Properties

Label 72.12.a.c.1.1
Level $72$
Weight $12$
Character 72.1
Self dual yes
Analytic conductor $55.321$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 72.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3207090003\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 72.1

$q$-expansion

\(f(q)\) \(=\) \(q+3490.00 q^{5} -55464.0 q^{7} +O(q^{10})\) \(q+3490.00 q^{5} -55464.0 q^{7} +597004. q^{11} +1.37388e6 q^{13} -1.01408e7 q^{17} -7.29740e6 q^{19} +3.20575e7 q^{23} -3.66480e7 q^{25} +1.36054e7 q^{29} +2.33161e8 q^{31} -1.93569e8 q^{35} -2.57786e8 q^{37} +2.21439e8 q^{41} -1.69776e9 q^{43} -5.27509e8 q^{47} +1.09893e9 q^{49} -3.27738e9 q^{53} +2.08354e9 q^{55} +3.00191e9 q^{59} -1.16300e10 q^{61} +4.79483e9 q^{65} -1.71890e10 q^{67} -2.61695e10 q^{71} -7.03902e9 q^{73} -3.31122e10 q^{77} -4.19991e9 q^{79} +3.97399e10 q^{83} -3.53916e10 q^{85} -1.05653e10 q^{89} -7.62008e10 q^{91} -2.54679e10 q^{95} -6.98516e10 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3490.00 0.499448 0.249724 0.968317i \(-0.419660\pi\)
0.249724 + 0.968317i \(0.419660\pi\)
\(6\) 0 0
\(7\) −55464.0 −1.24730 −0.623652 0.781703i \(-0.714352\pi\)
−0.623652 + 0.781703i \(0.714352\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 597004. 1.11768 0.558840 0.829276i \(-0.311247\pi\)
0.558840 + 0.829276i \(0.311247\pi\)
\(12\) 0 0
\(13\) 1.37388e6 1.02627 0.513133 0.858309i \(-0.328485\pi\)
0.513133 + 0.858309i \(0.328485\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.01408e7 −1.73223 −0.866114 0.499846i \(-0.833390\pi\)
−0.866114 + 0.499846i \(0.833390\pi\)
\(18\) 0 0
\(19\) −7.29740e6 −0.676119 −0.338059 0.941125i \(-0.609771\pi\)
−0.338059 + 0.941125i \(0.609771\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.20575e7 1.03855 0.519273 0.854608i \(-0.326203\pi\)
0.519273 + 0.854608i \(0.326203\pi\)
\(24\) 0 0
\(25\) −3.66480e7 −0.750552
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.36054e7 0.123175 0.0615875 0.998102i \(-0.480384\pi\)
0.0615875 + 0.998102i \(0.480384\pi\)
\(30\) 0 0
\(31\) 2.33161e8 1.46274 0.731368 0.681983i \(-0.238882\pi\)
0.731368 + 0.681983i \(0.238882\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.93569e8 −0.622963
\(36\) 0 0
\(37\) −2.57786e8 −0.611153 −0.305577 0.952167i \(-0.598849\pi\)
−0.305577 + 0.952167i \(0.598849\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.21439e8 0.298498 0.149249 0.988800i \(-0.452314\pi\)
0.149249 + 0.988800i \(0.452314\pi\)
\(42\) 0 0
\(43\) −1.69776e9 −1.76116 −0.880581 0.473895i \(-0.842848\pi\)
−0.880581 + 0.473895i \(0.842848\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.27509e8 −0.335500 −0.167750 0.985830i \(-0.553650\pi\)
−0.167750 + 0.985830i \(0.553650\pi\)
\(48\) 0 0
\(49\) 1.09893e9 0.555765
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.27738e9 −1.07649 −0.538244 0.842789i \(-0.680912\pi\)
−0.538244 + 0.842789i \(0.680912\pi\)
\(54\) 0 0
\(55\) 2.08354e9 0.558223
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00191e9 0.546653 0.273326 0.961921i \(-0.411876\pi\)
0.273326 + 0.961921i \(0.411876\pi\)
\(60\) 0 0
\(61\) −1.16300e10 −1.76306 −0.881529 0.472130i \(-0.843485\pi\)
−0.881529 + 0.472130i \(0.843485\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.79483e9 0.512566
\(66\) 0 0
\(67\) −1.71890e10 −1.55539 −0.777695 0.628642i \(-0.783611\pi\)
−0.777695 + 0.628642i \(0.783611\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.61695e10 −1.72137 −0.860687 0.509135i \(-0.829966\pi\)
−0.860687 + 0.509135i \(0.829966\pi\)
\(72\) 0 0
\(73\) −7.03902e9 −0.397408 −0.198704 0.980060i \(-0.563673\pi\)
−0.198704 + 0.980060i \(0.563673\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.31122e10 −1.39409
\(78\) 0 0
\(79\) −4.19991e9 −0.153565 −0.0767823 0.997048i \(-0.524465\pi\)
−0.0767823 + 0.997048i \(0.524465\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.97399e10 1.10738 0.553691 0.832722i \(-0.313219\pi\)
0.553691 + 0.832722i \(0.313219\pi\)
\(84\) 0 0
\(85\) −3.53916e10 −0.865159
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.05653e10 −0.200557 −0.100279 0.994959i \(-0.531973\pi\)
−0.100279 + 0.994959i \(0.531973\pi\)
\(90\) 0 0
\(91\) −7.62008e10 −1.28006
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.54679e10 −0.337686
\(96\) 0 0
\(97\) −6.98516e10 −0.825909 −0.412954 0.910752i \(-0.635503\pi\)
−0.412954 + 0.910752i \(0.635503\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.45194e10 −0.705508 −0.352754 0.935716i \(-0.614755\pi\)
−0.352754 + 0.935716i \(0.614755\pi\)
\(102\) 0 0
\(103\) 7.86642e10 0.668609 0.334304 0.942465i \(-0.391499\pi\)
0.334304 + 0.942465i \(0.391499\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.88490e10 −0.543482 −0.271741 0.962370i \(-0.587599\pi\)
−0.271741 + 0.962370i \(0.587599\pi\)
\(108\) 0 0
\(109\) −7.08680e10 −0.441169 −0.220584 0.975368i \(-0.570796\pi\)
−0.220584 + 0.975368i \(0.570796\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.96816e10 −0.100491 −0.0502457 0.998737i \(-0.516000\pi\)
−0.0502457 + 0.998737i \(0.516000\pi\)
\(114\) 0 0
\(115\) 1.11881e11 0.518700
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.62452e11 2.16061
\(120\) 0 0
\(121\) 7.11021e10 0.249209
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.98312e11 −0.874310
\(126\) 0 0
\(127\) −1.90252e11 −0.510985 −0.255492 0.966811i \(-0.582238\pi\)
−0.255492 + 0.966811i \(0.582238\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.14525e11 −0.259363 −0.129681 0.991556i \(-0.541395\pi\)
−0.129681 + 0.991556i \(0.541395\pi\)
\(132\) 0 0
\(133\) 4.04743e11 0.843325
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.17101e11 −0.207299 −0.103650 0.994614i \(-0.533052\pi\)
−0.103650 + 0.994614i \(0.533052\pi\)
\(138\) 0 0
\(139\) 9.89561e11 1.61756 0.808781 0.588110i \(-0.200128\pi\)
0.808781 + 0.588110i \(0.200128\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.20211e11 1.14704
\(144\) 0 0
\(145\) 4.74829e10 0.0615195
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.97579e11 0.889712 0.444856 0.895602i \(-0.353255\pi\)
0.444856 + 0.895602i \(0.353255\pi\)
\(150\) 0 0
\(151\) 1.18323e12 1.22658 0.613290 0.789858i \(-0.289846\pi\)
0.613290 + 0.789858i \(0.289846\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.13731e11 0.730561
\(156\) 0 0
\(157\) −7.53512e11 −0.630438 −0.315219 0.949019i \(-0.602078\pi\)
−0.315219 + 0.949019i \(0.602078\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.77804e12 −1.29538
\(162\) 0 0
\(163\) 8.34305e11 0.567928 0.283964 0.958835i \(-0.408350\pi\)
0.283964 + 0.958835i \(0.408350\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.01514e12 0.604761 0.302380 0.953187i \(-0.402219\pi\)
0.302380 + 0.953187i \(0.402219\pi\)
\(168\) 0 0
\(169\) 9.53804e10 0.0532209
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.89496e12 −1.42033 −0.710165 0.704035i \(-0.751380\pi\)
−0.710165 + 0.704035i \(0.751380\pi\)
\(174\) 0 0
\(175\) 2.03265e12 0.936165
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.09263e12 1.66461 0.832303 0.554322i \(-0.187022\pi\)
0.832303 + 0.554322i \(0.187022\pi\)
\(180\) 0 0
\(181\) 5.08615e11 0.194606 0.0973032 0.995255i \(-0.468978\pi\)
0.0973032 + 0.995255i \(0.468978\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.99674e11 −0.305239
\(186\) 0 0
\(187\) −6.05413e12 −1.93608
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.99951e12 −0.853822 −0.426911 0.904294i \(-0.640398\pi\)
−0.426911 + 0.904294i \(0.640398\pi\)
\(192\) 0 0
\(193\) −2.18193e12 −0.586509 −0.293255 0.956034i \(-0.594738\pi\)
−0.293255 + 0.956034i \(0.594738\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.77475e12 1.14653 0.573267 0.819369i \(-0.305676\pi\)
0.573267 + 0.819369i \(0.305676\pi\)
\(198\) 0 0
\(199\) 2.95210e11 0.0670563 0.0335281 0.999438i \(-0.489326\pi\)
0.0335281 + 0.999438i \(0.489326\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.54610e11 −0.153636
\(204\) 0 0
\(205\) 7.72821e11 0.149085
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.35657e12 −0.755685
\(210\) 0 0
\(211\) 4.76886e12 0.784984 0.392492 0.919755i \(-0.371613\pi\)
0.392492 + 0.919755i \(0.371613\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.92518e12 −0.879609
\(216\) 0 0
\(217\) −1.29320e13 −1.82448
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.39323e13 −1.77773
\(222\) 0 0
\(223\) −9.21455e12 −1.11892 −0.559458 0.828859i \(-0.688991\pi\)
−0.559458 + 0.828859i \(0.688991\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.85444e12 0.644678 0.322339 0.946624i \(-0.395531\pi\)
0.322339 + 0.946624i \(0.395531\pi\)
\(228\) 0 0
\(229\) −1.40273e13 −1.47191 −0.735953 0.677033i \(-0.763266\pi\)
−0.735953 + 0.677033i \(0.763266\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.29610e12 0.505241 0.252621 0.967565i \(-0.418708\pi\)
0.252621 + 0.967565i \(0.418708\pi\)
\(234\) 0 0
\(235\) −1.84101e12 −0.167565
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.31646e13 1.09199 0.545997 0.837787i \(-0.316151\pi\)
0.545997 + 0.837787i \(0.316151\pi\)
\(240\) 0 0
\(241\) 2.25897e13 1.78985 0.894924 0.446219i \(-0.147230\pi\)
0.894924 + 0.446219i \(0.147230\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.83526e12 0.277576
\(246\) 0 0
\(247\) −1.00257e13 −0.693878
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.61185e13 −1.65479 −0.827396 0.561619i \(-0.810179\pi\)
−0.827396 + 0.561619i \(0.810179\pi\)
\(252\) 0 0
\(253\) 1.91384e13 1.16076
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.26580e13 0.704259 0.352129 0.935951i \(-0.385458\pi\)
0.352129 + 0.935951i \(0.385458\pi\)
\(258\) 0 0
\(259\) 1.42979e13 0.762293
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.37023e12 −0.263170 −0.131585 0.991305i \(-0.542007\pi\)
−0.131585 + 0.991305i \(0.542007\pi\)
\(264\) 0 0
\(265\) −1.14381e13 −0.537650
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.60140e12 0.372333 0.186167 0.982518i \(-0.440394\pi\)
0.186167 + 0.982518i \(0.440394\pi\)
\(270\) 0 0
\(271\) −6.01371e12 −0.249926 −0.124963 0.992161i \(-0.539881\pi\)
−0.124963 + 0.992161i \(0.539881\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.18790e13 −0.838876
\(276\) 0 0
\(277\) −2.63312e13 −0.970134 −0.485067 0.874477i \(-0.661205\pi\)
−0.485067 + 0.874477i \(0.661205\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.11748e13 1.74250 0.871248 0.490843i \(-0.163311\pi\)
0.871248 + 0.490843i \(0.163311\pi\)
\(282\) 0 0
\(283\) 8.94896e12 0.293054 0.146527 0.989207i \(-0.453191\pi\)
0.146527 + 0.989207i \(0.453191\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.22819e13 −0.372318
\(288\) 0 0
\(289\) 6.85649e13 2.00062
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.68480e13 0.726339 0.363170 0.931723i \(-0.381695\pi\)
0.363170 + 0.931723i \(0.381695\pi\)
\(294\) 0 0
\(295\) 1.04767e13 0.273025
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.40430e13 1.06582
\(300\) 0 0
\(301\) 9.41645e13 2.19670
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −4.05888e13 −0.880556
\(306\) 0 0
\(307\) 6.62433e12 0.138638 0.0693188 0.997595i \(-0.477917\pi\)
0.0693188 + 0.997595i \(0.477917\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.02852e13 −0.785170 −0.392585 0.919716i \(-0.628419\pi\)
−0.392585 + 0.919716i \(0.628419\pi\)
\(312\) 0 0
\(313\) −2.15423e13 −0.405320 −0.202660 0.979249i \(-0.564959\pi\)
−0.202660 + 0.979249i \(0.564959\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.79479e12 0.0490369 0.0245184 0.999699i \(-0.492195\pi\)
0.0245184 + 0.999699i \(0.492195\pi\)
\(318\) 0 0
\(319\) 8.12248e12 0.137670
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.40018e13 1.17119
\(324\) 0 0
\(325\) −5.03499e13 −0.770265
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.92578e13 0.418470
\(330\) 0 0
\(331\) −4.48560e13 −0.620536 −0.310268 0.950649i \(-0.600419\pi\)
−0.310268 + 0.950649i \(0.600419\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.99896e13 −0.776836
\(336\) 0 0
\(337\) −5.23428e13 −0.655982 −0.327991 0.944681i \(-0.606372\pi\)
−0.327991 + 0.944681i \(0.606372\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.39198e14 1.63487
\(342\) 0 0
\(343\) 4.87195e13 0.554096
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.62475e14 −1.73370 −0.866849 0.498571i \(-0.833858\pi\)
−0.866849 + 0.498571i \(0.833858\pi\)
\(348\) 0 0
\(349\) 5.60706e13 0.579689 0.289845 0.957074i \(-0.406396\pi\)
0.289845 + 0.957074i \(0.406396\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.21433e14 −1.17917 −0.589585 0.807707i \(-0.700709\pi\)
−0.589585 + 0.807707i \(0.700709\pi\)
\(354\) 0 0
\(355\) −9.13317e13 −0.859737
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.93082e13 −0.878953 −0.439476 0.898254i \(-0.644836\pi\)
−0.439476 + 0.898254i \(0.644836\pi\)
\(360\) 0 0
\(361\) −6.32383e13 −0.542863
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.45662e13 −0.198485
\(366\) 0 0
\(367\) 5.80965e13 0.455498 0.227749 0.973720i \(-0.426863\pi\)
0.227749 + 0.973720i \(0.426863\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.81777e14 1.34271
\(372\) 0 0
\(373\) 1.06336e13 0.0762576 0.0381288 0.999273i \(-0.487860\pi\)
0.0381288 + 0.999273i \(0.487860\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.86922e13 0.126410
\(378\) 0 0
\(379\) −1.94564e14 −1.27805 −0.639023 0.769188i \(-0.720661\pi\)
−0.639023 + 0.769188i \(0.720661\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.19102e14 1.35848 0.679240 0.733916i \(-0.262309\pi\)
0.679240 + 0.733916i \(0.262309\pi\)
\(384\) 0 0
\(385\) −1.15562e14 −0.696273
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.58688e14 −1.47250 −0.736248 0.676712i \(-0.763404\pi\)
−0.736248 + 0.676712i \(0.763404\pi\)
\(390\) 0 0
\(391\) −3.25090e14 −1.79900
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.46577e13 −0.0766975
\(396\) 0 0
\(397\) −2.53970e14 −1.29251 −0.646256 0.763121i \(-0.723666\pi\)
−0.646256 + 0.763121i \(0.723666\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.76194e14 −0.848590 −0.424295 0.905524i \(-0.639478\pi\)
−0.424295 + 0.905524i \(0.639478\pi\)
\(402\) 0 0
\(403\) 3.20334e14 1.50116
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.53899e14 −0.683074
\(408\) 0 0
\(409\) 2.54652e14 1.10019 0.550097 0.835101i \(-0.314591\pi\)
0.550097 + 0.835101i \(0.314591\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.66498e14 −0.681841
\(414\) 0 0
\(415\) 1.38692e14 0.553080
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.09764e13 0.0415224 0.0207612 0.999784i \(-0.493391\pi\)
0.0207612 + 0.999784i \(0.493391\pi\)
\(420\) 0 0
\(421\) −3.06791e14 −1.13056 −0.565278 0.824901i \(-0.691231\pi\)
−0.565278 + 0.824901i \(0.691231\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.71642e14 1.30013
\(426\) 0 0
\(427\) 6.45048e14 2.19907
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.94137e14 0.628756 0.314378 0.949298i \(-0.398204\pi\)
0.314378 + 0.949298i \(0.398204\pi\)
\(432\) 0 0
\(433\) −1.97021e14 −0.622055 −0.311027 0.950401i \(-0.600673\pi\)
−0.311027 + 0.950401i \(0.600673\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.33936e14 −0.702181
\(438\) 0 0
\(439\) −4.81833e14 −1.41040 −0.705199 0.709009i \(-0.749143\pi\)
−0.705199 + 0.709009i \(0.749143\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.72499e14 1.31577 0.657886 0.753118i \(-0.271451\pi\)
0.657886 + 0.753118i \(0.271451\pi\)
\(444\) 0 0
\(445\) −3.68730e13 −0.100168
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.52414e13 0.0911377 0.0455688 0.998961i \(-0.485490\pi\)
0.0455688 + 0.998961i \(0.485490\pi\)
\(450\) 0 0
\(451\) 1.32200e14 0.333626
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.65941e14 −0.639326
\(456\) 0 0
\(457\) 5.41522e14 1.27080 0.635400 0.772183i \(-0.280835\pi\)
0.635400 + 0.772183i \(0.280835\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.63903e14 0.590322 0.295161 0.955447i \(-0.404627\pi\)
0.295161 + 0.955447i \(0.404627\pi\)
\(462\) 0 0
\(463\) −3.44968e13 −0.0753499 −0.0376750 0.999290i \(-0.511995\pi\)
−0.0376750 + 0.999290i \(0.511995\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 5.88838e14 1.22674 0.613370 0.789795i \(-0.289813\pi\)
0.613370 + 0.789795i \(0.289813\pi\)
\(468\) 0 0
\(469\) 9.53371e14 1.94004
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.01357e15 −1.96842
\(474\) 0 0
\(475\) 2.67435e14 0.507462
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.73690e14 0.314723 0.157362 0.987541i \(-0.449701\pi\)
0.157362 + 0.987541i \(0.449701\pi\)
\(480\) 0 0
\(481\) −3.54167e14 −0.627206
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.43782e14 −0.412499
\(486\) 0 0
\(487\) 8.42617e14 1.39387 0.696933 0.717136i \(-0.254547\pi\)
0.696933 + 0.717136i \(0.254547\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.38408e14 0.851458 0.425729 0.904851i \(-0.360018\pi\)
0.425729 + 0.904851i \(0.360018\pi\)
\(492\) 0 0
\(493\) −1.37970e14 −0.213367
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.45147e15 2.14707
\(498\) 0 0
\(499\) −1.36234e14 −0.197121 −0.0985603 0.995131i \(-0.531424\pi\)
−0.0985603 + 0.995131i \(0.531424\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.88252e14 −0.676115 −0.338057 0.941125i \(-0.609770\pi\)
−0.338057 + 0.941125i \(0.609770\pi\)
\(504\) 0 0
\(505\) −2.60073e14 −0.352365
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.88014e14 0.633117 0.316559 0.948573i \(-0.397473\pi\)
0.316559 + 0.948573i \(0.397473\pi\)
\(510\) 0 0
\(511\) 3.90412e14 0.495688
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.74538e14 0.333935
\(516\) 0 0
\(517\) −3.14925e14 −0.374981
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.10790e15 −1.26442 −0.632211 0.774796i \(-0.717852\pi\)
−0.632211 + 0.774796i \(0.717852\pi\)
\(522\) 0 0
\(523\) −6.18140e14 −0.690761 −0.345380 0.938463i \(-0.612250\pi\)
−0.345380 + 0.938463i \(0.612250\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.36445e15 −2.53380
\(528\) 0 0
\(529\) 7.48712e13 0.0785794
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.04230e14 0.306339
\(534\) 0 0
\(535\) −2.75183e14 −0.271441
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.56065e14 0.621167
\(540\) 0 0
\(541\) 7.61295e14 0.706265 0.353133 0.935573i \(-0.385116\pi\)
0.353133 + 0.935573i \(0.385116\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.47329e14 −0.220341
\(546\) 0 0
\(547\) −5.65709e14 −0.493927 −0.246964 0.969025i \(-0.579433\pi\)
−0.246964 + 0.969025i \(0.579433\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −9.92840e13 −0.0832809
\(552\) 0 0
\(553\) 2.32944e14 0.191542
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.88089e15 1.48649 0.743243 0.669021i \(-0.233287\pi\)
0.743243 + 0.669021i \(0.233287\pi\)
\(558\) 0 0
\(559\) −2.33251e15 −1.80742
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.76231e14 0.131306 0.0656531 0.997843i \(-0.479087\pi\)
0.0656531 + 0.997843i \(0.479087\pi\)
\(564\) 0 0
\(565\) −6.86888e13 −0.0501903
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.91855e14 0.345716 0.172858 0.984947i \(-0.444700\pi\)
0.172858 + 0.984947i \(0.444700\pi\)
\(570\) 0 0
\(571\) 9.18553e14 0.633295 0.316647 0.948543i \(-0.397443\pi\)
0.316647 + 0.948543i \(0.397443\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.17484e15 −0.779483
\(576\) 0 0
\(577\) −1.53842e15 −1.00140 −0.500701 0.865620i \(-0.666924\pi\)
−0.500701 + 0.865620i \(0.666924\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.20414e15 −1.38124
\(582\) 0 0
\(583\) −1.95661e15 −1.20317
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.66027e12 −0.00335218 −0.00167609 0.999999i \(-0.500534\pi\)
−0.00167609 + 0.999999i \(0.500534\pi\)
\(588\) 0 0
\(589\) −1.70147e15 −0.988984
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.11514e14 0.230454 0.115227 0.993339i \(-0.463240\pi\)
0.115227 + 0.993339i \(0.463240\pi\)
\(594\) 0 0
\(595\) 1.96296e15 1.07911
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.35375e15 −1.77698 −0.888492 0.458893i \(-0.848246\pi\)
−0.888492 + 0.458893i \(0.848246\pi\)
\(600\) 0 0
\(601\) 1.02502e15 0.533240 0.266620 0.963802i \(-0.414093\pi\)
0.266620 + 0.963802i \(0.414093\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.48146e14 0.124467
\(606\) 0 0
\(607\) −1.99079e15 −0.980591 −0.490296 0.871556i \(-0.663111\pi\)
−0.490296 + 0.871556i \(0.663111\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7.24734e14 −0.344312
\(612\) 0 0
\(613\) −5.16975e14 −0.241233 −0.120617 0.992699i \(-0.538487\pi\)
−0.120617 + 0.992699i \(0.538487\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.78133e14 0.305314 0.152657 0.988279i \(-0.451217\pi\)
0.152657 + 0.988279i \(0.451217\pi\)
\(618\) 0 0
\(619\) −6.55793e14 −0.290047 −0.145023 0.989428i \(-0.546326\pi\)
−0.145023 + 0.989428i \(0.546326\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.85996e14 0.250156
\(624\) 0 0
\(625\) 7.48346e14 0.313879
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.61417e15 1.05866
\(630\) 0 0
\(631\) 3.36388e14 0.133869 0.0669343 0.997757i \(-0.478678\pi\)
0.0669343 + 0.997757i \(0.478678\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.63978e14 −0.255210
\(636\) 0 0
\(637\) 1.50979e15 0.570362
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.68016e15 0.978233 0.489116 0.872218i \(-0.337319\pi\)
0.489116 + 0.872218i \(0.337319\pi\)
\(642\) 0 0
\(643\) 1.63908e14 0.0588085 0.0294043 0.999568i \(-0.490639\pi\)
0.0294043 + 0.999568i \(0.490639\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 3.11319e15 1.07952 0.539762 0.841818i \(-0.318514\pi\)
0.539762 + 0.841818i \(0.318514\pi\)
\(648\) 0 0
\(649\) 1.79215e15 0.610983
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.15674e15 −1.04044 −0.520218 0.854033i \(-0.674149\pi\)
−0.520218 + 0.854033i \(0.674149\pi\)
\(654\) 0 0
\(655\) −3.99691e14 −0.129538
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.21997e15 0.695787 0.347894 0.937534i \(-0.386897\pi\)
0.347894 + 0.937534i \(0.386897\pi\)
\(660\) 0 0
\(661\) 2.94533e15 0.907873 0.453937 0.891034i \(-0.350019\pi\)
0.453937 + 0.891034i \(0.350019\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.41255e15 0.421197
\(666\) 0 0
\(667\) 4.36155e14 0.127923
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.94317e15 −1.97053
\(672\) 0 0
\(673\) 2.03972e15 0.569492 0.284746 0.958603i \(-0.408091\pi\)
0.284746 + 0.958603i \(0.408091\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.47929e14 −0.0399775 −0.0199887 0.999800i \(-0.506363\pi\)
−0.0199887 + 0.999800i \(0.506363\pi\)
\(678\) 0 0
\(679\) 3.87425e15 1.03016
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.94016e15 1.27183 0.635913 0.771761i \(-0.280624\pi\)
0.635913 + 0.771761i \(0.280624\pi\)
\(684\) 0 0
\(685\) −4.08683e14 −0.103535
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.50272e15 −1.10476
\(690\) 0 0
\(691\) 2.96403e15 0.715737 0.357868 0.933772i \(-0.383504\pi\)
0.357868 + 0.933772i \(0.383504\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.45357e15 0.807888
\(696\) 0 0
\(697\) −2.24558e15 −0.517068
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −8.45018e14 −0.188546 −0.0942729 0.995546i \(-0.530053\pi\)
−0.0942729 + 0.995546i \(0.530053\pi\)
\(702\) 0 0
\(703\) 1.88117e15 0.413212
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.13314e15 0.879982
\(708\) 0 0
\(709\) −6.16194e15 −1.29171 −0.645853 0.763462i \(-0.723498\pi\)
−0.645853 + 0.763462i \(0.723498\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.47454e15 1.51912
\(714\) 0 0
\(715\) 2.86254e15 0.572885
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7.67467e15 −1.48954 −0.744768 0.667324i \(-0.767440\pi\)
−0.744768 + 0.667324i \(0.767440\pi\)
\(720\) 0 0
\(721\) −4.36303e15 −0.833958
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.98611e14 −0.0924491
\(726\) 0 0
\(727\) 2.55382e15 0.466391 0.233196 0.972430i \(-0.425082\pi\)
0.233196 + 0.972430i \(0.425082\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.72167e16 3.05074
\(732\) 0 0
\(733\) 1.06639e16 1.86142 0.930709 0.365760i \(-0.119191\pi\)
0.930709 + 0.365760i \(0.119191\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.02619e16 −1.73843
\(738\) 0 0
\(739\) 4.93967e15 0.824430 0.412215 0.911087i \(-0.364755\pi\)
0.412215 + 0.911087i \(0.364755\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.04838e16 −1.69855 −0.849277 0.527948i \(-0.822961\pi\)
−0.849277 + 0.527948i \(0.822961\pi\)
\(744\) 0 0
\(745\) 2.78355e15 0.444365
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.37328e15 0.677887
\(750\) 0 0
\(751\) −9.78697e15 −1.49496 −0.747478 0.664287i \(-0.768735\pi\)
−0.747478 + 0.664287i \(0.768735\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.12947e15 0.612613
\(756\) 0 0
\(757\) −4.93317e14 −0.0721271 −0.0360636 0.999349i \(-0.511482\pi\)
−0.0360636 + 0.999349i \(0.511482\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.97168e15 −1.27426 −0.637130 0.770756i \(-0.719879\pi\)
−0.637130 + 0.770756i \(0.719879\pi\)
\(762\) 0 0
\(763\) 3.93062e15 0.550271
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.12426e15 0.561011
\(768\) 0 0
\(769\) 3.72650e15 0.499697 0.249849 0.968285i \(-0.419619\pi\)
0.249849 + 0.968285i \(0.419619\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.12337e15 −0.146399 −0.0731993 0.997317i \(-0.523321\pi\)
−0.0731993 + 0.997317i \(0.523321\pi\)
\(774\) 0 0
\(775\) −8.54488e15 −1.09786
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.61593e15 −0.201820
\(780\) 0 0
\(781\) −1.56233e16 −1.92394
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.62976e15 −0.314871
\(786\) 0 0
\(787\) −1.39683e16 −1.64923 −0.824617 0.565692i \(-0.808609\pi\)
−0.824617 + 0.565692i \(0.808609\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.09162e15 0.125343
\(792\) 0 0
\(793\) −1.59782e16 −1.80937
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.77226e16 −1.95213 −0.976063 0.217490i \(-0.930213\pi\)
−0.976063 + 0.217490i \(0.930213\pi\)
\(798\) 0 0
\(799\) 5.34939e15 0.581162
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.20232e15 −0.444175
\(804\) 0 0
\(805\) −6.20534e15 −0.646976
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.35271e16 1.37242 0.686211 0.727402i \(-0.259273\pi\)
0.686211 + 0.727402i \(0.259273\pi\)
\(810\) 0 0
\(811\) 1.11634e16 1.11733 0.558666 0.829393i \(-0.311313\pi\)
0.558666 + 0.829393i \(0.311313\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.91173e15 0.283651
\(816\) 0 0
\(817\) 1.23892e16 1.19076
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.29639e16 1.21297 0.606483 0.795097i \(-0.292580\pi\)
0.606483 + 0.795097i \(0.292580\pi\)
\(822\) 0 0
\(823\) −3.20238e15 −0.295647 −0.147824 0.989014i \(-0.547227\pi\)
−0.147824 + 0.989014i \(0.547227\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.92879e15 −0.892516 −0.446258 0.894904i \(-0.647244\pi\)
−0.446258 + 0.894904i \(0.647244\pi\)
\(828\) 0 0
\(829\) 1.17935e16 1.04615 0.523073 0.852288i \(-0.324785\pi\)
0.523073 + 0.852288i \(0.324785\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.11441e16 −0.962712
\(834\) 0 0
\(835\) 3.54282e15 0.302047
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.90602e14 0.0241328 0.0120664 0.999927i \(-0.496159\pi\)
0.0120664 + 0.999927i \(0.496159\pi\)
\(840\) 0 0
\(841\) −1.20154e16 −0.984828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.32877e14 0.0265811
\(846\) 0 0
\(847\) −3.94361e15 −0.310839
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8.26397e15 −0.634711
\(852\) 0 0
\(853\) −1.57981e16 −1.19780 −0.598902 0.800823i \(-0.704396\pi\)
−0.598902 + 0.800823i \(0.704396\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.64826e16 −1.21796 −0.608978 0.793187i \(-0.708420\pi\)
−0.608978 + 0.793187i \(0.708420\pi\)
\(858\) 0 0
\(859\) −6.21838e13 −0.00453644 −0.00226822 0.999997i \(-0.500722\pi\)
−0.00226822 + 0.999997i \(0.500722\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.77134e16 1.25963 0.629815 0.776745i \(-0.283131\pi\)
0.629815 + 0.776745i \(0.283131\pi\)
\(864\) 0 0
\(865\) −1.01034e16 −0.709382
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.50736e15 −0.171636
\(870\) 0 0
\(871\) −2.36156e16 −1.59624
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.65456e16 1.09053
\(876\) 0 0
\(877\) 8.21556e15 0.534736 0.267368 0.963595i \(-0.413846\pi\)
0.267368 + 0.963595i \(0.413846\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.91858e16 1.21790 0.608950 0.793209i \(-0.291591\pi\)
0.608950 + 0.793209i \(0.291591\pi\)
\(882\) 0 0
\(883\) 1.04412e16 0.654584 0.327292 0.944923i \(-0.393864\pi\)
0.327292 + 0.944923i \(0.393864\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9.33912e15 −0.571118 −0.285559 0.958361i \(-0.592179\pi\)
−0.285559 + 0.958361i \(0.592179\pi\)
\(888\) 0 0
\(889\) 1.05521e16 0.637353
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.84944e15 0.226838
\(894\) 0 0
\(895\) 1.42833e16 0.831384
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.17225e15 0.180173
\(900\) 0 0
\(901\) 3.32354e16 1.86472
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.77507e15 0.0971958
\(906\) 0 0
\(907\) −1.35976e16 −0.735565 −0.367782 0.929912i \(-0.619883\pi\)
−0.367782 + 0.929912i \(0.619883\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −2.44918e16 −1.29321 −0.646607 0.762823i \(-0.723813\pi\)
−0.646607 + 0.762823i \(0.723813\pi\)
\(912\) 0 0
\(913\) 2.37249e16 1.23770
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.35200e15 0.323504
\(918\) 0 0
\(919\) 1.94715e16 0.979860 0.489930 0.871762i \(-0.337022\pi\)
0.489930 + 0.871762i \(0.337022\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3.59538e16 −1.76659
\(924\) 0 0
\(925\) 9.44735e15 0.458702
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.72465e16 1.76604 0.883018 0.469340i \(-0.155508\pi\)
0.883018 + 0.469340i \(0.155508\pi\)
\(930\) 0 0
\(931\) −8.01932e15 −0.375763
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.11289e16 −0.966970
\(936\) 0 0
\(937\) −3.42521e15 −0.154924 −0.0774622 0.996995i \(-0.524682\pi\)
−0.0774622 + 0.996995i \(0.524682\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.69228e16 1.18954 0.594768 0.803898i \(-0.297244\pi\)
0.594768 + 0.803898i \(0.297244\pi\)
\(942\) 0 0
\(943\) 7.09876e15 0.310005
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.87322e16 1.22587 0.612935 0.790133i \(-0.289989\pi\)
0.612935 + 0.790133i \(0.289989\pi\)
\(948\) 0 0
\(949\) −9.67076e15 −0.407846
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.77454e16 0.731265 0.365633 0.930759i \(-0.380853\pi\)
0.365633 + 0.930759i \(0.380853\pi\)
\(954\) 0 0
\(955\) −1.04683e16 −0.426440
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.49490e15 0.258565
\(960\) 0 0
\(961\) 2.89555e16 1.13960
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.61492e15 −0.292931
\(966\) 0 0
\(967\) 2.45602e16 0.934086 0.467043 0.884235i \(-0.345319\pi\)
0.467043 + 0.884235i \(0.345319\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.69158e15 0.174427 0.0872135 0.996190i \(-0.472204\pi\)
0.0872135 + 0.996190i \(0.472204\pi\)
\(972\) 0 0
\(973\) −5.48850e16 −2.01759
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.77872e16 0.998676 0.499338 0.866407i \(-0.333577\pi\)
0.499338 + 0.866407i \(0.333577\pi\)
\(978\) 0 0
\(979\) −6.30755e15 −0.224159
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.09519e16 0.380581 0.190290 0.981728i \(-0.439057\pi\)
0.190290 + 0.981728i \(0.439057\pi\)
\(984\) 0 0
\(985\) 1.66639e16 0.572634
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.44258e16 −1.82905
\(990\) 0 0
\(991\) 3.36280e16 1.11762 0.558811 0.829295i \(-0.311258\pi\)
0.558811 + 0.829295i \(0.311258\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.03028e15 0.0334911
\(996\) 0 0
\(997\) 5.27110e16 1.69464 0.847321 0.531082i \(-0.178214\pi\)
0.847321 + 0.531082i \(0.178214\pi\)
\(998\) 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 72.12.a.c.1.1 1
3.2 odd 2 8.12.a.a.1.1 1
4.3 odd 2 144.12.a.j.1.1 1
12.11 even 2 16.12.a.b.1.1 1
15.2 even 4 200.12.c.b.49.2 2
15.8 even 4 200.12.c.b.49.1 2
15.14 odd 2 200.12.a.b.1.1 1
24.5 odd 2 64.12.a.e.1.1 1
24.11 even 2 64.12.a.c.1.1 1
48.5 odd 4 256.12.b.g.129.2 2
48.11 even 4 256.12.b.a.129.1 2
48.29 odd 4 256.12.b.g.129.1 2
48.35 even 4 256.12.b.a.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.12.a.a.1.1 1 3.2 odd 2
16.12.a.b.1.1 1 12.11 even 2
64.12.a.c.1.1 1 24.11 even 2
64.12.a.e.1.1 1 24.5 odd 2
72.12.a.c.1.1 1 1.1 even 1 trivial
144.12.a.j.1.1 1 4.3 odd 2
200.12.a.b.1.1 1 15.14 odd 2
200.12.c.b.49.1 2 15.8 even 4
200.12.c.b.49.2 2 15.2 even 4
256.12.b.a.129.1 2 48.11 even 4
256.12.b.a.129.2 2 48.35 even 4
256.12.b.g.129.1 2 48.29 odd 4
256.12.b.g.129.2 2 48.5 odd 4