Properties

Label 43.9.b.a.42.1
Level 43
Weight 9
Character 43.42
Self dual yes
Analytic conductor 17.517
Analytic rank 0
Dimension 1
CM discriminant -43
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 43.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(17.5172802326\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 42.1
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 43.42

$q$-expansion

\(f(q)\) \(=\) \(q+256.000 q^{4} +6561.00 q^{9} +O(q^{10})\) \(q+256.000 q^{4} +6561.00 q^{9} +10319.0 q^{11} -54721.0 q^{13} +65536.0 q^{16} +79967.0 q^{17} +540719. q^{23} +390625. q^{25} +589679. q^{31} +1.67962e6 q^{36} -2.26224e6 q^{41} +3.41880e6 q^{43} +2.64166e6 q^{44} -6.98381e6 q^{47} +5.76480e6 q^{49} -1.40086e7 q^{52} -1.30618e7 q^{53} -7.86461e6 q^{59} +1.67772e7 q^{64} -3.98164e7 q^{67} +2.04716e7 q^{68} +7.30456e7 q^{79} +4.30467e7 q^{81} -9.30914e7 q^{83} +1.38424e8 q^{92} +1.62643e8 q^{97} +6.77030e7 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 256.000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 10319.0 0.704802 0.352401 0.935849i \(-0.385365\pi\)
0.352401 + 0.935849i \(0.385365\pi\)
\(12\) 0 0
\(13\) −54721.0 −1.91593 −0.957967 0.286878i \(-0.907383\pi\)
−0.957967 + 0.286878i \(0.907383\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 65536.0 1.00000
\(17\) 79967.0 0.957448 0.478724 0.877965i \(-0.341099\pi\)
0.478724 + 0.877965i \(0.341099\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 540719. 1.93224 0.966118 0.258100i \(-0.0830963\pi\)
0.966118 + 0.258100i \(0.0830963\pi\)
\(24\) 0 0
\(25\) 390625. 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 589679. 0.638512 0.319256 0.947669i \(-0.396567\pi\)
0.319256 + 0.947669i \(0.396567\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.67962e6 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.26224e6 −0.800578 −0.400289 0.916389i \(-0.631090\pi\)
−0.400289 + 0.916389i \(0.631090\pi\)
\(42\) 0 0
\(43\) 3.41880e6 1.00000
\(44\) 2.64166e6 0.704802
\(45\) 0 0
\(46\) 0 0
\(47\) −6.98381e6 −1.43120 −0.715601 0.698510i \(-0.753847\pi\)
−0.715601 + 0.698510i \(0.753847\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −1.40086e7 −1.91593
\(53\) −1.30618e7 −1.65538 −0.827691 0.561184i \(-0.810346\pi\)
−0.827691 + 0.561184i \(0.810346\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.86461e6 −0.649036 −0.324518 0.945879i \(-0.605202\pi\)
−0.324518 + 0.945879i \(0.605202\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.67772e7 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.98164e7 −1.97589 −0.987946 0.154800i \(-0.950527\pi\)
−0.987946 + 0.154800i \(0.950527\pi\)
\(68\) 2.04716e7 0.957448
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 7.30456e7 1.87537 0.937683 0.347493i \(-0.112967\pi\)
0.937683 + 0.347493i \(0.112967\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) −9.30914e7 −1.96154 −0.980770 0.195165i \(-0.937476\pi\)
−0.980770 + 0.195165i \(0.937476\pi\)
\(84\) 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
\(91\) 0 0
\(92\) 1.38424e8 1.93224
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.62643e8 1.83717 0.918584 0.395225i \(-0.129333\pi\)
0.918584 + 0.395225i \(0.129333\pi\)
\(98\) 0 0
\(99\) 6.77030e7 0.704802
\(100\) 1.00000e8 1.00000
\(101\) −1.84316e8 −1.77124 −0.885621 0.464409i \(-0.846267\pi\)
−0.885621 + 0.464409i \(0.846267\pi\)
\(102\) 0 0
\(103\) −9.18609e7 −0.816172 −0.408086 0.912943i \(-0.633804\pi\)
−0.408086 + 0.912943i \(0.633804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.84487e8 1.40744 0.703720 0.710477i \(-0.251521\pi\)
0.703720 + 0.710477i \(0.251521\pi\)
\(108\) 0 0
\(109\) −2.59286e8 −1.83685 −0.918423 0.395599i \(-0.870537\pi\)
−0.918423 + 0.395599i \(0.870537\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.59024e8 −1.91593
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1.07877e8 −0.503255
\(122\) 0 0
\(123\) 0 0
\(124\) 1.50958e8 0.638512
\(125\) 0 0
\(126\) 0 0
\(127\) −3.08034e8 −1.18409 −0.592043 0.805907i \(-0.701678\pi\)
−0.592043 + 0.805907i \(0.701678\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −3.04510e7 −0.0815721 −0.0407861 0.999168i \(-0.512986\pi\)
−0.0407861 + 0.999168i \(0.512986\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.64666e8 −1.35035
\(144\) 4.29982e8 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 5.24663e8 0.957448
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −5.79134e8 −0.800578
\(165\) 0 0
\(166\) 0 0
\(167\) −7.20209e8 −0.925961 −0.462981 0.886368i \(-0.653220\pi\)
−0.462981 + 0.886368i \(0.653220\pi\)
\(168\) 0 0
\(169\) 2.17866e9 2.67080
\(170\) 0 0
\(171\) 0 0
\(172\) 8.75213e8 1.00000
\(173\) 1.46961e9 1.64065 0.820326 0.571897i \(-0.193792\pi\)
0.820326 + 0.571897i \(0.193792\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.76266e8 0.704802
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −4.85678e8 −0.452516 −0.226258 0.974067i \(-0.572649\pi\)
−0.226258 + 0.974067i \(0.572649\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.25179e8 0.674811
\(188\) −1.78785e9 −1.43120
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −9.12967e8 −0.658000 −0.329000 0.944330i \(-0.606712\pi\)
−0.329000 + 0.944330i \(0.606712\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.47579e9 1.00000
\(197\) −2.93452e9 −1.94837 −0.974187 0.225744i \(-0.927519\pi\)
−0.974187 + 0.225744i \(0.927519\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.54766e9 1.93224
\(208\) −3.58620e9 −1.91593
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −3.34381e9 −1.65538
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.37587e9 −1.83441
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 2.56289e9 1.00000
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 4.46811e9 1.62473 0.812366 0.583148i \(-0.198179\pi\)
0.812366 + 0.583148i \(0.198179\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.01334e9 −0.649036
\(237\) 0 0
\(238\) 0 0
\(239\) −6.10101e9 −1.86986 −0.934932 0.354826i \(-0.884540\pi\)
−0.934932 + 0.354826i \(0.884540\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.67719e8 −0.193423 −0.0967113 0.995312i \(-0.530832\pi\)
−0.0967113 + 0.995312i \(0.530832\pi\)
\(252\) 0 0
\(253\) 5.57968e9 1.36184
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.01930e10 −1.97589
\(269\) 1.01622e10 1.94080 0.970398 0.241510i \(-0.0776428\pi\)
0.970398 + 0.241510i \(0.0776428\pi\)
\(270\) 0 0
\(271\) 8.03860e9 1.49040 0.745201 0.666840i \(-0.232354\pi\)
0.745201 + 0.666840i \(0.232354\pi\)
\(272\) 5.24072e9 0.957448
\(273\) 0 0
\(274\) 0 0
\(275\) 4.03086e9 0.704802
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 3.86888e9 0.638512
\(280\) 0 0
\(281\) −1.39275e9 −0.223382 −0.111691 0.993743i \(-0.535627\pi\)
−0.111691 + 0.993743i \(0.535627\pi\)
\(282\) 0 0
\(283\) 1.99534e7 0.00311079 0.00155539 0.999999i \(-0.499505\pi\)
0.00155539 + 0.999999i \(0.499505\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −5.81036e8 −0.0832937
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.12757e10 1.52993 0.764964 0.644073i \(-0.222757\pi\)
0.764964 + 0.644073i \(0.222757\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.95887e10 −3.70204
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.94759e8 0.111986 0.0559931 0.998431i \(-0.482168\pi\)
0.0559931 + 0.998431i \(0.482168\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.85714e10 −1.98520 −0.992600 0.121434i \(-0.961251\pi\)
−0.992600 + 0.121434i \(0.961251\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.86997e10 1.87537
\(317\) 1.79770e9 0.178025 0.0890124 0.996031i \(-0.471629\pi\)
0.0890124 + 0.996031i \(0.471629\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.10200e10 1.00000
\(325\) −2.13754e10 −1.91593
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −2.38314e10 −1.96154
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.83777e9 −0.375082 −0.187541 0.982257i \(-0.560052\pi\)
−0.187541 + 0.982257i \(0.560052\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.08490e9 0.450024
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.17870e9 0.333520 0.166760 0.985998i \(-0.446670\pi\)
0.166760 + 0.985998i \(0.446670\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.62391e10 −0.977654 −0.488827 0.872381i \(-0.662575\pi\)
−0.488827 + 0.872381i \(0.662575\pi\)
\(360\) 0 0
\(361\) 1.69836e10 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.41232e10 −1.88098 −0.940492 0.339816i \(-0.889635\pi\)
−0.940492 + 0.339816i \(0.889635\pi\)
\(368\) 3.54366e10 1.93224
\(369\) −1.48426e10 −0.800578
\(370\) 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
\(379\) −6.37385e9 −0.308919 −0.154460 0.987999i \(-0.549364\pi\)
−0.154460 + 0.987999i \(0.549364\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.24308e10 1.00000
\(388\) 4.16367e10 1.83717
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 4.32397e10 1.85002
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 1.73320e10 0.704802
\(397\) 4.27239e10 1.71992 0.859960 0.510361i \(-0.170488\pi\)
0.859960 + 0.510361i \(0.170488\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 2.56000e10 1.00000
\(401\) 9.33114e9 0.360876 0.180438 0.983586i \(-0.442249\pi\)
0.180438 + 0.983586i \(0.442249\pi\)
\(402\) 0 0
\(403\) −3.22678e10 −1.22335
\(404\) −4.71849e10 −1.77124
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2.35164e10 −0.816172
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −4.58208e10 −1.43120
\(424\) 0 0
\(425\) 3.12371e10 0.957448
\(426\) 0 0
\(427\) 0 0
\(428\) 4.72286e10 1.40744
\(429\) 0 0
\(430\) 0 0
\(431\) 2.05297e10 0.594940 0.297470 0.954731i \(-0.403857\pi\)
0.297470 + 0.954731i \(0.403857\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.63772e10 −1.83685
\(437\) 0 0
\(438\) 0 0
\(439\) −5.34427e10 −1.43890 −0.719449 0.694545i \(-0.755606\pi\)
−0.719449 + 0.694545i \(0.755606\pi\)
\(440\) 0 0
\(441\) 3.78229e10 1.00000
\(442\) 0 0
\(443\) −6.32685e10 −1.64275 −0.821377 0.570385i \(-0.806794\pi\)
−0.821377 + 0.570385i \(0.806794\pi\)
\(444\) 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
\(451\) −2.33441e10 −0.564248
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.67814e10 1.03578 0.517892 0.855446i \(-0.326717\pi\)
0.517892 + 0.855446i \(0.326717\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −9.19103e10 −1.91593
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.52786e10 0.704802
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.56982e10 −1.65538
\(478\) 0 0
\(479\) 9.29105e10 1.76491 0.882455 0.470397i \(-0.155889\pi\)
0.882455 + 0.470397i \(0.155889\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.76165e10 −0.503255
\(485\) 0 0
\(486\) 0 0
\(487\) −9.15133e10 −1.62693 −0.813464 0.581616i \(-0.802421\pi\)
−0.813464 + 0.581616i \(0.802421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 3.86452e10 0.638512
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 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
\(507\) 0 0
\(508\) −7.88566e10 −1.18409
\(509\) 1.30889e11 1.95000 0.974998 0.222215i \(-0.0713287\pi\)
0.974998 + 0.222215i \(0.0713287\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −7.20659e10 −1.00871
\(518\) 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
\(527\) 4.71549e10 0.611342
\(528\) 0 0
\(529\) 2.14066e11 2.73354
\(530\) 0 0
\(531\) −5.15997e10 −0.649036
\(532\) 0 0
\(533\) 1.23792e11 1.53385
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.94870e10 0.704802
\(540\) 0 0
\(541\) 1.71267e11 1.99933 0.999665 0.0258753i \(-0.00823728\pi\)
0.999665 + 0.0258753i \(0.00823728\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.91887e9 0.0437735 0.0218867 0.999760i \(-0.493033\pi\)
0.0218867 + 0.999760i \(0.493033\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −7.79545e9 −0.0815721
\(557\) −5.88814e10 −0.611726 −0.305863 0.952076i \(-0.598945\pi\)
−0.305863 + 0.952076i \(0.598945\pi\)
\(558\) 0 0
\(559\) −1.87080e11 −1.91593
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.95295e10 0.891112 0.445556 0.895254i \(-0.353006\pi\)
0.445556 + 0.895254i \(0.353006\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.24706e11 −1.18970 −0.594850 0.803837i \(-0.702788\pi\)
−0.594850 + 0.803837i \(0.702788\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) −1.44554e11 −1.35035
\(573\) 0 0
\(574\) 0 0
\(575\) 2.11218e11 1.93224
\(576\) 1.10075e11 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.34784e11 −1.16672
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 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
\(597\) 0 0
\(598\) 0 0
\(599\) −2.45665e11 −1.90825 −0.954127 0.299401i \(-0.903213\pi\)
−0.954127 + 0.299401i \(0.903213\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) −2.61236e11 −1.97589
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.82161e11 2.74209
\(612\) 1.34314e11 0.957448
\(613\) −6.88738e10 −0.487767 −0.243883 0.969805i \(-0.578421\pi\)
−0.243883 + 0.969805i \(0.578421\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.89848e11 −2.00000 −0.999998 0.00197274i \(-0.999372\pi\)
−0.999998 + 0.00197274i \(0.999372\pi\)
\(618\) 0 0
\(619\) −1.56700e11 −1.06735 −0.533673 0.845691i \(-0.679189\pi\)
−0.533673 + 0.845691i \(0.679189\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.52588e11 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.15456e11 −1.91593
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 2.33069e11 1.36345 0.681727 0.731607i \(-0.261229\pi\)
0.681727 + 0.731607i \(0.261229\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) −8.11549e10 −0.457442
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.48258e11 −0.800578
\(657\) 0 0
\(658\) 0 0
\(659\) −3.16664e11 −1.67902 −0.839512 0.543342i \(-0.817159\pi\)
−0.839512 + 0.543342i \(0.817159\pi\)
\(660\) 0 0
\(661\) 1.98843e11 1.04161 0.520803 0.853677i \(-0.325633\pi\)
0.520803 + 0.853677i \(0.325633\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −1.84374e11 −0.925961
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 5.57736e11 2.67080
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.87847e11 −1.32275 −0.661377 0.750054i \(-0.730028\pi\)
−0.661377 + 0.750054i \(0.730028\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 2.24055e11 1.00000
\(689\) 7.14753e11 3.17160
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 3.76219e11 1.64065
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.80905e11 −0.766511
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.59215e11 −1.07347 −0.536733 0.843752i \(-0.680342\pi\)
−0.536733 + 0.843752i \(0.680342\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.73124e11 0.704802
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.23797e11 −0.885663 −0.442831 0.896605i \(-0.646026\pi\)
−0.442831 + 0.896605i \(0.646026\pi\)
\(710\) 0 0
\(711\) 4.79252e11 1.87537
\(712\) 0 0
\(713\) 3.18851e11 1.23376
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.10925e11 −0.789247 −0.394624 0.918843i \(-0.629125\pi\)
−0.394624 + 0.918843i \(0.629125\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −1.24333e11 −0.452516
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.82430e11 1.00000
\(730\) 0 0
\(731\) 2.73391e11 0.957448
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.10866e11 −1.39261
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −6.10773e11 −1.96154
\(748\) 2.11246e11 0.674811
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −4.57691e11 −1.43120
\(753\) 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
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.30359e11 1.24351
\(768\) 0 0
\(769\) −6.14692e11 −1.75773 −0.878864 0.477071i \(-0.841698\pi\)
−0.878864 + 0.477071i \(0.841698\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −2.33720e11 −0.658000
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 2.30343e11 0.638512
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.77802e11 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 7.65562e11 1.99564 0.997818 0.0660286i \(-0.0210329\pi\)
0.997818 + 0.0660286i \(0.0210329\pi\)
\(788\) −7.51237e11 −1.94837
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.04857e11 −1.49906 −0.749531 0.661970i \(-0.769721\pi\)
−0.749531 + 0.661970i \(0.769721\pi\)
\(798\) 0 0
\(799\) −5.58474e11 −1.37030
\(800\) 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
\(809\) −8.56161e11 −1.99876 −0.999382 0.0351461i \(-0.988810\pi\)
−0.999382 + 0.0351461i \(0.988810\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.45231e11 0.319658 0.159829 0.987145i \(-0.448906\pi\)
0.159829 + 0.987145i \(0.448906\pi\)
\(822\) 0 0
\(823\) 5.58557e11 1.21750 0.608749 0.793363i \(-0.291672\pi\)
0.608749 + 0.793363i \(0.291672\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.49162e11 −0.532672 −0.266336 0.963880i \(-0.585813\pi\)
−0.266336 + 0.963880i \(0.585813\pi\)
\(828\) 9.08200e11 1.93224
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −9.18066e11 −1.91593
\(833\) 4.60994e11 0.957448
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −8.56016e11 −1.65538
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −9.77865e11 −1.84707 −0.923534 0.383517i \(-0.874713\pi\)
−0.923534 + 0.383517i \(0.874713\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.25594e11 −1.71592 −0.857961 0.513715i \(-0.828269\pi\)
−0.857961 + 0.513715i \(0.828269\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 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
\(867\) 0 0
\(868\) 0 0
\(869\) 7.53758e11 1.32176
\(870\) 0 0
\(871\) 2.17880e12 3.78568
\(872\) 0 0
\(873\) 1.06710e12 1.83717
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 9.22813e11 1.55997 0.779984 0.625800i \(-0.215227\pi\)
0.779984 + 0.625800i \(0.215227\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.62544e11 1.26579 0.632895 0.774238i \(-0.281867\pi\)
0.632895 + 0.774238i \(0.281867\pi\)
\(882\) 0 0
\(883\) 7.12688e11 1.17235 0.586174 0.810185i \(-0.300633\pi\)
0.586174 + 0.810185i \(0.300633\pi\)
\(884\) −1.12022e12 −1.83441
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.44199e11 0.704802
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 6.56100e11 1.00000
\(901\) −1.04451e12 −1.58494
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.07838e12 1.59346 0.796732 0.604333i \(-0.206560\pi\)
0.796732 + 0.604333i \(0.206560\pi\)
\(908\) 0 0
\(909\) −1.20930e12 −1.77124
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −9.60611e11 −1.38250
\(914\) 0 0
\(915\) 0 0
\(916\) 1.14384e12 1.62473
\(917\) 0 0
\(918\) 0 0
\(919\) −2.01222e11 −0.282106 −0.141053 0.990002i \(-0.545049\pi\)
−0.141053 + 0.990002i \(0.545049\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −6.02699e11 −0.816172
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.02704e12 1.30987 0.654933 0.755687i \(-0.272697\pi\)
0.654933 + 0.755687i \(0.272697\pi\)
\(942\) 0 0
\(943\) −1.22324e12 −1.54691
\(944\) −5.15415e11 −0.649036
\(945\) 0 0
\(946\) 0 0
\(947\) 1.44784e12 1.80020 0.900101 0.435681i \(-0.143493\pi\)
0.900101 + 0.435681i \(0.143493\pi\)
\(948\) 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
\(955\) 0 0
\(956\) −1.56186e12 −1.86986
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −5.05170e11 −0.592303
\(962\) 0 0
\(963\) 1.21042e12 1.40744
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.47522e12 −1.68714 −0.843570 0.537019i \(-0.819550\pi\)
−0.843570 + 0.537019i \(0.819550\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.91958e11 −0.553415 −0.276708 0.960954i \(-0.589243\pi\)
−0.276708 + 0.960954i \(0.589243\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.03621e11 −0.223483 −0.111741 0.993737i \(-0.535643\pi\)
−0.111741 + 0.993737i \(0.535643\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.70117e12 −1.83685
\(982\) 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
\(989\) 1.84861e12 1.93224
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 43.9.b.a.42.1 1
43.42 odd 2 CM 43.9.b.a.42.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.9.b.a.42.1 1 1.1 even 1 trivial
43.9.b.a.42.1 1 43.42 odd 2 CM