Properties

Label 216.3.h.b.53.2
Level $216$
Weight $3$
Character 216.53
Self dual yes
Analytic conductor $5.886$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 53.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 216.53

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} +9.48528 q^{5} -3.48528 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} +9.48528 q^{5} -3.48528 q^{7} +8.00000 q^{8} +18.9706 q^{10} -21.9706 q^{11} -6.97056 q^{14} +16.0000 q^{16} +37.9411 q^{20} -43.9411 q^{22} +64.9706 q^{25} -13.9411 q^{28} -50.0000 q^{29} +23.4264 q^{31} +32.0000 q^{32} -33.0589 q^{35} +75.8823 q^{40} -87.8823 q^{44} -36.8528 q^{49} +129.941 q^{50} -89.4264 q^{53} -208.397 q^{55} -27.8823 q^{56} -100.000 q^{58} +10.0000 q^{59} +46.8528 q^{62} +64.0000 q^{64} -66.1177 q^{70} +93.7939 q^{73} +76.5736 q^{77} -58.0000 q^{79} +151.765 q^{80} +151.853 q^{83} -175.765 q^{88} +61.0589 q^{97} -73.7056 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 2q^{5} + 10q^{7} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 2q^{5} + 10q^{7} + 16q^{8} + 4q^{10} - 10q^{11} + 20q^{14} + 32q^{16} + 8q^{20} - 20q^{22} + 96q^{25} + 40q^{28} - 100q^{29} - 38q^{31} + 64q^{32} - 134q^{35} + 16q^{40} - 40q^{44} + 96q^{49} + 192q^{50} - 94q^{53} - 298q^{55} + 80q^{56} - 200q^{58} + 20q^{59} - 76q^{62} + 128q^{64} - 268q^{70} - 50q^{73} + 238q^{77} - 116q^{79} + 32q^{80} + 134q^{83} - 80q^{88} + 190q^{97} + 192q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) 9.48528 1.89706 0.948528 0.316693i \(-0.102572\pi\)
0.948528 + 0.316693i \(0.102572\pi\)
\(6\) 0 0
\(7\) −3.48528 −0.497897 −0.248949 0.968517i \(-0.580085\pi\)
−0.248949 + 0.968517i \(0.580085\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) 18.9706 1.89706
\(11\) −21.9706 −1.99732 −0.998662 0.0517139i \(-0.983532\pi\)
−0.998662 + 0.0517139i \(0.983532\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) −6.97056 −0.497897
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 37.9411 1.89706
\(21\) 0 0
\(22\) −43.9411 −1.99732
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 64.9706 2.59882
\(26\) 0 0
\(27\) 0 0
\(28\) −13.9411 −0.497897
\(29\) −50.0000 −1.72414 −0.862069 0.506791i \(-0.830832\pi\)
−0.862069 + 0.506791i \(0.830832\pi\)
\(30\) 0 0
\(31\) 23.4264 0.755691 0.377845 0.925869i \(-0.376665\pi\)
0.377845 + 0.925869i \(0.376665\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −33.0589 −0.944539
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 75.8823 1.89706
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −87.8823 −1.99732
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −36.8528 −0.752098
\(50\) 129.941 2.59882
\(51\) 0 0
\(52\) 0 0
\(53\) −89.4264 −1.68729 −0.843645 0.536901i \(-0.819595\pi\)
−0.843645 + 0.536901i \(0.819595\pi\)
\(54\) 0 0
\(55\) −208.397 −3.78904
\(56\) −27.8823 −0.497897
\(57\) 0 0
\(58\) −100.000 −1.72414
\(59\) 10.0000 0.169492 0.0847458 0.996403i \(-0.472992\pi\)
0.0847458 + 0.996403i \(0.472992\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 46.8528 0.755691
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −66.1177 −0.944539
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 93.7939 1.28485 0.642424 0.766349i \(-0.277929\pi\)
0.642424 + 0.766349i \(0.277929\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 76.5736 0.994462
\(78\) 0 0
\(79\) −58.0000 −0.734177 −0.367089 0.930186i \(-0.619645\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) 151.765 1.89706
\(81\) 0 0
\(82\) 0 0
\(83\) 151.853 1.82955 0.914776 0.403962i \(-0.132367\pi\)
0.914776 + 0.403962i \(0.132367\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −175.765 −1.99732
\(89\) 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
\(97\) 61.0589 0.629473 0.314736 0.949179i \(-0.398084\pi\)
0.314736 + 0.949179i \(0.398084\pi\)
\(98\) −73.7056 −0.752098
\(99\) 0 0
\(100\) 259.882 2.59882
\(101\) −35.6030 −0.352505 −0.176253 0.984345i \(-0.556398\pi\)
−0.176253 + 0.984345i \(0.556398\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.0970874 −0.0485437 0.998821i \(-0.515458\pi\)
−0.0485437 + 0.998821i \(0.515458\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −178.853 −1.68729
\(107\) −126.706 −1.18416 −0.592082 0.805877i \(-0.701694\pi\)
−0.592082 + 0.805877i \(0.701694\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −416.794 −3.78904
\(111\) 0 0
\(112\) −55.7645 −0.497897
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −200.000 −1.72414
\(117\) 0 0
\(118\) 20.0000 0.169492
\(119\) 0 0
\(120\) 0 0
\(121\) 361.706 2.98930
\(122\) 0 0
\(123\) 0 0
\(124\) 93.7056 0.755691
\(125\) 379.132 3.03306
\(126\) 0 0
\(127\) −21.6619 −0.170566 −0.0852831 0.996357i \(-0.527179\pi\)
−0.0852831 + 0.996357i \(0.527179\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −192.882 −1.47238 −0.736192 0.676773i \(-0.763378\pi\)
−0.736192 + 0.676773i \(0.763378\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −132.235 −0.944539
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −474.264 −3.27079
\(146\) 187.588 1.28485
\(147\) 0 0
\(148\) 0 0
\(149\) 204.397 1.37179 0.685896 0.727700i \(-0.259411\pi\)
0.685896 + 0.727700i \(0.259411\pi\)
\(150\) 0 0
\(151\) 191.426 1.26772 0.633862 0.773446i \(-0.281469\pi\)
0.633862 + 0.773446i \(0.281469\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 153.147 0.994462
\(155\) 222.206 1.43359
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −116.000 −0.734177
\(159\) 0 0
\(160\) 303.529 1.89706
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 303.706 1.82955
\(167\) 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
\(173\) −319.985 −1.84962 −0.924812 0.380425i \(-0.875778\pi\)
−0.924812 + 0.380425i \(0.875778\pi\)
\(174\) 0 0
\(175\) −226.441 −1.29395
\(176\) −351.529 −1.99732
\(177\) 0 0
\(178\) 0 0
\(179\) 352.588 1.96976 0.984882 0.173225i \(-0.0554187\pi\)
0.984882 + 0.173225i \(0.0554187\pi\)
\(180\) 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 \(0.437237\pi\)
\(194\) 122.118 0.629473
\(195\) 0 0
\(196\) −147.411 −0.752098
\(197\) −199.985 −1.01515 −0.507576 0.861607i \(-0.669458\pi\)
−0.507576 + 0.861607i \(0.669458\pi\)
\(198\) 0 0
\(199\) 397.985 1.99992 0.999962 0.00872575i \(-0.00277753\pi\)
0.999962 + 0.00872575i \(0.00277753\pi\)
\(200\) 519.765 2.59882
\(201\) 0 0
\(202\) −71.2061 −0.352505
\(203\) 174.264 0.858444
\(204\) 0 0
\(205\) 0 0
\(206\) −20.0000 −0.0970874
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −357.706 −1.68729
\(213\) 0 0
\(214\) −253.411 −1.18416
\(215\) 0 0
\(216\) 0 0
\(217\) −81.6476 −0.376256
\(218\) 0 0
\(219\) 0 0
\(220\) −833.588 −3.78904
\(221\) 0 0
\(222\) 0 0
\(223\) 230.000 1.03139 0.515695 0.856772i \(-0.327534\pi\)
0.515695 + 0.856772i \(0.327534\pi\)
\(224\) −111.529 −0.497897
\(225\) 0 0
\(226\) 0 0
\(227\) 346.000 1.52423 0.762115 0.647442i \(-0.224161\pi\)
0.762115 + 0.647442i \(0.224161\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −400.000 −1.72414
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 40.0000 0.169492
\(237\) 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 \(-0.791237\pi\)
−0.792531 + 0.609831i \(0.791237\pi\)
\(242\) 723.411 2.98930
\(243\) 0 0
\(244\) 0 0
\(245\) −349.559 −1.42677
\(246\) 0 0
\(247\) 0 0
\(248\) 187.411 0.755691
\(249\) 0 0
\(250\) 758.264 3.03306
\(251\) −470.000 −1.87251 −0.936255 0.351321i \(-0.885733\pi\)
−0.936255 + 0.351321i \(0.885733\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −43.3238 −0.170566
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −385.765 −1.47238
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) −848.235 −3.20089
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 430.000 1.59851 0.799257 0.600990i \(-0.205227\pi\)
0.799257 + 0.600990i \(0.205227\pi\)
\(270\) 0 0
\(271\) −437.690 −1.61509 −0.807547 0.589803i \(-0.799205\pi\)
−0.807547 + 0.589803i \(0.799205\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1427.44 −5.19069
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −264.471 −0.944539
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) −948.528 −3.27079
\(291\) 0 0
\(292\) 375.176 1.28485
\(293\) −386.000 −1.31741 −0.658703 0.752403i \(-0.728895\pi\)
−0.658703 + 0.752403i \(0.728895\pi\)
\(294\) 0 0
\(295\) 94.8528 0.321535
\(296\) 0 0
\(297\) 0 0
\(298\) 408.794 1.37179
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 382.853 1.26772
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 306.294 0.994462
\(309\) 0 0
\(310\) 444.412 1.43359
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −553.500 −1.76837 −0.884185 0.467138i \(-0.845285\pi\)
−0.884185 + 0.467138i \(0.845285\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −232.000 −0.734177
\(317\) 299.690 0.945396 0.472698 0.881225i \(-0.343280\pi\)
0.472698 + 0.881225i \(0.343280\pi\)
\(318\) 0 0
\(319\) 1098.53 3.44366
\(320\) 607.058 1.89706
\(321\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 607.411 1.82955
\(333\) 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 \(0.590964\pi\)
\(338\) 338.000 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) −514.691 −1.50936
\(342\) 0 0
\(343\) 299.221 0.872365
\(344\) 0 0
\(345\) 0 0
\(346\) −639.970 −1.84962
\(347\) 540.970 1.55899 0.779495 0.626408i \(-0.215476\pi\)
0.779495 + 0.626408i \(0.215476\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) −452.881 −1.29395
\(351\) 0 0
\(352\) −703.058 −1.99732
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 705.176 1.96976
\(359\) 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
\(365\) 889.662 2.43743
\(366\) 0 0
\(367\) −516.220 −1.40659 −0.703297 0.710896i \(-0.748290\pi\)
−0.703297 + 0.710896i \(0.748290\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 311.676 0.840098
\(372\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 726.322 1.88655
\(386\) 151.235 0.391800
\(387\) 0 0
\(388\) 244.235 0.629473
\(389\) −748.367 −1.92382 −0.961911 0.273363i \(-0.911864\pi\)
−0.961911 + 0.273363i \(0.911864\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −294.823 −0.752098
\(393\) 0 0
\(394\) −399.970 −1.01515
\(395\) −550.146 −1.39278
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 795.970 1.99992
\(399\) 0 0
\(400\) 1039.53 2.59882
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −142.412 −0.352505
\(405\) 0 0
\(406\) 348.528 0.858444
\(407\) 0 0
\(408\) 0 0
\(409\) 19.5887 0.0478942 0.0239471 0.999713i \(-0.492377\pi\)
0.0239471 + 0.999713i \(0.492377\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −40.0000 −0.0970874
\(413\) −34.8528 −0.0843894
\(414\) 0 0
\(415\) 1440.37 3.47076
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 730.000 1.74224 0.871122 0.491067i \(-0.163393\pi\)
0.871122 + 0.491067i \(0.163393\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −715.411 −1.68729
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −506.823 −1.18416
\(429\) 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 \(-0.114916\pi\)
0.935538 + 0.353227i \(0.114916\pi\)
\(434\) −163.295 −0.376256
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −860.249 −1.95956 −0.979782 0.200066i \(-0.935884\pi\)
−0.979782 + 0.200066i \(0.935884\pi\)
\(440\) −1667.18 −3.78904
\(441\) 0 0
\(442\) 0 0
\(443\) −86.0000 −0.194131 −0.0970655 0.995278i \(-0.530946\pi\)
−0.0970655 + 0.995278i \(0.530946\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 460.000 1.03139
\(447\) 0 0
\(448\) −223.058 −0.497897
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 692.000 1.52423
\(455\) 0 0
\(456\) 0 0
\(457\) −909.881 −1.99099 −0.995494 0.0948261i \(-0.969771\pi\)
−0.995494 + 0.0948261i \(0.969771\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −388.367 −0.842444 −0.421222 0.906958i \(-0.638399\pi\)
−0.421222 + 0.906958i \(0.638399\pi\)
\(462\) 0 0
\(463\) −647.161 −1.39776 −0.698878 0.715241i \(-0.746317\pi\)
−0.698878 + 0.715241i \(0.746317\pi\)
\(464\) −800.000 −1.72414
\(465\) 0 0
\(466\) 0 0
\(467\) 276.970 0.593083 0.296541 0.955020i \(-0.404167\pi\)
0.296541 + 0.955020i \(0.404167\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 80.0000 0.169492
\(473\) 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
\(482\) −764.000 −1.58506
\(483\) 0 0
\(484\) 1446.82 2.98930
\(485\) 579.161 1.19415
\(486\) 0 0
\(487\) −970.000 −1.99179 −0.995893 0.0905356i \(-0.971142\pi\)
−0.995893 + 0.0905356i \(0.971142\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −699.119 −1.42677
\(491\) 981.705 1.99940 0.999699 0.0245196i \(-0.00780562\pi\)
0.999699 + 0.0245196i \(0.00780562\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 374.823 0.755691
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1516.53 3.03306
\(501\) 0 0
\(502\) −940.000 −1.87251
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) −337.705 −0.668722
\(506\) 0 0
\(507\) 0 0
\(508\) −86.6476 −0.170566
\(509\) 615.309 1.20886 0.604429 0.796659i \(-0.293401\pi\)
0.604429 + 0.796659i \(0.293401\pi\)
\(510\) 0 0
\(511\) −326.898 −0.639723
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) −94.8528 −0.184180
\(516\) 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
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −771.529 −1.47238
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) −1696.47 −3.20089
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1201.84 −2.24643
\(536\) 0 0
\(537\) 0 0
\(538\) 860.000 1.59851
\(539\) 809.677 1.50218
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −875.381 −1.61509
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −2854.88 −5.19069
\(551\) 0 0
\(552\) 0 0
\(553\) 202.146 0.365545
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −94.5433 −0.169737 −0.0848683 0.996392i \(-0.527047\pi\)
−0.0848683 + 0.996392i \(0.527047\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −528.942 −0.944539
\(561\) 0 0
\(562\) 0 0
\(563\) −770.381 −1.36835 −0.684175 0.729318i \(-0.739838\pi\)
−0.684175 + 0.729318i \(0.739838\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 290.000 0.502600 0.251300 0.967909i \(-0.419142\pi\)
0.251300 + 0.967909i \(0.419142\pi\)
\(578\) 578.000 1.00000
\(579\) 0 0
\(580\) −1897.06 −3.27079
\(581\) −529.250 −0.910929
\(582\) 0 0
\(583\) 1964.75 3.37007
\(584\) 750.352 1.28485
\(585\) 0 0
\(586\) −772.000 −1.31741
\(587\) −1115.82 −1.90089 −0.950445 0.310893i \(-0.899372\pi\)
−0.950445 + 0.310893i \(0.899372\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 189.706 0.321535
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 817.588 1.37179
\(597\) 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
\(604\) 765.706 1.26772
\(605\) 3430.88 5.67088
\(606\) 0 0
\(607\) −730.000 −1.20264 −0.601318 0.799010i \(-0.705357\pi\)
−0.601318 + 0.799010i \(0.705357\pi\)
\(608\) 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
\(616\) 612.589 0.994462
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 888.824 1.43359
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1971.91 3.15506
\(626\) −1107.00 −1.76837
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −831.132 −1.31717 −0.658583 0.752508i \(-0.728844\pi\)
−0.658583 + 0.752508i \(0.728844\pi\)
\(632\) −464.000 −0.734177
\(633\) 0 0
\(634\) 599.381 0.945396
\(635\) −205.469 −0.323574
\(636\) 0 0
\(637\) 0 0
\(638\) 2197.06 3.44366
\(639\) 0 0
\(640\) 1214.12 1.89706
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −219.706 −0.338529
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 218.249 0.334225 0.167112 0.985938i \(-0.446556\pi\)
0.167112 + 0.985938i \(0.446556\pi\)
\(654\) 0 0
\(655\) −1829.54 −2.79319
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 585.352 0.888242 0.444121 0.895967i \(-0.353516\pi\)
0.444121 + 0.895967i \(0.353516\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 1214.82 1.82955
\(665\) 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 \(-0.788250\pi\)
−0.786774 + 0.617241i \(0.788250\pi\)
\(674\) −380.000 −0.563798
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) −1346.00 −1.98818 −0.994092 0.108545i \(-0.965381\pi\)
−0.994092 + 0.108545i \(0.965381\pi\)
\(678\) 0 0
\(679\) −212.807 −0.313413
\(680\) 0 0
\(681\) 0 0
\(682\) −1029.38 −1.50936
\(683\) −1334.00 −1.95315 −0.976574 0.215182i \(-0.930965\pi\)
−0.976574 + 0.215182i \(0.930965\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 598.442 0.872365
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) −1279.94 −1.84962
\(693\) 0 0
\(694\) 1081.94 1.55899
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −905.763 −1.29395
\(701\) −1188.40 −1.69529 −0.847643 0.530567i \(-0.821979\pi\)
−0.847643 + 0.530567i \(0.821979\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1406.12 −1.99732
\(705\) 0 0
\(706\) 0 0
\(707\) 124.087 0.175511
\(708\) 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
\(716\) 1410.35 1.96976
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 34.8528 0.0483395
\(722\) 722.000 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) −3248.53 −4.48073
\(726\) 0 0
\(727\) 818.338 1.12564 0.562818 0.826581i \(-0.309717\pi\)
0.562818 + 0.826581i \(0.309717\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1779.32 2.43743
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −1032.44 −1.40659
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 623.352 0.840098
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1938.76 2.60237
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 441.605 0.589592
\(750\) 0 0
\(751\) 234.310 0.311997 0.155998 0.987757i \(-0.450141\pi\)
0.155998 + 0.987757i \(0.450141\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1815.73 2.40494
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 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 \(0.643955\pi\)
\(770\) 1452.64 1.88655
\(771\) 0 0
\(772\) 302.469 0.391800
\(773\) −1154.00 −1.49288 −0.746442 0.665450i \(-0.768240\pi\)
−0.746442 + 0.665450i \(0.768240\pi\)
\(774\) 0 0
\(775\) 1522.03 1.96391
\(776\) 488.471 0.629473
\(777\) 0 0
\(778\) −1496.73 −1.92382
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −589.645 −0.752098
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −799.939 −1.01515
\(789\) 0 0
\(790\) −1100.29 −1.39278
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 1591.94 1.99992
\(797\) −1453.10 −1.82321 −0.911607 0.411063i \(-0.865158\pi\)
−0.911607 + 0.411063i \(0.865158\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 2079.06 2.59882
\(801\) 0 0
\(802\) 0 0
\(803\) −2060.71 −2.56626
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −284.824 −0.352505
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 697.056 0.858444
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 39.1775 0.0478942
\(819\) 0 0
\(820\) 0 0
\(821\) 670.000 0.816078 0.408039 0.912965i \(-0.366213\pi\)
0.408039 + 0.912965i \(0.366213\pi\)
\(822\) 0 0
\(823\) 1131.13 1.37440 0.687199 0.726469i \(-0.258840\pi\)
0.687199 + 0.726469i \(0.258840\pi\)
\(824\) −80.0000 −0.0970874
\(825\) 0 0
\(826\) −69.7056 −0.0843894
\(827\) 1546.00 1.86941 0.934704 0.355428i \(-0.115665\pi\)
0.934704 + 0.355428i \(0.115665\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 2880.73 3.47076
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 1460.00 1.74224
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1659.00 1.97265
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1603.01 1.89706
\(846\) 0 0
\(847\) −1260.65 −1.48837
\(848\) −1430.82 −1.68729
\(849\) 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
\(856\) −1013.65 −1.18416
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −3035.15 −3.50884
\(866\) 1620.35 1.87108
\(867\) 0 0
\(868\) −326.590 −0.376256
\(869\) 1274.29 1.46639
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1321.38 −1.51015
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) −1720.50 −1.95956
\(879\) 0 0
\(880\) −3334.35 −3.78904
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −172.000 −0.194131
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 75.4978 0.0849244
\(890\) 0 0
\(891\) 0 0
\(892\) 920.000 1.03139
\(893\) 0 0
\(894\) 0 0
\(895\) 3344.40 3.73675
\(896\) −446.116 −0.497897
\(897\) 0 0
\(898\) 0 0
\(899\) −1171.32 −1.30291
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 1384.00 1.52423
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −3336.29 −3.65421
\(914\) −1819.76 −1.99099
\(915\) 0 0
\(916\) 0 0
\(917\) 672.249 0.733096
\(918\) 0 0
\(919\) −1815.92 −1.97598 −0.987989 0.154523i \(-0.950616\pi\)
−0.987989 + 0.154523i \(0.950616\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −776.733 −0.842444
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −1294.32 −1.39776
\(927\) 0 0
\(928\) −1600.00 −1.72414
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 553.939 0.593083
\(935\) 0 0
\(936\) 0 0
\(937\) 239.293 0.255382 0.127691 0.991814i \(-0.459243\pi\)
0.127691 + 0.991814i \(0.459243\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1371.75 1.45776 0.728878 0.684644i \(-0.240042\pi\)
0.728878 + 0.684644i \(0.240042\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 160.000 0.169492
\(945\) 0 0
\(946\) 0 0
\(947\) −1840.94 −1.94397 −0.971985 0.235043i \(-0.924477\pi\)
−0.971985 + 0.235043i \(0.924477\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 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
\(964\) −1528.00 −1.58506
\(965\) 717.252 0.743266
\(966\) 0 0
\(967\) −1218.04 −1.25961 −0.629805 0.776753i \(-0.716865\pi\)
−0.629805 + 0.776753i \(0.716865\pi\)
\(968\) 2893.65 2.98930
\(969\) 0 0
\(970\) 1158.32 1.19415
\(971\) −778.324 −0.801569 −0.400785 0.916172i \(-0.631262\pi\)
−0.400785 + 0.916172i \(0.631262\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1940.00 −1.99179
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1398.24 −1.42677
\(981\) 0 0
\(982\) 1963.41 1.99940
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −1896.91 −1.92580
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1921.37 −1.93882 −0.969408 0.245457i \(-0.921062\pi\)
−0.969408 + 0.245457i \(0.921062\pi\)
\(992\) 749.645 0.755691
\(993\) 0 0
\(994\) 0 0
\(995\) 3775.00 3.79397
\(996\) 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 216.3.h.b.53.2 yes 2
3.2 odd 2 216.3.h.a.53.1 2
4.3 odd 2 864.3.h.b.593.2 2
8.3 odd 2 864.3.h.a.593.1 2
8.5 even 2 216.3.h.a.53.1 2
12.11 even 2 864.3.h.a.593.1 2
24.5 odd 2 CM 216.3.h.b.53.2 yes 2
24.11 even 2 864.3.h.b.593.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.h.a.53.1 2 3.2 odd 2
216.3.h.a.53.1 2 8.5 even 2
216.3.h.b.53.2 yes 2 1.1 even 1 trivial
216.3.h.b.53.2 yes 2 24.5 odd 2 CM
864.3.h.a.593.1 2 8.3 odd 2
864.3.h.a.593.1 2 12.11 even 2
864.3.h.b.593.2 2 4.3 odd 2
864.3.h.b.593.2 2 24.11 even 2