Properties

Label 48.7.e.a.17.1
Level 48
Weight 7
Character 48.17
Self dual yes
Analytic conductor 11.043
Analytic rank 0
Dimension 1
CM discriminant -3
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 48.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0425960138\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 17.1
Character \(\chi\) \(=\) 48.17

$q$-expansion

\(f(q)\) \(=\) \(q+27.0000 q^{3} +286.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q+27.0000 q^{3} +286.000 q^{7} +729.000 q^{9} +506.000 q^{13} +10582.0 q^{19} +7722.00 q^{21} +15625.0 q^{25} +19683.0 q^{27} -35282.0 q^{31} -89206.0 q^{37} +13662.0 q^{39} -111386. q^{43} -35853.0 q^{49} +285714. q^{57} -420838. q^{61} +208494. q^{63} -172874. q^{67} +638066. q^{73} +421875. q^{75} +204622. q^{79} +531441. q^{81} +144716. q^{91} -952614. q^{93} -56446.0 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 27.0000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 286.000 0.833819 0.416910 0.908948i \(-0.363113\pi\)
0.416910 + 0.908948i \(0.363113\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 506.000 0.230314 0.115157 0.993347i \(-0.463263\pi\)
0.115157 + 0.993347i \(0.463263\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 10582.0 1.54279 0.771395 0.636356i \(-0.219559\pi\)
0.771395 + 0.636356i \(0.219559\pi\)
\(20\) 0 0
\(21\) 7722.00 0.833819
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 19683.0 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −35282.0 −1.18432 −0.592159 0.805821i \(-0.701724\pi\)
−0.592159 + 0.805821i \(0.701724\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −89206.0 −1.76112 −0.880560 0.473935i \(-0.842833\pi\)
−0.880560 + 0.473935i \(0.842833\pi\)
\(38\) 0 0
\(39\) 13662.0 0.230314
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −111386. −1.40096 −0.700479 0.713673i \(-0.747030\pi\)
−0.700479 + 0.713673i \(0.747030\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) −35853.0 −0.304745
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 285714. 1.54279
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −420838. −1.85407 −0.927034 0.374978i \(-0.877650\pi\)
−0.927034 + 0.374978i \(0.877650\pi\)
\(62\) 0 0
\(63\) 208494. 0.833819
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −172874. −0.574785 −0.287392 0.957813i \(-0.592788\pi\)
−0.287392 + 0.957813i \(0.592788\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 638066. 1.64020 0.820100 0.572220i \(-0.193918\pi\)
0.820100 + 0.572220i \(0.193918\pi\)
\(74\) 0 0
\(75\) 421875. 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 204622. 0.415022 0.207511 0.978233i \(-0.433464\pi\)
0.207511 + 0.978233i \(0.433464\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 144716. 0.192040
\(92\) 0 0
\(93\) −952614. −1.18432
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −56446.0 −0.0618469 −0.0309235 0.999522i \(-0.509845\pi\)
−0.0309235 + 0.999522i \(0.509845\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.12695e6 −1.03132 −0.515658 0.856795i \(-0.672452\pi\)
−0.515658 + 0.856795i \(0.672452\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −2.17274e6 −1.67776 −0.838878 0.544320i \(-0.816788\pi\)
−0.838878 + 0.544320i \(0.816788\pi\)
\(110\) 0 0
\(111\) −2.40856e6 −1.76112
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 368874. 0.230314
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.95237e6 1.92951 0.964753 0.263158i \(-0.0847641\pi\)
0.964753 + 0.263158i \(0.0847641\pi\)
\(128\) 0 0
\(129\) −3.00742e6 −1.40096
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 3.02645e6 1.28641
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −1.26454e6 −0.470855 −0.235428 0.971892i \(-0.575649\pi\)
−0.235428 + 0.971892i \(0.575649\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −968031. −0.304745
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 3.83040e6 1.11253 0.556267 0.831004i \(-0.312233\pi\)
0.556267 + 0.831004i \(0.312233\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.08271e6 1.83021 0.915105 0.403216i \(-0.132108\pi\)
0.915105 + 0.403216i \(0.132108\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.89851e6 −0.669285 −0.334643 0.942345i \(-0.608616\pi\)
−0.334643 + 0.942345i \(0.608616\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −4.57077e6 −0.946955
\(170\) 0 0
\(171\) 7.71428e6 1.54279
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.46875e6 0.833819
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 98282.0 0.0165744 0.00828721 0.999966i \(-0.497362\pi\)
0.00828721 + 0.999966i \(0.497362\pi\)
\(182\) 0 0
\(183\) −1.13626e7 −1.85407
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 5.62934e6 0.833819
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.30556e7 −1.81604 −0.908020 0.418927i \(-0.862406\pi\)
−0.908020 + 0.418927i \(0.862406\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.16545e7 −1.47888 −0.739442 0.673220i \(-0.764911\pi\)
−0.739442 + 0.673220i \(0.764911\pi\)
\(200\) 0 0
\(201\) −4.66760e6 −0.574785
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.75972e7 −1.87325 −0.936624 0.350336i \(-0.886067\pi\)
−0.936624 + 0.350336i \(0.886067\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.00907e7 −0.987507
\(218\) 0 0
\(219\) 1.72278e7 1.64020
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.18107e7 1.06503 0.532516 0.846420i \(-0.321247\pi\)
0.532516 + 0.846420i \(0.321247\pi\)
\(224\) 0 0
\(225\) 1.13906e7 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −4.07282e6 −0.339148 −0.169574 0.985517i \(-0.554239\pi\)
−0.169574 + 0.985517i \(0.554239\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 5.52479e6 0.415022
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 2.64398e7 1.88889 0.944447 0.328663i \(-0.106598\pi\)
0.944447 + 0.328663i \(0.106598\pi\)
\(242\) 0 0
\(243\) 1.43489e7 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.35449e6 0.355326
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 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
\(259\) −2.55129e7 −1.46846
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 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
\(271\) 3.91457e7 1.96687 0.983436 0.181258i \(-0.0580168\pi\)
0.983436 + 0.181258i \(0.0580168\pi\)
\(272\) 0 0
\(273\) 3.90733e6 0.192040
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.62670e7 −1.23586 −0.617932 0.786232i \(-0.712029\pi\)
−0.617932 + 0.786232i \(0.712029\pi\)
\(278\) 0 0
\(279\) −2.57206e7 −1.18432
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.39704e7 0.616380 0.308190 0.951325i \(-0.400277\pi\)
0.308190 + 0.951325i \(0.400277\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) −1.52404e6 −0.0618469
\(292\) 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
\(301\) −3.18564e7 −1.16815
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.53407e7 −1.91262 −0.956312 0.292348i \(-0.905564\pi\)
−0.956312 + 0.292348i \(0.905564\pi\)
\(308\) 0 0
\(309\) −3.04275e7 −1.03132
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.88715e7 1.26765 0.633824 0.773478i \(-0.281485\pi\)
0.633824 + 0.773478i \(0.281485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 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
\(323\) 0 0
\(324\) 0 0
\(325\) 7.90625e6 0.230314
\(326\) 0 0
\(327\) −5.86640e7 −1.67776
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.15453e7 1.97286 0.986432 0.164169i \(-0.0524943\pi\)
0.986432 + 0.164169i \(0.0524943\pi\)
\(332\) 0 0
\(333\) −6.50312e7 −1.76112
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5.22406e7 −1.36496 −0.682478 0.730906i \(-0.739098\pi\)
−0.682478 + 0.730906i \(0.739098\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.39016e7 −1.08792
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 5.69263e7 1.33917 0.669586 0.742734i \(-0.266471\pi\)
0.669586 + 0.742734i \(0.266471\pi\)
\(350\) 0 0
\(351\) 9.95960e6 0.230314
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 6.49328e7 1.38020
\(362\) 0 0
\(363\) 4.78321e7 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00261e7 1.61895 0.809475 0.587154i \(-0.199752\pi\)
0.809475 + 0.587154i \(0.199752\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 4.87323e7 0.939053 0.469527 0.882918i \(-0.344425\pi\)
0.469527 + 0.882918i \(0.344425\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.51948e7 0.646489 0.323245 0.946315i \(-0.395226\pi\)
0.323245 + 0.946315i \(0.395226\pi\)
\(380\) 0 0
\(381\) 1.06714e8 1.92951
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.12004e7 −1.40096
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.23725e8 −1.97737 −0.988684 0.150013i \(-0.952068\pi\)
−0.988684 + 0.150013i \(0.952068\pi\)
\(398\) 0 0
\(399\) 8.17142e7 1.28641
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.78527e7 −0.272765
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.88393e7 −1.00616 −0.503080 0.864240i \(-0.667800\pi\)
−0.503080 + 0.864240i \(0.667800\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −3.41425e7 −0.470855
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.44474e8 1.93617 0.968086 0.250620i \(-0.0806343\pi\)
0.968086 + 0.250620i \(0.0806343\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.20360e8 −1.54596
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −1.55657e8 −1.91737 −0.958685 0.284469i \(-0.908183\pi\)
−0.958685 + 0.284469i \(0.908183\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −5.35167e7 −0.632552 −0.316276 0.948667i \(-0.602433\pi\)
−0.316276 + 0.948667i \(0.602433\pi\)
\(440\) 0 0
\(441\) −2.61368e7 −0.304745
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 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
\(453\) 1.03421e8 1.11253
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.93439e7 −0.307446 −0.153723 0.988114i \(-0.549126\pi\)
−0.153723 + 0.988114i \(0.549126\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.92743e8 −1.94194 −0.970968 0.239209i \(-0.923112\pi\)
−0.970968 + 0.239209i \(0.923112\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) −4.94420e7 −0.479267
\(470\) 0 0
\(471\) 1.91233e8 1.83021
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.65344e8 1.54279
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −4.51382e7 −0.405611
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.05807e8 −1.78186 −0.890931 0.454139i \(-0.849947\pi\)
−0.890931 + 0.454139i \(0.849947\pi\)
\(488\) 0 0
\(489\) −7.82597e7 −0.669285
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.94045e7 0.156171 0.0780856 0.996947i \(-0.475119\pi\)
0.0780856 + 0.996947i \(0.475119\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.23411e8 −0.946955
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 1.82487e8 1.36763
\(512\) 0 0
\(513\) 2.08286e8 1.54279
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.41150e8 0.986677 0.493338 0.869837i \(-0.335776\pi\)
0.493338 + 0.869837i \(0.335776\pi\)
\(524\) 0 0
\(525\) 1.20656e8 0.833819
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 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
\(539\) 0 0
\(540\) 0 0
\(541\) 1.97611e8 1.24801 0.624006 0.781419i \(-0.285504\pi\)
0.624006 + 0.781419i \(0.285504\pi\)
\(542\) 0 0
\(543\) 2.65361e6 0.0165744
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.24645e8 1.98357 0.991783 0.127929i \(-0.0408330\pi\)
0.991783 + 0.127929i \(0.0408330\pi\)
\(548\) 0 0
\(549\) −3.06791e8 −1.85407
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 5.85219e7 0.346053
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −5.63613e7 −0.322660
\(560\) 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
\(567\) 1.51992e8 0.833819
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −1.71111e8 −0.919112 −0.459556 0.888149i \(-0.651991\pi\)
−0.459556 + 0.888149i \(0.651991\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.05560e7 −0.367288 −0.183644 0.982993i \(-0.558789\pi\)
−0.183644 + 0.982993i \(0.558789\pi\)
\(578\) 0 0
\(579\) −3.52502e8 −1.81604
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) −3.73354e8 −1.82715
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.14671e8 −1.47888
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 4.24444e8 1.95522 0.977612 0.210415i \(-0.0674816\pi\)
0.977612 + 0.210415i \(0.0674816\pi\)
\(602\) 0 0
\(603\) −1.26025e8 −0.574785
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.60399e8 −1.61145 −0.805727 0.592287i \(-0.798225\pi\)
−0.805727 + 0.592287i \(0.798225\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.58281e8 −0.687142 −0.343571 0.939127i \(-0.611637\pi\)
−0.343571 + 0.939127i \(0.611637\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −2.36189e8 −0.995836 −0.497918 0.867224i \(-0.665902\pi\)
−0.497918 + 0.867224i \(0.665902\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 4.98900e8 1.98575 0.992876 0.119152i \(-0.0380177\pi\)
0.992876 + 0.119152i \(0.0380177\pi\)
\(632\) 0 0
\(633\) −4.75123e8 −1.87325
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.81416e7 −0.0701872
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 3.58510e8 1.34855 0.674277 0.738479i \(-0.264456\pi\)
0.674277 + 0.738479i \(0.264456\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.72448e8 −0.987507
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 4.65150e8 1.64020
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.58097e8 −0.547419 −0.273710 0.961812i \(-0.588251\pi\)
−0.273710 + 0.961812i \(0.588251\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.18890e8 1.06503
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −3.12399e7 −0.102486 −0.0512430 0.998686i \(-0.516318\pi\)
−0.0512430 + 0.998686i \(0.516318\pi\)
\(674\) 0 0
\(675\) 3.07547e8 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1.61436e7 −0.0515691
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.09966e8 −0.339148
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 4.01570e8 1.21710 0.608551 0.793515i \(-0.291751\pi\)
0.608551 + 0.793515i \(0.291751\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −9.43978e8 −2.71704
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5.93732e8 1.66591 0.832955 0.553341i \(-0.186647\pi\)
0.832955 + 0.553341i \(0.186647\pi\)
\(710\) 0 0
\(711\) 1.49169e8 0.415022
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −3.22307e8 −0.859930
\(722\) 0 0
\(723\) 7.13876e8 1.88889
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6.52273e8 1.69756 0.848782 0.528743i \(-0.177337\pi\)
0.848782 + 0.528743i \(0.177337\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.61163e8 −1.42488 −0.712438 0.701735i \(-0.752409\pi\)
−0.712438 + 0.701735i \(0.752409\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.77287e8 −0.439281 −0.219641 0.975581i \(-0.570488\pi\)
−0.219641 + 0.975581i \(0.570488\pi\)
\(740\) 0 0
\(741\) 1.44571e8 0.355326
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.97133e8 0.701506 0.350753 0.936468i \(-0.385926\pi\)
0.350753 + 0.936468i \(0.385926\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.52165e8 1.96443 0.982214 0.187767i \(-0.0601251\pi\)
0.982214 + 0.187767i \(0.0601251\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −6.21404e8 −1.39894
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −8.88298e8 −1.95335 −0.976674 0.214727i \(-0.931114\pi\)
−0.976674 + 0.214727i \(0.931114\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −5.51281e8 −1.18432
\(776\) 0 0
\(777\) −6.88849e8 −1.46846
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −9.08673e8 −1.86416 −0.932081 0.362251i \(-0.882008\pi\)
−0.932081 + 0.362251i \(0.882008\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.12944e8 −0.427018
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −4.83016e8 −0.905522 −0.452761 0.891632i \(-0.649561\pi\)
−0.452761 + 0.891632i \(0.649561\pi\)
\(812\) 0 0
\(813\) 1.05693e9 1.96687
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.17869e9 −2.16139
\(818\) 0 0
\(819\) 1.05498e8 0.192040
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 9.65555e8 1.73212 0.866059 0.499941i \(-0.166645\pi\)
0.866059 + 0.499941i \(0.166645\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −8.48197e8 −1.48879 −0.744395 0.667740i \(-0.767262\pi\)
−0.744395 + 0.667740i \(0.767262\pi\)
\(830\) 0 0
\(831\) −7.09208e8 −1.23586
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.94456e8 −1.18432
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.06666e8 0.833819
\(848\) 0 0
\(849\) 3.77200e8 0.616380
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 9.38655e8 1.51237 0.756187 0.654356i \(-0.227060\pi\)
0.756187 + 0.654356i \(0.227060\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 2.87739e8 0.453962 0.226981 0.973899i \(-0.427114\pi\)
0.226981 + 0.973899i \(0.427114\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.51714e8 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −8.74742e7 −0.132381
\(872\) 0 0
\(873\) −4.11491e7 −0.0618469
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.16597e9 1.72858 0.864288 0.502997i \(-0.167769\pi\)
0.864288 + 0.502997i \(0.167769\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 9.49268e8 1.37882 0.689409 0.724372i \(-0.257870\pi\)
0.689409 + 0.724372i \(0.257870\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.13038e9 1.60886
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 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
\(903\) −8.60123e8 −1.16815
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −5.18340e8 −0.694693 −0.347347 0.937737i \(-0.612917\pi\)
−0.347347 + 0.937737i \(0.612917\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 1.54471e9 1.99021 0.995107 0.0988039i \(-0.0315017\pi\)
0.995107 + 0.0988039i \(0.0315017\pi\)
\(920\) 0 0
\(921\) −1.49420e9 −1.91262
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.39384e9 −1.76112
\(926\) 0 0
\(927\) −8.21544e8 −1.03132
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −3.79396e8 −0.470158
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.43605e9 1.74562 0.872810 0.488060i \(-0.162295\pi\)
0.872810 + 0.488060i \(0.162295\pi\)
\(938\) 0 0
\(939\) 1.04953e9 1.26765
\(940\) 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
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 3.22861e8 0.377761
\(950\) 0 0
\(951\) 0 0
\(952\) 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
\(959\) 0 0
\(960\) 0 0
\(961\) 3.57316e8 0.402608
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −6.93538e8 −0.766992 −0.383496 0.923542i \(-0.625280\pi\)
−0.383496 + 0.923542i \(0.625280\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −3.61658e8 −0.392608
\(974\) 0 0
\(975\) 2.13469e8 0.230314
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.58393e9 −1.67776
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −1.35430e8 −0.139153 −0.0695766 0.997577i \(-0.522165\pi\)
−0.0695766 + 0.997577i \(0.522165\pi\)
\(992\) 0 0
\(993\) 1.93172e9 1.97286
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.14627e9 −1.15664 −0.578322 0.815808i \(-0.696292\pi\)
−0.578322 + 0.815808i \(0.696292\pi\)
\(998\) 0 0
\(999\) −1.75584e9 −1.76112
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 48.7.e.a.17.1 1
3.2 odd 2 CM 48.7.e.a.17.1 1
4.3 odd 2 3.7.b.a.2.1 1
8.3 odd 2 192.7.e.b.65.1 1
8.5 even 2 192.7.e.a.65.1 1
12.11 even 2 3.7.b.a.2.1 1
20.3 even 4 75.7.d.a.74.1 2
20.7 even 4 75.7.d.a.74.2 2
20.19 odd 2 75.7.c.a.26.1 1
24.5 odd 2 192.7.e.a.65.1 1
24.11 even 2 192.7.e.b.65.1 1
28.27 even 2 147.7.b.a.50.1 1
36.7 odd 6 81.7.d.a.53.1 2
36.11 even 6 81.7.d.a.53.1 2
36.23 even 6 81.7.d.a.26.1 2
36.31 odd 6 81.7.d.a.26.1 2
60.23 odd 4 75.7.d.a.74.1 2
60.47 odd 4 75.7.d.a.74.2 2
60.59 even 2 75.7.c.a.26.1 1
84.83 odd 2 147.7.b.a.50.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.7.b.a.2.1 1 4.3 odd 2
3.7.b.a.2.1 1 12.11 even 2
48.7.e.a.17.1 1 1.1 even 1 trivial
48.7.e.a.17.1 1 3.2 odd 2 CM
75.7.c.a.26.1 1 20.19 odd 2
75.7.c.a.26.1 1 60.59 even 2
75.7.d.a.74.1 2 20.3 even 4
75.7.d.a.74.1 2 60.23 odd 4
75.7.d.a.74.2 2 20.7 even 4
75.7.d.a.74.2 2 60.47 odd 4
81.7.d.a.26.1 2 36.23 even 6
81.7.d.a.26.1 2 36.31 odd 6
81.7.d.a.53.1 2 36.7 odd 6
81.7.d.a.53.1 2 36.11 even 6
147.7.b.a.50.1 1 28.27 even 2
147.7.b.a.50.1 1 84.83 odd 2
192.7.e.a.65.1 1 8.5 even 2
192.7.e.a.65.1 1 24.5 odd 2
192.7.e.b.65.1 1 8.3 odd 2
192.7.e.b.65.1 1 24.11 even 2