Properties

Label 72.9.b.a.19.1
Level $72$
Weight $9$
Character 72.19
Self dual yes
Analytic conductor $29.331$
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: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3312599244\)
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

\(f(q)\) \(=\) \(q-16.0000 q^{2} +256.000 q^{4} -4096.00 q^{8} +O(q^{10})\) \(q-16.0000 q^{2} +256.000 q^{4} -4096.00 q^{8} +27166.0 q^{11} +65536.0 q^{16} -162434. q^{17} -72286.0 q^{19} -434656. q^{22} +390625. q^{25} -1.04858e6 q^{32} +2.59894e6 q^{34} +1.15658e6 q^{38} +4.09901e6 q^{41} +5.42640e6 q^{43} +6.95450e6 q^{44} +5.76480e6 q^{49} -6.25000e6 q^{50} +2.41781e7 q^{59} +1.67772e7 q^{64} -1.39443e7 q^{67} -4.15831e7 q^{68} +3.35676e7 q^{73} -1.85052e7 q^{76} -6.55841e7 q^{82} -3.02100e7 q^{83} -8.68224e7 q^{86} -1.11272e8 q^{88} +9.55198e7 q^{89} -7.74182e7 q^{97} -9.22368e7 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −1.00000
\(3\) 0 0
\(4\) 256.000 1.00000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −4096.00 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 27166.0 1.85547 0.927737 0.373234i \(-0.121751\pi\)
0.927737 + 0.373234i \(0.121751\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 65536.0 1.00000
\(17\) −162434. −1.94483 −0.972414 0.233261i \(-0.925060\pi\)
−0.972414 + 0.233261i \(0.925060\pi\)
\(18\) 0 0
\(19\) −72286.0 −0.554677 −0.277338 0.960772i \(-0.589452\pi\)
−0.277338 + 0.960772i \(0.589452\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −434656. −1.85547
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 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
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −1.04858e6 −1.00000
\(33\) 0 0
\(34\) 2.59894e6 1.94483
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.15658e6 0.554677
\(39\) 0 0
\(40\) 0 0
\(41\) 4.09901e6 1.45058 0.725292 0.688441i \(-0.241705\pi\)
0.725292 + 0.688441i \(0.241705\pi\)
\(42\) 0 0
\(43\) 5.42640e6 1.58722 0.793612 0.608424i \(-0.208198\pi\)
0.793612 + 0.608424i \(0.208198\pi\)
\(44\) 6.95450e6 1.85547
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) −6.25000e6 −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
\(59\) 2.41781e7 1.99533 0.997663 0.0683312i \(-0.0217675\pi\)
0.997663 + 0.0683312i \(0.0217675\pi\)
\(60\) 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
\(67\) −1.39443e7 −0.691986 −0.345993 0.938237i \(-0.612458\pi\)
−0.345993 + 0.938237i \(0.612458\pi\)
\(68\) −4.15831e7 −1.94483
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 3.35676e7 1.18203 0.591015 0.806661i \(-0.298728\pi\)
0.591015 + 0.806661i \(0.298728\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1.85052e7 −0.554677
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.55841e7 −1.45058
\(83\) −3.02100e7 −0.636558 −0.318279 0.947997i \(-0.603105\pi\)
−0.318279 + 0.947997i \(0.603105\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.68224e7 −1.58722
\(87\) 0 0
\(88\) −1.11272e8 −1.85547
\(89\) 9.55198e7 1.52242 0.761208 0.648508i \(-0.224607\pi\)
0.761208 + 0.648508i \(0.224607\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.74182e7 −0.874493 −0.437247 0.899342i \(-0.644046\pi\)
−0.437247 + 0.899342i \(0.644046\pi\)
\(98\) −9.22368e7 −1.00000
\(99\) 0 0
\(100\) 1.00000e8 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
\(107\) 1.84965e8 1.41109 0.705546 0.708664i \(-0.250702\pi\)
0.705546 + 0.708664i \(0.250702\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.22024e7 0.442831 0.221415 0.975180i \(-0.428932\pi\)
0.221415 + 0.975180i \(0.428932\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −3.86849e8 −1.99533
\(119\) 0 0
\(120\) 0 0
\(121\) 5.23633e8 2.44279
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −2.68435e8 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −3.39909e8 −1.15419 −0.577095 0.816677i \(-0.695814\pi\)
−0.577095 + 0.816677i \(0.695814\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 2.23109e8 0.691986
\(135\) 0 0
\(136\) 6.65330e8 1.94483
\(137\) 3.39511e8 0.963767 0.481884 0.876235i \(-0.339953\pi\)
0.481884 + 0.876235i \(0.339953\pi\)
\(138\) 0 0
\(139\) −7.32208e8 −1.96144 −0.980720 0.195418i \(-0.937394\pi\)
−0.980720 + 0.195418i \(0.937394\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −5.37081e8 −1.18203
\(147\) 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 \(\pi\)
\(152\) 2.96083e8 0.554677
\(153\) 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
\(163\) 1.14307e9 1.61929 0.809644 0.586921i \(-0.199660\pi\)
0.809644 + 0.586921i \(0.199660\pi\)
\(164\) 1.04935e9 1.45058
\(165\) 0 0
\(166\) 4.83359e8 0.636558
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 1.38916e9 1.58722
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.78035e9 1.85547
\(177\) 0 0
\(178\) −1.52832e9 −1.52242
\(179\) −1.90643e9 −1.85699 −0.928493 0.371349i \(-0.878895\pi\)
−0.928493 + 0.371349i \(0.878895\pi\)
\(180\) 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
\(187\) −4.41268e9 −3.60858
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.43626e9 1.03515 0.517574 0.855638i \(-0.326835\pi\)
0.517574 + 0.855638i \(0.326835\pi\)
\(194\) 1.23869e9 0.874493
\(195\) 0 0
\(196\) 1.47579e9 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.60000e9 −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
\(209\) −1.96372e9 −1.02919
\(210\) 0 0
\(211\) −2.52282e9 −1.27279 −0.636395 0.771363i \(-0.719575\pi\)
−0.636395 + 0.771363i \(0.719575\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −2.95945e9 −1.41109
\(215\) 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
\(226\) −1.15524e9 −0.442831
\(227\) −3.87828e9 −1.46062 −0.730308 0.683118i \(-0.760623\pi\)
−0.730308 + 0.683118i \(0.760623\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.69938e8 −0.159447 −0.0797236 0.996817i \(-0.525404\pi\)
−0.0797236 + 0.996817i \(0.525404\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.18959e9 1.99533
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −5.80472e8 −0.172073 −0.0860366 0.996292i \(-0.527420\pi\)
−0.0860366 + 0.996292i \(0.527420\pi\)
\(242\) −8.37812e9 −2.44279
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.70800e8 −0.0934210 −0.0467105 0.998908i \(-0.514874\pi\)
−0.0467105 + 0.998908i \(0.514874\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 8.43940e9 1.93455 0.967273 0.253738i \(-0.0816603\pi\)
0.967273 + 0.253738i \(0.0816603\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 5.43854e9 1.15419
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −3.56974e9 −0.691986
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −1.06453e10 −1.94483
\(273\) 0 0
\(274\) −5.43218e9 −0.963767
\(275\) 1.06117e10 1.85547
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 1.17153e10 1.96144
\(279\) 0 0
\(280\) 0 0
\(281\) 2.21245e9 0.354852 0.177426 0.984134i \(-0.443223\pi\)
0.177426 + 0.984134i \(0.443223\pi\)
\(282\) 0 0
\(283\) 1.07196e10 1.67122 0.835611 0.549321i \(-0.185114\pi\)
0.835611 + 0.549321i \(0.185114\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.94090e10 2.78236
\(290\) 0 0
\(291\) 0 0
\(292\) 8.59329e9 1.18203
\(293\) 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
\(304\) −4.73734e9 −0.554677
\(305\) 0 0
\(306\) 0 0
\(307\) −6.68482e9 −0.752551 −0.376276 0.926508i \(-0.622795\pi\)
−0.376276 + 0.926508i \(0.622795\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −1.26772e10 −1.32083 −0.660415 0.750901i \(-0.729620\pi\)
−0.660415 + 0.750901i \(0.729620\pi\)
\(314\) 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
\(323\) 1.17417e10 1.07875
\(324\) 0 0
\(325\) 0 0
\(326\) −1.82892e10 −1.61929
\(327\) 0 0
\(328\) −1.67895e10 −1.45058
\(329\) 0 0
\(330\) 0 0
\(331\) 2.39214e10 1.99285 0.996424 0.0844976i \(-0.0269285\pi\)
0.996424 + 0.0844976i \(0.0269285\pi\)
\(332\) −7.73375e9 −0.636558
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.57940e10 −1.99986 −0.999930 0.0118516i \(-0.996227\pi\)
−0.999930 + 0.0118516i \(0.996227\pi\)
\(338\) −1.30517e10 −1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −2.22265e10 −1.58722
\(345\) 0 0
\(346\) 0 0
\(347\) −7.92343e9 −0.546507 −0.273253 0.961942i \(-0.588100\pi\)
−0.273253 + 0.961942i \(0.588100\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.84856e10 −1.85547
\(353\) −1.37121e10 −0.883093 −0.441546 0.897238i \(-0.645570\pi\)
−0.441546 + 0.897238i \(0.645570\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.44531e10 1.52242
\(357\) 0 0
\(358\) 3.05029e10 1.85699
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −1.17583e10 −0.692334
\(362\) 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 \(\pi\)
\(374\) 7.06029e10 3.60858
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −7.66145e9 −0.371325 −0.185663 0.982614i \(-0.559443\pi\)
−0.185663 + 0.982614i \(0.559443\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.29801e10 −1.03515
\(387\) 0 0
\(388\) −1.98191e10 −0.874493
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.36126e10 −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
\(400\) 2.56000e10 1.00000
\(401\) −1.85927e10 −0.719061 −0.359530 0.933133i \(-0.617063\pi\)
−0.359530 + 0.933133i \(0.617063\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.23419e9 −0.222785 −0.111393 0.993776i \(-0.535531\pi\)
−0.111393 + 0.993776i \(0.535531\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 3.14195e10 1.02919
\(419\) 5.40872e10 1.75484 0.877422 0.479720i \(-0.159262\pi\)
0.877422 + 0.479720i \(0.159262\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 4.03651e10 1.27279
\(423\) 0 0
\(424\) 0 0
\(425\) −6.34508e10 −1.94483
\(426\) 0 0
\(427\) 0 0
\(428\) 4.73511e10 1.41109
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −6.86319e10 −1.95243 −0.976213 0.216813i \(-0.930434\pi\)
−0.976213 + 0.216813i \(0.930434\pi\)
\(434\) 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
\(443\) −6.61486e10 −1.71754 −0.858768 0.512364i \(-0.828770\pi\)
−0.858768 + 0.512364i \(0.828770\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.89524e10 −0.958405 −0.479202 0.877704i \(-0.659074\pi\)
−0.479202 + 0.877704i \(0.659074\pi\)
\(450\) 0 0
\(451\) 1.11354e11 2.69152
\(452\) 1.84838e10 0.442831
\(453\) 0 0
\(454\) 6.20525e10 1.46062
\(455\) 0 0
\(456\) 0 0
\(457\) 4.31242e10 0.988681 0.494340 0.869268i \(-0.335410\pi\)
0.494340 + 0.869268i \(0.335410\pi\)
\(458\) 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
\(466\) 7.51901e9 0.159447
\(467\) −9.41185e10 −1.97883 −0.989413 0.145128i \(-0.953641\pi\)
−0.989413 + 0.145128i \(0.953641\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −9.90334e10 −1.99533
\(473\) 1.47414e11 2.94505
\(474\) 0 0
\(475\) −2.82367e10 −0.554677
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 9.28756e9 0.172073
\(483\) 0 0
\(484\) 1.34050e11 2.44279
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.95056e10 1.71207 0.856035 0.516918i \(-0.172921\pi\)
0.856035 + 0.516918i \(0.172921\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.02919e11 −1.65995 −0.829975 0.557801i \(-0.811645\pi\)
−0.829975 + 0.557801i \(0.811645\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.93280e9 0.0934210
\(503\) 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
\(512\) −6.87195e10 −1.00000
\(513\) 0 0
\(514\) −1.35030e11 −1.93455
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.27450e10 −0.987308 −0.493654 0.869658i \(-0.664339\pi\)
−0.493654 + 0.869658i \(0.664339\pi\)
\(522\) 0 0
\(523\) 1.41570e9 0.0189219 0.00946097 0.999955i \(-0.496988\pi\)
0.00946097 + 0.999955i \(0.496988\pi\)
\(524\) −8.70166e10 −1.15419
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.83110e10 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 5.71158e10 0.691986
\(537\) 0 0
\(538\) 0 0
\(539\) 1.56607e11 1.85547
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.70324e11 1.94483
\(545\) 0 0
\(546\) 0 0
\(547\) 1.99228e10 0.222537 0.111268 0.993790i \(-0.464509\pi\)
0.111268 + 0.993790i \(0.464509\pi\)
\(548\) 8.69149e10 0.963767
\(549\) 0 0
\(550\) −1.69788e11 −1.85547
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.87445e11 −1.96144
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −3.53991e10 −0.354852
\(563\) −1.38789e11 −1.38141 −0.690705 0.723136i \(-0.742700\pi\)
−0.690705 + 0.723136i \(0.742700\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.71514e11 −1.67122
\(567\) 0 0
\(568\) 0 0
\(569\) 1.44287e11 1.37651 0.688255 0.725468i \(-0.258377\pi\)
0.688255 + 0.725468i \(0.258377\pi\)
\(570\) 0 0
\(571\) −1.64593e11 −1.54834 −0.774170 0.632978i \(-0.781832\pi\)
−0.774170 + 0.632978i \(0.781832\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2.21678e11 1.99995 0.999976 0.00693236i \(-0.00220666\pi\)
0.999976 + 0.00693236i \(0.00220666\pi\)
\(578\) −3.10545e11 −2.78236
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −1.37493e11 −1.18203
\(585\) 0 0
\(586\) 0 0
\(587\) −1.29666e11 −1.09213 −0.546066 0.837742i \(-0.683875\pi\)
−0.546066 + 0.837742i \(0.683875\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 2.45734e11 1.98722 0.993612 0.112847i \(-0.0359969\pi\)
0.993612 + 0.112847i \(0.0359969\pi\)
\(594\) 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
\(601\) −2.48165e11 −1.90214 −0.951069 0.308978i \(-0.900013\pi\)
−0.951069 + 0.308978i \(0.900013\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 7.57974e10 0.554677
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.06957e11 0.752551
\(615\) 0 0
\(616\) 0 0
\(617\) −1.32420e11 −0.913722 −0.456861 0.889538i \(-0.651026\pi\)
−0.456861 + 0.889538i \(0.651026\pi\)
\(618\) 0 0
\(619\) −9.06943e10 −0.617756 −0.308878 0.951102i \(-0.599954\pi\)
−0.308878 + 0.951102i \(0.599954\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.52588e11 1.00000
\(626\) 2.02836e11 1.32083
\(627\) 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
\(641\) −9.79076e9 −0.0579942 −0.0289971 0.999579i \(-0.509231\pi\)
−0.0289971 + 0.999579i \(0.509231\pi\)
\(642\) 0 0
\(643\) 7.65969e10 0.448092 0.224046 0.974579i \(-0.428073\pi\)
0.224046 + 0.974579i \(0.428073\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −1.87867e11 −1.07875
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 6.56822e11 3.70228
\(650\) 0 0
\(651\) 0 0
\(652\) 2.92627e11 1.61929
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.68632e11 1.45058
\(657\) 0 0
\(658\) 0 0
\(659\) 3.62926e11 1.92432 0.962158 0.272493i \(-0.0878480\pi\)
0.962158 + 0.272493i \(0.0878480\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −3.82742e11 −1.99285
\(663\) 0 0
\(664\) 1.23740e11 0.636558
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 7.87721e9 0.0383983 0.0191992 0.999816i \(-0.493888\pi\)
0.0191992 + 0.999816i \(0.493888\pi\)
\(674\) 4.12704e11 1.99986
\(675\) 0 0
\(676\) 2.08827e11 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
\(683\) −1.64741e11 −0.757040 −0.378520 0.925593i \(-0.623567\pi\)
−0.378520 + 0.925593i \(0.623567\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 3.55625e11 1.58722
\(689\) 0 0
\(690\) 0 0
\(691\) −2.82749e11 −1.24019 −0.620095 0.784526i \(-0.712906\pi\)
−0.620095 + 0.784526i \(0.712906\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.26775e11 0.546507
\(695\) 0 0
\(696\) 0 0
\(697\) −6.65818e11 −2.82114
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 4.55770e11 1.85547
\(705\) 0 0
\(706\) 2.19394e11 0.883093
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.91249e11 −1.52242
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −4.88046e11 −1.85699
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.88133e11 0.692334
\(723\) 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
\(731\) −8.81432e11 −3.08688
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.78810e11 −1.28396
\(738\) 0 0
\(739\) 3.80751e11 1.27662 0.638312 0.769778i \(-0.279633\pi\)
0.638312 + 0.769778i \(0.279633\pi\)
\(740\) 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
\(748\) −1.12965e12 −3.60858
\(749\) 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 \(\pi\)
\(758\) 1.22583e11 0.371325
\(759\) 0 0
\(760\) 0 0
\(761\) 5.45467e10 0.162641 0.0813204 0.996688i \(-0.474086\pi\)
0.0813204 + 0.996688i \(0.474086\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −6.94260e11 −1.98526 −0.992628 0.121201i \(-0.961325\pi\)
−0.992628 + 0.121201i \(0.961325\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.67681e11 1.03515
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.17105e11 0.874493
\(777\) 0 0
\(778\) 0 0
\(779\) −2.96301e11 −0.804605
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.77802e11 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −6.21875e11 −1.62108 −0.810540 0.585684i \(-0.800826\pi\)
−0.810540 + 0.585684i \(0.800826\pi\)
\(788\) 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
\(800\) −4.09600e11 −1.00000
\(801\) 0 0
\(802\) 2.97484e11 0.719061
\(803\) 9.11896e11 2.19323
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.68377e11 −1.32691 −0.663456 0.748215i \(-0.730911\pi\)
−0.663456 + 0.748215i \(0.730911\pi\)
\(810\) 0 0
\(811\) 7.82841e11 1.80963 0.904815 0.425804i \(-0.140009\pi\)
0.904815 + 0.425804i \(0.140009\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.92253e11 −0.880396
\(818\) 9.97470e10 0.222785
\(819\) 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
\(827\) 8.84989e11 1.89198 0.945988 0.324200i \(-0.105095\pi\)
0.945988 + 0.324200i \(0.105095\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −9.36400e11 −1.94483
\(834\) 0 0
\(835\) 0 0
\(836\) −5.02713e11 −1.02919
\(837\) 0 0
\(838\) −8.65395e11 −1.75484
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −6.45842e11 −1.27279
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 1.01521e12 1.94483
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.57618e11 −1.41109
\(857\) 1.07825e12 1.99892 0.999462 0.0328011i \(-0.0104428\pi\)
0.999462 + 0.0328011i \(0.0104428\pi\)
\(858\) 0 0
\(859\) 4.32719e11 0.794755 0.397377 0.917655i \(-0.369920\pi\)
0.397377 + 0.917655i \(0.369920\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.09811e12 1.95243
\(867\) 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
\(881\) 9.39089e11 1.55885 0.779423 0.626498i \(-0.215512\pi\)
0.779423 + 0.626498i \(0.215512\pi\)
\(882\) 0 0
\(883\) 1.17202e12 1.92793 0.963965 0.266030i \(-0.0857120\pi\)
0.963965 + 0.266030i \(0.0857120\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.05838e12 1.71754
\(887\) 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
\(898\) 6.23239e11 0.958405
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −1.78166e12 −2.69152
\(903\) 0 0
\(904\) −2.95741e11 −0.442831
\(905\) 0 0
\(906\) 0 0
\(907\) 3.17048e11 0.468486 0.234243 0.972178i \(-0.424739\pi\)
0.234243 + 0.972178i \(0.424739\pi\)
\(908\) −9.92840e11 −1.46062
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −8.20684e11 −1.18112
\(914\) −6.89987e11 −0.988681
\(915\) 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
\(929\) 1.12157e12 1.50579 0.752895 0.658140i \(-0.228657\pi\)
0.752895 + 0.658140i \(0.228657\pi\)
\(930\) 0 0
\(931\) −4.16714e11 −0.554677
\(932\) −1.20304e11 −0.159447
\(933\) 0 0
\(934\) 1.50590e12 1.97883
\(935\) 0 0
\(936\) 0 0
\(937\) −3.03224e9 −0.00393373 −0.00196687 0.999998i \(-0.500626\pi\)
−0.00196687 + 0.999998i \(0.500626\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.58453e12 1.99533
\(945\) 0 0
\(946\) −2.35862e12 −2.94505
\(947\) 9.59570e11 1.19310 0.596550 0.802576i \(-0.296538\pi\)
0.596550 + 0.802576i \(0.296538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.51787e11 0.554677
\(951\) 0 0
\(952\) 0 0
\(953\) −1.57684e12 −1.91169 −0.955843 0.293879i \(-0.905054\pi\)
−0.955843 + 0.293879i \(0.905054\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 8.52891e11 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −1.48601e11 −0.172073
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −2.14480e12 −2.44279
\(969\) 0 0
\(970\) 0 0
\(971\) 8.99918e11 1.01234 0.506170 0.862434i \(-0.331061\pi\)
0.506170 + 0.862434i \(0.331061\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.30947e12 −1.43720 −0.718598 0.695425i \(-0.755216\pi\)
−0.718598 + 0.695425i \(0.755216\pi\)
\(978\) 0 0
\(979\) 2.59489e12 2.82480
\(980\) 0 0
\(981\) 0 0
\(982\) −1.59209e12 −1.71207
\(983\) 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 \(\pi\)
\(998\) 1.64671e12 1.65995
\(999\) 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 72.9.b.a.19.1 1
3.2 odd 2 8.9.d.a.3.1 1
4.3 odd 2 288.9.b.a.271.1 1
8.3 odd 2 CM 72.9.b.a.19.1 1
8.5 even 2 288.9.b.a.271.1 1
12.11 even 2 32.9.d.a.15.1 1
24.5 odd 2 32.9.d.a.15.1 1
24.11 even 2 8.9.d.a.3.1 1
48.5 odd 4 256.9.c.f.255.1 2
48.11 even 4 256.9.c.f.255.2 2
48.29 odd 4 256.9.c.f.255.2 2
48.35 even 4 256.9.c.f.255.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.9.d.a.3.1 1 3.2 odd 2
8.9.d.a.3.1 1 24.11 even 2
32.9.d.a.15.1 1 12.11 even 2
32.9.d.a.15.1 1 24.5 odd 2
72.9.b.a.19.1 1 1.1 even 1 trivial
72.9.b.a.19.1 1 8.3 odd 2 CM
256.9.c.f.255.1 2 48.5 odd 4
256.9.c.f.255.1 2 48.35 even 4
256.9.c.f.255.2 2 48.11 even 4
256.9.c.f.255.2 2 48.29 odd 4
288.9.b.a.271.1 1 4.3 odd 2
288.9.b.a.271.1 1 8.5 even 2