Properties

Label 1216.2.h.b.1215.4
Level $1216$
Weight $2$
Character 1216.1215
Analytic conductor $9.710$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1215.4
Root \(2.13746 + 0.656712i\) of defining polynomial
Character \(\chi\) \(=\) 1216.1215
Dual form 1216.2.h.b.1215.3

$q$-expansion

\(f(q)\) \(=\) \(q+3.27492 q^{5} +0.418627i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+3.27492 q^{5} +0.418627i q^{7} -3.00000 q^{9} +6.50958i q^{11} -7.27492 q^{17} +4.35890i q^{19} +8.71780i q^{23} +5.72508 q^{25} +1.37097i q^{35} -5.67232i q^{43} -9.82475 q^{45} -13.4378i q^{47} +6.82475 q^{49} +21.3183i q^{55} +11.2749 q^{61} -1.25588i q^{63} +5.82475 q^{73} -2.72508 q^{77} +9.00000 q^{81} +8.71780i q^{83} -23.8248 q^{85} +14.2750i q^{95} -19.5287i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{5} - 12q^{9} + O(q^{10}) \) \( 4q - 2q^{5} - 12q^{9} - 14q^{17} + 38q^{25} + 6q^{45} - 18q^{49} + 30q^{61} - 22q^{73} - 26q^{77} + 36q^{81} - 50q^{85} + O(q^{100}) \)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 3.27492 1.46459 0.732294 0.680989i \(-0.238450\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) 0.418627i 0.158226i 0.996866 + 0.0791130i \(0.0252088\pi\)
−0.996866 + 0.0791130i \(0.974791\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 6.50958i 1.96271i 0.192201 + 0.981356i \(0.438437\pi\)
−0.192201 + 0.981356i \(0.561563\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 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
\(19\) 4.35890i 1.00000i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.71780i 1.81779i 0.417029 + 0.908893i \(0.363071\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 5.72508 1.14502
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.37097i 0.231736i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 5.67232i − 0.865021i −0.901629 0.432511i \(-0.857628\pi\)
0.901629 0.432511i \(-0.142372\pi\)
\(44\) 0 0
\(45\) −9.82475 −1.46459
\(46\) 0 0
\(47\) − 13.4378i − 1.96010i −0.198747 0.980051i \(-0.563687\pi\)
0.198747 0.980051i \(-0.436313\pi\)
\(48\) 0 0
\(49\) 6.82475 0.974965
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 21.3183i 2.87456i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 11.2749 1.44361 0.721803 0.692099i \(-0.243314\pi\)
0.721803 + 0.692099i \(0.243314\pi\)
\(62\) 0 0
\(63\) − 1.25588i − 0.158226i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 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 \(-0.389279\pi\)
0.340868 + 0.940111i \(0.389279\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.72508 −0.310552
\(78\) 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
\(83\) 8.71780i 0.956903i 0.878114 + 0.478451i \(0.158802\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −23.8248 −2.58416
\(86\) 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
\(95\) 14.2750i 1.46459i
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) − 19.5287i − 1.96271i
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 28.5501i 2.66231i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 3.04547i − 0.279178i
\(120\) 0 0
\(121\) −31.3746 −2.85224
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.37459 0.212389
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.83507i 0.422442i 0.977438 + 0.211221i \(0.0677440\pi\)
−0.977438 + 0.211221i \(0.932256\pi\)
\(132\) 0 0
\(133\) −1.82475 −0.158226
\(134\) 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
\(139\) 18.6915i 1.58539i 0.609618 + 0.792695i \(0.291323\pi\)
−0.609618 + 0.792695i \(0.708677\pi\)
\(140\) 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
\(149\) 24.3746 1.99684 0.998422 0.0561570i \(-0.0178847\pi\)
0.998422 + 0.0561570i \(0.0178847\pi\)
\(150\) 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
\(157\) 18.0000 1.43656 0.718278 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.64950 −0.287621
\(162\) 0 0
\(163\) 8.71780i 0.682831i 0.939913 + 0.341415i \(0.110906\pi\)
−0.939913 + 0.341415i \(0.889094\pi\)
\(164\) 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
\(171\) − 13.0767i − 1.00000i
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 2.39667i 0.181171i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 47.3566i − 3.46306i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 25.6197i − 1.85377i −0.375339 0.926887i \(-0.622474\pi\)
0.375339 0.926887i \(-0.377526\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) − 15.1123i − 1.07128i −0.844446 0.535641i \(-0.820070\pi\)
0.844446 0.535641i \(-0.179930\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 26.1534i − 1.81779i
\(208\) 0 0
\(209\) −28.3746 −1.96271
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 18.5764i − 1.26690i
\(216\) 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
\(225\) −17.1752 −1.14502
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 8.37459 0.553408 0.276704 0.960955i \(-0.410758\pi\)
0.276704 + 0.960955i \(0.410758\pi\)
\(230\) 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
\(235\) − 44.0076i − 2.87074i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 10.9260i 0.706745i 0.935483 + 0.353373i \(0.114965\pi\)
−0.935483 + 0.353373i \(0.885035\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 22.3505 1.42792
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 9.02134i − 0.569422i −0.958613 0.284711i \(-0.908102\pi\)
0.958613 0.284711i \(-0.0918976\pi\)
\(252\) 0 0
\(253\) −56.7492 −3.56779
\(254\) 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
\(263\) 24.7824i 1.52815i 0.645128 + 0.764075i \(0.276804\pi\)
−0.645128 + 0.764075i \(0.723196\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) − 26.1534i − 1.58871i −0.607457 0.794353i \(-0.707810\pi\)
0.607457 0.794353i \(-0.292190\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 37.2679i 2.24734i
\(276\) 0 0
\(277\) −12.7251 −0.764576 −0.382288 0.924043i \(-0.624864\pi\)
−0.382288 + 0.924043i \(0.624864\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) − 33.3851i − 1.98454i −0.124096 0.992270i \(-0.539603\pi\)
0.124096 0.992270i \(-0.460397\pi\)
\(284\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.37459 0.136869
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 36.9244 2.11429
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.1314i 1.59519i 0.603195 + 0.797594i \(0.293894\pi\)
−0.603195 + 0.797594i \(0.706106\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) − 4.11290i − 0.231736i
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 31.7106i − 1.76443i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 5.62541 0.310139
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 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
\(343\) 5.78741i 0.312491i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 34.2224i 1.83715i 0.395242 + 0.918577i \(0.370661\pi\)
−0.395242 + 0.918577i \(0.629339\pi\)
\(348\) 0 0
\(349\) 6.17525 0.330553 0.165277 0.986247i \(-0.447148\pi\)
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 \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.9495i 0.841785i 0.907111 + 0.420892i \(0.138283\pi\)
−0.907111 + 0.420892i \(0.861717\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 19.0756 0.998461
\(366\) 0 0
\(367\) − 26.1534i − 1.36520i −0.730794 0.682598i \(-0.760850\pi\)
0.730794 0.682598i \(-0.239150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −8.92442 −0.454831
\(386\) 0 0
\(387\) 17.0170i 0.865021i
\(388\) 0 0
\(389\) −38.9244 −1.97355 −0.986773 0.162107i \(-0.948171\pi\)
−0.986773 + 0.162107i \(0.948171\pi\)
\(390\) 0 0
\(391\) − 63.4213i − 3.20735i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.4743 1.88078 0.940389 0.340099i \(-0.110461\pi\)
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
\(405\) 29.4743 1.46459
\(406\) 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
\(415\) 28.5501i 1.40147i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.71780i 0.425892i 0.977064 + 0.212946i \(0.0683059\pi\)
−0.977064 + 0.212946i \(0.931694\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 40.3133i 1.96010i
\(424\) 0 0
\(425\) −41.6495 −2.02030
\(426\) 0 0
\(427\) 4.71998i 0.228416i
\(428\) 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
\(437\) −38.0000 −1.81779
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −20.4743 −0.974965
\(442\) 0 0
\(443\) 9.85859i 0.468396i 0.972189 + 0.234198i \(0.0752464\pi\)
−0.972189 + 0.234198i \(0.924754\pi\)
\(444\) 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
\(461\) 0.374586 0.0174462 0.00872311 0.999962i \(-0.497223\pi\)
0.00872311 + 0.999962i \(0.497223\pi\)
\(462\) 0 0
\(463\) − 28.9687i − 1.34629i −0.739511 0.673145i \(-0.764943\pi\)
0.739511 0.673145i \(-0.235057\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 35.0596i − 1.62237i −0.584792 0.811183i \(-0.698824\pi\)
0.584792 0.811183i \(-0.301176\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 36.9244 1.69779
\(474\) 0 0
\(475\) 24.9551i 1.14502i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 43.5890i 1.99163i 0.0913823 + 0.995816i \(0.470871\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 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
\(491\) 43.5890i 1.96714i 0.180517 + 0.983572i \(0.442223\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) − 63.9550i − 2.87456i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 22.8777i − 1.02415i −0.858941 0.512074i \(-0.828877\pi\)
0.858941 0.512074i \(-0.171123\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.71780i 0.388707i 0.980932 + 0.194354i \(0.0622609\pi\)
−0.980932 + 0.194354i \(0.937739\pi\)
\(504\) 0 0
\(505\) 32.7492 1.45732
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 2.43840i 0.107868i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 87.4743 3.84711
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −53.0000 −2.30435
\(530\) 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
\(539\) 44.4262i 1.91357i
\(540\) 0 0
\(541\) 21.4743 0.923250 0.461625 0.887075i \(-0.347267\pi\)
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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) −33.8248 −1.44361
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.2749 1.15568 0.577838 0.816152i \(-0.303897\pi\)
0.577838 + 0.816152i \(0.303897\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.76764i 0.158226i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) − 26.1534i − 1.09449i −0.836974 0.547243i \(-0.815677\pi\)
0.836974 0.547243i \(-0.184323\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 49.9101i 2.08140i
\(576\) 0 0
\(577\) 40.0241 1.66622 0.833112 0.553104i \(-0.186557\pi\)
0.833112 + 0.553104i \(0.186557\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.64950 −0.151407
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 30.0361i − 1.23972i −0.784711 0.619862i \(-0.787189\pi\)
0.784711 0.619862i \(-0.212811\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 34.0000 1.39621 0.698106 0.715994i \(-0.254026\pi\)
0.698106 + 0.715994i \(0.254026\pi\)
\(594\) 0 0
\(595\) − 9.97368i − 0.408881i
\(596\) 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
\(605\) −102.749 −4.17735
\(606\) 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
\(613\) −6.92442 −0.279675 −0.139837 0.990174i \(-0.544658\pi\)
−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 \(-0.0823901\pi\)
0.966689 + 0.255956i \(0.0823901\pi\)
\(618\) 0 0
\(619\) 43.5890i 1.75199i 0.482321 + 0.875995i \(0.339794\pi\)
−0.482321 + 0.875995i \(0.660206\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −20.8488 −0.833954
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 36.9643i 1.47153i 0.677239 + 0.735763i \(0.263176\pi\)
−0.677239 + 0.735763i \(0.736824\pi\)
\(632\) 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
\(643\) 35.8969i 1.41564i 0.706395 + 0.707818i \(0.250320\pi\)
−0.706395 + 0.707818i \(0.749680\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.25151i 0.363714i 0.983325 + 0.181857i \(0.0582109\pi\)
−0.983325 + 0.181857i \(0.941789\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −46.9244 −1.83629 −0.918147 0.396239i \(-0.870315\pi\)
−0.918147 + 0.396239i \(0.870315\pi\)
\(654\) 0 0
\(655\) 15.8345i 0.618703i
\(656\) 0 0
\(657\) −17.4743 −0.681736
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −5.97591 −0.231736
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 73.3949i 2.83338i
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −50.0241 −1.91132
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) − 10.6958i − 0.406889i −0.979086 0.203445i \(-0.934786\pi\)
0.979086 0.203445i \(-0.0652137\pi\)
\(692\) 0 0
\(693\) 8.17525 0.310552
\(694\) 0 0
\(695\) 61.2130i 2.32194i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.18627i 0.157441i
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
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
\(719\) − 53.3325i − 1.98897i −0.104896 0.994483i \(-0.533451\pi\)
0.104896 0.994483i \(-0.466549\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 51.6580i − 1.91589i −0.286954 0.957944i \(-0.592643\pi\)
0.286954 0.957944i \(-0.407357\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 41.2657i 1.52627i
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 23.7150i 0.872370i 0.899857 + 0.436185i \(0.143671\pi\)
−0.899857 + 0.436185i \(0.856329\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 79.8248 2.92455
\(746\) 0 0
\(747\) − 26.1534i − 0.956903i
\(748\) 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
\(757\) −28.0241 −1.01855 −0.509276 0.860603i \(-0.670087\pi\)
−0.509276 + 0.860603i \(0.670087\pi\)
\(758\) 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
\(765\) 71.4743 2.58416
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −44.3746 −1.60019 −0.800094 0.599874i \(-0.795217\pi\)
−0.800094 + 0.599874i \(0.795217\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 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
\(785\) 58.9485 2.10396
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 97.7587i 3.45846i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 37.9167i 1.33805i
\(804\) 0 0
\(805\) −11.9518 −0.421246
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −46.5739 −1.63745 −0.818726 0.574184i \(-0.805319\pi\)
−0.818726 + 0.574184i \(0.805319\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 28.5501i 1.00007i
\(816\) 0 0
\(817\) 24.7251 0.865021
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −7.62541 −0.266129 −0.133064 0.991107i \(-0.542482\pi\)
−0.133064 + 0.991107i \(0.542482\pi\)
\(822\) 0 0
\(823\) − 36.1271i − 1.25931i −0.776875 0.629655i \(-0.783196\pi\)
0.776875 0.629655i \(-0.216804\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −49.6495 −1.72025
\(834\) 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
\(845\) 42.5739 1.46459
\(846\) 0 0
\(847\) − 13.1342i − 0.451298i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) − 42.8251i − 1.46459i
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 15.3425i 0.523478i 0.965139 + 0.261739i \(0.0842960\pi\)
−0.965139 + 0.261739i \(0.915704\pi\)
\(860\) 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
\(875\) 0.994065i 0.0336055i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0241 0.809392 0.404696 0.914451i \(-0.367377\pi\)
0.404696 + 0.914451i \(0.367377\pi\)
\(882\) 0 0
\(883\) − 59.4234i − 1.99976i −0.0155546 0.999879i \(-0.504951\pi\)
0.0155546 0.999879i \(-0.495049\pi\)
\(884\) 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
\(891\) 58.5862i 1.96271i
\(892\) 0 0
\(893\) 58.5739 1.96010
\(894\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) −30.0000 −0.995037
\(910\) 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
\(917\) −2.02409 −0.0668413
\(918\) 0 0
\(919\) 8.71780i 0.287574i 0.989609 + 0.143787i \(0.0459280\pi\)
−0.989609 + 0.143787i \(0.954072\pi\)
\(920\) 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 \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 29.7484i 0.974965i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 155.089i − 5.07195i
\(936\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 61.0246i − 1.98303i −0.129983 0.991516i \(-0.541492\pi\)
0.129983 0.991516i \(-0.458508\pi\)
\(948\) 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
\(955\) − 83.9023i − 2.71502i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 6.39449i − 0.206489i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 61.0246i − 1.96242i −0.192947 0.981209i \(-0.561805\pi\)
0.192947 0.981209i \(-0.438195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) −7.82475 −0.250850
\(974\) 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
\(985\) −72.0482 −2.29565
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 49.4502 1.57242
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 49.4915i − 1.56899i
\(996\) 0 0
\(997\) 35.2749 1.11717 0.558584 0.829448i \(-0.311345\pi\)
0.558584 + 0.829448i \(0.311345\pi\)
\(998\) 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.h.b.1215.4 4
4.3 odd 2 inner 1216.2.h.b.1215.3 4
8.3 odd 2 304.2.h.c.303.1 4
8.5 even 2 304.2.h.c.303.2 yes 4
19.18 odd 2 CM 1216.2.h.b.1215.4 4
24.5 odd 2 2736.2.k.j.2431.4 4
24.11 even 2 2736.2.k.j.2431.3 4
76.75 even 2 inner 1216.2.h.b.1215.3 4
152.37 odd 2 304.2.h.c.303.2 yes 4
152.75 even 2 304.2.h.c.303.1 4
456.227 odd 2 2736.2.k.j.2431.3 4
456.341 even 2 2736.2.k.j.2431.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
304.2.h.c.303.1 4 8.3 odd 2
304.2.h.c.303.1 4 152.75 even 2
304.2.h.c.303.2 yes 4 8.5 even 2
304.2.h.c.303.2 yes 4 152.37 odd 2
1216.2.h.b.1215.3 4 4.3 odd 2 inner
1216.2.h.b.1215.3 4 76.75 even 2 inner
1216.2.h.b.1215.4 4 1.1 even 1 trivial
1216.2.h.b.1215.4 4 19.18 odd 2 CM
2736.2.k.j.2431.3 4 24.11 even 2
2736.2.k.j.2431.3 4 456.227 odd 2
2736.2.k.j.2431.4 4 24.5 odd 2
2736.2.k.j.2431.4 4 456.341 even 2