Properties

Label 1216.2.b.f.607.10
Level $1216$
Weight $2$
Character 1216.607
Analytic conductor $9.710$
Analytic rank $0$
Dimension $12$
CM discriminant -152
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 18 x^{10} + 207 x^{8} + 1014 x^{6} + 1065 x^{4} - 5508 x^{2} + 8464\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 607.10
Root \(-1.13617 - 0.418247i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.f.607.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.27234i q^{3} +4.53419i q^{7} -2.16351 q^{9} +O(q^{10})\) \(q+2.27234i q^{3} +4.53419i q^{7} -2.16351 q^{9} +6.13008 q^{13} +8.23189 q^{17} +4.35890i q^{19} -10.3032 q^{21} -2.86121i q^{23} +5.00000 q^{25} +1.90079i q^{27} -10.6747 q^{29} -8.71780 q^{37} +13.9296i q^{39} -6.00000i q^{47} -13.5589 q^{49} +18.7056i q^{51} -2.95926 q^{53} -9.90488 q^{57} -14.5325i q^{59} -9.80976i q^{63} -5.44315i q^{67} +6.50162 q^{69} +15.6273 q^{73} +11.3617i q^{75} -10.8098 q^{81} -24.2566i q^{87} +27.7950i q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 36q^{9} + O(q^{10}) \) \( 12q - 36q^{9} + 60q^{25} - 84q^{49} + 108q^{81} + O(q^{100}) \)

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
\(3\) 2.27234i 1.31193i 0.754790 + 0.655967i \(0.227739\pi\)
−0.754790 + 0.655967i \(0.772261\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.53419i 1.71376i 0.515513 + 0.856882i \(0.327602\pi\)
−0.515513 + 0.856882i \(0.672398\pi\)
\(8\) 0 0
\(9\) −2.16351 −0.721169
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 6.13008 1.70018 0.850089 0.526639i \(-0.176548\pi\)
0.850089 + 0.526639i \(0.176548\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.23189 1.99653 0.998264 0.0589035i \(-0.0187604\pi\)
0.998264 + 0.0589035i \(0.0187604\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) −10.3032 −2.24834
\(22\) 0 0
\(23\) − 2.86121i − 0.596603i −0.954472 0.298301i \(-0.903580\pi\)
0.954472 0.298301i \(-0.0964201\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 1.90079i 0.365808i
\(28\) 0 0
\(29\) −10.6747 −1.98225 −0.991126 0.132929i \(-0.957562\pi\)
−0.991126 + 0.132929i \(0.957562\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.71780 −1.43320 −0.716599 0.697486i \(-0.754302\pi\)
−0.716599 + 0.697486i \(0.754302\pi\)
\(38\) 0 0
\(39\) 13.9296i 2.23052i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 0 0
\(49\) −13.5589 −1.93699
\(50\) 0 0
\(51\) 18.7056i 2.61931i
\(52\) 0 0
\(53\) −2.95926 −0.406486 −0.203243 0.979128i \(-0.565148\pi\)
−0.203243 + 0.979128i \(0.565148\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −9.90488 −1.31193
\(58\) 0 0
\(59\) − 14.5325i − 1.89197i −0.324211 0.945985i \(-0.605099\pi\)
0.324211 0.945985i \(-0.394901\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 9.80976i − 1.23591i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.44315i − 0.664987i −0.943106 0.332493i \(-0.892110\pi\)
0.943106 0.332493i \(-0.107890\pi\)
\(68\) 0 0
\(69\) 6.50162 0.782703
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 15.6273 1.82904 0.914518 0.404545i \(-0.132570\pi\)
0.914518 + 0.404545i \(0.132570\pi\)
\(74\) 0 0
\(75\) 11.3617i 1.31193i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −10.8098 −1.20108
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 24.2566i − 2.60058i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 27.7950i 2.91370i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.898482i 0.0868595i 0.999056 + 0.0434298i \(0.0138285\pi\)
−0.999056 + 0.0434298i \(0.986172\pi\)
\(108\) 0 0
\(109\) 7.50393 0.718746 0.359373 0.933194i \(-0.382991\pi\)
0.359373 + 0.933194i \(0.382991\pi\)
\(110\) 0 0
\(111\) − 19.8098i − 1.88026i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −13.2625 −1.22612
\(118\) 0 0
\(119\) 37.3250i 3.42158i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −19.7641 −1.71376
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.9543 −1.19220 −0.596098 0.802911i \(-0.703283\pi\)
−0.596098 + 0.802911i \(0.703283\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 13.6340 1.14819
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 30.8104i − 2.54120i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −17.8098 −1.43983
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) − 6.72443i − 0.533282i
\(160\) 0 0
\(161\) 12.9733 1.02244
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 24.5779 1.89061
\(170\) 0 0
\(171\) − 9.43051i − 0.721169i
\(172\) 0 0
\(173\) 26.1534 1.98841 0.994203 0.107521i \(-0.0342912\pi\)
0.994203 + 0.107521i \(0.0342912\pi\)
\(174\) 0 0
\(175\) 22.6710i 1.71376i
\(176\) 0 0
\(177\) 33.0227 2.48214
\(178\) 0 0
\(179\) − 26.1534i − 1.95480i −0.211407 0.977398i \(-0.567804\pi\)
0.211407 0.977398i \(-0.432196\pi\)
\(180\) 0 0
\(181\) −8.71780 −0.647989 −0.323994 0.946059i \(-0.605026\pi\)
−0.323994 + 0.946059i \(0.605026\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −8.61856 −0.626908
\(190\) 0 0
\(191\) 7.88017i 0.570189i 0.958499 + 0.285094i \(0.0920249\pi\)
−0.958499 + 0.285094i \(0.907975\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) − 17.6520i − 1.25132i −0.780097 0.625659i \(-0.784830\pi\)
0.780097 0.625659i \(-0.215170\pi\)
\(200\) 0 0
\(201\) 12.3687 0.872418
\(202\) 0 0
\(203\) − 48.4014i − 3.39711i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.19024i 0.430252i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 17.7033i 1.21875i 0.792884 + 0.609373i \(0.208579\pi\)
−0.792884 + 0.609373i \(0.791421\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 35.5104i 2.39957i
\(220\) 0 0
\(221\) 50.4622 3.39445
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −10.8175 −0.721169
\(226\) 0 0
\(227\) 28.1665i 1.86948i 0.355337 + 0.934738i \(0.384366\pi\)
−0.355337 + 0.934738i \(0.615634\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 30.0664i 1.94483i 0.233254 + 0.972416i \(0.425063\pi\)
−0.233254 + 0.972416i \(0.574937\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) − 18.8610i − 1.20993i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 26.7204i 1.70018i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) − 39.5282i − 2.45616i
\(260\) 0 0
\(261\) 23.0949 1.42954
\(262\) 0 0
\(263\) − 30.0000i − 1.84988i −0.380114 0.924940i \(-0.624115\pi\)
0.380114 0.924940i \(-0.375885\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 26.1534 1.59460 0.797300 0.603583i \(-0.206261\pi\)
0.797300 + 0.603583i \(0.206261\pi\)
\(270\) 0 0
\(271\) 0.484765i 0.0294474i 0.999892 + 0.0147237i \(0.00468686\pi\)
−0.999892 + 0.0147237i \(0.995313\pi\)
\(272\) 0 0
\(273\) −63.1595 −3.82258
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 50.7641 2.98612
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 16.5933 0.969389 0.484695 0.874683i \(-0.338931\pi\)
0.484695 + 0.874683i \(0.338931\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 17.5394i − 1.01433i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.71780i 0.497551i 0.968561 + 0.248776i \(0.0800281\pi\)
−0.968561 + 0.248776i \(0.919972\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 24.3440i − 1.38042i −0.723610 0.690209i \(-0.757518\pi\)
0.723610 0.690209i \(-0.242482\pi\)
\(312\) 0 0
\(313\) −20.6463 −1.16700 −0.583498 0.812115i \(-0.698316\pi\)
−0.583498 + 0.812115i \(0.698316\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −26.1057 −1.46624 −0.733122 0.680097i \(-0.761937\pi\)
−0.733122 + 0.680097i \(0.761937\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −2.04165 −0.113954
\(322\) 0 0
\(323\) 35.8820i 1.99653i
\(324\) 0 0
\(325\) 30.6504 1.70018
\(326\) 0 0
\(327\) 17.0514i 0.942947i
\(328\) 0 0
\(329\) 27.2052 1.49987
\(330\) 0 0
\(331\) − 32.7112i − 1.79797i −0.437981 0.898984i \(-0.644306\pi\)
0.437981 0.898984i \(-0.355694\pi\)
\(332\) 0 0
\(333\) 18.8610 1.03358
\(334\) 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
\(343\) − 29.7394i − 1.60577i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 11.6520i 0.621938i
\(352\) 0 0
\(353\) −13.2509 −0.705272 −0.352636 0.935761i \(-0.614715\pi\)
−0.352636 + 0.935761i \(0.614715\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −84.8149 −4.48888
\(358\) 0 0
\(359\) − 25.0474i − 1.32195i −0.750407 0.660976i \(-0.770143\pi\)
0.750407 0.660976i \(-0.229857\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) − 24.9957i − 1.31193i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 10.0000i 0.521996i 0.965339 + 0.260998i \(0.0840516\pi\)
−0.965339 + 0.260998i \(0.915948\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 13.4179i − 0.696621i
\(372\) 0 0
\(373\) −17.0164 −0.881075 −0.440537 0.897734i \(-0.645212\pi\)
−0.440537 + 0.897734i \(0.645212\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −65.4371 −3.37018
\(378\) 0 0
\(379\) − 4.06930i − 0.209026i −0.994524 0.104513i \(-0.966672\pi\)
0.994524 0.104513i \(-0.0333284\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) − 23.5532i − 1.19113i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) − 44.9106i − 2.24834i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) − 31.7089i − 1.56408i
\(412\) 0 0
\(413\) 65.8931 3.24239
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −21.1379 −1.03020 −0.515100 0.857130i \(-0.672245\pi\)
−0.515100 + 0.857130i \(0.672245\pi\)
\(422\) 0 0
\(423\) 12.9810i 0.631160i
\(424\) 0 0
\(425\) 41.1595 1.99653
\(426\) 0 0
\(427\) 0 0
\(428\) 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
\(437\) 12.4717 0.596603
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 29.3348 1.39689
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 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
\(457\) −28.7451 −1.34464 −0.672320 0.740261i \(-0.734702\pi\)
−0.672320 + 0.740261i \(0.734702\pi\)
\(458\) 0 0
\(459\) 15.6471i 0.730345i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) − 22.0000i − 1.02243i −0.859454 0.511213i \(-0.829196\pi\)
0.859454 0.511213i \(-0.170804\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 24.6803 1.13963
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 21.7945i 1.00000i
\(476\) 0 0
\(477\) 6.40238 0.293145
\(478\) 0 0
\(479\) 42.0000i 1.91903i 0.281659 + 0.959514i \(0.409115\pi\)
−0.281659 + 0.959514i \(0.590885\pi\)
\(480\) 0 0
\(481\) −53.4408 −2.43669
\(482\) 0 0
\(483\) 29.4796i 1.34137i
\(484\) 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
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −87.8734 −3.95762
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 2.15775i − 0.0962094i −0.998842 0.0481047i \(-0.984682\pi\)
0.998842 0.0481047i \(-0.0153181\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 55.8491i 2.48035i
\(508\) 0 0
\(509\) 26.1534 1.15923 0.579614 0.814891i \(-0.303203\pi\)
0.579614 + 0.814891i \(0.303203\pi\)
\(510\) 0 0
\(511\) 70.8572i 3.13454i
\(512\) 0 0
\(513\) −8.28536 −0.365808
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 59.4293i 2.60866i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 44.9713i 1.96646i 0.182373 + 0.983230i \(0.441622\pi\)
−0.182373 + 0.983230i \(0.558378\pi\)
\(524\) 0 0
\(525\) −51.5160 −2.24834
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 14.8135 0.644065
\(530\) 0 0
\(531\) 31.4411i 1.36443i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 59.4293 2.56456
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) − 19.8098i − 0.850118i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.5890i 1.86373i 0.362804 + 0.931865i \(0.381819\pi\)
−0.362804 + 0.931865i \(0.618181\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 46.5302i − 1.98225i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 26.1534i − 1.10223i −0.834428 0.551117i \(-0.814202\pi\)
0.834428 0.551117i \(-0.185798\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 49.0135i − 2.05837i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) −17.9064 −0.748050
\(574\) 0 0
\(575\) − 14.3060i − 0.596603i
\(576\) 0 0
\(577\) 1.53995 0.0641089 0.0320545 0.999486i \(-0.489795\pi\)
0.0320545 + 0.999486i \(0.489795\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 40.1113 1.64165
\(598\) 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
\(603\) 11.7763i 0.479568i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 109.984 4.45678
\(610\) 0 0
\(611\) − 36.7805i − 1.48798i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 5.43856 0.218242
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −71.7640 −2.86142
\(630\) 0 0
\(631\) 34.0000i 1.35352i 0.736204 + 0.676759i \(0.236616\pi\)
−0.736204 + 0.676759i \(0.763384\pi\)
\(632\) 0 0
\(633\) −40.2279 −1.59891
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −83.1172 −3.29322
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 41.5112i 1.63197i 0.578071 + 0.815987i \(0.303806\pi\)
−0.578071 + 0.815987i \(0.696194\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −33.8098 −1.31904
\(658\) 0 0
\(659\) − 49.5160i − 1.92887i −0.264321 0.964435i \(-0.585148\pi\)
0.264321 0.964435i \(-0.414852\pi\)
\(660\) 0 0
\(661\) 29.2765 1.13873 0.569363 0.822086i \(-0.307190\pi\)
0.569363 + 0.822086i \(0.307190\pi\)
\(662\) 0 0
\(663\) 114.667i 4.45329i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 30.5427i 1.18262i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 9.50396i 0.365808i
\(676\) 0 0
\(677\) 47.4552 1.82385 0.911926 0.410354i \(-0.134595\pi\)
0.911926 + 0.410354i \(0.134595\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −64.0037 −2.45263
\(682\) 0 0
\(683\) − 26.1534i − 1.00073i −0.865814 0.500366i \(-0.833199\pi\)
0.865814 0.500366i \(-0.166801\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −18.1405 −0.691098
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 40.9020i 1.54706i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) − 38.0000i − 1.43320i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −68.3209 −2.55149
\(718\) 0 0
\(719\) 52.2526i 1.94869i 0.225055 + 0.974346i \(0.427744\pi\)
−0.225055 + 0.974346i \(0.572256\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −53.3737 −1.98225
\(726\) 0 0
\(727\) − 53.9256i − 1.99999i −0.00345931 0.999994i \(-0.501101\pi\)
0.00345931 0.999994i \(-0.498899\pi\)
\(728\) 0 0
\(729\) 10.4293 0.386269
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) −60.7177 −2.23052
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.07389 −0.148857
\(750\) 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 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8.93535 0.323906 0.161953 0.986798i \(-0.448221\pi\)
0.161953 + 0.986798i \(0.448221\pi\)
\(762\) 0 0
\(763\) 34.0243i 1.23176i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 89.0853i − 3.21668i
\(768\) 0 0
\(769\) −38.7830 −1.39855 −0.699276 0.714852i \(-0.746494\pi\)
−0.699276 + 0.714852i \(0.746494\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.16230 0.0418050 0.0209025 0.999782i \(-0.493346\pi\)
0.0209025 + 0.999782i \(0.493346\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 89.8213 3.22232
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 20.2905i − 0.725123i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.3452i 1.65203i 0.563650 + 0.826014i \(0.309397\pi\)
−0.563650 + 0.826014i \(0.690603\pi\)
\(788\) 0 0
\(789\) 68.1701 2.42692
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.1872 0.715067 0.357534 0.933900i \(-0.383618\pi\)
0.357534 + 0.933900i \(0.383618\pi\)
\(798\) 0 0
\(799\) − 49.3914i − 1.74734i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 59.4293i 2.09201i
\(808\) 0 0
\(809\) −46.1784 −1.62355 −0.811773 0.583972i \(-0.801498\pi\)
−0.811773 + 0.583972i \(0.801498\pi\)
\(810\) 0 0
\(811\) 42.2236i 1.48267i 0.671134 + 0.741336i \(0.265807\pi\)
−0.671134 + 0.741336i \(0.734193\pi\)
\(812\) 0 0
\(813\) −1.10155 −0.0386330
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) − 60.1346i − 2.10127i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) − 39.8382i − 1.38867i −0.719651 0.694336i \(-0.755698\pi\)
0.719651 0.694336i \(-0.244302\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 37.6790i − 1.31023i −0.755531 0.655113i \(-0.772621\pi\)
0.755531 0.655113i \(-0.227379\pi\)
\(828\) 0 0
\(829\) 56.5446 1.96387 0.981937 0.189209i \(-0.0605923\pi\)
0.981937 + 0.189209i \(0.0605923\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −111.615 −3.86725
\(834\) 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
\(841\) 84.9503 2.92932
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 49.8761i − 1.71376i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.9434i 0.855050i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 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
\(867\) 115.353i 3.91759i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) − 33.3670i − 1.13060i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 57.9184 1.95577 0.977883 0.209153i \(-0.0670706\pi\)
0.977883 + 0.209153i \(0.0670706\pi\)
\(878\) 0 0
\(879\) 37.7055i 1.27177i
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) 26.1534 0.875190
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 39.8555 1.33073
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −24.3603 −0.811560
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 55.8576i − 1.85472i −0.374167 0.927361i \(-0.622071\pi\)
0.374167 0.927361i \(-0.377929\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 48.9066i 1.61328i 0.591043 + 0.806640i \(0.298716\pi\)
−0.591043 + 0.806640i \(0.701284\pi\)
\(920\) 0 0
\(921\) −19.8098 −0.652754
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −43.5890 −1.43320
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −25.3991 −0.833319 −0.416659 0.909063i \(-0.636799\pi\)
−0.416659 + 0.909063i \(0.636799\pi\)
\(930\) 0 0
\(931\) − 59.1019i − 1.93699i
\(932\) 0 0
\(933\) 55.3176 1.81102
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −16.5968 −0.542195 −0.271097 0.962552i \(-0.587387\pi\)
−0.271097 + 0.962552i \(0.587387\pi\)
\(938\) 0 0
\(939\) − 46.9152i − 1.53102i
\(940\) 0 0
\(941\) −61.0892 −1.99145 −0.995726 0.0923562i \(-0.970560\pi\)
−0.995726 + 0.0923562i \(0.970560\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 95.7965 3.10969
\(950\) 0 0
\(951\) − 59.3210i − 1.92361i
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 63.2715i − 2.04314i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 1.94387i − 0.0626404i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 50.0000i 1.60789i 0.594703 + 0.803946i \(0.297270\pi\)
−0.594703 + 0.803946i \(0.702730\pi\)
\(968\) 0 0
\(969\) −81.5359 −2.61931
\(970\) 0 0
\(971\) − 26.1534i − 0.839302i −0.907685 0.419651i \(-0.862152\pi\)
0.907685 0.419651i \(-0.137848\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 69.6480i 2.23052i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −16.2348 −0.518338
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 61.8192i 1.96773i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 74.3307 2.35882
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) − 16.5707i − 0.524275i
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.f.607.10 yes 12
4.3 odd 2 inner 1216.2.b.f.607.3 12
8.3 odd 2 inner 1216.2.b.f.607.9 yes 12
8.5 even 2 inner 1216.2.b.f.607.4 yes 12
19.18 odd 2 inner 1216.2.b.f.607.4 yes 12
76.75 even 2 inner 1216.2.b.f.607.9 yes 12
152.37 odd 2 CM 1216.2.b.f.607.10 yes 12
152.75 even 2 inner 1216.2.b.f.607.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.b.f.607.3 12 4.3 odd 2 inner
1216.2.b.f.607.3 12 152.75 even 2 inner
1216.2.b.f.607.4 yes 12 8.5 even 2 inner
1216.2.b.f.607.4 yes 12 19.18 odd 2 inner
1216.2.b.f.607.9 yes 12 8.3 odd 2 inner
1216.2.b.f.607.9 yes 12 76.75 even 2 inner
1216.2.b.f.607.10 yes 12 1.1 even 1 trivial
1216.2.b.f.607.10 yes 12 152.37 odd 2 CM