Properties

Label 2304.3.g.l.1279.1
Level $2304$
Weight $3$
Character 2304.1279
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
Defining polynomial: \(x^{2} + 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1279.1
Root \(2.44949i\) of defining polynomial
Character \(\chi\) \(=\) 2304.1279
Dual form 2304.3.g.l.1279.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{5} -9.79796i q^{7} +O(q^{10})\) \(q+2.00000 q^{5} -9.79796i q^{7} -19.5959i q^{11} -21.0000 q^{25} -50.0000 q^{29} -48.9898i q^{31} -19.5959i q^{35} -47.0000 q^{49} +94.0000 q^{53} -39.1918i q^{55} +117.576i q^{59} -50.0000 q^{73} -192.000 q^{77} +146.969i q^{79} +97.9796i q^{83} -190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} + O(q^{10}) \) \( 2q + 4q^{5} - 42q^{25} - 100q^{29} - 94q^{49} + 188q^{53} - 100q^{73} - 384q^{77} - 380q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.00000 0.400000 0.200000 0.979796i \(-0.435906\pi\)
0.200000 + 0.979796i \(0.435906\pi\)
\(6\) 0 0
\(7\) − 9.79796i − 1.39971i −0.714286 0.699854i \(-0.753248\pi\)
0.714286 0.699854i \(-0.246752\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 19.5959i − 1.78145i −0.454545 0.890724i \(-0.650198\pi\)
0.454545 0.890724i \(-0.349802\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −21.0000 −0.840000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −50.0000 −1.72414 −0.862069 0.506791i \(-0.830832\pi\)
−0.862069 + 0.506791i \(0.830832\pi\)
\(30\) 0 0
\(31\) − 48.9898i − 1.58032i −0.612903 0.790158i \(-0.709998\pi\)
0.612903 0.790158i \(-0.290002\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 19.5959i − 0.559883i
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −47.0000 −0.959184
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 94.0000 1.77358 0.886792 0.462168i \(-0.152928\pi\)
0.886792 + 0.462168i \(0.152928\pi\)
\(54\) 0 0
\(55\) − 39.1918i − 0.712579i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 117.576i 1.99281i 0.0847458 + 0.996403i \(0.472992\pi\)
−0.0847458 + 0.996403i \(0.527008\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 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
\(73\) −50.0000 −0.684932 −0.342466 0.939530i \(-0.611262\pi\)
−0.342466 + 0.939530i \(0.611262\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −192.000 −2.49351
\(78\) 0 0
\(79\) 146.969i 1.86037i 0.367089 + 0.930186i \(0.380355\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 97.9796i 1.18048i 0.807229 + 0.590238i \(0.200966\pi\)
−0.807229 + 0.590238i \(0.799034\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −190.000 −1.95876 −0.979381 0.202020i \(-0.935249\pi\)
−0.979381 + 0.202020i \(0.935249\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 190.000 1.88119 0.940594 0.339533i \(-0.110269\pi\)
0.940594 + 0.339533i \(0.110269\pi\)
\(102\) 0 0
\(103\) 205.757i 1.99764i 0.0485437 + 0.998821i \(0.484542\pi\)
−0.0485437 + 0.998821i \(0.515458\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 195.959i − 1.83139i −0.401869 0.915697i \(-0.631639\pi\)
0.401869 0.915697i \(-0.368361\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −263.000 −2.17355
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −92.0000 −0.736000
\(126\) 0 0
\(127\) − 107.778i − 0.848642i −0.905512 0.424321i \(-0.860513\pi\)
0.905512 0.424321i \(-0.139487\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 78.3837i − 0.598349i −0.954198 0.299174i \(-0.903289\pi\)
0.954198 0.299174i \(-0.0967112\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −100.000 −0.689655
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 290.000 1.94631 0.973154 0.230153i \(-0.0739228\pi\)
0.973154 + 0.230153i \(0.0739228\pi\)
\(150\) 0 0
\(151\) − 48.9898i − 0.324436i −0.986755 0.162218i \(-0.948135\pi\)
0.986755 0.162218i \(-0.0518647\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 97.9796i − 0.632126i
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −169.000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −46.0000 −0.265896 −0.132948 0.991123i \(-0.542444\pi\)
−0.132948 + 0.991123i \(0.542444\pi\)
\(174\) 0 0
\(175\) 205.757i 1.17576i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 274.343i − 1.53264i −0.642458 0.766321i \(-0.722085\pi\)
0.642458 0.766321i \(-0.277915\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −290.000 −1.50259 −0.751295 0.659966i \(-0.770571\pi\)
−0.751295 + 0.659966i \(0.770571\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −194.000 −0.984772 −0.492386 0.870377i \(-0.663875\pi\)
−0.492386 + 0.870377i \(0.663875\pi\)
\(198\) 0 0
\(199\) − 342.929i − 1.72326i −0.507538 0.861630i \(-0.669444\pi\)
0.507538 0.861630i \(-0.330556\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 489.898i 2.41329i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −480.000 −2.21198
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 382.120i − 1.71354i −0.515695 0.856772i \(-0.672466\pi\)
0.515695 0.856772i \(-0.327534\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 293.939i 1.29488i 0.762115 + 0.647442i \(0.224161\pi\)
−0.762115 + 0.647442i \(0.775839\pi\)
\(228\) 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 \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −382.000 −1.58506 −0.792531 0.609831i \(-0.791237\pi\)
−0.792531 + 0.609831i \(0.791237\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −94.0000 −0.383673
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 176.363i − 0.702642i −0.936255 0.351321i \(-0.885733\pi\)
0.936255 0.351321i \(-0.114267\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 188.000 0.709434
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 430.000 1.59851 0.799257 0.600990i \(-0.205227\pi\)
0.799257 + 0.600990i \(0.205227\pi\)
\(270\) 0 0
\(271\) − 538.888i − 1.98852i −0.107011 0.994258i \(-0.534128\pi\)
0.107011 0.994258i \(-0.465872\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 411.514i 1.49642i
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −289.000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −386.000 −1.31741 −0.658703 0.752403i \(-0.728895\pi\)
−0.658703 + 0.752403i \(0.728895\pi\)
\(294\) 0 0
\(295\) 235.151i 0.797122i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −530.000 −1.69329 −0.846645 0.532158i \(-0.821381\pi\)
−0.846645 + 0.532158i \(0.821381\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 334.000 1.05363 0.526814 0.849981i \(-0.323386\pi\)
0.526814 + 0.849981i \(0.323386\pi\)
\(318\) 0 0
\(319\) 979.796i 3.07146i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 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
\(336\) 0 0
\(337\) 190.000 0.563798 0.281899 0.959444i \(-0.409036\pi\)
0.281899 + 0.959444i \(0.409036\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −960.000 −2.81525
\(342\) 0 0
\(343\) − 19.5959i − 0.0571310i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 685.857i − 1.97653i −0.152738 0.988267i \(-0.548809\pi\)
0.152738 0.988267i \(-0.451191\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −100.000 −0.273973
\(366\) 0 0
\(367\) 186.161i 0.507251i 0.967302 + 0.253626i \(0.0816231\pi\)
−0.967302 + 0.253626i \(0.918377\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 921.008i − 2.48250i
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 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
\(385\) −384.000 −0.997403
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −190.000 −0.488432 −0.244216 0.969721i \(-0.578531\pi\)
−0.244216 + 0.969721i \(0.578531\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 293.939i 0.744149i
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −718.000 −1.75550 −0.877751 0.479118i \(-0.840957\pi\)
−0.877751 + 0.479118i \(0.840957\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1152.00 2.78935
\(414\) 0 0
\(415\) 195.959i 0.472191i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 411.514i − 0.982134i −0.871122 0.491067i \(-0.836607\pi\)
0.871122 0.491067i \(-0.163393\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 670.000 1.54734 0.773672 0.633586i \(-0.218418\pi\)
0.773672 + 0.633586i \(0.218418\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 832.827i 1.89710i 0.316629 + 0.948550i \(0.397449\pi\)
−0.316629 + 0.948550i \(0.602551\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 881.816i 1.99056i 0.0970655 + 0.995278i \(0.469054\pi\)
−0.0970655 + 0.995278i \(0.530946\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 530.000 1.15974 0.579869 0.814710i \(-0.303104\pi\)
0.579869 + 0.814710i \(0.303104\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 530.000 1.14967 0.574837 0.818268i \(-0.305065\pi\)
0.574837 + 0.818268i \(0.305065\pi\)
\(462\) 0 0
\(463\) − 891.614i − 1.92573i −0.269978 0.962866i \(-0.587017\pi\)
0.269978 0.962866i \(-0.412983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 685.857i 1.46864i 0.678801 + 0.734322i \(0.262500\pi\)
−0.678801 + 0.734322i \(0.737500\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −380.000 −0.783505
\(486\) 0 0
\(487\) − 88.1816i − 0.181071i −0.995893 0.0905356i \(-0.971142\pi\)
0.995893 0.0905356i \(-0.0288579\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 862.220i − 1.75605i −0.478615 0.878025i \(-0.658861\pi\)
0.478615 0.878025i \(-0.341139\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 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
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 380.000 0.752475
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1010.00 −1.98428 −0.992141 0.125121i \(-0.960068\pi\)
−0.992141 + 0.125121i \(0.960068\pi\)
\(510\) 0 0
\(511\) 489.898i 0.958704i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 411.514i 0.799057i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 391.918i − 0.732558i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 921.008i 1.70873i
\(540\) 0 0
\(541\) 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
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1440.00 2.60398
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 914.000 1.64093 0.820467 0.571694i \(-0.193714\pi\)
0.820467 + 0.571694i \(0.193714\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1077.78i 1.91434i 0.289520 + 0.957172i \(0.406504\pi\)
−0.289520 + 0.957172i \(0.593496\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 290.000 0.502600 0.251300 0.967909i \(-0.419142\pi\)
0.251300 + 0.967909i \(0.419142\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 960.000 1.65232
\(582\) 0 0
\(583\) − 1842.02i − 3.15955i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 783.837i − 1.33533i −0.744463 0.667663i \(-0.767295\pi\)
0.744463 0.667663i \(-0.232705\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1198.00 −1.99334 −0.996672 0.0815138i \(-0.974025\pi\)
−0.996672 + 0.0815138i \(0.974025\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −526.000 −0.869421
\(606\) 0 0
\(607\) − 969.998i − 1.59802i −0.601318 0.799010i \(-0.705357\pi\)
0.601318 0.799010i \(-0.294643\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 244.949i − 0.388192i −0.980983 0.194096i \(-0.937823\pi\)
0.980983 0.194096i \(-0.0621773\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 215.555i − 0.339457i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 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 \(\pi\)
\(648\) 0 0
\(649\) 2304.00 3.55008
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1006.00 −1.54058 −0.770291 0.637693i \(-0.779889\pi\)
−0.770291 + 0.637693i \(0.779889\pi\)
\(654\) 0 0
\(655\) − 156.767i − 0.239339i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1097.37i − 1.66521i −0.553869 0.832603i \(-0.686849\pi\)
0.553869 0.832603i \(-0.313151\pi\)
\(660\) 0 0
\(661\) 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
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 190.000 0.282318 0.141159 0.989987i \(-0.454917\pi\)
0.141159 + 0.989987i \(0.454917\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1346.00 1.98818 0.994092 0.108545i \(-0.0346190\pi\)
0.994092 + 0.108545i \(0.0346190\pi\)
\(678\) 0 0
\(679\) 1861.61i 2.74170i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 293.939i 0.430364i 0.976574 + 0.215182i \(0.0690345\pi\)
−0.976574 + 0.215182i \(0.930965\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 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 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −50.0000 −0.0713267 −0.0356633 0.999364i \(-0.511354\pi\)
−0.0356633 + 0.999364i \(0.511354\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1861.61i − 2.63311i
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 2016.00 2.79612
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1050.00 1.44828
\(726\) 0 0
\(727\) 107.778i 0.148250i 0.997249 + 0.0741249i \(0.0236163\pi\)
−0.997249 + 0.0741249i \(0.976384\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 580.000 0.778523
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1920.00 −2.56342
\(750\) 0 0
\(751\) 538.888i 0.717560i 0.933422 + 0.358780i \(0.116807\pi\)
−0.933422 + 0.358780i \(0.883193\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 97.9796i − 0.129774i
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 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
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 862.000 1.12094 0.560468 0.828176i \(-0.310621\pi\)
0.560468 + 0.828176i \(0.310621\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1154.00 −1.49288 −0.746442 0.665450i \(-0.768240\pi\)
−0.746442 + 0.665450i \(0.768240\pi\)
\(774\) 0 0
\(775\) 1028.79i 1.32747i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1294.00 −1.62359 −0.811794 0.583944i \(-0.801509\pi\)
−0.811794 + 0.583944i \(0.801509\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 979.796i 1.22017i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 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
\(821\) 670.000 0.816078 0.408039 0.912965i \(-0.366213\pi\)
0.408039 + 0.912965i \(0.366213\pi\)
\(822\) 0 0
\(823\) − 1577.47i − 1.91673i −0.285541 0.958367i \(-0.592173\pi\)
0.285541 0.958367i \(-0.407827\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 587.878i − 0.710856i −0.934704 0.355428i \(-0.884335\pi\)
0.934704 0.355428i \(-0.115665\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1659.00 1.97265
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −338.000 −0.400000
\(846\) 0 0
\(847\) 2576.86i 3.04234i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −92.0000 −0.106358
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2880.00 3.31415
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 901.412i 1.03019i
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 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
\(889\) −1056.00 −1.18785
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) − 548.686i − 0.613057i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 2449.49i 2.72468i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1920.00 2.10296
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −768.000 −0.837514
\(918\) 0 0
\(919\) 1714.64i 1.86577i 0.360174 + 0.932885i \(0.382717\pi\)
−0.360174 + 0.932885i \(0.617283\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 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
\(936\) 0 0
\(937\) 1490.00 1.59018 0.795091 0.606491i \(-0.207423\pi\)
0.795091 + 0.606491i \(0.207423\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −430.000 −0.456961 −0.228480 0.973549i \(-0.573376\pi\)
−0.228480 + 0.973549i \(0.573376\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1371.71i 1.44848i 0.689546 + 0.724242i \(0.257810\pi\)
−0.689546 + 0.724242i \(0.742190\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1439.00 −1.49740
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −580.000 −0.601036
\(966\) 0 0
\(967\) − 303.737i − 0.314102i −0.987590 0.157051i \(-0.949801\pi\)
0.987590 0.157051i \(-0.0501987\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 215.555i 0.221993i 0.993821 + 0.110996i \(0.0354042\pi\)
−0.993821 + 0.110996i \(0.964596\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −388.000 −0.393909
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) − 1420.70i − 1.43361i −0.697275 0.716803i \(-0.745605\pi\)
0.697275 0.716803i \(-0.254395\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 685.857i − 0.689304i
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 2304.3.g.l.1279.1 2
3.2 odd 2 2304.3.g.i.1279.1 2
4.3 odd 2 inner 2304.3.g.l.1279.2 2
8.3 odd 2 2304.3.g.i.1279.2 2
8.5 even 2 2304.3.g.i.1279.1 2
12.11 even 2 2304.3.g.i.1279.2 2
16.3 odd 4 1152.3.b.f.703.1 4
16.5 even 4 1152.3.b.f.703.4 yes 4
16.11 odd 4 1152.3.b.f.703.3 yes 4
16.13 even 4 1152.3.b.f.703.2 yes 4
24.5 odd 2 CM 2304.3.g.l.1279.1 2
24.11 even 2 inner 2304.3.g.l.1279.2 2
48.5 odd 4 1152.3.b.f.703.2 yes 4
48.11 even 4 1152.3.b.f.703.1 4
48.29 odd 4 1152.3.b.f.703.4 yes 4
48.35 even 4 1152.3.b.f.703.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.b.f.703.1 4 16.3 odd 4
1152.3.b.f.703.1 4 48.11 even 4
1152.3.b.f.703.2 yes 4 16.13 even 4
1152.3.b.f.703.2 yes 4 48.5 odd 4
1152.3.b.f.703.3 yes 4 16.11 odd 4
1152.3.b.f.703.3 yes 4 48.35 even 4
1152.3.b.f.703.4 yes 4 16.5 even 4
1152.3.b.f.703.4 yes 4 48.29 odd 4
2304.3.g.i.1279.1 2 3.2 odd 2
2304.3.g.i.1279.1 2 8.5 even 2
2304.3.g.i.1279.2 2 8.3 odd 2
2304.3.g.i.1279.2 2 12.11 even 2
2304.3.g.l.1279.1 2 1.1 even 1 trivial
2304.3.g.l.1279.1 2 24.5 odd 2 CM
2304.3.g.l.1279.2 2 4.3 odd 2 inner
2304.3.g.l.1279.2 2 24.11 even 2 inner