\(704\) − 2.00000i − 2.00000i
\(705\) 2.00000 2.00000
\(706\) 2.00000 2.00000
\(707\) 0 0
\(708\) 2.00000i 2.00000i
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 2.00000 2.00000
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 2.00000 2.00000
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.00000i 1.00000i
\(721\) 0 0
\(722\) 1.00000i 1.00000i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 3.00000i 3.00000i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) − 1.00000i − 1.00000i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −2.00000 −2.00000
\(746\) − 2.00000i − 2.00000i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −1.00000 −1.00000
\(751\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(752\) − 2.00000i − 2.00000i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −2.00000 −2.00000
\(767\) 2.00000i 2.00000i
\(768\) 1.00000 1.00000
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) − 2.00000i − 2.00000i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 1.00000i 1.00000i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −1.00000 −1.00000
\(785\) − 2.00000i − 2.00000i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 2.00000 2.00000
\(791\) 0 0
\(792\) −2.00000 −2.00000
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −2.00000 −2.00000
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000i 1.00000i
\(801\) 0 0
\(802\) 0 0
\(803\) 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 \(\pi\)
\(810\) 1.00000 1.00000
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) −2.00000 −2.00000
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) − 2.00000i − 2.00000i
\(826\) 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
\(831\) −2.00000 −2.00000
\(832\) 1.00000 1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) −2.00000 −2.00000
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.00000i 1.00000i
\(846\) −2.00000 −2.00000
\(847\) 0 0
\(848\) 0 0
\(849\) −2.00000 −2.00000
\(850\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) −2.00000 −2.00000
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) − 2.00000i − 2.00000i
\(861\) 0 0
\(862\) 0 0
\(863\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(864\) − 1.00000i − 1.00000i
\(865\) 0 0
\(866\) 0 0
\(867\) −1.00000 −1.00000
\(868\) 0 0
\(869\) 4.00000i 4.00000i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) − 2.00000i − 2.00000i
\(879\) 0 0
\(880\) −2.00000 −2.00000
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 1.00000i 1.00000i
\(883\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(884\) 0 0
\(885\) 2.00000 2.00000
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000i 2.00000i
\(892\) 0 0
\(893\) 0 0
\(894\) 2.00000 2.00000
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 1.00000 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −2.00000 −2.00000 −1.00000 \(\pi\)
−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
\(917\) 0 0
\(918\) 0 0
\(919\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.00000 −2.00000
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) − 1.00000i − 1.00000i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −2.00000 −2.00000
\(941\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(942\) 2.00000i 2.00000i
\(943\) 0 0
\(944\) − 2.00000i − 2.00000i
\(945\) 0 0
\(946\) 4.00000 4.00000
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) −2.00000 −2.00000
\(949\) 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
\(960\) − 1.00000i − 1.00000i
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) − 3.00000i − 3.00000i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −1.00000 −1.00000
\(973\) 0 0
\(974\) 0 0
\(975\) 1.00000 1.00000
\(976\) 0 0
\(977\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.00000i 1.00000i
\(981\) 0 0
\(982\) 0 0
\(983\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 2.00000i 2.00000i
\(991\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.00000i 2.00000i
\(996\) 0 0
\(997\) −2.00000 −2.00000 −1.00000 \(\pi\)
−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 1560.1.y.f.389.1 yes 2
3.2 odd 2 inner 1560.1.y.f.389.2 yes 2
5.4 even 2 1560.1.y.e.389.2 yes 2
8.5 even 2 1560.1.y.e.389.1 2
13.12 even 2 inner 1560.1.y.f.389.2 yes 2
15.14 odd 2 1560.1.y.e.389.1 2
24.5 odd 2 1560.1.y.e.389.2 yes 2
39.38 odd 2 CM 1560.1.y.f.389.1 yes 2
40.29 even 2 inner 1560.1.y.f.389.2 yes 2
65.64 even 2 1560.1.y.e.389.1 2
104.77 even 2 1560.1.y.e.389.2 yes 2
120.29 odd 2 CM 1560.1.y.f.389.1 yes 2
195.194 odd 2 1560.1.y.e.389.2 yes 2
312.77 odd 2 1560.1.y.e.389.1 2
520.389 even 2 RM 1560.1.y.f.389.1 yes 2
1560.389 odd 2 inner 1560.1.y.f.389.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1560.1.y.e.389.1 2 8.5 even 2
1560.1.y.e.389.1 2 15.14 odd 2
1560.1.y.e.389.1 2 65.64 even 2
1560.1.y.e.389.1 2 312.77 odd 2
1560.1.y.e.389.2 yes 2 5.4 even 2
1560.1.y.e.389.2 yes 2 24.5 odd 2
1560.1.y.e.389.2 yes 2 104.77 even 2
1560.1.y.e.389.2 yes 2 195.194 odd 2
1560.1.y.f.389.1 yes 2 1.1 even 1 trivial
1560.1.y.f.389.1 yes 2 39.38 odd 2 CM
1560.1.y.f.389.1 yes 2 120.29 odd 2 CM
1560.1.y.f.389.1 yes 2 520.389 even 2 RM
1560.1.y.f.389.2 yes 2 3.2 odd 2 inner
1560.1.y.f.389.2 yes 2 13.12 even 2 inner
1560.1.y.f.389.2 yes 2 40.29 even 2 inner
1560.1.y.f.389.2 yes 2 1560.389 odd 2 inner