Properties

Label 384.3.h.a.65.1
Level $384$
Weight $3$
Character 384.65
Self dual yes
Analytic conductor $10.463$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 65.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 384.65

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -9.79796 q^{5} -9.79796 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -9.79796 q^{5} -9.79796 q^{7} +9.00000 q^{9} -10.0000 q^{11} +29.3939 q^{15} +29.3939 q^{21} +71.0000 q^{25} -27.0000 q^{27} +29.3939 q^{29} -48.9898 q^{31} +30.0000 q^{33} +96.0000 q^{35} -88.1816 q^{45} +47.0000 q^{49} -48.9898 q^{53} +97.9796 q^{55} +10.0000 q^{59} -88.1816 q^{63} -50.0000 q^{73} -213.000 q^{75} +97.9796 q^{77} +146.969 q^{79} +81.0000 q^{81} +134.000 q^{83} -88.1816 q^{87} +146.969 q^{93} -190.000 q^{97} -90.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} + 18q^{9} + O(q^{10}) \) \( 2q - 6q^{3} + 18q^{9} - 20q^{11} + 142q^{25} - 54q^{27} + 60q^{33} + 192q^{35} + 94q^{49} + 20q^{59} - 100q^{73} - 426q^{75} + 162q^{81} + 268q^{83} - 380q^{97} - 180q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) −3.00000 −1.00000
\(4\) 0 0
\(5\) −9.79796 −1.95959 −0.979796 0.200000i \(-0.935906\pi\)
−0.979796 + 0.200000i \(0.935906\pi\)
\(6\) 0 0
\(7\) −9.79796 −1.39971 −0.699854 0.714286i \(-0.746752\pi\)
−0.699854 + 0.714286i \(0.746752\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) −10.0000 −0.909091 −0.454545 0.890724i \(-0.650198\pi\)
−0.454545 + 0.890724i \(0.650198\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 29.3939 1.95959
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 29.3939 1.39971
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 71.0000 2.84000
\(26\) 0 0
\(27\) −27.0000 −1.00000
\(28\) 0 0
\(29\) 29.3939 1.01358 0.506791 0.862069i \(-0.330832\pi\)
0.506791 + 0.862069i \(0.330832\pi\)
\(30\) 0 0
\(31\) −48.9898 −1.58032 −0.790158 0.612903i \(-0.790002\pi\)
−0.790158 + 0.612903i \(0.790002\pi\)
\(32\) 0 0
\(33\) 30.0000 0.909091
\(34\) 0 0
\(35\) 96.0000 2.74286
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −88.1816 −1.95959
\(46\) 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
\(53\) −48.9898 −0.924336 −0.462168 0.886792i \(-0.652928\pi\)
−0.462168 + 0.886792i \(0.652928\pi\)
\(54\) 0 0
\(55\) 97.9796 1.78145
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0000 0.169492 0.0847458 0.996403i \(-0.472992\pi\)
0.0847458 + 0.996403i \(0.472992\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −88.1816 −1.39971
\(64\) 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
\(75\) −213.000 −2.84000
\(76\) 0 0
\(77\) 97.9796 1.27246
\(78\) 0 0
\(79\) 146.969 1.86037 0.930186 0.367089i \(-0.119645\pi\)
0.930186 + 0.367089i \(0.119645\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) 134.000 1.61446 0.807229 0.590238i \(-0.200966\pi\)
0.807229 + 0.590238i \(0.200966\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −88.1816 −1.01358
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 146.969 1.58032
\(94\) 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
\(99\) −90.0000 −0.909091
\(100\) 0 0
\(101\) 68.5857 0.679066 0.339533 0.940594i \(-0.389731\pi\)
0.339533 + 0.940594i \(0.389731\pi\)
\(102\) 0 0
\(103\) −205.757 −1.99764 −0.998821 0.0485437i \(-0.984542\pi\)
−0.998821 + 0.0485437i \(0.984542\pi\)
\(104\) 0 0
\(105\) −288.000 −2.74286
\(106\) 0 0
\(107\) −86.0000 −0.803738 −0.401869 0.915697i \(-0.631639\pi\)
−0.401869 + 0.915697i \(0.631639\pi\)
\(108\) 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
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −21.0000 −0.173554
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −450.706 −3.60565
\(126\) 0 0
\(127\) 107.778 0.848642 0.424321 0.905512i \(-0.360513\pi\)
0.424321 + 0.905512i \(0.360513\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 250.000 1.90840 0.954198 0.299174i \(-0.0967112\pi\)
0.954198 + 0.299174i \(0.0967112\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 264.545 1.95959
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −288.000 −1.98621
\(146\) 0 0
\(147\) −141.000 −0.959184
\(148\) 0 0
\(149\) 68.5857 0.460307 0.230153 0.973154i \(-0.426077\pi\)
0.230153 + 0.973154i \(0.426077\pi\)
\(150\) 0 0
\(151\) −48.9898 −0.324436 −0.162218 0.986755i \(-0.551865\pi\)
−0.162218 + 0.986755i \(0.551865\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 480.000 3.09677
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 146.969 0.924336
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) −293.939 −1.78145
\(166\) 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
\(173\) 342.929 1.98225 0.991123 0.132948i \(-0.0424443\pi\)
0.991123 + 0.132948i \(0.0424443\pi\)
\(174\) 0 0
\(175\) −695.655 −3.97517
\(176\) 0 0
\(177\) −30.0000 −0.169492
\(178\) 0 0
\(179\) −230.000 −1.28492 −0.642458 0.766321i \(-0.722085\pi\)
−0.642458 + 0.766321i \(0.722085\pi\)
\(180\) 0 0
\(181\) 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
\(188\) 0 0
\(189\) 264.545 1.39971
\(190\) 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
\(197\) 342.929 1.74075 0.870377 0.492386i \(-0.163875\pi\)
0.870377 + 0.492386i \(0.163875\pi\)
\(198\) 0 0
\(199\) 342.929 1.72326 0.861630 0.507538i \(-0.169444\pi\)
0.861630 + 0.507538i \(0.169444\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −288.000 −1.41872
\(204\) 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
\(219\) 150.000 0.684932
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 382.120 1.71354 0.856772 0.515695i \(-0.172466\pi\)
0.856772 + 0.515695i \(0.172466\pi\)
\(224\) 0 0
\(225\) 639.000 2.84000
\(226\) 0 0
\(227\) −346.000 −1.52423 −0.762115 0.647442i \(-0.775839\pi\)
−0.762115 + 0.647442i \(0.775839\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −293.939 −1.27246
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −440.908 −1.86037
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 382.000 1.58506 0.792531 0.609831i \(-0.208763\pi\)
0.792531 + 0.609831i \(0.208763\pi\)
\(242\) 0 0
\(243\) −243.000 −1.00000
\(244\) 0 0
\(245\) −460.504 −1.87961
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −402.000 −1.61446
\(250\) 0 0
\(251\) 470.000 1.87251 0.936255 0.351321i \(-0.114267\pi\)
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 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 264.545 1.01358
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 480.000 1.81132
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −323.333 −1.20198 −0.600990 0.799257i \(-0.705227\pi\)
−0.600990 + 0.799257i \(0.705227\pi\)
\(270\) 0 0
\(271\) 538.888 1.98852 0.994258 0.107011i \(-0.0341280\pi\)
0.994258 + 0.107011i \(0.0341280\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −710.000 −2.58182
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −440.908 −1.58032
\(280\) 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
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 570.000 1.95876
\(292\) 0 0
\(293\) −440.908 −1.50481 −0.752403 0.658703i \(-0.771105\pi\)
−0.752403 + 0.658703i \(0.771105\pi\)
\(294\) 0 0
\(295\) −97.9796 −0.332134
\(296\) 0 0
\(297\) 270.000 0.909091
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −205.757 −0.679066
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 617.271 1.99764
\(310\) 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
\(315\) 864.000 2.74286
\(316\) 0 0
\(317\) 538.888 1.69996 0.849981 0.526814i \(-0.176614\pi\)
0.849981 + 0.526814i \(0.176614\pi\)
\(318\) 0 0
\(319\) −293.939 −0.921438
\(320\) 0 0
\(321\) 258.000 0.803738
\(322\) 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
\(341\) 489.898 1.43665
\(342\) 0 0
\(343\) 19.5959 0.0571310
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −106.000 −0.305476 −0.152738 0.988267i \(-0.548809\pi\)
−0.152738 + 0.988267i \(0.548809\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 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
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 63.0000 0.173554
\(364\) 0 0
\(365\) 489.898 1.34219
\(366\) 0 0
\(367\) 186.161 0.507251 0.253626 0.967302i \(-0.418377\pi\)
0.253626 + 0.967302i \(0.418377\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 480.000 1.29380
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 1352.12 3.60565
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −323.333 −0.848642
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −960.000 −2.49351
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −754.443 −1.93944 −0.969721 0.244216i \(-0.921469\pi\)
−0.969721 + 0.244216i \(0.921469\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −750.000 −1.90840
\(394\) 0 0
\(395\) −1440.00 −3.64557
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −793.635 −1.95959
\(406\) 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
\(413\) −97.9796 −0.237239
\(414\) 0 0
\(415\) −1312.93 −3.16368
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −730.000 −1.74224 −0.871122 0.491067i \(-0.836607\pi\)
−0.871122 + 0.491067i \(0.836607\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 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 \(-0.781582\pi\)
−0.773672 + 0.633586i \(0.781582\pi\)
\(434\) 0 0
\(435\) 864.000 1.98621
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −832.827 −1.89710 −0.948550 0.316629i \(-0.897449\pi\)
−0.948550 + 0.316629i \(0.897449\pi\)
\(440\) 0 0
\(441\) 423.000 0.959184
\(442\) 0 0
\(443\) 86.0000 0.194131 0.0970655 0.995278i \(-0.469054\pi\)
0.0970655 + 0.995278i \(0.469054\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −205.757 −0.460307
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 146.969 0.324436
\(454\) 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
\(461\) −754.443 −1.63654 −0.818268 0.574837i \(-0.805065\pi\)
−0.818268 + 0.574837i \(0.805065\pi\)
\(462\) 0 0
\(463\) 891.614 1.92573 0.962866 0.269978i \(-0.0870166\pi\)
0.962866 + 0.269978i \(0.0870166\pi\)
\(464\) 0 0
\(465\) −1440.00 −3.09677
\(466\) 0 0
\(467\) −634.000 −1.35760 −0.678801 0.734322i \(-0.737500\pi\)
−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
\(477\) −440.908 −0.924336
\(478\) 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
\(485\) 1861.61 3.83838
\(486\) 0 0
\(487\) −88.1816 −0.181071 −0.0905356 0.995893i \(-0.528858\pi\)
−0.0905356 + 0.995893i \(0.528858\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −470.000 −0.957230 −0.478615 0.878025i \(-0.658861\pi\)
−0.478615 + 0.878025i \(0.658861\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 881.816 1.78145
\(496\) 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
\(505\) −672.000 −1.33069
\(506\) 0 0
\(507\) −507.000 −1.00000
\(508\) 0 0
\(509\) −127.373 −0.250243 −0.125121 0.992141i \(-0.539932\pi\)
−0.125121 + 0.992141i \(0.539932\pi\)
\(510\) 0 0
\(511\) 489.898 0.958704
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2016.00 3.91456
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1028.79 −1.98225
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 2086.97 3.97517
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 90.0000 0.169492
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 842.624 1.57500
\(536\) 0 0
\(537\) 690.000 1.28492
\(538\) 0 0
\(539\) −470.000 −0.871985
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 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
\(557\) −636.867 −1.14339 −0.571694 0.820467i \(-0.693714\pi\)
−0.571694 + 0.820467i \(0.693714\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 326.000 0.579041 0.289520 0.957172i \(-0.406504\pi\)
0.289520 + 0.957172i \(0.406504\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −793.635 −1.39971
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −290.000 −0.502600 −0.251300 0.967909i \(-0.580858\pi\)
−0.251300 + 0.967909i \(0.580858\pi\)
\(578\) 0 0
\(579\) 870.000 1.50259
\(580\) 0 0
\(581\) −1312.93 −2.25977
\(582\) 0 0
\(583\) 489.898 0.840305
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 874.000 1.48893 0.744463 0.667663i \(-0.232705\pi\)
0.744463 + 0.667663i \(0.232705\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −1028.79 −1.74075
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1028.79 −1.72326
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1198.00 1.99334 0.996672 0.0815138i \(-0.0259755\pi\)
0.996672 + 0.0815138i \(0.0259755\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 205.757 0.340094
\(606\) 0 0
\(607\) 969.998 1.59802 0.799010 0.601318i \(-0.205357\pi\)
0.799010 + 0.601318i \(0.205357\pi\)
\(608\) 0 0
\(609\) 864.000 1.41872
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2641.00 4.22560
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −244.949 −0.388192 −0.194096 0.980983i \(-0.562177\pi\)
−0.194096 + 0.980983i \(0.562177\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1056.00 −1.66299
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −100.000 −0.154083
\(650\) 0 0
\(651\) −1440.00 −2.21198
\(652\) 0 0
\(653\) −832.827 −1.27539 −0.637693 0.770291i \(-0.720111\pi\)
−0.637693 + 0.770291i \(0.720111\pi\)
\(654\) 0 0
\(655\) −2449.49 −3.73968
\(656\) 0 0
\(657\) −450.000 −0.684932
\(658\) 0 0
\(659\) 730.000 1.10774 0.553869 0.832603i \(-0.313151\pi\)
0.553869 + 0.832603i \(0.313151\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1146.36 −1.71354
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −190.000 −0.282318 −0.141159 0.989987i \(-0.545083\pi\)
−0.141159 + 0.989987i \(0.545083\pi\)
\(674\) 0 0
\(675\) −1917.00 −2.84000
\(676\) 0 0
\(677\) 146.969 0.217089 0.108545 0.994092i \(-0.465381\pi\)
0.108545 + 0.994092i \(0.465381\pi\)
\(678\) 0 0
\(679\) 1861.61 2.74170
\(680\) 0 0
\(681\) 1038.00 1.52423
\(682\) 0 0
\(683\) 1334.00 1.95315 0.976574 0.215182i \(-0.0690345\pi\)
0.976574 + 0.215182i \(0.0690345\pi\)
\(684\) 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
\(693\) 881.816 1.27246
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1401.11 1.99873 0.999364 0.0356633i \(-0.0113544\pi\)
0.999364 + 0.0356633i \(0.0113544\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −672.000 −0.950495
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 1322.72 1.86037
\(712\) 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
\(723\) −1146.00 −1.58506
\(724\) 0 0
\(725\) 2086.97 2.87857
\(726\) 0 0
\(727\) 107.778 0.148250 0.0741249 0.997249i \(-0.476384\pi\)
0.0741249 + 0.997249i \(0.476384\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 1381.51 1.87961
\(736\) 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
\(745\) −672.000 −0.902013
\(746\) 0 0
\(747\) 1206.00 1.61446
\(748\) 0 0
\(749\) 842.624 1.12500
\(750\) 0 0
\(751\) 538.888 0.717560 0.358780 0.933422i \(-0.383193\pi\)
0.358780 + 0.933422i \(0.383193\pi\)
\(752\) 0 0
\(753\) −1410.00 −1.87251
\(754\) 0 0
\(755\) 480.000 0.635762
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 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
\(767\) 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
\(773\) −1028.79 −1.33090 −0.665450 0.746442i \(-0.731760\pi\)
−0.665450 + 0.746442i \(0.731760\pi\)
\(774\) 0 0
\(775\) −3478.28 −4.48810
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −793.635 −1.01358
\(784\) 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
\(795\) −1440.00 −1.81132
\(796\) 0 0
\(797\) 930.806 1.16789 0.583944 0.811794i \(-0.301509\pi\)
0.583944 + 0.811794i \(0.301509\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 500.000 0.622665
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 969.998 1.20198
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −1616.66 −1.98852
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1499.09 −1.82593 −0.912965 0.408039i \(-0.866213\pi\)
−0.912965 + 0.408039i \(0.866213\pi\)
\(822\) 0 0
\(823\) −1577.47 −1.91673 −0.958367 0.285541i \(-0.907827\pi\)
−0.958367 + 0.285541i \(0.907827\pi\)
\(824\) 0 0
\(825\) 2130.00 2.58182
\(826\) 0 0
\(827\) 1546.00 1.86941 0.934704 0.355428i \(-0.115665\pi\)
0.934704 + 0.355428i \(0.115665\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1322.72 1.58032
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 23.0000 0.0273484
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1655.86 −1.95959
\(846\) 0 0
\(847\) 205.757 0.242925
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −3360.00 −3.88439
\(866\) 0 0
\(867\) −867.000 −1.00000
\(868\) 0 0
\(869\) −1469.69 −1.69125
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −1710.00 −1.95876
\(874\) 0 0
\(875\) 4416.00 5.04686
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 1322.72 1.50481
\(880\) 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
\(885\) 293.939 0.332134
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1056.00 −1.18785
\(890\) 0 0
\(891\) −810.000 −0.909091
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 2253.53 2.51791
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1440.00 −1.60178
\(900\) 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
\(909\) 617.271 0.679066
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1340.00 −1.46769
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −2449.49 −2.67120
\(918\) 0 0
\(919\) 1714.64 1.86577 0.932885 0.360174i \(-0.117283\pi\)
0.932885 + 0.360174i \(0.117283\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1851.81 −1.99764
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 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
\(939\) 1590.00 1.69329
\(940\) 0 0
\(941\) 1832.22 1.94710 0.973549 0.228480i \(-0.0733757\pi\)
0.973549 + 0.228480i \(0.0733757\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −2592.00 −2.74286
\(946\) 0 0
\(947\) 1306.00 1.37909 0.689546 0.724242i \(-0.257810\pi\)
0.689546 + 0.724242i \(0.257810\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1616.66 −1.69996
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 881.816 0.921438
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1439.00 1.49740
\(962\) 0 0
\(963\) −774.000 −0.803738
\(964\) 0 0
\(965\) 2841.41 2.94446
\(966\) 0 0
\(967\) 303.737 0.314102 0.157051 0.987590i \(-0.449801\pi\)
0.157051 + 0.987590i \(0.449801\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1930.00 −1.98764 −0.993821 0.110996i \(-0.964596\pi\)
−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 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 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
\(985\) −3360.00 −3.41117
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1420.70 −1.43361 −0.716803 0.697275i \(-0.754395\pi\)
−0.716803 + 0.697275i \(0.754395\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3360.00 −3.37688
\(996\) 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 384.3.h.a.65.1 2
3.2 odd 2 384.3.h.d.65.2 yes 2
4.3 odd 2 384.3.h.d.65.1 yes 2
8.3 odd 2 inner 384.3.h.a.65.2 yes 2
8.5 even 2 384.3.h.d.65.2 yes 2
12.11 even 2 inner 384.3.h.a.65.2 yes 2
16.3 odd 4 768.3.e.j.257.4 4
16.5 even 4 768.3.e.j.257.3 4
16.11 odd 4 768.3.e.j.257.1 4
16.13 even 4 768.3.e.j.257.2 4
24.5 odd 2 CM 384.3.h.a.65.1 2
24.11 even 2 384.3.h.d.65.1 yes 2
48.5 odd 4 768.3.e.j.257.2 4
48.11 even 4 768.3.e.j.257.4 4
48.29 odd 4 768.3.e.j.257.3 4
48.35 even 4 768.3.e.j.257.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.h.a.65.1 2 1.1 even 1 trivial
384.3.h.a.65.1 2 24.5 odd 2 CM
384.3.h.a.65.2 yes 2 8.3 odd 2 inner
384.3.h.a.65.2 yes 2 12.11 even 2 inner
384.3.h.d.65.1 yes 2 4.3 odd 2
384.3.h.d.65.1 yes 2 24.11 even 2
384.3.h.d.65.2 yes 2 3.2 odd 2
384.3.h.d.65.2 yes 2 8.5 even 2
768.3.e.j.257.1 4 16.11 odd 4
768.3.e.j.257.1 4 48.35 even 4
768.3.e.j.257.2 4 16.13 even 4
768.3.e.j.257.2 4 48.5 odd 4
768.3.e.j.257.3 4 16.5 even 4
768.3.e.j.257.3 4 48.29 odd 4
768.3.e.j.257.4 4 16.3 odd 4
768.3.e.j.257.4 4 48.11 even 4