Properties

Label 327.1.d.a.326.1
Level $327$
Weight $1$
Character 327.326
Self dual yes
Analytic conductor $0.163$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -327
Inner twists $2$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 327 = 3 \cdot 109 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 327.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.163194259131\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.327.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.327.1

Embedding invariants

Embedding label 326.1
Character \(\chi\) \(=\) 327.326

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{14} -1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -1.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{31} +2.00000 q^{33} +1.00000 q^{34} -1.00000 q^{41} +1.00000 q^{42} -1.00000 q^{43} +1.00000 q^{46} -1.00000 q^{47} -1.00000 q^{48} -1.00000 q^{50} -1.00000 q^{51} +2.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -1.00000 q^{59} -1.00000 q^{61} +1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} -1.00000 q^{69} +1.00000 q^{72} -1.00000 q^{73} +1.00000 q^{75} -2.00000 q^{77} +1.00000 q^{81} +1.00000 q^{82} +1.00000 q^{86} +2.00000 q^{88} -1.00000 q^{93} +1.00000 q^{94} -1.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/327\mathbb{Z}\right)^\times\).

\(n\) \(110\) \(115\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 1.00000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −1.00000 −1.00000
\(7\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 1.00000 1.00000
\(9\) 1.00000 1.00000
\(10\) 0 0
\(11\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.00000 1.00000
\(15\) 0 0
\(16\) −1.00000 −1.00000
\(17\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) −1.00000 −1.00000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −1.00000 −1.00000
\(22\) −2.00000 −2.00000
\(23\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 1.00000 1.00000
\(25\) 1.00000 1.00000
\(26\) 0 0
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) 0 0
\(33\) 2.00000 2.00000
\(34\) 1.00000 1.00000
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(42\) 1.00000 1.00000
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.00000 1.00000
\(47\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) −1.00000 −1.00000
\(49\) 0 0
\(50\) −1.00000 −1.00000
\(51\) −1.00000 −1.00000
\(52\) 0 0
\(53\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 1.00000 1.00000
\(63\) −1.00000 −1.00000
\(64\) 1.00000 1.00000
\(65\) 0 0
\(66\) −2.00000 −2.00000
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −1.00000 −1.00000
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.00000 1.00000
\(73\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 0 0
\(75\) 1.00000 1.00000
\(76\) 0 0
\(77\) −2.00000 −2.00000
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.00000 1.00000
\(82\) 1.00000 1.00000
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.00000 1.00000
\(87\) 0 0
\(88\) 2.00000 2.00000
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.00000 −1.00000
\(94\) 1.00000 1.00000
\(95\) 0 0
\(96\) 0 0
\(97\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 0 0
\(99\) 2.00000 2.00000
\(100\) 0 0
\(101\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(102\) 1.00000 1.00000
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −2.00000
\(107\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) 1.00000 1.00000
\(110\) 0 0
\(111\) 0 0
\(112\) 1.00000 1.00000
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 1.00000 1.00000
\(119\) 1.00000 1.00000
\(120\) 0 0
\(121\) 3.00000 3.00000
\(122\) 1.00000 1.00000
\(123\) −1.00000 −1.00000
\(124\) 0 0
\(125\) 0 0
\(126\) 1.00000 1.00000
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.00000 −1.00000
\(129\) −1.00000 −1.00000
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1.00000 −1.00000
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 1.00000 1.00000
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −1.00000 −1.00000
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −1.00000
\(145\) 0 0
\(146\) 1.00000 1.00000
\(147\) 0 0
\(148\) 0 0
\(149\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) −1.00000 −1.00000
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −1.00000 −1.00000
\(154\) 2.00000 2.00000
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(158\) 0 0
\(159\) 2.00000 2.00000
\(160\) 0 0
\(161\) 1.00000 1.00000
\(162\) −1.00000 −1.00000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(168\) −1.00000 −1.00000
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.00000 −1.00000
\(176\) −2.00000 −2.00000
\(177\) −1.00000 −1.00000
\(178\) 0 0
\(179\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) −1.00000 −1.00000
\(184\) −1.00000 −1.00000
\(185\) 0 0
\(186\) 1.00000 1.00000
\(187\) −2.00000 −2.00000
\(188\) 0 0
\(189\) −1.00000 −1.00000
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.00000 1.00000
\(193\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 1.00000 1.00000
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −2.00000 −2.00000
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 1.00000 1.00000
\(201\) 0 0
\(202\) −2.00000 −2.00000
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.00000 −1.00000
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 1.00000 1.00000
\(215\) 0 0
\(216\) 1.00000 1.00000
\(217\) 1.00000 1.00000
\(218\) −1.00000 −1.00000
\(219\) −1.00000 −1.00000
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 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
\(231\) −2.00000 −2.00000
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) −1.00000 −1.00000
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −3.00000 −3.00000
\(243\) 1.00000 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 1.00000 1.00000
\(247\) 0 0
\(248\) −1.00000 −1.00000
\(249\) 0 0
\(250\) 0 0
\(251\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(252\) 0 0
\(253\) −2.00000 −2.00000
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(258\) 1.00000 1.00000
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 2.00000 2.00000
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 1.00000 1.00000
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 2.00000
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −1.00000 −1.00000
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 1.00000 1.00000
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00000 1.00000
\(288\) 0 0
\(289\) 0 0
\(290\) 0 0
\(291\) −1.00000 −1.00000
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000 2.00000
\(298\) 1.00000 1.00000
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 1.00000
\(302\) 0 0
\(303\) 2.00000 2.00000
\(304\) 0 0
\(305\) 0 0
\(306\) 1.00000 1.00000
\(307\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −2.00000 −2.00000
\(315\) 0 0
\(316\) 0 0
\(317\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(318\) −2.00000 −2.00000
\(319\) 0 0
\(320\) 0 0
\(321\) −1.00000 −1.00000
\(322\) −1.00000 −1.00000
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.00000 1.00000
\(328\) −1.00000 −1.00000
\(329\) 1.00000 1.00000
\(330\) 0 0
\(331\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(332\) 0 0
\(333\) 0 0
\(334\) −2.00000 −2.00000
\(335\) 0 0
\(336\) 1.00000 1.00000
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −1.00000 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) −2.00000 −2.00000
\(342\) 0 0
\(343\) 1.00000 1.00000
\(344\) −1.00000 −1.00000
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(350\) 1.00000 1.00000
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 1.00000 1.00000
\(355\) 0 0
\(356\) 0 0
\(357\) 1.00000 1.00000
\(358\) 1.00000 1.00000
\(359\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 3.00000 3.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 1.00000 1.00000
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 1.00000 1.00000
\(369\) −1.00000 −1.00000
\(370\) 0 0
\(371\) −2.00000 −2.00000
\(372\) 0 0
\(373\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 2.00000 2.00000
\(375\) 0 0
\(376\) −1.00000 −1.00000
\(377\) 0 0
\(378\) 1.00000 1.00000
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) −1.00000 −1.00000
\(385\) 0 0
\(386\) 1.00000 1.00000
\(387\) −1.00000 −1.00000
\(388\) 0 0
\(389\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(390\) 0 0
\(391\) 1.00000 1.00000
\(392\) 0 0
\(393\) 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
\(400\) −1.00000 −1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −1.00000 −1.00000
\(409\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.00000 1.00000
\(414\) 1.00000 1.00000
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(420\) 0 0
\(421\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 1.00000 1.00000
\(423\) −1.00000 −1.00000
\(424\) 2.00000 2.00000
\(425\) −1.00000 −1.00000
\(426\) 0 0
\(427\) 1.00000 1.00000
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −1.00000 −1.00000
\(433\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(434\) −1.00000 −1.00000
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 1.00000 1.00000
\(439\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.00000 1.00000
\(447\) −1.00000 −1.00000
\(448\) −1.00000 −1.00000
\(449\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(450\) −1.00000 −1.00000
\(451\) −2.00000 −2.00000
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) −1.00000 −1.00000
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 2.00000 2.00000
\(463\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 2.00000
\(472\) −1.00000 −1.00000
\(473\) −2.00000 −2.00000
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.00000 2.00000
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 1.00000 1.00000
\(484\) 0 0
\(485\) 0 0
\(486\) −1.00000 −1.00000
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) −1.00000 −1.00000
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.00000 1.00000
\(497\) 0 0
\(498\) 0 0
\(499\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0 0
\(501\) 2.00000 2.00000
\(502\) 1.00000 1.00000
\(503\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(504\) −1.00000 −1.00000
\(505\) 0 0
\(506\) 2.00000 2.00000
\(507\) 1.00000 1.00000
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.00000 1.00000
\(512\) 1.00000 1.00000
\(513\) 0 0
\(514\) −2.00000 −2.00000
\(515\) 0 0
\(516\) 0 0
\(517\) −2.00000 −2.00000
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(524\) 0 0
\(525\) −1.00000 −1.00000
\(526\) 0 0
\(527\) 1.00000 1.00000
\(528\) −2.00000 −2.00000
\(529\) 0 0
\(530\) 0 0
\(531\) −1.00000 −1.00000
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −1.00000 −1.00000
\(538\) −2.00000 −2.00000
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) −1.00000 −1.00000
\(550\) −2.00000 −2.00000
\(551\) 0 0
\(552\) −1.00000 −1.00000
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 1.00000 1.00000
\(559\) 0 0
\(560\) 0 0
\(561\) −2.00000 −2.00000
\(562\) 0 0
\(563\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.00000 −1.00000
\(568\) 0 0
\(569\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.00000 −1.00000
\(575\) −1.00000 −1.00000
\(576\) 1.00000 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) −1.00000 −1.00000
\(580\) 0 0
\(581\) 0 0
\(582\) 1.00000 1.00000
\(583\) 4.00000 4.00000
\(584\) −1.00000 −1.00000
\(585\) 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 \(\pi\)
\(594\) −2.00000 −2.00000
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 1.00000 1.00000
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −1.00000 −1.00000
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) −2.00000 −2.00000
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.00000 1.00000
\(615\) 0 0
\(616\) −2.00000 −2.00000
\(617\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(618\) 0 0
\(619\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(620\) 0 0
\(621\) −1.00000 −1.00000
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 1.00000
\(626\) 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
\(633\) −1.00000 −1.00000
\(634\) 1.00000 1.00000
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(642\) 1.00000 1.00000
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 1.00000 1.00000
\(649\) −2.00000 −2.00000
\(650\) 0 0
\(651\) 1.00000 1.00000
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) −1.00000 −1.00000
\(655\) 0 0
\(656\) 1.00000 1.00000
\(657\) −1.00000 −1.00000
\(658\) −1.00000 −1.00000
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) −2.00000 −2.00000
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.00000 −1.00000
\(670\) 0 0
\(671\) −2.00000 −2.00000
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 1.00000 1.00000
\(676\) 0 0
\(677\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 1.00000 1.00000
\(680\) 0 0
\(681\) 0 0
\(682\) 2.00000 2.00000
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1.00000 −1.00000
\(687\) 0 0
\(688\) 1.00000 1.00000
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) −2.00000 −2.00000
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.00000 1.00000
\(698\) 1.00000 1.00000
\(699\) 0 0
\(700\) 0 0
\(701\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 2.00000 2.00000
\(705\) 0 0
\(706\) 0 0
\(707\) −2.00000 −2.00000
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.00000 1.00000
\(714\) −1.00000 −1.00000
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 1.00000 1.00000
\(719\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −3.00000 −3.00000
\(727\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 1.00000 1.00000
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 1.00000 1.00000
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.00000 2.00000
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −1.00000 −1.00000
\(745\) 0 0
\(746\) 1.00000 1.00000
\(747\) 0 0
\(748\) 0 0
\(749\) 1.00000 1.00000
\(750\) 0 0
\(751\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(752\) 1.00000 1.00000
\(753\) −1.00000 −1.00000
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) −2.00000 −2.00000
\(760\) 0 0
\(761\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(762\) 0 0
\(763\) −1.00000 −1.00000
\(764\) 0 0
\(765\) 0 0
\(766\) 1.00000 1.00000
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 2.00000 2.00000
\(772\) 0 0
\(773\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(774\) 1.00000 1.00000
\(775\) −1.00000 −1.00000
\(776\) −1.00000 −1.00000
\(777\) 0 0
\(778\) 1.00000 1.00000
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −1.00000 −1.00000
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 2.00000 2.00000
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 1.00000 1.00000
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.00000 −2.00000
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.00000 2.00000
\(808\) 2.00000 2.00000
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 1.00000 1.00000
\(817\) 0 0
\(818\) −2.00000 −2.00000
\(819\) 0 0
\(820\) 0 0
\(821\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(824\) 0 0
\(825\) 2.00000 2.00000
\(826\) −1.00000 −1.00000
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.00000 −1.00000
\(838\) 1.00000 1.00000
\(839\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 1.00000 1.00000
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 1.00000 1.00000
\(847\) −3.00000 −3.00000
\(848\) −2.00000 −2.00000
\(849\) 0 0
\(850\) 1.00000 1.00000
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) −1.00000 −1.00000
\(855\) 0 0
\(856\) −1.00000 −1.00000
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 1.00000 1.00000
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.00000 1.00000
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.00000 1.00000
\(873\) −1.00000 −1.00000
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(878\) −2.00000 −2.00000
\(879\) 0 0
\(880\) 0 0
\(881\) 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 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 2.00000 2.00000
\(892\) 0 0
\(893\) 0 0
\(894\) 1.00000 1.00000
\(895\) 0 0
\(896\) 1.00000 1.00000
\(897\) 0 0
\(898\) 1.00000 1.00000
\(899\) 0 0
\(900\) 0 0
\(901\) −2.00000 −2.00000
\(902\) 2.00000 2.00000
\(903\) 1.00000 1.00000
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(908\) 0 0
\(909\) 2.00000 2.00000
\(910\) 0 0
\(911\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.00000 1.00000
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 1.00000 1.00000
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.00000 −1.00000
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −2.00000 −2.00000
\(927\) 0 0
\(928\) 0 0
\(929\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(942\) −2.00000 −2.00000
\(943\) 1.00000 1.00000
\(944\) 1.00000 1.00000
\(945\) 0 0
\(946\) 2.00000 2.00000
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.00000 −1.00000
\(952\) 1.00000 1.00000
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) −2.00000 −2.00000
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) −1.00000 −1.00000
\(964\) 0 0
\(965\) 0 0
\(966\) −1.00000 −1.00000
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 3.00000 3.00000
\(969\) 0 0
\(970\) 0 0
\(971\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 1.00000 1.00000
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.00000 1.00000
\(982\) −2.00000 −2.00000
\(983\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(984\) −1.00000 −1.00000
\(985\) 0 0
\(986\) 0 0
\(987\) 1.00000 1.00000
\(988\) 0 0
\(989\) 1.00000 1.00000
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 2.00000 2.00000
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 1.00000 1.00000
\(999\) 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 327.1.d.a.326.1 1
3.2 odd 2 327.1.d.c.326.1 yes 1
109.108 even 2 327.1.d.c.326.1 yes 1
327.326 odd 2 CM 327.1.d.a.326.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
327.1.d.a.326.1 1 1.1 even 1 trivial
327.1.d.a.326.1 1 327.326 odd 2 CM
327.1.d.c.326.1 yes 1 3.2 odd 2
327.1.d.c.326.1 yes 1 109.108 even 2