Properties

Label 72.21.b.a.19.1
Level $72$
Weight $21$
Character 72.19
Self dual yes
Analytic conductor $182.530$
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: \( k \) \(=\) \( 21 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(182.529910874\)
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

\(f(q)\) \(=\) \(q-1024.00 q^{2} +1.04858e6 q^{4} -1.07374e9 q^{8} +O(q^{10})\) \(q-1024.00 q^{2} +1.04858e6 q^{4} -1.07374e9 q^{8} +4.23830e10 q^{11} +1.09951e12 q^{16} +3.35354e12 q^{17} -1.01465e12 q^{19} -4.34002e13 q^{22} +9.53674e13 q^{25} -1.12590e15 q^{32} -3.43402e15 q^{34} +1.03901e15 q^{38} +2.54181e16 q^{41} +2.78111e15 q^{43} +4.44418e16 q^{44} +7.97923e16 q^{49} -9.76562e16 q^{50} +1.73912e17 q^{59} +1.15292e18 q^{64} -3.56138e17 q^{67} +3.51644e18 q^{68} -6.01672e18 q^{73} -1.06394e18 q^{76} -2.60281e19 q^{82} +3.10229e19 q^{83} -2.84786e18 q^{86} -4.55084e19 q^{88} -6.12024e19 q^{89} -5.00091e19 q^{97} -8.17073e19 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −1.00000
\(3\) 0 0
\(4\) 1.04858e6 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.07374e9 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 4.23830e10 1.63405 0.817025 0.576603i \(-0.195622\pi\)
0.817025 + 0.576603i \(0.195622\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.09951e12 1.00000
\(17\) 3.35354e12 1.66347 0.831733 0.555176i \(-0.187349\pi\)
0.831733 + 0.555176i \(0.187349\pi\)
\(18\) 0 0
\(19\) −1.01465e12 −0.165494 −0.0827470 0.996571i \(-0.526369\pi\)
−0.0827470 + 0.996571i \(0.526369\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.34002e13 −1.63405
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 9.53674e13 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 \(\pi\)
\(32\) −1.12590e15 −1.00000
\(33\) 0 0
\(34\) −3.43402e15 −1.66347
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.03901e15 0.165494
\(39\) 0 0
\(40\) 0 0
\(41\) 2.54181e16 1.89367 0.946834 0.321721i \(-0.104261\pi\)
0.946834 + 0.321721i \(0.104261\pi\)
\(42\) 0 0
\(43\) 2.78111e15 0.128687 0.0643434 0.997928i \(-0.479505\pi\)
0.0643434 + 0.997928i \(0.479505\pi\)
\(44\) 4.44418e16 1.63405
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.97923e16 1.00000
\(50\) −9.76562e16 −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
\(59\) 1.73912e17 0.340259 0.170130 0.985422i \(-0.445581\pi\)
0.170130 + 0.985422i \(0.445581\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.15292e18 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −3.56138e17 −0.195375 −0.0976877 0.995217i \(-0.531145\pi\)
−0.0976877 + 0.995217i \(0.531145\pi\)
\(68\) 3.51644e18 1.66347
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −6.01672e18 −1.40001 −0.700005 0.714138i \(-0.746819\pi\)
−0.700005 + 0.714138i \(0.746819\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.06394e18 −0.165494
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.60281e19 −1.89367
\(83\) 3.10229e19 1.99941 0.999703 0.0243827i \(-0.00776202\pi\)
0.999703 + 0.0243827i \(0.00776202\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.84786e18 −0.128687
\(87\) 0 0
\(88\) −4.55084e19 −1.63405
\(89\) −6.12024e19 −1.96277 −0.981383 0.192059i \(-0.938484\pi\)
−0.981383 + 0.192059i \(0.938484\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.00091e19 −0.678160 −0.339080 0.940758i \(-0.610116\pi\)
−0.339080 + 0.940758i \(0.610116\pi\)
\(98\) −8.17073e19 −1.00000
\(99\) 0 0
\(100\) 1.00000e20 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
\(107\) −3.62647e20 −1.84351 −0.921756 0.387770i \(-0.873246\pi\)
−0.921756 + 0.387770i \(0.873246\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.52948e20 −0.450567 −0.225284 0.974293i \(-0.572331\pi\)
−0.225284 + 0.974293i \(0.572331\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −1.78086e20 −0.340259
\(119\) 0 0
\(120\) 0 0
\(121\) 1.12357e21 1.67012
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −1.18059e21 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −2.17294e21 −1.45994 −0.729970 0.683479i \(-0.760466\pi\)
−0.729970 + 0.683479i \(0.760466\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.64685e20 0.195375
\(135\) 0 0
\(136\) −3.60083e21 −1.66347
\(137\) −2.11640e21 −0.908642 −0.454321 0.890838i \(-0.650118\pi\)
−0.454321 + 0.890838i \(0.650118\pi\)
\(138\) 0 0
\(139\) 2.54239e21 0.944266 0.472133 0.881527i \(-0.343484\pi\)
0.472133 + 0.881527i \(0.343484\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 6.16112e21 1.40001
\(147\) 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 \(\pi\)
\(152\) 1.08948e21 0.165494
\(153\) 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
\(163\) 7.06792e19 0.00533845 0.00266923 0.999996i \(-0.499150\pi\)
0.00266923 + 0.999996i \(0.499150\pi\)
\(164\) 2.66528e22 1.89367
\(165\) 0 0
\(166\) −3.17674e22 −1.99941
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.90050e22 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 2.91621e21 0.128687
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.66006e22 1.63405
\(177\) 0 0
\(178\) 6.26713e22 1.96277
\(179\) 3.92234e22 1.16149 0.580744 0.814086i \(-0.302762\pi\)
0.580744 + 0.814086i \(0.302762\pi\)
\(180\) 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
\(187\) 1.42133e23 2.71818
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.20384e23 1.67878 0.839391 0.543528i \(-0.182912\pi\)
0.839391 + 0.543528i \(0.182912\pi\)
\(194\) 5.12093e22 0.678160
\(195\) 0 0
\(196\) 8.36683e22 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.02400e23 −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
\(209\) −4.30041e22 −0.270425
\(210\) 0 0
\(211\) −2.82329e23 −1.61410 −0.807051 0.590482i \(-0.798938\pi\)
−0.807051 + 0.590482i \(0.798938\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 3.71350e23 1.84351
\(215\) 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
\(226\) 1.56619e23 0.450567
\(227\) 2.21143e23 0.608716 0.304358 0.952558i \(-0.401558\pi\)
0.304358 + 0.952558i \(0.401558\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.85874e23 1.66645 0.833227 0.552931i \(-0.186491\pi\)
0.833227 + 0.552931i \(0.186491\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.82360e23 0.340259
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 7.13387e23 1.07933 0.539666 0.841879i \(-0.318551\pi\)
0.539666 + 0.841879i \(0.318551\pi\)
\(242\) −1.15054e24 −1.67012
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.55766e24 1.56941 0.784703 0.619872i \(-0.212815\pi\)
0.784703 + 0.619872i \(0.212815\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.20893e24 1.00000
\(257\) −1.50407e24 −1.19657 −0.598284 0.801284i \(-0.704151\pi\)
−0.598284 + 0.801284i \(0.704151\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 2.22509e24 1.45994
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3.73437e23 −0.195375
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 3.68725e24 1.66347
\(273\) 0 0
\(274\) 2.16720e24 0.908642
\(275\) 4.04196e24 1.63405
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) −2.60341e24 −0.944266
\(279\) 0 0
\(280\) 0 0
\(281\) −2.04420e24 −0.665979 −0.332989 0.942931i \(-0.608057\pi\)
−0.332989 + 0.942931i \(0.608057\pi\)
\(282\) 0 0
\(283\) −7.68742e23 −0.233301 −0.116650 0.993173i \(-0.537216\pi\)
−0.116650 + 0.993173i \(0.537216\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 7.18197e24 1.76712
\(290\) 0 0
\(291\) 0 0
\(292\) −6.30899e24 −1.40001
\(293\) 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
\(304\) −1.11562e24 −0.165494
\(305\) 0 0
\(306\) 0 0
\(307\) 2.64867e24 0.356160 0.178080 0.984016i \(-0.443011\pi\)
0.178080 + 0.984016i \(0.443011\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 1.53618e25 1.70215 0.851075 0.525044i \(-0.175951\pi\)
0.851075 + 0.525044i \(0.175951\pi\)
\(314\) 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
\(323\) −3.40268e24 −0.275293
\(324\) 0 0
\(325\) 0 0
\(326\) −7.23755e22 −0.00533845
\(327\) 0 0
\(328\) −2.72924e25 −1.89367
\(329\) 0 0
\(330\) 0 0
\(331\) −3.08691e25 −1.95544 −0.977718 0.209923i \(-0.932679\pi\)
−0.977718 + 0.209923i \(0.932679\pi\)
\(332\) 3.25298e25 1.99941
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.11942e24 0.0592508 0.0296254 0.999561i \(-0.490569\pi\)
0.0296254 + 0.999561i \(0.490569\pi\)
\(338\) −1.94611e25 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −2.98620e24 −0.128687
\(345\) 0 0
\(346\) 0 0
\(347\) 5.03994e25 1.99127 0.995635 0.0933286i \(-0.0297507\pi\)
0.995635 + 0.0933286i \(0.0297507\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.77190e25 −1.63405
\(353\) −5.62790e25 −1.87327 −0.936633 0.350313i \(-0.886075\pi\)
−0.936633 + 0.350313i \(0.886075\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.41754e25 −1.96277
\(357\) 0 0
\(358\) −4.01647e25 −1.16149
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −3.65604e25 −0.972612
\(362\) 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 \(\pi\)
\(374\) −1.45544e26 −2.71818
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −3.83008e25 −0.626347 −0.313174 0.949696i \(-0.601392\pi\)
−0.313174 + 0.949696i \(0.601392\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.23273e26 −1.67878
\(387\) 0 0
\(388\) −5.24384e25 −0.678160
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −8.56763e25 −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
\(400\) 1.04858e26 1.00000
\(401\) 2.13090e26 1.98207 0.991034 0.133612i \(-0.0426576\pi\)
0.991034 + 0.133612i \(0.0426576\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.27053e26 0.969955 0.484977 0.874527i \(-0.338828\pi\)
0.484977 + 0.874527i \(0.338828\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 4.40362e25 0.270425
\(419\) 3.16629e26 1.89850 0.949249 0.314527i \(-0.101846\pi\)
0.949249 + 0.314527i \(0.101846\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 2.89105e26 1.61410
\(423\) 0 0
\(424\) 0 0
\(425\) 3.19818e26 1.66347
\(426\) 0 0
\(427\) 0 0
\(428\) −3.80263e26 −1.84351
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −2.40752e26 −1.03918 −0.519591 0.854415i \(-0.673916\pi\)
−0.519591 + 0.854415i \(0.673916\pi\)
\(434\) 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
\(443\) −1.30434e26 −0.448078 −0.224039 0.974580i \(-0.571924\pi\)
−0.224039 + 0.974580i \(0.571924\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.95620e26 1.78857 0.894285 0.447499i \(-0.147685\pi\)
0.894285 + 0.447499i \(0.147685\pi\)
\(450\) 0 0
\(451\) 1.07729e27 3.09435
\(452\) −1.60378e26 −0.450567
\(453\) 0 0
\(454\) −2.26450e26 −0.608716
\(455\) 0 0
\(456\) 0 0
\(457\) 6.94600e26 1.74813 0.874064 0.485812i \(-0.161476\pi\)
0.874064 + 0.485812i \(0.161476\pi\)
\(458\) 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
\(466\) −8.04735e26 −1.66645
\(467\) 9.22043e26 1.86888 0.934442 0.356114i \(-0.115899\pi\)
0.934442 + 0.356114i \(0.115899\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.86737e26 −0.340259
\(473\) 1.17872e26 0.210281
\(474\) 0 0
\(475\) −9.67650e25 −0.165494
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −7.30509e26 −1.07933
\(483\) 0 0
\(484\) 1.17815e27 1.67012
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.59202e27 −1.95494 −0.977469 0.211079i \(-0.932302\pi\)
−0.977469 + 0.211079i \(0.932302\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1.90641e27 1.99164 0.995820 0.0913348i \(-0.0291133\pi\)
0.995820 + 0.0913348i \(0.0291133\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −1.59504e27 −1.56941
\(503\) 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
\(512\) −1.23794e27 −1.00000
\(513\) 0 0
\(514\) 1.54017e27 1.19657
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.57885e27 1.75004 0.875021 0.484085i \(-0.160848\pi\)
0.875021 + 0.484085i \(0.160848\pi\)
\(522\) 0 0
\(523\) 2.21598e27 1.44726 0.723632 0.690186i \(-0.242471\pi\)
0.723632 + 0.690186i \(0.242471\pi\)
\(524\) −2.27849e27 −1.45994
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.71616e27 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 3.82400e26 0.195375
\(537\) 0 0
\(538\) 0 0
\(539\) 3.38184e27 1.63405
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −3.77575e27 −1.66347
\(545\) 0 0
\(546\) 0 0
\(547\) −4.19373e27 −1.74875 −0.874375 0.485251i \(-0.838728\pi\)
−0.874375 + 0.485251i \(0.838728\pi\)
\(548\) −2.21921e27 −0.908642
\(549\) 0 0
\(550\) −4.13897e27 −1.63405
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 2.66589e27 0.944266
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 2.09326e27 0.665979
\(563\) −2.78356e27 −0.869994 −0.434997 0.900432i \(-0.643251\pi\)
−0.434997 + 0.900432i \(0.643251\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.87192e26 0.233301
\(567\) 0 0
\(568\) 0 0
\(569\) 6.37985e27 1.79344 0.896722 0.442595i \(-0.145942\pi\)
0.896722 + 0.442595i \(0.145942\pi\)
\(570\) 0 0
\(571\) −7.29382e27 −1.97967 −0.989837 0.142204i \(-0.954581\pi\)
−0.989837 + 0.142204i \(0.954581\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.17943e27 −1.99970 −0.999850 0.0173302i \(-0.994483\pi\)
−0.999850 + 0.0173302i \(0.994483\pi\)
\(578\) −7.35434e27 −1.76712
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 6.46040e27 1.40001
\(585\) 0 0
\(586\) 0 0
\(587\) 7.68162e27 1.58151 0.790755 0.612133i \(-0.209688\pi\)
0.790755 + 0.612133i \(0.209688\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.00007e27 −0.557936 −0.278968 0.960300i \(-0.589992\pi\)
−0.278968 + 0.960300i \(0.589992\pi\)
\(594\) 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
\(601\) 8.69391e27 1.41407 0.707035 0.707179i \(-0.250033\pi\)
0.707035 + 0.707179i \(0.250033\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 1.14240e27 0.165494
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) −2.71224e27 −0.356160
\(615\) 0 0
\(616\) 0 0
\(617\) −1.47242e28 −1.84153 −0.920765 0.390119i \(-0.872434\pi\)
−0.920765 + 0.390119i \(0.872434\pi\)
\(618\) 0 0
\(619\) 6.02524e24 0.000729572 0 0.000364786 1.00000i \(-0.499884\pi\)
0.000364786 1.00000i \(0.499884\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9.09495e27 1.00000
\(626\) −1.57305e28 −1.70215
\(627\) 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
\(641\) −1.77175e28 −1.51294 −0.756471 0.654027i \(-0.773078\pi\)
−0.756471 + 0.654027i \(0.773078\pi\)
\(642\) 0 0
\(643\) −2.35774e28 −1.95158 −0.975791 0.218706i \(-0.929816\pi\)
−0.975791 + 0.218706i \(0.929816\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.48434e27 0.275293
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 7.37092e27 0.556000
\(650\) 0 0
\(651\) 0 0
\(652\) 7.41125e25 0.00533845
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.79475e28 1.89367
\(657\) 0 0
\(658\) 0 0
\(659\) −1.96646e28 −1.27301 −0.636504 0.771274i \(-0.719620\pi\)
−0.636504 + 0.771274i \(0.719620\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 3.16099e28 1.95544
\(663\) 0 0
\(664\) −3.33105e28 −1.99941
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.82190e28 −1.48044 −0.740221 0.672364i \(-0.765279\pi\)
−0.740221 + 0.672364i \(0.765279\pi\)
\(674\) −1.14629e27 −0.0592508
\(675\) 0 0
\(676\) 1.99281e28 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
\(683\) −4.34265e28 −1.96584 −0.982919 0.184038i \(-0.941083\pi\)
−0.982919 + 0.184038i \(0.941083\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 3.05787e27 0.128687
\(689\) 0 0
\(690\) 0 0
\(691\) −3.84706e28 −1.55006 −0.775030 0.631924i \(-0.782265\pi\)
−0.775030 + 0.631924i \(0.782265\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −5.16089e28 −1.99127
\(695\) 0 0
\(696\) 0 0
\(697\) 8.52404e28 3.15005
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.88643e28 1.63405
\(705\) 0 0
\(706\) 5.76297e28 1.87327
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.57156e28 1.96277
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 4.11287e28 1.16149
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 3.74379e28 0.972612
\(723\) 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
\(731\) 9.32657e27 0.214066
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.50942e28 −0.319253
\(738\) 0 0
\(739\) 5.68807e28 1.17090 0.585451 0.810708i \(-0.300917\pi\)
0.585451 + 0.810708i \(0.300917\pi\)
\(740\) 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
\(748\) 1.49037e29 2.71818
\(749\) 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 \(\pi\)
\(758\) 3.92200e28 0.626347
\(759\) 0 0
\(760\) 0 0
\(761\) 7.16003e28 1.09918 0.549589 0.835435i \(-0.314784\pi\)
0.549589 + 0.835435i \(0.314784\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −4.32623e28 −0.598200 −0.299100 0.954222i \(-0.596686\pi\)
−0.299100 + 0.954222i \(0.596686\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.26231e29 1.67878
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.36969e28 0.678160
\(777\) 0 0
\(778\) 0 0
\(779\) −2.57906e28 −0.313391
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.77325e28 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −1.82292e29 −1.99996 −0.999979 0.00648819i \(-0.997935\pi\)
−0.999979 + 0.00648819i \(0.997935\pi\)
\(788\) 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
\(800\) −1.07374e29 −1.00000
\(801\) 0 0
\(802\) −2.18204e29 −1.98207
\(803\) −2.55007e29 −2.28768
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.24019e29 1.03277 0.516383 0.856357i \(-0.327278\pi\)
0.516383 + 0.856357i \(0.327278\pi\)
\(810\) 0 0
\(811\) −1.11745e29 −0.907860 −0.453930 0.891037i \(-0.649978\pi\)
−0.453930 + 0.891037i \(0.649978\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.82187e27 −0.0212969
\(818\) −1.30102e29 −0.969955
\(819\) 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
\(827\) −2.19924e29 −1.46966 −0.734830 0.678252i \(-0.762738\pi\)
−0.734830 + 0.678252i \(0.762738\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.67586e29 1.66347
\(834\) 0 0
\(835\) 0 0
\(836\) −4.50931e28 −0.270425
\(837\) 0 0
\(838\) −3.24228e29 −1.89850
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.76995e29 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −2.96043e29 −1.61410
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −3.27494e29 −1.66347
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 3.89389e29 1.84351
\(857\) 3.50153e28 0.163851 0.0819256 0.996638i \(-0.473893\pi\)
0.0819256 + 0.996638i \(0.473893\pi\)
\(858\) 0 0
\(859\) −4.25332e29 −1.94445 −0.972224 0.234053i \(-0.924801\pi\)
−0.972224 + 0.234053i \(0.924801\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.46530e29 1.03918
\(867\) 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
\(881\) −5.59188e29 −1.98517 −0.992587 0.121534i \(-0.961219\pi\)
−0.992587 + 0.121534i \(0.961219\pi\)
\(882\) 0 0
\(883\) 4.50564e29 1.56369 0.781843 0.623476i \(-0.214280\pi\)
0.781843 + 0.623476i \(0.214280\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.33564e29 0.448078
\(887\) 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
\(898\) −6.09915e29 −1.78857
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −1.10315e30 −3.09435
\(903\) 0 0
\(904\) 1.64227e29 0.450567
\(905\) 0 0
\(906\) 0 0
\(907\) −7.39355e29 −1.96237 −0.981183 0.193079i \(-0.938153\pi\)
−0.981183 + 0.193079i \(0.938153\pi\)
\(908\) 2.31885e29 0.608716
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1.31484e30 3.26713
\(914\) −7.11271e29 −1.74813
\(915\) 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
\(929\) 9.33455e29 1.94956 0.974779 0.223173i \(-0.0716415\pi\)
0.974779 + 0.223173i \(0.0716415\pi\)
\(930\) 0 0
\(931\) −8.09616e28 −0.165494
\(932\) 8.24048e29 1.66645
\(933\) 0 0
\(934\) −9.44172e29 −1.86888
\(935\) 0 0
\(936\) 0 0
\(937\) 7.34116e29 1.40724 0.703621 0.710575i \(-0.251565\pi\)
0.703621 + 0.710575i \(0.251565\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.91218e29 0.340259
\(945\) 0 0
\(946\) −1.20701e29 −0.210281
\(947\) 8.42382e29 1.45214 0.726071 0.687619i \(-0.241344\pi\)
0.726071 + 0.687619i \(0.241344\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 9.90873e28 0.165494
\(951\) 0 0
\(952\) 0 0
\(953\) 9.07852e29 1.46922 0.734609 0.678491i \(-0.237366\pi\)
0.734609 + 0.678491i \(0.237366\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 6.71791e29 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 7.48041e29 1.07933
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −1.20642e30 −1.67012
\(969\) 0 0
\(970\) 0 0
\(971\) 7.67973e29 1.03075 0.515376 0.856964i \(-0.327652\pi\)
0.515376 + 0.856964i \(0.327652\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 5.46005e29 0.689051 0.344525 0.938777i \(-0.388040\pi\)
0.344525 + 0.938777i \(0.388040\pi\)
\(978\) 0 0
\(979\) −2.59394e30 −3.20726
\(980\) 0 0
\(981\) 0 0
\(982\) 1.63023e30 1.95494
\(983\) 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 \(\pi\)
\(998\) −1.95216e30 −1.99164
\(999\) 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 72.21.b.a.19.1 1
3.2 odd 2 8.21.d.a.3.1 1
8.3 odd 2 CM 72.21.b.a.19.1 1
12.11 even 2 32.21.d.a.15.1 1
24.5 odd 2 32.21.d.a.15.1 1
24.11 even 2 8.21.d.a.3.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.21.d.a.3.1 1 3.2 odd 2
8.21.d.a.3.1 1 24.11 even 2
32.21.d.a.15.1 1 12.11 even 2
32.21.d.a.15.1 1 24.5 odd 2
72.21.b.a.19.1 1 1.1 even 1 trivial
72.21.b.a.19.1 1 8.3 odd 2 CM