Properties

Label 1216.2.b.e.607.1
Level $1216$
Weight $2$
Character 1216.607
Analytic conductor $9.710$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.2702336256.1
Defining polynomial: \(x^{8} + 9 x^{6} + 56 x^{4} + 225 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 607.1
Root \(-0.656712 + 2.13746i\) of defining polynomial
Character \(\chi\) \(=\) 1216.607
Dual form 1216.2.b.e.607.8

$q$-expansion

\(f(q)\) \(=\) \(q-3.04547i q^{5} -5.27492i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-3.04547i q^{5} -5.27492i q^{7} +3.00000 q^{9} +6.50958 q^{11} -7.27492 q^{17} +4.35890 q^{19} +4.00000i q^{23} -4.27492 q^{25} -16.0646 q^{35} +5.67232 q^{43} -9.13642i q^{45} +2.72508i q^{47} -20.8248 q^{49} -19.8248i q^{55} +10.8109i q^{61} -15.8248i q^{63} -5.82475 q^{73} -34.3375i q^{77} +9.00000 q^{81} +8.71780 q^{83} +22.1556i q^{85} -13.2749i q^{95} +19.5287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 24q^{9} + O(q^{10}) \) \( 8q + 24q^{9} - 28q^{17} - 4q^{25} - 76q^{49} + 44q^{73} + 72q^{81} + O(q^{100}) \)

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
\(5\) − 3.04547i − 1.36198i −0.732294 0.680989i \(-0.761550\pi\)
0.732294 0.680989i \(-0.238450\pi\)
\(6\) 0 0
\(7\) − 5.27492i − 1.99373i −0.0791130 0.996866i \(-0.525209\pi\)
0.0791130 0.996866i \(-0.474791\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 6.50958 1.96271 0.981356 0.192201i \(-0.0615626\pi\)
0.981356 + 0.192201i \(0.0615626\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −7.27492 −1.76443 −0.882213 0.470850i \(-0.843947\pi\)
−0.882213 + 0.470850i \(0.843947\pi\)
\(18\) 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
\(25\) −4.27492 −0.854983
\(26\) 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
\(35\) −16.0646 −2.71542
\(36\) 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
\(43\) 5.67232 0.865021 0.432511 0.901629i \(-0.357628\pi\)
0.432511 + 0.901629i \(0.357628\pi\)
\(44\) 0 0
\(45\) − 9.13642i − 1.36198i
\(46\) 0 0
\(47\) 2.72508i 0.397494i 0.980051 + 0.198747i \(0.0636872\pi\)
−0.980051 + 0.198747i \(0.936313\pi\)
\(48\) 0 0
\(49\) −20.8248 −2.97496
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) − 19.8248i − 2.67317i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 10.8109i 1.38420i 0.721803 + 0.692099i \(0.243314\pi\)
−0.721803 + 0.692099i \(0.756686\pi\)
\(62\) 0 0
\(63\) − 15.8248i − 1.99373i
\(64\) 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
\(73\) −5.82475 −0.681736 −0.340868 0.940111i \(-0.610721\pi\)
−0.340868 + 0.940111i \(0.610721\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 34.3375i − 3.91312i
\(78\) 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 \(-0.341198\pi\)
0.478451 + 0.878114i \(0.341198\pi\)
\(84\) 0 0
\(85\) 22.1556i 2.40311i
\(86\) 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
\(95\) − 13.2749i − 1.36198i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 19.5287 1.96271
\(100\) 0 0
\(101\) 17.4356i 1.73491i 0.497519 + 0.867453i \(0.334245\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 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
\(115\) 12.1819 1.13597
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 38.3746i 3.51779i
\(120\) 0 0
\(121\) 31.3746 2.85224
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.20822i − 0.197509i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.83507 −0.422442 −0.211221 0.977438i \(-0.567744\pi\)
−0.211221 + 0.977438i \(0.567744\pi\)
\(132\) 0 0
\(133\) − 22.9928i − 1.99373i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.2749 −1.30502 −0.652512 0.757778i \(-0.726285\pi\)
−0.652512 + 0.757778i \(0.726285\pi\)
\(138\) 0 0
\(139\) −18.6915 −1.58539 −0.792695 0.609618i \(-0.791323\pi\)
−0.792695 + 0.609618i \(0.791323\pi\)
\(140\) 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
\(149\) 1.37097i 0.112314i 0.998422 + 0.0561570i \(0.0178847\pi\)
−0.998422 + 0.0561570i \(0.982115\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) −21.8248 −1.76443
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 17.4356i − 1.39151i −0.718278 0.695756i \(-0.755069\pi\)
0.718278 0.695756i \(-0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 21.0997 1.66289
\(162\) 0 0
\(163\) −8.71780 −0.682831 −0.341415 0.939913i \(-0.610906\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(164\) 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
\(175\) 22.5498i 1.70461i
\(176\) 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
\(187\) −47.3566 −3.46306
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.3746i 0.750679i 0.926887 + 0.375339i \(0.122474\pi\)
−0.926887 + 0.375339i \(0.877526\pi\)
\(192\) 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 \(0.213321\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) − 23.8248i − 1.68889i −0.535641 0.844446i \(-0.679930\pi\)
0.535641 0.844446i \(-0.320070\pi\)
\(200\) 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
\(209\) 28.3746 1.96271
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 17.2749i − 1.17814i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −12.8248 −0.854983
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 29.0838i 1.92191i 0.276704 + 0.960955i \(0.410758\pi\)
−0.276704 + 0.960955i \(0.589242\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.9244 1.76388 0.881939 0.471364i \(-0.156238\pi\)
0.881939 + 0.471364i \(0.156238\pi\)
\(234\) 0 0
\(235\) 8.29917 0.541378
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 28.9244i 1.87097i 0.353373 + 0.935483i \(0.385035\pi\)
−0.353373 + 0.935483i \(0.614965\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 63.4213i 4.05184i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 9.02134 0.569422 0.284711 0.958613i \(-0.408102\pi\)
0.284711 + 0.958613i \(0.408102\pi\)
\(252\) 0 0
\(253\) 26.0383i 1.63701i
\(254\) 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
\(263\) 20.9244i 1.29026i 0.764075 + 0.645128i \(0.223196\pi\)
−0.764075 + 0.645128i \(0.776804\pi\)
\(264\) 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
\(275\) −27.8279 −1.67809
\(276\) 0 0
\(277\) − 30.7583i − 1.84809i −0.382288 0.924043i \(-0.624864\pi\)
0.382288 0.924043i \(-0.375136\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 33.3851 1.98454 0.992270 0.124096i \(-0.0396032\pi\)
0.992270 + 0.124096i \(0.0396032\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 35.9244 2.11320
\(290\) 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
\(301\) − 29.9210i − 1.72462i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 32.9244 1.88525
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 21.2749i − 1.20639i −0.797594 0.603195i \(-0.793894\pi\)
0.797594 0.603195i \(-0.206106\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −48.1939 −2.71542
\(316\) 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
\(323\) −31.7106 −1.76443
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.3746 0.792497
\(330\) 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
\(343\) 72.9244i 3.93755i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 34.2224 1.83715 0.918577 0.395242i \(-0.129339\pi\)
0.918577 + 0.395242i \(0.129339\pi\)
\(348\) 0 0
\(349\) − 36.8492i − 1.97249i −0.165277 0.986247i \(-0.552852\pi\)
0.165277 0.986247i \(-0.447148\pi\)
\(350\) 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
\(359\) 34.3746i 1.81422i 0.420892 + 0.907111i \(0.361717\pi\)
−0.420892 + 0.907111i \(0.638283\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 17.7391i 0.928509i
\(366\) 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
\(385\) −104.574 −5.32958
\(386\) 0 0
\(387\) 17.0170 0.865021
\(388\) 0 0
\(389\) − 6.39449i − 0.324213i −0.986773 0.162107i \(-0.948171\pi\)
0.986773 0.162107i \(-0.0518289\pi\)
\(390\) 0 0
\(391\) − 29.0997i − 1.47163i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 13.5529i − 0.680199i −0.940389 0.340099i \(-0.889539\pi\)
0.940389 0.340099i \(-0.110461\pi\)
\(398\) 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
\(405\) − 27.4093i − 1.36198i
\(406\) 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
\(415\) − 26.5498i − 1.30328i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8.71780 −0.425892 −0.212946 0.977064i \(-0.568306\pi\)
−0.212946 + 0.977064i \(0.568306\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 8.17525i 0.397494i
\(424\) 0 0
\(425\) 31.0997 1.50856
\(426\) 0 0
\(427\) 57.0268 2.75972
\(428\) 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
\(441\) −62.4743 −2.97496
\(442\) 0 0
\(443\) 9.85859 0.468396 0.234198 0.972189i \(-0.424754\pi\)
0.234198 + 0.972189i \(0.424754\pi\)
\(444\) 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
\(457\) −25.4743 −1.19164 −0.595818 0.803120i \(-0.703172\pi\)
−0.595818 + 0.803120i \(0.703172\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 42.9402i 1.99992i 0.00872311 + 0.999962i \(0.497223\pi\)
−0.00872311 + 0.999962i \(0.502777\pi\)
\(462\) 0 0
\(463\) − 31.8248i − 1.47902i −0.673145 0.739511i \(-0.735057\pi\)
0.673145 0.739511i \(-0.264943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −35.0596 −1.62237 −0.811183 0.584792i \(-0.801176\pi\)
−0.811183 + 0.584792i \(0.801176\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 36.9244 1.69779
\(474\) 0 0
\(475\) −18.6339 −0.854983
\(476\) 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 \(-0.0577772\pi\)
0.983572 + 0.180517i \(0.0577772\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) − 59.4743i − 2.67317i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 22.8777 1.02415 0.512074 0.858941i \(-0.328877\pi\)
0.512074 + 0.858941i \(0.328877\pi\)
\(500\) 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
\(505\) 53.0997 2.36290
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 30.7251i 1.35920i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 17.7391i 0.780166i
\(518\) 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
\(539\) −135.560 −5.83900
\(540\) 0 0
\(541\) − 41.2657i − 1.77415i −0.461625 0.887075i \(-0.652733\pi\)
0.461625 0.887075i \(-0.347267\pi\)
\(542\) 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
\(549\) 32.4328i 1.38420i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.5237i 1.63230i 0.577838 + 0.816152i \(0.303897\pi\)
−0.577838 + 0.816152i \(0.696103\pi\)
\(558\) 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
\(567\) − 47.4743i − 1.99373i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −26.1534 −1.09449 −0.547243 0.836974i \(-0.684323\pi\)
−0.547243 + 0.836974i \(0.684323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 17.0997i − 0.713105i
\(576\) 0 0
\(577\) −40.0241 −1.66622 −0.833112 0.553104i \(-0.813443\pi\)
−0.833112 + 0.553104i \(0.813443\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 45.9857i − 1.90781i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 30.0361 1.23972 0.619862 0.784711i \(-0.287189\pi\)
0.619862 + 0.784711i \(0.287189\pi\)
\(588\) 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
\(595\) 116.869 4.79116
\(596\) 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
\(605\) − 95.5505i − 3.88468i
\(606\) 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
\(613\) 49.0311i 1.98035i 0.139837 + 0.990174i \(0.455342\pi\)
−0.139837 + 0.990174i \(0.544658\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −48.0241 −1.93338 −0.966689 0.255956i \(-0.917610\pi\)
−0.966689 + 0.255956i \(0.917610\pi\)
\(618\) 0 0
\(619\) 43.5890 1.75199 0.875995 0.482321i \(-0.160206\pi\)
0.875995 + 0.482321i \(0.160206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −28.0997 −1.12399
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 34.0241i − 1.35448i −0.735763 0.677239i \(-0.763176\pi\)
0.735763 0.677239i \(-0.236824\pi\)
\(632\) 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
\(643\) −35.8969 −1.41564 −0.707818 0.706395i \(-0.750320\pi\)
−0.707818 + 0.706395i \(0.750320\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 50.0241i − 1.96665i −0.181857 0.983325i \(-0.558211\pi\)
0.181857 0.983325i \(-0.441789\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 20.2509i − 0.792479i −0.918147 0.396239i \(-0.870315\pi\)
0.918147 0.396239i \(-0.129685\pi\)
\(654\) 0 0
\(655\) 14.7251i 0.575357i
\(656\) 0 0
\(657\) −17.4743 −0.681736
\(658\) 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
\(665\) −70.0241 −2.71542
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 70.3746i 2.71678i
\(672\) 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
\(685\) 46.5194i 1.77741i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −10.6958 −0.406889 −0.203445 0.979086i \(-0.565214\pi\)
−0.203445 + 0.979086i \(0.565214\pi\)
\(692\) 0 0
\(693\) − 103.012i − 3.91312i
\(694\) 0 0
\(695\) 56.9244i 2.15927i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 17.4356i − 0.658533i −0.944237 0.329267i \(-0.893198\pi\)
0.944237 0.329267i \(-0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 91.9713 3.45894
\(708\) 0 0
\(709\) − 52.3068i − 1.96442i −0.187779 0.982211i \(-0.560129\pi\)
0.187779 0.982211i \(-0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 5.62541i − 0.209793i −0.994483 0.104896i \(-0.966549\pi\)
0.994483 0.104896i \(-0.0334511\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 15.4743i − 0.573908i −0.957944 0.286954i \(-0.907357\pi\)
0.957944 0.286954i \(-0.0926427\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −41.2657 −1.52627
\(732\) 0 0
\(733\) 52.3068i 1.93200i 0.258551 + 0.965998i \(0.416755\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.7150 0.872370 0.436185 0.899857i \(-0.356329\pi\)
0.436185 + 0.899857i \(0.356329\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 4.17525 0.152969
\(746\) 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
\(757\) − 47.3566i − 1.72121i −0.509276 0.860603i \(-0.670087\pi\)
0.509276 0.860603i \(-0.329913\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.2749 −1.13371 −0.566857 0.823816i \(-0.691841\pi\)
−0.566857 + 0.823816i \(0.691841\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 66.4667i 2.40311i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 44.3746 1.60019 0.800094 0.599874i \(-0.204783\pi\)
0.800094 + 0.599874i \(0.204783\pi\)
\(770\) 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
\(785\) −53.0997 −1.89521
\(786\) 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
\(799\) − 19.8248i − 0.701349i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −37.9167 −1.33805
\(804\) 0 0
\(805\) − 64.2585i − 2.26481i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 46.5739 1.63745 0.818726 0.574184i \(-0.194681\pi\)
0.818726 + 0.574184i \(0.194681\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 26.5498i 0.930000i
\(816\) 0 0
\(817\) 24.7251 0.865021
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 56.7966i 1.98221i 0.133064 + 0.991107i \(0.457518\pi\)
−0.133064 + 0.991107i \(0.542482\pi\)
\(822\) 0 0
\(823\) 44.5739i 1.55375i 0.629655 + 0.776875i \(0.283196\pi\)
−0.629655 + 0.776875i \(0.716804\pi\)
\(824\) 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
\(833\) 151.498 5.24911
\(834\) 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
\(845\) 39.5912i 1.36198i
\(846\) 0 0
\(847\) − 165.498i − 5.68659i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 52.3068i − 1.79095i −0.445112 0.895475i \(-0.646836\pi\)
0.445112 0.895475i \(-0.353164\pi\)
\(854\) 0 0
\(855\) − 39.8248i − 1.36198i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −15.3425 −0.523478 −0.261739 0.965139i \(-0.584296\pi\)
−0.261739 + 0.965139i \(0.584296\pi\)
\(860\) 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
\(875\) −11.6482 −0.393781
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −24.0241 −0.809392 −0.404696 0.914451i \(-0.632623\pi\)
−0.404696 + 0.914451i \(0.632623\pi\)
\(882\) 0 0
\(883\) −59.4234 −1.99976 −0.999879 0.0155546i \(-0.995049\pi\)
−0.999879 + 0.0155546i \(0.995049\pi\)
\(884\) 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
\(891\) 58.5862 1.96271
\(892\) 0 0
\(893\) 11.8784i 0.397494i
\(894\) 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
\(913\) 56.7492 1.87812
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 25.5046i 0.842236i
\(918\) 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
\(931\) −90.7730 −2.97496
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 144.223i 4.71661i
\(936\) 0 0
\(937\) −57.4743 −1.87760 −0.938801 0.344460i \(-0.888062\pi\)
−0.938801 + 0.344460i \(0.888062\pi\)
\(938\) 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 \(-0.0414921\pi\)
0.991516 + 0.129983i \(0.0414921\pi\)
\(948\) 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
\(955\) 31.5955 1.02241
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 80.5739i 2.60187i
\(960\) 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
\(973\) 98.5960i 3.16084i
\(974\) 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
\(985\) 53.0997 1.69190
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 22.6893i 0.721478i
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −72.5577 −2.30023
\(996\) 0 0
\(997\) 52.3802i 1.65890i 0.558584 + 0.829448i \(0.311345\pi\)
−0.558584 + 0.829448i \(0.688655\pi\)
\(998\) 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.e.607.1 8
4.3 odd 2 inner 1216.2.b.e.607.2 yes 8
8.3 odd 2 inner 1216.2.b.e.607.8 yes 8
8.5 even 2 inner 1216.2.b.e.607.7 yes 8
19.18 odd 2 CM 1216.2.b.e.607.1 8
76.75 even 2 inner 1216.2.b.e.607.2 yes 8
152.37 odd 2 inner 1216.2.b.e.607.7 yes 8
152.75 even 2 inner 1216.2.b.e.607.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1216.2.b.e.607.1 8 1.1 even 1 trivial
1216.2.b.e.607.1 8 19.18 odd 2 CM
1216.2.b.e.607.2 yes 8 4.3 odd 2 inner
1216.2.b.e.607.2 yes 8 76.75 even 2 inner
1216.2.b.e.607.7 yes 8 8.5 even 2 inner
1216.2.b.e.607.7 yes 8 152.37 odd 2 inner
1216.2.b.e.607.8 yes 8 8.3 odd 2 inner
1216.2.b.e.607.8 yes 8 152.75 even 2 inner