Properties

Label 72.17.b.a.19.1
Level $72$
Weight $17$
Character 72.19
Self dual yes
Analytic conductor $116.874$
Analytic rank $0$
Dimension $1$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 17 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(116.873671577\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.1
Character \(\chi\) \(=\) 72.19

$q$-expansion

\(f(q)\) \(=\) \(q-256.000 q^{2} +65536.0 q^{4} -1.67772e7 q^{8} +O(q^{10})\) \(q-256.000 q^{2} +65536.0 q^{4} -1.67772e7 q^{8} -3.09274e8 q^{11} +4.29497e9 q^{16} -1.24333e10 q^{17} -2.87419e10 q^{19} +7.91741e10 q^{22} +1.52588e11 q^{25} -1.09951e12 q^{32} +3.18292e12 q^{34} +7.35792e12 q^{38} -8.32000e11 q^{41} +6.06944e12 q^{43} -2.02686e13 q^{44} +3.32329e13 q^{49} -3.90625e13 q^{50} -2.90919e14 q^{59} +2.81475e14 q^{64} -6.17692e14 q^{67} -8.14828e14 q^{68} -4.86140e14 q^{73} -1.88363e15 q^{76} +2.12992e14 q^{82} +3.59194e15 q^{83} -1.55378e15 q^{86} +5.18875e15 q^{88} -1.25086e15 q^{89} -9.68128e15 q^{97} -8.50763e15 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −256.000 −1.00000
\(3\) 0 0
\(4\) 65536.0 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.67772e7 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) −3.09274e8 −1.44279 −0.721393 0.692526i \(-0.756498\pi\)
−0.721393 + 0.692526i \(0.756498\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.29497e9 1.00000
\(17\) −1.24333e10 −1.78236 −0.891178 0.453653i \(-0.850121\pi\)
−0.891178 + 0.453653i \(0.850121\pi\)
\(18\) 0 0
\(19\) −2.87419e10 −1.69233 −0.846167 0.532918i \(-0.821095\pi\)
−0.846167 + 0.532918i \(0.821095\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 7.91741e10 1.44279
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 1.52588e11 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.09951e12 −1.00000
\(33\) 0 0
\(34\) 3.18292e12 1.78236
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 7.35792e12 1.69233
\(39\) 0 0
\(40\) 0 0
\(41\) −8.32000e11 −0.104196 −0.0520982 0.998642i \(-0.516591\pi\)
−0.0520982 + 0.998642i \(0.516591\pi\)
\(42\) 0 0
\(43\) 6.06944e12 0.519279 0.259640 0.965706i \(-0.416396\pi\)
0.259640 + 0.965706i \(0.416396\pi\)
\(44\) −2.02686e13 −1.44279
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.32329e13 1.00000
\(50\) −3.90625e13 −1.00000
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.90919e14 −1.98132 −0.990662 0.136343i \(-0.956465\pi\)
−0.990662 + 0.136343i \(0.956465\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 2.81475e14 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.17692e14 −1.52116 −0.760578 0.649247i \(-0.775084\pi\)
−0.760578 + 0.649247i \(0.775084\pi\)
\(68\) −8.14828e14 −1.78236
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −4.86140e14 −0.602807 −0.301403 0.953497i \(-0.597455\pi\)
−0.301403 + 0.953497i \(0.597455\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.88363e15 −1.69233
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.12992e14 0.104196
\(83\) 3.59194e15 1.59479 0.797397 0.603455i \(-0.206210\pi\)
0.797397 + 0.603455i \(0.206210\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.55378e15 −0.519279
\(87\) 0 0
\(88\) 5.18875e15 1.44279
\(89\) −1.25086e15 −0.317751 −0.158876 0.987299i \(-0.550787\pi\)
−0.158876 + 0.987299i \(0.550787\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.68128e15 −1.23526 −0.617631 0.786468i \(-0.711907\pi\)
−0.617631 + 0.786468i \(0.711907\pi\)
\(98\) −8.50763e15 −1.00000
\(99\) 0 0
\(100\) 1.00000e16 1.00000
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.51522e14 0.00881873 0.00440936 0.999990i \(-0.498596\pi\)
0.00440936 + 0.999990i \(0.498596\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.79557e16 1.80390 0.901951 0.431839i \(-0.142135\pi\)
0.901951 + 0.431839i \(0.142135\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 7.44752e16 1.98132
\(119\) 0 0
\(120\) 0 0
\(121\) 4.97005e16 1.08163
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −7.20576e16 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 5.79225e16 0.667847 0.333924 0.942600i \(-0.391627\pi\)
0.333924 + 0.942600i \(0.391627\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.58129e17 1.52116
\(135\) 0 0
\(136\) 2.08596e17 1.78236
\(137\) 1.32928e17 1.07115 0.535576 0.844487i \(-0.320094\pi\)
0.535576 + 0.844487i \(0.320094\pi\)
\(138\) 0 0
\(139\) 2.57421e17 1.84725 0.923624 0.383300i \(-0.125213\pi\)
0.923624 + 0.383300i \(0.125213\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 1.24452e17 0.602807
\(147\) 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 \(\pi\)
\(152\) 4.82208e17 1.69233
\(153\) 0 0
\(154\) 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
\(163\) 3.09997e17 0.622094 0.311047 0.950394i \(-0.399320\pi\)
0.311047 + 0.950394i \(0.399320\pi\)
\(164\) −5.45259e16 −0.104196
\(165\) 0 0
\(166\) −9.19537e17 −1.59479
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 6.65417e17 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 3.97767e17 0.519279
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.32832e18 −1.44279
\(177\) 0 0
\(178\) 3.20219e17 0.317751
\(179\) −1.52656e18 −1.44840 −0.724200 0.689590i \(-0.757791\pi\)
−0.724200 + 0.689590i \(0.757791\pi\)
\(180\) 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
\(187\) 3.84529e18 2.57156
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.78742e18 −0.928469 −0.464234 0.885712i \(-0.653670\pi\)
−0.464234 + 0.885712i \(0.653670\pi\)
\(194\) 2.47841e18 1.23526
\(195\) 0 0
\(196\) 2.17795e18 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −2.56000e18 −1.00000
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.88910e18 2.44167
\(210\) 0 0
\(211\) −1.49297e18 −0.380006 −0.190003 0.981784i \(-0.560850\pi\)
−0.190003 + 0.981784i \(0.560850\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −3.87897e16 −0.00881873
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −1.22767e19 −1.80390
\(227\) −9.40484e17 −0.133396 −0.0666982 0.997773i \(-0.521246\pi\)
−0.0666982 + 0.997773i \(0.521246\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.71523e19 1.97458 0.987288 0.158940i \(-0.0508076\pi\)
0.987288 + 0.158940i \(0.0508076\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1.90656e19 −1.98132
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −2.24227e19 −1.97039 −0.985195 0.171435i \(-0.945160\pi\)
−0.985195 + 0.171435i \(0.945160\pi\)
\(242\) −1.27233e19 −1.08163
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.13704e19 1.99127 0.995636 0.0933190i \(-0.0297477\pi\)
0.995636 + 0.0933190i \(0.0297477\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.84467e19 1.00000
\(257\) −3.31611e19 −1.74247 −0.871234 0.490869i \(-0.836679\pi\)
−0.871234 + 0.490869i \(0.836679\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −1.48282e19 −0.667847
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −4.04811e19 −1.52116
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −5.34006e19 −1.78236
\(273\) 0 0
\(274\) −3.40295e19 −1.07115
\(275\) −4.71914e19 −1.44279
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −6.58997e19 −1.84725
\(279\) 0 0
\(280\) 0 0
\(281\) 7.28515e19 1.87408 0.937040 0.349222i \(-0.113554\pi\)
0.937040 + 0.349222i \(0.113554\pi\)
\(282\) 0 0
\(283\) 3.26254e19 0.792984 0.396492 0.918038i \(-0.370227\pi\)
0.396492 + 0.918038i \(0.370227\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.05925e20 2.17680
\(290\) 0 0
\(291\) 0 0
\(292\) −3.18596e19 −0.602807
\(293\) 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
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −1.23445e20 −1.69233
\(305\) 0 0
\(306\) 0 0
\(307\) −1.13124e20 −1.43367 −0.716833 0.697245i \(-0.754409\pi\)
−0.716833 + 0.697245i \(0.754409\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −2.35283e19 −0.255409 −0.127704 0.991812i \(-0.540761\pi\)
−0.127704 + 0.991812i \(0.540761\pi\)
\(314\) 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
\(323\) 3.57356e20 3.01634
\(324\) 0 0
\(325\) 0 0
\(326\) −7.93592e19 −0.622094
\(327\) 0 0
\(328\) 1.39586e19 0.104196
\(329\) 0 0
\(330\) 0 0
\(331\) 2.84058e20 1.97144 0.985720 0.168391i \(-0.0538570\pi\)
0.985720 + 0.168391i \(0.0538570\pi\)
\(332\) 2.35402e20 1.59479
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.32619e20 1.99944 0.999719 0.0237016i \(-0.00754515\pi\)
0.999719 + 0.0237016i \(0.00754515\pi\)
\(338\) −1.70347e20 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.01828e20 −0.519279
\(345\) 0 0
\(346\) 0 0
\(347\) 3.57622e20 1.70133 0.850665 0.525708i \(-0.176199\pi\)
0.850665 + 0.525708i \(0.176199\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.40050e20 1.44279
\(353\) 2.94178e20 1.22015 0.610074 0.792345i \(-0.291140\pi\)
0.610074 + 0.792345i \(0.291140\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −8.19761e19 −0.317751
\(357\) 0 0
\(358\) 3.90798e20 1.44840
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 5.37653e20 1.86399
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) −9.84394e20 −2.57156
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −7.92722e20 −1.86212 −0.931059 0.364869i \(-0.881114\pi\)
−0.931059 + 0.364869i \(0.881114\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.57579e20 0.928469
\(387\) 0 0
\(388\) −6.34473e20 −1.23526
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −5.57556e20 −1.00000
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.55360e20 1.00000
\(401\) 9.91476e20 1.48295 0.741476 0.670979i \(-0.234126\pi\)
0.741476 + 0.670979i \(0.234126\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.52722e21 −1.95037 −0.975183 0.221399i \(-0.928938\pi\)
−0.975183 + 0.221399i \(0.928938\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −2.27561e21 −2.44167
\(419\) −1.02548e21 −1.07948 −0.539738 0.841833i \(-0.681477\pi\)
−0.539738 + 0.841833i \(0.681477\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 3.82199e20 0.380006
\(423\) 0 0
\(424\) 0 0
\(425\) −1.89717e21 −1.78236
\(426\) 0 0
\(427\) 0 0
\(428\) 9.93016e18 0.00881873
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 2.23900e21 1.81197 0.905984 0.423312i \(-0.139132\pi\)
0.905984 + 0.423312i \(0.139132\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.40904e21 −0.949931 −0.474965 0.880004i \(-0.657539\pi\)
−0.474965 + 0.880004i \(0.657539\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.78641e21 1.08146 0.540730 0.841196i \(-0.318148\pi\)
0.540730 + 0.841196i \(0.318148\pi\)
\(450\) 0 0
\(451\) 2.57316e20 0.150333
\(452\) 3.14282e21 1.80390
\(453\) 0 0
\(454\) 2.40764e20 0.133396
\(455\) 0 0
\(456\) 0 0
\(457\) −1.94535e21 −1.02251 −0.511255 0.859429i \(-0.670819\pi\)
−0.511255 + 0.859429i \(0.670819\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −4.39098e21 −1.97458
\(467\) −4.33385e21 −1.91575 −0.957876 0.287183i \(-0.907281\pi\)
−0.957876 + 0.287183i \(0.907281\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 4.88080e21 1.98132
\(473\) −1.87712e21 −0.749208
\(474\) 0 0
\(475\) −4.38566e21 −1.69233
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 5.74022e21 1.97039
\(483\) 0 0
\(484\) 3.25718e21 1.08163
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.14548e21 −0.931182 −0.465591 0.885000i \(-0.654158\pi\)
−0.465591 + 0.885000i \(0.654158\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.90403e21 0.755433 0.377717 0.925921i \(-0.376709\pi\)
0.377717 + 0.925921i \(0.376709\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −8.03083e21 −1.99127
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −4.72237e21 −1.00000
\(513\) 0 0
\(514\) 8.48925e21 1.74247
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.56571e21 1.02522 0.512612 0.858620i \(-0.328678\pi\)
0.512612 + 0.858620i \(0.328678\pi\)
\(522\) 0 0
\(523\) −1.11935e22 −1.99964 −0.999821 0.0189211i \(-0.993977\pi\)
−0.999821 + 0.0189211i \(0.993977\pi\)
\(524\) 3.79601e21 0.667847
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 6.13261e21 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 1.03632e22 1.52116
\(537\) 0 0
\(538\) 0 0
\(539\) −1.02781e22 −1.44279
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.36705e22 1.78236
\(545\) 0 0
\(546\) 0 0
\(547\) −1.56329e22 −1.95048 −0.975239 0.221155i \(-0.929017\pi\)
−0.975239 + 0.221155i \(0.929017\pi\)
\(548\) 8.71156e21 1.07115
\(549\) 0 0
\(550\) 1.20810e22 1.44279
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 1.68703e22 1.84725
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −1.86500e22 −1.87408
\(563\) 9.25677e20 0.0917049 0.0458524 0.998948i \(-0.485400\pi\)
0.0458524 + 0.998948i \(0.485400\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.35210e21 −0.792984
\(567\) 0 0
\(568\) 0 0
\(569\) 1.15608e21 0.105218 0.0526090 0.998615i \(-0.483246\pi\)
0.0526090 + 0.998615i \(0.483246\pi\)
\(570\) 0 0
\(571\) 4.49024e21 0.397357 0.198678 0.980065i \(-0.436335\pi\)
0.198678 + 0.980065i \(0.436335\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.45694e22 1.99981 0.999904 0.0138644i \(-0.00441331\pi\)
0.999904 + 0.0138644i \(0.00441331\pi\)
\(578\) −2.71169e22 −2.17680
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 8.15607e21 0.602807
\(585\) 0 0
\(586\) 0 0
\(587\) 1.13792e22 0.807247 0.403623 0.914925i \(-0.367751\pi\)
0.403623 + 0.914925i \(0.367751\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.98032e22 −1.94906 −0.974531 0.224252i \(-0.928006\pi\)
−0.974531 + 0.224252i \(0.928006\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 2.75429e22 1.61813 0.809065 0.587719i \(-0.199974\pi\)
0.809065 + 0.587719i \(0.199974\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 3.16020e22 1.69233
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 2.89598e22 1.43367
\(615\) 0 0
\(616\) 0 0
\(617\) 2.44708e22 1.16511 0.582556 0.812791i \(-0.302053\pi\)
0.582556 + 0.812791i \(0.302053\pi\)
\(618\) 0 0
\(619\) −3.48823e22 −1.61838 −0.809188 0.587549i \(-0.800093\pi\)
−0.809188 + 0.587549i \(0.800093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.32831e22 1.00000
\(626\) 6.02324e21 0.255409
\(627\) 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
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.69067e22 1.99664 0.998318 0.0579698i \(-0.0184627\pi\)
0.998318 + 0.0579698i \(0.0184627\pi\)
\(642\) 0 0
\(643\) −5.25739e22 −1.79921 −0.899607 0.436701i \(-0.856147\pi\)
−0.899607 + 0.436701i \(0.856147\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −9.14831e22 −3.01634
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 8.99735e22 2.85862
\(650\) 0 0
\(651\) 0 0
\(652\) 2.03159e22 0.622094
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.57341e21 −0.104196
\(657\) 0 0
\(658\) 0 0
\(659\) −6.05753e22 −1.70299 −0.851496 0.524362i \(-0.824304\pi\)
−0.851496 + 0.524362i \(0.824304\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −7.27190e22 −1.97144
\(663\) 0 0
\(664\) −6.02628e22 −1.59479
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.41066e22 −1.99853 −0.999263 0.0383912i \(-0.987777\pi\)
−0.999263 + 0.0383912i \(0.987777\pi\)
\(674\) −8.51505e22 −1.99944
\(675\) 0 0
\(676\) 4.36087e22 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.75704e22 1.42689 0.713445 0.700711i \(-0.247134\pi\)
0.713445 + 0.700711i \(0.247134\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 2.60680e22 0.519279
\(689\) 0 0
\(690\) 0 0
\(691\) −2.40103e22 −0.461927 −0.230963 0.972962i \(-0.574188\pi\)
−0.230963 + 0.972962i \(0.574188\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −9.15513e22 −1.70133
\(695\) 0 0
\(696\) 0 0
\(697\) 1.03445e22 0.185715
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −8.70528e22 −1.44279
\(705\) 0 0
\(706\) −7.53095e22 −1.22015
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.09859e22 0.317751
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −1.00044e23 −1.44840
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.37639e23 −1.86399
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.54631e22 −0.925541
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.91036e23 2.19470
\(738\) 0 0
\(739\) −3.29329e22 −0.370232 −0.185116 0.982717i \(-0.559266\pi\)
−0.185116 + 0.982717i \(0.559266\pi\)
\(740\) 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
\(747\) 0 0
\(748\) 2.52005e23 2.57156
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 2.02937e23 1.86212
\(759\) 0 0
\(760\) 0 0
\(761\) 2.21986e23 1.97355 0.986774 0.162102i \(-0.0518274\pi\)
0.986774 + 0.162102i \(0.0518274\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.37405e23 1.94124 0.970621 0.240615i \(-0.0773491\pi\)
0.970621 + 0.240615i \(0.0773491\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.17140e23 −0.928469
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.62425e23 1.23526
\(777\) 0 0
\(778\) 0 0
\(779\) 2.39132e22 0.176335
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.42734e23 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) 9.24033e22 0.627899 0.313949 0.949440i \(-0.398348\pi\)
0.313949 + 0.949440i \(0.398348\pi\)
\(788\) 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
\(799\) 0 0
\(800\) −1.67772e23 −1.00000
\(801\) 0 0
\(802\) −2.53818e23 −1.48295
\(803\) 1.50350e23 0.869720
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.39072e22 0.239303 0.119651 0.992816i \(-0.461822\pi\)
0.119651 + 0.992816i \(0.461822\pi\)
\(810\) 0 0
\(811\) 2.38559e23 1.27476 0.637382 0.770548i \(-0.280017\pi\)
0.637382 + 0.770548i \(0.280017\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.74447e23 −0.878794
\(818\) 3.90969e23 1.95037
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.45609e23 −1.57958 −0.789788 0.613379i \(-0.789810\pi\)
−0.789788 + 0.613379i \(0.789810\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.13195e23 −1.78236
\(834\) 0 0
\(835\) 0 0
\(836\) 5.82556e23 2.44167
\(837\) 0 0
\(838\) 2.62522e23 1.07948
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.50246e23 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −9.78430e22 −0.380006
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 4.85675e23 1.78236
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.54212e21 −0.00881873
\(857\) −5.80686e23 −1.99570 −0.997848 0.0655670i \(-0.979114\pi\)
−0.997848 + 0.0655670i \(0.979114\pi\)
\(858\) 0 0
\(859\) −4.05646e23 −1.36837 −0.684183 0.729311i \(-0.739841\pi\)
−0.684183 + 0.729311i \(0.739841\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −5.73184e23 −1.81197
\(867\) 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 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.56054e23 −0.429999 −0.215000 0.976614i \(-0.568975\pi\)
−0.215000 + 0.976614i \(0.568975\pi\)
\(882\) 0 0
\(883\) 6.34503e23 1.71691 0.858456 0.512887i \(-0.171424\pi\)
0.858456 + 0.512887i \(0.171424\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 3.60713e23 0.949931
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −4.57321e23 −1.08146
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −6.58728e22 −0.150333
\(903\) 0 0
\(904\) −8.04563e23 −1.80390
\(905\) 0 0
\(906\) 0 0
\(907\) −8.15465e23 −1.78052 −0.890261 0.455452i \(-0.849478\pi\)
−0.890261 + 0.455452i \(0.849478\pi\)
\(908\) −6.16355e22 −0.133396
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −1.11089e24 −2.30095
\(914\) 4.98009e23 1.02251
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 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
\(929\) −1.48353e23 −0.267406 −0.133703 0.991021i \(-0.542687\pi\)
−0.133703 + 0.991021i \(0.542687\pi\)
\(930\) 0 0
\(931\) −9.55176e23 −1.69233
\(932\) 1.12409e24 1.97458
\(933\) 0 0
\(934\) 1.10947e24 1.91575
\(935\) 0 0
\(936\) 0 0
\(937\) −1.18835e24 −1.99998 −0.999992 0.00393373i \(-0.998748\pi\)
−0.999992 + 0.00393373i \(0.998748\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −1.24949e24 −1.98132
\(945\) 0 0
\(946\) 4.80542e23 0.749208
\(947\) 3.72913e23 0.576512 0.288256 0.957553i \(-0.406925\pi\)
0.288256 + 0.957553i \(0.406925\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1.12273e24 1.69233
\(951\) 0 0
\(952\) 0 0
\(953\) −1.12569e24 −1.65454 −0.827270 0.561804i \(-0.810107\pi\)
−0.827270 + 0.561804i \(0.810107\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.27423e23 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −1.46950e24 −1.97039
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −8.33837e23 −1.08163
\(969\) 0 0
\(970\) 0 0
\(971\) 7.70609e23 0.975169 0.487585 0.873076i \(-0.337878\pi\)
0.487585 + 0.873076i \(0.337878\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.44028e22 −0.0655337 −0.0327669 0.999463i \(-0.510432\pi\)
−0.0327669 + 0.999463i \(0.510432\pi\)
\(978\) 0 0
\(979\) 3.86857e23 0.458447
\(980\) 0 0
\(981\) 0 0
\(982\) 8.05242e23 0.931182
\(983\) 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
\(991\) 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 \(\pi\)
\(998\) −7.43431e23 −0.755433
\(999\) 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 72.17.b.a.19.1 1
3.2 odd 2 8.17.d.a.3.1 1
8.3 odd 2 CM 72.17.b.a.19.1 1
12.11 even 2 32.17.d.a.15.1 1
24.5 odd 2 32.17.d.a.15.1 1
24.11 even 2 8.17.d.a.3.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.17.d.a.3.1 1 3.2 odd 2
8.17.d.a.3.1 1 24.11 even 2
32.17.d.a.15.1 1 12.11 even 2
32.17.d.a.15.1 1 24.5 odd 2
72.17.b.a.19.1 1 1.1 even 1 trivial
72.17.b.a.19.1 1 8.3 odd 2 CM