Properties

Label 1152.3.b.f.703.1
Level $1152$
Weight $3$
Character 1152.703
Analytic conductor $31.390$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 703.1
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1152.703
Dual form 1152.3.b.f.703.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.00000i q^{5} -9.79796i q^{7} +O(q^{10})\) \(q-2.00000i q^{5} -9.79796i q^{7} +19.5959 q^{11} +21.0000 q^{25} -50.0000i q^{29} +48.9898i q^{31} -19.5959 q^{35} -47.0000 q^{49} -94.0000i q^{53} -39.1918i q^{55} -117.576 q^{59} +50.0000 q^{73} -192.000i q^{77} -146.969i q^{79} +97.9796 q^{83} -190.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 84q^{25} - 188q^{49} + 200q^{73} - 760q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) − 2.00000i − 0.400000i −0.979796 0.200000i \(-0.935906\pi\)
0.979796 0.200000i \(-0.0640942\pi\)
\(6\) 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
\(11\) 19.5959 1.78145 0.890724 0.454545i \(-0.150198\pi\)
0.890724 + 0.454545i \(0.150198\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 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
\(29\) − 50.0000i − 1.72414i −0.506791 0.862069i \(-0.669168\pi\)
0.506791 0.862069i \(-0.330832\pi\)
\(30\) 0 0
\(31\) 48.9898i 1.58032i 0.612903 + 0.790158i \(0.290002\pi\)
−0.612903 + 0.790158i \(0.709998\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −19.5959 −0.559883
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 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
\(53\) − 94.0000i − 1.77358i −0.462168 0.886792i \(-0.652928\pi\)
0.462168 0.886792i \(-0.347072\pi\)
\(54\) 0 0
\(55\) − 39.1918i − 0.712579i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −117.576 −1.99281 −0.996403 0.0847458i \(-0.972992\pi\)
−0.996403 + 0.0847458i \(0.972992\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 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 \(-0.388738\pi\)
0.342466 + 0.939530i \(0.388738\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 192.000i − 2.49351i
\(78\) 0 0
\(79\) − 146.969i − 1.86037i −0.367089 0.930186i \(-0.619645\pi\)
0.367089 0.930186i \(-0.380355\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 97.9796 1.18048 0.590238 0.807229i \(-0.299034\pi\)
0.590238 + 0.807229i \(0.299034\pi\)
\(84\) 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
\(101\) − 190.000i − 1.88119i −0.339533 0.940594i \(-0.610269\pi\)
0.339533 0.940594i \(-0.389731\pi\)
\(102\) 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
\(107\) 195.959 1.83139 0.915697 0.401869i \(-0.131639\pi\)
0.915697 + 0.401869i \(0.131639\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
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
\(125\) − 92.0000i − 0.736000i
\(126\) 0 0
\(127\) 107.778i 0.848642i 0.905512 + 0.424321i \(0.139487\pi\)
−0.905512 + 0.424321i \(0.860513\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −78.3837 −0.598349 −0.299174 0.954198i \(-0.596711\pi\)
−0.299174 + 0.954198i \(0.596711\pi\)
\(132\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 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
\(149\) − 290.000i − 1.94631i −0.230153 0.973154i \(-0.573923\pi\)
0.230153 0.973154i \(-0.426077\pi\)
\(150\) 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
\(155\) 97.9796 0.632126
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 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
\(173\) − 46.0000i − 0.265896i −0.991123 0.132948i \(-0.957556\pi\)
0.991123 0.132948i \(-0.0424443\pi\)
\(174\) 0 0
\(175\) − 205.757i − 1.17576i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −274.343 −1.53264 −0.766321 0.642458i \(-0.777915\pi\)
−0.766321 + 0.642458i \(0.777915\pi\)
\(180\) 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
\(189\) 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
\(197\) 194.000i 0.984772i 0.870377 + 0.492386i \(0.163875\pi\)
−0.870377 + 0.492386i \(0.836125\pi\)
\(198\) 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
\(203\) −489.898 −2.41329
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 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 \(0.327534\pi\)
−0.515695 + 0.856772i \(0.672466\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 293.939 1.29488 0.647442 0.762115i \(-0.275839\pi\)
0.647442 + 0.762115i \(0.275839\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 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
\(245\) 94.0000i 0.383673i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 176.363 0.702642 0.351321 0.936255i \(-0.385733\pi\)
0.351321 + 0.936255i \(0.385733\pi\)
\(252\) 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
\(269\) 430.000i 1.59851i 0.600990 + 0.799257i \(0.294773\pi\)
−0.600990 + 0.799257i \(0.705227\pi\)
\(270\) 0 0
\(271\) 538.888i 1.98852i 0.107011 + 0.994258i \(0.465872\pi\)
−0.107011 + 0.994258i \(0.534128\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 411.514 1.49642
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 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
\(293\) 386.000i 1.31741i 0.752403 + 0.658703i \(0.228895\pi\)
−0.752403 + 0.658703i \(0.771105\pi\)
\(294\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 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 \(-0.178619\pi\)
0.846645 + 0.532158i \(0.178619\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 334.000i 1.05363i 0.849981 + 0.526814i \(0.176614\pi\)
−0.849981 + 0.526814i \(0.823386\pi\)
\(318\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 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
\(341\) 960.000i 2.81525i
\(342\) 0 0
\(343\) − 19.5959i − 0.0571310i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 685.857 1.97653 0.988267 0.152738i \(-0.0488090\pi\)
0.988267 + 0.152738i \(0.0488090\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
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
\(365\) − 100.000i − 0.273973i
\(366\) 0 0
\(367\) − 186.161i − 0.507251i −0.967302 0.253626i \(-0.918377\pi\)
0.967302 0.253626i \(-0.0816231\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −921.008 −2.48250
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 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
\(383\) 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
\(389\) 190.000i 0.488432i 0.969721 + 0.244216i \(0.0785306\pi\)
−0.969721 + 0.244216i \(0.921469\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −293.939 −0.744149
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
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 \(-0.159043\pi\)
0.877751 + 0.479118i \(0.159043\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1152.00i 2.78935i
\(414\) 0 0
\(415\) − 195.959i − 0.472191i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −411.514 −0.982134 −0.491067 0.871122i \(-0.663393\pi\)
−0.491067 + 0.871122i \(0.663393\pi\)
\(420\) 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 \(-0.218418\pi\)
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
\(443\) −881.816 −1.99056 −0.995278 0.0970655i \(-0.969054\pi\)
−0.995278 + 0.0970655i \(0.969054\pi\)
\(444\) 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 \(-0.696896\pi\)
−0.579869 + 0.814710i \(0.696896\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 530.000i 1.14967i 0.818268 + 0.574837i \(0.194935\pi\)
−0.818268 + 0.574837i \(0.805065\pi\)
\(462\) 0 0
\(463\) 891.614i 1.92573i 0.269978 + 0.962866i \(0.412983\pi\)
−0.269978 + 0.962866i \(0.587017\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 685.857 1.46864 0.734322 0.678801i \(-0.237500\pi\)
0.734322 + 0.678801i \(0.237500\pi\)
\(468\) 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
\(485\) 380.000i 0.783505i
\(486\) 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
\(491\) 862.220 1.75605 0.878025 0.478615i \(-0.158861\pi\)
0.878025 + 0.478615i \(0.158861\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 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
\(509\) − 1010.00i − 1.98428i −0.125121 0.992141i \(-0.539932\pi\)
0.125121 0.992141i \(-0.460068\pi\)
\(510\) 0 0
\(511\) − 489.898i − 0.958704i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 411.514 0.799057
\(516\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 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
\(539\) −921.008 −1.70873
\(540\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 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
\(557\) 914.000i 1.64093i 0.571694 + 0.820467i \(0.306286\pi\)
−0.571694 + 0.820467i \(0.693714\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1077.78 1.91434 0.957172 0.289520i \(-0.0934958\pi\)
0.957172 + 0.289520i \(0.0934958\pi\)
\(564\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 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
\(581\) − 960.000i − 1.65232i
\(582\) 0 0
\(583\) − 1842.02i − 3.15955i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 783.837 1.33533 0.667663 0.744463i \(-0.267295\pi\)
0.667663 + 0.744463i \(0.267295\pi\)
\(588\) 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 \(-0.0259755\pi\)
0.996672 + 0.0815138i \(0.0259755\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 526.000i − 0.869421i
\(606\) 0 0
\(607\) 969.998i 1.59802i 0.601318 + 0.799010i \(0.294643\pi\)
−0.601318 + 0.799010i \(0.705357\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 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
\(635\) 215.555 0.339457
\(636\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 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
\(653\) − 1006.00i − 1.54058i −0.637693 0.770291i \(-0.720111\pi\)
0.637693 0.770291i \(-0.279889\pi\)
\(654\) 0 0
\(655\) 156.767i 0.239339i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1097.37 −1.66521 −0.832603 0.553869i \(-0.813151\pi\)
−0.832603 + 0.553869i \(0.813151\pi\)
\(660\) 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
\(669\) 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
\(677\) − 1346.00i − 1.98818i −0.108545 0.994092i \(-0.534619\pi\)
0.108545 0.994092i \(-0.465381\pi\)
\(678\) 0 0
\(679\) 1861.61i 2.74170i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −293.939 −0.430364 −0.215182 0.976574i \(-0.569035\pi\)
−0.215182 + 0.976574i \(0.569035\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
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
\(701\) − 50.0000i − 0.0713267i −0.999364 0.0356633i \(-0.988646\pi\)
0.999364 0.0356633i \(-0.0113544\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1861.61 −2.63311
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 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
\(725\) − 1050.00i − 1.44828i
\(726\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 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
\(743\) 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
\(749\) − 1920.00i − 2.56342i
\(750\) 0 0
\(751\) − 538.888i − 0.717560i −0.933422 0.358780i \(-0.883193\pi\)
0.933422 0.358780i \(-0.116807\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −97.9796 −0.129774
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
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
\(773\) 1154.00i 1.49288i 0.665450 + 0.746442i \(0.268240\pi\)
−0.665450 + 0.746442i \(0.731760\pi\)
\(774\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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
\(797\) − 1294.00i − 1.62359i −0.583944 0.811794i \(-0.698491\pi\)
0.583944 0.811794i \(-0.301509\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 979.796 1.22017
\(804\) 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 1.00000i \(-0.5\pi\)
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
\(821\) − 670.000i − 0.816078i −0.912965 0.408039i \(-0.866213\pi\)
0.912965 0.408039i \(-0.133787\pi\)
\(822\) 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
\(827\) 587.878 0.710856 0.355428 0.934704i \(-0.384335\pi\)
0.355428 + 0.934704i \(0.384335\pi\)
\(828\) 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
\(837\) 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
\(845\) − 338.000i − 0.400000i
\(846\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 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
\(869\) − 2880.00i − 3.31415i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −901.412 −1.03019
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 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
\(899\) 2449.49 2.72468
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 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
\(917\) 768.000i 0.837514i
\(918\) 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 \(-0.792577\pi\)
−0.795091 + 0.606491i \(0.792577\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 430.000i − 0.456961i −0.973549 0.228480i \(-0.926624\pi\)
0.973549 0.228480i \(-0.0733757\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1371.71 1.44848 0.724242 0.689546i \(-0.242190\pi\)
0.724242 + 0.689546i \(0.242190\pi\)
\(948\) 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
\(965\) 580.000i 0.601036i
\(966\) 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
\(971\) −215.555 −0.221993 −0.110996 0.993821i \(-0.535404\pi\)
−0.110996 + 0.993821i \(0.535404\pi\)
\(972\) 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 \(0.254395\pi\)
−0.697275 + 0.716803i \(0.745605\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −685.857 −0.689304
\(996\) 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 1152.3.b.f.703.1 4
3.2 odd 2 inner 1152.3.b.f.703.3 yes 4
4.3 odd 2 inner 1152.3.b.f.703.2 yes 4
8.3 odd 2 inner 1152.3.b.f.703.4 yes 4
8.5 even 2 inner 1152.3.b.f.703.3 yes 4
12.11 even 2 inner 1152.3.b.f.703.4 yes 4
16.3 odd 4 2304.3.g.i.1279.1 2
16.5 even 4 2304.3.g.l.1279.2 2
16.11 odd 4 2304.3.g.l.1279.1 2
16.13 even 4 2304.3.g.i.1279.2 2
24.5 odd 2 CM 1152.3.b.f.703.1 4
24.11 even 2 inner 1152.3.b.f.703.2 yes 4
48.5 odd 4 2304.3.g.i.1279.2 2
48.11 even 4 2304.3.g.i.1279.1 2
48.29 odd 4 2304.3.g.l.1279.2 2
48.35 even 4 2304.3.g.l.1279.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.3.b.f.703.1 4 1.1 even 1 trivial
1152.3.b.f.703.1 4 24.5 odd 2 CM
1152.3.b.f.703.2 yes 4 4.3 odd 2 inner
1152.3.b.f.703.2 yes 4 24.11 even 2 inner
1152.3.b.f.703.3 yes 4 3.2 odd 2 inner
1152.3.b.f.703.3 yes 4 8.5 even 2 inner
1152.3.b.f.703.4 yes 4 8.3 odd 2 inner
1152.3.b.f.703.4 yes 4 12.11 even 2 inner
2304.3.g.i.1279.1 2 16.3 odd 4
2304.3.g.i.1279.1 2 48.11 even 4
2304.3.g.i.1279.2 2 16.13 even 4
2304.3.g.i.1279.2 2 48.5 odd 4
2304.3.g.l.1279.1 2 16.11 odd 4
2304.3.g.l.1279.1 2 48.35 even 4
2304.3.g.l.1279.2 2 16.5 even 4
2304.3.g.l.1279.2 2 48.29 odd 4