Properties

Label 1840.2.m.e.1839.7
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $8$
CM discriminant -115
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14166950625.1
Defining polynomial: \(x^{8} - 9 x^{6} + 32 x^{4} - 441 x^{2} + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1839.7
Root \(1.51764 + 2.16720i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.e.1839.6

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{5} +0.461424i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{5} +0.461424i q^{7} -3.00000 q^{9} +0.799210 q^{17} +4.79583i q^{23} +5.00000 q^{25} +9.78709 q^{29} +7.95998i q^{31} +1.03178i q^{35} +9.74348 q^{37} +5.78709 q^{41} -9.59166i q^{43} -6.70820 q^{45} +6.78709 q^{49} -11.3419 q^{53} +14.8882i q^{59} -1.38427i q^{63} +15.9534i q^{67} -5.89643i q^{71} +9.00000 q^{81} -16.8762i q^{83} +1.78709 q^{85} -4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 24q^{9} + O(q^{10}) \) \( 8q - 24q^{9} + 40q^{25} + 4q^{29} - 28q^{41} - 20q^{49} + 72q^{81} - 60q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 2.23607 1.00000
\(6\) 0 0
\(7\) 0.461424i 0.174402i 0.996191 + 0.0872010i \(0.0277922\pi\)
−0.996191 + 0.0872010i \(0.972208\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.799210 0.193837 0.0969184 0.995292i \(-0.469101\pi\)
0.0969184 + 0.995292i \(0.469101\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.79583i 1.00000i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 9.78709 1.81742 0.908708 0.417432i \(-0.137070\pi\)
0.908708 + 0.417432i \(0.137070\pi\)
\(30\) 0 0
\(31\) 7.95998i 1.42965i 0.699301 + 0.714827i \(0.253495\pi\)
−0.699301 + 0.714827i \(0.746505\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.03178i 0.174402i
\(36\) 0 0
\(37\) 9.74348 1.60182 0.800909 0.598786i \(-0.204350\pi\)
0.800909 + 0.598786i \(0.204350\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.78709 0.903791 0.451896 0.892071i \(-0.350748\pi\)
0.451896 + 0.892071i \(0.350748\pi\)
\(42\) 0 0
\(43\) − 9.59166i − 1.46271i −0.681994 0.731357i \(-0.738887\pi\)
0.681994 0.731357i \(-0.261113\pi\)
\(44\) 0 0
\(45\) −6.70820 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 6.78709 0.969584
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −11.3419 −1.55793 −0.778965 0.627067i \(-0.784255\pi\)
−0.778965 + 0.627067i \(0.784255\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.8882i 1.93828i 0.246518 + 0.969138i \(0.420713\pi\)
−0.246518 + 0.969138i \(0.579287\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 1.38427i − 0.174402i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 15.9534i 1.94901i 0.224360 + 0.974506i \(0.427971\pi\)
−0.224360 + 0.974506i \(0.572029\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.89643i − 0.699777i −0.936791 0.349889i \(-0.886219\pi\)
0.936791 0.349889i \(-0.113781\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 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
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 16.8762i − 1.85240i −0.377027 0.926202i \(-0.623054\pi\)
0.377027 0.926202i \(-0.376946\pi\)
\(84\) 0 0
\(85\) 1.78709 0.193837
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −4.47214 −0.454077 −0.227038 0.973886i \(-0.572904\pi\)
−0.227038 + 0.973886i \(0.572904\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 17.7871 1.76988 0.884941 0.465704i \(-0.154199\pi\)
0.884941 + 0.465704i \(0.154199\pi\)
\(102\) 0 0
\(103\) 9.59166i 0.945095i 0.881305 + 0.472547i \(0.156665\pi\)
−0.881305 + 0.472547i \(0.843335\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 15.0305i − 1.45305i −0.687138 0.726527i \(-0.741133\pi\)
0.687138 0.726527i \(-0.258867\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −20.2862 −1.90836 −0.954181 0.299229i \(-0.903271\pi\)
−0.954181 + 0.299229i \(0.903271\pi\)
\(114\) 0 0
\(115\) 10.7238i 1.00000i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.368775i 0.0338055i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.4476i 1.87389i 0.349482 + 0.936943i \(0.386358\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.4164 1.14624 0.573121 0.819471i \(-0.305733\pi\)
0.573121 + 0.819471i \(0.305733\pi\)
\(138\) 0 0
\(139\) − 12.8246i − 1.08777i −0.839159 0.543885i \(-0.816953\pi\)
0.839159 0.543885i \(-0.183047\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 21.8846 1.81742
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) − 21.4476i − 1.74538i −0.488273 0.872691i \(-0.662373\pi\)
0.488273 0.872691i \(-0.337627\pi\)
\(152\) 0 0
\(153\) −2.39763 −0.193837
\(154\) 0 0
\(155\) 17.7991i 1.42965i
\(156\) 0 0
\(157\) −8.14506 −0.650047 −0.325023 0.945706i \(-0.605372\pi\)
−0.325023 + 0.945706i \(0.605372\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.21291 −0.174402
\(162\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 2.30712i 0.174402i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.4476i 1.60307i 0.597948 + 0.801535i \(0.295983\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 21.7871 1.60182
\(186\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.51600i 0.316961i
\(204\) 0 0
\(205\) 12.9403 0.903791
\(206\) 0 0
\(207\) − 14.3875i − 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 28.7446i 1.97886i 0.145014 + 0.989430i \(0.453677\pi\)
−0.145014 + 0.989430i \(0.546323\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 21.4476i − 1.46271i
\(216\) 0 0
\(217\) −3.67293 −0.249334
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 0 0
\(227\) 28.7750i 1.90986i 0.296826 + 0.954932i \(0.404072\pi\)
−0.296826 + 0.954932i \(0.595928\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 19.7528i − 1.27770i −0.769329 0.638852i \(-0.779409\pi\)
0.769329 0.638852i \(-0.220591\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 15.1764 0.969584
\(246\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 4.49588i 0.279360i
\(260\) 0 0
\(261\) −29.3613 −1.81742
\(262\) 0 0
\(263\) − 32.3681i − 1.99590i −0.0639598 0.997952i \(-0.520373\pi\)
0.0639598 0.997952i \(-0.479627\pi\)
\(264\) 0 0
\(265\) −25.3613 −1.55793
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.21291 −0.378808 −0.189404 0.981899i \(-0.560656\pi\)
−0.189404 + 0.981899i \(0.560656\pi\)
\(270\) 0 0
\(271\) − 10.0235i − 0.608886i −0.952531 0.304443i \(-0.901530\pi\)
0.952531 0.304443i \(-0.0984703\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) − 23.8799i − 1.42965i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) − 18.7219i − 1.11290i −0.830881 0.556451i \(-0.812163\pi\)
0.830881 0.556451i \(-0.187837\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.67030i 0.157623i
\(288\) 0 0
\(289\) −16.3613 −0.962427
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −32.4273 −1.89442 −0.947211 0.320610i \(-0.896112\pi\)
−0.947211 + 0.320610i \(0.896112\pi\)
\(294\) 0 0
\(295\) 33.2910i 1.93828i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 4.42582 0.255100
\(302\) 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
\(311\) − 21.4476i − 1.21618i −0.793867 0.608091i \(-0.791935\pi\)
0.793867 0.608091i \(-0.208065\pi\)
\(312\) 0 0
\(313\) −23.4830 −1.32734 −0.663669 0.748026i \(-0.731002\pi\)
−0.663669 + 0.748026i \(0.731002\pi\)
\(314\) 0 0
\(315\) − 3.09533i − 0.174402i
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 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
\(329\) 0 0
\(330\) 0 0
\(331\) − 30.8081i − 1.69337i −0.532096 0.846684i \(-0.678595\pi\)
0.532096 0.846684i \(-0.321405\pi\)
\(332\) 0 0
\(333\) −29.2304 −1.60182
\(334\) 0 0
\(335\) 35.6728i 1.94901i
\(336\) 0 0
\(337\) 31.3050 1.70529 0.852645 0.522491i \(-0.174997\pi\)
0.852645 + 0.522491i \(0.174997\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 6.36169i 0.343499i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −37.3613 −1.99990 −0.999951 0.00987003i \(-0.996858\pi\)
−0.999951 + 0.00987003i \(0.996858\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) − 13.1848i − 0.699777i
\(356\) 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
\(367\) − 30.5224i − 1.59326i −0.604468 0.796629i \(-0.706614\pi\)
0.604468 0.796629i \(-0.293386\pi\)
\(368\) 0 0
\(369\) −17.3613 −0.903791
\(370\) 0 0
\(371\) − 5.23343i − 0.271706i
\(372\) 0 0
\(373\) 4.47214 0.231558 0.115779 0.993275i \(-0.463063\pi\)
0.115779 + 0.993275i \(0.463063\pi\)
\(374\) 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
\(383\) − 34.2138i − 1.74825i −0.485705 0.874123i \(-0.661437\pi\)
0.485705 0.874123i \(-0.338563\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 28.7750i 1.46271i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 3.83288i 0.193837i
\(392\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 20.1246 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −10.2129 −0.504996 −0.252498 0.967597i \(-0.581252\pi\)
−0.252498 + 0.967597i \(0.581252\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.86977 −0.338039
\(414\) 0 0
\(415\) − 37.7363i − 1.85240i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.99605 0.193837
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −41.3716 −1.98819 −0.994095 0.108513i \(-0.965391\pi\)
−0.994095 + 0.108513i \(0.965391\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) − 21.4476i − 1.02364i −0.859093 0.511819i \(-0.828972\pi\)
0.859093 0.511819i \(-0.171028\pi\)
\(440\) 0 0
\(441\) −20.3613 −0.969584
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.7871 1.40574 0.702870 0.711319i \(-0.251902\pi\)
0.702870 + 0.711319i \(0.251902\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −17.0893 −0.799405 −0.399703 0.916645i \(-0.630887\pi\)
−0.399703 + 0.916645i \(0.630887\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.6448i 0.909051i 0.890734 + 0.454525i \(0.150191\pi\)
−0.890734 + 0.454525i \(0.849809\pi\)
\(468\) 0 0
\(469\) −7.36126 −0.339912
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 34.0257 1.55793
\(478\) 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
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 42.6010i 1.92256i 0.275581 + 0.961278i \(0.411130\pi\)
−0.275581 + 0.961278i \(0.588870\pi\)
\(492\) 0 0
\(493\) 7.82194 0.352282
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.72075 0.122042
\(498\) 0 0
\(499\) 10.7611i 0.481732i 0.970558 + 0.240866i \(0.0774314\pi\)
−0.970558 + 0.240866i \(0.922569\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 29.5996i 1.31978i 0.751362 + 0.659890i \(0.229397\pi\)
−0.751362 + 0.659890i \(0.770603\pi\)
\(504\) 0 0
\(505\) 39.7731 1.76988
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.4476i 0.945095i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) − 9.59166i − 0.419414i −0.977764 0.209707i \(-0.932749\pi\)
0.977764 0.209707i \(-0.0672510\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.36169i 0.277120i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) − 44.6645i − 1.93828i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 33.6092i − 1.45305i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 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
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.6320 1.17081 0.585403 0.810742i \(-0.300936\pi\)
0.585403 + 0.810742i \(0.300936\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.2620i 0.516780i 0.966041 + 0.258390i \(0.0831920\pi\)
−0.966041 + 0.258390i \(0.916808\pi\)
\(564\) 0 0
\(565\) −45.3613 −1.90836
\(566\) 0 0
\(567\) 4.15282i 0.174402i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 23.9792i 1.00000i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.78709 0.323063
\(582\) 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
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0.824605i 0.0338055i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 21.4476i − 0.876326i −0.898896 0.438163i \(-0.855629\pi\)
0.898896 0.438163i \(-0.144371\pi\)
\(600\) 0 0
\(601\) −9.36126 −0.381854 −0.190927 0.981604i \(-0.561149\pi\)
−0.190927 + 0.981604i \(0.561149\pi\)
\(602\) 0 0
\(603\) − 47.8601i − 1.94901i
\(604\) 0 0
\(605\) −24.5967 −1.00000
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −31.3050 −1.26440 −0.632198 0.774807i \(-0.717847\pi\)
−0.632198 + 0.774807i \(0.717847\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 25.0814 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(618\) 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
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.78709 0.310492
\(630\) 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
\(639\) 17.6893i 0.699777i
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) − 49.7058i − 1.96020i −0.198494 0.980102i \(-0.563605\pi\)
0.198494 0.980102i \(-0.436395\pi\)
\(644\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 47.9583i 1.87389i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 46.9372i 1.81742i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 51.9142 1.99523 0.997613 0.0690480i \(-0.0219962\pi\)
0.997613 + 0.0690480i \(0.0219962\pi\)
\(678\) 0 0
\(679\) − 2.06355i − 0.0791918i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 21.4476i 0.815906i 0.913003 + 0.407953i \(0.133757\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 28.6767i − 1.08777i
\(696\) 0 0
\(697\) 4.62510 0.175188
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.20739i 0.308671i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −38.1747 −1.42965
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 25.9435i 0.967529i 0.875198 + 0.483764i \(0.160731\pi\)
−0.875198 + 0.483764i \(0.839269\pi\)
\(720\) 0 0
\(721\) −4.42582 −0.164826
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 48.9354 1.81742
\(726\) 0 0
\(727\) − 36.0595i − 1.33737i −0.743544 0.668687i \(-0.766857\pi\)
0.743544 0.668687i \(-0.233143\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) − 7.66575i − 0.283528i
\(732\) 0 0
\(733\) −50.3158 −1.85846 −0.929229 0.369505i \(-0.879527\pi\)
−0.929229 + 0.369505i \(0.879527\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 19.0153i 0.699489i 0.936845 + 0.349744i \(0.113732\pi\)
−0.936845 + 0.349744i \(0.886268\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.59166i 0.351884i 0.984401 + 0.175942i \(0.0562971\pi\)
−0.984401 + 0.175942i \(0.943703\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 50.6286i 1.85240i
\(748\) 0 0
\(749\) 6.93544 0.253415
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 47.9583i − 1.74538i
\(756\) 0 0
\(757\) −53.5127 −1.94495 −0.972476 0.233005i \(-0.925144\pi\)
−0.972476 + 0.233005i \(0.925144\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 52.9354 1.91891 0.959454 0.281865i \(-0.0909530\pi\)
0.959454 + 0.281865i \(0.0909530\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −5.36126 −0.193837
\(766\) 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
\(773\) 40.2492 1.44766 0.723832 0.689976i \(-0.242379\pi\)
0.723832 + 0.689976i \(0.242379\pi\)
\(774\) 0 0
\(775\) 39.7999i 1.42965i
\(776\) 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
\(785\) −18.2129 −0.650047
\(786\) 0 0
\(787\) − 46.0144i − 1.64024i −0.572195 0.820118i \(-0.693908\pi\)
0.572195 0.820118i \(-0.306092\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 9.36053i − 0.332822i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −35.6241 −1.26187 −0.630936 0.775835i \(-0.717329\pi\)
−0.630936 + 0.775835i \(0.717329\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −4.94822 −0.174402
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36.9354 1.29858 0.649290 0.760541i \(-0.275066\pi\)
0.649290 + 0.760541i \(0.275066\pi\)
\(810\) 0 0
\(811\) 24.6175i 0.864437i 0.901769 + 0.432218i \(0.142269\pi\)
−0.901769 + 0.432218i \(0.857731\pi\)
\(812\) 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
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 11.3391i − 0.394300i −0.980373 0.197150i \(-0.936831\pi\)
0.980373 0.197150i \(-0.0631686\pi\)
\(828\) 0 0
\(829\) 56.9354 1.97745 0.988725 0.149744i \(-0.0478450\pi\)
0.988725 + 0.149744i \(0.0478450\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.42431 0.187941
\(834\) 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
\(841\) 66.7871 2.30300
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 29.0689 1.00000
\(846\) 0 0
\(847\) − 5.07566i − 0.174402i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 46.7281i 1.60182i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) − 58.5210i − 1.99671i −0.0573424 0.998355i \(-0.518263\pi\)
0.0573424 0.998355i \(-0.481737\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 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
\(873\) 13.4164 0.454077
\(874\) 0 0
\(875\) 5.15888i 0.174402i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 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 1.00000i \(-0.5\pi\)
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
\(895\) 47.9583i 1.60307i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 77.9050i 2.59828i
\(900\) 0 0
\(901\) −9.06456 −0.301984
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 51.5515i − 1.71174i −0.517192 0.855869i \(-0.673023\pi\)
0.517192 0.855869i \(-0.326977\pi\)
\(908\) 0 0
\(909\) −53.3613 −1.76988
\(910\) 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
\(917\) −9.89644 −0.326809
\(918\) 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
\(925\) 48.7174 1.60182
\(926\) 0 0
\(927\) − 28.7750i − 0.945095i
\(928\) 0 0
\(929\) 60.9354 1.99923 0.999613 0.0278019i \(-0.00885076\pi\)
0.999613 + 0.0278019i \(0.00885076\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −58.1378 −1.89928 −0.949639 0.313346i \(-0.898550\pi\)
−0.949639 + 0.313346i \(0.898550\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 27.7539i 0.903791i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −22.3607 −0.724333 −0.362167 0.932113i \(-0.617963\pi\)
−0.362167 + 0.932113i \(0.617963\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.19065i 0.199907i
\(960\) 0 0
\(961\) −32.3613 −1.04391
\(962\) 0 0
\(963\) 45.0915i 1.45305i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 5.91759 0.189709
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −62.4569 −1.99817 −0.999087 0.0427153i \(-0.986399\pi\)
−0.999087 + 0.0427153i \(0.986399\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.9824i 1.17955i 0.807566 + 0.589777i \(0.200785\pi\)
−0.807566 + 0.589777i \(0.799215\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.0000 1.46271
\(990\) 0 0
\(991\) − 1.76933i − 0.0562045i −0.999605 0.0281022i \(-0.991054\pi\)
0.999605 0.0281022i \(-0.00894640\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 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 1840.2.m.e.1839.7 yes 8
4.3 odd 2 inner 1840.2.m.e.1839.6 yes 8
5.4 even 2 inner 1840.2.m.e.1839.2 8
20.19 odd 2 inner 1840.2.m.e.1839.3 yes 8
23.22 odd 2 inner 1840.2.m.e.1839.2 8
92.91 even 2 inner 1840.2.m.e.1839.3 yes 8
115.114 odd 2 CM 1840.2.m.e.1839.7 yes 8
460.459 even 2 inner 1840.2.m.e.1839.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.e.1839.2 8 5.4 even 2 inner
1840.2.m.e.1839.2 8 23.22 odd 2 inner
1840.2.m.e.1839.3 yes 8 20.19 odd 2 inner
1840.2.m.e.1839.3 yes 8 92.91 even 2 inner
1840.2.m.e.1839.6 yes 8 4.3 odd 2 inner
1840.2.m.e.1839.6 yes 8 460.459 even 2 inner
1840.2.m.e.1839.7 yes 8 1.1 even 1 trivial
1840.2.m.e.1839.7 yes 8 115.114 odd 2 CM