Properties

Label 192.9.g.f.127.1
Level $192$
Weight $9$
Character 192.127
Analytic conductor $78.217$
Analytic rank $0$
Dimension $8$
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] = [192,9,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, 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("192.127");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 192.g (of order \(2\), degree \(1\), not 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: \(78.2166931317\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(9.62695 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 192.127
Dual form 192.9.g.f.127.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-46.7654i q^{3} -455.790 q^{5} -1689.76i q^{7} -2187.00 q^{9} +O(q^{10})\) \(q-46.7654i q^{3} -455.790 q^{5} -1689.76i q^{7} -2187.00 q^{9} +6803.31i q^{11} +15357.7 q^{13} +21315.2i q^{15} -93787.7 q^{17} +14546.9i q^{19} -79022.4 q^{21} +336401. i q^{23} -182880. q^{25} +102276. i q^{27} +10707.6 q^{29} -740903. i q^{31} +318159. q^{33} +770177. i q^{35} +1.33541e6 q^{37} -718209. i q^{39} -3.17759e6 q^{41} +4.00670e6i q^{43} +996813. q^{45} -1.57345e6i q^{47} +2.90950e6 q^{49} +4.38601e6i q^{51} +7.67006e6 q^{53} -3.10088e6i q^{55} +680289. q^{57} +1.53199e7i q^{59} +1.93832e7 q^{61} +3.69551e6i q^{63} -6.99990e6 q^{65} -1.37063e7i q^{67} +1.57319e7 q^{69} -1.95800e7i q^{71} -3.69806e7 q^{73} +8.55247e6i q^{75} +1.14960e7 q^{77} -6.14961e7i q^{79} +4.78297e6 q^{81} +1.79749e7i q^{83} +4.27475e7 q^{85} -500744. i q^{87} +7.04735e7 q^{89} -2.59509e7i q^{91} -3.46486e7 q^{93} -6.63031e6i q^{95} +1.04776e8 q^{97} -1.48788e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1232 q^{5} - 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 1232 q^{5} - 17496 q^{9} + 54256 q^{13} - 204336 q^{17} + 298080 q^{21} - 781416 q^{25} + 333712 q^{29} - 775008 q^{33} - 3945168 q^{37} + 11803088 q^{41} - 2694384 q^{45} - 3977208 q^{49} - 788592 q^{53} - 1194912 q^{57} + 40252336 q^{61} - 20747936 q^{65} - 26687232 q^{69} - 56712048 q^{73} + 52014208 q^{77} + 38263752 q^{81} - 52697568 q^{85} + 61660176 q^{89} - 81738720 q^{93} + 27830288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 46.7654i − 0.577350i
\(4\) 0 0
\(5\) −455.790 −0.729264 −0.364632 0.931152i \(-0.618805\pi\)
−0.364632 + 0.931152i \(0.618805\pi\)
\(6\) 0 0
\(7\) − 1689.76i − 0.703775i −0.936042 0.351887i \(-0.885540\pi\)
0.936042 0.351887i \(-0.114460\pi\)
\(8\) 0 0
\(9\) −2187.00 −0.333333
\(10\) 0 0
\(11\) 6803.31i 0.464675i 0.972635 + 0.232338i \(0.0746374\pi\)
−0.972635 + 0.232338i \(0.925363\pi\)
\(12\) 0 0
\(13\) 15357.7 0.537716 0.268858 0.963180i \(-0.413354\pi\)
0.268858 + 0.963180i \(0.413354\pi\)
\(14\) 0 0
\(15\) 21315.2i 0.421041i
\(16\) 0 0
\(17\) −93787.7 −1.12292 −0.561462 0.827503i \(-0.689761\pi\)
−0.561462 + 0.827503i \(0.689761\pi\)
\(18\) 0 0
\(19\) 14546.9i 0.111623i 0.998441 + 0.0558117i \(0.0177746\pi\)
−0.998441 + 0.0558117i \(0.982225\pi\)
\(20\) 0 0
\(21\) −79022.4 −0.406325
\(22\) 0 0
\(23\) 336401.i 1.20212i 0.799205 + 0.601058i \(0.205254\pi\)
−0.799205 + 0.601058i \(0.794746\pi\)
\(24\) 0 0
\(25\) −182880. −0.468174
\(26\) 0 0
\(27\) 102276.i 0.192450i
\(28\) 0 0
\(29\) 10707.6 0.0151391 0.00756953 0.999971i \(-0.497591\pi\)
0.00756953 + 0.999971i \(0.497591\pi\)
\(30\) 0 0
\(31\) − 740903.i − 0.802259i −0.916021 0.401130i \(-0.868618\pi\)
0.916021 0.401130i \(-0.131382\pi\)
\(32\) 0 0
\(33\) 318159. 0.268280
\(34\) 0 0
\(35\) 770177.i 0.513238i
\(36\) 0 0
\(37\) 1.33541e6 0.712540 0.356270 0.934383i \(-0.384048\pi\)
0.356270 + 0.934383i \(0.384048\pi\)
\(38\) 0 0
\(39\) − 718209.i − 0.310451i
\(40\) 0 0
\(41\) −3.17759e6 −1.12451 −0.562253 0.826965i \(-0.690065\pi\)
−0.562253 + 0.826965i \(0.690065\pi\)
\(42\) 0 0
\(43\) 4.00670e6i 1.17196i 0.810325 + 0.585981i \(0.199291\pi\)
−0.810325 + 0.585981i \(0.800709\pi\)
\(44\) 0 0
\(45\) 996813. 0.243088
\(46\) 0 0
\(47\) − 1.57345e6i − 0.322450i −0.986918 0.161225i \(-0.948456\pi\)
0.986918 0.161225i \(-0.0515445\pi\)
\(48\) 0 0
\(49\) 2.90950e6 0.504701
\(50\) 0 0
\(51\) 4.38601e6i 0.648320i
\(52\) 0 0
\(53\) 7.67006e6 0.972065 0.486033 0.873941i \(-0.338444\pi\)
0.486033 + 0.873941i \(0.338444\pi\)
\(54\) 0 0
\(55\) − 3.10088e6i − 0.338871i
\(56\) 0 0
\(57\) 680289. 0.0644457
\(58\) 0 0
\(59\) 1.53199e7i 1.26430i 0.774848 + 0.632148i \(0.217827\pi\)
−0.774848 + 0.632148i \(0.782173\pi\)
\(60\) 0 0
\(61\) 1.93832e7 1.39993 0.699965 0.714178i \(-0.253199\pi\)
0.699965 + 0.714178i \(0.253199\pi\)
\(62\) 0 0
\(63\) 3.69551e6i 0.234592i
\(64\) 0 0
\(65\) −6.99990e6 −0.392137
\(66\) 0 0
\(67\) − 1.37063e7i − 0.680176i −0.940394 0.340088i \(-0.889543\pi\)
0.940394 0.340088i \(-0.110457\pi\)
\(68\) 0 0
\(69\) 1.57319e7 0.694042
\(70\) 0 0
\(71\) − 1.95800e7i − 0.770510i −0.922810 0.385255i \(-0.874113\pi\)
0.922810 0.385255i \(-0.125887\pi\)
\(72\) 0 0
\(73\) −3.69806e7 −1.30221 −0.651107 0.758986i \(-0.725695\pi\)
−0.651107 + 0.758986i \(0.725695\pi\)
\(74\) 0 0
\(75\) 8.55247e6i 0.270300i
\(76\) 0 0
\(77\) 1.14960e7 0.327027
\(78\) 0 0
\(79\) − 6.14961e7i − 1.57884i −0.613851 0.789422i \(-0.710381\pi\)
0.613851 0.789422i \(-0.289619\pi\)
\(80\) 0 0
\(81\) 4.78297e6 0.111111
\(82\) 0 0
\(83\) 1.79749e7i 0.378751i 0.981905 + 0.189375i \(0.0606463\pi\)
−0.981905 + 0.189375i \(0.939354\pi\)
\(84\) 0 0
\(85\) 4.27475e7 0.818908
\(86\) 0 0
\(87\) − 500744.i − 0.00874055i
\(88\) 0 0
\(89\) 7.04735e7 1.12322 0.561611 0.827401i \(-0.310182\pi\)
0.561611 + 0.827401i \(0.310182\pi\)
\(90\) 0 0
\(91\) − 2.59509e7i − 0.378431i
\(92\) 0 0
\(93\) −3.46486e7 −0.463185
\(94\) 0 0
\(95\) − 6.63031e6i − 0.0814029i
\(96\) 0 0
\(97\) 1.04776e8 1.18351 0.591757 0.806116i \(-0.298434\pi\)
0.591757 + 0.806116i \(0.298434\pi\)
\(98\) 0 0
\(99\) − 1.48788e7i − 0.154892i
\(100\) 0 0
\(101\) 5.62286e7 0.540346 0.270173 0.962812i \(-0.412919\pi\)
0.270173 + 0.962812i \(0.412919\pi\)
\(102\) 0 0
\(103\) 1.22106e8i 1.08490i 0.840090 + 0.542448i \(0.182502\pi\)
−0.840090 + 0.542448i \(0.817498\pi\)
\(104\) 0 0
\(105\) 3.60176e7 0.296318
\(106\) 0 0
\(107\) 2.28826e6i 0.0174570i 0.999962 + 0.00872850i \(0.00277840\pi\)
−0.999962 + 0.00872850i \(0.997222\pi\)
\(108\) 0 0
\(109\) 1.50447e8 1.06580 0.532901 0.846177i \(-0.321102\pi\)
0.532901 + 0.846177i \(0.321102\pi\)
\(110\) 0 0
\(111\) − 6.24512e7i − 0.411385i
\(112\) 0 0
\(113\) 2.57360e8 1.57844 0.789220 0.614111i \(-0.210485\pi\)
0.789220 + 0.614111i \(0.210485\pi\)
\(114\) 0 0
\(115\) − 1.53328e8i − 0.876660i
\(116\) 0 0
\(117\) −3.35873e7 −0.179239
\(118\) 0 0
\(119\) 1.58479e8i 0.790285i
\(120\) 0 0
\(121\) 1.68074e8 0.784077
\(122\) 0 0
\(123\) 1.48601e8i 0.649234i
\(124\) 0 0
\(125\) 2.61398e8 1.07069
\(126\) 0 0
\(127\) 3.20725e8i 1.23287i 0.787405 + 0.616436i \(0.211424\pi\)
−0.787405 + 0.616436i \(0.788576\pi\)
\(128\) 0 0
\(129\) 1.87375e8 0.676633
\(130\) 0 0
\(131\) − 1.62608e8i − 0.552149i −0.961136 0.276075i \(-0.910966\pi\)
0.961136 0.276075i \(-0.0890337\pi\)
\(132\) 0 0
\(133\) 2.45808e7 0.0785577
\(134\) 0 0
\(135\) − 4.66163e7i − 0.140347i
\(136\) 0 0
\(137\) 3.73201e8 1.05940 0.529700 0.848185i \(-0.322304\pi\)
0.529700 + 0.848185i \(0.322304\pi\)
\(138\) 0 0
\(139\) − 2.91334e8i − 0.780428i −0.920724 0.390214i \(-0.872401\pi\)
0.920724 0.390214i \(-0.127599\pi\)
\(140\) 0 0
\(141\) −7.35831e7 −0.186167
\(142\) 0 0
\(143\) 1.04483e8i 0.249864i
\(144\) 0 0
\(145\) −4.88041e6 −0.0110404
\(146\) 0 0
\(147\) − 1.36064e8i − 0.291389i
\(148\) 0 0
\(149\) −6.00711e8 −1.21877 −0.609383 0.792876i \(-0.708583\pi\)
−0.609383 + 0.792876i \(0.708583\pi\)
\(150\) 0 0
\(151\) 5.53985e8i 1.06559i 0.846244 + 0.532795i \(0.178858\pi\)
−0.846244 + 0.532795i \(0.821142\pi\)
\(152\) 0 0
\(153\) 2.05114e8 0.374308
\(154\) 0 0
\(155\) 3.37696e8i 0.585059i
\(156\) 0 0
\(157\) 9.83548e8 1.61881 0.809407 0.587248i \(-0.199789\pi\)
0.809407 + 0.587248i \(0.199789\pi\)
\(158\) 0 0
\(159\) − 3.58693e8i − 0.561222i
\(160\) 0 0
\(161\) 5.68439e8 0.846019
\(162\) 0 0
\(163\) − 7.01240e6i − 0.00993382i −0.999988 0.00496691i \(-0.998419\pi\)
0.999988 0.00496691i \(-0.00158102\pi\)
\(164\) 0 0
\(165\) −1.45014e8 −0.195647
\(166\) 0 0
\(167\) − 1.07398e9i − 1.38079i −0.723431 0.690396i \(-0.757436\pi\)
0.723431 0.690396i \(-0.242564\pi\)
\(168\) 0 0
\(169\) −5.79871e8 −0.710861
\(170\) 0 0
\(171\) − 3.18140e7i − 0.0372078i
\(172\) 0 0
\(173\) −3.38472e7 −0.0377867 −0.0188933 0.999822i \(-0.506014\pi\)
−0.0188933 + 0.999822i \(0.506014\pi\)
\(174\) 0 0
\(175\) 3.09025e8i 0.329489i
\(176\) 0 0
\(177\) 7.16442e8 0.729942
\(178\) 0 0
\(179\) 1.55294e9i 1.51266i 0.654188 + 0.756332i \(0.273010\pi\)
−0.654188 + 0.756332i \(0.726990\pi\)
\(180\) 0 0
\(181\) 1.27084e9 1.18407 0.592033 0.805913i \(-0.298325\pi\)
0.592033 + 0.805913i \(0.298325\pi\)
\(182\) 0 0
\(183\) − 9.06462e8i − 0.808249i
\(184\) 0 0
\(185\) −6.08669e8 −0.519630
\(186\) 0 0
\(187\) − 6.38067e8i − 0.521795i
\(188\) 0 0
\(189\) 1.72822e8 0.135442
\(190\) 0 0
\(191\) 3.90488e8i 0.293409i 0.989180 + 0.146705i \(0.0468667\pi\)
−0.989180 + 0.146705i \(0.953133\pi\)
\(192\) 0 0
\(193\) −1.80145e9 −1.29836 −0.649178 0.760637i \(-0.724887\pi\)
−0.649178 + 0.760637i \(0.724887\pi\)
\(194\) 0 0
\(195\) 3.27353e8i 0.226401i
\(196\) 0 0
\(197\) 5.96764e8 0.396221 0.198110 0.980180i \(-0.436520\pi\)
0.198110 + 0.980180i \(0.436520\pi\)
\(198\) 0 0
\(199\) − 4.63439e8i − 0.295516i −0.989024 0.147758i \(-0.952794\pi\)
0.989024 0.147758i \(-0.0472056\pi\)
\(200\) 0 0
\(201\) −6.40980e8 −0.392700
\(202\) 0 0
\(203\) − 1.80933e7i − 0.0106545i
\(204\) 0 0
\(205\) 1.44831e9 0.820062
\(206\) 0 0
\(207\) − 7.35710e8i − 0.400705i
\(208\) 0 0
\(209\) −9.89668e7 −0.0518686
\(210\) 0 0
\(211\) − 3.26587e9i − 1.64767i −0.566833 0.823833i \(-0.691832\pi\)
0.566833 0.823833i \(-0.308168\pi\)
\(212\) 0 0
\(213\) −9.15664e8 −0.444854
\(214\) 0 0
\(215\) − 1.82622e9i − 0.854670i
\(216\) 0 0
\(217\) −1.25195e9 −0.564610
\(218\) 0 0
\(219\) 1.72941e9i 0.751834i
\(220\) 0 0
\(221\) −1.44036e9 −0.603814
\(222\) 0 0
\(223\) − 8.00369e8i − 0.323647i −0.986820 0.161823i \(-0.948263\pi\)
0.986820 0.161823i \(-0.0517375\pi\)
\(224\) 0 0
\(225\) 3.99959e8 0.156058
\(226\) 0 0
\(227\) 3.56089e9i 1.34108i 0.741872 + 0.670541i \(0.233938\pi\)
−0.741872 + 0.670541i \(0.766062\pi\)
\(228\) 0 0
\(229\) −1.50719e9 −0.548058 −0.274029 0.961721i \(-0.588356\pi\)
−0.274029 + 0.961721i \(0.588356\pi\)
\(230\) 0 0
\(231\) − 5.37614e8i − 0.188809i
\(232\) 0 0
\(233\) −3.73819e9 −1.26835 −0.634173 0.773191i \(-0.718659\pi\)
−0.634173 + 0.773191i \(0.718659\pi\)
\(234\) 0 0
\(235\) 7.17164e8i 0.235151i
\(236\) 0 0
\(237\) −2.87589e9 −0.911546
\(238\) 0 0
\(239\) 4.67896e9i 1.43403i 0.697058 + 0.717014i \(0.254492\pi\)
−0.697058 + 0.717014i \(0.745508\pi\)
\(240\) 0 0
\(241\) −4.10707e9 −1.21748 −0.608742 0.793368i \(-0.708326\pi\)
−0.608742 + 0.793368i \(0.708326\pi\)
\(242\) 0 0
\(243\) − 2.23677e8i − 0.0641500i
\(244\) 0 0
\(245\) −1.32612e9 −0.368060
\(246\) 0 0
\(247\) 2.23407e8i 0.0600217i
\(248\) 0 0
\(249\) 8.40602e8 0.218672
\(250\) 0 0
\(251\) 3.48115e9i 0.877057i 0.898717 + 0.438529i \(0.144500\pi\)
−0.898717 + 0.438529i \(0.855500\pi\)
\(252\) 0 0
\(253\) −2.28864e9 −0.558594
\(254\) 0 0
\(255\) − 1.99910e9i − 0.472796i
\(256\) 0 0
\(257\) 3.32719e9 0.762685 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(258\) 0 0
\(259\) − 2.25653e9i − 0.501468i
\(260\) 0 0
\(261\) −2.34175e7 −0.00504636
\(262\) 0 0
\(263\) 4.91281e9i 1.02685i 0.858135 + 0.513425i \(0.171623\pi\)
−0.858135 + 0.513425i \(0.828377\pi\)
\(264\) 0 0
\(265\) −3.49594e9 −0.708892
\(266\) 0 0
\(267\) − 3.29572e9i − 0.648493i
\(268\) 0 0
\(269\) −7.37829e9 −1.40912 −0.704558 0.709646i \(-0.748855\pi\)
−0.704558 + 0.709646i \(0.748855\pi\)
\(270\) 0 0
\(271\) 3.89984e9i 0.723052i 0.932362 + 0.361526i \(0.117744\pi\)
−0.932362 + 0.361526i \(0.882256\pi\)
\(272\) 0 0
\(273\) −1.21360e9 −0.218487
\(274\) 0 0
\(275\) − 1.24419e9i − 0.217549i
\(276\) 0 0
\(277\) −9.38474e9 −1.59405 −0.797027 0.603943i \(-0.793595\pi\)
−0.797027 + 0.603943i \(0.793595\pi\)
\(278\) 0 0
\(279\) 1.62036e9i 0.267420i
\(280\) 0 0
\(281\) 4.48974e9 0.720106 0.360053 0.932932i \(-0.382759\pi\)
0.360053 + 0.932932i \(0.382759\pi\)
\(282\) 0 0
\(283\) − 5.99019e9i − 0.933888i −0.884287 0.466944i \(-0.845355\pi\)
0.884287 0.466944i \(-0.154645\pi\)
\(284\) 0 0
\(285\) −3.10069e8 −0.0469980
\(286\) 0 0
\(287\) 5.36937e9i 0.791399i
\(288\) 0 0
\(289\) 1.82037e9 0.260956
\(290\) 0 0
\(291\) − 4.89987e9i − 0.683302i
\(292\) 0 0
\(293\) −1.09529e9 −0.148613 −0.0743066 0.997235i \(-0.523674\pi\)
−0.0743066 + 0.997235i \(0.523674\pi\)
\(294\) 0 0
\(295\) − 6.98267e9i − 0.922006i
\(296\) 0 0
\(297\) −6.95815e8 −0.0894268
\(298\) 0 0
\(299\) 5.16636e9i 0.646398i
\(300\) 0 0
\(301\) 6.77038e9 0.824797
\(302\) 0 0
\(303\) − 2.62955e9i − 0.311969i
\(304\) 0 0
\(305\) −8.83467e9 −1.02092
\(306\) 0 0
\(307\) − 1.16757e10i − 1.31440i −0.753714 0.657202i \(-0.771740\pi\)
0.753714 0.657202i \(-0.228260\pi\)
\(308\) 0 0
\(309\) 5.71033e9 0.626365
\(310\) 0 0
\(311\) 5.41321e9i 0.578647i 0.957231 + 0.289323i \(0.0934303\pi\)
−0.957231 + 0.289323i \(0.906570\pi\)
\(312\) 0 0
\(313\) 1.50436e10 1.56738 0.783691 0.621151i \(-0.213335\pi\)
0.783691 + 0.621151i \(0.213335\pi\)
\(314\) 0 0
\(315\) − 1.68438e9i − 0.171079i
\(316\) 0 0
\(317\) −1.36257e9 −0.134934 −0.0674672 0.997721i \(-0.521492\pi\)
−0.0674672 + 0.997721i \(0.521492\pi\)
\(318\) 0 0
\(319\) 7.28470e7i 0.00703475i
\(320\) 0 0
\(321\) 1.07011e8 0.0100788
\(322\) 0 0
\(323\) − 1.36432e9i − 0.125344i
\(324\) 0 0
\(325\) −2.80863e9 −0.251745
\(326\) 0 0
\(327\) − 7.03570e9i − 0.615342i
\(328\) 0 0
\(329\) −2.65876e9 −0.226932
\(330\) 0 0
\(331\) 1.66087e10i 1.38364i 0.722069 + 0.691821i \(0.243191\pi\)
−0.722069 + 0.691821i \(0.756809\pi\)
\(332\) 0 0
\(333\) −2.92055e9 −0.237513
\(334\) 0 0
\(335\) 6.24720e9i 0.496028i
\(336\) 0 0
\(337\) −1.92166e9 −0.148990 −0.0744948 0.997221i \(-0.523734\pi\)
−0.0744948 + 0.997221i \(0.523734\pi\)
\(338\) 0 0
\(339\) − 1.20356e10i − 0.911313i
\(340\) 0 0
\(341\) 5.04060e9 0.372790
\(342\) 0 0
\(343\) − 1.46575e10i − 1.05897i
\(344\) 0 0
\(345\) −7.17046e9 −0.506140
\(346\) 0 0
\(347\) 1.88173e10i 1.29789i 0.760833 + 0.648947i \(0.224791\pi\)
−0.760833 + 0.648947i \(0.775209\pi\)
\(348\) 0 0
\(349\) −1.56042e10 −1.05182 −0.525910 0.850540i \(-0.676275\pi\)
−0.525910 + 0.850540i \(0.676275\pi\)
\(350\) 0 0
\(351\) 1.57072e9i 0.103484i
\(352\) 0 0
\(353\) −6.78029e9 −0.436666 −0.218333 0.975874i \(-0.570062\pi\)
−0.218333 + 0.975874i \(0.570062\pi\)
\(354\) 0 0
\(355\) 8.92435e9i 0.561906i
\(356\) 0 0
\(357\) 7.41133e9 0.456271
\(358\) 0 0
\(359\) 1.33198e9i 0.0801900i 0.999196 + 0.0400950i \(0.0127661\pi\)
−0.999196 + 0.0400950i \(0.987234\pi\)
\(360\) 0 0
\(361\) 1.67720e10 0.987540
\(362\) 0 0
\(363\) − 7.86003e9i − 0.452687i
\(364\) 0 0
\(365\) 1.68554e10 0.949658
\(366\) 0 0
\(367\) − 1.87117e10i − 1.03145i −0.856754 0.515725i \(-0.827523\pi\)
0.856754 0.515725i \(-0.172477\pi\)
\(368\) 0 0
\(369\) 6.94938e9 0.374835
\(370\) 0 0
\(371\) − 1.29606e10i − 0.684115i
\(372\) 0 0
\(373\) 1.69388e10 0.875081 0.437541 0.899199i \(-0.355850\pi\)
0.437541 + 0.899199i \(0.355850\pi\)
\(374\) 0 0
\(375\) − 1.22244e10i − 0.618161i
\(376\) 0 0
\(377\) 1.64444e8 0.00814053
\(378\) 0 0
\(379\) 3.28410e10i 1.59169i 0.605498 + 0.795847i \(0.292974\pi\)
−0.605498 + 0.795847i \(0.707026\pi\)
\(380\) 0 0
\(381\) 1.49988e10 0.711799
\(382\) 0 0
\(383\) 3.74844e10i 1.74203i 0.491255 + 0.871016i \(0.336538\pi\)
−0.491255 + 0.871016i \(0.663462\pi\)
\(384\) 0 0
\(385\) −5.23976e9 −0.238489
\(386\) 0 0
\(387\) − 8.76266e9i − 0.390654i
\(388\) 0 0
\(389\) 4.42667e10 1.93321 0.966604 0.256274i \(-0.0824948\pi\)
0.966604 + 0.256274i \(0.0824948\pi\)
\(390\) 0 0
\(391\) − 3.15503e10i − 1.34988i
\(392\) 0 0
\(393\) −7.60442e9 −0.318783
\(394\) 0 0
\(395\) 2.80293e10i 1.15139i
\(396\) 0 0
\(397\) 2.16236e10 0.870495 0.435248 0.900311i \(-0.356661\pi\)
0.435248 + 0.900311i \(0.356661\pi\)
\(398\) 0 0
\(399\) − 1.14953e9i − 0.0453553i
\(400\) 0 0
\(401\) 5.59819e9 0.216506 0.108253 0.994123i \(-0.465474\pi\)
0.108253 + 0.994123i \(0.465474\pi\)
\(402\) 0 0
\(403\) − 1.13786e10i − 0.431388i
\(404\) 0 0
\(405\) −2.18003e9 −0.0810293
\(406\) 0 0
\(407\) 9.08524e9i 0.331100i
\(408\) 0 0
\(409\) 2.03559e10 0.727440 0.363720 0.931508i \(-0.381506\pi\)
0.363720 + 0.931508i \(0.381506\pi\)
\(410\) 0 0
\(411\) − 1.74529e10i − 0.611645i
\(412\) 0 0
\(413\) 2.58871e10 0.889780
\(414\) 0 0
\(415\) − 8.19277e9i − 0.276209i
\(416\) 0 0
\(417\) −1.36244e10 −0.450580
\(418\) 0 0
\(419\) 4.45043e10i 1.44393i 0.691930 + 0.721965i \(0.256761\pi\)
−0.691930 + 0.721965i \(0.743239\pi\)
\(420\) 0 0
\(421\) −6.02330e9 −0.191737 −0.0958685 0.995394i \(-0.530563\pi\)
−0.0958685 + 0.995394i \(0.530563\pi\)
\(422\) 0 0
\(423\) 3.44114e9i 0.107483i
\(424\) 0 0
\(425\) 1.71519e10 0.525723
\(426\) 0 0
\(427\) − 3.27530e10i − 0.985235i
\(428\) 0 0
\(429\) 4.88620e9 0.144259
\(430\) 0 0
\(431\) 3.56553e10i 1.03327i 0.856205 + 0.516637i \(0.172816\pi\)
−0.856205 + 0.516637i \(0.827184\pi\)
\(432\) 0 0
\(433\) 4.62137e10 1.31468 0.657338 0.753596i \(-0.271682\pi\)
0.657338 + 0.753596i \(0.271682\pi\)
\(434\) 0 0
\(435\) 2.28234e8i 0.00637417i
\(436\) 0 0
\(437\) −4.89358e9 −0.134184
\(438\) 0 0
\(439\) − 3.76751e10i − 1.01437i −0.861837 0.507185i \(-0.830686\pi\)
0.861837 0.507185i \(-0.169314\pi\)
\(440\) 0 0
\(441\) −6.36308e9 −0.168234
\(442\) 0 0
\(443\) 4.34035e10i 1.12696i 0.826128 + 0.563482i \(0.190539\pi\)
−0.826128 + 0.563482i \(0.809461\pi\)
\(444\) 0 0
\(445\) −3.21211e10 −0.819126
\(446\) 0 0
\(447\) 2.80925e10i 0.703655i
\(448\) 0 0
\(449\) 4.97918e10 1.22510 0.612551 0.790431i \(-0.290143\pi\)
0.612551 + 0.790431i \(0.290143\pi\)
\(450\) 0 0
\(451\) − 2.16181e10i − 0.522530i
\(452\) 0 0
\(453\) 2.59073e10 0.615219
\(454\) 0 0
\(455\) 1.18282e10i 0.275976i
\(456\) 0 0
\(457\) −4.97522e9 −0.114064 −0.0570319 0.998372i \(-0.518164\pi\)
−0.0570319 + 0.998372i \(0.518164\pi\)
\(458\) 0 0
\(459\) − 9.59221e9i − 0.216107i
\(460\) 0 0
\(461\) −2.43202e10 −0.538472 −0.269236 0.963074i \(-0.586771\pi\)
−0.269236 + 0.963074i \(0.586771\pi\)
\(462\) 0 0
\(463\) 4.95954e10i 1.07924i 0.841909 + 0.539619i \(0.181432\pi\)
−0.841909 + 0.539619i \(0.818568\pi\)
\(464\) 0 0
\(465\) 1.57925e10 0.337784
\(466\) 0 0
\(467\) − 8.32165e10i − 1.74961i −0.484474 0.874806i \(-0.660989\pi\)
0.484474 0.874806i \(-0.339011\pi\)
\(468\) 0 0
\(469\) −2.31604e10 −0.478691
\(470\) 0 0
\(471\) − 4.59960e10i − 0.934623i
\(472\) 0 0
\(473\) −2.72589e10 −0.544582
\(474\) 0 0
\(475\) − 2.66034e9i − 0.0522591i
\(476\) 0 0
\(477\) −1.67744e10 −0.324022
\(478\) 0 0
\(479\) 4.26328e10i 0.809844i 0.914351 + 0.404922i \(0.132701\pi\)
−0.914351 + 0.404922i \(0.867299\pi\)
\(480\) 0 0
\(481\) 2.05089e10 0.383144
\(482\) 0 0
\(483\) − 2.65833e10i − 0.488449i
\(484\) 0 0
\(485\) −4.77557e10 −0.863094
\(486\) 0 0
\(487\) − 6.66609e10i − 1.18510i −0.805533 0.592550i \(-0.798121\pi\)
0.805533 0.592550i \(-0.201879\pi\)
\(488\) 0 0
\(489\) −3.27938e8 −0.00573530
\(490\) 0 0
\(491\) 4.56152e10i 0.784845i 0.919785 + 0.392422i \(0.128363\pi\)
−0.919785 + 0.392422i \(0.871637\pi\)
\(492\) 0 0
\(493\) −1.00424e9 −0.0170000
\(494\) 0 0
\(495\) 6.78163e9i 0.112957i
\(496\) 0 0
\(497\) −3.30855e10 −0.542266
\(498\) 0 0
\(499\) − 4.30599e10i − 0.694497i −0.937773 0.347249i \(-0.887116\pi\)
0.937773 0.347249i \(-0.112884\pi\)
\(500\) 0 0
\(501\) −5.02249e10 −0.797201
\(502\) 0 0
\(503\) − 3.00633e10i − 0.469639i −0.972039 0.234819i \(-0.924550\pi\)
0.972039 0.234819i \(-0.0754499\pi\)
\(504\) 0 0
\(505\) −2.56285e10 −0.394055
\(506\) 0 0
\(507\) 2.71179e10i 0.410416i
\(508\) 0 0
\(509\) −4.28086e10 −0.637763 −0.318882 0.947795i \(-0.603307\pi\)
−0.318882 + 0.947795i \(0.603307\pi\)
\(510\) 0 0
\(511\) 6.24884e10i 0.916465i
\(512\) 0 0
\(513\) −1.48779e9 −0.0214819
\(514\) 0 0
\(515\) − 5.56547e10i − 0.791175i
\(516\) 0 0
\(517\) 1.07047e10 0.149835
\(518\) 0 0
\(519\) 1.58288e9i 0.0218162i
\(520\) 0 0
\(521\) 2.00634e10 0.272304 0.136152 0.990688i \(-0.456526\pi\)
0.136152 + 0.990688i \(0.456526\pi\)
\(522\) 0 0
\(523\) − 4.87294e10i − 0.651304i −0.945490 0.325652i \(-0.894416\pi\)
0.945490 0.325652i \(-0.105584\pi\)
\(524\) 0 0
\(525\) 1.44517e10 0.190231
\(526\) 0 0
\(527\) 6.94876e10i 0.900876i
\(528\) 0 0
\(529\) −3.48549e10 −0.445083
\(530\) 0 0
\(531\) − 3.35047e10i − 0.421432i
\(532\) 0 0
\(533\) −4.88005e10 −0.604666
\(534\) 0 0
\(535\) − 1.04296e9i − 0.0127308i
\(536\) 0 0
\(537\) 7.26238e10 0.873337
\(538\) 0 0
\(539\) 1.97942e10i 0.234522i
\(540\) 0 0
\(541\) 6.88063e10 0.803229 0.401614 0.915809i \(-0.368449\pi\)
0.401614 + 0.915809i \(0.368449\pi\)
\(542\) 0 0
\(543\) − 5.94313e10i − 0.683621i
\(544\) 0 0
\(545\) −6.85722e10 −0.777252
\(546\) 0 0
\(547\) − 8.25362e10i − 0.921924i −0.887420 0.460962i \(-0.847504\pi\)
0.887420 0.460962i \(-0.152496\pi\)
\(548\) 0 0
\(549\) −4.23910e10 −0.466643
\(550\) 0 0
\(551\) 1.55762e8i 0.00168987i
\(552\) 0 0
\(553\) −1.03914e11 −1.11115
\(554\) 0 0
\(555\) 2.84646e10i 0.300008i
\(556\) 0 0
\(557\) −3.74890e9 −0.0389478 −0.0194739 0.999810i \(-0.506199\pi\)
−0.0194739 + 0.999810i \(0.506199\pi\)
\(558\) 0 0
\(559\) 6.15338e10i 0.630183i
\(560\) 0 0
\(561\) −2.98394e10 −0.301258
\(562\) 0 0
\(563\) 5.87145e10i 0.584402i 0.956357 + 0.292201i \(0.0943876\pi\)
−0.956357 + 0.292201i \(0.905612\pi\)
\(564\) 0 0
\(565\) −1.17302e11 −1.15110
\(566\) 0 0
\(567\) − 8.08209e9i − 0.0781972i
\(568\) 0 0
\(569\) −5.06599e10 −0.483299 −0.241649 0.970364i \(-0.577688\pi\)
−0.241649 + 0.970364i \(0.577688\pi\)
\(570\) 0 0
\(571\) 2.04586e11i 1.92456i 0.272067 + 0.962278i \(0.412293\pi\)
−0.272067 + 0.962278i \(0.587707\pi\)
\(572\) 0 0
\(573\) 1.82613e10 0.169400
\(574\) 0 0
\(575\) − 6.15212e10i − 0.562799i
\(576\) 0 0
\(577\) −5.83766e10 −0.526667 −0.263333 0.964705i \(-0.584822\pi\)
−0.263333 + 0.964705i \(0.584822\pi\)
\(578\) 0 0
\(579\) 8.42456e10i 0.749606i
\(580\) 0 0
\(581\) 3.03733e10 0.266555
\(582\) 0 0
\(583\) 5.21818e10i 0.451695i
\(584\) 0 0
\(585\) 1.53088e10 0.130712
\(586\) 0 0
\(587\) 1.89365e11i 1.59495i 0.603349 + 0.797477i \(0.293833\pi\)
−0.603349 + 0.797477i \(0.706167\pi\)
\(588\) 0 0
\(589\) 1.07778e10 0.0895508
\(590\) 0 0
\(591\) − 2.79079e10i − 0.228758i
\(592\) 0 0
\(593\) −1.98001e10 −0.160121 −0.0800605 0.996790i \(-0.525511\pi\)
−0.0800605 + 0.996790i \(0.525511\pi\)
\(594\) 0 0
\(595\) − 7.22331e10i − 0.576327i
\(596\) 0 0
\(597\) −2.16729e10 −0.170616
\(598\) 0 0
\(599\) − 2.18765e11i − 1.69930i −0.527343 0.849652i \(-0.676812\pi\)
0.527343 0.849652i \(-0.323188\pi\)
\(600\) 0 0
\(601\) −2.58060e11 −1.97798 −0.988992 0.147972i \(-0.952725\pi\)
−0.988992 + 0.147972i \(0.952725\pi\)
\(602\) 0 0
\(603\) 2.99757e10i 0.226725i
\(604\) 0 0
\(605\) −7.66064e10 −0.571799
\(606\) 0 0
\(607\) 8.74343e10i 0.644061i 0.946729 + 0.322030i \(0.104365\pi\)
−0.946729 + 0.322030i \(0.895635\pi\)
\(608\) 0 0
\(609\) −8.46138e8 −0.00615138
\(610\) 0 0
\(611\) − 2.41646e10i − 0.173387i
\(612\) 0 0
\(613\) 1.79871e11 1.27385 0.636925 0.770926i \(-0.280206\pi\)
0.636925 + 0.770926i \(0.280206\pi\)
\(614\) 0 0
\(615\) − 6.77309e10i − 0.473463i
\(616\) 0 0
\(617\) 1.01331e11 0.699202 0.349601 0.936899i \(-0.386317\pi\)
0.349601 + 0.936899i \(0.386317\pi\)
\(618\) 0 0
\(619\) 1.04565e11i 0.712234i 0.934441 + 0.356117i \(0.115900\pi\)
−0.934441 + 0.356117i \(0.884100\pi\)
\(620\) 0 0
\(621\) −3.44057e10 −0.231347
\(622\) 0 0
\(623\) − 1.19084e11i − 0.790496i
\(624\) 0 0
\(625\) −4.77050e10 −0.312639
\(626\) 0 0
\(627\) 4.62822e9i 0.0299464i
\(628\) 0 0
\(629\) −1.25245e11 −0.800128
\(630\) 0 0
\(631\) 1.38666e11i 0.874687i 0.899295 + 0.437344i \(0.144081\pi\)
−0.899295 + 0.437344i \(0.855919\pi\)
\(632\) 0 0
\(633\) −1.52730e11 −0.951280
\(634\) 0 0
\(635\) − 1.46183e11i − 0.899089i
\(636\) 0 0
\(637\) 4.46833e10 0.271386
\(638\) 0 0
\(639\) 4.28214e10i 0.256837i
\(640\) 0 0
\(641\) −6.29564e10 −0.372913 −0.186457 0.982463i \(-0.559700\pi\)
−0.186457 + 0.982463i \(0.559700\pi\)
\(642\) 0 0
\(643\) − 1.36211e11i − 0.796834i −0.917204 0.398417i \(-0.869560\pi\)
0.917204 0.398417i \(-0.130440\pi\)
\(644\) 0 0
\(645\) −8.54037e10 −0.493444
\(646\) 0 0
\(647\) − 2.53047e11i − 1.44405i −0.691864 0.722027i \(-0.743210\pi\)
0.691864 0.722027i \(-0.256790\pi\)
\(648\) 0 0
\(649\) −1.04226e11 −0.587487
\(650\) 0 0
\(651\) 5.85480e10i 0.325978i
\(652\) 0 0
\(653\) 8.08800e10 0.444824 0.222412 0.974953i \(-0.428607\pi\)
0.222412 + 0.974953i \(0.428607\pi\)
\(654\) 0 0
\(655\) 7.41151e10i 0.402663i
\(656\) 0 0
\(657\) 8.08765e10 0.434071
\(658\) 0 0
\(659\) − 8.97635e10i − 0.475946i −0.971272 0.237973i \(-0.923517\pi\)
0.971272 0.237973i \(-0.0764830\pi\)
\(660\) 0 0
\(661\) 1.12878e11 0.591296 0.295648 0.955297i \(-0.404464\pi\)
0.295648 + 0.955297i \(0.404464\pi\)
\(662\) 0 0
\(663\) 6.73592e10i 0.348612i
\(664\) 0 0
\(665\) −1.12037e10 −0.0572893
\(666\) 0 0
\(667\) 3.60204e9i 0.0181989i
\(668\) 0 0
\(669\) −3.74296e10 −0.186857
\(670\) 0 0
\(671\) 1.31870e11i 0.650513i
\(672\) 0 0
\(673\) 1.26871e10 0.0618444 0.0309222 0.999522i \(-0.490156\pi\)
0.0309222 + 0.999522i \(0.490156\pi\)
\(674\) 0 0
\(675\) − 1.87043e10i − 0.0901001i
\(676\) 0 0
\(677\) 1.20385e11 0.573082 0.286541 0.958068i \(-0.407495\pi\)
0.286541 + 0.958068i \(0.407495\pi\)
\(678\) 0 0
\(679\) − 1.77046e11i − 0.832927i
\(680\) 0 0
\(681\) 1.66526e11 0.774274
\(682\) 0 0
\(683\) − 8.09582e10i − 0.372030i −0.982547 0.186015i \(-0.940443\pi\)
0.982547 0.186015i \(-0.0595573\pi\)
\(684\) 0 0
\(685\) −1.70101e11 −0.772583
\(686\) 0 0
\(687\) 7.04844e10i 0.316422i
\(688\) 0 0
\(689\) 1.17795e11 0.522695
\(690\) 0 0
\(691\) − 5.81335e10i − 0.254985i −0.991840 0.127492i \(-0.959307\pi\)
0.991840 0.127492i \(-0.0406928\pi\)
\(692\) 0 0
\(693\) −2.51417e10 −0.109009
\(694\) 0 0
\(695\) 1.32787e11i 0.569138i
\(696\) 0 0
\(697\) 2.98018e11 1.26273
\(698\) 0 0
\(699\) 1.74818e11i 0.732280i
\(700\) 0 0
\(701\) 1.42599e10 0.0590531 0.0295266 0.999564i \(-0.490600\pi\)
0.0295266 + 0.999564i \(0.490600\pi\)
\(702\) 0 0
\(703\) 1.94261e10i 0.0795361i
\(704\) 0 0
\(705\) 3.35384e10 0.135765
\(706\) 0 0
\(707\) − 9.50131e10i − 0.380282i
\(708\) 0 0
\(709\) 3.59144e11 1.42130 0.710648 0.703548i \(-0.248402\pi\)
0.710648 + 0.703548i \(0.248402\pi\)
\(710\) 0 0
\(711\) 1.34492e11i 0.526281i
\(712\) 0 0
\(713\) 2.49241e11 0.964409
\(714\) 0 0
\(715\) − 4.76225e10i − 0.182217i
\(716\) 0 0
\(717\) 2.18813e11 0.827937
\(718\) 0 0
\(719\) − 1.87449e11i − 0.701404i −0.936487 0.350702i \(-0.885943\pi\)
0.936487 0.350702i \(-0.114057\pi\)
\(720\) 0 0
\(721\) 2.06330e11 0.763522
\(722\) 0 0
\(723\) 1.92068e11i 0.702915i
\(724\) 0 0
\(725\) −1.95821e9 −0.00708772
\(726\) 0 0
\(727\) 1.12950e11i 0.404341i 0.979350 + 0.202171i \(0.0647996\pi\)
−0.979350 + 0.202171i \(0.935200\pi\)
\(728\) 0 0
\(729\) −1.04604e10 −0.0370370
\(730\) 0 0
\(731\) − 3.75779e11i − 1.31602i
\(732\) 0 0
\(733\) −2.07098e11 −0.717397 −0.358699 0.933453i \(-0.616779\pi\)
−0.358699 + 0.933453i \(0.616779\pi\)
\(734\) 0 0
\(735\) 6.20166e10i 0.212500i
\(736\) 0 0
\(737\) 9.32483e10 0.316061
\(738\) 0 0
\(739\) 5.08051e10i 0.170345i 0.996366 + 0.0851726i \(0.0271442\pi\)
−0.996366 + 0.0851726i \(0.972856\pi\)
\(740\) 0 0
\(741\) 1.04477e10 0.0346535
\(742\) 0 0
\(743\) − 5.69279e10i − 0.186797i −0.995629 0.0933985i \(-0.970227\pi\)
0.995629 0.0933985i \(-0.0297731\pi\)
\(744\) 0 0
\(745\) 2.73798e11 0.888803
\(746\) 0 0
\(747\) − 3.93111e10i − 0.126250i
\(748\) 0 0
\(749\) 3.86661e9 0.0122858
\(750\) 0 0
\(751\) 2.75630e10i 0.0866497i 0.999061 + 0.0433248i \(0.0137950\pi\)
−0.999061 + 0.0433248i \(0.986205\pi\)
\(752\) 0 0
\(753\) 1.62797e11 0.506369
\(754\) 0 0
\(755\) − 2.52501e11i − 0.777097i
\(756\) 0 0
\(757\) −2.84625e11 −0.866743 −0.433371 0.901215i \(-0.642676\pi\)
−0.433371 + 0.901215i \(0.642676\pi\)
\(758\) 0 0
\(759\) 1.07029e11i 0.322504i
\(760\) 0 0
\(761\) 3.24323e11 0.967028 0.483514 0.875337i \(-0.339360\pi\)
0.483514 + 0.875337i \(0.339360\pi\)
\(762\) 0 0
\(763\) − 2.54219e11i − 0.750085i
\(764\) 0 0
\(765\) −9.34887e10 −0.272969
\(766\) 0 0
\(767\) 2.35279e11i 0.679833i
\(768\) 0 0
\(769\) 4.03018e11 1.15244 0.576221 0.817294i \(-0.304527\pi\)
0.576221 + 0.817294i \(0.304527\pi\)
\(770\) 0 0
\(771\) − 1.55597e11i − 0.440336i
\(772\) 0 0
\(773\) 3.04590e11 0.853097 0.426548 0.904465i \(-0.359729\pi\)
0.426548 + 0.904465i \(0.359729\pi\)
\(774\) 0 0
\(775\) 1.35497e11i 0.375597i
\(776\) 0 0
\(777\) −1.05528e11 −0.289522
\(778\) 0 0
\(779\) − 4.62239e10i − 0.125521i
\(780\) 0 0
\(781\) 1.33209e11 0.358037
\(782\) 0 0
\(783\) 1.09513e9i 0.00291352i
\(784\) 0 0
\(785\) −4.48292e11 −1.18054
\(786\) 0 0
\(787\) 5.85835e11i 1.52713i 0.645730 + 0.763565i \(0.276553\pi\)
−0.645730 + 0.763565i \(0.723447\pi\)
\(788\) 0 0
\(789\) 2.29749e11 0.592852
\(790\) 0 0
\(791\) − 4.34878e11i − 1.11087i
\(792\) 0 0
\(793\) 2.97682e11 0.752765
\(794\) 0 0
\(795\) 1.63489e11i 0.409279i
\(796\) 0 0
\(797\) −4.31674e11 −1.06985 −0.534925 0.844900i \(-0.679660\pi\)
−0.534925 + 0.844900i \(0.679660\pi\)
\(798\) 0 0
\(799\) 1.47570e11i 0.362086i
\(800\) 0 0
\(801\) −1.54126e11 −0.374408
\(802\) 0 0
\(803\) − 2.51590e11i − 0.605107i
\(804\) 0 0
\(805\) −2.59089e11 −0.616971
\(806\) 0 0
\(807\) 3.45049e11i 0.813554i
\(808\) 0 0
\(809\) −1.05205e10 −0.0245608 −0.0122804 0.999925i \(-0.503909\pi\)
−0.0122804 + 0.999925i \(0.503909\pi\)
\(810\) 0 0
\(811\) 4.65876e11i 1.07693i 0.842648 + 0.538464i \(0.180995\pi\)
−0.842648 + 0.538464i \(0.819005\pi\)
\(812\) 0 0
\(813\) 1.82377e11 0.417454
\(814\) 0 0
\(815\) 3.19618e9i 0.00724438i
\(816\) 0 0
\(817\) −5.82850e10 −0.130818
\(818\) 0 0
\(819\) 5.67546e10i 0.126144i
\(820\) 0 0
\(821\) 5.66593e11 1.24709 0.623546 0.781786i \(-0.285691\pi\)
0.623546 + 0.781786i \(0.285691\pi\)
\(822\) 0 0
\(823\) − 4.94664e10i − 0.107823i −0.998546 0.0539114i \(-0.982831\pi\)
0.998546 0.0539114i \(-0.0171689\pi\)
\(824\) 0 0
\(825\) −5.81851e10 −0.125602
\(826\) 0 0
\(827\) − 3.83529e11i − 0.819928i −0.912102 0.409964i \(-0.865541\pi\)
0.912102 0.409964i \(-0.134459\pi\)
\(828\) 0 0
\(829\) 5.61549e11 1.18897 0.594483 0.804108i \(-0.297357\pi\)
0.594483 + 0.804108i \(0.297357\pi\)
\(830\) 0 0
\(831\) 4.38881e11i 0.920328i
\(832\) 0 0
\(833\) −2.72875e11 −0.566740
\(834\) 0 0
\(835\) 4.89507e11i 1.00696i
\(836\) 0 0
\(837\) 7.57765e10 0.154395
\(838\) 0 0
\(839\) − 2.21865e11i − 0.447755i −0.974617 0.223877i \(-0.928129\pi\)
0.974617 0.223877i \(-0.0718715\pi\)
\(840\) 0 0
\(841\) −5.00132e11 −0.999771
\(842\) 0 0
\(843\) − 2.09965e11i − 0.415753i
\(844\) 0 0
\(845\) 2.64300e11 0.518405
\(846\) 0 0
\(847\) − 2.84005e11i − 0.551814i
\(848\) 0 0
\(849\) −2.80134e11 −0.539181
\(850\) 0 0
\(851\) 4.49235e11i 0.856556i
\(852\) 0 0
\(853\) −2.73278e11 −0.516189 −0.258095 0.966120i \(-0.583095\pi\)
−0.258095 + 0.966120i \(0.583095\pi\)
\(854\) 0 0
\(855\) 1.45005e10i 0.0271343i
\(856\) 0 0
\(857\) 2.78664e11 0.516605 0.258302 0.966064i \(-0.416837\pi\)
0.258302 + 0.966064i \(0.416837\pi\)
\(858\) 0 0
\(859\) 2.77264e11i 0.509239i 0.967041 + 0.254619i \(0.0819502\pi\)
−0.967041 + 0.254619i \(0.918050\pi\)
\(860\) 0 0
\(861\) 2.51101e11 0.456915
\(862\) 0 0
\(863\) − 8.42997e11i − 1.51979i −0.650047 0.759894i \(-0.725251\pi\)
0.650047 0.759894i \(-0.274749\pi\)
\(864\) 0 0
\(865\) 1.54272e10 0.0275565
\(866\) 0 0
\(867\) − 8.51302e10i − 0.150663i
\(868\) 0 0
\(869\) 4.18377e11 0.733650
\(870\) 0 0
\(871\) − 2.10498e11i − 0.365742i
\(872\) 0 0
\(873\) −2.29144e11 −0.394505
\(874\) 0 0
\(875\) − 4.41701e11i − 0.753522i
\(876\) 0 0
\(877\) 1.59168e11 0.269066 0.134533 0.990909i \(-0.457047\pi\)
0.134533 + 0.990909i \(0.457047\pi\)
\(878\) 0 0
\(879\) 5.12215e10i 0.0858019i
\(880\) 0 0
\(881\) 2.76086e11 0.458290 0.229145 0.973392i \(-0.426407\pi\)
0.229145 + 0.973392i \(0.426407\pi\)
\(882\) 0 0
\(883\) 7.01305e11i 1.15362i 0.816877 + 0.576811i \(0.195703\pi\)
−0.816877 + 0.576811i \(0.804297\pi\)
\(884\) 0 0
\(885\) −3.26547e11 −0.532320
\(886\) 0 0
\(887\) 7.11458e11i 1.14936i 0.818379 + 0.574678i \(0.194873\pi\)
−0.818379 + 0.574678i \(0.805127\pi\)
\(888\) 0 0
\(889\) 5.41949e11 0.867664
\(890\) 0 0
\(891\) 3.25400e10i 0.0516306i
\(892\) 0 0
\(893\) 2.28888e10 0.0359929
\(894\) 0 0
\(895\) − 7.07814e11i − 1.10313i
\(896\) 0 0
\(897\) 2.41607e11 0.373198
\(898\) 0 0
\(899\) − 7.93328e9i − 0.0121455i
\(900\) 0 0
\(901\) −7.19357e11 −1.09155
\(902\) 0 0
\(903\) − 3.16619e11i − 0.476197i
\(904\) 0 0
\(905\) −5.79236e11 −0.863497
\(906\) 0 0
\(907\) 5.42602e10i 0.0801774i 0.999196 + 0.0400887i \(0.0127641\pi\)
−0.999196 + 0.0400887i \(0.987236\pi\)
\(908\) 0 0
\(909\) −1.22972e11 −0.180115
\(910\) 0 0
\(911\) − 7.14023e11i − 1.03667i −0.855179 0.518333i \(-0.826553\pi\)
0.855179 0.518333i \(-0.173447\pi\)
\(912\) 0 0
\(913\) −1.22289e11 −0.175996
\(914\) 0 0
\(915\) 4.13157e11i 0.589427i
\(916\) 0 0
\(917\) −2.74769e11 −0.388589
\(918\) 0 0
\(919\) 1.58012e11i 0.221527i 0.993847 + 0.110764i \(0.0353296\pi\)
−0.993847 + 0.110764i \(0.964670\pi\)
\(920\) 0 0
\(921\) −5.46018e11 −0.758872
\(922\) 0 0
\(923\) − 3.00704e11i − 0.414316i
\(924\) 0 0
\(925\) −2.44221e11 −0.333593
\(926\) 0 0
\(927\) − 2.67046e11i − 0.361632i
\(928\) 0 0
\(929\) −1.26528e12 −1.69873 −0.849365 0.527805i \(-0.823015\pi\)
−0.849365 + 0.527805i \(0.823015\pi\)
\(930\) 0 0
\(931\) 4.23241e10i 0.0563364i
\(932\) 0 0
\(933\) 2.53151e11 0.334082
\(934\) 0 0
\(935\) 2.90824e11i 0.380526i
\(936\) 0 0
\(937\) −6.74824e11 −0.875452 −0.437726 0.899108i \(-0.644216\pi\)
−0.437726 + 0.899108i \(0.644216\pi\)
\(938\) 0 0
\(939\) − 7.03520e11i − 0.904928i
\(940\) 0 0
\(941\) −3.19026e11 −0.406881 −0.203441 0.979087i \(-0.565212\pi\)
−0.203441 + 0.979087i \(0.565212\pi\)
\(942\) 0 0
\(943\) − 1.06894e12i − 1.35179i
\(944\) 0 0
\(945\) −7.87706e10 −0.0987726
\(946\) 0 0
\(947\) − 6.73069e11i − 0.836873i −0.908246 0.418437i \(-0.862578\pi\)
0.908246 0.418437i \(-0.137422\pi\)
\(948\) 0 0
\(949\) −5.67937e11 −0.700222
\(950\) 0 0
\(951\) 6.37212e10i 0.0779044i
\(952\) 0 0
\(953\) −5.41655e11 −0.656676 −0.328338 0.944560i \(-0.606489\pi\)
−0.328338 + 0.944560i \(0.606489\pi\)
\(954\) 0 0
\(955\) − 1.77980e11i − 0.213973i
\(956\) 0 0
\(957\) 3.40672e9 0.00406152
\(958\) 0 0
\(959\) − 6.30621e11i − 0.745580i
\(960\) 0 0
\(961\) 3.03953e11 0.356380
\(962\) 0 0
\(963\) − 5.00441e9i − 0.00581900i
\(964\) 0 0
\(965\) 8.21084e11 0.946844
\(966\) 0 0
\(967\) 1.61996e12i 1.85267i 0.376697 + 0.926336i \(0.377060\pi\)
−0.376697 + 0.926336i \(0.622940\pi\)
\(968\) 0 0
\(969\) −6.38027e10 −0.0723676
\(970\) 0 0
\(971\) 5.73599e11i 0.645255i 0.946526 + 0.322627i \(0.104566\pi\)
−0.946526 + 0.322627i \(0.895434\pi\)
\(972\) 0 0
\(973\) −4.92286e11 −0.549245
\(974\) 0 0
\(975\) 1.31346e11i 0.145345i
\(976\) 0 0
\(977\) 1.32544e11 0.145473 0.0727366 0.997351i \(-0.476827\pi\)
0.0727366 + 0.997351i \(0.476827\pi\)
\(978\) 0 0
\(979\) 4.79453e11i 0.521934i
\(980\) 0 0
\(981\) −3.29027e11 −0.355268
\(982\) 0 0
\(983\) 1.02796e12i 1.10093i 0.834857 + 0.550466i \(0.185550\pi\)
−0.834857 + 0.550466i \(0.814450\pi\)
\(984\) 0 0
\(985\) −2.71999e11 −0.288950
\(986\) 0 0
\(987\) 1.24338e11i 0.131019i
\(988\) 0 0
\(989\) −1.34786e12 −1.40883
\(990\) 0 0
\(991\) 4.84738e11i 0.502588i 0.967911 + 0.251294i \(0.0808561\pi\)
−0.967911 + 0.251294i \(0.919144\pi\)
\(992\) 0 0
\(993\) 7.76712e11 0.798846
\(994\) 0 0
\(995\) 2.11231e11i 0.215509i
\(996\) 0 0
\(997\) −9.70160e11 −0.981890 −0.490945 0.871191i \(-0.663348\pi\)
−0.490945 + 0.871191i \(0.663348\pi\)
\(998\) 0 0
\(999\) 1.36581e11i 0.137128i
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 192.9.g.f.127.1 8
4.3 odd 2 inner 192.9.g.f.127.5 8
8.3 odd 2 96.9.g.a.31.4 8
8.5 even 2 96.9.g.a.31.8 yes 8
24.5 odd 2 288.9.g.f.127.1 8
24.11 even 2 288.9.g.f.127.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.9.g.a.31.4 8 8.3 odd 2
96.9.g.a.31.8 yes 8 8.5 even 2
192.9.g.f.127.1 8 1.1 even 1 trivial
192.9.g.f.127.5 8 4.3 odd 2 inner
288.9.g.f.127.1 8 24.5 odd 2
288.9.g.f.127.2 8 24.11 even 2