Properties

Label 144.9.g.g.127.2
Level $144$
Weight $9$
Character 144.127
Analytic conductor $58.663$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,9,Mod(127,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 144.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(58.6625198488\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(0.500000 - 2.95804i\) of defining polynomial
Character \(\chi\) \(=\) 144.127
Dual form 144.9.g.g.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+510.000 q^{5} +2555.75i q^{7} +O(q^{10})\) \(q+510.000 q^{5} +2555.75i q^{7} +19168.1i q^{11} -27710.0 q^{13} -50370.0 q^{17} -108619. i q^{19} +176347. i q^{23} -130525. q^{25} -54978.0 q^{29} -1.17564e6i q^{31} +1.30343e6i q^{35} +793730. q^{37} +75582.0 q^{41} -499648. i q^{43} +2.86755e6i q^{47} -767039. q^{49} -1.11662e7 q^{53} +9.77573e6i q^{55} -2.18325e7i q^{59} -2.38266e7 q^{61} -1.41321e7 q^{65} -7.49473e6i q^{67} -1.00824e7i q^{71} +6.51661e6 q^{73} -4.89888e7 q^{77} +4.87892e7i q^{79} +7.34483e7i q^{83} -2.56887e7 q^{85} -8.67958e7 q^{89} -7.08197e7i q^{91} -5.53958e7i q^{95} -4.66703e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1020 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1020 q^{5} - 55420 q^{13} - 100740 q^{17} - 261050 q^{25} - 109956 q^{29} + 1587460 q^{37} + 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} - 47653244 q^{61} - 28264200 q^{65} + 13033220 q^{73} - 97977600 q^{77} - 51377400 q^{85} - 173591556 q^{89} - 93340540 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 510.000 0.816000 0.408000 0.912982i \(-0.366226\pi\)
0.408000 + 0.912982i \(0.366226\pi\)
\(6\) 0 0
\(7\) 2555.75i 1.06445i 0.846603 + 0.532225i \(0.178644\pi\)
−0.846603 + 0.532225i \(0.821356\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 19168.1i 1.30921i 0.755972 + 0.654603i \(0.227164\pi\)
−0.755972 + 0.654603i \(0.772836\pi\)
\(12\) 0 0
\(13\) −27710.0 −0.970204 −0.485102 0.874458i \(-0.661218\pi\)
−0.485102 + 0.874458i \(0.661218\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −50370.0 −0.603082 −0.301541 0.953453i \(-0.597501\pi\)
−0.301541 + 0.953453i \(0.597501\pi\)
\(18\) 0 0
\(19\) − 108619.i − 0.833474i −0.909027 0.416737i \(-0.863174\pi\)
0.909027 0.416737i \(-0.136826\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 176347.i 0.630167i 0.949064 + 0.315083i \(0.102032\pi\)
−0.949064 + 0.315083i \(0.897968\pi\)
\(24\) 0 0
\(25\) −130525. −0.334144
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −54978.0 −0.0777315 −0.0388657 0.999244i \(-0.512374\pi\)
−0.0388657 + 0.999244i \(0.512374\pi\)
\(30\) 0 0
\(31\) − 1.17564e6i − 1.27300i −0.771276 0.636501i \(-0.780381\pi\)
0.771276 0.636501i \(-0.219619\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.30343e6i 0.868592i
\(36\) 0 0
\(37\) 793730. 0.423512 0.211756 0.977323i \(-0.432082\pi\)
0.211756 + 0.977323i \(0.432082\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 75582.0 0.0267475 0.0133737 0.999911i \(-0.495743\pi\)
0.0133737 + 0.999911i \(0.495743\pi\)
\(42\) 0 0
\(43\) − 499648.i − 0.146147i −0.997327 0.0730736i \(-0.976719\pi\)
0.997327 0.0730736i \(-0.0232808\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.86755e6i 0.587651i 0.955859 + 0.293825i \(0.0949284\pi\)
−0.955859 + 0.293825i \(0.905072\pi\)
\(48\) 0 0
\(49\) −767039. −0.133056
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.11662e7 −1.41515 −0.707575 0.706639i \(-0.750211\pi\)
−0.707575 + 0.706639i \(0.750211\pi\)
\(54\) 0 0
\(55\) 9.77573e6i 1.06831i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.18325e7i − 1.80175i −0.434078 0.900875i \(-0.642926\pi\)
0.434078 0.900875i \(-0.357074\pi\)
\(60\) 0 0
\(61\) −2.38266e7 −1.72085 −0.860425 0.509577i \(-0.829802\pi\)
−0.860425 + 0.509577i \(0.829802\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.41321e7 −0.791687
\(66\) 0 0
\(67\) − 7.49473e6i − 0.371926i −0.982557 0.185963i \(-0.940460\pi\)
0.982557 0.185963i \(-0.0595405\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1.00824e7i − 0.396763i −0.980125 0.198382i \(-0.936431\pi\)
0.980125 0.198382i \(-0.0635685\pi\)
\(72\) 0 0
\(73\) 6.51661e6 0.229472 0.114736 0.993396i \(-0.463398\pi\)
0.114736 + 0.993396i \(0.463398\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89888e7 −1.39359
\(78\) 0 0
\(79\) 4.87892e7i 1.25261i 0.779579 + 0.626304i \(0.215433\pi\)
−0.779579 + 0.626304i \(0.784567\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.34483e7i 1.54764i 0.633407 + 0.773819i \(0.281656\pi\)
−0.633407 + 0.773819i \(0.718344\pi\)
\(84\) 0 0
\(85\) −2.56887e7 −0.492115
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.67958e7 −1.38337 −0.691685 0.722199i \(-0.743131\pi\)
−0.691685 + 0.722199i \(0.743131\pi\)
\(90\) 0 0
\(91\) − 7.08197e7i − 1.03273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 5.53958e7i − 0.680115i
\(96\) 0 0
\(97\) −4.66703e7 −0.527173 −0.263587 0.964636i \(-0.584905\pi\)
−0.263587 + 0.964636i \(0.584905\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.59910e7 −0.634161 −0.317080 0.948399i \(-0.602703\pi\)
−0.317080 + 0.948399i \(0.602703\pi\)
\(102\) 0 0
\(103\) − 1.64884e8i − 1.46497i −0.680782 0.732486i \(-0.738360\pi\)
0.680782 0.732486i \(-0.261640\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.27326e8i 0.971364i 0.874136 + 0.485682i \(0.161429\pi\)
−0.874136 + 0.485682i \(0.838571\pi\)
\(108\) 0 0
\(109\) −1.56119e8 −1.10598 −0.552992 0.833186i \(-0.686514\pi\)
−0.552992 + 0.833186i \(0.686514\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −2.36346e8 −1.44955 −0.724776 0.688984i \(-0.758057\pi\)
−0.724776 + 0.688984i \(0.758057\pi\)
\(114\) 0 0
\(115\) 8.99367e7i 0.514216i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 1.28733e8i − 0.641951i
\(120\) 0 0
\(121\) −1.53057e8 −0.714023
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −2.65786e8 −1.08866
\(126\) 0 0
\(127\) 3.67741e8i 1.41360i 0.707412 + 0.706802i \(0.249863\pi\)
−0.707412 + 0.706802i \(0.750137\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.16350e8i 0.734636i 0.930095 + 0.367318i \(0.119724\pi\)
−0.930095 + 0.367318i \(0.880276\pi\)
\(132\) 0 0
\(133\) 2.77603e8 0.887193
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.86442e8 1.09699 0.548494 0.836155i \(-0.315201\pi\)
0.548494 + 0.836155i \(0.315201\pi\)
\(138\) 0 0
\(139\) − 3.51077e8i − 0.940465i −0.882543 0.470232i \(-0.844170\pi\)
0.882543 0.470232i \(-0.155830\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 5.31148e8i − 1.27020i
\(144\) 0 0
\(145\) −2.80388e7 −0.0634289
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.54099e8 0.921308 0.460654 0.887580i \(-0.347615\pi\)
0.460654 + 0.887580i \(0.347615\pi\)
\(150\) 0 0
\(151\) 6.60188e8i 1.26987i 0.772565 + 0.634936i \(0.218973\pi\)
−0.772565 + 0.634936i \(0.781027\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 5.99578e8i − 1.03877i
\(156\) 0 0
\(157\) 4.35318e8 0.716486 0.358243 0.933628i \(-0.383376\pi\)
0.358243 + 0.933628i \(0.383376\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.50697e8 −0.670782
\(162\) 0 0
\(163\) − 2.44065e8i − 0.345744i −0.984944 0.172872i \(-0.944695\pi\)
0.984944 0.172872i \(-0.0553047\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.71351e8i 0.863145i 0.902078 + 0.431573i \(0.142041\pi\)
−0.902078 + 0.431573i \(0.857959\pi\)
\(168\) 0 0
\(169\) −4.78866e7 −0.0587040
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.76764e9 1.97337 0.986685 0.162644i \(-0.0520022\pi\)
0.986685 + 0.162644i \(0.0520022\pi\)
\(174\) 0 0
\(175\) − 3.33589e8i − 0.355680i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.56967e8i 0.737335i 0.929561 + 0.368668i \(0.120186\pi\)
−0.929561 + 0.368668i \(0.879814\pi\)
\(180\) 0 0
\(181\) 6.27094e8 0.584277 0.292138 0.956376i \(-0.405633\pi\)
0.292138 + 0.956376i \(0.405633\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.04802e8 0.345586
\(186\) 0 0
\(187\) − 9.65497e8i − 0.789559i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.07924e9i 0.810933i 0.914110 + 0.405466i \(0.132891\pi\)
−0.914110 + 0.405466i \(0.867109\pi\)
\(192\) 0 0
\(193\) −2.96757e7 −0.0213881 −0.0106940 0.999943i \(-0.503404\pi\)
−0.0106940 + 0.999943i \(0.503404\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.12484e9 0.746837 0.373419 0.927663i \(-0.378186\pi\)
0.373419 + 0.927663i \(0.378186\pi\)
\(198\) 0 0
\(199\) − 1.04718e9i − 0.667742i −0.942619 0.333871i \(-0.891645\pi\)
0.942619 0.333871i \(-0.108355\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.40510e8i − 0.0827413i
\(204\) 0 0
\(205\) 3.85468e7 0.0218259
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.08202e9 1.09119
\(210\) 0 0
\(211\) 2.77676e9i 1.40090i 0.713699 + 0.700452i \(0.247018\pi\)
−0.713699 + 0.700452i \(0.752982\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 2.54821e8i − 0.119256i
\(216\) 0 0
\(217\) 3.00465e9 1.35505
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.39575e9 0.585113
\(222\) 0 0
\(223\) − 2.27822e9i − 0.921248i −0.887595 0.460624i \(-0.847626\pi\)
0.887595 0.460624i \(-0.152374\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.48863e9i − 0.560639i −0.959907 0.280319i \(-0.909560\pi\)
0.959907 0.280319i \(-0.0904404\pi\)
\(228\) 0 0
\(229\) 1.69447e9 0.616157 0.308079 0.951361i \(-0.400314\pi\)
0.308079 + 0.951361i \(0.400314\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.21423e8 −0.176916 −0.0884580 0.996080i \(-0.528194\pi\)
−0.0884580 + 0.996080i \(0.528194\pi\)
\(234\) 0 0
\(235\) 1.46245e9i 0.479523i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 4.40690e9i − 1.35065i −0.737522 0.675323i \(-0.764004\pi\)
0.737522 0.675323i \(-0.235996\pi\)
\(240\) 0 0
\(241\) −1.62148e9 −0.480666 −0.240333 0.970691i \(-0.577257\pi\)
−0.240333 + 0.970691i \(0.577257\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.91190e8 −0.108573
\(246\) 0 0
\(247\) 3.00984e9i 0.808640i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 1.31321e8i − 0.0330855i −0.999863 0.0165428i \(-0.994734\pi\)
0.999863 0.0165428i \(-0.00526597\pi\)
\(252\) 0 0
\(253\) −3.38023e9 −0.825019
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.27789e9 −1.20984 −0.604920 0.796287i \(-0.706795\pi\)
−0.604920 + 0.796287i \(0.706795\pi\)
\(258\) 0 0
\(259\) 2.02857e9i 0.450808i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.38745e9i 1.54409i 0.635570 + 0.772044i \(0.280765\pi\)
−0.635570 + 0.772044i \(0.719235\pi\)
\(264\) 0 0
\(265\) −5.69477e9 −1.15476
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.46450e8 −0.0661655 −0.0330828 0.999453i \(-0.510532\pi\)
−0.0330828 + 0.999453i \(0.510532\pi\)
\(270\) 0 0
\(271\) − 6.51715e6i − 0.00120832i −1.00000 0.000604158i \(-0.999808\pi\)
1.00000 0.000604158i \(-0.000192310\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 2.50192e9i − 0.437464i
\(276\) 0 0
\(277\) 2.15061e9 0.365293 0.182647 0.983179i \(-0.441534\pi\)
0.182647 + 0.983179i \(0.441534\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.04256e10 −1.67215 −0.836074 0.548616i \(-0.815155\pi\)
−0.836074 + 0.548616i \(0.815155\pi\)
\(282\) 0 0
\(283\) − 1.28042e9i − 0.199622i −0.995006 0.0998108i \(-0.968176\pi\)
0.995006 0.0998108i \(-0.0318238\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.93168e8i 0.0284714i
\(288\) 0 0
\(289\) −4.43862e9 −0.636292
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.13786e9 0.290074 0.145037 0.989426i \(-0.453670\pi\)
0.145037 + 0.989426i \(0.453670\pi\)
\(294\) 0 0
\(295\) − 1.11346e10i − 1.47023i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 4.88656e9i − 0.611390i
\(300\) 0 0
\(301\) 1.27697e9 0.155567
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.21516e10 −1.40421
\(306\) 0 0
\(307\) − 5.45140e9i − 0.613698i −0.951758 0.306849i \(-0.900725\pi\)
0.951758 0.306849i \(-0.0992747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.07550e10i 1.14965i 0.818275 + 0.574827i \(0.194931\pi\)
−0.818275 + 0.574827i \(0.805069\pi\)
\(312\) 0 0
\(313\) −2.99804e8 −0.0312364 −0.0156182 0.999878i \(-0.504972\pi\)
−0.0156182 + 0.999878i \(0.504972\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.31172e9 −0.526015 −0.263007 0.964794i \(-0.584714\pi\)
−0.263007 + 0.964794i \(0.584714\pi\)
\(318\) 0 0
\(319\) − 1.05382e9i − 0.101767i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.47115e9i 0.502653i
\(324\) 0 0
\(325\) 3.61685e9 0.324188
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.32872e9 −0.625525
\(330\) 0 0
\(331\) 1.01004e10i 0.841446i 0.907189 + 0.420723i \(0.138224\pi\)
−0.907189 + 0.420723i \(0.861776\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 3.82231e9i − 0.303492i
\(336\) 0 0
\(337\) −1.84359e10 −1.42937 −0.714684 0.699448i \(-0.753429\pi\)
−0.714684 + 0.699448i \(0.753429\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.25348e10 1.66662
\(342\) 0 0
\(343\) 1.27730e10i 0.922820i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.27822e10i 0.881636i 0.897597 + 0.440818i \(0.145312\pi\)
−0.897597 + 0.440818i \(0.854688\pi\)
\(348\) 0 0
\(349\) −6.39381e8 −0.0430981 −0.0215490 0.999768i \(-0.506860\pi\)
−0.0215490 + 0.999768i \(0.506860\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.59837e10 −1.67341 −0.836705 0.547653i \(-0.815521\pi\)
−0.836705 + 0.547653i \(0.815521\pi\)
\(354\) 0 0
\(355\) − 5.14203e9i − 0.323759i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 2.22541e10i − 1.33978i −0.742461 0.669889i \(-0.766342\pi\)
0.742461 0.669889i \(-0.233658\pi\)
\(360\) 0 0
\(361\) 5.18543e9 0.305320
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.32347e9 0.187249
\(366\) 0 0
\(367\) 1.86229e10i 1.02656i 0.858221 + 0.513280i \(0.171570\pi\)
−0.858221 + 0.513280i \(0.828430\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 2.85380e10i − 1.50636i
\(372\) 0 0
\(373\) 1.19680e10 0.618283 0.309141 0.951016i \(-0.399958\pi\)
0.309141 + 0.951016i \(0.399958\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.52344e9 0.0754154
\(378\) 0 0
\(379\) − 2.30787e10i − 1.11855i −0.828982 0.559275i \(-0.811080\pi\)
0.828982 0.559275i \(-0.188920\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 1.43419e10i − 0.666518i −0.942835 0.333259i \(-0.891852\pi\)
0.942835 0.333259i \(-0.108148\pi\)
\(384\) 0 0
\(385\) −2.49843e10 −1.13717
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.73457e10 1.19424 0.597119 0.802152i \(-0.296312\pi\)
0.597119 + 0.802152i \(0.296312\pi\)
\(390\) 0 0
\(391\) − 8.88257e9i − 0.380042i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.48825e10i 1.02213i
\(396\) 0 0
\(397\) 3.99456e10 1.60808 0.804039 0.594576i \(-0.202680\pi\)
0.804039 + 0.594576i \(0.202680\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.12767e10 0.822863 0.411431 0.911441i \(-0.365029\pi\)
0.411431 + 0.911441i \(0.365029\pi\)
\(402\) 0 0
\(403\) 3.25771e10i 1.23507i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.52143e10i 0.554465i
\(408\) 0 0
\(409\) 1.14283e10 0.408404 0.204202 0.978929i \(-0.434540\pi\)
0.204202 + 0.978929i \(0.434540\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.57982e10 1.91788
\(414\) 0 0
\(415\) 3.74586e10i 1.26287i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.10009e10i 0.356922i 0.983947 + 0.178461i \(0.0571119\pi\)
−0.983947 + 0.178461i \(0.942888\pi\)
\(420\) 0 0
\(421\) 2.28766e10 0.728220 0.364110 0.931356i \(-0.381373\pi\)
0.364110 + 0.931356i \(0.381373\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.57454e9 0.201516
\(426\) 0 0
\(427\) − 6.08948e10i − 1.83176i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 9.55108e8i − 0.0276786i −0.999904 0.0138393i \(-0.995595\pi\)
0.999904 0.0138393i \(-0.00440532\pi\)
\(432\) 0 0
\(433\) 3.82225e10 1.08735 0.543673 0.839297i \(-0.317033\pi\)
0.543673 + 0.839297i \(0.317033\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.91546e10 0.525228
\(438\) 0 0
\(439\) − 6.40288e10i − 1.72392i −0.506976 0.861960i \(-0.669237\pi\)
0.506976 0.861960i \(-0.330763\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 7.47659e10i 1.94128i 0.240533 + 0.970641i \(0.422678\pi\)
−0.240533 + 0.970641i \(0.577322\pi\)
\(444\) 0 0
\(445\) −4.42658e10 −1.12883
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.51987e10 −0.620001 −0.310000 0.950736i \(-0.600329\pi\)
−0.310000 + 0.950736i \(0.600329\pi\)
\(450\) 0 0
\(451\) 1.44876e9i 0.0350180i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 3.61181e10i − 0.842711i
\(456\) 0 0
\(457\) 4.66828e9 0.107027 0.0535133 0.998567i \(-0.482958\pi\)
0.0535133 + 0.998567i \(0.482958\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.88096e10 0.859281 0.429641 0.903000i \(-0.358640\pi\)
0.429641 + 0.903000i \(0.358640\pi\)
\(462\) 0 0
\(463\) 3.23432e10i 0.703817i 0.936034 + 0.351908i \(0.114467\pi\)
−0.936034 + 0.351908i \(0.885533\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 2.31902e10i − 0.487570i −0.969829 0.243785i \(-0.921611\pi\)
0.969829 0.243785i \(-0.0783891\pi\)
\(468\) 0 0
\(469\) 1.91546e10 0.395897
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.57731e9 0.191337
\(474\) 0 0
\(475\) 1.41775e10i 0.278500i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 4.83542e9i − 0.0918528i −0.998945 0.0459264i \(-0.985376\pi\)
0.998945 0.0459264i \(-0.0146240\pi\)
\(480\) 0 0
\(481\) −2.19943e10 −0.410893
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.38018e10 −0.430173
\(486\) 0 0
\(487\) − 3.03878e10i − 0.540236i −0.962827 0.270118i \(-0.912937\pi\)
0.962827 0.270118i \(-0.0870628\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 5.56483e10i − 0.957472i −0.877959 0.478736i \(-0.841095\pi\)
0.877959 0.478736i \(-0.158905\pi\)
\(492\) 0 0
\(493\) 2.76924e9 0.0468784
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.57681e10 0.422335
\(498\) 0 0
\(499\) 7.88458e10i 1.27168i 0.771822 + 0.635838i \(0.219345\pi\)
−0.771822 + 0.635838i \(0.780655\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.41092e10i 0.689061i 0.938775 + 0.344530i \(0.111962\pi\)
−0.938775 + 0.344530i \(0.888038\pi\)
\(504\) 0 0
\(505\) −3.36554e10 −0.517475
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.05927e10 −0.157811 −0.0789055 0.996882i \(-0.525143\pi\)
−0.0789055 + 0.996882i \(0.525143\pi\)
\(510\) 0 0
\(511\) 1.66548e10i 0.244262i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 8.40908e10i − 1.19542i
\(516\) 0 0
\(517\) −5.49654e10 −0.769356
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.24958e11 −1.69595 −0.847973 0.530039i \(-0.822177\pi\)
−0.847973 + 0.530039i \(0.822177\pi\)
\(522\) 0 0
\(523\) − 2.80408e10i − 0.374786i −0.982285 0.187393i \(-0.939996\pi\)
0.982285 0.187393i \(-0.0600038\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 5.92172e10i 0.767724i
\(528\) 0 0
\(529\) 4.72129e10 0.602890
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.09438e9 −0.0259505
\(534\) 0 0
\(535\) 6.49363e10i 0.792633i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.47027e10i − 0.174197i
\(540\) 0 0
\(541\) 1.44659e11 1.68871 0.844356 0.535782i \(-0.179983\pi\)
0.844356 + 0.535782i \(0.179983\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −7.96205e10 −0.902483
\(546\) 0 0
\(547\) − 1.03774e10i − 0.115915i −0.998319 0.0579573i \(-0.981541\pi\)
0.998319 0.0579573i \(-0.0184587\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.97167e9i 0.0647872i
\(552\) 0 0
\(553\) −1.24693e11 −1.33334
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 5.47312e9 0.0568610 0.0284305 0.999596i \(-0.490949\pi\)
0.0284305 + 0.999596i \(0.490949\pi\)
\(558\) 0 0
\(559\) 1.38453e10i 0.141793i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 4.36118e10i − 0.434081i −0.976163 0.217040i \(-0.930360\pi\)
0.976163 0.217040i \(-0.0696403\pi\)
\(564\) 0 0
\(565\) −1.20536e11 −1.18283
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.27822e10 −0.121943 −0.0609716 0.998140i \(-0.519420\pi\)
−0.0609716 + 0.998140i \(0.519420\pi\)
\(570\) 0 0
\(571\) 7.59455e10i 0.714427i 0.934023 + 0.357213i \(0.116273\pi\)
−0.934023 + 0.357213i \(0.883727\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 2.30176e10i − 0.210566i
\(576\) 0 0
\(577\) 2.13827e10 0.192912 0.0964560 0.995337i \(-0.469249\pi\)
0.0964560 + 0.995337i \(0.469249\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1.87715e11 −1.64738
\(582\) 0 0
\(583\) − 2.14035e11i − 1.85272i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.07298e10i 0.511504i 0.966742 + 0.255752i \(0.0823231\pi\)
−0.966742 + 0.255752i \(0.917677\pi\)
\(588\) 0 0
\(589\) −1.27697e11 −1.06101
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.15978e11 −0.937899 −0.468949 0.883225i \(-0.655367\pi\)
−0.468949 + 0.883225i \(0.655367\pi\)
\(594\) 0 0
\(595\) − 6.56538e10i − 0.523832i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.40647e11i 1.86927i 0.355607 + 0.934636i \(0.384274\pi\)
−0.355607 + 0.934636i \(0.615726\pi\)
\(600\) 0 0
\(601\) −1.92942e11 −1.47887 −0.739434 0.673229i \(-0.764907\pi\)
−0.739434 + 0.673229i \(0.764907\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.80591e10 −0.582643
\(606\) 0 0
\(607\) − 1.62042e11i − 1.19364i −0.802376 0.596819i \(-0.796431\pi\)
0.802376 0.596819i \(-0.203569\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 7.94597e10i − 0.570141i
\(612\) 0 0
\(613\) 1.76424e11 1.24944 0.624722 0.780847i \(-0.285212\pi\)
0.624722 + 0.780847i \(0.285212\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.84986e10 0.679656 0.339828 0.940488i \(-0.389631\pi\)
0.339828 + 0.940488i \(0.389631\pi\)
\(618\) 0 0
\(619\) 1.28596e10i 0.0875923i 0.999040 + 0.0437961i \(0.0139452\pi\)
−0.999040 + 0.0437961i \(0.986055\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 2.21828e11i − 1.47253i
\(624\) 0 0
\(625\) −8.45648e10 −0.554204
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.99802e10 −0.255413
\(630\) 0 0
\(631\) 1.73463e11i 1.09418i 0.837073 + 0.547091i \(0.184265\pi\)
−0.837073 + 0.547091i \(0.815735\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.87548e11i 1.15350i
\(636\) 0 0
\(637\) 2.12547e10 0.129091
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.13903e11 0.674690 0.337345 0.941381i \(-0.390471\pi\)
0.337345 + 0.941381i \(0.390471\pi\)
\(642\) 0 0
\(643\) 7.04067e10i 0.411879i 0.978565 + 0.205940i \(0.0660250\pi\)
−0.978565 + 0.205940i \(0.933975\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.99175e11i 1.13663i 0.822812 + 0.568314i \(0.192404\pi\)
−0.822812 + 0.568314i \(0.807596\pi\)
\(648\) 0 0
\(649\) 4.18487e11 2.35886
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.49972e9 −0.0357472 −0.0178736 0.999840i \(-0.505690\pi\)
−0.0178736 + 0.999840i \(0.505690\pi\)
\(654\) 0 0
\(655\) 1.10339e11i 0.599463i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.20982e11i 1.17170i 0.810421 + 0.585848i \(0.199239\pi\)
−0.810421 + 0.585848i \(0.800761\pi\)
\(660\) 0 0
\(661\) −2.69549e11 −1.41199 −0.705995 0.708217i \(-0.749500\pi\)
−0.705995 + 0.708217i \(0.749500\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.41578e11 0.723949
\(666\) 0 0
\(667\) − 9.69518e9i − 0.0489838i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 4.56711e11i − 2.25295i
\(672\) 0 0
\(673\) 9.44470e10 0.460392 0.230196 0.973144i \(-0.426063\pi\)
0.230196 + 0.973144i \(0.426063\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.02735e10 −0.382136 −0.191068 0.981577i \(-0.561195\pi\)
−0.191068 + 0.981577i \(0.561195\pi\)
\(678\) 0 0
\(679\) − 1.19277e11i − 0.561150i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 3.00783e11i − 1.38220i −0.722760 0.691099i \(-0.757127\pi\)
0.722760 0.691099i \(-0.242873\pi\)
\(684\) 0 0
\(685\) 1.97085e11 0.895142
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.09416e11 1.37298
\(690\) 0 0
\(691\) 3.06208e11i 1.34309i 0.740964 + 0.671544i \(0.234369\pi\)
−0.740964 + 0.671544i \(0.765631\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 1.79049e11i − 0.767419i
\(696\) 0 0
\(697\) −3.80707e9 −0.0161309
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.73603e11 1.13305 0.566524 0.824045i \(-0.308288\pi\)
0.566524 + 0.824045i \(0.308288\pi\)
\(702\) 0 0
\(703\) − 8.62143e10i − 0.352987i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.68656e11i − 0.675033i
\(708\) 0 0
\(709\) −1.76662e11 −0.699129 −0.349564 0.936912i \(-0.613670\pi\)
−0.349564 + 0.936912i \(0.613670\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.07321e11 0.802203
\(714\) 0 0
\(715\) − 2.70885e11i − 1.03648i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.25510e11i 0.843821i 0.906637 + 0.421911i \(0.138640\pi\)
−0.906637 + 0.421911i \(0.861360\pi\)
\(720\) 0 0
\(721\) 4.21402e11 1.55939
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.17600e9 0.0259735
\(726\) 0 0
\(727\) 2.87080e11i 1.02770i 0.857881 + 0.513849i \(0.171781\pi\)
−0.857881 + 0.513849i \(0.828219\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.51673e10i 0.0881388i
\(732\) 0 0
\(733\) −2.94176e11 −1.01904 −0.509520 0.860459i \(-0.670177\pi\)
−0.509520 + 0.860459i \(0.670177\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.43660e11 0.486928
\(738\) 0 0
\(739\) − 9.69888e10i − 0.325195i −0.986692 0.162598i \(-0.948013\pi\)
0.986692 0.162598i \(-0.0519872\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.01567e11i 0.333271i 0.986019 + 0.166635i \(0.0532902\pi\)
−0.986019 + 0.166635i \(0.946710\pi\)
\(744\) 0 0
\(745\) 2.31590e11 0.751788
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −3.25413e11 −1.03397
\(750\) 0 0
\(751\) 4.17899e11i 1.31375i 0.754001 + 0.656873i \(0.228121\pi\)
−0.754001 + 0.656873i \(0.771879\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.36696e11i 1.03621i
\(756\) 0 0
\(757\) 1.82006e11 0.554244 0.277122 0.960835i \(-0.410619\pi\)
0.277122 + 0.960835i \(0.410619\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.27419e11 1.27443 0.637213 0.770687i \(-0.280087\pi\)
0.637213 + 0.770687i \(0.280087\pi\)
\(762\) 0 0
\(763\) − 3.99000e11i − 1.17727i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.04978e11i 1.74807i
\(768\) 0 0
\(769\) −5.09969e11 −1.45827 −0.729136 0.684368i \(-0.760078\pi\)
−0.729136 + 0.684368i \(0.760078\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.49408e11 0.418462 0.209231 0.977866i \(-0.432904\pi\)
0.209231 + 0.977866i \(0.432904\pi\)
\(774\) 0 0
\(775\) 1.53451e11i 0.425366i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 8.20966e9i − 0.0222933i
\(780\) 0 0
\(781\) 1.93261e11 0.519445
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 2.22012e11 0.584652
\(786\) 0 0
\(787\) − 7.33252e11i − 1.91141i −0.294323 0.955706i \(-0.595094\pi\)
0.294323 0.955706i \(-0.404906\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 6.04040e11i − 1.54298i
\(792\) 0 0
\(793\) 6.60236e11 1.66958
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.02703e11 −0.750212 −0.375106 0.926982i \(-0.622394\pi\)
−0.375106 + 0.926982i \(0.622394\pi\)
\(798\) 0 0
\(799\) − 1.44438e11i − 0.354401i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.24911e11i 0.300427i
\(804\) 0 0
\(805\) −2.29855e11 −0.547358
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.84316e11 1.36412 0.682062 0.731295i \(-0.261084\pi\)
0.682062 + 0.731295i \(0.261084\pi\)
\(810\) 0 0
\(811\) 1.21470e11i 0.280793i 0.990095 + 0.140396i \(0.0448377\pi\)
−0.990095 + 0.140396i \(0.955162\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1.24473e11i − 0.282127i
\(816\) 0 0
\(817\) −5.42714e10 −0.121810
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.52470e11 −0.995903 −0.497952 0.867205i \(-0.665914\pi\)
−0.497952 + 0.867205i \(0.665914\pi\)
\(822\) 0 0
\(823\) − 3.06704e11i − 0.668528i −0.942479 0.334264i \(-0.891512\pi\)
0.942479 0.334264i \(-0.108488\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.93276e11i 0.626982i 0.949591 + 0.313491i \(0.101499\pi\)
−0.949591 + 0.313491i \(0.898501\pi\)
\(828\) 0 0
\(829\) 3.35532e11 0.710421 0.355210 0.934786i \(-0.384409\pi\)
0.355210 + 0.934786i \(0.384409\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.86358e10 0.0802434
\(834\) 0 0
\(835\) 3.42389e11i 0.704326i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 3.42844e11i − 0.691908i −0.938252 0.345954i \(-0.887555\pi\)
0.938252 0.345954i \(-0.112445\pi\)
\(840\) 0 0
\(841\) −4.97224e11 −0.993958
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.44222e10 −0.0479024
\(846\) 0 0
\(847\) − 3.91175e11i − 0.760042i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.39972e11i 0.266883i
\(852\) 0 0
\(853\) −5.08662e11 −0.960801 −0.480400 0.877049i \(-0.659509\pi\)
−0.480400 + 0.877049i \(0.659509\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.06764e11 −1.12486 −0.562428 0.826846i \(-0.690133\pi\)
−0.562428 + 0.826846i \(0.690133\pi\)
\(858\) 0 0
\(859\) 9.49431e11i 1.74378i 0.489705 + 0.871888i \(0.337105\pi\)
−0.489705 + 0.871888i \(0.662895\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 2.99836e10i − 0.0540556i −0.999635 0.0270278i \(-0.991396\pi\)
0.999635 0.0270278i \(-0.00860426\pi\)
\(864\) 0 0
\(865\) 9.01494e11 1.61027
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.35196e11 −1.63992
\(870\) 0 0
\(871\) 2.07679e11i 0.360844i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 6.79283e11i − 1.15883i
\(876\) 0 0
\(877\) −8.80195e11 −1.48792 −0.743961 0.668223i \(-0.767055\pi\)
−0.743961 + 0.668223i \(0.767055\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.04085e12 −1.72776 −0.863879 0.503699i \(-0.831972\pi\)
−0.863879 + 0.503699i \(0.831972\pi\)
\(882\) 0 0
\(883\) 7.60446e11i 1.25091i 0.780261 + 0.625454i \(0.215086\pi\)
−0.780261 + 0.625454i \(0.784914\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.34097e11i 0.539732i 0.962898 + 0.269866i \(0.0869794\pi\)
−0.962898 + 0.269866i \(0.913021\pi\)
\(888\) 0 0
\(889\) −9.39853e11 −1.50471
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.11471e11 0.489792
\(894\) 0 0
\(895\) 3.86053e11i 0.601666i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.46345e10i 0.0989523i
\(900\) 0 0
\(901\) 5.62442e11 0.853451
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.19818e11 0.476770
\(906\) 0 0
\(907\) − 7.65213e11i − 1.13071i −0.824846 0.565357i \(-0.808738\pi\)
0.824846 0.565357i \(-0.191262\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 3.83541e11i − 0.556851i −0.960458 0.278425i \(-0.910188\pi\)
0.960458 0.278425i \(-0.0898125\pi\)
\(912\) 0 0
\(913\) −1.40786e12 −2.02618
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −5.52937e11 −0.781984
\(918\) 0 0
\(919\) − 6.82775e11i − 0.957229i −0.878025 0.478615i \(-0.841139\pi\)
0.878025 0.478615i \(-0.158861\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.79384e11i 0.384941i
\(924\) 0 0
\(925\) −1.03602e11 −0.141514
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.94973e11 −0.396021 −0.198011 0.980200i \(-0.563448\pi\)
−0.198011 + 0.980200i \(0.563448\pi\)
\(930\) 0 0
\(931\) 8.33152e10i 0.110898i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 4.92404e11i − 0.644280i
\(936\) 0 0
\(937\) 1.03941e12 1.34843 0.674217 0.738533i \(-0.264481\pi\)
0.674217 + 0.738533i \(0.264481\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.26891e11 −0.161836 −0.0809178 0.996721i \(-0.525785\pi\)
−0.0809178 + 0.996721i \(0.525785\pi\)
\(942\) 0 0
\(943\) 1.33286e10i 0.0168554i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 6.13064e11i − 0.762265i −0.924520 0.381133i \(-0.875534\pi\)
0.924520 0.381133i \(-0.124466\pi\)
\(948\) 0 0
\(949\) −1.80575e11 −0.222635
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −6.58227e11 −0.798002 −0.399001 0.916951i \(-0.630643\pi\)
−0.399001 + 0.916951i \(0.630643\pi\)
\(954\) 0 0
\(955\) 5.50413e11i 0.661721i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.87647e11i 1.16769i
\(960\) 0 0
\(961\) −5.29246e11 −0.620532
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.51346e10 −0.0174527
\(966\) 0 0
\(967\) 5.41485e11i 0.619271i 0.950855 + 0.309635i \(0.100207\pi\)
−0.950855 + 0.309635i \(0.899793\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 3.54981e11i − 0.399327i −0.979865 0.199663i \(-0.936015\pi\)
0.979865 0.199663i \(-0.0639849\pi\)
\(972\) 0 0
\(973\) 8.97263e11 1.00108
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −6.02238e11 −0.660982 −0.330491 0.943809i \(-0.607214\pi\)
−0.330491 + 0.943809i \(0.607214\pi\)
\(978\) 0 0
\(979\) − 1.66371e12i − 1.81112i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.37465e12i 1.47223i 0.676854 + 0.736117i \(0.263343\pi\)
−0.676854 + 0.736117i \(0.736657\pi\)
\(984\) 0 0
\(985\) 5.73668e11 0.609419
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.81113e10 0.0920972
\(990\) 0 0
\(991\) − 1.01081e12i − 1.04803i −0.851709 0.524015i \(-0.824433\pi\)
0.851709 0.524015i \(-0.175567\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 5.34061e11i − 0.544877i
\(996\) 0 0
\(997\) −3.28556e11 −0.332528 −0.166264 0.986081i \(-0.553170\pi\)
−0.166264 + 0.986081i \(0.553170\pi\)
\(998\) 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 144.9.g.g.127.2 2
3.2 odd 2 16.9.c.a.15.1 2
4.3 odd 2 inner 144.9.g.g.127.1 2
12.11 even 2 16.9.c.a.15.2 yes 2
15.2 even 4 400.9.h.b.399.1 4
15.8 even 4 400.9.h.b.399.3 4
15.14 odd 2 400.9.b.c.351.2 2
24.5 odd 2 64.9.c.d.63.2 2
24.11 even 2 64.9.c.d.63.1 2
48.5 odd 4 256.9.d.f.127.1 4
48.11 even 4 256.9.d.f.127.3 4
48.29 odd 4 256.9.d.f.127.4 4
48.35 even 4 256.9.d.f.127.2 4
60.23 odd 4 400.9.h.b.399.2 4
60.47 odd 4 400.9.h.b.399.4 4
60.59 even 2 400.9.b.c.351.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.9.c.a.15.1 2 3.2 odd 2
16.9.c.a.15.2 yes 2 12.11 even 2
64.9.c.d.63.1 2 24.11 even 2
64.9.c.d.63.2 2 24.5 odd 2
144.9.g.g.127.1 2 4.3 odd 2 inner
144.9.g.g.127.2 2 1.1 even 1 trivial
256.9.d.f.127.1 4 48.5 odd 4
256.9.d.f.127.2 4 48.35 even 4
256.9.d.f.127.3 4 48.11 even 4
256.9.d.f.127.4 4 48.29 odd 4
400.9.b.c.351.1 2 60.59 even 2
400.9.b.c.351.2 2 15.14 odd 2
400.9.h.b.399.1 4 15.2 even 4
400.9.h.b.399.2 4 60.23 odd 4
400.9.h.b.399.3 4 15.8 even 4
400.9.h.b.399.4 4 60.47 odd 4