Properties

Label 1728.3.b.h.1567.1
Level $1728$
Weight $3$
Character 1728.1567
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

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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 1567.1
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1567
Dual form 1728.3.b.h.1567.8

$q$-expansion

\(f(q)\) \(=\) \(q-6.63103i q^{5} -13.4853i q^{7} +O(q^{10})\) \(q-6.63103i q^{5} -13.4853i q^{7} -18.4582 q^{11} -18.9706 q^{25} -29.3939i q^{29} -61.4264i q^{31} -89.4213 q^{35} -132.853 q^{49} +105.901i q^{53} +122.397i q^{55} +117.576 q^{59} +143.794 q^{73} +248.914i q^{77} -58.0000i q^{79} -165.037 q^{83} +128.941 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) 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
\(5\) − 6.63103i − 1.32621i −0.748528 0.663103i \(-0.769239\pi\)
0.748528 0.663103i \(-0.230761\pi\)
\(6\) 0 0
\(7\) − 13.4853i − 1.92647i −0.268662 0.963234i \(-0.586582\pi\)
0.268662 0.963234i \(-0.413418\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −18.4582 −1.67802 −0.839010 0.544116i \(-0.816865\pi\)
−0.839010 + 0.544116i \(0.816865\pi\)
\(12\) 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
\(25\) −18.9706 −0.758823
\(26\) 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
\(31\) − 61.4264i − 1.98150i −0.135711 0.990748i \(-0.543332\pi\)
0.135711 0.990748i \(-0.456668\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −89.4213 −2.55489
\(36\) 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
\(49\) −132.853 −2.71128
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 105.901i 1.99814i 0.0431471 + 0.999069i \(0.486262\pi\)
−0.0431471 + 0.999069i \(0.513738\pi\)
\(54\) 0 0
\(55\) 122.397i 2.22540i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 117.576 1.99281 0.996403 0.0847458i \(-0.0270078\pi\)
0.996403 + 0.0847458i \(0.0270078\pi\)
\(60\) 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
\(73\) 143.794 1.96978 0.984890 0.173181i \(-0.0554046\pi\)
0.984890 + 0.173181i \(0.0554046\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 248.914i 3.23265i
\(78\) 0 0
\(79\) − 58.0000i − 0.734177i −0.930186 0.367089i \(-0.880355\pi\)
0.930186 0.367089i \(-0.119645\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −165.037 −1.98840 −0.994200 0.107547i \(-0.965700\pi\)
−0.994200 + 0.107547i \(0.965700\pi\)
\(84\) 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
\(97\) 128.941 1.32929 0.664645 0.747159i \(-0.268583\pi\)
0.664645 + 0.747159i \(0.268583\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 130.252i 1.28962i 0.764341 + 0.644812i \(0.223064\pi\)
−0.764341 + 0.644812i \(0.776936\pi\)
\(102\) 0 0
\(103\) − 10.0000i − 0.0970874i −0.998821 0.0485437i \(-0.984542\pi\)
0.998821 0.0485437i \(-0.0154580\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 23.5014 0.219639 0.109820 0.993952i \(-0.464973\pi\)
0.109820 + 0.993952i \(0.464973\pi\)
\(108\) 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
\(121\) 219.706 1.81575
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 39.9814i − 0.319851i
\(126\) 0 0
\(127\) 208.338i 1.64046i 0.572036 + 0.820229i \(0.306154\pi\)
−0.572036 + 0.820229i \(0.693846\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −255.698 −1.95189 −0.975947 0.218007i \(-0.930045\pi\)
−0.975947 + 0.218007i \(0.930045\pi\)
\(132\) 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
\(145\) −194.912 −1.34422
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 285.440i 1.91571i 0.287258 + 0.957853i \(0.407256\pi\)
−0.287258 + 0.957853i \(0.592744\pi\)
\(150\) 0 0
\(151\) − 106.574i − 0.705785i −0.935664 0.352893i \(-0.885198\pi\)
0.935664 0.352893i \(-0.114802\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −407.320 −2.62787
\(156\) 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
\(173\) − 211.301i − 1.22140i −0.791864 0.610698i \(-0.790889\pi\)
0.791864 0.610698i \(-0.209111\pi\)
\(174\) 0 0
\(175\) 255.823i 1.46185i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −336.357 −1.87909 −0.939545 0.342424i \(-0.888752\pi\)
−0.939545 + 0.342424i \(0.888752\pi\)
\(180\) 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
\(193\) −365.617 −1.89439 −0.947195 0.320658i \(-0.896096\pi\)
−0.947195 + 0.320658i \(0.896096\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 3.45535i − 0.0175399i −0.999962 0.00876993i \(-0.997208\pi\)
0.999962 0.00876993i \(-0.00279159\pi\)
\(198\) 0 0
\(199\) − 195.985i − 0.984848i −0.870355 0.492424i \(-0.836111\pi\)
0.870355 0.492424i \(-0.163889\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −396.385 −1.95263
\(204\) 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
\(217\) −828.352 −3.81729
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 230.000i − 1.03139i −0.856772 0.515695i \(-0.827534\pi\)
0.856772 0.515695i \(-0.172466\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 293.939 1.29488 0.647442 0.762115i \(-0.275839\pi\)
0.647442 + 0.762115i \(0.275839\pi\)
\(228\) 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
\(245\) 880.951i 3.59572i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 176.363 0.702642 0.351321 0.936255i \(-0.385733\pi\)
0.351321 + 0.936255i \(0.385733\pi\)
\(252\) 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
\(265\) 702.235 2.64994
\(266\) 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
\(271\) − 495.690i − 1.82912i −0.404455 0.914558i \(-0.632539\pi\)
0.404455 0.914558i \(-0.367461\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 350.163 1.27332
\(276\) 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
\(295\) − 779.647i − 2.64287i
\(296\) 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
\(313\) −23.4996 −0.0750785 −0.0375392 0.999295i \(-0.511952\pi\)
−0.0375392 + 0.999295i \(0.511952\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 19.8086i − 0.0624877i −0.999512 0.0312439i \(-0.990053\pi\)
0.999512 0.0312439i \(-0.00994685\pi\)
\(318\) 0 0
\(319\) 542.558i 1.70081i
\(320\) 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
\(341\) 1133.82i 3.32499i
\(342\) 0 0
\(343\) 1130.78i 3.29673i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −251.130 −0.723717 −0.361859 0.932233i \(-0.617858\pi\)
−0.361859 + 0.932233i \(0.617858\pi\)
\(348\) 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
\(365\) − 953.502i − 2.61233i
\(366\) 0 0
\(367\) − 193.780i − 0.528010i −0.964521 0.264005i \(-0.914956\pi\)
0.964521 0.264005i \(-0.0850435\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1428.11 3.84935
\(372\) 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
\(385\) 1650.56 4.28716
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 541.766i − 1.39272i −0.717695 0.696358i \(-0.754803\pi\)
0.717695 0.696358i \(-0.245197\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −384.600 −0.973670
\(396\) 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
\(409\) −698.411 −1.70761 −0.853803 0.520595i \(-0.825710\pi\)
−0.853803 + 0.520595i \(0.825710\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1585.54i − 3.83908i
\(414\) 0 0
\(415\) 1094.37i 2.63703i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −411.514 −0.982134 −0.491067 0.871122i \(-0.663393\pi\)
−0.491067 + 0.871122i \(0.663393\pi\)
\(420\) 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
\(433\) 140.176 0.323732 0.161866 0.986813i \(-0.448249\pi\)
0.161866 + 0.986813i \(0.448249\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 582.249i 1.32631i 0.748483 + 0.663154i \(0.230782\pi\)
−0.748483 + 0.663154i \(0.769218\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −881.816 −1.99056 −0.995278 0.0970655i \(-0.969054\pi\)
−0.995278 + 0.0970655i \(0.969054\pi\)
\(444\) 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
\(457\) −379.881 −0.831250 −0.415625 0.909536i \(-0.636437\pi\)
−0.415625 + 0.909536i \(0.636437\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 81.7720i − 0.177380i −0.996059 0.0886899i \(-0.971732\pi\)
0.996059 0.0886899i \(-0.0282680\pi\)
\(462\) 0 0
\(463\) − 897.161i − 1.93771i −0.247625 0.968856i \(-0.579650\pi\)
0.247625 0.968856i \(-0.420350\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 206.132 0.441395 0.220698 0.975342i \(-0.429167\pi\)
0.220698 + 0.975342i \(0.429167\pi\)
\(468\) 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
\(485\) − 855.013i − 1.76291i
\(486\) 0 0
\(487\) 970.000i 1.99179i 0.0905356 + 0.995893i \(0.471142\pi\)
−0.0905356 + 0.995893i \(0.528858\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 838.142 1.70701 0.853505 0.521084i \(-0.174472\pi\)
0.853505 + 0.521084i \(0.174472\pi\)
\(492\) 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
\(505\) 863.705 1.71031
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 938.372i − 1.84356i −0.387713 0.921780i \(-0.626735\pi\)
0.387713 0.921780i \(-0.373265\pi\)
\(510\) 0 0
\(511\) − 1939.10i − 3.79472i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −66.3103 −0.128758
\(516\) 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
\(535\) − 155.839i − 0.291287i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2452.23 4.54958
\(540\) 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
\(553\) −782.146 −1.41437
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 473.114i − 0.849396i −0.905335 0.424698i \(-0.860380\pi\)
0.905335 0.424698i \(-0.139620\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −256.563 −0.455708 −0.227854 0.973695i \(-0.573171\pi\)
−0.227854 + 0.973695i \(0.573171\pi\)
\(564\) 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
\(581\) 2225.57i 3.83059i
\(582\) 0 0
\(583\) − 1954.75i − 3.35291i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1148.82 1.95711 0.978556 0.205982i \(-0.0660387\pi\)
0.978556 + 0.205982i \(0.0660387\pi\)
\(588\) 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
\(601\) 683.853 1.13786 0.568929 0.822387i \(-0.307358\pi\)
0.568929 + 0.822387i \(0.307358\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 1456.87i − 2.40806i
\(606\) 0 0
\(607\) 730.000i 1.20264i 0.799010 + 0.601318i \(0.205357\pi\)
−0.799010 + 0.601318i \(0.794643\pi\)
\(608\) 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
\(625\) −739.382 −1.18301
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 406.868i 0.644799i 0.946604 + 0.322399i \(0.104489\pi\)
−0.946604 + 0.322399i \(0.895511\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1381.50 2.17558
\(636\) 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
\(649\) −2170.23 −3.34397
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 454.808i − 0.696491i −0.937403 0.348245i \(-0.886778\pi\)
0.937403 0.348245i \(-0.113222\pi\)
\(654\) 0 0
\(655\) 1695.54i 2.58861i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 83.5128 0.126727 0.0633633 0.997991i \(-0.479817\pi\)
0.0633633 + 0.997991i \(0.479817\pi\)
\(660\) 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
\(673\) −1249.00 −1.85587 −0.927933 0.372746i \(-0.878416\pi\)
−0.927933 + 0.372746i \(0.878416\pi\)
\(674\) 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
\(679\) − 1738.81i − 2.56084i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −293.939 −0.430364 −0.215182 0.976574i \(-0.569035\pi\)
−0.215182 + 0.976574i \(0.569035\pi\)
\(684\) 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
\(701\) 657.253i 0.937593i 0.883306 + 0.468797i \(0.155312\pi\)
−0.883306 + 0.468797i \(0.844688\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1756.48 2.48442
\(708\) 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
\(721\) −134.853 −0.187036
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 557.618i 0.769129i
\(726\) 0 0
\(727\) − 631.662i − 0.868861i −0.900705 0.434430i \(-0.856950\pi\)
0.900705 0.434430i \(-0.143050\pi\)
\(728\) 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
\(745\) 1892.76 2.54062
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 316.923i − 0.423128i
\(750\) 0 0
\(751\) 1167.69i 1.55485i 0.628977 + 0.777424i \(0.283474\pi\)
−0.628977 + 0.777424i \(0.716526\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −706.693 −0.936017
\(756\) 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
\(769\) 1534.09 1.99491 0.997456 0.0712917i \(-0.0227121\pi\)
0.997456 + 0.0712917i \(0.0227121\pi\)
\(770\) 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
\(775\) 1165.29i 1.50360i
\(776\) 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
\(797\) − 1586.04i − 1.99001i −0.0998129 0.995006i \(-0.531824\pi\)
0.0998129 0.995006i \(-0.468176\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2654.18 −3.30533
\(804\) 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
\(823\) 1601.13i 1.94548i 0.231898 + 0.972740i \(0.425507\pi\)
−0.231898 + 0.972740i \(0.574493\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −587.878 −0.710856 −0.355428 0.934704i \(-0.615665\pi\)
−0.355428 + 0.934704i \(0.615665\pi\)
\(828\) 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
\(845\) − 1120.64i − 1.32621i
\(846\) 0 0
\(847\) − 2962.79i − 3.49798i
\(848\) 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
\(865\) −1401.15 −1.61982
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1070.58i 1.23196i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −539.160 −0.616183
\(876\) 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
\(889\) 2809.50 3.16029
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 2230.40i 2.49206i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1805.56 −2.00841
\(900\) 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
\(913\) 3046.29 3.33657
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3448.16i 3.76026i
\(918\) 0 0
\(919\) − 1153.92i − 1.25563i −0.778362 0.627815i \(-0.783949\pi\)
0.778362 0.627815i \(-0.216051\pi\)
\(920\) 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
\(937\) 1729.29 1.84556 0.922782 0.385323i \(-0.125910\pi\)
0.922782 + 0.385323i \(0.125910\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 543.718i − 0.577809i −0.957358 0.288904i \(-0.906709\pi\)
0.957358 0.288904i \(-0.0932910\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1816.89 1.91857 0.959285 0.282439i \(-0.0911435\pi\)
0.959285 + 0.282439i \(0.0911435\pi\)
\(948\) 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
\(961\) −2812.20 −2.92633
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2424.42i 2.51235i
\(966\) 0 0
\(967\) − 691.956i − 0.715570i −0.933804 0.357785i \(-0.883532\pi\)
0.933804 0.357785i \(-0.116468\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1563.65 −1.61035 −0.805176 0.593036i \(-0.797929\pi\)
−0.805176 + 0.593036i \(0.797929\pi\)
\(972\) 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
\(985\) −22.9126 −0.0232615
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 539.366i 0.544264i 0.962260 + 0.272132i \(0.0877288\pi\)
−0.962260 + 0.272132i \(0.912271\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1299.58 −1.30611
\(996\) 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.1567.1 8
3.2 odd 2 inner 1728.3.b.h.1567.7 yes 8
4.3 odd 2 inner 1728.3.b.h.1567.2 yes 8
8.3 odd 2 inner 1728.3.b.h.1567.8 yes 8
8.5 even 2 inner 1728.3.b.h.1567.7 yes 8
12.11 even 2 inner 1728.3.b.h.1567.8 yes 8
24.5 odd 2 CM 1728.3.b.h.1567.1 8
24.11 even 2 inner 1728.3.b.h.1567.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.h.1567.1 8 1.1 even 1 trivial
1728.3.b.h.1567.1 8 24.5 odd 2 CM
1728.3.b.h.1567.2 yes 8 4.3 odd 2 inner
1728.3.b.h.1567.2 yes 8 24.11 even 2 inner
1728.3.b.h.1567.7 yes 8 3.2 odd 2 inner
1728.3.b.h.1567.7 yes 8 8.5 even 2 inner
1728.3.b.h.1567.8 yes 8 8.3 odd 2 inner
1728.3.b.h.1567.8 yes 8 12.11 even 2 inner