Properties

Label 1216.2.h.a.1215.2
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Defining polynomial: \(x^{2} - x + 5\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1215.2
Root \(0.500000 + 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.a.1215.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +4.35890i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{5} +4.35890i q^{7} -3.00000 q^{9} -4.35890i q^{11} +7.00000 q^{17} +4.35890i q^{19} +8.71780i q^{23} -4.00000 q^{25} +4.35890i q^{35} +13.0767i q^{43} -3.00000 q^{45} +4.35890i q^{47} -12.0000 q^{49} -4.35890i q^{55} -15.0000 q^{61} -13.0767i q^{63} +11.0000 q^{73} +19.0000 q^{77} +9.00000 q^{81} +8.71780i q^{83} +7.00000 q^{85} +4.35890i q^{95} +13.0767i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 6q^{9} + 14q^{17} - 8q^{25} - 6q^{45} - 24q^{49} - 30q^{61} + 22q^{73} + 38q^{77} + 18q^{81} + 14q^{85} + O(q^{100}) \)

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
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) 4.35890i 1.64751i 0.566947 + 0.823754i \(0.308125\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) − 4.35890i − 1.31426i −0.753778 0.657129i \(-0.771771\pi\)
0.753778 0.657129i \(-0.228229\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 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 \(0.363071\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 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
\(35\) 4.35890i 0.736788i
\(36\) 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
\(43\) 13.0767i 1.99418i 0.0762493 + 0.997089i \(0.475706\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) −3.00000 −0.447214
\(46\) 0 0
\(47\) 4.35890i 0.635811i 0.948122 + 0.317905i \(0.102979\pi\)
−0.948122 + 0.317905i \(0.897021\pi\)
\(48\) 0 0
\(49\) −12.0000 −1.71429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) − 4.35890i − 0.587754i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) 0 0
\(63\) − 13.0767i − 1.64751i
\(64\) 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
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 19.0000 2.16525
\(78\) 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 \(0.158802\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) 7.00000 0.759257
\(86\) 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
\(95\) 4.35890i 0.447214i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 13.0767i 1.31426i
\(100\) 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
\(115\) 8.71780i 0.812939i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 30.5123i 2.79706i
\(120\) 0 0
\(121\) −8.00000 −0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 21.7945i − 1.90419i −0.305796 0.952097i \(-0.598923\pi\)
0.305796 0.952097i \(-0.401077\pi\)
\(132\) 0 0
\(133\) −19.0000 −1.64751
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 23.0000 1.96502 0.982511 0.186203i \(-0.0596182\pi\)
0.982511 + 0.186203i \(0.0596182\pi\)
\(138\) 0 0
\(139\) − 21.7945i − 1.84858i −0.381685 0.924292i \(-0.624656\pi\)
0.381685 0.924292i \(-0.375344\pi\)
\(140\) 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
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −21.0000 −1.69775
\(154\) 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
\(161\) −38.0000 −2.99482
\(162\) 0 0
\(163\) 8.71780i 0.682831i 0.939913 + 0.341415i \(0.110906\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 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
\(175\) − 17.4356i − 1.31801i
\(176\) 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
\(187\) − 30.5123i − 2.23128i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.7945i 1.57699i 0.615038 + 0.788497i \(0.289140\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) 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
\(199\) − 13.0767i − 0.926982i −0.886102 0.463491i \(-0.846597\pi\)
0.886102 0.463491i \(-0.153403\pi\)
\(200\) 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
\(209\) 19.0000 1.31426
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 13.0767i 0.891823i
\(216\) 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
\(225\) 12.0000 0.800000
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 21.0000 1.38772 0.693860 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.00000 −0.0655122 −0.0327561 0.999463i \(-0.510428\pi\)
−0.0327561 + 0.999463i \(0.510428\pi\)
\(234\) 0 0
\(235\) 4.35890i 0.284343i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 30.5123i − 1.97368i −0.161712 0.986838i \(-0.551701\pi\)
0.161712 0.986838i \(-0.448299\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −12.0000 −0.766652
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 21.7945i − 1.37566i −0.725874 0.687828i \(-0.758564\pi\)
0.725874 0.687828i \(-0.241436\pi\)
\(252\) 0 0
\(253\) 38.0000 2.38904
\(254\) 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
\(263\) − 30.5123i − 1.88147i −0.339145 0.940734i \(-0.610138\pi\)
0.339145 0.940734i \(-0.389862\pi\)
\(264\) 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 \(-0.707810\pi\)
0.607457 0.794353i \(-0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.4356i 1.05141i
\(276\) 0 0
\(277\) 33.0000 1.98278 0.991389 0.130950i \(-0.0418029\pi\)
0.991389 + 0.130950i \(0.0418029\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 13.0767i 0.777329i 0.921379 + 0.388664i \(0.127063\pi\)
−0.921379 + 0.388664i \(0.872937\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 32.0000 1.88235
\(290\) 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
\(301\) −57.0000 −3.28543
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −15.0000 −0.858898
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.35890i 0.247170i 0.992334 + 0.123585i \(0.0394392\pi\)
−0.992334 + 0.123585i \(0.960561\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) − 13.0767i − 0.736788i
\(316\) 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
\(323\) 30.5123i 1.69775i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −19.0000 −1.04750
\(330\) 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
\(343\) − 21.7945i − 1.17679i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 4.35890i − 0.233998i −0.993132 0.116999i \(-0.962673\pi\)
0.993132 0.116999i \(-0.0373274\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 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
\(359\) 21.7945i 1.15027i 0.818059 + 0.575135i \(0.195050\pi\)
−0.818059 + 0.575135i \(0.804950\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) − 26.1534i − 1.36520i −0.730794 0.682598i \(-0.760850\pi\)
0.730794 0.682598i \(-0.239150\pi\)
\(368\) 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
\(385\) 19.0000 0.968330
\(386\) 0 0
\(387\) − 39.2301i − 1.99418i
\(388\) 0 0
\(389\) 25.0000 1.26755 0.633775 0.773517i \(-0.281504\pi\)
0.633775 + 0.773517i \(0.281504\pi\)
\(390\) 0 0
\(391\) 61.0246i 3.08615i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 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
\(405\) 9.00000 0.447214
\(406\) 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
\(415\) 8.71780i 0.427940i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i 0.977064 + 0.212946i \(0.0683059\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) − 13.0767i − 0.635811i
\(424\) 0 0
\(425\) −28.0000 −1.35820
\(426\) 0 0
\(427\) − 65.3835i − 3.16413i
\(428\) 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
\(441\) 36.0000 1.71429
\(442\) 0 0
\(443\) 30.5123i 1.44968i 0.688916 + 0.724841i \(0.258087\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) 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
\(457\) −17.0000 −0.795226 −0.397613 0.917553i \(-0.630161\pi\)
−0.397613 + 0.917553i \(0.630161\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 37.0000 1.72326 0.861631 0.507535i \(-0.169443\pi\)
0.861631 + 0.507535i \(0.169443\pi\)
\(462\) 0 0
\(463\) − 13.0767i − 0.607726i −0.952716 0.303863i \(-0.901724\pi\)
0.952716 0.303863i \(-0.0982765\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4.35890i − 0.201706i −0.994901 0.100853i \(-0.967843\pi\)
0.994901 0.100853i \(-0.0321571\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 57.0000 2.62086
\(474\) 0 0
\(475\) − 17.4356i − 0.800000i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 43.5890i 1.99163i 0.0913823 + 0.995816i \(0.470871\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 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 \(0.442223\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 13.0767i 0.587754i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 21.7945i − 0.975656i −0.872940 0.487828i \(-0.837789\pi\)
0.872940 0.487828i \(-0.162211\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.71780i 0.388707i 0.980932 + 0.194354i \(0.0622609\pi\)
−0.980932 + 0.194354i \(0.937739\pi\)
\(504\) 0 0
\(505\) 10.0000 0.444994
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 47.9479i 2.12109i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 19.0000 0.835619
\(518\) 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
\(539\) 52.3068i 2.25301i
\(540\) 0 0
\(541\) 25.0000 1.07483 0.537417 0.843317i \(-0.319400\pi\)
0.537417 + 0.843317i \(0.319400\pi\)
\(542\) 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
\(549\) 45.0000 1.92055
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −47.0000 −1.99145 −0.995727 0.0923462i \(-0.970563\pi\)
−0.995727 + 0.0923462i \(0.970563\pi\)
\(558\) 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
\(567\) 39.2301i 1.64751i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) − 26.1534i − 1.09449i −0.836974 0.547243i \(-0.815677\pi\)
0.836974 0.547243i \(-0.184323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 34.8712i − 1.45423i
\(576\) 0 0
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −38.0000 −1.57651
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 47.9479i 1.97902i 0.144460 + 0.989511i \(0.453855\pi\)
−0.144460 + 0.989511i \(0.546145\pi\)
\(588\) 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
\(595\) 30.5123i 1.25088i
\(596\) 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
\(605\) −8.00000 −0.325246
\(606\) 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
\(613\) −39.0000 −1.57520 −0.787598 0.616190i \(-0.788675\pi\)
−0.787598 + 0.616190i \(0.788675\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.0000 −0.523360 −0.261680 0.965155i \(-0.584277\pi\)
−0.261680 + 0.965155i \(0.584277\pi\)
\(618\) 0 0
\(619\) 43.5890i 1.75199i 0.482321 + 0.875995i \(0.339794\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 47.9479i − 1.90878i −0.298570 0.954388i \(-0.596510\pi\)
0.298570 0.954388i \(-0.403490\pi\)
\(632\) 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
\(643\) 13.0767i 0.515695i 0.966186 + 0.257847i \(0.0830131\pi\)
−0.966186 + 0.257847i \(0.916987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 47.9479i − 1.88503i −0.334169 0.942513i \(-0.608456\pi\)
0.334169 0.942513i \(-0.391544\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 41.0000 1.60445 0.802227 0.597019i \(-0.203648\pi\)
0.802227 + 0.597019i \(0.203648\pi\)
\(654\) 0 0
\(655\) − 21.7945i − 0.851581i
\(656\) 0 0
\(657\) −33.0000 −1.28745
\(658\) 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
\(665\) −19.0000 −0.736788
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 65.3835i 2.52410i
\(672\) 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
\(685\) 23.0000 0.878785
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 39.2301i − 1.49238i −0.665731 0.746191i \(-0.731880\pi\)
0.665731 0.746191i \(-0.268120\pi\)
\(692\) 0 0
\(693\) −57.0000 −2.16525
\(694\) 0 0
\(695\) − 21.7945i − 0.826712i
\(696\) 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
\(707\) 43.5890i 1.63933i
\(708\) 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
\(719\) 21.7945i 0.812798i 0.913696 + 0.406399i \(0.133216\pi\)
−0.913696 + 0.406399i \(0.866784\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 39.2301i 1.45496i 0.686127 + 0.727482i \(0.259309\pi\)
−0.686127 + 0.727482i \(0.740691\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 91.5369i 3.38561i
\(732\) 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
\(739\) 30.5123i 1.12241i 0.827676 + 0.561206i \(0.189663\pi\)
−0.827676 + 0.561206i \(0.810337\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −11.0000 −0.403009
\(746\) 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
\(757\) −27.0000 −0.981332 −0.490666 0.871348i \(-0.663246\pi\)
−0.490666 + 0.871348i \(0.663246\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 55.0000 1.99375 0.996874 0.0790050i \(-0.0251743\pi\)
0.996874 + 0.0790050i \(0.0251743\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −21.0000 −0.759257
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 51.0000 1.83911 0.919554 0.392965i \(-0.128551\pi\)
0.919554 + 0.392965i \(0.128551\pi\)
\(770\) 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
\(785\) 18.0000 0.642448
\(786\) 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
\(799\) 30.5123i 1.07945i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 47.9479i − 1.69204i
\(804\) 0 0
\(805\) −38.0000 −1.33932
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.71780i 0.305371i
\(816\) 0 0
\(817\) −57.0000 −1.99418
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 53.0000 1.84971 0.924856 0.380317i \(-0.124185\pi\)
0.924856 + 0.380317i \(0.124185\pi\)
\(822\) 0 0
\(823\) 56.6657i 1.97524i 0.156860 + 0.987621i \(0.449863\pi\)
−0.156860 + 0.987621i \(0.550137\pi\)
\(824\) 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
\(833\) −84.0000 −2.91043
\(834\) 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
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) − 34.8712i − 1.19819i
\(848\) 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
\(855\) − 13.0767i − 0.447214i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) − 56.6657i − 1.93341i −0.255897 0.966704i \(-0.582371\pi\)
0.255897 0.966704i \(-0.417629\pi\)
\(860\) 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
\(875\) − 39.2301i − 1.32622i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 35.0000 1.17918 0.589590 0.807703i \(-0.299289\pi\)
0.589590 + 0.807703i \(0.299289\pi\)
\(882\) 0 0
\(883\) 30.5123i 1.02682i 0.858143 + 0.513410i \(0.171618\pi\)
−0.858143 + 0.513410i \(0.828382\pi\)
\(884\) 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
\(891\) − 39.2301i − 1.31426i
\(892\) 0 0
\(893\) −19.0000 −0.635811
\(894\) 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
\(913\) 38.0000 1.25762
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 95.0000 3.13718
\(918\) 0 0
\(919\) 8.71780i 0.287574i 0.989609 + 0.143787i \(0.0459280\pi\)
−0.989609 + 0.143787i \(0.954072\pi\)
\(920\) 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
\(931\) − 52.3068i − 1.71429i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 30.5123i − 0.997859i
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) 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 \(-0.541492\pi\)
0.129983 0.991516i \(-0.458508\pi\)
\(948\) 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
\(955\) 21.7945i 0.705253i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 100.255i 3.23739i
\(960\) 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 \(-0.561805\pi\)
0.192947 0.981209i \(-0.438195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 95.0000 3.04556
\(974\) 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
\(985\) −22.0000 −0.700978
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −114.000 −3.62499
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 13.0767i − 0.414559i
\(996\) 0 0
\(997\) −63.0000 −1.99523 −0.997615 0.0690239i \(-0.978012\pi\)
−0.997615 + 0.0690239i \(0.978012\pi\)
\(998\) 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.a.1215.2 2
4.3 odd 2 inner 1216.2.h.a.1215.1 2
8.3 odd 2 304.2.h.a.303.1 2
8.5 even 2 304.2.h.a.303.2 yes 2
19.18 odd 2 CM 1216.2.h.a.1215.2 2
24.5 odd 2 2736.2.k.g.2431.2 2
24.11 even 2 2736.2.k.g.2431.1 2
76.75 even 2 inner 1216.2.h.a.1215.1 2
152.37 odd 2 304.2.h.a.303.2 yes 2
152.75 even 2 304.2.h.a.303.1 2
456.227 odd 2 2736.2.k.g.2431.1 2
456.341 even 2 2736.2.k.g.2431.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
304.2.h.a.303.1 2 8.3 odd 2
304.2.h.a.303.1 2 152.75 even 2
304.2.h.a.303.2 yes 2 8.5 even 2
304.2.h.a.303.2 yes 2 152.37 odd 2
1216.2.h.a.1215.1 2 4.3 odd 2 inner
1216.2.h.a.1215.1 2 76.75 even 2 inner
1216.2.h.a.1215.2 2 1.1 even 1 trivial
1216.2.h.a.1215.2 2 19.18 odd 2 CM
2736.2.k.g.2431.1 2 24.11 even 2
2736.2.k.g.2431.1 2 456.227 odd 2
2736.2.k.g.2431.2 2 24.5 odd 2
2736.2.k.g.2431.2 2 456.341 even 2