Properties

Label 144.9.g.i.127.1
Level $144$
Weight $9$
Character 144.127
Analytic conductor $58.663$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{1801})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 451x^{2} + 450x + 202500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(10.8595 + 18.8093i\) of defining polynomial
Character \(\chi\) \(=\) 144.127
Dual form 144.9.g.i.127.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-952.517 q^{5} -3101.27i q^{7} +O(q^{10})\) \(q-952.517 q^{5} -3101.27i q^{7} +7175.83i q^{11} -26897.5 q^{13} -146374. q^{17} -220215. i q^{19} -96574.1i q^{23} +516663. q^{25} -169321. q^{29} -429668. i q^{31} +2.95401e6i q^{35} +2.94894e6 q^{37} -3.15721e6 q^{41} +4.89907e6i q^{43} +808105. i q^{47} -3.85305e6 q^{49} +1.21447e7 q^{53} -6.83510e6i q^{55} -2.47774e6i q^{59} -6.21934e6 q^{61} +2.56203e7 q^{65} +1.52139e7i q^{67} +1.92087e7i q^{71} -3.17283e7 q^{73} +2.22542e7 q^{77} +5.67932e7i q^{79} +8.52789e7i q^{83} +1.39424e8 q^{85} -3.06662e7 q^{89} +8.34163e7i q^{91} +2.09758e8i q^{95} +5.15412e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 264 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 264 q^{5} + 14632 q^{13} - 332904 q^{17} + 2604428 q^{25} - 2343576 q^{29} + 4315784 q^{37} - 9035496 q^{41} + 3458884 q^{49} + 42186600 q^{53} - 48148408 q^{61} + 125450832 q^{65} - 21215480 q^{73} + 32354496 q^{77} + 235297584 q^{85} + 12675576 q^{89} + 263153800 q^{97}+O(q^{100}) \) 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
\(5\) −952.517 −1.52403 −0.762013 0.647561i \(-0.775789\pi\)
−0.762013 + 0.647561i \(0.775789\pi\)
\(6\) 0 0
\(7\) − 3101.27i − 1.29166i −0.763483 0.645828i \(-0.776512\pi\)
0.763483 0.645828i \(-0.223488\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 7175.83i 0.490119i 0.969508 + 0.245059i \(0.0788075\pi\)
−0.969508 + 0.245059i \(0.921193\pi\)
\(12\) 0 0
\(13\) −26897.5 −0.941756 −0.470878 0.882198i \(-0.656063\pi\)
−0.470878 + 0.882198i \(0.656063\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −146374. −1.75254 −0.876271 0.481819i \(-0.839976\pi\)
−0.876271 + 0.481819i \(0.839976\pi\)
\(18\) 0 0
\(19\) − 220215.i − 1.68979i −0.534935 0.844893i \(-0.679664\pi\)
0.534935 0.844893i \(-0.320336\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 96574.1i − 0.345104i −0.985000 0.172552i \(-0.944799\pi\)
0.985000 0.172552i \(-0.0552012\pi\)
\(24\) 0 0
\(25\) 516663. 1.32266
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −169321. −0.239397 −0.119698 0.992810i \(-0.538193\pi\)
−0.119698 + 0.992810i \(0.538193\pi\)
\(30\) 0 0
\(31\) − 429668.i − 0.465250i −0.972567 0.232625i \(-0.925269\pi\)
0.972567 0.232625i \(-0.0747314\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.95401e6i 1.96852i
\(36\) 0 0
\(37\) 2.94894e6 1.57347 0.786737 0.617289i \(-0.211769\pi\)
0.786737 + 0.617289i \(0.211769\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.15721e6 −1.11729 −0.558647 0.829406i \(-0.688679\pi\)
−0.558647 + 0.829406i \(0.688679\pi\)
\(42\) 0 0
\(43\) 4.89907e6i 1.43298i 0.697597 + 0.716490i \(0.254252\pi\)
−0.697597 + 0.716490i \(0.745748\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 808105.i 0.165606i 0.996566 + 0.0828031i \(0.0263873\pi\)
−0.996566 + 0.0828031i \(0.973613\pi\)
\(48\) 0 0
\(49\) −3.85305e6 −0.668375
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.21447e7 1.53916 0.769579 0.638551i \(-0.220466\pi\)
0.769579 + 0.638551i \(0.220466\pi\)
\(54\) 0 0
\(55\) − 6.83510e6i − 0.746954i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.47774e6i − 0.204478i −0.994760 0.102239i \(-0.967399\pi\)
0.994760 0.102239i \(-0.0326007\pi\)
\(60\) 0 0
\(61\) −6.21934e6 −0.449184 −0.224592 0.974453i \(-0.572105\pi\)
−0.224592 + 0.974453i \(0.572105\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.56203e7 1.43526
\(66\) 0 0
\(67\) 1.52139e7i 0.754989i 0.926012 + 0.377494i \(0.123214\pi\)
−0.926012 + 0.377494i \(0.876786\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.92087e7i 0.755899i 0.925826 + 0.377949i \(0.123371\pi\)
−0.925826 + 0.377949i \(0.876629\pi\)
\(72\) 0 0
\(73\) −3.17283e7 −1.11726 −0.558631 0.829416i \(-0.688673\pi\)
−0.558631 + 0.829416i \(0.688673\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.22542e7 0.633065
\(78\) 0 0
\(79\) 5.67932e7i 1.45810i 0.684460 + 0.729051i \(0.260038\pi\)
−0.684460 + 0.729051i \(0.739962\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.52789e7i 1.79692i 0.439053 + 0.898461i \(0.355314\pi\)
−0.439053 + 0.898461i \(0.644686\pi\)
\(84\) 0 0
\(85\) 1.39424e8 2.67092
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.06662e7 −0.488765 −0.244383 0.969679i \(-0.578585\pi\)
−0.244383 + 0.969679i \(0.578585\pi\)
\(90\) 0 0
\(91\) 8.34163e7i 1.21642i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.09758e8i 2.57528i
\(96\) 0 0
\(97\) 5.15412e6 0.0582194 0.0291097 0.999576i \(-0.490733\pi\)
0.0291097 + 0.999576i \(0.490733\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.01284e8 0.973316 0.486658 0.873593i \(-0.338216\pi\)
0.486658 + 0.873593i \(0.338216\pi\)
\(102\) 0 0
\(103\) − 2.04461e8i − 1.81661i −0.418310 0.908304i \(-0.637377\pi\)
0.418310 0.908304i \(-0.362623\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.61055e8i − 1.22868i −0.789041 0.614340i \(-0.789422\pi\)
0.789041 0.614340i \(-0.210578\pi\)
\(108\) 0 0
\(109\) 5.73094e7 0.405994 0.202997 0.979179i \(-0.434932\pi\)
0.202997 + 0.979179i \(0.434932\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.49424e7 −0.0916446 −0.0458223 0.998950i \(-0.514591\pi\)
−0.0458223 + 0.998950i \(0.514591\pi\)
\(114\) 0 0
\(115\) 9.19884e7i 0.525947i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.53945e8i 2.26368i
\(120\) 0 0
\(121\) 1.62866e8 0.759784
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.20053e8 −0.491738
\(126\) 0 0
\(127\) 8.68934e7i 0.334019i 0.985955 + 0.167010i \(0.0534111\pi\)
−0.985955 + 0.167010i \(0.946589\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 2.93118e8i − 0.995308i −0.867376 0.497654i \(-0.834195\pi\)
0.867376 0.497654i \(-0.165805\pi\)
\(132\) 0 0
\(133\) −6.82944e8 −2.18262
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.93337e8 1.40043 0.700215 0.713932i \(-0.253087\pi\)
0.700215 + 0.713932i \(0.253087\pi\)
\(138\) 0 0
\(139\) 5.21073e8i 1.39585i 0.716170 + 0.697926i \(0.245894\pi\)
−0.716170 + 0.697926i \(0.754106\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 1.93012e8i − 0.461572i
\(144\) 0 0
\(145\) 1.61281e8 0.364847
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.09228e8 0.830271 0.415135 0.909760i \(-0.363734\pi\)
0.415135 + 0.909760i \(0.363734\pi\)
\(150\) 0 0
\(151\) − 8.92557e6i − 0.0171683i −0.999963 0.00858417i \(-0.997268\pi\)
0.999963 0.00858417i \(-0.00273246\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.09266e8i 0.709053i
\(156\) 0 0
\(157\) 2.58929e8 0.426169 0.213085 0.977034i \(-0.431649\pi\)
0.213085 + 0.977034i \(0.431649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.99502e8 −0.445755
\(162\) 0 0
\(163\) − 2.01640e8i − 0.285645i −0.989748 0.142823i \(-0.954382\pi\)
0.989748 0.142823i \(-0.0456179\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1.45117e9i − 1.86575i −0.360202 0.932874i \(-0.617292\pi\)
0.360202 0.932874i \(-0.382708\pi\)
\(168\) 0 0
\(169\) −9.22554e7 −0.113095
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.36420e8 −0.710492 −0.355246 0.934773i \(-0.615603\pi\)
−0.355246 + 0.934773i \(0.615603\pi\)
\(174\) 0 0
\(175\) − 1.60231e9i − 1.70842i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 1.84104e9i − 1.79329i −0.442751 0.896644i \(-0.645998\pi\)
0.442751 0.896644i \(-0.354002\pi\)
\(180\) 0 0
\(181\) −7.25192e8 −0.675676 −0.337838 0.941204i \(-0.609696\pi\)
−0.337838 + 0.941204i \(0.609696\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.80892e9 −2.39802
\(186\) 0 0
\(187\) − 1.05036e9i − 0.858954i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.59853e9i 1.20112i 0.799579 + 0.600561i \(0.205056\pi\)
−0.799579 + 0.600561i \(0.794944\pi\)
\(192\) 0 0
\(193\) −5.05071e8 −0.364018 −0.182009 0.983297i \(-0.558260\pi\)
−0.182009 + 0.983297i \(0.558260\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.23833e9 −1.48614 −0.743070 0.669214i \(-0.766631\pi\)
−0.743070 + 0.669214i \(0.766631\pi\)
\(198\) 0 0
\(199\) − 1.11582e9i − 0.711510i −0.934579 0.355755i \(-0.884224\pi\)
0.934579 0.355755i \(-0.115776\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.25109e8i 0.309218i
\(204\) 0 0
\(205\) 3.00729e9 1.70279
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.58022e9 0.828196
\(210\) 0 0
\(211\) 2.94087e7i 0.0148370i 0.999972 + 0.00741850i \(0.00236140\pi\)
−0.999972 + 0.00741850i \(0.997639\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 4.66645e9i − 2.18390i
\(216\) 0 0
\(217\) −1.33251e9 −0.600942
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.93709e9 1.65047
\(222\) 0 0
\(223\) 3.32442e9i 1.34430i 0.740414 + 0.672151i \(0.234629\pi\)
−0.740414 + 0.672151i \(0.765371\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.68532e8i − 0.138794i −0.997589 0.0693971i \(-0.977892\pi\)
0.997589 0.0693971i \(-0.0221076\pi\)
\(228\) 0 0
\(229\) −1.32990e9 −0.483591 −0.241795 0.970327i \(-0.577736\pi\)
−0.241795 + 0.970327i \(0.577736\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.49591e9 −0.846846 −0.423423 0.905932i \(-0.639172\pi\)
−0.423423 + 0.905932i \(0.639172\pi\)
\(234\) 0 0
\(235\) − 7.69734e8i − 0.252388i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.77198e7i 0.0176902i 0.999961 + 0.00884511i \(0.00281552\pi\)
−0.999961 + 0.00884511i \(0.997184\pi\)
\(240\) 0 0
\(241\) −1.57199e9 −0.465996 −0.232998 0.972477i \(-0.574854\pi\)
−0.232998 + 0.972477i \(0.574854\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.67009e9 1.01862
\(246\) 0 0
\(247\) 5.92322e9i 1.59137i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.63479e9i 0.663822i 0.943311 + 0.331911i \(0.107693\pi\)
−0.943311 + 0.331911i \(0.892307\pi\)
\(252\) 0 0
\(253\) 6.92999e8 0.169142
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.67876e9 −0.843276 −0.421638 0.906764i \(-0.638545\pi\)
−0.421638 + 0.906764i \(0.638545\pi\)
\(258\) 0 0
\(259\) − 9.14545e9i − 2.03239i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 1.21643e9i − 0.254252i −0.991887 0.127126i \(-0.959425\pi\)
0.991887 0.127126i \(-0.0405753\pi\)
\(264\) 0 0
\(265\) −1.15680e10 −2.34572
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.26695e9 1.76981 0.884907 0.465768i \(-0.154222\pi\)
0.884907 + 0.465768i \(0.154222\pi\)
\(270\) 0 0
\(271\) 3.54288e7i 0.00656871i 0.999995 + 0.00328435i \(0.00104544\pi\)
−0.999995 + 0.00328435i \(0.998955\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.70748e9i 0.648259i
\(276\) 0 0
\(277\) 5.73085e9 0.973419 0.486709 0.873564i \(-0.338197\pi\)
0.486709 + 0.873564i \(0.338197\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.79380e9 −1.08965 −0.544825 0.838550i \(-0.683404\pi\)
−0.544825 + 0.838550i \(0.683404\pi\)
\(282\) 0 0
\(283\) 5.67140e9i 0.884188i 0.896969 + 0.442094i \(0.145764\pi\)
−0.896969 + 0.442094i \(0.854236\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.79133e9i 1.44316i
\(288\) 0 0
\(289\) 1.44496e10 2.07140
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.52415e9 0.206803 0.103402 0.994640i \(-0.467027\pi\)
0.103402 + 0.994640i \(0.467027\pi\)
\(294\) 0 0
\(295\) 2.36009e9i 0.311631i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.59760e9i 0.325003i
\(300\) 0 0
\(301\) 1.51933e10 1.85092
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.92402e9 0.684569
\(306\) 0 0
\(307\) 5.36906e9i 0.604428i 0.953240 + 0.302214i \(0.0977257\pi\)
−0.953240 + 0.302214i \(0.902274\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.21354e9i 0.664198i 0.943245 + 0.332099i \(0.107757\pi\)
−0.943245 + 0.332099i \(0.892243\pi\)
\(312\) 0 0
\(313\) 2.95026e8 0.0307385 0.0153693 0.999882i \(-0.495108\pi\)
0.0153693 + 0.999882i \(0.495108\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.22463e9 −0.121274 −0.0606369 0.998160i \(-0.519313\pi\)
−0.0606369 + 0.998160i \(0.519313\pi\)
\(318\) 0 0
\(319\) − 1.21502e9i − 0.117333i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.22337e10i 2.96142i
\(324\) 0 0
\(325\) −1.38969e10 −1.24562
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.50615e9 0.213906
\(330\) 0 0
\(331\) 1.49808e10i 1.24802i 0.781415 + 0.624012i \(0.214498\pi\)
−0.781415 + 0.624012i \(0.785502\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 1.44915e10i − 1.15062i
\(336\) 0 0
\(337\) 1.22500e10 0.949768 0.474884 0.880049i \(-0.342490\pi\)
0.474884 + 0.880049i \(0.342490\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.08322e9 0.228028
\(342\) 0 0
\(343\) − 5.92885e9i − 0.428346i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.66765e9i − 0.666811i −0.942783 0.333406i \(-0.891802\pi\)
0.942783 0.333406i \(-0.108198\pi\)
\(348\) 0 0
\(349\) −8.41057e9 −0.566923 −0.283461 0.958984i \(-0.591483\pi\)
−0.283461 + 0.958984i \(0.591483\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.96120e10 −1.26305 −0.631527 0.775354i \(-0.717572\pi\)
−0.631527 + 0.775354i \(0.717572\pi\)
\(354\) 0 0
\(355\) − 1.82966e10i − 1.15201i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 3.86050e9i − 0.232416i −0.993225 0.116208i \(-0.962926\pi\)
0.993225 0.116208i \(-0.0370739\pi\)
\(360\) 0 0
\(361\) −3.15109e10 −1.85538
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.02217e10 1.70274
\(366\) 0 0
\(367\) − 4.07167e9i − 0.224444i −0.993683 0.112222i \(-0.964203\pi\)
0.993683 0.112222i \(-0.0357968\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 3.76639e10i − 1.98806i
\(372\) 0 0
\(373\) 9.73315e8 0.0502827 0.0251413 0.999684i \(-0.491996\pi\)
0.0251413 + 0.999684i \(0.491996\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.55430e9 0.225453
\(378\) 0 0
\(379\) 1.03474e10i 0.501506i 0.968051 + 0.250753i \(0.0806781\pi\)
−0.968051 + 0.250753i \(0.919322\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.47254e10i 0.684342i 0.939638 + 0.342171i \(0.111162\pi\)
−0.939638 + 0.342171i \(0.888838\pi\)
\(384\) 0 0
\(385\) −2.11974e10 −0.964808
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.12190e10 0.926673 0.463336 0.886183i \(-0.346652\pi\)
0.463336 + 0.886183i \(0.346652\pi\)
\(390\) 0 0
\(391\) 1.41359e10i 0.604808i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 5.40964e10i − 2.22219i
\(396\) 0 0
\(397\) 2.05017e10 0.825331 0.412665 0.910883i \(-0.364598\pi\)
0.412665 + 0.910883i \(0.364598\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.20594e10 −1.23987 −0.619937 0.784652i \(-0.712842\pi\)
−0.619937 + 0.784652i \(0.712842\pi\)
\(402\) 0 0
\(403\) 1.15570e10i 0.438152i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.11611e10i 0.771189i
\(408\) 0 0
\(409\) 2.67550e10 0.956117 0.478058 0.878328i \(-0.341341\pi\)
0.478058 + 0.878328i \(0.341341\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −7.68413e9 −0.264116
\(414\) 0 0
\(415\) − 8.12296e10i − 2.73856i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.08653e10i 0.352521i 0.984344 + 0.176261i \(0.0564002\pi\)
−0.984344 + 0.176261i \(0.943600\pi\)
\(420\) 0 0
\(421\) 2.03721e9 0.0648495 0.0324247 0.999474i \(-0.489677\pi\)
0.0324247 + 0.999474i \(0.489677\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −7.56260e10 −2.31801
\(426\) 0 0
\(427\) 1.92878e10i 0.580192i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.19963e10i 0.927236i 0.886035 + 0.463618i \(0.153449\pi\)
−0.886035 + 0.463618i \(0.846551\pi\)
\(432\) 0 0
\(433\) −2.78313e10 −0.791738 −0.395869 0.918307i \(-0.629557\pi\)
−0.395869 + 0.918307i \(0.629557\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.12670e10 −0.583151
\(438\) 0 0
\(439\) 3.88247e10i 1.04532i 0.852540 + 0.522661i \(0.175061\pi\)
−0.852540 + 0.522661i \(0.824939\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 4.68689e10i − 1.21694i −0.793576 0.608471i \(-0.791783\pi\)
0.793576 0.608471i \(-0.208217\pi\)
\(444\) 0 0
\(445\) 2.92101e10 0.744891
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3.48851e10 −0.858331 −0.429166 0.903226i \(-0.641192\pi\)
−0.429166 + 0.903226i \(0.641192\pi\)
\(450\) 0 0
\(451\) − 2.26556e10i − 0.547607i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 7.94554e10i − 1.85386i
\(456\) 0 0
\(457\) −7.46508e9 −0.171147 −0.0855736 0.996332i \(-0.527272\pi\)
−0.0855736 + 0.996332i \(0.527272\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.60087e10 1.24008 0.620042 0.784568i \(-0.287115\pi\)
0.620042 + 0.784568i \(0.287115\pi\)
\(462\) 0 0
\(463\) − 4.51649e10i − 0.982826i −0.870927 0.491413i \(-0.836481\pi\)
0.870927 0.491413i \(-0.163519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.30108e10i 0.273550i 0.990602 + 0.136775i \(0.0436737\pi\)
−0.990602 + 0.136775i \(0.956326\pi\)
\(468\) 0 0
\(469\) 4.71822e10 0.975186
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.51549e10 −0.702330
\(474\) 0 0
\(475\) − 1.13777e11i − 2.23501i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.63024e9i 0.106951i 0.998569 + 0.0534755i \(0.0170299\pi\)
−0.998569 + 0.0534755i \(0.982970\pi\)
\(480\) 0 0
\(481\) −7.93192e10 −1.48183
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.90939e9 −0.0887279
\(486\) 0 0
\(487\) − 2.21903e10i − 0.394501i −0.980353 0.197250i \(-0.936799\pi\)
0.980353 0.197250i \(-0.0632012\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.38353e10i − 0.410104i −0.978751 0.205052i \(-0.934264\pi\)
0.978751 0.205052i \(-0.0657364\pi\)
\(492\) 0 0
\(493\) 2.47842e10 0.419553
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.95712e10 0.976361
\(498\) 0 0
\(499\) 8.49679e10i 1.37042i 0.728347 + 0.685209i \(0.240289\pi\)
−0.728347 + 0.685209i \(0.759711\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 8.35912e10i − 1.30584i −0.757428 0.652918i \(-0.773544\pi\)
0.757428 0.652918i \(-0.226456\pi\)
\(504\) 0 0
\(505\) −9.64744e10 −1.48336
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.04693e10 0.602912 0.301456 0.953480i \(-0.402527\pi\)
0.301456 + 0.953480i \(0.402527\pi\)
\(510\) 0 0
\(511\) 9.83978e10i 1.44312i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.94752e11i 2.76856i
\(516\) 0 0
\(517\) −5.79883e9 −0.0811667
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.68115e10 1.17822 0.589110 0.808053i \(-0.299478\pi\)
0.589110 + 0.808053i \(0.299478\pi\)
\(522\) 0 0
\(523\) 1.28187e11i 1.71332i 0.515881 + 0.856660i \(0.327465\pi\)
−0.515881 + 0.856660i \(0.672535\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.28922e10i 0.815369i
\(528\) 0 0
\(529\) 6.89844e10 0.880904
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.49209e10 1.05222
\(534\) 0 0
\(535\) 1.53408e11i 1.87254i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 2.76488e10i − 0.327583i
\(540\) 0 0
\(541\) −9.63677e10 −1.12497 −0.562487 0.826806i \(-0.690155\pi\)
−0.562487 + 0.826806i \(0.690155\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.45882e10 −0.618746
\(546\) 0 0
\(547\) − 1.13025e11i − 1.26248i −0.775587 0.631241i \(-0.782546\pi\)
0.775587 0.631241i \(-0.217454\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.72869e10i 0.404529i
\(552\) 0 0
\(553\) 1.76131e11 1.88337
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.42898e11 −1.48458 −0.742291 0.670078i \(-0.766261\pi\)
−0.742291 + 0.670078i \(0.766261\pi\)
\(558\) 0 0
\(559\) − 1.31773e11i − 1.34952i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.62444e10i 0.858415i 0.903206 + 0.429207i \(0.141207\pi\)
−0.903206 + 0.429207i \(0.858793\pi\)
\(564\) 0 0
\(565\) 1.42329e10 0.139669
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.08653e10 0.676059 0.338029 0.941136i \(-0.390240\pi\)
0.338029 + 0.941136i \(0.390240\pi\)
\(570\) 0 0
\(571\) 1.62595e11i 1.52955i 0.644298 + 0.764774i \(0.277150\pi\)
−0.644298 + 0.764774i \(0.722850\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 4.98963e10i − 0.456453i
\(576\) 0 0
\(577\) 7.57766e10 0.683647 0.341824 0.939764i \(-0.388955\pi\)
0.341824 + 0.939764i \(0.388955\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.64473e11 2.32101
\(582\) 0 0
\(583\) 8.71483e10i 0.754371i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.92106e11i 1.61804i 0.587784 + 0.809018i \(0.300000\pi\)
−0.587784 + 0.809018i \(0.700000\pi\)
\(588\) 0 0
\(589\) −9.46191e10 −0.786172
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.12974e10 0.414836 0.207418 0.978252i \(-0.433494\pi\)
0.207418 + 0.978252i \(0.433494\pi\)
\(594\) 0 0
\(595\) − 4.32390e11i − 3.44991i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.95617e10i 0.540335i 0.962813 + 0.270167i \(0.0870790\pi\)
−0.962813 + 0.270167i \(0.912921\pi\)
\(600\) 0 0
\(601\) 1.46315e11 1.12148 0.560739 0.827993i \(-0.310517\pi\)
0.560739 + 0.827993i \(0.310517\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.55133e11 −1.15793
\(606\) 0 0
\(607\) 6.48265e10i 0.477527i 0.971078 + 0.238763i \(0.0767420\pi\)
−0.971078 + 0.238763i \(0.923258\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.17360e10i − 0.155961i
\(612\) 0 0
\(613\) 5.62833e9 0.0398600 0.0199300 0.999801i \(-0.493656\pi\)
0.0199300 + 0.999801i \(0.493656\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.70885e11 −1.17913 −0.589567 0.807719i \(-0.700702\pi\)
−0.589567 + 0.807719i \(0.700702\pi\)
\(618\) 0 0
\(619\) 2.59388e10i 0.176680i 0.996090 + 0.0883401i \(0.0281562\pi\)
−0.996090 + 0.0883401i \(0.971844\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.51041e10i 0.631316i
\(624\) 0 0
\(625\) −8.74688e10 −0.573236
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.31649e11 −2.75758
\(630\) 0 0
\(631\) − 3.47218e9i − 0.0219021i −0.999940 0.0109510i \(-0.996514\pi\)
0.999940 0.0109510i \(-0.00348589\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 8.27674e10i − 0.509055i
\(636\) 0 0
\(637\) 1.03637e11 0.629446
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.20527e11 0.713927 0.356963 0.934118i \(-0.383812\pi\)
0.356963 + 0.934118i \(0.383812\pi\)
\(642\) 0 0
\(643\) 1.04525e11i 0.611470i 0.952117 + 0.305735i \(0.0989021\pi\)
−0.952117 + 0.305735i \(0.901098\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.75874e10i 0.271566i 0.990739 + 0.135783i \(0.0433550\pi\)
−0.990739 + 0.135783i \(0.956645\pi\)
\(648\) 0 0
\(649\) 1.77798e10 0.100219
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.53529e11 0.844381 0.422190 0.906507i \(-0.361261\pi\)
0.422190 + 0.906507i \(0.361261\pi\)
\(654\) 0 0
\(655\) 2.79200e11i 1.51688i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1.08583e11i − 0.575731i −0.957671 0.287865i \(-0.907054\pi\)
0.957671 0.287865i \(-0.0929456\pi\)
\(660\) 0 0
\(661\) 1.09736e11 0.574834 0.287417 0.957806i \(-0.407204\pi\)
0.287417 + 0.957806i \(0.407204\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.50515e11 3.32637
\(666\) 0 0
\(667\) 1.63520e10i 0.0826166i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 4.46289e10i − 0.220154i
\(672\) 0 0
\(673\) −2.53333e11 −1.23490 −0.617449 0.786611i \(-0.711834\pi\)
−0.617449 + 0.786611i \(0.711834\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.20339e10 −0.390516 −0.195258 0.980752i \(-0.562554\pi\)
−0.195258 + 0.980752i \(0.562554\pi\)
\(678\) 0 0
\(679\) − 1.59843e10i − 0.0751994i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 3.43876e10i 0.158022i 0.996874 + 0.0790112i \(0.0251763\pi\)
−0.996874 + 0.0790112i \(0.974824\pi\)
\(684\) 0 0
\(685\) −4.69912e11 −2.13429
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.26662e11 −1.44951
\(690\) 0 0
\(691\) 1.89939e11i 0.833111i 0.909110 + 0.416556i \(0.136763\pi\)
−0.909110 + 0.416556i \(0.863237\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 4.96331e11i − 2.12732i
\(696\) 0 0
\(697\) 4.62133e11 1.95810
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.49808e11 0.620388 0.310194 0.950673i \(-0.399606\pi\)
0.310194 + 0.950673i \(0.399606\pi\)
\(702\) 0 0
\(703\) − 6.49400e11i − 2.65883i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 3.14107e11i − 1.25719i
\(708\) 0 0
\(709\) −1.29962e11 −0.514317 −0.257158 0.966369i \(-0.582786\pi\)
−0.257158 + 0.966369i \(0.582786\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.14948e10 −0.160559
\(714\) 0 0
\(715\) 1.83847e11i 0.703449i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.22515e11i 1.20680i 0.797440 + 0.603398i \(0.206187\pi\)
−0.797440 + 0.603398i \(0.793813\pi\)
\(720\) 0 0
\(721\) −6.34088e11 −2.34643
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.74817e10 −0.316640
\(726\) 0 0
\(727\) − 4.13304e11i − 1.47956i −0.672850 0.739779i \(-0.734930\pi\)
0.672850 0.739779i \(-0.265070\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 7.17097e11i − 2.51136i
\(732\) 0 0
\(733\) 3.98880e11 1.38174 0.690871 0.722978i \(-0.257227\pi\)
0.690871 + 0.722978i \(0.257227\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.09172e11 −0.370034
\(738\) 0 0
\(739\) − 1.18702e11i − 0.397998i −0.980000 0.198999i \(-0.936231\pi\)
0.980000 0.198999i \(-0.0637691\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 2.31061e11i − 0.758179i −0.925360 0.379089i \(-0.876237\pi\)
0.925360 0.379089i \(-0.123763\pi\)
\(744\) 0 0
\(745\) −3.89796e11 −1.26535
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.99474e11 −1.58703
\(750\) 0 0
\(751\) 1.94612e11i 0.611801i 0.952064 + 0.305900i \(0.0989574\pi\)
−0.952064 + 0.305900i \(0.901043\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.50175e9i 0.0261650i
\(756\) 0 0
\(757\) 3.63936e10 0.110826 0.0554130 0.998464i \(-0.482352\pi\)
0.0554130 + 0.998464i \(0.482352\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.62354e10 0.108043 0.0540213 0.998540i \(-0.482796\pi\)
0.0540213 + 0.998540i \(0.482796\pi\)
\(762\) 0 0
\(763\) − 1.77732e11i − 0.524405i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.66450e10i 0.192569i
\(768\) 0 0
\(769\) 4.96871e11 1.42082 0.710409 0.703789i \(-0.248510\pi\)
0.710409 + 0.703789i \(0.248510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.63957e11 −1.85961 −0.929806 0.368051i \(-0.880025\pi\)
−0.929806 + 0.368051i \(0.880025\pi\)
\(774\) 0 0
\(775\) − 2.21993e11i − 0.615366i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.95263e11i 1.88799i
\(780\) 0 0
\(781\) −1.37838e11 −0.370480
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.46634e11 −0.649493
\(786\) 0 0
\(787\) 6.50846e10i 0.169660i 0.996395 + 0.0848299i \(0.0270347\pi\)
−0.996395 + 0.0848299i \(0.972965\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 4.63404e10i 0.118373i
\(792\) 0 0
\(793\) 1.67285e11 0.423022
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.57879e10 −0.138263 −0.0691316 0.997608i \(-0.522023\pi\)
−0.0691316 + 0.997608i \(0.522023\pi\)
\(798\) 0 0
\(799\) − 1.18286e11i − 0.290232i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 2.27677e11i − 0.547591i
\(804\) 0 0
\(805\) 2.85281e11 0.679342
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.06402e11 1.41568 0.707842 0.706370i \(-0.249669\pi\)
0.707842 + 0.706370i \(0.249669\pi\)
\(810\) 0 0
\(811\) − 4.98163e11i − 1.15156i −0.817603 0.575782i \(-0.804698\pi\)
0.817603 0.575782i \(-0.195302\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.92066e11i 0.435331i
\(816\) 0 0
\(817\) 1.07885e12 2.42143
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.16956e11 0.697632 0.348816 0.937191i \(-0.386584\pi\)
0.348816 + 0.937191i \(0.386584\pi\)
\(822\) 0 0
\(823\) 6.55634e11i 1.42910i 0.699585 + 0.714549i \(0.253368\pi\)
−0.699585 + 0.714549i \(0.746632\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 8.69774e10i 0.185945i 0.995669 + 0.0929724i \(0.0296368\pi\)
−0.995669 + 0.0929724i \(0.970363\pi\)
\(828\) 0 0
\(829\) −3.79641e11 −0.803813 −0.401906 0.915681i \(-0.631652\pi\)
−0.401906 + 0.915681i \(0.631652\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.63986e11 1.17135
\(834\) 0 0
\(835\) 1.38227e12i 2.84345i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.07825e11i 0.419421i 0.977764 + 0.209710i \(0.0672521\pi\)
−0.977764 + 0.209710i \(0.932748\pi\)
\(840\) 0 0
\(841\) −4.71577e11 −0.942689
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.78748e10 0.172360
\(846\) 0 0
\(847\) − 5.05092e11i − 0.981379i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 2.84791e11i − 0.543011i
\(852\) 0 0
\(853\) −7.32516e11 −1.38363 −0.691817 0.722073i \(-0.743189\pi\)
−0.691817 + 0.722073i \(0.743189\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.23294e11 −0.413955 −0.206978 0.978346i \(-0.566363\pi\)
−0.206978 + 0.978346i \(0.566363\pi\)
\(858\) 0 0
\(859\) 2.41807e11i 0.444116i 0.975034 + 0.222058i \(0.0712774\pi\)
−0.975034 + 0.222058i \(0.928723\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1.52874e11i − 0.275607i −0.990460 0.137804i \(-0.955996\pi\)
0.990460 0.137804i \(-0.0440043\pi\)
\(864\) 0 0
\(865\) 6.06200e11 1.08281
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.07538e11 −0.714643
\(870\) 0 0
\(871\) − 4.09215e11i − 0.711015i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.72317e11i 0.635156i
\(876\) 0 0
\(877\) −1.94466e11 −0.328735 −0.164367 0.986399i \(-0.552558\pi\)
−0.164367 + 0.986399i \(0.552558\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.23735e11 1.53336 0.766679 0.642030i \(-0.221908\pi\)
0.766679 + 0.642030i \(0.221908\pi\)
\(882\) 0 0
\(883\) − 1.15008e12i − 1.89184i −0.324405 0.945918i \(-0.605164\pi\)
0.324405 0.945918i \(-0.394836\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.00865e12i 1.62947i 0.579831 + 0.814737i \(0.303119\pi\)
−0.579831 + 0.814737i \(0.696881\pi\)
\(888\) 0 0
\(889\) 2.69479e11 0.431438
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.77957e11 0.279839
\(894\) 0 0
\(895\) 1.75362e12i 2.73302i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 7.27517e10i 0.111379i
\(900\) 0 0
\(901\) −1.77767e12 −2.69744
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.90758e11 1.02975
\(906\) 0 0
\(907\) 1.63003e11i 0.240861i 0.992722 + 0.120430i \(0.0384274\pi\)
−0.992722 + 0.120430i \(0.961573\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 2.86326e11i − 0.415707i −0.978160 0.207854i \(-0.933352\pi\)
0.978160 0.207854i \(-0.0666478\pi\)
\(912\) 0 0
\(913\) −6.11947e11 −0.880705
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −9.09037e11 −1.28560
\(918\) 0 0
\(919\) − 4.28536e11i − 0.600793i −0.953814 0.300396i \(-0.902881\pi\)
0.953814 0.300396i \(-0.0971190\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 5.16665e11i − 0.711872i
\(924\) 0 0
\(925\) 1.52361e12 2.08117
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.91477e11 1.06261 0.531307 0.847180i \(-0.321701\pi\)
0.531307 + 0.847180i \(0.321701\pi\)
\(930\) 0 0
\(931\) 8.48497e11i 1.12941i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.00048e12i 1.30907i
\(936\) 0 0
\(937\) −8.10675e11 −1.05169 −0.525846 0.850580i \(-0.676251\pi\)
−0.525846 + 0.850580i \(0.676251\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.14478e12 1.46004 0.730021 0.683425i \(-0.239510\pi\)
0.730021 + 0.683425i \(0.239510\pi\)
\(942\) 0 0
\(943\) 3.04904e11i 0.385582i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.36874e12i 1.70185i 0.525284 + 0.850927i \(0.323959\pi\)
−0.525284 + 0.850927i \(0.676041\pi\)
\(948\) 0 0
\(949\) 8.53411e11 1.05219
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.59249e11 −0.193065 −0.0965327 0.995330i \(-0.530775\pi\)
−0.0965327 + 0.995330i \(0.530775\pi\)
\(954\) 0 0
\(955\) − 1.52263e12i − 1.83054i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1.52997e12i − 1.80887i
\(960\) 0 0
\(961\) 6.68277e11 0.783543
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.81088e11 0.554773
\(966\) 0 0
\(967\) 9.07631e11i 1.03801i 0.854770 + 0.519007i \(0.173698\pi\)
−0.854770 + 0.519007i \(0.826302\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.07988e12i 1.21478i 0.794405 + 0.607389i \(0.207783\pi\)
−0.794405 + 0.607389i \(0.792217\pi\)
\(972\) 0 0
\(973\) 1.61599e12 1.80296
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.39047e12 −1.52610 −0.763049 0.646341i \(-0.776298\pi\)
−0.763049 + 0.646341i \(0.776298\pi\)
\(978\) 0 0
\(979\) − 2.20056e11i − 0.239553i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.23832e12i 1.32623i 0.748516 + 0.663117i \(0.230767\pi\)
−0.748516 + 0.663117i \(0.769233\pi\)
\(984\) 0 0
\(985\) 2.13205e12 2.26492
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.73124e11 0.494526
\(990\) 0 0
\(991\) 1.64879e12i 1.70951i 0.519033 + 0.854754i \(0.326292\pi\)
−0.519033 + 0.854754i \(0.673708\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.06284e12i 1.08436i
\(996\) 0 0
\(997\) 1.01638e10 0.0102867 0.00514335 0.999987i \(-0.498363\pi\)
0.00514335 + 0.999987i \(0.498363\pi\)
\(998\) 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.i.127.1 4
3.2 odd 2 48.9.g.c.31.4 yes 4
4.3 odd 2 inner 144.9.g.i.127.2 4
12.11 even 2 48.9.g.c.31.2 4
24.5 odd 2 192.9.g.c.127.1 4
24.11 even 2 192.9.g.c.127.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.9.g.c.31.2 4 12.11 even 2
48.9.g.c.31.4 yes 4 3.2 odd 2
144.9.g.i.127.1 4 1.1 even 1 trivial
144.9.g.i.127.2 4 4.3 odd 2 inner
192.9.g.c.127.1 4 24.5 odd 2
192.9.g.c.127.3 4 24.11 even 2