Properties

Label 1216.2.b.e.607.4
Level $1216$
Weight $2$
Character 1216.607
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
CM discriminant -19
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 607.4
Root \(-1.52274 - 1.63746i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.e.607.5

$q$-expansion

\(f(q)\) \(=\) \(q-1.31342i q^{5} +2.27492i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.31342i q^{5} +2.27492i q^{7} +3.00000 q^{9} -2.15068 q^{11} +0.274917 q^{17} +4.35890 q^{19} +4.00000i q^{23} +3.27492 q^{25} +2.98793 q^{35} +7.40437 q^{43} -3.94027i q^{45} +10.2749i q^{47} +1.82475 q^{49} +2.82475i q^{55} -15.1698i q^{61} +6.82475i q^{63} +16.8248 q^{73} -4.89261i q^{77} +9.00000 q^{81} +8.71780 q^{83} -0.361083i q^{85} -5.72508i q^{95} -6.45203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 28q^{17} - 4q^{25} - 76q^{49} + 44q^{73} + 72q^{81} + O(q^{100}) \)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) − 1.31342i − 0.587381i −0.955901 0.293691i \(-0.905116\pi\)
0.955901 0.293691i \(-0.0948835\pi\)
\(6\) 0 0
\(7\) 2.27492i 0.859838i 0.902867 + 0.429919i \(0.141458\pi\)
−0.902867 + 0.429919i \(0.858542\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −2.15068 −0.648454 −0.324227 0.945979i \(-0.605104\pi\)
−0.324227 + 0.945979i \(0.605104\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 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
\(19\) 4.35890 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 3.27492 0.654983
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.98793 0.505053
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 7.40437 1.12916 0.564578 0.825380i \(-0.309039\pi\)
0.564578 + 0.825380i \(0.309039\pi\)
\(44\) 0 0
\(45\) − 3.94027i − 0.587381i
\(46\) 0 0
\(47\) 10.2749i 1.49875i 0.662145 + 0.749375i \(0.269646\pi\)
−0.662145 + 0.749375i \(0.730354\pi\)
\(48\) 0 0
\(49\) 1.82475 0.260679
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 2.82475i 0.380889i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 15.1698i − 1.94230i −0.238474 0.971149i \(-0.576647\pi\)
0.238474 0.971149i \(-0.423353\pi\)
\(62\) 0 0
\(63\) 6.82475i 0.859838i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 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 \(-0.0559458\pi\)
0.984594 + 0.174855i \(0.0559458\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 4.89261i − 0.557565i
\(78\) 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
\(83\) 8.71780 0.956903 0.478451 0.878114i \(-0.341198\pi\)
0.478451 + 0.878114i \(0.341198\pi\)
\(84\) 0 0
\(85\) − 0.361083i − 0.0391649i
\(86\) 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
\(95\) − 5.72508i − 0.587381i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) −6.45203 −0.648454
\(100\) 0 0
\(101\) 17.4356i 1.73491i 0.497519 + 0.867453i \(0.334245\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 5.25370 0.489910
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.625414i 0.0573316i
\(120\) 0 0
\(121\) −6.37459 −0.579508
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 10.8685i − 0.972106i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −16.9594 −1.48175 −0.740876 0.671642i \(-0.765589\pi\)
−0.740876 + 0.671642i \(0.765589\pi\)
\(132\) 0 0
\(133\) 9.91613i 0.859838i
\(134\) 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
\(139\) −3.10302 −0.263195 −0.131597 0.991303i \(-0.542011\pi\)
−0.131597 + 0.991303i \(0.542011\pi\)
\(140\) 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
\(149\) 20.4235i 1.67316i 0.547844 + 0.836580i \(0.315449\pi\)
−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
\(157\) − 17.4356i − 1.39151i −0.718278 0.695756i \(-0.755069\pi\)
0.718278 0.695756i \(-0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.09967 −0.717154
\(162\) 0 0
\(163\) −8.71780 −0.682831 −0.341415 0.939913i \(-0.610906\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 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
\(171\) 13.0767 1.00000
\(172\) 0 0
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 7.45017i 0.563180i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.591258 −0.0432371
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 27.3746i − 1.98076i −0.138390 0.990378i \(-0.544193\pi\)
0.138390 0.990378i \(-0.455807\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.4356i 1.24223i 0.783718 + 0.621117i \(0.213321\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) − 1.17525i − 0.0833111i −0.999132 0.0416556i \(-0.986737\pi\)
0.999132 0.0416556i \(-0.0132632\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 12.0000i 0.834058i
\(208\) 0 0
\(209\) −9.37459 −0.648454
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 9.72508i − 0.663245i
\(216\) 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
\(225\) 9.82475 0.654983
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) − 7.28929i − 0.481690i −0.970564 0.240845i \(-0.922576\pi\)
0.970564 0.240845i \(-0.0774245\pi\)
\(230\) 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
\(235\) 13.4953 0.880338
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 23.9244i − 1.54754i −0.633465 0.773771i \(-0.718368\pi\)
0.633465 0.773771i \(-0.281632\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.39667i − 0.153118i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −30.8158 −1.94508 −0.972539 0.232740i \(-0.925231\pi\)
−0.972539 + 0.232740i \(0.925231\pi\)
\(252\) 0 0
\(253\) − 8.60271i − 0.540848i
\(254\) 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
\(263\) − 31.9244i − 1.96854i −0.176659 0.984272i \(-0.556529\pi\)
0.176659 0.984272i \(-0.443471\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) − 20.0000i − 1.21491i −0.794353 0.607457i \(-0.792190\pi\)
0.794353 0.607457i \(-0.207810\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.04329 −0.424726
\(276\) 0 0
\(277\) 26.3994i 1.58619i 0.609101 + 0.793093i \(0.291530\pi\)
−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
\(283\) −20.3084 −1.20721 −0.603606 0.797283i \(-0.706270\pi\)
−0.603606 + 0.797283i \(0.706270\pi\)
\(284\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 16.8443i 0.970891i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −19.9244 −1.14087
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 13.7251i − 0.778278i −0.921179 0.389139i \(-0.872773\pi\)
0.921179 0.389139i \(-0.127227\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 8.96379 0.505053
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.19834 0.0666772
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −23.3746 −1.28868
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 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
\(343\) 20.0756i 1.08398i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −29.8635 −1.60316 −0.801578 0.597890i \(-0.796006\pi\)
−0.801578 + 0.597890i \(0.796006\pi\)
\(348\) 0 0
\(349\) 23.7725i 1.27251i 0.771477 + 0.636257i \(0.219518\pi\)
−0.771477 + 0.636257i \(0.780482\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 3.37459i − 0.178104i −0.996027 0.0890519i \(-0.971616\pi\)
0.996027 0.0890519i \(-0.0283837\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 22.0980i − 1.15666i
\(366\) 0 0
\(367\) 28.0000i 1.46159i 0.682598 + 0.730794i \(0.260850\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −6.42608 −0.327503
\(386\) 0 0
\(387\) 22.2131 1.12916
\(388\) 0 0
\(389\) 36.9068i 1.87125i 0.352998 + 0.935624i \(0.385162\pi\)
−0.352998 + 0.935624i \(0.614838\pi\)
\(390\) 0 0
\(391\) 1.09967i 0.0556126i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 25.6772i − 1.28870i −0.764730 0.644351i \(-0.777127\pi\)
0.764730 0.644351i \(-0.222873\pi\)
\(398\) 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
\(405\) − 11.8208i − 0.587381i
\(406\) 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
\(415\) − 11.4502i − 0.562067i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8.71780 −0.425892 −0.212946 0.977064i \(-0.568306\pi\)
−0.212946 + 0.977064i \(0.568306\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 30.8248i 1.49875i
\(424\) 0 0
\(425\) 0.900331 0.0436725
\(426\) 0 0
\(427\) 34.5101 1.67006
\(428\) 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
\(437\) 17.4356i 0.834058i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 5.47425 0.260679
\(442\) 0 0
\(443\) −40.3709 −1.91808 −0.959039 0.283273i \(-0.908580\pi\)
−0.959039 + 0.283273i \(0.908580\pi\)
\(444\) 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
\(461\) − 21.1457i − 0.984853i −0.870354 0.492427i \(-0.836110\pi\)
0.870354 0.492427i \(-0.163890\pi\)
\(462\) 0 0
\(463\) − 9.17525i − 0.426410i −0.977007 0.213205i \(-0.931610\pi\)
0.977007 0.213205i \(-0.0683902\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 39.4185 1.82407 0.912036 0.410110i \(-0.134510\pi\)
0.912036 + 0.410110i \(0.134510\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −15.9244 −0.732206
\(474\) 0 0
\(475\) 14.2750 0.654983
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 4.00000i − 0.182765i −0.995816 0.0913823i \(-0.970871\pi\)
0.995816 0.0913823i \(-0.0291285\pi\)
\(480\) 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
\(491\) 43.5890 1.96714 0.983572 0.180517i \(-0.0577772\pi\)
0.983572 + 0.180517i \(0.0577772\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 8.47425i 0.380889i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −44.6722 −1.99980 −0.999902 0.0139987i \(-0.995544\pi\)
−0.999902 + 0.0139987i \(0.995544\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 44.0000i − 1.96186i −0.194354 0.980932i \(-0.562261\pi\)
0.194354 0.980932i \(-0.437739\pi\)
\(504\) 0 0
\(505\) 22.9003 1.01905
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 38.2749i 1.69318i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 22.0980i − 0.971870i
\(518\) 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 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 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
\(539\) −3.92445 −0.169038
\(540\) 0 0
\(541\) 2.03559i 0.0875168i 0.999042 + 0.0437584i \(0.0139332\pi\)
−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 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) − 45.5095i − 1.94230i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 42.8826i − 1.81700i −0.417889 0.908498i \(-0.637230\pi\)
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 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 20.4743i 0.859838i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −26.1534 −1.09449 −0.547243 0.836974i \(-0.684323\pi\)
−0.547243 + 0.836974i \(0.684323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.0997i 0.546294i
\(576\) 0 0
\(577\) 43.0241 1.79112 0.895558 0.444945i \(-0.146777\pi\)
0.895558 + 0.444945i \(0.146777\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 19.8323i 0.822781i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.9118 0.739298 0.369649 0.929172i \(-0.379478\pi\)
0.369649 + 0.929172i \(0.379478\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 0.821434 0.0336755
\(596\) 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
\(605\) 8.37253i 0.340392i
\(606\) 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
\(613\) − 18.5188i − 0.747969i −0.927435 0.373985i \(-0.877991\pi\)
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 \(-0.250943\pi\)
0.705008 + 0.709199i \(0.250943\pi\)
\(618\) 0 0
\(619\) 43.5890 1.75199 0.875995 0.482321i \(-0.160206\pi\)
0.875995 + 0.482321i \(0.160206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.09967 0.0839868
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 49.0241i 1.95162i 0.218624 + 0.975809i \(0.429843\pi\)
−0.218624 + 0.975809i \(0.570157\pi\)
\(632\) 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
\(643\) 48.9736 1.93133 0.965665 0.259791i \(-0.0836535\pi\)
0.965665 + 0.259791i \(0.0836535\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0241i 1.29831i 0.760656 + 0.649155i \(0.224878\pi\)
−0.760656 + 0.649155i \(0.775122\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 50.7632i 1.98652i 0.115920 + 0.993259i \(0.463018\pi\)
−0.115920 + 0.993259i \(0.536982\pi\)
\(654\) 0 0
\(655\) 22.2749i 0.870353i
\(656\) 0 0
\(657\) 50.4743 1.96919
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13.0241 0.505053
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 32.6254i 1.25949i
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 10.1463i 0.387671i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 49.9259 1.89927 0.949636 0.313355i \(-0.101453\pi\)
0.949636 + 0.313355i \(0.101453\pi\)
\(692\) 0 0
\(693\) − 14.6778i − 0.557565i
\(694\) 0 0
\(695\) 4.07558i 0.154596i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 17.4356i − 0.658533i −0.944237 0.329267i \(-0.893198\pi\)
0.944237 0.329267i \(-0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −39.6645 −1.49174
\(708\) 0 0
\(709\) − 52.3068i − 1.96442i −0.187779 0.982211i \(-0.560129\pi\)
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
\(719\) − 43.3746i − 1.61760i −0.588084 0.808800i \(-0.700118\pi\)
0.588084 0.808800i \(-0.299882\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 52.4743i 1.94616i 0.230463 + 0.973081i \(0.425976\pi\)
−0.230463 + 0.973081i \(0.574024\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 2.03559 0.0752890
\(732\) 0 0
\(733\) 52.3068i 1.93200i 0.258551 + 0.965998i \(0.416755\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −54.2273 −1.99478 −0.997392 0.0721811i \(-0.977004\pi\)
−0.997392 + 0.0721811i \(0.977004\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 26.8248 0.982783
\(746\) 0 0
\(747\) 26.1534 0.956903
\(748\) 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
\(757\) − 0.591258i − 0.0214896i −0.999942 0.0107448i \(-0.996580\pi\)
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
\(765\) − 1.08325i − 0.0391649i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.62541 0.238919 0.119459 0.992839i \(-0.461884\pi\)
0.119459 + 0.992839i \(0.461884\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 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
\(785\) −22.9003 −0.817348
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 2.82475i 0.0999325i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −36.1846 −1.27693
\(804\) 0 0
\(805\) 11.9517i 0.421243i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −51.5739 −1.81324 −0.906621 0.421945i \(-0.861347\pi\)
−0.906621 + 0.421945i \(0.861347\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11.4502i 0.401082i
\(816\) 0 0
\(817\) 32.2749 1.12916
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 35.0021i − 1.22158i −0.791792 0.610791i \(-0.790852\pi\)
0.791792 0.610791i \(-0.209148\pi\)
\(822\) 0 0
\(823\) − 53.5739i − 1.86747i −0.357966 0.933735i \(-0.616529\pi\)
0.357966 0.933735i \(-0.383471\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.501656 0.0173813
\(834\) 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
\(845\) 17.0745i 0.587381i
\(846\) 0 0
\(847\) − 14.5017i − 0.498283i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 52.3068i − 1.79095i −0.445112 0.895475i \(-0.646836\pi\)
0.445112 0.895475i \(-0.353164\pi\)
\(854\) 0 0
\(855\) − 17.1752i − 0.587381i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −41.3232 −1.40993 −0.704965 0.709242i \(-0.749037\pi\)
−0.704965 + 0.709242i \(0.749037\pi\)
\(860\) 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
\(875\) 24.7249 0.835854
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 59.0241 1.98857 0.994286 0.106749i \(-0.0340440\pi\)
0.994286 + 0.106749i \(0.0340440\pi\)
\(882\) 0 0
\(883\) 28.9111 0.972938 0.486469 0.873698i \(-0.338285\pi\)
0.486469 + 0.873698i \(0.338285\pi\)
\(884\) 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
\(891\) −19.3561 −0.648454
\(892\) 0 0
\(893\) 44.7873i 1.49875i
\(894\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 52.3068i 1.73491i
\(910\) 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
\(917\) − 38.5813i − 1.27407i
\(918\) 0 0
\(919\) − 60.0000i − 1.97922i −0.143787 0.989609i \(-0.545928\pi\)
0.143787 0.989609i \(-0.454072\pi\)
\(920\) 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 \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 7.95391 0.260679
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0.776573i 0.0253966i
\(936\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 61.0246 1.98303 0.991516 0.129983i \(-0.0414921\pi\)
0.991516 + 0.129983i \(0.0414921\pi\)
\(948\) 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
\(955\) −35.9544 −1.16346
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 17.5739i − 0.567492i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 12.0000i − 0.385894i −0.981209 0.192947i \(-0.938195\pi\)
0.981209 0.192947i \(-0.0618045\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) − 7.05911i − 0.226305i
\(974\) 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
\(985\) 22.9003 0.729665
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29.6175i 0.941782i
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.54360 −0.0489354
\(996\) 0 0
\(997\) − 56.7390i − 1.79694i −0.439031 0.898472i \(-0.644678\pi\)
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.b.e.607.4 yes 8
4.3 odd 2 inner 1216.2.b.e.607.3 8
8.3 odd 2 inner 1216.2.b.e.607.5 yes 8
8.5 even 2 inner 1216.2.b.e.607.6 yes 8
19.18 odd 2 CM 1216.2.b.e.607.4 yes 8
76.75 even 2 inner 1216.2.b.e.607.3 8
152.37 odd 2 inner 1216.2.b.e.607.6 yes 8
152.75 even 2 inner 1216.2.b.e.607.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.b.e.607.3 8 4.3 odd 2 inner
1216.2.b.e.607.3 8 76.75 even 2 inner
1216.2.b.e.607.4 yes 8 1.1 even 1 trivial
1216.2.b.e.607.4 yes 8 19.18 odd 2 CM
1216.2.b.e.607.5 yes 8 8.3 odd 2 inner
1216.2.b.e.607.5 yes 8 152.75 even 2 inner
1216.2.b.e.607.6 yes 8 8.5 even 2 inner
1216.2.b.e.607.6 yes 8 152.37 odd 2 inner