Properties

Label 1216.2.b.f.607.5
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 607.5
Root \(-0.564101 + 2.36358i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.f.607.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.12820i q^{3} -0.0952793i q^{7} +1.72716 q^{9} +O(q^{10})\) \(q-1.12820i q^{3} -0.0952793i q^{7} +1.72716 q^{9} -6.35390 q^{13} -4.53660 q^{17} -4.35890i q^{19} -0.107494 q^{21} -9.35904i q^{23} +5.00000 q^{25} -5.33319i q^{27} +4.09750 q^{29} -8.71780 q^{37} +7.16848i q^{39} +6.00000i q^{47} +6.99092 q^{49} +5.11820i q^{51} -10.8667 q^{53} -4.91772 q^{57} -11.5796i q^{59} -0.164563i q^{63} -16.0924i q^{67} -10.5589 q^{69} -13.8004 q^{73} -5.64101i q^{75} -0.835437 q^{81} -4.62280i q^{87} +0.605395i q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 36q^{9} + O(q^{10}) \) \( 12q - 36q^{9} + 60q^{25} - 84q^{49} + 108q^{81} + O(q^{100}) \)

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
\(3\) − 1.12820i − 0.651368i −0.945479 0.325684i \(-0.894405\pi\)
0.945479 0.325684i \(-0.105595\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) − 0.0952793i − 0.0360122i −0.999838 0.0180061i \(-0.994268\pi\)
0.999838 0.0180061i \(-0.00573183\pi\)
\(8\) 0 0
\(9\) 1.72716 0.575720
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −6.35390 −1.76225 −0.881127 0.472879i \(-0.843215\pi\)
−0.881127 + 0.472879i \(0.843215\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.53660 −1.10029 −0.550144 0.835070i \(-0.685427\pi\)
−0.550144 + 0.835070i \(0.685427\pi\)
\(18\) 0 0
\(19\) − 4.35890i − 1.00000i
\(20\) 0 0
\(21\) −0.107494 −0.0234572
\(22\) 0 0
\(23\) − 9.35904i − 1.95149i −0.218899 0.975747i \(-0.570247\pi\)
0.218899 0.975747i \(-0.429753\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 5.33319i − 1.02637i
\(28\) 0 0
\(29\) 4.09750 0.760886 0.380443 0.924804i \(-0.375772\pi\)
0.380443 + 0.924804i \(0.375772\pi\)
\(30\) 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
\(39\) 7.16848i 1.14788i
\(40\) 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
\(49\) 6.99092 0.998703
\(50\) 0 0
\(51\) 5.11820i 0.716692i
\(52\) 0 0
\(53\) −10.8667 −1.49266 −0.746329 0.665578i \(-0.768185\pi\)
−0.746329 + 0.665578i \(0.768185\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.91772 −0.651368
\(58\) 0 0
\(59\) − 11.5796i − 1.50754i −0.657141 0.753768i \(-0.728234\pi\)
0.657141 0.753768i \(-0.271766\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 0.164563i − 0.0207329i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 16.0924i − 1.96600i −0.183605 0.983000i \(-0.558777\pi\)
0.183605 0.983000i \(-0.441223\pi\)
\(68\) 0 0
\(69\) −10.5589 −1.27114
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −13.8004 −1.61521 −0.807605 0.589724i \(-0.799237\pi\)
−0.807605 + 0.589724i \(0.799237\pi\)
\(74\) 0 0
\(75\) − 5.64101i − 0.651368i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) −0.835437 −0.0928264
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.62280i − 0.495616i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0.605395i 0.0634626i
\(92\) 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
\(107\) 18.3488i 1.77385i 0.461917 + 0.886923i \(0.347162\pi\)
−0.461917 + 0.886923i \(0.652838\pi\)
\(108\) 0 0
\(109\) 13.1231 1.25697 0.628483 0.777823i \(-0.283676\pi\)
0.628483 + 0.777823i \(0.283676\pi\)
\(110\) 0 0
\(111\) 9.83544i 0.933538i
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.9742 −1.01457
\(118\) 0 0
\(119\) 0.432244i 0.0396238i
\(120\) 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
\(133\) −0.415313 −0.0360122
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 23.2547 1.98678 0.993391 0.114781i \(-0.0366166\pi\)
0.993391 + 0.114781i \(0.0366166\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 6.76921 0.570071
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 7.88717i − 0.650523i
\(148\) 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
\(153\) −7.83544 −0.633458
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 12.2598i 0.972269i
\(160\) 0 0
\(161\) −0.891723 −0.0702776
\(162\) 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
\(169\) 27.3720 2.10554
\(170\) 0 0
\(171\) − 7.52852i − 0.575720i
\(172\) 0 0
\(173\) 26.1534 1.98841 0.994203 0.107521i \(-0.0342912\pi\)
0.994203 + 0.107521i \(0.0342912\pi\)
\(174\) 0 0
\(175\) − 0.476396i − 0.0360122i
\(176\) 0 0
\(177\) −13.0641 −0.981960
\(178\) 0 0
\(179\) 26.1534i 1.95480i 0.211407 + 0.977398i \(0.432196\pi\)
−0.211407 + 0.977398i \(0.567804\pi\)
\(180\) 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
\(189\) −0.508143 −0.0369619
\(190\) 0 0
\(191\) − 19.0039i − 1.37508i −0.726149 0.687538i \(-0.758692\pi\)
0.726149 0.687538i \(-0.241308\pi\)
\(192\) 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
\(199\) − 27.8866i − 1.97683i −0.151788 0.988413i \(-0.548503\pi\)
0.151788 0.988413i \(-0.451497\pi\)
\(200\) 0 0
\(201\) −18.1555 −1.28059
\(202\) 0 0
\(203\) − 0.390406i − 0.0274012i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 16.1646i − 1.12351i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 28.8002i 1.98269i 0.131291 + 0.991344i \(0.458088\pi\)
−0.131291 + 0.991344i \(0.541912\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 15.5696i 1.05210i
\(220\) 0 0
\(221\) 28.8251 1.93899
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 8.63580 0.575720
\(226\) 0 0
\(227\) 4.81038i 0.319276i 0.987176 + 0.159638i \(0.0510328\pi\)
−0.987176 + 0.159638i \(0.948967\pi\)
\(228\) 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
\(239\) 8.78737i 0.568407i 0.958764 + 0.284204i \(0.0917292\pi\)
−0.958764 + 0.284204i \(0.908271\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) − 15.0570i − 0.965909i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 27.6960i 1.76225i
\(248\) 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
\(259\) 0.830626i 0.0516126i
\(260\) 0 0
\(261\) 7.07703 0.438057
\(262\) 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 \(-0.206261\pi\)
0.797300 + 0.603583i \(0.206261\pi\)
\(270\) 0 0
\(271\) − 28.2677i − 1.71714i −0.512697 0.858570i \(-0.671354\pi\)
0.512697 0.858570i \(-0.328646\pi\)
\(272\) 0 0
\(273\) 0.683008 0.0413375
\(274\) 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
\(289\) 3.58075 0.210633
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.6359 1.03030 0.515151 0.857100i \(-0.327736\pi\)
0.515151 + 0.857100i \(0.327736\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 59.4664i 3.43903i
\(300\) 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 \(-0.919972\pi\)
0.968561 0.248776i \(-0.0800281\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.93072i 0.563119i 0.959544 + 0.281560i \(0.0908517\pi\)
−0.959544 + 0.281560i \(0.909148\pi\)
\(312\) 0 0
\(313\) −14.5626 −0.823127 −0.411563 0.911381i \(-0.635017\pi\)
−0.411563 + 0.911381i \(0.635017\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.0259 1.91109 0.955543 0.294853i \(-0.0952708\pi\)
0.955543 + 0.294853i \(0.0952708\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 20.7012 1.15543
\(322\) 0 0
\(323\) 19.7746i 1.10029i
\(324\) 0 0
\(325\) −31.7695 −1.76225
\(326\) 0 0
\(327\) − 14.8055i − 0.818747i
\(328\) 0 0
\(329\) 0.571676 0.0315175
\(330\) 0 0
\(331\) − 2.55398i − 0.140379i −0.997534 0.0701897i \(-0.977640\pi\)
0.997534 0.0701897i \(-0.0223605\pi\)
\(332\) 0 0
\(333\) −15.0570 −0.825120
\(334\) 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
\(343\) − 1.33305i − 0.0719777i
\(344\) 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
\(351\) 33.8866i 1.80873i
\(352\) 0 0
\(353\) −23.8264 −1.26815 −0.634075 0.773272i \(-0.718619\pi\)
−0.634075 + 0.773272i \(0.718619\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0.487659 0.0258096
\(358\) 0 0
\(359\) − 37.1503i − 1.96072i −0.197218 0.980360i \(-0.563191\pi\)
0.197218 0.980360i \(-0.436809\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 12.4102i 0.651368i
\(364\) 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
\(371\) 1.03537i 0.0537538i
\(372\) 0 0
\(373\) 38.5387 1.99546 0.997729 0.0673505i \(-0.0214546\pi\)
0.997729 + 0.0673505i \(0.0214546\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −26.0351 −1.34087
\(378\) 0 0
\(379\) − 35.5694i − 1.82708i −0.406751 0.913539i \(-0.633338\pi\)
0.406751 0.913539i \(-0.366662\pi\)
\(380\) 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
\(391\) 42.4582i 2.14721i
\(392\) 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
\(399\) 0.468557i 0.0234572i
\(400\) 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
\(411\) − 26.2360i − 1.29413i
\(412\) 0 0
\(413\) −1.10330 −0.0542896
\(414\) 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
\(421\) −19.8923 −0.969493 −0.484746 0.874655i \(-0.661088\pi\)
−0.484746 + 0.874655i \(0.661088\pi\)
\(422\) 0 0
\(423\) 10.3630i 0.503864i
\(424\) 0 0
\(425\) −22.6830 −1.10029
\(426\) 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
\(437\) −40.7951 −1.95149
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 12.0744 0.574973
\(442\) 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
\(457\) 41.7822 1.95449 0.977245 0.212116i \(-0.0680353\pi\)
0.977245 + 0.212116i \(0.0680353\pi\)
\(458\) 0 0
\(459\) 24.1946i 1.12931i
\(460\) 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
\(469\) −1.53327 −0.0708000
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) − 21.7945i − 1.00000i
\(476\) 0 0
\(477\) −18.7685 −0.859353
\(478\) 0 0
\(479\) − 42.0000i − 1.91903i −0.281659 0.959514i \(-0.590885\pi\)
0.281659 0.959514i \(-0.409115\pi\)
\(480\) 0 0
\(481\) 55.3920 2.52566
\(482\) 0 0
\(483\) 1.00604i 0.0457766i
\(484\) 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
\(493\) −18.5887 −0.837193
\(494\) 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
\(503\) 37.7220i 1.68194i 0.541081 + 0.840970i \(0.318015\pi\)
−0.541081 + 0.840970i \(0.681985\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 30.8812i − 1.37148i
\(508\) 0 0
\(509\) 26.1534 1.15923 0.579614 0.814891i \(-0.303203\pi\)
0.579614 + 0.814891i \(0.303203\pi\)
\(510\) 0 0
\(511\) 1.31489i 0.0581673i
\(512\) 0 0
\(513\) −23.2468 −1.02637
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) − 29.5063i − 1.29518i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 15.2618i 0.667351i 0.942688 + 0.333676i \(0.108289\pi\)
−0.942688 + 0.333676i \(0.891711\pi\)
\(524\) 0 0
\(525\) −0.537471 −0.0234572
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −64.5916 −2.80833
\(530\) 0 0
\(531\) − 19.9998i − 0.867918i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 29.5063 1.27329
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 9.83544i 0.422079i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 43.5890i − 1.86373i −0.362804 0.931865i \(-0.618181\pi\)
0.362804 0.931865i \(-0.381819\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 17.8606i − 0.760886i
\(552\) 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 \(0.185798\pi\)
−0.834428 + 0.551117i \(0.814202\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.0795999i 0.00334288i
\(568\) 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
\(573\) −21.4403 −0.895680
\(574\) 0 0
\(575\) − 46.7952i − 1.95149i
\(576\) 0 0
\(577\) −42.3539 −1.76322 −0.881608 0.471983i \(-0.843538\pi\)
−0.881608 + 0.471983i \(0.843538\pi\)
\(578\) 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
\(597\) −31.4617 −1.28764
\(598\) 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
\(603\) − 27.7942i − 1.13187i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −0.440457 −0.0178482
\(610\) 0 0
\(611\) − 38.1234i − 1.54231i
\(612\) 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
\(621\) −49.9136 −2.00296
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 39.5492 1.57693
\(630\) 0 0
\(631\) − 34.0000i − 1.35352i −0.736204 0.676759i \(-0.763384\pi\)
0.736204 0.676759i \(-0.236616\pi\)
\(632\) 0 0
\(633\) 32.4924 1.29146
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −44.4196 −1.75997
\(638\) 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
\(647\) 46.2235i 1.81723i 0.417630 + 0.908617i \(0.362861\pi\)
−0.417630 + 0.908617i \(0.637139\pi\)
\(648\) 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
\(657\) −23.8354 −0.929909
\(658\) 0 0
\(659\) − 13.0054i − 0.506618i −0.967385 0.253309i \(-0.918481\pi\)
0.967385 0.253309i \(-0.0815189\pi\)
\(660\) 0 0
\(661\) −51.2465 −1.99326 −0.996629 0.0820394i \(-0.973857\pi\)
−0.996629 + 0.0820394i \(0.973857\pi\)
\(662\) 0 0
\(663\) − 32.5205i − 1.26299i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 38.3486i − 1.48486i
\(668\) 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
\(675\) − 26.6660i − 1.02637i
\(676\) 0 0
\(677\) −42.2209 −1.62268 −0.811340 0.584574i \(-0.801262\pi\)
−0.811340 + 0.584574i \(0.801262\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 5.42709 0.207966
\(682\) 0 0
\(683\) 26.1534i 1.00073i 0.865814 + 0.500366i \(0.166801\pi\)
−0.865814 + 0.500366i \(0.833199\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 69.0460 2.63044
\(690\) 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
\(699\) − 20.3076i − 0.768105i
\(700\) 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
\(717\) 9.91392 0.370242
\(718\) 0 0
\(719\) 36.5787i 1.36415i 0.731281 + 0.682077i \(0.238923\pi\)
−0.731281 + 0.682077i \(0.761077\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 20.4875 0.760886
\(726\) 0 0
\(727\) − 27.1243i − 1.00599i −0.864291 0.502993i \(-0.832232\pi\)
0.864291 0.502993i \(-0.167768\pi\)
\(728\) 0 0
\(729\) −19.4937 −0.721988
\(730\) 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
\(741\) 31.2467 1.14788
\(742\) 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
\(749\) 1.74826 0.0638801
\(750\) 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
\(761\) −51.6176 −1.87114 −0.935569 0.353144i \(-0.885113\pi\)
−0.935569 + 0.353144i \(0.885113\pi\)
\(762\) 0 0
\(763\) − 1.25036i − 0.0452661i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 73.5756i 2.65666i
\(768\) 0 0
\(769\) −14.9437 −0.538884 −0.269442 0.963017i \(-0.586839\pi\)
−0.269442 + 0.963017i \(0.586839\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 47.5643 1.71077 0.855385 0.517993i \(-0.173321\pi\)
0.855385 + 0.517993i \(0.173321\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.937113 0.0336188
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 21.8527i − 0.780953i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.21523i − 0.150257i −0.997174 0.0751284i \(-0.976063\pi\)
0.997174 0.0751284i \(-0.0239367\pi\)
\(788\) 0 0
\(789\) 33.8461 1.20495
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −55.7593 −1.97510 −0.987548 0.157317i \(-0.949716\pi\)
−0.987548 + 0.157317i \(0.949716\pi\)
\(798\) 0 0
\(799\) − 27.2196i − 0.962961i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 29.5063i − 1.03867i
\(808\) 0 0
\(809\) −5.67995 −0.199697 −0.0998483 0.995003i \(-0.531836\pi\)
−0.0998483 + 0.995003i \(0.531836\pi\)
\(810\) 0 0
\(811\) 54.2158i 1.90377i 0.306449 + 0.951887i \(0.400859\pi\)
−0.306449 + 0.951887i \(0.599141\pi\)
\(812\) 0 0
\(813\) −31.8917 −1.11849
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 1.04561i 0.0365367i
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) − 55.6778i − 1.94081i −0.241488 0.970404i \(-0.577635\pi\)
0.241488 0.970404i \(-0.422365\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 56.4722i − 1.96373i −0.189579 0.981866i \(-0.560712\pi\)
0.189579 0.981866i \(-0.439288\pi\)
\(828\) 0 0
\(829\) −37.7081 −1.30966 −0.654828 0.755778i \(-0.727259\pi\)
−0.654828 + 0.755778i \(0.727259\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −31.7150 −1.09886
\(834\) 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
\(841\) −12.2105 −0.421053
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.04807i 0.0360122i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 81.5902i 2.79688i
\(852\) 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
\(867\) − 4.03981i − 0.137199i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 102.250i 3.46459i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −18.2311 −0.615620 −0.307810 0.951448i \(-0.599596\pi\)
−0.307810 + 0.951448i \(0.599596\pi\)
\(878\) 0 0
\(879\) − 19.8969i − 0.671105i
\(880\) 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
\(897\) 67.0901 2.24007
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 49.2979 1.64235
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 47.4466i − 1.57544i −0.616035 0.787719i \(-0.711262\pi\)
0.616035 0.787719i \(-0.288738\pi\)
\(908\) 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
\(919\) 55.4873i 1.83036i 0.403049 + 0.915178i \(0.367950\pi\)
−0.403049 + 0.915178i \(0.632050\pi\)
\(920\) 0 0
\(921\) −9.83544 −0.324089
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −43.5890 −1.43320
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 60.6908 1.99120 0.995601 0.0936938i \(-0.0298675\pi\)
0.995601 + 0.0936938i \(0.0298675\pi\)
\(930\) 0 0
\(931\) − 30.4727i − 0.998703i
\(932\) 0 0
\(933\) 11.2039 0.366798
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −42.7350 −1.39609 −0.698046 0.716053i \(-0.745947\pi\)
−0.698046 + 0.716053i \(0.745947\pi\)
\(938\) 0 0
\(939\) 16.4296i 0.536158i
\(940\) 0 0
\(941\) 35.4517 1.15569 0.577846 0.816146i \(-0.303893\pi\)
0.577846 + 0.816146i \(0.303893\pi\)
\(942\) 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
\(949\) 87.6861 2.84641
\(950\) 0 0
\(951\) − 38.3881i − 1.24482i
\(952\) 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
\(959\) − 2.21569i − 0.0715483i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 31.6913i 1.02124i
\(964\) 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
\(969\) 22.3097 0.716692
\(970\) 0 0
\(971\) 26.1534i 0.839302i 0.907685 + 0.419651i \(0.137848\pi\)
−0.907685 + 0.419651i \(0.862152\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 35.8424i 1.14788i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 22.6657 0.723661
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 0.644966i − 0.0205295i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) −2.88141 −0.0914387
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) 46.4937i 1.47100i
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.607.5 12
4.3 odd 2 inner 1216.2.b.f.607.8 yes 12
8.3 odd 2 inner 1216.2.b.f.607.6 yes 12
8.5 even 2 inner 1216.2.b.f.607.7 yes 12
19.18 odd 2 inner 1216.2.b.f.607.7 yes 12
76.75 even 2 inner 1216.2.b.f.607.6 yes 12
152.37 odd 2 CM 1216.2.b.f.607.5 12
152.75 even 2 inner 1216.2.b.f.607.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.b.f.607.5 12 1.1 even 1 trivial
1216.2.b.f.607.5 12 152.37 odd 2 CM
1216.2.b.f.607.6 yes 12 8.3 odd 2 inner
1216.2.b.f.607.6 yes 12 76.75 even 2 inner
1216.2.b.f.607.7 yes 12 8.5 even 2 inner
1216.2.b.f.607.7 yes 12 19.18 odd 2 inner
1216.2.b.f.607.8 yes 12 4.3 odd 2 inner
1216.2.b.f.607.8 yes 12 152.75 even 2 inner