Properties

Label 7200.2.a.cb.1.1
Level $7200$
Weight $2$
Character 7200.1
Self dual yes
Analytic conductor $57.492$
Analytic rank $1$
Dimension $2$
CM discriminant -20
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(57.4922894553\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 7200.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.23607 q^{7} +O(q^{10})\) \(q-5.23607 q^{7} +7.70820 q^{23} +6.00000 q^{29} -4.47214 q^{41} -6.76393 q^{43} +0.291796 q^{47} +20.4164 q^{49} +13.4164 q^{61} -14.1803 q^{67} -4.29180 q^{83} -6.00000 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{7} + O(q^{10}) \) \( 2q - 6q^{7} + 2q^{23} + 12q^{29} - 18q^{43} + 14q^{47} + 14q^{49} - 6q^{67} - 22q^{83} - 12q^{89} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 0 0
\(7\) −5.23607 −1.97905 −0.989524 0.144370i \(-0.953885\pi\)
−0.989524 + 0.144370i \(0.953885\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.70820 1.60727 0.803636 0.595121i \(-0.202896\pi\)
0.803636 + 0.595121i \(0.202896\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.47214 −0.698430 −0.349215 0.937043i \(-0.613552\pi\)
−0.349215 + 0.937043i \(0.613552\pi\)
\(42\) 0 0
\(43\) −6.76393 −1.03149 −0.515745 0.856742i \(-0.672485\pi\)
−0.515745 + 0.856742i \(0.672485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.291796 0.0425628 0.0212814 0.999774i \(-0.493225\pi\)
0.0212814 + 0.999774i \(0.493225\pi\)
\(48\) 0 0
\(49\) 20.4164 2.91663
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 13.4164 1.71780 0.858898 0.512148i \(-0.171150\pi\)
0.858898 + 0.512148i \(0.171150\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −14.1803 −1.73240 −0.866202 0.499694i \(-0.833446\pi\)
−0.866202 + 0.499694i \(0.833446\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.29180 −0.471086 −0.235543 0.971864i \(-0.575687\pi\)
−0.235543 + 0.971864i \(0.575687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 2.18034 0.214835 0.107418 0.994214i \(-0.465742\pi\)
0.107418 + 0.994214i \(0.465742\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −19.7082 −1.90526 −0.952632 0.304125i \(-0.901636\pi\)
−0.952632 + 0.304125i \(0.901636\pi\)
\(108\) 0 0
\(109\) −13.4164 −1.28506 −0.642529 0.766261i \(-0.722115\pi\)
−0.642529 + 0.766261i \(0.722115\pi\)
\(110\) 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
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.6525 1.12273 0.561363 0.827570i \(-0.310277\pi\)
0.561363 + 0.827570i \(0.310277\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 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
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −4.47214 −0.366372 −0.183186 0.983078i \(-0.558641\pi\)
−0.183186 + 0.983078i \(0.558641\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −40.3607 −3.18087
\(162\) 0 0
\(163\) −24.6525 −1.93093 −0.965465 0.260531i \(-0.916102\pi\)
−0.965465 + 0.260531i \(0.916102\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.7082 −1.83460 −0.917298 0.398202i \(-0.869634\pi\)
−0.917298 + 0.398202i \(0.869634\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −31.4164 −2.20500
\(204\) 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
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −18.7639 −1.25653 −0.628263 0.778001i \(-0.716234\pi\)
−0.628263 + 0.778001i \(0.716234\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.1246 1.80032 0.900162 0.435556i \(-0.143448\pi\)
0.900162 + 0.435556i \(0.143448\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −13.4164 −0.864227 −0.432113 0.901819i \(-0.642232\pi\)
−0.432113 + 0.901819i \(0.642232\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 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
\(253\) 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
\(263\) −31.1246 −1.91923 −0.959613 0.281324i \(-0.909226\pi\)
−0.959613 + 0.281324i \(0.909226\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −22.3607 −1.36335 −0.681677 0.731653i \(-0.738749\pi\)
−0.681677 + 0.731653i \(0.738749\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 31.3050 1.86750 0.933748 0.357930i \(-0.116517\pi\)
0.933748 + 0.357930i \(0.116517\pi\)
\(282\) 0 0
\(283\) −9.81966 −0.583718 −0.291859 0.956461i \(-0.594274\pi\)
−0.291859 + 0.956461i \(0.594274\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.4164 1.38223
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 35.4164 2.04137
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 21.5967 1.23259 0.616296 0.787515i \(-0.288633\pi\)
0.616296 + 0.787515i \(0.288633\pi\)
\(308\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 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
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.52786 −0.0842339
\(330\) 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
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −70.2492 −3.79310
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.12461 0.167738 0.0838690 0.996477i \(-0.473272\pi\)
0.0838690 + 0.996477i \(0.473272\pi\)
\(348\) 0 0
\(349\) 26.0000 1.39175 0.695874 0.718164i \(-0.255017\pi\)
0.695874 + 0.718164i \(0.255017\pi\)
\(350\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −29.2361 −1.52611 −0.763055 0.646333i \(-0.776302\pi\)
−0.763055 + 0.646333i \(0.776302\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −39.1246 −1.99917 −0.999587 0.0287325i \(-0.990853\pi\)
−0.999587 + 0.0287325i \(0.990853\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 31.3050 1.58722 0.793612 0.608424i \(-0.208198\pi\)
0.793612 + 0.608424i \(0.208198\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −40.2492 −1.99020 −0.995098 0.0988936i \(-0.968470\pi\)
−0.995098 + 0.0988936i \(0.968470\pi\)
\(410\) 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
\(421\) −40.2492 −1.96163 −0.980814 0.194948i \(-0.937546\pi\)
−0.980814 + 0.194948i \(0.937546\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −70.2492 −3.39960
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.7082 1.69655 0.848274 0.529558i \(-0.177642\pi\)
0.848274 + 0.529558i \(0.177642\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 22.3607 1.05527 0.527633 0.849473i \(-0.323080\pi\)
0.527633 + 0.849473i \(0.323080\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 42.0000 1.95614 0.978068 0.208288i \(-0.0667892\pi\)
0.978068 + 0.208288i \(0.0667892\pi\)
\(462\) 0 0
\(463\) 20.0689 0.932680 0.466340 0.884606i \(-0.345572\pi\)
0.466340 + 0.884606i \(0.345572\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 43.1246 1.99557 0.997785 0.0665285i \(-0.0211923\pi\)
0.997785 + 0.0665285i \(0.0211923\pi\)
\(468\) 0 0
\(469\) 74.2492 3.42851
\(470\) 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 1.00000i \(-0.5\pi\)
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
\(487\) −11.3475 −0.514205 −0.257103 0.966384i \(-0.582768\pi\)
−0.257103 + 0.966384i \(0.582768\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 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
\(503\) −24.2918 −1.08312 −0.541559 0.840663i \(-0.682166\pi\)
−0.541559 + 0.840663i \(0.682166\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) −45.5967 −1.99381 −0.996903 0.0786374i \(-0.974943\pi\)
−0.996903 + 0.0786374i \(0.974943\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 36.4164 1.58332
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 30.7639 1.31537 0.657685 0.753293i \(-0.271536\pi\)
0.657685 + 0.753293i \(0.271536\pi\)
\(548\) 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
\(563\) 34.5410 1.45573 0.727865 0.685720i \(-0.240513\pi\)
0.727865 + 0.685720i \(0.240513\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −31.3050 −1.31237 −0.656186 0.754599i \(-0.727831\pi\)
−0.656186 + 0.754599i \(0.727831\pi\)
\(570\) 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
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 22.4721 0.932301
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −26.5410 −1.09547 −0.547733 0.836653i \(-0.684509\pi\)
−0.547733 + 0.836653i \(0.684509\pi\)
\(588\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 40.2492 1.64180 0.820900 0.571072i \(-0.193472\pi\)
0.820900 + 0.571072i \(0.193472\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 21.8197 0.885633 0.442816 0.896612i \(-0.353979\pi\)
0.442816 + 0.896612i \(0.353979\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 31.4164 1.25867
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 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
\(641\) 49.1935 1.94303 0.971513 0.236986i \(-0.0761595\pi\)
0.971513 + 0.236986i \(0.0761595\pi\)
\(642\) 0 0
\(643\) 8.06888 0.318206 0.159103 0.987262i \(-0.449140\pi\)
0.159103 + 0.987262i \(0.449140\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −46.5410 −1.82972 −0.914858 0.403775i \(-0.867698\pi\)
−0.914858 + 0.403775i \(0.867698\pi\)
\(648\) 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
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −40.2492 −1.56551 −0.782757 0.622328i \(-0.786187\pi\)
−0.782757 + 0.622328i \(0.786187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 46.2492 1.79078
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −51.1246 −1.95623 −0.978114 0.208068i \(-0.933283\pi\)
−0.978114 + 0.208068i \(0.933283\pi\)
\(684\) 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
\(701\) 22.3607 0.844551 0.422276 0.906467i \(-0.361231\pi\)
0.422276 + 0.906467i \(0.361231\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −94.2492 −3.54461
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −11.4164 −0.425169
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −41.0132 −1.52109 −0.760547 0.649283i \(-0.775069\pi\)
−0.760547 + 0.649283i \(0.775069\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.5410 0.533458 0.266729 0.963772i \(-0.414057\pi\)
0.266729 + 0.963772i \(0.414057\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 103.193 3.77061
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) 0 0
\(763\) 70.2492 2.54319
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 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
\(787\) −56.0689 −1.99864 −0.999320 0.0368739i \(-0.988260\pi\)
−0.999320 + 0.0368739i \(0.988260\pi\)
\(788\) 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
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 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
\(821\) −31.3050 −1.09255 −0.546275 0.837606i \(-0.683955\pi\)
−0.546275 + 0.837606i \(0.683955\pi\)
\(822\) 0 0
\(823\) 50.1803 1.74918 0.874588 0.484866i \(-0.161132\pi\)
0.874588 + 0.484866i \(0.161132\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.5410 −0.366547 −0.183274 0.983062i \(-0.558669\pi\)
−0.183274 + 0.983062i \(0.558669\pi\)
\(828\) 0 0
\(829\) −13.4164 −0.465971 −0.232986 0.972480i \(-0.574849\pi\)
−0.232986 + 0.972480i \(0.574849\pi\)
\(830\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 57.5967 1.97905
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 47.7082 1.62401 0.812003 0.583653i \(-0.198377\pi\)
0.812003 + 0.583653i \(0.198377\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −58.1378 −1.95871 −0.979356 0.202145i \(-0.935209\pi\)
−0.979356 + 0.202145i \(0.935209\pi\)
\(882\) 0 0
\(883\) 23.3475 0.785707 0.392853 0.919601i \(-0.371488\pi\)
0.392853 + 0.919601i \(0.371488\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −16.8754 −0.566620 −0.283310 0.959028i \(-0.591433\pi\)
−0.283310 + 0.959028i \(0.591433\pi\)
\(888\) 0 0
\(889\) −66.2492 −2.22193
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 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
\(907\) 39.4853 1.31109 0.655544 0.755157i \(-0.272439\pi\)
0.655544 + 0.755157i \(0.272439\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 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
\(929\) −49.1935 −1.61399 −0.806993 0.590561i \(-0.798907\pi\)
−0.806993 + 0.590561i \(0.798907\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 0 0
\(943\) −34.4721 −1.12257
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.2918 1.17932 0.589662 0.807650i \(-0.299261\pi\)
0.589662 + 0.807650i \(0.299261\pi\)
\(948\) 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
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 3.93112 0.126416 0.0632081 0.998000i \(-0.479867\pi\)
0.0632081 + 0.998000i \(0.479867\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 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
\(983\) 62.5410 1.99475 0.997374 0.0724180i \(-0.0230716\pi\)
0.997374 + 0.0724180i \(0.0230716\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −52.1378 −1.65788
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7200.2.a.cb.1.1 2
3.2 odd 2 800.2.a.j.1.2 2
4.3 odd 2 7200.2.a.cr.1.2 2
5.2 odd 4 1440.2.f.i.289.3 4
5.3 odd 4 1440.2.f.i.289.4 4
5.4 even 2 7200.2.a.cr.1.2 2
12.11 even 2 800.2.a.n.1.1 2
15.2 even 4 160.2.c.b.129.2 4
15.8 even 4 160.2.c.b.129.3 yes 4
15.14 odd 2 800.2.a.n.1.1 2
20.3 even 4 1440.2.f.i.289.3 4
20.7 even 4 1440.2.f.i.289.4 4
20.19 odd 2 CM 7200.2.a.cb.1.1 2
24.5 odd 2 1600.2.a.bd.1.1 2
24.11 even 2 1600.2.a.z.1.2 2
40.3 even 4 2880.2.f.w.1729.1 4
40.13 odd 4 2880.2.f.w.1729.2 4
40.27 even 4 2880.2.f.w.1729.2 4
40.37 odd 4 2880.2.f.w.1729.1 4
60.23 odd 4 160.2.c.b.129.2 4
60.47 odd 4 160.2.c.b.129.3 yes 4
60.59 even 2 800.2.a.j.1.2 2
120.29 odd 2 1600.2.a.z.1.2 2
120.53 even 4 320.2.c.d.129.2 4
120.59 even 2 1600.2.a.bd.1.1 2
120.77 even 4 320.2.c.d.129.3 4
120.83 odd 4 320.2.c.d.129.3 4
120.107 odd 4 320.2.c.d.129.2 4
240.53 even 4 1280.2.f.g.129.3 4
240.77 even 4 1280.2.f.g.129.4 4
240.83 odd 4 1280.2.f.g.129.4 4
240.107 odd 4 1280.2.f.g.129.3 4
240.173 even 4 1280.2.f.h.129.2 4
240.197 even 4 1280.2.f.h.129.1 4
240.203 odd 4 1280.2.f.h.129.1 4
240.227 odd 4 1280.2.f.h.129.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.c.b.129.2 4 15.2 even 4
160.2.c.b.129.2 4 60.23 odd 4
160.2.c.b.129.3 yes 4 15.8 even 4
160.2.c.b.129.3 yes 4 60.47 odd 4
320.2.c.d.129.2 4 120.53 even 4
320.2.c.d.129.2 4 120.107 odd 4
320.2.c.d.129.3 4 120.77 even 4
320.2.c.d.129.3 4 120.83 odd 4
800.2.a.j.1.2 2 3.2 odd 2
800.2.a.j.1.2 2 60.59 even 2
800.2.a.n.1.1 2 12.11 even 2
800.2.a.n.1.1 2 15.14 odd 2
1280.2.f.g.129.3 4 240.53 even 4
1280.2.f.g.129.3 4 240.107 odd 4
1280.2.f.g.129.4 4 240.77 even 4
1280.2.f.g.129.4 4 240.83 odd 4
1280.2.f.h.129.1 4 240.197 even 4
1280.2.f.h.129.1 4 240.203 odd 4
1280.2.f.h.129.2 4 240.173 even 4
1280.2.f.h.129.2 4 240.227 odd 4
1440.2.f.i.289.3 4 5.2 odd 4
1440.2.f.i.289.3 4 20.3 even 4
1440.2.f.i.289.4 4 5.3 odd 4
1440.2.f.i.289.4 4 20.7 even 4
1600.2.a.z.1.2 2 24.11 even 2
1600.2.a.z.1.2 2 120.29 odd 2
1600.2.a.bd.1.1 2 24.5 odd 2
1600.2.a.bd.1.1 2 120.59 even 2
2880.2.f.w.1729.1 4 40.3 even 4
2880.2.f.w.1729.1 4 40.37 odd 4
2880.2.f.w.1729.2 4 40.13 odd 4
2880.2.f.w.1729.2 4 40.27 even 4
7200.2.a.cb.1.1 2 1.1 even 1 trivial
7200.2.a.cb.1.1 2 20.19 odd 2 CM
7200.2.a.cr.1.2 2 4.3 odd 2
7200.2.a.cr.1.2 2 5.4 even 2