Properties

Label 72.12.a.b.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 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1190.00 q^{5} +18480.0 q^{7} +O(q^{10})\) \(q-1190.00 q^{5} +18480.0 q^{7} -135884. q^{11} -848186. q^{13} +7.12461e6 q^{17} -5.04632e6 q^{19} +1.48912e7 q^{23} -4.74120e7 q^{25} +1.15001e8 q^{29} -1.63991e8 q^{31} -2.19912e7 q^{35} -2.23622e8 q^{37} -1.05358e8 q^{41} +1.41948e9 q^{43} -2.46928e9 q^{47} -1.63582e9 q^{49} +4.83705e8 q^{53} +1.61702e8 q^{55} -6.15184e9 q^{59} -7.53273e9 q^{61} +1.00934e9 q^{65} -8.76495e9 q^{67} +1.04016e10 q^{71} -3.17384e10 q^{73} -2.51114e9 q^{77} -3.98800e10 q^{79} -1.35133e10 q^{83} -8.47828e9 q^{85} -8.15145e10 q^{89} -1.56745e10 q^{91} +6.00512e9 q^{95} +3.07830e10 q^{97} +O(q^{100})\)

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
\(5\) −1190.00 −0.170299 −0.0851495 0.996368i \(-0.527137\pi\)
−0.0851495 + 0.996368i \(0.527137\pi\)
\(6\) 0 0
\(7\) 18480.0 0.415588 0.207794 0.978173i \(-0.433372\pi\)
0.207794 + 0.978173i \(0.433372\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −135884. −0.254395 −0.127197 0.991877i \(-0.540598\pi\)
−0.127197 + 0.991877i \(0.540598\pi\)
\(12\) 0 0
\(13\) −848186. −0.633582 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.12461e6 1.21700 0.608502 0.793553i \(-0.291771\pi\)
0.608502 + 0.793553i \(0.291771\pi\)
\(18\) 0 0
\(19\) −5.04632e6 −0.467552 −0.233776 0.972291i \(-0.575108\pi\)
−0.233776 + 0.972291i \(0.575108\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.48912e7 0.482422 0.241211 0.970473i \(-0.422455\pi\)
0.241211 + 0.970473i \(0.422455\pi\)
\(24\) 0 0
\(25\) −4.74120e7 −0.970998
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.15001e8 1.04115 0.520576 0.853815i \(-0.325717\pi\)
0.520576 + 0.853815i \(0.325717\pi\)
\(30\) 0 0
\(31\) −1.63991e8 −1.02880 −0.514398 0.857551i \(-0.671985\pi\)
−0.514398 + 0.857551i \(0.671985\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.19912e7 −0.0707742
\(36\) 0 0
\(37\) −2.23622e8 −0.530158 −0.265079 0.964227i \(-0.585398\pi\)
−0.265079 + 0.964227i \(0.585398\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.05358e8 −0.142023 −0.0710113 0.997476i \(-0.522623\pi\)
−0.0710113 + 0.997476i \(0.522623\pi\)
\(42\) 0 0
\(43\) 1.41948e9 1.47249 0.736244 0.676717i \(-0.236598\pi\)
0.736244 + 0.676717i \(0.236598\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.46928e9 −1.57048 −0.785239 0.619193i \(-0.787460\pi\)
−0.785239 + 0.619193i \(0.787460\pi\)
\(48\) 0 0
\(49\) −1.63582e9 −0.827287
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.83705e8 0.158878 0.0794389 0.996840i \(-0.474687\pi\)
0.0794389 + 0.996840i \(0.474687\pi\)
\(54\) 0 0
\(55\) 1.61702e8 0.0433232
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.15184e9 −1.12026 −0.560130 0.828404i \(-0.689249\pi\)
−0.560130 + 0.828404i \(0.689249\pi\)
\(60\) 0 0
\(61\) −7.53273e9 −1.14193 −0.570964 0.820975i \(-0.693430\pi\)
−0.570964 + 0.820975i \(0.693430\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00934e9 0.107898
\(66\) 0 0
\(67\) −8.76495e9 −0.793118 −0.396559 0.918009i \(-0.629796\pi\)
−0.396559 + 0.918009i \(0.629796\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.04016e10 0.684196 0.342098 0.939664i \(-0.388863\pi\)
0.342098 + 0.939664i \(0.388863\pi\)
\(72\) 0 0
\(73\) −3.17384e10 −1.79188 −0.895941 0.444174i \(-0.853497\pi\)
−0.895941 + 0.444174i \(0.853497\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.51114e9 −0.105723
\(78\) 0 0
\(79\) −3.98800e10 −1.45816 −0.729082 0.684426i \(-0.760053\pi\)
−0.729082 + 0.684426i \(0.760053\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.35133e10 −0.376559 −0.188279 0.982116i \(-0.560291\pi\)
−0.188279 + 0.982116i \(0.560291\pi\)
\(84\) 0 0
\(85\) −8.47828e9 −0.207254
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.15145e10 −1.54735 −0.773677 0.633580i \(-0.781585\pi\)
−0.773677 + 0.633580i \(0.781585\pi\)
\(90\) 0 0
\(91\) −1.56745e10 −0.263309
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.00512e9 0.0796236
\(96\) 0 0
\(97\) 3.07830e10 0.363971 0.181986 0.983301i \(-0.441748\pi\)
0.181986 + 0.983301i \(0.441748\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.61449e11 −1.52850 −0.764252 0.644918i \(-0.776892\pi\)
−0.764252 + 0.644918i \(0.776892\pi\)
\(102\) 0 0
\(103\) −5.18595e10 −0.440782 −0.220391 0.975412i \(-0.570733\pi\)
−0.220391 + 0.975412i \(0.570733\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.20810e11 0.832709 0.416355 0.909202i \(-0.363307\pi\)
0.416355 + 0.909202i \(0.363307\pi\)
\(108\) 0 0
\(109\) 4.25401e10 0.264821 0.132410 0.991195i \(-0.457728\pi\)
0.132410 + 0.991195i \(0.457728\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.25294e10 −0.166090 −0.0830452 0.996546i \(-0.526465\pi\)
−0.0830452 + 0.996546i \(0.526465\pi\)
\(114\) 0 0
\(115\) −1.77206e10 −0.0821560
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.31663e11 0.505772
\(120\) 0 0
\(121\) −2.66847e11 −0.935283
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.14526e11 0.335659
\(126\) 0 0
\(127\) 3.71230e11 0.997063 0.498532 0.866872i \(-0.333873\pi\)
0.498532 + 0.866872i \(0.333873\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.37142e11 1.21646 0.608229 0.793762i \(-0.291880\pi\)
0.608229 + 0.793762i \(0.291880\pi\)
\(132\) 0 0
\(133\) −9.32559e10 −0.194309
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.62059e11 0.640938 0.320469 0.947259i \(-0.396159\pi\)
0.320469 + 0.947259i \(0.396159\pi\)
\(138\) 0 0
\(139\) 7.19954e11 1.17686 0.588428 0.808550i \(-0.299747\pi\)
0.588428 + 0.808550i \(0.299747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.15255e11 0.161180
\(144\) 0 0
\(145\) −1.36852e11 −0.177307
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.19007e12 −1.32754 −0.663769 0.747938i \(-0.731044\pi\)
−0.663769 + 0.747938i \(0.731044\pi\)
\(150\) 0 0
\(151\) 1.50709e12 1.56231 0.781153 0.624339i \(-0.214632\pi\)
0.781153 + 0.624339i \(0.214632\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.95149e11 0.175203
\(156\) 0 0
\(157\) 1.52866e12 1.27898 0.639490 0.768800i \(-0.279146\pi\)
0.639490 + 0.768800i \(0.279146\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.75190e11 0.200489
\(162\) 0 0
\(163\) 1.53695e12 1.04623 0.523116 0.852261i \(-0.324769\pi\)
0.523116 + 0.852261i \(0.324769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.64628e11 0.574671 0.287336 0.957830i \(-0.407231\pi\)
0.287336 + 0.957830i \(0.407231\pi\)
\(168\) 0 0
\(169\) −1.07274e12 −0.598574
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.93014e12 −0.946966 −0.473483 0.880803i \(-0.657003\pi\)
−0.473483 + 0.880803i \(0.657003\pi\)
\(174\) 0 0
\(175\) −8.76174e11 −0.403535
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.70894e12 0.695080 0.347540 0.937665i \(-0.387017\pi\)
0.347540 + 0.937665i \(0.387017\pi\)
\(180\) 0 0
\(181\) 1.30945e11 0.0501021 0.0250511 0.999686i \(-0.492025\pi\)
0.0250511 + 0.999686i \(0.492025\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.66110e11 0.0902854
\(186\) 0 0
\(187\) −9.68120e11 −0.309600
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.05880e12 1.44000 0.720002 0.693972i \(-0.244141\pi\)
0.720002 + 0.693972i \(0.244141\pi\)
\(192\) 0 0
\(193\) −4.84580e12 −1.30257 −0.651283 0.758835i \(-0.725769\pi\)
−0.651283 + 0.758835i \(0.725769\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −7.83197e12 −1.88064 −0.940322 0.340286i \(-0.889476\pi\)
−0.940322 + 0.340286i \(0.889476\pi\)
\(198\) 0 0
\(199\) −5.02421e12 −1.14124 −0.570618 0.821216i \(-0.693296\pi\)
−0.570618 + 0.821216i \(0.693296\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.12522e12 0.432690
\(204\) 0 0
\(205\) 1.25376e11 0.0241863
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.85714e11 0.118943
\(210\) 0 0
\(211\) −5.40292e12 −0.889355 −0.444677 0.895691i \(-0.646682\pi\)
−0.444677 + 0.895691i \(0.646682\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.68918e12 −0.250763
\(216\) 0 0
\(217\) −3.03055e12 −0.427555
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.04299e12 −0.771071
\(222\) 0 0
\(223\) 2.65520e12 0.322419 0.161210 0.986920i \(-0.448460\pi\)
0.161210 + 0.986920i \(0.448460\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.00927e13 −1.11138 −0.555691 0.831389i \(-0.687546\pi\)
−0.555691 + 0.831389i \(0.687546\pi\)
\(228\) 0 0
\(229\) 8.75264e12 0.918425 0.459213 0.888326i \(-0.348132\pi\)
0.459213 + 0.888326i \(0.348132\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.29016e12 0.313877 0.156938 0.987608i \(-0.449838\pi\)
0.156938 + 0.987608i \(0.449838\pi\)
\(234\) 0 0
\(235\) 2.93844e12 0.267451
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.73649e13 −1.44040 −0.720200 0.693767i \(-0.755950\pi\)
−0.720200 + 0.693767i \(0.755950\pi\)
\(240\) 0 0
\(241\) −1.88851e13 −1.49632 −0.748162 0.663516i \(-0.769063\pi\)
−0.748162 + 0.663516i \(0.769063\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.94662e12 0.140886
\(246\) 0 0
\(247\) 4.28021e12 0.296232
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.73484e12 0.236628 0.118314 0.992976i \(-0.462251\pi\)
0.118314 + 0.992976i \(0.462251\pi\)
\(252\) 0 0
\(253\) −2.02348e12 −0.122726
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.55617e13 1.42219 0.711096 0.703095i \(-0.248199\pi\)
0.711096 + 0.703095i \(0.248199\pi\)
\(258\) 0 0
\(259\) −4.13254e12 −0.220327
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.71976e12 −0.182288 −0.0911441 0.995838i \(-0.529052\pi\)
−0.0911441 + 0.995838i \(0.529052\pi\)
\(264\) 0 0
\(265\) −5.75609e11 −0.0270567
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.32714e13 1.00736 0.503680 0.863890i \(-0.331979\pi\)
0.503680 + 0.863890i \(0.331979\pi\)
\(270\) 0 0
\(271\) −1.69620e12 −0.0704931 −0.0352465 0.999379i \(-0.511222\pi\)
−0.0352465 + 0.999379i \(0.511222\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.44254e12 0.247017
\(276\) 0 0
\(277\) 3.22077e13 1.18664 0.593322 0.804965i \(-0.297816\pi\)
0.593322 + 0.804965i \(0.297816\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.19732e13 0.748185 0.374092 0.927391i \(-0.377954\pi\)
0.374092 + 0.927391i \(0.377954\pi\)
\(282\) 0 0
\(283\) 6.26529e12 0.205171 0.102585 0.994724i \(-0.467289\pi\)
0.102585 + 0.994724i \(0.467289\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.94702e12 −0.0590229
\(288\) 0 0
\(289\) 1.64881e13 0.481097
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.30466e12 −0.116457 −0.0582286 0.998303i \(-0.518545\pi\)
−0.0582286 + 0.998303i \(0.518545\pi\)
\(294\) 0 0
\(295\) 7.32069e12 0.190779
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.26305e13 −0.305654
\(300\) 0 0
\(301\) 2.62319e13 0.611948
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 8.96395e12 0.194469
\(306\) 0 0
\(307\) 1.49844e13 0.313601 0.156800 0.987630i \(-0.449882\pi\)
0.156800 + 0.987630i \(0.449882\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.92805e13 −0.375783 −0.187891 0.982190i \(-0.560165\pi\)
−0.187891 + 0.982190i \(0.560165\pi\)
\(312\) 0 0
\(313\) 9.38955e13 1.76665 0.883325 0.468760i \(-0.155299\pi\)
0.883325 + 0.468760i \(0.155299\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.75293e13 0.307566 0.153783 0.988105i \(-0.450854\pi\)
0.153783 + 0.988105i \(0.450854\pi\)
\(318\) 0 0
\(319\) −1.56268e13 −0.264864
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.59530e13 −0.569012
\(324\) 0 0
\(325\) 4.02142e13 0.615207
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.56322e13 −0.652671
\(330\) 0 0
\(331\) −2.01778e13 −0.279139 −0.139569 0.990212i \(-0.544572\pi\)
−0.139569 + 0.990212i \(0.544572\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.04303e13 0.135067
\(336\) 0 0
\(337\) 2.21846e13 0.278027 0.139014 0.990290i \(-0.455607\pi\)
0.139014 + 0.990290i \(0.455607\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.22837e13 0.261721
\(342\) 0 0
\(343\) −6.67709e13 −0.759398
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.47701e13 0.797840 0.398920 0.916986i \(-0.369385\pi\)
0.398920 + 0.916986i \(0.369385\pi\)
\(348\) 0 0
\(349\) −1.57514e14 −1.62847 −0.814237 0.580533i \(-0.802844\pi\)
−0.814237 + 0.580533i \(0.802844\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.54892e13 0.247512 0.123756 0.992313i \(-0.460506\pi\)
0.123756 + 0.992313i \(0.460506\pi\)
\(354\) 0 0
\(355\) −1.23779e13 −0.116518
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.41138e12 −0.0832979 −0.0416489 0.999132i \(-0.513261\pi\)
−0.0416489 + 0.999132i \(0.513261\pi\)
\(360\) 0 0
\(361\) −9.10250e13 −0.781395
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.77687e13 0.305155
\(366\) 0 0
\(367\) −1.24601e14 −0.976922 −0.488461 0.872586i \(-0.662441\pi\)
−0.488461 + 0.872586i \(0.662441\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.93887e12 0.0660277
\(372\) 0 0
\(373\) −2.05385e14 −1.47289 −0.736446 0.676496i \(-0.763498\pi\)
−0.736446 + 0.676496i \(0.763498\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.75425e13 −0.659655
\(378\) 0 0
\(379\) 7.18102e13 0.471705 0.235852 0.971789i \(-0.424212\pi\)
0.235852 + 0.971789i \(0.424212\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2.94282e14 −1.82461 −0.912307 0.409508i \(-0.865700\pi\)
−0.912307 + 0.409508i \(0.865700\pi\)
\(384\) 0 0
\(385\) 2.98825e12 0.0180046
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.34095e14 0.763291 0.381646 0.924309i \(-0.375357\pi\)
0.381646 + 0.924309i \(0.375357\pi\)
\(390\) 0 0
\(391\) 1.06094e14 0.587109
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.74572e13 0.248324
\(396\) 0 0
\(397\) −1.17524e14 −0.598109 −0.299054 0.954236i \(-0.596671\pi\)
−0.299054 + 0.954236i \(0.596671\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.56980e14 −1.23767 −0.618835 0.785521i \(-0.712395\pi\)
−0.618835 + 0.785521i \(0.712395\pi\)
\(402\) 0 0
\(403\) 1.39094e14 0.651827
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.03867e13 0.134870
\(408\) 0 0
\(409\) −1.40372e13 −0.0606460 −0.0303230 0.999540i \(-0.509654\pi\)
−0.0303230 + 0.999540i \(0.509654\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.13686e14 −0.465567
\(414\) 0 0
\(415\) 1.60809e13 0.0641275
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 9.05803e13 0.342655 0.171327 0.985214i \(-0.445194\pi\)
0.171327 + 0.985214i \(0.445194\pi\)
\(420\) 0 0
\(421\) −4.25794e13 −0.156909 −0.0784546 0.996918i \(-0.524999\pi\)
−0.0784546 + 0.996918i \(0.524999\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.37792e14 −1.18171
\(426\) 0 0
\(427\) −1.39205e14 −0.474571
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.45393e14 0.794763 0.397381 0.917654i \(-0.369919\pi\)
0.397381 + 0.917654i \(0.369919\pi\)
\(432\) 0 0
\(433\) 3.45088e14 1.08955 0.544775 0.838583i \(-0.316615\pi\)
0.544775 + 0.838583i \(0.316615\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −7.51458e13 −0.225557
\(438\) 0 0
\(439\) 1.66129e14 0.486283 0.243142 0.969991i \(-0.421822\pi\)
0.243142 + 0.969991i \(0.421822\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.76583e14 1.60562 0.802808 0.596238i \(-0.203338\pi\)
0.802808 + 0.596238i \(0.203338\pi\)
\(444\) 0 0
\(445\) 9.70023e13 0.263513
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.12410e13 −0.0807923 −0.0403962 0.999184i \(-0.512862\pi\)
−0.0403962 + 0.999184i \(0.512862\pi\)
\(450\) 0 0
\(451\) 1.43165e13 0.0361298
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.86526e13 0.0448412
\(456\) 0 0
\(457\) −8.36591e13 −0.196324 −0.0981622 0.995170i \(-0.531296\pi\)
−0.0981622 + 0.995170i \(0.531296\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.12974e14 0.700090 0.350045 0.936733i \(-0.386166\pi\)
0.350045 + 0.936733i \(0.386166\pi\)
\(462\) 0 0
\(463\) 1.73544e13 0.0379065 0.0189532 0.999820i \(-0.493967\pi\)
0.0189532 + 0.999820i \(0.493967\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.41703e14 1.33688 0.668439 0.743767i \(-0.266963\pi\)
0.668439 + 0.743767i \(0.266963\pi\)
\(468\) 0 0
\(469\) −1.61976e14 −0.329610
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.92884e14 −0.374593
\(474\) 0 0
\(475\) 2.39256e14 0.453992
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.54411e14 0.642187 0.321094 0.947047i \(-0.395950\pi\)
0.321094 + 0.947047i \(0.395950\pi\)
\(480\) 0 0
\(481\) 1.89673e14 0.335898
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.66318e13 −0.0619839
\(486\) 0 0
\(487\) −1.20917e14 −0.200022 −0.100011 0.994986i \(-0.531888\pi\)
−0.100011 + 0.994986i \(0.531888\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.44195e14 0.386179 0.193090 0.981181i \(-0.438149\pi\)
0.193090 + 0.981181i \(0.438149\pi\)
\(492\) 0 0
\(493\) 8.19339e14 1.26708
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.92222e14 0.284343
\(498\) 0 0
\(499\) 8.60223e14 1.24468 0.622340 0.782747i \(-0.286182\pi\)
0.622340 + 0.782747i \(0.286182\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1.21648e15 −1.68453 −0.842266 0.539062i \(-0.818779\pi\)
−0.842266 + 0.539062i \(0.818779\pi\)
\(504\) 0 0
\(505\) 1.92124e14 0.260303
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.57278e14 −1.24191 −0.620955 0.783846i \(-0.713255\pi\)
−0.620955 + 0.783846i \(0.713255\pi\)
\(510\) 0 0
\(511\) −5.86525e14 −0.744684
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.17129e13 0.0750647
\(516\) 0 0
\(517\) 3.35535e14 0.399521
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.39668e15 1.59400 0.797000 0.603979i \(-0.206419\pi\)
0.797000 + 0.603979i \(0.206419\pi\)
\(522\) 0 0
\(523\) 1.14451e15 1.27897 0.639483 0.768805i \(-0.279148\pi\)
0.639483 + 0.768805i \(0.279148\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.16837e15 −1.25205
\(528\) 0 0
\(529\) −7.31061e14 −0.767269
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.93634e13 0.0899830
\(534\) 0 0
\(535\) −1.43764e14 −0.141809
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2.22281e14 0.210458
\(540\) 0 0
\(541\) −9.32189e14 −0.864807 −0.432403 0.901680i \(-0.642334\pi\)
−0.432403 + 0.901680i \(0.642334\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.06227e13 −0.0450987
\(546\) 0 0
\(547\) −6.90223e13 −0.0602642 −0.0301321 0.999546i \(-0.509593\pi\)
−0.0301321 + 0.999546i \(0.509593\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.80333e14 −0.486792
\(552\) 0 0
\(553\) −7.36983e14 −0.605995
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.24654e14 0.177546 0.0887728 0.996052i \(-0.471706\pi\)
0.0887728 + 0.996052i \(0.471706\pi\)
\(558\) 0 0
\(559\) −1.20398e15 −0.932941
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.12470e15 −1.58307 −0.791537 0.611122i \(-0.790719\pi\)
−0.791537 + 0.611122i \(0.790719\pi\)
\(564\) 0 0
\(565\) 3.87100e13 0.0282850
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 2.52164e15 1.77241 0.886206 0.463291i \(-0.153331\pi\)
0.886206 + 0.463291i \(0.153331\pi\)
\(570\) 0 0
\(571\) −2.83532e15 −1.95481 −0.977403 0.211386i \(-0.932202\pi\)
−0.977403 + 0.211386i \(0.932202\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.06023e14 −0.468431
\(576\) 0 0
\(577\) −1.62921e15 −1.06050 −0.530250 0.847842i \(-0.677902\pi\)
−0.530250 + 0.847842i \(0.677902\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.49726e14 −0.156493
\(582\) 0 0
\(583\) −6.57278e13 −0.0404177
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.99007e15 1.77081 0.885404 0.464822i \(-0.153882\pi\)
0.885404 + 0.464822i \(0.153882\pi\)
\(588\) 0 0
\(589\) 8.27548e14 0.481016
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.26562e14 −0.126878 −0.0634389 0.997986i \(-0.520207\pi\)
−0.0634389 + 0.997986i \(0.520207\pi\)
\(594\) 0 0
\(595\) −1.56679e14 −0.0861324
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.50028e15 −1.85462 −0.927311 0.374293i \(-0.877886\pi\)
−0.927311 + 0.374293i \(0.877886\pi\)
\(600\) 0 0
\(601\) −1.20578e15 −0.627275 −0.313638 0.949543i \(-0.601548\pi\)
−0.313638 + 0.949543i \(0.601548\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.17548e14 0.159278
\(606\) 0 0
\(607\) −2.22942e15 −1.09813 −0.549067 0.835779i \(-0.685017\pi\)
−0.549067 + 0.835779i \(0.685017\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.09441e15 0.995026
\(612\) 0 0
\(613\) 3.15369e15 1.47159 0.735795 0.677204i \(-0.236809\pi\)
0.735795 + 0.677204i \(0.236809\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.64703e15 −0.741539 −0.370770 0.928725i \(-0.620906\pi\)
−0.370770 + 0.928725i \(0.620906\pi\)
\(618\) 0 0
\(619\) −1.94862e14 −0.0861845 −0.0430922 0.999071i \(-0.513721\pi\)
−0.0430922 + 0.999071i \(0.513721\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.50639e15 −0.643062
\(624\) 0 0
\(625\) 2.17875e15 0.913836
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.59322e15 −0.645204
\(630\) 0 0
\(631\) −1.38596e15 −0.551555 −0.275777 0.961222i \(-0.588935\pi\)
−0.275777 + 0.961222i \(0.588935\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.41764e14 −0.169799
\(636\) 0 0
\(637\) 1.38748e15 0.524154
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.58508e15 0.943529 0.471764 0.881725i \(-0.343617\pi\)
0.471764 + 0.881725i \(0.343617\pi\)
\(642\) 0 0
\(643\) 4.96300e15 1.78067 0.890337 0.455303i \(-0.150469\pi\)
0.890337 + 0.455303i \(0.150469\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3.12484e15 −1.08356 −0.541782 0.840519i \(-0.682250\pi\)
−0.541782 + 0.840519i \(0.682250\pi\)
\(648\) 0 0
\(649\) 8.35937e14 0.284989
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −4.29342e15 −1.41508 −0.707540 0.706674i \(-0.750195\pi\)
−0.707540 + 0.706674i \(0.750195\pi\)
\(654\) 0 0
\(655\) −6.39199e14 −0.207161
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.93625e15 1.54713 0.773566 0.633716i \(-0.218471\pi\)
0.773566 + 0.633716i \(0.218471\pi\)
\(660\) 0 0
\(661\) −9.32360e14 −0.287393 −0.143696 0.989622i \(-0.545899\pi\)
−0.143696 + 0.989622i \(0.545899\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.10975e14 0.0330906
\(666\) 0 0
\(667\) 1.71251e15 0.502275
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.02358e15 0.290501
\(672\) 0 0
\(673\) 1.51685e15 0.423507 0.211753 0.977323i \(-0.432083\pi\)
0.211753 + 0.977323i \(0.432083\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.42102e15 1.73527 0.867634 0.497203i \(-0.165640\pi\)
0.867634 + 0.497203i \(0.165640\pi\)
\(678\) 0 0
\(679\) 5.68870e14 0.151262
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.67035e15 0.944916 0.472458 0.881353i \(-0.343367\pi\)
0.472458 + 0.881353i \(0.343367\pi\)
\(684\) 0 0
\(685\) −4.30850e14 −0.109151
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.10272e14 −0.100662
\(690\) 0 0
\(691\) 2.15581e15 0.520573 0.260286 0.965531i \(-0.416183\pi\)
0.260286 + 0.965531i \(0.416183\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.56745e14 −0.200417
\(696\) 0 0
\(697\) −7.50636e14 −0.172842
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.84907e15 1.52821 0.764104 0.645094i \(-0.223182\pi\)
0.764104 + 0.645094i \(0.223182\pi\)
\(702\) 0 0
\(703\) 1.12847e15 0.247876
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.98357e15 −0.635227
\(708\) 0 0
\(709\) −9.12308e14 −0.191244 −0.0956218 0.995418i \(-0.530484\pi\)
−0.0956218 + 0.995418i \(0.530484\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.44202e15 −0.496314
\(714\) 0 0
\(715\) −1.37153e14 −0.0274488
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.99411e15 0.775195 0.387597 0.921829i \(-0.373305\pi\)
0.387597 + 0.921829i \(0.373305\pi\)
\(720\) 0 0
\(721\) −9.58364e14 −0.183184
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.45245e15 −1.01096
\(726\) 0 0
\(727\) 3.31749e15 0.605858 0.302929 0.953013i \(-0.402036\pi\)
0.302929 + 0.953013i \(0.402036\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.01132e16 1.79202
\(732\) 0 0
\(733\) −2.60662e15 −0.454994 −0.227497 0.973779i \(-0.573054\pi\)
−0.227497 + 0.973779i \(0.573054\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.19102e15 0.201765
\(738\) 0 0
\(739\) 7.46259e15 1.24550 0.622752 0.782419i \(-0.286015\pi\)
0.622752 + 0.782419i \(0.286015\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.12334e16 −1.82001 −0.910003 0.414602i \(-0.863921\pi\)
−0.910003 + 0.414602i \(0.863921\pi\)
\(744\) 0 0
\(745\) 1.41618e15 0.226078
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.23258e15 0.346064
\(750\) 0 0
\(751\) −4.15071e15 −0.634020 −0.317010 0.948422i \(-0.602679\pi\)
−0.317010 + 0.948422i \(0.602679\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.79344e15 −0.266059
\(756\) 0 0
\(757\) −9.93212e15 −1.45216 −0.726080 0.687610i \(-0.758660\pi\)
−0.726080 + 0.687610i \(0.758660\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.37276e16 −1.94974 −0.974872 0.222765i \(-0.928492\pi\)
−0.974872 + 0.222765i \(0.928492\pi\)
\(762\) 0 0
\(763\) 7.86140e14 0.110056
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.21791e15 0.709777
\(768\) 0 0
\(769\) 2.85267e15 0.382522 0.191261 0.981539i \(-0.438742\pi\)
0.191261 + 0.981539i \(0.438742\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.44730e15 −0.709895 −0.354948 0.934886i \(-0.615501\pi\)
−0.354948 + 0.934886i \(0.615501\pi\)
\(774\) 0 0
\(775\) 7.77512e15 0.998960
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.31671e14 0.0664029
\(780\) 0 0
\(781\) −1.41341e15 −0.174056
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.81911e15 −0.217809
\(786\) 0 0
\(787\) 3.33904e15 0.394240 0.197120 0.980379i \(-0.436841\pi\)
0.197120 + 0.980379i \(0.436841\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −6.01143e14 −0.0690251
\(792\) 0 0
\(793\) 6.38916e15 0.723504
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.31632e15 −0.365289 −0.182644 0.983179i \(-0.558466\pi\)
−0.182644 + 0.983179i \(0.558466\pi\)
\(798\) 0 0
\(799\) −1.75926e16 −1.91128
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.31274e15 0.455846
\(804\) 0 0
\(805\) −3.27476e14 −0.0341430
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.29315e15 −0.841400 −0.420700 0.907200i \(-0.638215\pi\)
−0.420700 + 0.907200i \(0.638215\pi\)
\(810\) 0 0
\(811\) −1.67813e16 −1.67962 −0.839811 0.542879i \(-0.817334\pi\)
−0.839811 + 0.542879i \(0.817334\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.82897e15 −0.178172
\(816\) 0 0
\(817\) −7.16312e15 −0.688464
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.03242e15 0.190163 0.0950813 0.995470i \(-0.469689\pi\)
0.0950813 + 0.995470i \(0.469689\pi\)
\(822\) 0 0
\(823\) −5.59910e15 −0.516915 −0.258457 0.966023i \(-0.583214\pi\)
−0.258457 + 0.966023i \(0.583214\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.09682e16 1.88486 0.942432 0.334399i \(-0.108533\pi\)
0.942432 + 0.334399i \(0.108533\pi\)
\(828\) 0 0
\(829\) −1.75589e16 −1.55757 −0.778783 0.627293i \(-0.784163\pi\)
−0.778783 + 0.627293i \(0.784163\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.16545e16 −1.00681
\(834\) 0 0
\(835\) −1.14791e15 −0.0978659
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6.17140e14 −0.0512499 −0.0256249 0.999672i \(-0.508158\pi\)
−0.0256249 + 0.999672i \(0.508158\pi\)
\(840\) 0 0
\(841\) 1.02480e15 0.0839965
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.27656e15 0.101937
\(846\) 0 0
\(847\) −4.93134e15 −0.388692
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.33001e15 −0.255760
\(852\) 0 0
\(853\) 7.46292e15 0.565834 0.282917 0.959144i \(-0.408698\pi\)
0.282917 + 0.959144i \(0.408698\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.16990e16 1.60341 0.801707 0.597717i \(-0.203925\pi\)
0.801707 + 0.597717i \(0.203925\pi\)
\(858\) 0 0
\(859\) −5.63489e15 −0.411077 −0.205538 0.978649i \(-0.565895\pi\)
−0.205538 + 0.978649i \(0.565895\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.74062e15 −0.621560 −0.310780 0.950482i \(-0.600590\pi\)
−0.310780 + 0.950482i \(0.600590\pi\)
\(864\) 0 0
\(865\) 2.29686e15 0.161267
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 5.41906e15 0.370950
\(870\) 0 0
\(871\) 7.43431e15 0.502505
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.11644e15 0.139496
\(876\) 0 0
\(877\) 5.97195e15 0.388704 0.194352 0.980932i \(-0.437740\pi\)
0.194352 + 0.980932i \(0.437740\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.75775e15 −0.238540 −0.119270 0.992862i \(-0.538055\pi\)
−0.119270 + 0.992862i \(0.538055\pi\)
\(882\) 0 0
\(883\) 1.34954e16 0.846061 0.423030 0.906115i \(-0.360966\pi\)
0.423030 + 0.906115i \(0.360966\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.17242e16 −1.94004 −0.970020 0.243025i \(-0.921860\pi\)
−0.970020 + 0.243025i \(0.921860\pi\)
\(888\) 0 0
\(889\) 6.86033e15 0.414367
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.24608e16 0.734279
\(894\) 0 0
\(895\) −2.03364e15 −0.118371
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.88591e16 −1.07113
\(900\) 0 0
\(901\) 3.44621e15 0.193355
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.55824e14 −0.00853234
\(906\) 0 0
\(907\) 1.10011e16 0.595110 0.297555 0.954705i \(-0.403829\pi\)
0.297555 + 0.954705i \(0.403829\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 2.68524e16 1.41786 0.708928 0.705281i \(-0.249179\pi\)
0.708928 + 0.705281i \(0.249179\pi\)
\(912\) 0 0
\(913\) 1.83624e15 0.0957946
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 9.92638e15 0.505545
\(918\) 0 0
\(919\) 3.69307e16 1.85846 0.929228 0.369506i \(-0.120473\pi\)
0.929228 + 0.369506i \(0.120473\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −8.82252e15 −0.433494
\(924\) 0 0
\(925\) 1.06024e16 0.514783
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.35447e16 0.642221 0.321110 0.947042i \(-0.395944\pi\)
0.321110 + 0.947042i \(0.395944\pi\)
\(930\) 0 0
\(931\) 8.25485e15 0.386799
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.15206e15 0.0527245
\(936\) 0 0
\(937\) −6.01961e15 −0.272270 −0.136135 0.990690i \(-0.543468\pi\)
−0.136135 + 0.990690i \(0.543468\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −8.63146e15 −0.381365 −0.190683 0.981652i \(-0.561070\pi\)
−0.190683 + 0.981652i \(0.561070\pi\)
\(942\) 0 0
\(943\) −1.56891e15 −0.0685149
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −3.74754e16 −1.59890 −0.799450 0.600733i \(-0.794876\pi\)
−0.799450 + 0.600733i \(0.794876\pi\)
\(948\) 0 0
\(949\) 2.69201e16 1.13530
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.58712e16 1.47821 0.739104 0.673592i \(-0.235249\pi\)
0.739104 + 0.673592i \(0.235249\pi\)
\(954\) 0 0
\(955\) −6.01997e15 −0.245231
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.69085e15 0.266366
\(960\) 0 0
\(961\) 1.48442e15 0.0584224
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.76650e15 0.221826
\(966\) 0 0
\(967\) −1.66314e16 −0.632534 −0.316267 0.948670i \(-0.602429\pi\)
−0.316267 + 0.948670i \(0.602429\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.14015e15 0.116747 0.0583734 0.998295i \(-0.481409\pi\)
0.0583734 + 0.998295i \(0.481409\pi\)
\(972\) 0 0
\(973\) 1.33047e16 0.489087
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.43733e16 0.875980 0.437990 0.898980i \(-0.355690\pi\)
0.437990 + 0.898980i \(0.355690\pi\)
\(978\) 0 0
\(979\) 1.10765e16 0.393639
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.37201e16 0.476775 0.238387 0.971170i \(-0.423381\pi\)
0.238387 + 0.971170i \(0.423381\pi\)
\(984\) 0 0
\(985\) 9.32004e15 0.320272
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.11377e16 0.710360
\(990\) 0 0
\(991\) −2.95853e16 −0.983267 −0.491633 0.870802i \(-0.663600\pi\)
−0.491633 + 0.870802i \(0.663600\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 5.97880e15 0.194351
\(996\) 0 0
\(997\) −1.90504e16 −0.612465 −0.306233 0.951957i \(-0.599069\pi\)
−0.306233 + 0.951957i \(0.599069\pi\)
\(998\) 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.b.1.1 1
3.2 odd 2 24.12.a.b.1.1 1
4.3 odd 2 144.12.a.h.1.1 1
12.11 even 2 48.12.a.g.1.1 1
24.5 odd 2 192.12.a.p.1.1 1
24.11 even 2 192.12.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.12.a.b.1.1 1 3.2 odd 2
48.12.a.g.1.1 1 12.11 even 2
72.12.a.b.1.1 1 1.1 even 1 trivial
144.12.a.h.1.1 1 4.3 odd 2
192.12.a.f.1.1 1 24.11 even 2
192.12.a.p.1.1 1 24.5 odd 2