Properties

Label 3024.2.b.t.1567.2
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 1567.2
Root \(-2.01563i\) of defining polynomial
Character \(\chi\) \(=\) 3024.1567
Dual form 3024.2.b.t.1567.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.01563i q^{5} +2.64575 q^{7} +O(q^{10})\) \(q-2.01563i q^{5} +2.64575 q^{7} -2.01563i q^{11} +7.34847i q^{17} +3.35425 q^{19} +9.36409i q^{23} +0.937254 q^{25} -2.29150 q^{31} -5.33284i q^{35} +11.9373 q^{37} +9.36409i q^{41} +7.00000 q^{49} -4.06275 q^{55} -16.7126i q^{71} -5.33284i q^{77} +14.8118 q^{85} -17.4266i q^{89} -6.76091i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 24q^{19} - 28q^{25} + 12q^{31} + 16q^{37} + 28q^{49} - 48q^{55} - 36q^{85} + O(q^{100}) \)

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
\(5\) − 2.01563i − 0.901415i −0.892672 0.450708i \(-0.851172\pi\)
0.892672 0.450708i \(-0.148828\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 2.01563i − 0.607734i −0.952714 0.303867i \(-0.901722\pi\)
0.952714 0.303867i \(-0.0982778\pi\)
\(12\) 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
\(19\) 3.35425 0.769517 0.384759 0.923017i \(-0.374285\pi\)
0.384759 + 0.923017i \(0.374285\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 9.36409i 1.95255i 0.216537 + 0.976274i \(0.430524\pi\)
−0.216537 + 0.976274i \(0.569476\pi\)
\(24\) 0 0
\(25\) 0.937254 0.187451
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −2.29150 −0.411566 −0.205783 0.978598i \(-0.565974\pi\)
−0.205783 + 0.978598i \(0.565974\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 5.33284i − 0.901415i
\(36\) 0 0
\(37\) 11.9373 1.96247 0.981236 0.192809i \(-0.0617599\pi\)
0.981236 + 0.192809i \(0.0617599\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.36409i 1.46243i 0.682149 + 0.731213i \(0.261045\pi\)
−0.682149 + 0.731213i \(0.738955\pi\)
\(42\) 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
\(55\) −4.06275 −0.547821
\(56\) 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
\(71\) − 16.7126i − 1.98342i −0.128510 0.991708i \(-0.541019\pi\)
0.128510 0.991708i \(-0.458981\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.33284i − 0.607734i
\(78\) 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
\(85\) 14.8118 1.60656
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 17.4266i − 1.84722i −0.383339 0.923608i \(-0.625226\pi\)
0.383339 0.923608i \(-0.374774\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 6.76091i − 0.693655i
\(96\) 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
\(103\) −20.2915 −1.99938 −0.999691 0.0248745i \(-0.992081\pi\)
−0.999691 + 0.0248745i \(0.992081\pi\)
\(104\) 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
\(109\) 20.8745 1.99942 0.999708 0.0241802i \(-0.00769755\pi\)
0.999708 + 0.0241802i \(0.00769755\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 18.8745 1.76006
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 19.4422i 1.78227i
\(120\) 0 0
\(121\) 6.93725 0.630659
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.9673i − 1.07039i
\(126\) 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
\(133\) 8.87451 0.769517
\(134\) 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
\(155\) 4.61881i 0.370992i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 24.7751i 1.95255i
\(162\) 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
\(173\) 21.4578i 1.63141i 0.578468 + 0.815705i \(0.303651\pi\)
−0.578468 + 0.815705i \(0.696349\pi\)
\(174\) 0 0
\(175\) 2.47974 0.187451
\(176\) 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
\(185\) − 24.0610i − 1.76900i
\(186\) 0 0
\(187\) 14.8118 1.08314
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 20.0298i − 1.44930i −0.689115 0.724652i \(-0.742000\pi\)
0.689115 0.724652i \(-0.258000\pi\)
\(192\) 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
\(199\) 21.3542 1.51376 0.756881 0.653552i \(-0.226722\pi\)
0.756881 + 0.653552i \(0.226722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 18.8745 1.31825
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 6.76091i − 0.467662i
\(210\) 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
\(217\) −6.06275 −0.411566
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.22876 0.0822836 0.0411418 0.999153i \(-0.486900\pi\)
0.0411418 + 0.999153i \(0.486900\pi\)
\(224\) 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
\(245\) − 14.1094i − 0.901415i
\(246\) 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
\(253\) 18.8745 1.18663
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.6813i 0.791039i 0.918458 + 0.395519i \(0.129435\pi\)
−0.918458 + 0.395519i \(0.870565\pi\)
\(258\) 0 0
\(259\) 31.5830 1.96247
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 17.4266i − 1.07457i −0.843401 0.537285i \(-0.819450\pi\)
0.843401 0.537285i \(-0.180550\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 2.72966i − 0.166430i −0.996532 0.0832151i \(-0.973481\pi\)
0.996532 0.0832151i \(-0.0265189\pi\)
\(270\) 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
\(275\) − 1.88915i − 0.113920i
\(276\) 0 0
\(277\) 23.9373 1.43825 0.719125 0.694881i \(-0.244543\pi\)
0.719125 + 0.694881i \(0.244543\pi\)
\(278\) 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
\(287\) 24.7751i 1.46243i
\(288\) 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
\(307\) −32.6458 −1.86319 −0.931596 0.363496i \(-0.881583\pi\)
−0.931596 + 0.363496i \(0.881583\pi\)
\(308\) 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
\(323\) 24.6486i 1.37148i
\(324\) 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
\(337\) −32.7490 −1.78395 −0.891976 0.452082i \(-0.850681\pi\)
−0.891976 + 0.452082i \(0.850681\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.61881i 0.250123i
\(342\) 0 0
\(343\) 18.5203 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 32.1235i − 1.72448i −0.506498 0.862241i \(-0.669060\pi\)
0.506498 0.862241i \(-0.330940\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.1548i 1.92433i 0.272476 + 0.962163i \(0.412157\pi\)
−0.272476 + 0.962163i \(0.587843\pi\)
\(354\) 0 0
\(355\) −33.6863 −1.78788
\(356\) 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
\(361\) −7.74902 −0.407843
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 37.2288 1.94333 0.971663 0.236372i \(-0.0759584\pi\)
0.971663 + 0.236372i \(0.0759584\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 32.8745 1.70218 0.851089 0.525022i \(-0.175943\pi\)
0.851089 + 0.525022i \(0.175943\pi\)
\(374\) 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
\(385\) −10.7490 −0.547821
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −68.8118 −3.47996
\(392\) 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
\(407\) − 24.0610i − 1.19266i
\(408\) 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
\(421\) −12.0627 −0.587902 −0.293951 0.955820i \(-0.594970\pi\)
−0.293951 + 0.955820i \(0.594970\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.88738i 0.334087i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.7580i 1.86691i 0.358700 + 0.933453i \(0.383220\pi\)
−0.358700 + 0.933453i \(0.616780\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.4095i 1.50252i
\(438\) 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
\(443\) 39.4720i 1.87537i 0.347484 + 0.937686i \(0.387036\pi\)
−0.347484 + 0.937686i \(0.612964\pi\)
\(444\) 0 0
\(445\) −35.1255 −1.66511
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 18.8745 0.888766
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −20.7490 −0.970598 −0.485299 0.874348i \(-0.661289\pi\)
−0.485299 + 0.874348i \(0.661289\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 29.5203i − 1.37490i −0.726232 0.687450i \(-0.758730\pi\)
0.726232 0.687450i \(-0.241270\pi\)
\(462\) 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
\(475\) 3.14378 0.144247
\(476\) 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
\(491\) − 34.7267i − 1.56719i −0.621269 0.783597i \(-0.713383\pi\)
0.621269 0.783597i \(-0.286617\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 44.2173i − 1.98342i
\(498\) 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
\(505\) −14.8118 −0.659115
\(506\) 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
\(515\) 40.9001i 1.80227i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 4.61881i − 0.202354i −0.994868 0.101177i \(-0.967739\pi\)
0.994868 0.101177i \(-0.0322608\pi\)
\(522\) 0 0
\(523\) −36.1660 −1.58143 −0.790715 0.612185i \(-0.790291\pi\)
−0.790715 + 0.612185i \(0.790291\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 16.8390i − 0.733520i
\(528\) 0 0
\(529\) −64.6863 −2.81245
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 39.1882 1.69426
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 14.1094i − 0.607734i
\(540\) 0 0
\(541\) −35.6863 −1.53427 −0.767136 0.641484i \(-0.778319\pi\)
−0.767136 + 0.641484i \(0.778319\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 42.0752i − 1.80230i
\(546\) 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
\(575\) 8.77653i 0.366007i
\(576\) 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
\(589\) −7.68627 −0.316707
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 40.9001i − 1.67956i −0.542923 0.839782i \(-0.682683\pi\)
0.542923 0.839782i \(-0.317317\pi\)
\(594\) 0 0
\(595\) 39.1882 1.60656
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.76091i 0.276243i 0.990415 + 0.138122i \(0.0441065\pi\)
−0.990415 + 0.138122i \(0.955894\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 13.9829i − 0.568486i
\(606\) 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
\(613\) 38.8745 1.57013 0.785063 0.619416i \(-0.212630\pi\)
0.785063 + 0.619416i \(0.212630\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 49.5830 1.99291 0.996455 0.0841320i \(-0.0268117\pi\)
0.996455 + 0.0841320i \(0.0268117\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 46.1064i − 1.84722i
\(624\) 0 0
\(625\) −19.4353 −0.777411
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 87.7205i 3.49765i
\(630\) 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
\(643\) −48.5203 −1.91345 −0.956726 0.290990i \(-0.906015\pi\)
−0.956726 + 0.290990i \(0.906015\pi\)
\(644\) 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
\(659\) − 43.5033i − 1.69465i −0.531078 0.847323i \(-0.678213\pi\)
0.531078 0.847323i \(-0.321787\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 17.8877i − 0.693655i
\(666\) 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
\(677\) 51.5658i 1.98183i 0.134478 + 0.990917i \(0.457064\pi\)
−0.134478 + 0.990917i \(0.542936\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 19.3157i − 0.739097i −0.929212 0.369548i \(-0.879512\pi\)
0.929212 0.369548i \(-0.120488\pi\)
\(684\) 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
\(695\) 42.6628i 1.61829i
\(696\) 0 0
\(697\) −68.8118 −2.60643
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 40.0405 1.51016
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 19.4422i − 0.731200i
\(708\) 0 0
\(709\) −9.12549 −0.342715 −0.171358 0.985209i \(-0.554815\pi\)
−0.171358 + 0.985209i \(0.554815\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 21.4578i − 0.803603i
\(714\) 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
\(721\) −53.6863 −1.99938
\(722\) 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
\(743\) 54.1689i 1.98727i 0.112666 + 0.993633i \(0.464061\pi\)
−0.112666 + 0.993633i \(0.535939\pi\)
\(744\) 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
\(763\) 55.2288 1.99942
\(764\) 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
\(773\) − 55.5970i − 1.99969i −0.0177365 0.999843i \(-0.505646\pi\)
0.0177365 0.999843i \(-0.494354\pi\)
\(774\) 0 0
\(775\) −2.14772 −0.0771484
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 31.4095i 1.12536i
\(780\) 0 0
\(781\) −33.6863 −1.20539
\(782\) 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
\(797\) − 31.4095i − 1.11258i −0.830988 0.556291i \(-0.812224\pi\)
0.830988 0.556291i \(-0.187776\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 49.9373 1.76006
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −54.1660 −1.90203 −0.951013 0.309151i \(-0.899955\pi\)
−0.951013 + 0.309151i \(0.899955\pi\)
\(812\) 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
\(827\) − 7.93603i − 0.275963i −0.990435 0.137981i \(-0.955939\pi\)
0.990435 0.137981i \(-0.0440614\pi\)
\(828\) 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
\(845\) − 26.2031i − 0.901415i
\(846\) 0 0
\(847\) 18.3542 0.630659
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 111.782i 3.83182i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 14.8234i − 0.506358i −0.967419 0.253179i \(-0.918524\pi\)
0.967419 0.253179i \(-0.0814762\pi\)
\(858\) 0 0
\(859\) 35.1033 1.19771 0.598854 0.800858i \(-0.295623\pi\)
0.598854 + 0.800858i \(0.295623\pi\)
\(860\) 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
\(865\) 43.2510 1.47058
\(866\) 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
\(875\) − 31.6624i − 1.07039i
\(876\) 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
\(881\) 11.9673i 0.403188i 0.979469 + 0.201594i \(0.0646121\pi\)
−0.979469 + 0.201594i \(0.935388\pi\)
\(882\) 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
\(895\) −39.1882 −1.30992
\(896\) 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
\(925\) 11.1882 0.367867
\(926\) 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
\(931\) 23.4797 0.769517
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 29.8550i − 0.976362i
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.3783i 0.892505i 0.894907 + 0.446253i \(0.147242\pi\)
−0.894907 + 0.446253i \(0.852758\pi\)
\(942\) 0 0
\(943\) −87.6863 −2.85546
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 61.5174i − 1.99905i −0.0308624 0.999524i \(-0.509825\pi\)
0.0308624 0.999524i \(-0.490175\pi\)
\(948\) 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
\(955\) −40.3725 −1.30642
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25.7490 −0.830613
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 8.06250i − 0.259541i
\(966\) 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
\(979\) −35.1255 −1.12262
\(980\) 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
\(995\) − 43.0422i − 1.36453i
\(996\) 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.t.1567.2 yes 4
3.2 odd 2 inner 3024.2.b.t.1567.3 yes 4
4.3 odd 2 3024.2.b.q.1567.2 4
7.6 odd 2 3024.2.b.q.1567.3 yes 4
12.11 even 2 3024.2.b.q.1567.3 yes 4
21.20 even 2 3024.2.b.q.1567.2 4
28.27 even 2 inner 3024.2.b.t.1567.3 yes 4
84.83 odd 2 CM 3024.2.b.t.1567.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.q.1567.2 4 4.3 odd 2
3024.2.b.q.1567.2 4 21.20 even 2
3024.2.b.q.1567.3 yes 4 7.6 odd 2
3024.2.b.q.1567.3 yes 4 12.11 even 2
3024.2.b.t.1567.2 yes 4 1.1 even 1 trivial
3024.2.b.t.1567.2 yes 4 84.83 odd 2 CM
3024.2.b.t.1567.3 yes 4 3.2 odd 2 inner
3024.2.b.t.1567.3 yes 4 28.27 even 2 inner