Properties

Label 1216.2.h.b.1215.2
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1215.2
Root \(-1.63746 - 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.b.1215.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.27492 q^{5} +4.77753i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-4.27492 q^{5} +4.77753i q^{7} -3.00000 q^{9} +2.15068i q^{11} +0.274917 q^{17} -4.35890i q^{19} -8.71780i q^{23} +13.2749 q^{25} -20.4235i q^{35} +7.40437i q^{43} +12.8248 q^{45} -9.07888i q^{47} -15.8248 q^{49} -9.19397i q^{55} +3.72508 q^{61} -14.3326i q^{63} -16.8248 q^{73} -10.2749 q^{77} +9.00000 q^{81} -8.71780i q^{83} -1.17525 q^{85} +18.6339i q^{95} -6.45203i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 2q^{5} - 12q^{9} - 14q^{17} + 38q^{25} + 6q^{45} - 18q^{49} + 30q^{61} - 22q^{73} - 26q^{77} + 36q^{81} - 50q^{85} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −4.27492 −1.91180 −0.955901 0.293691i \(-0.905116\pi\)
−0.955901 + 0.293691i \(0.905116\pi\)
\(6\) 0 0
\(7\) 4.77753i 1.80573i 0.429919 + 0.902867i \(0.358542\pi\)
−0.429919 + 0.902867i \(0.641458\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 2.15068i 0.648454i 0.945979 + 0.324227i \(0.105104\pi\)
−0.945979 + 0.324227i \(0.894896\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.274917 0.0666772 0.0333386 0.999444i \(-0.489386\pi\)
0.0333386 + 0.999444i \(0.489386\pi\)
\(18\) 0 0
\(19\) − 4.35890i − 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 8.71780i − 1.81779i −0.417029 0.908893i \(-0.636929\pi\)
0.417029 0.908893i \(-0.363071\pi\)
\(24\) 0 0
\(25\) 13.2749 2.65498
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 20.4235i − 3.45221i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 7.40437i 1.12916i 0.825380 + 0.564578i \(0.190961\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) 0 0
\(45\) 12.8248 1.91180
\(46\) 0 0
\(47\) − 9.07888i − 1.32429i −0.749375 0.662145i \(-0.769646\pi\)
0.749375 0.662145i \(-0.230354\pi\)
\(48\) 0 0
\(49\) −15.8248 −2.26068
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) − 9.19397i − 1.23971i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 3.72508 0.476948 0.238474 0.971149i \(-0.423353\pi\)
0.238474 + 0.971149i \(0.423353\pi\)
\(62\) 0 0
\(63\) − 14.3326i − 1.80573i
\(64\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.8248 −1.96919 −0.984594 0.174855i \(-0.944054\pi\)
−0.984594 + 0.174855i \(0.944054\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −10.2749 −1.17094
\(78\) 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
\(83\) − 8.71780i − 0.956903i −0.878114 0.478451i \(-0.841198\pi\)
0.878114 0.478451i \(-0.158802\pi\)
\(84\) 0 0
\(85\) −1.17525 −0.127474
\(86\) 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
\(95\) 18.6339i 1.91180i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) − 6.45203i − 0.648454i
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 37.2679i 3.47525i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.31342i 0.120401i
\(120\) 0 0
\(121\) 6.37459 0.579508
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −35.3746 −3.16400
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 16.9594i − 1.48175i −0.671642 0.740876i \(-0.734411\pi\)
0.671642 0.740876i \(-0.265589\pi\)
\(132\) 0 0
\(133\) 20.8248 1.80573
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.72508 −0.659998 −0.329999 0.943981i \(-0.607048\pi\)
−0.329999 + 0.943981i \(0.607048\pi\)
\(138\) 0 0
\(139\) − 3.10302i − 0.263195i −0.991303 0.131597i \(-0.957989\pi\)
0.991303 0.131597i \(-0.0420106\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −13.3746 −1.09569 −0.547844 0.836580i \(-0.684551\pi\)
−0.547844 + 0.836580i \(0.684551\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −0.824752 −0.0666772
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 41.6495 3.28244
\(162\) 0 0
\(163\) − 8.71780i − 0.682831i −0.939913 0.341415i \(-0.889094\pi\)
0.939913 0.341415i \(-0.110906\pi\)
\(164\) 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
\(171\) 13.0767i 1.00000i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 63.4213i 4.79420i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.591258i 0.0432371i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 3.82518i − 0.276781i −0.990378 0.138390i \(-0.955807\pi\)
0.990378 0.138390i \(-0.0441928\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) − 28.1890i − 1.99826i −0.0416556 0.999132i \(-0.513263\pi\)
0.0416556 0.999132i \(-0.486737\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 26.1534i 1.81779i
\(208\) 0 0
\(209\) 9.37459 0.648454
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 31.6531i − 2.15872i
\(216\) 0 0
\(217\) 0 0
\(218\) 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
\(225\) −39.8248 −2.65498
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −29.3746 −1.94113 −0.970564 0.240845i \(-0.922576\pi\)
−0.970564 + 0.240845i \(0.922576\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −25.9244 −1.69837 −0.849183 0.528099i \(-0.822905\pi\)
−0.849183 + 0.528099i \(0.822905\pi\)
\(234\) 0 0
\(235\) 38.8115i 2.53178i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 19.5863i − 1.26693i −0.773771 0.633465i \(-0.781632\pi\)
0.773771 0.633465i \(-0.218368\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 67.6495 4.32197
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 30.8158i − 1.94508i −0.232740 0.972539i \(-0.574769\pi\)
0.232740 0.972539i \(-0.425231\pi\)
\(252\) 0 0
\(253\) 18.7492 1.17875
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 5.72987i − 0.353319i −0.984272 0.176659i \(-0.943471\pi\)
0.984272 0.176659i \(-0.0565291\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 26.1534i 1.58871i 0.607457 + 0.794353i \(0.292190\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 28.5501i 1.72163i
\(276\) 0 0
\(277\) −20.2749 −1.21820 −0.609101 0.793093i \(-0.708470\pi\)
−0.609101 + 0.793093i \(0.708470\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) − 20.3084i − 1.20721i −0.797283 0.603606i \(-0.793730\pi\)
0.797283 0.603606i \(-0.206270\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.9244 −0.995554
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −35.3746 −2.03896
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.9244 −0.911830
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 32.4903i 1.84236i 0.389139 + 0.921179i \(0.372773\pi\)
−0.389139 + 0.921179i \(0.627227\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 61.2706i 3.45221i
\(316\) 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
\(323\) − 1.19834i − 0.0666772i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 43.3746 2.39132
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 42.1605i − 2.27645i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 29.8635i 1.60316i 0.597890 + 0.801578i \(0.296006\pi\)
−0.597890 + 0.801578i \(0.703994\pi\)
\(348\) 0 0
\(349\) 28.8248 1.54295 0.771477 0.636257i \(-0.219518\pi\)
0.771477 + 0.636257i \(0.219518\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 37.7440i 1.99205i 0.0890519 + 0.996027i \(0.471616\pi\)
−0.0890519 + 0.996027i \(0.528384\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 71.9244 3.76470
\(366\) 0 0
\(367\) 26.1534i 1.36520i 0.730794 + 0.682598i \(0.239150\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 43.9244 2.23860
\(386\) 0 0
\(387\) − 22.2131i − 1.12916i
\(388\) 0 0
\(389\) 13.9244 0.705996 0.352998 0.935624i \(-0.385162\pi\)
0.352998 + 0.935624i \(0.385162\pi\)
\(390\) 0 0
\(391\) − 2.39667i − 0.121205i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −30.4743 −1.52946 −0.764730 0.644351i \(-0.777127\pi\)
−0.764730 + 0.644351i \(0.777127\pi\)
\(398\) 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
\(405\) −38.4743 −1.91180
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 37.2679i 1.82941i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 8.71780i − 0.425892i −0.977064 0.212946i \(-0.931694\pi\)
0.977064 0.212946i \(-0.0683059\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 27.2366i 1.32429i
\(424\) 0 0
\(425\) 3.64950 0.177027
\(426\) 0 0
\(427\) 17.7967i 0.861242i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −38.0000 −1.81779
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 47.4743 2.26068
\(442\) 0 0
\(443\) 40.3709i 1.91808i 0.283273 + 0.959039i \(0.408580\pi\)
−0.283273 + 0.959039i \(0.591420\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 42.4743 1.98686 0.993431 0.114433i \(-0.0365053\pi\)
0.993431 + 0.114433i \(0.0365053\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −37.3746 −1.74071 −0.870354 0.492427i \(-0.836110\pi\)
−0.870354 + 0.492427i \(0.836110\pi\)
\(462\) 0 0
\(463\) − 42.0454i − 1.95401i −0.213205 0.977007i \(-0.568390\pi\)
0.213205 0.977007i \(-0.431610\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 39.4185i − 1.82407i −0.410110 0.912036i \(-0.634510\pi\)
0.410110 0.912036i \(-0.365490\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −15.9244 −0.732206
\(474\) 0 0
\(475\) − 57.8640i − 2.65498i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 43.5890i − 1.99163i −0.0913823 0.995816i \(-0.529129\pi\)
0.0913823 0.995816i \(-0.470871\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 43.5890i − 1.96714i −0.180517 0.983572i \(-0.557777\pi\)
0.180517 0.983572i \(-0.442223\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 27.5819i 1.23971i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 44.6722i − 1.99980i −0.0139987 0.999902i \(-0.504456\pi\)
0.0139987 0.999902i \(-0.495544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 8.71780i − 0.388707i −0.980932 0.194354i \(-0.937739\pi\)
0.980932 0.194354i \(-0.0622609\pi\)
\(504\) 0 0
\(505\) −42.7492 −1.90231
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) − 80.3807i − 3.55583i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 19.5257 0.858741
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −53.0000 −2.30435
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 34.0339i − 1.46595i
\(540\) 0 0
\(541\) −46.4743 −1.99808 −0.999042 0.0437584i \(-0.986067\pi\)
−0.999042 + 0.0437584i \(0.986067\pi\)
\(542\) 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
\(549\) −11.1752 −0.476948
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 19.7251 0.835779 0.417889 0.908498i \(-0.362770\pi\)
0.417889 + 0.908498i \(0.362770\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 42.9977i 1.80573i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 26.1534i 1.09449i 0.836974 + 0.547243i \(0.184323\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 115.728i − 4.82619i
\(576\) 0 0
\(577\) −43.0241 −1.79112 −0.895558 0.444945i \(-0.853223\pi\)
−0.895558 + 0.444945i \(0.853223\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 41.6495 1.72791
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.9118i 0.739298i 0.929172 + 0.369649i \(0.120522\pi\)
−0.929172 + 0.369649i \(0.879478\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) − 5.61478i − 0.230184i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.2508 −1.10790
\(606\) 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
\(613\) 45.9244 1.85487 0.927435 0.373985i \(-0.122009\pi\)
0.927435 + 0.373985i \(0.122009\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.0241 −1.41002 −0.705008 0.709199i \(-0.749057\pi\)
−0.705008 + 0.709199i \(0.749057\pi\)
\(618\) 0 0
\(619\) − 43.5890i − 1.75199i −0.482321 0.875995i \(-0.660206\pi\)
0.482321 0.875995i \(-0.339794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 84.8488 3.39395
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 10.9836i − 0.437249i −0.975809 0.218624i \(-0.929843\pi\)
0.975809 0.218624i \(-0.0701569\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 48.9736i 1.93133i 0.259791 + 0.965665i \(0.416346\pi\)
−0.259791 + 0.965665i \(0.583654\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 38.6964i − 1.52131i −0.649155 0.760656i \(-0.724878\pi\)
0.649155 0.760656i \(-0.275122\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.92442 0.231840 0.115920 0.993259i \(-0.463018\pi\)
0.115920 + 0.993259i \(0.463018\pi\)
\(654\) 0 0
\(655\) 72.5001i 2.83281i
\(656\) 0 0
\(657\) 50.4743 1.96919
\(658\) 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
\(665\) −89.0241 −3.45221
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 8.01145i 0.309279i
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 33.0241 1.26179
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 49.9259i − 1.89927i −0.313355 0.949636i \(-0.601453\pi\)
0.313355 0.949636i \(-0.398547\pi\)
\(692\) 0 0
\(693\) 30.8248 1.17094
\(694\) 0 0
\(695\) 13.2651i 0.503176i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 47.7753i 1.79677i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 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
\(719\) − 31.5380i − 1.17617i −0.808800 0.588084i \(-0.799882\pi\)
0.808800 0.588084i \(-0.200118\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 12.4279i − 0.460925i −0.973081 0.230463i \(-0.925976\pi\)
0.973081 0.230463i \(-0.0740239\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 2.03559i 0.0752890i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 54.2273i 1.99478i 0.0721811 + 0.997392i \(0.477004\pi\)
−0.0721811 + 0.997392i \(0.522996\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 57.1752 2.09474
\(746\) 0 0
\(747\) 26.1534i 0.956903i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 55.0241 1.99988 0.999942 0.0107448i \(-0.00342025\pi\)
0.999942 + 0.0107448i \(0.00342025\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −23.7251 −0.860034 −0.430017 0.902821i \(-0.641492\pi\)
−0.430017 + 0.902821i \(0.641492\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 3.52575 0.127474
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.62541 −0.238919 −0.119459 0.992839i \(-0.538116\pi\)
−0.119459 + 0.992839i \(0.538116\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −76.9485 −2.74641
\(786\) 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
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) − 2.49594i − 0.0883000i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 36.1846i − 1.27693i
\(804\) 0 0
\(805\) −178.048 −6.27537
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 51.5739 1.81324 0.906621 0.421945i \(-0.138653\pi\)
0.906621 + 0.421945i \(0.138653\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 37.2679i 1.30544i
\(816\) 0 0
\(817\) 32.2749 1.12916
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.3746 −1.58358 −0.791792 0.610791i \(-0.790852\pi\)
−0.791792 + 0.610791i \(0.790852\pi\)
\(822\) 0 0
\(823\) 20.5386i 0.715931i 0.933735 + 0.357966i \(0.116529\pi\)
−0.933735 + 0.357966i \(0.883471\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.35050 −0.150736
\(834\) 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
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −55.5739 −1.91180
\(846\) 0 0
\(847\) 30.4547i 1.04644i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) − 55.9018i − 1.91180i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) − 41.3232i − 1.40993i −0.709242 0.704965i \(-0.750963\pi\)
0.709242 0.704965i \(-0.249037\pi\)
\(860\) 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
\(873\) 0 0
\(874\) 0 0
\(875\) − 169.003i − 5.71334i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −59.0241 −1.98857 −0.994286 0.106749i \(-0.965956\pi\)
−0.994286 + 0.106749i \(0.965956\pi\)
\(882\) 0 0
\(883\) − 28.9111i − 0.972938i −0.873698 0.486469i \(-0.838285\pi\)
0.873698 0.486469i \(-0.161715\pi\)
\(884\) 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
\(891\) 19.3561i 0.648454i
\(892\) 0 0
\(893\) −39.5739 −1.32429
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 18.7492 0.620507
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 81.0241 2.67565
\(918\) 0 0
\(919\) − 8.71780i − 0.287574i −0.989609 0.143787i \(-0.954072\pi\)
0.989609 0.143787i \(-0.0459280\pi\)
\(920\) 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
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 68.9785i 2.26068i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 2.52758i − 0.0826607i
\(936\) 0 0
\(937\) 10.4743 0.342179 0.171089 0.985255i \(-0.445271\pi\)
0.171089 + 0.985255i \(0.445271\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 61.0246i 1.98303i 0.129983 + 0.991516i \(0.458508\pi\)
−0.129983 + 0.991516i \(0.541492\pi\)
\(948\) 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
\(955\) 16.3523i 0.529150i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 36.9068i − 1.19178i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 61.0246i 1.96242i 0.192947 + 0.981209i \(0.438195\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 14.8248 0.475260
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 94.0482 2.99662
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 64.5498 2.05257
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 120.506i 3.82028i
\(996\) 0 0
\(997\) 27.7251 0.878062 0.439031 0.898472i \(-0.355322\pi\)
0.439031 + 0.898472i \(0.355322\pi\)
\(998\) 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 1216.2.h.b.1215.2 4
4.3 odd 2 inner 1216.2.h.b.1215.1 4
8.3 odd 2 304.2.h.c.303.3 4
8.5 even 2 304.2.h.c.303.4 yes 4
19.18 odd 2 CM 1216.2.h.b.1215.2 4
24.5 odd 2 2736.2.k.j.2431.2 4
24.11 even 2 2736.2.k.j.2431.1 4
76.75 even 2 inner 1216.2.h.b.1215.1 4
152.37 odd 2 304.2.h.c.303.4 yes 4
152.75 even 2 304.2.h.c.303.3 4
456.227 odd 2 2736.2.k.j.2431.1 4
456.341 even 2 2736.2.k.j.2431.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
304.2.h.c.303.3 4 8.3 odd 2
304.2.h.c.303.3 4 152.75 even 2
304.2.h.c.303.4 yes 4 8.5 even 2
304.2.h.c.303.4 yes 4 152.37 odd 2
1216.2.h.b.1215.1 4 4.3 odd 2 inner
1216.2.h.b.1215.1 4 76.75 even 2 inner
1216.2.h.b.1215.2 4 1.1 even 1 trivial
1216.2.h.b.1215.2 4 19.18 odd 2 CM
2736.2.k.j.2431.1 4 24.11 even 2
2736.2.k.j.2431.1 4 456.227 odd 2
2736.2.k.j.2431.2 4 24.5 odd 2
2736.2.k.j.2431.2 4 456.341 even 2