Properties

Label 1216.2.b.f.607.2
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 607.2
Root \(-1.70027 - 2.78183i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.f.607.11

$q$-expansion

\(f(q)\) \(=\) \(q-3.40054i q^{3} +4.62947i q^{7} -8.56365 q^{9} +O(q^{10})\) \(q-3.40054i q^{3} +4.62947i q^{7} -8.56365 q^{9} -0.223821 q^{13} -3.69529 q^{17} +4.35890i q^{19} +15.7427 q^{21} +6.49783i q^{23} +5.00000 q^{25} +18.9194i q^{27} -6.57725 q^{29} +8.71780 q^{37} +0.761111i q^{39} +6.00000i q^{47} -14.4320 q^{49} +12.5660i q^{51} -13.8260 q^{53} +14.8226 q^{57} +2.95290i q^{59} -39.6452i q^{63} -10.6493i q^{67} +22.0961 q^{69} -1.82693 q^{73} -17.0027i q^{75} +38.6452 q^{81} +22.3662i q^{87} -1.03617i q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 36q^{9} + O(q^{10}) \) \( 12q - 36q^{9} + 60q^{25} - 84q^{49} + 108q^{81} + O(q^{100}) \)

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
\(3\) − 3.40054i − 1.96330i −0.190689 0.981651i \(-0.561072\pi\)
0.190689 0.981651i \(-0.438928\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 4.62947i 1.74978i 0.484325 + 0.874888i \(0.339065\pi\)
−0.484325 + 0.874888i \(0.660935\pi\)
\(8\) 0 0
\(9\) −8.56365 −2.85455
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −0.223821 −0.0620767 −0.0310384 0.999518i \(-0.509881\pi\)
−0.0310384 + 0.999518i \(0.509881\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.69529 −0.896240 −0.448120 0.893973i \(-0.647906\pi\)
−0.448120 + 0.893973i \(0.647906\pi\)
\(18\) 0 0
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 15.7427 3.43534
\(22\) 0 0
\(23\) 6.49783i 1.35489i 0.735572 + 0.677446i \(0.236913\pi\)
−0.735572 + 0.677446i \(0.763087\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 18.9194i 3.64104i
\(28\) 0 0
\(29\) −6.57725 −1.22137 −0.610683 0.791875i \(-0.709105\pi\)
−0.610683 + 0.791875i \(0.709105\pi\)
\(30\) 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 \(-0.245698\pi\)
0.716599 + 0.697486i \(0.245698\pi\)
\(38\) 0 0
\(39\) 0.761111i 0.121875i
\(40\) 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 \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) −14.4320 −2.06172
\(50\) 0 0
\(51\) 12.5660i 1.75959i
\(52\) 0 0
\(53\) −13.8260 −1.89914 −0.949571 0.313551i \(-0.898481\pi\)
−0.949571 + 0.313551i \(0.898481\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 14.8226 1.96330
\(58\) 0 0
\(59\) 2.95290i 0.384434i 0.981352 + 0.192217i \(0.0615678\pi\)
−0.981352 + 0.192217i \(0.938432\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 39.6452i − 4.99483i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.6493i − 1.30101i −0.759501 0.650507i \(-0.774557\pi\)
0.759501 0.650507i \(-0.225443\pi\)
\(68\) 0 0
\(69\) 22.0961 2.66006
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.82693 −0.213826 −0.106913 0.994268i \(-0.534097\pi\)
−0.106913 + 0.994268i \(0.534097\pi\)
\(74\) 0 0
\(75\) − 17.0027i − 1.96330i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 38.6452 4.29391
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 22.3662i 2.39791i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) − 1.03617i − 0.108620i
\(92\) 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
\(107\) 17.4503i 1.68699i 0.537140 + 0.843493i \(0.319505\pi\)
−0.537140 + 0.843493i \(0.680495\pi\)
\(108\) 0 0
\(109\) 20.6270 1.97571 0.987856 0.155371i \(-0.0496572\pi\)
0.987856 + 0.155371i \(0.0496572\pi\)
\(110\) 0 0
\(111\) − 29.6452i − 2.81380i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.91672 0.177201
\(118\) 0 0
\(119\) − 17.1073i − 1.56822i
\(120\) 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
\(133\) −20.1794 −1.74978
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.30038 −0.794585 −0.397292 0.917692i \(-0.630050\pi\)
−0.397292 + 0.917692i \(0.630050\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 20.4032 1.71826
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 49.0766i 4.04777i
\(148\) 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
\(153\) 31.6452 2.55836
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 47.0157i 3.72859i
\(160\) 0 0
\(161\) −30.0815 −2.37076
\(162\) 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
\(169\) −12.9499 −0.996146
\(170\) 0 0
\(171\) − 37.3281i − 2.85455i
\(172\) 0 0
\(173\) −26.1534 −1.98841 −0.994203 0.107521i \(-0.965709\pi\)
−0.994203 + 0.107521i \(0.965709\pi\)
\(174\) 0 0
\(175\) 23.1474i 1.74978i
\(176\) 0 0
\(177\) 10.0414 0.754760
\(178\) 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 \(-0.394974\pi\)
0.323994 + 0.946059i \(0.394974\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −87.5869 −6.37101
\(190\) 0 0
\(191\) 26.8841i 1.94526i 0.232351 + 0.972632i \(0.425358\pi\)
−0.232351 + 0.972632i \(0.574642\pi\)
\(192\) 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
\(199\) 10.2346i 0.725509i 0.931885 + 0.362754i \(0.118164\pi\)
−0.931885 + 0.362754i \(0.881836\pi\)
\(200\) 0 0
\(201\) −36.2132 −2.55428
\(202\) 0 0
\(203\) − 30.4492i − 2.13712i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 55.6452i − 3.86761i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.0969i 0.763942i 0.924174 + 0.381971i \(0.124755\pi\)
−0.924174 + 0.381971i \(0.875245\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.21254i 0.419805i
\(220\) 0 0
\(221\) 0.827083 0.0556356
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −42.8183 −2.85455
\(226\) 0 0
\(227\) − 23.3561i − 1.55020i −0.631839 0.775100i \(-0.717699\pi\)
0.631839 0.775100i \(-0.282301\pi\)
\(228\) 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
\(239\) 21.2790i 1.37642i 0.725510 + 0.688212i \(0.241604\pi\)
−0.725510 + 0.688212i \(0.758396\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) − 74.6562i − 4.78920i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 0.975612i − 0.0620767i
\(248\) 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
\(259\) 40.3588i 2.50777i
\(260\) 0 0
\(261\) 56.3253 3.48645
\(262\) 0 0
\(263\) 30.0000i 1.84988i 0.380114 + 0.924940i \(0.375885\pi\)
−0.380114 + 0.924940i \(0.624115\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −26.1534 −1.59460 −0.797300 0.603583i \(-0.793739\pi\)
−0.797300 + 0.603583i \(0.793739\pi\)
\(270\) 0 0
\(271\) 28.7524i 1.74659i 0.487195 + 0.873293i \(0.338020\pi\)
−0.487195 + 0.873293i \(0.661980\pi\)
\(272\) 0 0
\(273\) −3.52354 −0.213254
\(274\) 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
\(289\) −3.34482 −0.196754
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 34.2292 1.99969 0.999845 0.0175838i \(-0.00559739\pi\)
0.999845 + 0.0175838i \(0.00559739\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.45435i − 0.0841072i
\(300\) 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
\(311\) − 34.2747i − 1.94354i −0.235934 0.971769i \(-0.575815\pi\)
0.235934 0.971769i \(-0.424185\pi\)
\(312\) 0 0
\(313\) 35.2088 1.99012 0.995061 0.0992663i \(-0.0316496\pi\)
0.995061 + 0.0992663i \(0.0316496\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.92018 0.444842 0.222421 0.974951i \(-0.428604\pi\)
0.222421 + 0.974951i \(0.428604\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 59.3405 3.31206
\(322\) 0 0
\(323\) − 16.1074i − 0.896240i
\(324\) 0 0
\(325\) −1.11910 −0.0620767
\(326\) 0 0
\(327\) − 70.1430i − 3.87892i
\(328\) 0 0
\(329\) −27.7768 −1.53139
\(330\) 0 0
\(331\) 30.1572i 1.65759i 0.559553 + 0.828795i \(0.310973\pi\)
−0.559553 + 0.828795i \(0.689027\pi\)
\(332\) 0 0
\(333\) −74.6562 −4.09113
\(334\) 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
\(343\) − 34.4063i − 1.85777i
\(344\) 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
\(351\) − 4.23456i − 0.226024i
\(352\) 0 0
\(353\) 37.0772 1.97342 0.986710 0.162489i \(-0.0519521\pi\)
0.986710 + 0.162489i \(0.0519521\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −58.1738 −3.07889
\(358\) 0 0
\(359\) 12.1029i 0.638768i 0.947625 + 0.319384i \(0.103476\pi\)
−0.947625 + 0.319384i \(0.896524\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 37.4059i 1.96330i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 10.0000i − 0.521996i −0.965339 0.260998i \(-0.915948\pi\)
0.965339 0.260998i \(-0.0840516\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 64.0069i − 3.32307i
\(372\) 0 0
\(373\) 21.5223 1.11438 0.557192 0.830384i \(-0.311879\pi\)
0.557192 + 0.830384i \(0.311879\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.47213 0.0758183
\(378\) 0 0
\(379\) − 31.5001i − 1.61805i −0.587773 0.809026i \(-0.699995\pi\)
0.587773 0.809026i \(-0.300005\pi\)
\(380\) 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
\(391\) − 24.0114i − 1.21431i
\(392\) 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
\(399\) 68.6208i 3.43534i
\(400\) 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
\(411\) 31.6263i 1.56001i
\(412\) 0 0
\(413\) −13.6703 −0.672674
\(414\) 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
\(421\) −41.0303 −1.99969 −0.999846 0.0175248i \(-0.994421\pi\)
−0.999846 + 0.0175248i \(0.994421\pi\)
\(422\) 0 0
\(423\) − 51.3819i − 2.49827i
\(424\) 0 0
\(425\) −18.4765 −0.896240
\(426\) 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
\(437\) −28.3234 −1.35489
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 123.591 5.88527
\(442\) 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
\(457\) −13.0371 −0.609850 −0.304925 0.952376i \(-0.598631\pi\)
−0.304925 + 0.952376i \(0.598631\pi\)
\(458\) 0 0
\(459\) − 69.9127i − 3.26325i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 22.0000i 1.02243i 0.859454 + 0.511213i \(0.170804\pi\)
−0.859454 + 0.511213i \(0.829196\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 49.3004 2.27648
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 21.7945i 1.00000i
\(476\) 0 0
\(477\) 118.401 5.42120
\(478\) 0 0
\(479\) − 42.0000i − 1.91903i −0.281659 0.959514i \(-0.590885\pi\)
0.281659 0.959514i \(-0.409115\pi\)
\(480\) 0 0
\(481\) −1.95122 −0.0889682
\(482\) 0 0
\(483\) 102.293i 4.65451i
\(484\) 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
\(493\) 24.3049 1.09464
\(494\) 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
\(503\) − 39.8798i − 1.77815i −0.457761 0.889075i \(-0.651349\pi\)
0.457761 0.889075i \(-0.348651\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 44.0366i 1.95574i
\(508\) 0 0
\(509\) −26.1534 −1.15923 −0.579614 0.814891i \(-0.696797\pi\)
−0.579614 + 0.814891i \(0.696797\pi\)
\(510\) 0 0
\(511\) − 8.45772i − 0.374148i
\(512\) 0 0
\(513\) −82.4678 −3.64104
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 88.9356i 3.90384i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) − 29.7096i − 1.29911i −0.760315 0.649554i \(-0.774956\pi\)
0.760315 0.649554i \(-0.225044\pi\)
\(524\) 0 0
\(525\) 78.7135 3.43534
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.2218 −0.835732
\(530\) 0 0
\(531\) − 25.2876i − 1.09739i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −88.9356 −3.83785
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) − 29.6452i − 1.27220i
\(544\) 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
\(551\) − 28.6696i − 1.22137i
\(552\) 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
\(567\) 178.907i 7.51338i
\(568\) 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
\(573\) 91.4203 3.81914
\(574\) 0 0
\(575\) 32.4892i 1.35489i
\(576\) 0 0
\(577\) 40.8139 1.69911 0.849553 0.527503i \(-0.176872\pi\)
0.849553 + 0.527503i \(0.176872\pi\)
\(578\) 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
\(597\) 34.8030 1.42439
\(598\) 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
\(603\) 91.1965i 3.71381i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −103.544 −4.19580
\(610\) 0 0
\(611\) − 1.34292i − 0.0543289i
\(612\) 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
\(621\) −122.935 −4.93322
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −32.2148 −1.28449
\(630\) 0 0
\(631\) − 34.0000i − 1.35352i −0.736204 0.676759i \(-0.763384\pi\)
0.736204 0.676759i \(-0.236616\pi\)
\(632\) 0 0
\(633\) 37.7354 1.49985
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3.23018 0.127985
\(638\) 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
\(647\) − 4.71234i − 0.185261i −0.995701 0.0926305i \(-0.970472\pi\)
0.995701 0.0926305i \(-0.0295275\pi\)
\(648\) 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
\(657\) 15.6452 0.610377
\(658\) 0 0
\(659\) 36.5106i 1.42225i 0.703065 + 0.711126i \(0.251814\pi\)
−0.703065 + 0.711126i \(0.748186\pi\)
\(660\) 0 0
\(661\) −21.9700 −0.854533 −0.427266 0.904126i \(-0.640523\pi\)
−0.427266 + 0.904126i \(0.640523\pi\)
\(662\) 0 0
\(663\) − 2.81253i − 0.109229i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 42.7379i − 1.65482i
\(668\) 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
\(675\) 94.5970i 3.64104i
\(676\) 0 0
\(677\) 5.23433 0.201172 0.100586 0.994928i \(-0.467928\pi\)
0.100586 + 0.994928i \(0.467928\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −79.4234 −3.04351
\(682\) 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
\(689\) 3.09454 0.117893
\(690\) 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
\(699\) − 61.2097i − 2.31516i
\(700\) 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
\(717\) 72.3600 2.70233
\(718\) 0 0
\(719\) 15.6739i 0.584538i 0.956336 + 0.292269i \(0.0944103\pi\)
−0.956336 + 0.292269i \(0.905590\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −32.8863 −1.22137
\(726\) 0 0
\(727\) − 26.8012i − 0.994002i −0.867750 0.497001i \(-0.834434\pi\)
0.867750 0.497001i \(-0.165566\pi\)
\(728\) 0 0
\(729\) −137.936 −5.10873
\(730\) 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
\(741\) −3.31761 −0.121875
\(742\) 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
\(749\) −80.7858 −2.95185
\(750\) 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
\(761\) 42.6823 1.54723 0.773616 0.633655i \(-0.218446\pi\)
0.773616 + 0.633655i \(0.218446\pi\)
\(762\) 0 0
\(763\) 95.4923i 3.45705i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 0.660919i − 0.0238644i
\(768\) 0 0
\(769\) 53.7267 1.93744 0.968718 0.248165i \(-0.0798274\pi\)
0.968718 + 0.248165i \(0.0798274\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 48.7266 1.75257 0.876287 0.481789i \(-0.160013\pi\)
0.876287 + 0.481789i \(0.160013\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 137.242 4.92352
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 124.438i − 4.44704i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 50.5604i − 1.80228i −0.433524 0.901142i \(-0.642730\pi\)
0.433524 0.901142i \(-0.357270\pi\)
\(788\) 0 0
\(789\) 102.016 3.63187
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −35.5721 −1.26003 −0.630015 0.776583i \(-0.716951\pi\)
−0.630015 + 0.776583i \(0.716951\pi\)
\(798\) 0 0
\(799\) − 22.1717i − 0.784380i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 88.9356i 3.13068i
\(808\) 0 0
\(809\) 51.8584 1.82324 0.911622 0.411030i \(-0.134831\pi\)
0.911622 + 0.411030i \(0.134831\pi\)
\(810\) 0 0
\(811\) 11.9922i 0.421102i 0.977583 + 0.210551i \(0.0675259\pi\)
−0.977583 + 0.210551i \(0.932474\pi\)
\(812\) 0 0
\(813\) 97.7738 3.42908
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 8.87342i 0.310062i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 15.8396i 0.552135i 0.961138 + 0.276068i \(0.0890314\pi\)
−0.961138 + 0.276068i \(0.910969\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 18.7933i − 0.653505i −0.945110 0.326753i \(-0.894046\pi\)
0.945110 0.326753i \(-0.105954\pi\)
\(828\) 0 0
\(829\) 18.8365 0.654218 0.327109 0.944987i \(-0.393926\pi\)
0.327109 + 0.944987i \(0.393926\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 53.3305 1.84779
\(834\) 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
\(841\) 14.2603 0.491733
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 50.9242i − 1.74978i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 56.6468i 1.94183i
\(852\) 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
\(867\) 11.3742i 0.386288i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.38352i 0.0807626i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 39.6873 1.34015 0.670073 0.742295i \(-0.266263\pi\)
0.670073 + 0.742295i \(0.266263\pi\)
\(878\) 0 0
\(879\) − 116.398i − 3.92600i
\(880\) 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
\(897\) −4.94557 −0.165128
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 51.0910 1.70209
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 8.41105i 0.279284i 0.990202 + 0.139642i \(0.0445952\pi\)
−0.990202 + 0.139642i \(0.955405\pi\)
\(908\) 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
\(919\) − 6.58070i − 0.217077i −0.994092 0.108539i \(-0.965383\pi\)
0.994092 0.108539i \(-0.0346171\pi\)
\(920\) 0 0
\(921\) 29.6452 0.976842
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 43.5890 1.43320
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −35.2917 −1.15788 −0.578942 0.815369i \(-0.696534\pi\)
−0.578942 + 0.815369i \(0.696534\pi\)
\(930\) 0 0
\(931\) − 62.9077i − 2.06172i
\(932\) 0 0
\(933\) −116.552 −3.81575
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 59.3318 1.93829 0.969143 0.246499i \(-0.0792802\pi\)
0.969143 + 0.246499i \(0.0792802\pi\)
\(938\) 0 0
\(939\) − 119.729i − 3.90721i
\(940\) 0 0
\(941\) −25.6376 −0.835760 −0.417880 0.908502i \(-0.637227\pi\)
−0.417880 + 0.908502i \(0.637227\pi\)
\(942\) 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
\(949\) 0.408905 0.0132736
\(950\) 0 0
\(951\) − 26.9329i − 0.873358i
\(952\) 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
\(959\) − 43.0558i − 1.39035i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 149.439i − 4.81559i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 50.0000i − 1.60789i −0.594703 0.803946i \(-0.702730\pi\)
0.594703 0.803946i \(-0.297270\pi\)
\(968\) 0 0
\(969\) −54.7738 −1.75959
\(970\) 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
\(975\) 3.80555i 0.121875i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −176.643 −5.63977
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 94.4562i 3.00657i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 102.551 3.25435
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 164.936i 5.21833i
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.607.2 yes 12
4.3 odd 2 inner 1216.2.b.f.607.11 yes 12
8.3 odd 2 inner 1216.2.b.f.607.1 12
8.5 even 2 inner 1216.2.b.f.607.12 yes 12
19.18 odd 2 inner 1216.2.b.f.607.12 yes 12
76.75 even 2 inner 1216.2.b.f.607.1 12
152.37 odd 2 CM 1216.2.b.f.607.2 yes 12
152.75 even 2 inner 1216.2.b.f.607.11 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.b.f.607.1 12 8.3 odd 2 inner
1216.2.b.f.607.1 12 76.75 even 2 inner
1216.2.b.f.607.2 yes 12 1.1 even 1 trivial
1216.2.b.f.607.2 yes 12 152.37 odd 2 CM
1216.2.b.f.607.11 yes 12 4.3 odd 2 inner
1216.2.b.f.607.11 yes 12 152.75 even 2 inner
1216.2.b.f.607.12 yes 12 8.5 even 2 inner
1216.2.b.f.607.12 yes 12 19.18 odd 2 inner