Properties

Label 1728.3.b.h.1567.3
Level $1728$
Weight $3$
Character 1728.1567
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1567.3
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.h.1567.6

$q$-expansion

\(f(q)\) \(=\) \(q-3.16693i q^{5} -3.48528i q^{7} +O(q^{10})\) \(q-3.16693i q^{5} -3.48528i q^{7} +1.13770 q^{11} +14.9706 q^{25} -29.3939i q^{29} -23.4264i q^{31} -11.0376 q^{35} +36.8528 q^{49} -56.9115i q^{53} -3.60303i q^{55} -117.576 q^{59} -93.7939 q^{73} -3.96522i q^{77} +58.0000i q^{79} -67.0576 q^{83} +61.0589 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{25} - 384 q^{49} + 200 q^{73} + 760 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 3.16693i − 0.633386i −0.948528 0.316693i \(-0.897428\pi\)
0.948528 0.316693i \(-0.102572\pi\)
\(6\) 0 0
\(7\) − 3.48528i − 0.497897i −0.968517 0.248949i \(-0.919915\pi\)
0.968517 0.248949i \(-0.0800850\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.13770 0.103428 0.0517139 0.998662i \(-0.483532\pi\)
0.0517139 + 0.998662i \(0.483532\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 14.9706 0.598823
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 29.3939i − 1.01358i −0.862069 0.506791i \(-0.830832\pi\)
0.862069 0.506791i \(-0.169168\pi\)
\(30\) 0 0
\(31\) − 23.4264i − 0.755691i −0.925869 0.377845i \(-0.876665\pi\)
0.925869 0.377845i \(-0.123335\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −11.0376 −0.315361
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 36.8528 0.752098
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 56.9115i − 1.07380i −0.843645 0.536901i \(-0.819595\pi\)
0.843645 0.536901i \(-0.180405\pi\)
\(54\) 0 0
\(55\) − 3.60303i − 0.0655096i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −117.576 −1.99281 −0.996403 0.0847458i \(-0.972992\pi\)
−0.996403 + 0.0847458i \(0.972992\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −93.7939 −1.28485 −0.642424 0.766349i \(-0.722071\pi\)
−0.642424 + 0.766349i \(0.722071\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.96522i − 0.0514964i
\(78\) 0 0
\(79\) 58.0000i 0.734177i 0.930186 + 0.367089i \(0.119645\pi\)
−0.930186 + 0.367089i \(0.880355\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −67.0576 −0.807923 −0.403962 0.914776i \(-0.632367\pi\)
−0.403962 + 0.914776i \(0.632367\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 61.0589 0.629473 0.314736 0.949179i \(-0.398084\pi\)
0.314736 + 0.949179i \(0.398084\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 198.838i − 1.96869i −0.176253 0.984345i \(-0.556398\pi\)
0.176253 0.984345i \(-0.443602\pi\)
\(102\) 0 0
\(103\) 10.0000i 0.0970874i 0.998821 + 0.0485437i \(0.0154580\pi\)
−0.998821 + 0.0485437i \(0.984542\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −172.458 −1.61175 −0.805877 0.592082i \(-0.798306\pi\)
−0.805877 + 0.592082i \(0.798306\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −119.706 −0.989303
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 126.584i − 1.01267i
\(126\) 0 0
\(127\) − 21.6619i − 0.170566i −0.996357 0.0852831i \(-0.972821\pi\)
0.996357 0.0852831i \(-0.0271795\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −177.315 −1.35355 −0.676773 0.736192i \(-0.736622\pi\)
−0.676773 + 0.736192i \(0.736622\pi\)
\(132\) 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
\(145\) −93.0883 −0.641988
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 216.855i − 1.45540i −0.685896 0.727700i \(-0.740589\pi\)
0.685896 0.727700i \(-0.259411\pi\)
\(150\) 0 0
\(151\) 191.426i 1.26772i 0.773446 + 0.633862i \(0.218531\pi\)
−0.773446 + 0.633862i \(0.781469\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −74.1898 −0.478644
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 131.627i − 0.760850i −0.924812 0.380425i \(-0.875778\pi\)
0.924812 0.380425i \(-0.124222\pi\)
\(174\) 0 0
\(175\) − 52.1766i − 0.298152i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −62.0144 −0.346449 −0.173225 0.984882i \(-0.555419\pi\)
−0.173225 + 0.984882i \(0.555419\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 75.6173 0.391800 0.195900 0.980624i \(-0.437237\pi\)
0.195900 + 0.980624i \(0.437237\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 339.473i − 1.72321i −0.507576 0.861607i \(-0.669458\pi\)
0.507576 0.861607i \(-0.330542\pi\)
\(198\) 0 0
\(199\) − 397.985i − 1.99992i −0.00872575 0.999962i \(-0.502778\pi\)
0.00872575 0.999962i \(-0.497222\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −102.446 −0.504660
\(204\) 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
\(217\) −81.6476 −0.376256
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 230.000i 1.03139i 0.856772 + 0.515695i \(0.172466\pi\)
−0.856772 + 0.515695i \(0.827534\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −293.939 −1.29488 −0.647442 0.762115i \(-0.724161\pi\)
−0.647442 + 0.762115i \(0.724161\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 382.000 1.58506 0.792531 0.609831i \(-0.208763\pi\)
0.792531 + 0.609831i \(0.208763\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 116.710i − 0.476368i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −176.363 −0.702642 −0.351321 0.936255i \(-0.614267\pi\)
−0.351321 + 0.936255i \(0.614267\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −180.235 −0.680131
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 323.333i 1.20198i 0.799257 + 0.600990i \(0.205227\pi\)
−0.799257 + 0.600990i \(0.794773\pi\)
\(270\) 0 0
\(271\) − 437.690i − 1.61509i −0.589803 0.807547i \(-0.700795\pi\)
0.589803 0.807547i \(-0.299205\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.0321 0.0619348
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −289.000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 440.908i − 1.50481i −0.658703 0.752403i \(-0.728895\pi\)
0.658703 0.752403i \(-0.271105\pi\)
\(294\) 0 0
\(295\) 372.353i 1.26221i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 553.500 1.76837 0.884185 0.467138i \(-0.154715\pi\)
0.884185 + 0.467138i \(0.154715\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 558.696i 1.76245i 0.472698 + 0.881225i \(0.343280\pi\)
−0.472698 + 0.881225i \(0.656720\pi\)
\(318\) 0 0
\(319\) − 33.4416i − 0.104832i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −190.000 −0.563798 −0.281899 0.959444i \(-0.590964\pi\)
−0.281899 + 0.959444i \(0.590964\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 26.6523i − 0.0781594i
\(342\) 0 0
\(343\) − 299.221i − 0.872365i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 434.727 1.25282 0.626408 0.779495i \(-0.284524\pi\)
0.626408 + 0.779495i \(0.284524\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −361.000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 297.039i 0.813805i
\(366\) 0 0
\(367\) 516.220i 1.40659i 0.710896 + 0.703297i \(0.248290\pi\)
−0.710896 + 0.703297i \(0.751710\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −198.353 −0.534643
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −12.5576 −0.0326171
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 212.677i − 0.546726i −0.961911 0.273363i \(-0.911864\pi\)
0.961911 0.273363i \(-0.0881361\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 183.682 0.465017
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −19.5887 −0.0478942 −0.0239471 0.999713i \(-0.507623\pi\)
−0.0239471 + 0.999713i \(0.507623\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 409.784i 0.992212i
\(414\) 0 0
\(415\) 212.367i 0.511727i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 411.514 0.982134 0.491067 0.871122i \(-0.336607\pi\)
0.491067 + 0.871122i \(0.336607\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −810.176 −1.87108 −0.935538 0.353227i \(-0.885084\pi\)
−0.935538 + 0.353227i \(0.885084\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 860.249i 1.95956i 0.200066 + 0.979782i \(0.435884\pi\)
−0.200066 + 0.979782i \(0.564116\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 881.816 1.99056 0.995278 0.0970655i \(-0.0309456\pi\)
0.995278 + 0.0970655i \(0.0309456\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 909.881 1.99099 0.995494 0.0948261i \(-0.0302295\pi\)
0.995494 + 0.0948261i \(0.0302295\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 836.215i 1.81392i 0.421222 + 0.906958i \(0.361601\pi\)
−0.421222 + 0.906958i \(0.638399\pi\)
\(462\) 0 0
\(463\) − 647.161i − 1.39776i −0.715241 0.698878i \(-0.753683\pi\)
0.715241 0.698878i \(-0.246317\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 891.989 1.91004 0.955020 0.296541i \(-0.0958333\pi\)
0.955020 + 0.296541i \(0.0958333\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 193.369i − 0.398699i
\(486\) 0 0
\(487\) − 970.000i − 1.99179i −0.0905356 0.995893i \(-0.528858\pi\)
0.0905356 0.995893i \(-0.471142\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.0783 −0.0490392 −0.0245196 0.999699i \(-0.507806\pi\)
−0.0245196 + 0.999699i \(0.507806\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −629.705 −1.24694
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 810.999i 1.59332i 0.604429 + 0.796659i \(0.293401\pi\)
−0.604429 + 0.796659i \(0.706599\pi\)
\(510\) 0 0
\(511\) 326.898i 0.639723i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 31.6693 0.0614938
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 546.161i 1.02086i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 41.9276 0.0777878
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 202.146 0.365545
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1109.98i 1.99278i 0.0848683 + 0.996392i \(0.472953\pi\)
−0.0848683 + 0.996392i \(0.527047\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 821.212 1.45864 0.729318 0.684175i \(-0.239838\pi\)
0.729318 + 0.684175i \(0.239838\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −290.000 −0.502600 −0.251300 0.967909i \(-0.580858\pi\)
−0.251300 + 0.967909i \(0.580858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 233.715i 0.402263i
\(582\) 0 0
\(583\) − 64.7485i − 0.111061i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 364.988 0.621785 0.310893 0.950445i \(-0.399372\pi\)
0.310893 + 0.950445i \(0.399372\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 514.147 0.855486 0.427743 0.903900i \(-0.359309\pi\)
0.427743 + 0.903900i \(0.359309\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 379.099i 0.626610i
\(606\) 0 0
\(607\) − 730.000i − 1.20264i −0.799010 0.601318i \(-0.794643\pi\)
0.799010 0.601318i \(-0.205357\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −26.6182 −0.0425891
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 831.132i − 1.31717i −0.752508 0.658583i \(-0.771156\pi\)
0.752508 0.658583i \(-0.228844\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −68.6017 −0.108034
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −133.766 −0.206111
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1287.63i 1.97188i 0.167112 + 0.985938i \(0.446556\pi\)
−0.167112 + 0.985938i \(0.553444\pi\)
\(654\) 0 0
\(655\) 561.542i 0.857317i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1180.88 1.79193 0.895967 0.444121i \(-0.146484\pi\)
0.895967 + 0.444121i \(0.146484\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1059.00 1.57355 0.786774 0.617241i \(-0.211750\pi\)
0.786774 + 0.617241i \(0.211750\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 146.969i − 0.217089i −0.994092 0.108545i \(-0.965381\pi\)
0.994092 0.108545i \(-0.0346190\pi\)
\(678\) 0 0
\(679\) − 212.807i − 0.313413i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 293.939 0.430364 0.215182 0.976574i \(-0.430965\pi\)
0.215182 + 0.976574i \(0.430965\pi\)
\(684\) 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
\(701\) 743.855i 1.06113i 0.847643 + 0.530567i \(0.178021\pi\)
−0.847643 + 0.530567i \(0.821979\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −693.005 −0.980205
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 34.8528 0.0483395
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 440.043i − 0.606956i
\(726\) 0 0
\(727\) 818.338i 1.12564i 0.826581 + 0.562818i \(0.190283\pi\)
−0.826581 + 0.562818i \(0.809717\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −686.763 −0.921829
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 601.064i 0.802488i
\(750\) 0 0
\(751\) − 234.310i − 0.311997i −0.987757 0.155998i \(-0.950141\pi\)
0.987757 0.155998i \(-0.0498595\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 606.234 0.802959
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −672.087 −0.873975 −0.436987 0.899468i \(-0.643955\pi\)
−0.436987 + 0.899468i \(0.643955\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1028.79i − 1.33090i −0.746442 0.665450i \(-0.768240\pi\)
0.746442 0.665450i \(-0.231760\pi\)
\(774\) 0 0
\(775\) − 350.706i − 0.452525i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 655.234i 0.822125i 0.911607 + 0.411063i \(0.134842\pi\)
−0.911607 + 0.411063i \(0.865158\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −106.710 −0.132889
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 1499.09i − 1.82593i −0.408039 0.912965i \(-0.633787\pi\)
0.408039 0.912965i \(-0.366213\pi\)
\(822\) 0 0
\(823\) 1131.13i 1.37440i 0.726469 + 0.687199i \(0.241160\pi\)
−0.726469 + 0.687199i \(0.758840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 587.878 0.710856 0.355428 0.934704i \(-0.384335\pi\)
0.355428 + 0.934704i \(0.384335\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −23.0000 −0.0273484
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 535.211i − 0.633386i
\(846\) 0 0
\(847\) 417.208i 0.492571i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −416.854 −0.481912
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 65.9869i 0.0759343i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −441.181 −0.504206
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −75.4978 −0.0849244
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 196.395i 0.219436i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −688.593 −0.765954
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −76.2918 −0.0835616
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 617.991i 0.673927i
\(918\) 0 0
\(919\) − 1815.92i − 1.97598i −0.154523 0.987989i \(-0.549384\pi\)
0.154523 0.987989i \(-0.450616\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −239.293 −0.255382 −0.127691 0.991814i \(-0.540757\pi\)
−0.127691 + 0.991814i \(0.540757\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1288.50i − 1.36929i −0.728878 0.684644i \(-0.759958\pi\)
0.728878 0.684644i \(-0.240042\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 445.172 0.470087 0.235043 0.971985i \(-0.424477\pi\)
0.235043 + 0.971985i \(0.424477\pi\)
\(948\) 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
\(961\) 412.203 0.428932
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 239.475i − 0.248160i
\(966\) 0 0
\(967\) 1218.04i 1.25961i 0.776753 + 0.629805i \(0.216865\pi\)
−0.776753 + 0.629805i \(0.783135\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1779.21 −1.83234 −0.916172 0.400785i \(-0.868738\pi\)
−0.916172 + 0.400785i \(0.868738\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −1075.09 −1.09146
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1921.37i 1.93882i 0.245457 + 0.969408i \(0.421062\pi\)
−0.245457 + 0.969408i \(0.578938\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1260.39 −1.26672
\(996\) 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 1728.3.b.h.1567.3 8
3.2 odd 2 inner 1728.3.b.h.1567.5 yes 8
4.3 odd 2 inner 1728.3.b.h.1567.4 yes 8
8.3 odd 2 inner 1728.3.b.h.1567.6 yes 8
8.5 even 2 inner 1728.3.b.h.1567.5 yes 8
12.11 even 2 inner 1728.3.b.h.1567.6 yes 8
24.5 odd 2 CM 1728.3.b.h.1567.3 8
24.11 even 2 inner 1728.3.b.h.1567.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.h.1567.3 8 1.1 even 1 trivial
1728.3.b.h.1567.3 8 24.5 odd 2 CM
1728.3.b.h.1567.4 yes 8 4.3 odd 2 inner
1728.3.b.h.1567.4 yes 8 24.11 even 2 inner
1728.3.b.h.1567.5 yes 8 3.2 odd 2 inner
1728.3.b.h.1567.5 yes 8 8.5 even 2 inner
1728.3.b.h.1567.6 yes 8 8.3 odd 2 inner
1728.3.b.h.1567.6 yes 8 12.11 even 2 inner