Properties

Label 1216.2.b.c.607.2
Level $1216$
Weight $2$
Character 1216.607
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 607.2
Root \(2.17945 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.c.607.3

$q$-expansion

\(f(q)\) \(=\) \(q-4.35890i q^{5} +3.00000i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-4.35890i q^{5} +3.00000i q^{7} +3.00000 q^{9} +4.35890 q^{11} +7.00000 q^{17} -4.35890 q^{19} +4.00000i q^{23} -14.0000 q^{25} +13.0767 q^{35} +13.0767 q^{43} -13.0767i q^{45} -13.0000i q^{47} -2.00000 q^{49} -19.0000i q^{55} -4.35890i q^{61} +9.00000i q^{63} -11.0000 q^{73} +13.0767i q^{77} +9.00000 q^{81} -8.71780 q^{83} -30.5123i q^{85} +19.0000i q^{95} +13.0767 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} + 28q^{17} - 56q^{25} - 8q^{49} - 44q^{73} + 36q^{81} + O(q^{100}) \)

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
\(5\) − 4.35890i − 1.94936i −0.223607 0.974679i \(-0.571783\pi\)
0.223607 0.974679i \(-0.428217\pi\)
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 4.35890 1.31426 0.657129 0.753778i \(-0.271771\pi\)
0.657129 + 0.753778i \(0.271771\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000 1.69775 0.848875 0.528594i \(-0.177281\pi\)
0.848875 + 0.528594i \(0.177281\pi\)
\(18\) 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
\(25\) −14.0000 −2.80000
\(26\) 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
\(35\) 13.0767 2.21037
\(36\) 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
\(43\) 13.0767 1.99418 0.997089 0.0762493i \(-0.0242945\pi\)
0.997089 + 0.0762493i \(0.0242945\pi\)
\(44\) 0 0
\(45\) − 13.0767i − 1.94936i
\(46\) 0 0
\(47\) − 13.0000i − 1.89624i −0.317905 0.948122i \(-0.602979\pi\)
0.317905 0.948122i \(-0.397021\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) − 19.0000i − 2.56196i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) − 4.35890i − 0.558100i −0.960277 0.279050i \(-0.909981\pi\)
0.960277 0.279050i \(-0.0900195\pi\)
\(62\) 0 0
\(63\) 9.00000i 1.13389i
\(64\) 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
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.0767i 1.49023i
\(78\) 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 \(-0.658802\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) − 30.5123i − 3.30952i
\(86\) 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
\(95\) 19.0000i 1.94936i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 13.0767 1.31426
\(100\) 0 0
\(101\) − 17.4356i − 1.73491i −0.497519 0.867453i \(-0.665755\pi\)
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 1.00000 \(0\)
−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
\(115\) 17.4356 1.62588
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 21.0000i 1.92507i
\(120\) 0 0
\(121\) 8.00000 0.727273
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 39.2301i 3.50885i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −21.7945 −1.90419 −0.952097 0.305796i \(-0.901077\pi\)
−0.952097 + 0.305796i \(0.901077\pi\)
\(132\) 0 0
\(133\) − 13.0767i − 1.13389i
\(134\) 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
\(139\) −21.7945 −1.84858 −0.924292 0.381685i \(-0.875344\pi\)
−0.924292 + 0.381685i \(0.875344\pi\)
\(140\) 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
\(149\) 21.7945i 1.78547i 0.450578 + 0.892737i \(0.351218\pi\)
−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
\(157\) 17.4356i 1.39151i 0.718278 + 0.695756i \(0.244931\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 8.71780 0.682831 0.341415 0.939913i \(-0.389094\pi\)
0.341415 + 0.939913i \(0.389094\pi\)
\(164\) 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
\(175\) − 42.0000i − 3.17490i
\(176\) 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
\(187\) 30.5123 2.23128
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.0000i 1.23008i 0.788497 + 0.615038i \(0.210860\pi\)
−0.788497 + 0.615038i \(0.789140\pi\)
\(192\) 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 \(-0.786679\pi\)
0.783718 0.621117i \(-0.213321\pi\)
\(198\) 0 0
\(199\) 25.0000i 1.77220i 0.463491 + 0.886102i \(0.346597\pi\)
−0.463491 + 0.886102i \(0.653403\pi\)
\(200\) 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
\(209\) −19.0000 −1.31426
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 57.0000i − 3.88737i
\(216\) 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
\(225\) −42.0000 −2.80000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 21.7945i 1.44022i 0.693860 + 0.720110i \(0.255909\pi\)
−0.693860 + 0.720110i \(0.744091\pi\)
\(230\) 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
\(235\) −56.6657 −3.69646
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 5.00000i − 0.323423i −0.986838 0.161712i \(-0.948299\pi\)
0.986838 0.161712i \(-0.0517014\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.71780i 0.556960i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.7945 −1.37566 −0.687828 0.725874i \(-0.741436\pi\)
−0.687828 + 0.725874i \(0.741436\pi\)
\(252\) 0 0
\(253\) 17.4356i 1.09617i
\(254\) 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
\(263\) 11.0000i 0.678289i 0.940734 + 0.339145i \(0.110138\pi\)
−0.940734 + 0.339145i \(0.889862\pi\)
\(264\) 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
\(275\) −61.0246 −3.67992
\(276\) 0 0
\(277\) − 4.35890i − 0.261901i −0.991389 0.130950i \(-0.958197\pi\)
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
\(283\) 13.0767 0.777329 0.388664 0.921379i \(-0.372937\pi\)
0.388664 + 0.921379i \(0.372937\pi\)
\(284\) 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 1.00000i \(-0.5\pi\)
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
\(301\) 39.2301i 2.26118i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −19.0000 −1.08794
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 35.0000i 1.98467i 0.123585 + 0.992334i \(0.460561\pi\)
−0.123585 + 0.992334i \(0.539439\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) 39.2301 2.21037
\(316\) 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
\(323\) −30.5123 −1.69775
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 39.0000 2.15014
\(330\) 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
\(343\) 15.0000i 0.809924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.35890 0.233998 0.116999 0.993132i \(-0.462673\pi\)
0.116999 + 0.993132i \(0.462673\pi\)
\(348\) 0 0
\(349\) − 13.0767i − 0.699980i −0.936754 0.349990i \(-0.886185\pi\)
0.936754 0.349990i \(-0.113815\pi\)
\(350\) 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
\(359\) − 31.0000i − 1.63612i −0.575135 0.818059i \(-0.695050\pi\)
0.575135 0.818059i \(-0.304950\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 47.9479i 2.50971i
\(366\) 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
\(385\) 57.0000 2.90499
\(386\) 0 0
\(387\) 39.2301 1.99418
\(388\) 0 0
\(389\) 30.5123i 1.54703i 0.633775 + 0.773517i \(0.281504\pi\)
−0.633775 + 0.773517i \(0.718496\pi\)
\(390\) 0 0
\(391\) 28.0000i 1.41602i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 39.2301i − 1.96890i −0.175660 0.984451i \(-0.556206\pi\)
0.175660 0.984451i \(-0.443794\pi\)
\(398\) 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
\(405\) − 39.2301i − 1.94936i
\(406\) 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
\(415\) 38.0000i 1.86535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780 0.425892 0.212946 0.977064i \(-0.431694\pi\)
0.212946 + 0.977064i \(0.431694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) − 39.0000i − 1.89624i
\(424\) 0 0
\(425\) −98.0000 −4.75370
\(426\) 0 0
\(427\) 13.0767 0.632826
\(428\) 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
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −30.5123 −1.44968 −0.724841 0.688916i \(-0.758087\pi\)
−0.724841 + 0.688916i \(0.758087\pi\)
\(444\) 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
\(461\) 21.7945i 1.01507i 0.861631 + 0.507535i \(0.169443\pi\)
−0.861631 + 0.507535i \(0.830557\pi\)
\(462\) 0 0
\(463\) 41.0000i 1.90543i 0.303863 + 0.952716i \(0.401724\pi\)
−0.303863 + 0.952716i \(0.598276\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.35890 0.201706 0.100853 0.994901i \(-0.467843\pi\)
0.100853 + 0.994901i \(0.467843\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 57.0000 2.62086
\(474\) 0 0
\(475\) 61.0246 2.80000
\(476\) 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 \(-0.942223\pi\)
−0.983572 + 0.180517i \(0.942223\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) − 57.0000i − 2.56196i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −21.7945 −0.975656 −0.487828 0.872940i \(-0.662211\pi\)
−0.487828 + 0.872940i \(0.662211\pi\)
\(500\) 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
\(505\) −76.0000 −3.38196
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) − 33.0000i − 1.45983i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 56.6657i − 2.49215i
\(518\) 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
\(539\) −8.71780 −0.375502
\(540\) 0 0
\(541\) − 39.2301i − 1.68663i −0.537417 0.843317i \(-0.680600\pi\)
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 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) − 13.0767i − 0.558100i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 4.35890i − 0.184692i −0.995727 0.0923462i \(-0.970563\pi\)
0.995727 0.0923462i \(-0.0294367\pi\)
\(558\) 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
\(567\) 27.0000i 1.13389i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 26.1534 1.09449 0.547243 0.836974i \(-0.315677\pi\)
0.547243 + 0.836974i \(0.315677\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 56.0000i − 2.33536i
\(576\) 0 0
\(577\) −3.00000 −0.124892 −0.0624458 0.998048i \(-0.519890\pi\)
−0.0624458 + 0.998048i \(0.519890\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 26.1534i − 1.08503i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 47.9479 1.97902 0.989511 0.144460i \(-0.0461446\pi\)
0.989511 + 0.144460i \(0.0461446\pi\)
\(588\) 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
\(595\) 91.5369 3.75265
\(596\) 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
\(605\) − 34.8712i − 1.41772i
\(606\) 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
\(613\) 30.5123i 1.23238i 0.787598 + 0.616190i \(0.211325\pi\)
−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 \(-0.415723\pi\)
0.261680 + 0.965155i \(0.415723\pi\)
\(618\) 0 0
\(619\) −43.5890 −1.75199 −0.875995 0.482321i \(-0.839794\pi\)
−0.875995 + 0.482321i \(0.839794\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 101.000 4.04000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 15.0000i − 0.597141i −0.954388 0.298570i \(-0.903490\pi\)
0.954388 0.298570i \(-0.0965097\pi\)
\(632\) 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
\(643\) 13.0767 0.515695 0.257847 0.966186i \(-0.416987\pi\)
0.257847 + 0.966186i \(0.416987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.0000i 0.668339i 0.942513 + 0.334169i \(0.108456\pi\)
−0.942513 + 0.334169i \(0.891544\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 30.5123i 1.19404i 0.802227 + 0.597019i \(0.203648\pi\)
−0.802227 + 0.597019i \(0.796352\pi\)
\(654\) 0 0
\(655\) 95.0000i 3.71196i
\(656\) 0 0
\(657\) −33.0000 −1.28745
\(658\) 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
\(665\) −57.0000 −2.21037
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 19.0000i − 0.733487i
\(672\) 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
\(685\) − 100.255i − 3.83053i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 39.2301 1.49238 0.746191 0.665731i \(-0.231880\pi\)
0.746191 + 0.665731i \(0.231880\pi\)
\(692\) 0 0
\(693\) 39.2301i 1.49023i
\(694\) 0 0
\(695\) 95.0000i 3.60356i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 17.4356i 0.658533i 0.944237 + 0.329267i \(0.106802\pi\)
−0.944237 + 0.329267i \(0.893198\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 52.3068 1.96720
\(708\) 0 0
\(709\) 52.3068i 1.96442i 0.187779 + 0.982211i \(0.439871\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 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
\(719\) 49.0000i 1.82739i 0.406399 + 0.913696i \(0.366784\pi\)
−0.406399 + 0.913696i \(0.633216\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 37.0000i − 1.37225i −0.727482 0.686127i \(-0.759309\pi\)
0.727482 0.686127i \(-0.240691\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 91.5369 3.38561
\(732\) 0 0
\(733\) − 52.3068i − 1.93200i −0.258551 0.965998i \(-0.583245\pi\)
0.258551 0.965998i \(-0.416755\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −30.5123 −1.12241 −0.561206 0.827676i \(-0.689663\pi\)
−0.561206 + 0.827676i \(0.689663\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 95.0000 3.48053
\(746\) 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
\(757\) − 47.9479i − 1.74270i −0.490666 0.871348i \(-0.663246\pi\)
0.490666 0.871348i \(-0.336754\pi\)
\(758\) 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
\(765\) − 91.5369i − 3.30952i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −51.0000 −1.83911 −0.919554 0.392965i \(-0.871449\pi\)
−0.919554 + 0.392965i \(0.871449\pi\)
\(770\) 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
\(785\) 76.0000 2.71256
\(786\) 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
\(799\) − 91.0000i − 3.21935i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −47.9479 −1.69204
\(804\) 0 0
\(805\) 52.3068i 1.84357i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.00000 0.175791 0.0878953 0.996130i \(-0.471986\pi\)
0.0878953 + 0.996130i \(0.471986\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 38.0000i − 1.33108i
\(816\) 0 0
\(817\) −57.0000 −1.99418
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 21.7945i 0.760633i 0.924856 + 0.380317i \(0.124185\pi\)
−0.924856 + 0.380317i \(0.875815\pi\)
\(822\) 0 0
\(823\) 9.00000i 0.313720i 0.987621 + 0.156860i \(0.0501372\pi\)
−0.987621 + 0.156860i \(0.949863\pi\)
\(824\) 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
\(833\) −14.0000 −0.485071
\(834\) 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
\(845\) 56.6657i 1.94936i
\(846\) 0 0
\(847\) 24.0000i 0.824650i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 52.3068i 1.79095i 0.445112 + 0.895475i \(0.353164\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) 0 0
\(855\) 57.0000i 1.94936i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −56.6657 −1.93341 −0.966704 0.255897i \(-0.917629\pi\)
−0.966704 + 0.255897i \(0.917629\pi\)
\(860\) 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
\(875\) −117.690 −3.97866
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −35.0000 −1.17918 −0.589590 0.807703i \(-0.700711\pi\)
−0.589590 + 0.807703i \(0.700711\pi\)
\(882\) 0 0
\(883\) −30.5123 −1.02682 −0.513410 0.858143i \(-0.671618\pi\)
−0.513410 + 0.858143i \(0.671618\pi\)
\(884\) 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
\(891\) 39.2301 1.31426
\(892\) 0 0
\(893\) 56.6657i 1.89624i
\(894\) 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
\(913\) −38.0000 −1.25762
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 65.3835i − 2.15915i
\(918\) 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
\(931\) 8.71780 0.285714
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 133.000i − 4.34957i
\(936\) 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 1.00000i \(-0.5\pi\)
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 \(-0.958508\pi\)
−0.991516 + 0.129983i \(0.958508\pi\)
\(948\) 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
\(955\) 74.1013 2.39786
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 69.0000i 2.22813i
\(960\) 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
\(973\) − 65.3835i − 2.09610i
\(974\) 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
\(985\) −76.0000 −2.42156
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 52.3068i 1.66326i
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 108.972 3.45466
\(996\) 0 0
\(997\) − 4.35890i − 0.138048i −0.997615 0.0690239i \(-0.978012\pi\)
0.997615 0.0690239i \(-0.0219885\pi\)
\(998\) 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.c.607.2 yes 4
4.3 odd 2 inner 1216.2.b.c.607.1 4
8.3 odd 2 inner 1216.2.b.c.607.3 yes 4
8.5 even 2 inner 1216.2.b.c.607.4 yes 4
19.18 odd 2 CM 1216.2.b.c.607.2 yes 4
76.75 even 2 inner 1216.2.b.c.607.1 4
152.37 odd 2 inner 1216.2.b.c.607.4 yes 4
152.75 even 2 inner 1216.2.b.c.607.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.b.c.607.1 4 4.3 odd 2 inner
1216.2.b.c.607.1 4 76.75 even 2 inner
1216.2.b.c.607.2 yes 4 1.1 even 1 trivial
1216.2.b.c.607.2 yes 4 19.18 odd 2 CM
1216.2.b.c.607.3 yes 4 8.3 odd 2 inner
1216.2.b.c.607.3 yes 4 152.75 even 2 inner
1216.2.b.c.607.4 yes 4 8.5 even 2 inner
1216.2.b.c.607.4 yes 4 152.37 odd 2 inner