Properties

Label 3024.2.b.q.1567.4
Level 3024
Weight 2
Character 3024.1567
Analytic conductor 24.147
Analytic rank 0
Dimension 4
CM discriminant -84
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-6}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 24 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1567.4
Root \(4.46512i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.q.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.46512i q^{5} +2.64575 q^{7} +O(q^{10})\) \(q+4.46512i q^{5} +2.64575 q^{7} -4.46512i q^{11} +7.34847i q^{17} -8.64575 q^{19} -2.88335i q^{23} -14.9373 q^{25} -8.29150 q^{31} +11.8136i q^{35} -3.93725 q^{37} +2.88335i q^{41} +7.00000 q^{49} +19.9373 q^{55} +10.2318i q^{71} -11.8136i q^{77} -32.8118 q^{85} +14.9771i q^{89} -38.6043i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 24q^{19} - 28q^{25} - 12q^{31} + 16q^{37} + 28q^{49} + 48q^{55} - 36q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 4.46512i 1.99686i 0.0560116 + 0.998430i \(0.482162\pi\)
−0.0560116 + 0.998430i \(0.517838\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.46512i − 1.34628i −0.739514 0.673141i \(-0.764945\pi\)
0.739514 0.673141i \(-0.235055\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.34847i 1.78227i 0.453743 + 0.891133i \(0.350089\pi\)
−0.453743 + 0.891133i \(0.649911\pi\)
\(18\) 0 0
\(19\) −8.64575 −1.98347 −0.991736 0.128298i \(-0.959049\pi\)
−0.991736 + 0.128298i \(0.959049\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.88335i − 0.601221i −0.953747 0.300610i \(-0.902810\pi\)
0.953747 0.300610i \(-0.0971904\pi\)
\(24\) 0 0
\(25\) −14.9373 −2.98745
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −8.29150 −1.48920 −0.744599 0.667512i \(-0.767359\pi\)
−0.744599 + 0.667512i \(0.767359\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 11.8136i 1.99686i
\(36\) 0 0
\(37\) −3.93725 −0.647281 −0.323640 0.946180i \(-0.604907\pi\)
−0.323640 + 0.946180i \(0.604907\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.88335i 0.450304i 0.974324 + 0.225152i \(0.0722879\pi\)
−0.974324 + 0.225152i \(0.927712\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 19.9373 2.68834
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.2318i 1.21429i 0.794590 + 0.607147i \(0.207686\pi\)
−0.794590 + 0.607147i \(0.792314\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 11.8136i − 1.34628i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −32.8118 −3.55894
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.9771i 1.58757i 0.608198 + 0.793785i \(0.291893\pi\)
−0.608198 + 0.793785i \(0.708107\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 38.6043i − 3.96072i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 7.34847i − 0.731200i −0.930772 0.365600i \(-0.880864\pi\)
0.930772 0.365600i \(-0.119136\pi\)
\(102\) 0 0
\(103\) 9.70850 0.956607 0.478303 0.878195i \(-0.341252\pi\)
0.478303 + 0.878195i \(0.341252\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.4422i 1.87955i 0.341793 + 0.939775i \(0.388966\pi\)
−0.341793 + 0.939775i \(0.611034\pi\)
\(108\) 0 0
\(109\) −10.8745 −1.04159 −0.520794 0.853682i \(-0.674364\pi\)
−0.520794 + 0.853682i \(0.674364\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 12.8745 1.20055
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 19.4422i 1.78227i
\(120\) 0 0
\(121\) −8.93725 −0.812478
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 44.3710i − 3.96866i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −22.8745 −1.98347
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −21.1660 −1.79528 −0.897639 0.440732i \(-0.854719\pi\)
−0.897639 + 0.440732i \(0.854719\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 37.0225i − 2.97372i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 7.62864i − 0.601221i
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 23.9073i − 1.81764i −0.417187 0.908821i \(-0.636984\pi\)
0.417187 0.908821i \(-0.363016\pi\)
\(174\) 0 0
\(175\) −39.5203 −2.98745
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 19.4422i − 1.45318i −0.687071 0.726590i \(-0.741104\pi\)
0.687071 0.726590i \(-0.258896\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 17.5803i − 1.29253i
\(186\) 0 0
\(187\) 32.8118 2.39943
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.5105i 1.91823i 0.283010 + 0.959117i \(0.408667\pi\)
−0.283010 + 0.959117i \(0.591333\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) −26.6458 −1.88887 −0.944434 0.328702i \(-0.893389\pi\)
−0.944434 + 0.328702i \(0.893389\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −12.8745 −0.899195
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 38.6043i 2.67031i
\(210\) 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
\(217\) −21.9373 −1.48920
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 25.2288 1.68944 0.844721 0.535207i \(-0.179766\pi\)
0.844721 + 0.535207i \(0.179766\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 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
\(239\) 19.4422i 1.25761i 0.777562 + 0.628806i \(0.216456\pi\)
−0.777562 + 0.628806i \(0.783544\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 31.2558i 1.99686i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −12.8745 −0.809413
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.1621i 1.19530i 0.801759 + 0.597648i \(0.203898\pi\)
−0.801759 + 0.597648i \(0.796102\pi\)
\(258\) 0 0
\(259\) −10.4170 −0.647281
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 14.9771i − 0.923528i −0.887003 0.461764i \(-0.847217\pi\)
0.887003 0.461764i \(-0.152783\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 29.6740i 1.80926i 0.426199 + 0.904629i \(0.359852\pi\)
−0.426199 + 0.904629i \(0.640148\pi\)
\(270\) 0 0
\(271\) 10.5830 0.642872 0.321436 0.946931i \(-0.395835\pi\)
0.321436 + 0.946931i \(0.395835\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 66.6966i 4.02195i
\(276\) 0 0
\(277\) 8.06275 0.484443 0.242222 0.970221i \(-0.422124\pi\)
0.242222 + 0.970221i \(0.422124\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −26.4575 −1.57274 −0.786368 0.617758i \(-0.788041\pi\)
−0.786368 + 0.617758i \(0.788041\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.62864i 0.450304i
\(288\) 0 0
\(289\) −37.0000 −2.17647
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.0454i 1.28791i 0.765065 + 0.643953i \(0.222707\pi\)
−0.765065 + 0.643953i \(0.777293\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 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
\(307\) 27.3542 1.56119 0.780595 0.625038i \(-0.214916\pi\)
0.780595 + 0.625038i \(0.214916\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 63.5330i − 3.53507i
\(324\) 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
\(337\) 30.7490 1.67501 0.837503 0.546433i \(-0.184015\pi\)
0.837503 + 0.546433i \(0.184015\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 37.0225i 2.00488i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 0.280168i − 0.0150402i −0.999972 0.00752011i \(-0.997606\pi\)
0.999972 0.00752011i \(-0.00239375\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 9.21040i − 0.490220i −0.969495 0.245110i \(-0.921176\pi\)
0.969495 0.245110i \(-0.0788241\pi\)
\(354\) 0 0
\(355\) −45.6863 −2.42478
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 19.4422i − 1.02612i −0.858352 0.513061i \(-0.828512\pi\)
0.858352 0.513061i \(-0.171488\pi\)
\(360\) 0 0
\(361\) 55.7490 2.93416
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −10.7712 −0.562254 −0.281127 0.959671i \(-0.590708\pi\)
−0.281127 + 0.959671i \(0.590708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.12549 0.0582758 0.0291379 0.999575i \(-0.490724\pi\)
0.0291379 + 0.999575i \(0.490724\pi\)
\(374\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 52.7490 2.68834
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 21.1882 1.07154
\(392\) 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
\(400\) 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
\(407\) 17.5803i 0.871423i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −27.9373 −1.36158 −0.680789 0.732479i \(-0.738363\pi\)
−0.680789 + 0.732479i \(0.738363\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 109.766i − 5.32443i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 32.2772i − 1.55474i −0.629044 0.777370i \(-0.716553\pi\)
0.629044 0.777370i \(-0.283447\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.9288i 1.19250i
\(438\) 0 0
\(439\) 5.29150 0.252550 0.126275 0.991995i \(-0.459698\pi\)
0.126275 + 0.991995i \(0.459698\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 7.06830i − 0.335825i −0.985802 0.167913i \(-0.946297\pi\)
0.985802 0.167913i \(-0.0537026\pi\)
\(444\) 0 0
\(445\) −66.8745 −3.17016
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 12.8745 0.606237
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 42.7490 1.99971 0.999857 0.0168929i \(-0.00537742\pi\)
0.999857 + 0.0168929i \(0.00537742\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 41.7678i 1.94532i 0.232233 + 0.972660i \(0.425397\pi\)
−0.232233 + 0.972660i \(0.574603\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 129.144 5.92552
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 41.2075i 1.85967i 0.367981 + 0.929833i \(0.380049\pi\)
−0.367981 + 0.929833i \(0.619951\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 27.0709i 1.21429i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 32.8118 1.46010
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 36.7423i − 1.62858i −0.580461 0.814288i \(-0.697128\pi\)
0.580461 0.814288i \(-0.302872\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 43.3496i 1.91021i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 37.0225i − 1.62199i −0.585056 0.810993i \(-0.698927\pi\)
0.585056 0.810993i \(-0.301073\pi\)
\(522\) 0 0
\(523\) −6.16601 −0.269621 −0.134810 0.990871i \(-0.543043\pi\)
−0.134810 + 0.990871i \(0.543043\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 60.9299i − 2.65415i
\(528\) 0 0
\(529\) 14.6863 0.638533
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −86.8118 −3.75320
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 31.2558i − 1.34628i
\(540\) 0 0
\(541\) 43.6863 1.87822 0.939110 0.343617i \(-0.111652\pi\)
0.939110 + 0.343617i \(0.111652\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 48.5559i − 2.07991i
\(546\) 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
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 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
\(575\) 43.0694i 1.79612i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 71.6863 2.95378
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 43.3496i 1.78015i 0.455811 + 0.890077i \(0.349349\pi\)
−0.455811 + 0.890077i \(0.650651\pi\)
\(594\) 0 0
\(595\) −86.8118 −3.55894
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 38.6043i 1.57733i 0.614824 + 0.788664i \(0.289227\pi\)
−0.614824 + 0.788664i \(0.710773\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 39.9059i − 1.62240i
\(606\) 0 0
\(607\) −26.4575 −1.07388 −0.536939 0.843621i \(-0.680419\pi\)
−0.536939 + 0.843621i \(0.680419\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 7.12549 0.287796 0.143898 0.989593i \(-0.454036\pi\)
0.143898 + 0.989593i \(0.454036\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −28.4170 −1.14218 −0.571088 0.820889i \(-0.693478\pi\)
−0.571088 + 0.820889i \(0.693478\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 39.6257i 1.58757i
\(624\) 0 0
\(625\) 123.435 4.93741
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 28.9328i − 1.15363i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 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
\(643\) 11.4797 0.452717 0.226358 0.974044i \(-0.427318\pi\)
0.226358 + 0.974044i \(0.427318\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.86193i − 0.0725305i −0.999342 0.0362652i \(-0.988454\pi\)
0.999342 0.0362652i \(-0.0115461\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 102.137i − 3.96072i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 19.7224i − 0.757993i −0.925398 0.378997i \(-0.876269\pi\)
0.925398 0.378997i \(-0.123731\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 51.7195i 1.97899i 0.144568 + 0.989495i \(0.453821\pi\)
−0.144568 + 0.989495i \(0.546179\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 42.3320 1.61039 0.805193 0.593013i \(-0.202062\pi\)
0.805193 + 0.593013i \(0.202062\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 94.5087i − 3.58492i
\(696\) 0 0
\(697\) −21.1882 −0.802562
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 34.0405 1.28386
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 19.4422i − 0.731200i
\(708\) 0 0
\(709\) −40.8745 −1.53507 −0.767537 0.641004i \(-0.778518\pi\)
−0.767537 + 0.641004i \(0.778518\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 23.9073i 0.895337i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 25.6863 0.956607
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −52.9150 −1.96251 −0.981255 0.192715i \(-0.938271\pi\)
−0.981255 + 0.192715i \(0.938271\pi\)
\(728\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 21.7652i − 0.798489i −0.916844 0.399245i \(-0.869272\pi\)
0.916844 0.399245i \(-0.130728\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 51.4393i 1.87955i
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.7423i 1.33191i 0.745992 + 0.665955i \(0.231976\pi\)
−0.745992 + 0.665955i \(0.768024\pi\)
\(762\) 0 0
\(763\) −28.7712 −1.04159
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 28.6526i 1.03056i 0.857021 + 0.515282i \(0.172313\pi\)
−0.857021 + 0.515282i \(0.827687\pi\)
\(774\) 0 0
\(775\) 123.852 4.44891
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 24.9288i − 0.893166i
\(780\) 0 0
\(781\) 45.6863 1.63478
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −21.1660 −0.754487 −0.377243 0.926114i \(-0.623128\pi\)
−0.377243 + 0.926114i \(0.623128\pi\)
\(788\) 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
\(797\) − 24.9288i − 0.883022i −0.897256 0.441511i \(-0.854443\pi\)
0.897256 0.441511i \(-0.145557\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 34.0627 1.20055
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 11.8340 0.415548 0.207774 0.978177i \(-0.433378\pi\)
0.207774 + 0.978177i \(0.433378\pi\)
\(812\) 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
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 53.3012i 1.85346i 0.375722 + 0.926732i \(0.377395\pi\)
−0.375722 + 0.926732i \(0.622605\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 51.4393i 1.78227i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 58.0465i 1.99686i
\(846\) 0 0
\(847\) −23.6458 −0.812478
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3525i 0.389159i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 56.4647i 1.92880i 0.264450 + 0.964399i \(0.414810\pi\)
−0.264450 + 0.964399i \(0.585190\pi\)
\(858\) 0 0
\(859\) 23.1033 0.788273 0.394137 0.919052i \(-0.371044\pi\)
0.394137 + 0.919052i \(0.371044\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 58.3267i − 1.98546i −0.120351 0.992731i \(-0.538402\pi\)
0.120351 0.992731i \(-0.461598\pi\)
\(864\) 0 0
\(865\) 106.749 3.62958
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 117.395i − 3.96866i
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 44.3710i 1.49490i 0.664320 + 0.747448i \(0.268721\pi\)
−0.664320 + 0.747448i \(0.731279\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 86.8118 2.90180
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 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
\(911\) 58.3267i 1.93245i 0.257702 + 0.966224i \(0.417035\pi\)
−0.257702 + 0.966224i \(0.582965\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 58.8118 1.93372
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 36.7423i 1.20548i 0.797939 + 0.602739i \(0.205924\pi\)
−0.797939 + 0.602739i \(0.794076\pi\)
\(930\) 0 0
\(931\) −60.5203 −1.98347
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 146.508i 4.79133i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 33.8590i 1.10377i 0.833920 + 0.551886i \(0.186092\pi\)
−0.833920 + 0.551886i \(0.813908\pi\)
\(942\) 0 0
\(943\) 8.31373 0.270732
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.1137i 0.946068i 0.881044 + 0.473034i \(0.156841\pi\)
−0.881044 + 0.473034i \(0.843159\pi\)
\(948\) 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
\(955\) −118.373 −3.83045
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 37.7490 1.21771
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 17.8605i 0.574949i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −56.0000 −1.79528
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 66.8745 2.13732
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 118.976i − 3.77180i
\(996\) 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 3024.2.b.q.1567.4 yes 4
3.2 odd 2 inner 3024.2.b.q.1567.1 4
4.3 odd 2 3024.2.b.t.1567.4 yes 4
7.6 odd 2 3024.2.b.t.1567.1 yes 4
12.11 even 2 3024.2.b.t.1567.1 yes 4
21.20 even 2 3024.2.b.t.1567.4 yes 4
28.27 even 2 inner 3024.2.b.q.1567.1 4
84.83 odd 2 CM 3024.2.b.q.1567.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.q.1567.1 4 3.2 odd 2 inner
3024.2.b.q.1567.1 4 28.27 even 2 inner
3024.2.b.q.1567.4 yes 4 1.1 even 1 trivial
3024.2.b.q.1567.4 yes 4 84.83 odd 2 CM
3024.2.b.t.1567.1 yes 4 7.6 odd 2
3024.2.b.t.1567.1 yes 4 12.11 even 2
3024.2.b.t.1567.4 yes 4 4.3 odd 2
3024.2.b.t.1567.4 yes 4 21.20 even 2