Properties

Label 225.8.a.b.1.1
Level $225$
Weight $8$
Character 225.1
Self dual yes
Analytic conductor $70.287$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,8,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

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: yes
Analytic conductor: \(70.2866307339\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-14.0000 q^{2} +68.0000 q^{4} +1644.00 q^{7} +840.000 q^{8} +O(q^{10})\) \(q-14.0000 q^{2} +68.0000 q^{4} +1644.00 q^{7} +840.000 q^{8} -172.000 q^{11} -3862.00 q^{13} -23016.0 q^{14} -20464.0 q^{16} -12254.0 q^{17} -25940.0 q^{19} +2408.00 q^{22} +12972.0 q^{23} +54068.0 q^{26} +111792. q^{28} +81610.0 q^{29} -156888. q^{31} +178976. q^{32} +171556. q^{34} -110126. q^{37} +363160. q^{38} -467882. q^{41} +499208. q^{43} -11696.0 q^{44} -181608. q^{46} -396884. q^{47} +1.87919e6 q^{49} -262616. q^{52} -1.28050e6 q^{53} +1.38096e6 q^{56} -1.14254e6 q^{58} +1.33742e6 q^{59} -923978. q^{61} +2.19643e6 q^{62} +113728. q^{64} +797304. q^{67} -833272. q^{68} -5.10339e6 q^{71} +4.26748e6 q^{73} +1.54176e6 q^{74} -1.76392e6 q^{76} -282768. q^{77} -960.000 q^{79} +6.55035e6 q^{82} +6.14083e6 q^{83} -6.98891e6 q^{86} -144480. q^{88} -2.01057e6 q^{89} -6.34913e6 q^{91} +882096. q^{92} +5.55638e6 q^{94} +4.88193e6 q^{97} -2.63087e7 q^{98} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.0000 −1.23744 −0.618718 0.785613i \(-0.712348\pi\)
−0.618718 + 0.785613i \(0.712348\pi\)
\(3\) 0 0
\(4\) 68.0000 0.531250
\(5\) 0 0
\(6\) 0 0
\(7\) 1644.00 1.81158 0.905792 0.423722i \(-0.139277\pi\)
0.905792 + 0.423722i \(0.139277\pi\)
\(8\) 840.000 0.580049
\(9\) 0 0
\(10\) 0 0
\(11\) −172.000 −0.0389631 −0.0194816 0.999810i \(-0.506202\pi\)
−0.0194816 + 0.999810i \(0.506202\pi\)
\(12\) 0 0
\(13\) −3862.00 −0.487540 −0.243770 0.969833i \(-0.578384\pi\)
−0.243770 + 0.969833i \(0.578384\pi\)
\(14\) −23016.0 −2.24172
\(15\) 0 0
\(16\) −20464.0 −1.24902
\(17\) −12254.0 −0.604932 −0.302466 0.953160i \(-0.597810\pi\)
−0.302466 + 0.953160i \(0.597810\pi\)
\(18\) 0 0
\(19\) −25940.0 −0.867626 −0.433813 0.901003i \(-0.642832\pi\)
−0.433813 + 0.901003i \(0.642832\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2408.00 0.0482144
\(23\) 12972.0 0.222310 0.111155 0.993803i \(-0.464545\pi\)
0.111155 + 0.993803i \(0.464545\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 54068.0 0.603300
\(27\) 0 0
\(28\) 111792. 0.962404
\(29\) 81610.0 0.621370 0.310685 0.950513i \(-0.399442\pi\)
0.310685 + 0.950513i \(0.399442\pi\)
\(30\) 0 0
\(31\) −156888. −0.945853 −0.472927 0.881102i \(-0.656802\pi\)
−0.472927 + 0.881102i \(0.656802\pi\)
\(32\) 178976. 0.965539
\(33\) 0 0
\(34\) 171556. 0.748565
\(35\) 0 0
\(36\) 0 0
\(37\) −110126. −0.357424 −0.178712 0.983901i \(-0.557193\pi\)
−0.178712 + 0.983901i \(0.557193\pi\)
\(38\) 363160. 1.07363
\(39\) 0 0
\(40\) 0 0
\(41\) −467882. −1.06021 −0.530106 0.847931i \(-0.677848\pi\)
−0.530106 + 0.847931i \(0.677848\pi\)
\(42\) 0 0
\(43\) 499208. 0.957507 0.478753 0.877949i \(-0.341089\pi\)
0.478753 + 0.877949i \(0.341089\pi\)
\(44\) −11696.0 −0.0206992
\(45\) 0 0
\(46\) −181608. −0.275095
\(47\) −396884. −0.557598 −0.278799 0.960349i \(-0.589936\pi\)
−0.278799 + 0.960349i \(0.589936\pi\)
\(48\) 0 0
\(49\) 1.87919e6 2.28184
\(50\) 0 0
\(51\) 0 0
\(52\) −262616. −0.259006
\(53\) −1.28050e6 −1.18144 −0.590722 0.806875i \(-0.701157\pi\)
−0.590722 + 0.806875i \(0.701157\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.38096e6 1.05081
\(57\) 0 0
\(58\) −1.14254e6 −0.768906
\(59\) 1.33742e6 0.847785 0.423893 0.905712i \(-0.360663\pi\)
0.423893 + 0.905712i \(0.360663\pi\)
\(60\) 0 0
\(61\) −923978. −0.521203 −0.260602 0.965446i \(-0.583921\pi\)
−0.260602 + 0.965446i \(0.583921\pi\)
\(62\) 2.19643e6 1.17043
\(63\) 0 0
\(64\) 113728. 0.0542297
\(65\) 0 0
\(66\) 0 0
\(67\) 797304. 0.323864 0.161932 0.986802i \(-0.448228\pi\)
0.161932 + 0.986802i \(0.448228\pi\)
\(68\) −833272. −0.321370
\(69\) 0 0
\(70\) 0 0
\(71\) −5.10339e6 −1.69221 −0.846106 0.533015i \(-0.821059\pi\)
−0.846106 + 0.533015i \(0.821059\pi\)
\(72\) 0 0
\(73\) 4.26748e6 1.28393 0.641965 0.766734i \(-0.278119\pi\)
0.641965 + 0.766734i \(0.278119\pi\)
\(74\) 1.54176e6 0.442290
\(75\) 0 0
\(76\) −1.76392e6 −0.460926
\(77\) −282768. −0.0705850
\(78\) 0 0
\(79\) −960.000 −0.000219067 0 −0.000109533 1.00000i \(-0.500035\pi\)
−0.000109533 1.00000i \(0.500035\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.55035e6 1.31195
\(83\) 6.14083e6 1.17884 0.589419 0.807828i \(-0.299357\pi\)
0.589419 + 0.807828i \(0.299357\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.98891e6 −1.18485
\(87\) 0 0
\(88\) −144480. −0.0226005
\(89\) −2.01057e6 −0.302311 −0.151156 0.988510i \(-0.548299\pi\)
−0.151156 + 0.988510i \(0.548299\pi\)
\(90\) 0 0
\(91\) −6.34913e6 −0.883221
\(92\) 882096. 0.118102
\(93\) 0 0
\(94\) 5.55638e6 0.689992
\(95\) 0 0
\(96\) 0 0
\(97\) 4.88193e6 0.543114 0.271557 0.962422i \(-0.412461\pi\)
0.271557 + 0.962422i \(0.412461\pi\)
\(98\) −2.63087e7 −2.82363
\(99\) 0 0
\(100\) 0 0
\(101\) −9.72670e6 −0.939379 −0.469689 0.882832i \(-0.655634\pi\)
−0.469689 + 0.882832i \(0.655634\pi\)
\(102\) 0 0
\(103\) −1.63151e7 −1.47115 −0.735577 0.677441i \(-0.763089\pi\)
−0.735577 + 0.677441i \(0.763089\pi\)
\(104\) −3.24408e6 −0.282797
\(105\) 0 0
\(106\) 1.79270e7 1.46196
\(107\) −4.08974e6 −0.322740 −0.161370 0.986894i \(-0.551591\pi\)
−0.161370 + 0.986894i \(0.551591\pi\)
\(108\) 0 0
\(109\) −2.68318e7 −1.98453 −0.992263 0.124158i \(-0.960377\pi\)
−0.992263 + 0.124158i \(0.960377\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.36428e7 −2.26271
\(113\) −1.74810e7 −1.13971 −0.569853 0.821747i \(-0.693000\pi\)
−0.569853 + 0.821747i \(0.693000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.54948e6 0.330103
\(117\) 0 0
\(118\) −1.87239e7 −1.04908
\(119\) −2.01456e7 −1.09589
\(120\) 0 0
\(121\) −1.94576e7 −0.998482
\(122\) 1.29357e7 0.644956
\(123\) 0 0
\(124\) −1.06684e7 −0.502485
\(125\) 0 0
\(126\) 0 0
\(127\) 1.25018e7 0.541575 0.270787 0.962639i \(-0.412716\pi\)
0.270787 + 0.962639i \(0.412716\pi\)
\(128\) −2.45011e7 −1.03264
\(129\) 0 0
\(130\) 0 0
\(131\) 7.75619e6 0.301439 0.150719 0.988577i \(-0.451841\pi\)
0.150719 + 0.988577i \(0.451841\pi\)
\(132\) 0 0
\(133\) −4.26454e7 −1.57178
\(134\) −1.11623e7 −0.400761
\(135\) 0 0
\(136\) −1.02934e7 −0.350890
\(137\) 3.61720e7 1.20185 0.600926 0.799305i \(-0.294799\pi\)
0.600926 + 0.799305i \(0.294799\pi\)
\(138\) 0 0
\(139\) 1.09092e7 0.344542 0.172271 0.985050i \(-0.444890\pi\)
0.172271 + 0.985050i \(0.444890\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.14475e7 2.09401
\(143\) 664264. 0.0189961
\(144\) 0 0
\(145\) 0 0
\(146\) −5.97447e7 −1.58878
\(147\) 0 0
\(148\) −7.48857e6 −0.189882
\(149\) 3.64580e7 0.902904 0.451452 0.892295i \(-0.350906\pi\)
0.451452 + 0.892295i \(0.350906\pi\)
\(150\) 0 0
\(151\) 7.18955e6 0.169935 0.0849674 0.996384i \(-0.472921\pi\)
0.0849674 + 0.996384i \(0.472921\pi\)
\(152\) −2.17896e7 −0.503265
\(153\) 0 0
\(154\) 3.95875e6 0.0873445
\(155\) 0 0
\(156\) 0 0
\(157\) −8.79932e7 −1.81468 −0.907341 0.420396i \(-0.861891\pi\)
−0.907341 + 0.420396i \(0.861891\pi\)
\(158\) 13440.0 0.000271081 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.13260e7 0.402734
\(162\) 0 0
\(163\) 5.48875e7 0.992697 0.496349 0.868123i \(-0.334674\pi\)
0.496349 + 0.868123i \(0.334674\pi\)
\(164\) −3.18160e7 −0.563238
\(165\) 0 0
\(166\) −8.59716e7 −1.45874
\(167\) 8.61460e6 0.143129 0.0715644 0.997436i \(-0.477201\pi\)
0.0715644 + 0.997436i \(0.477201\pi\)
\(168\) 0 0
\(169\) −4.78335e7 −0.762304
\(170\) 0 0
\(171\) 0 0
\(172\) 3.39461e7 0.508676
\(173\) −5.12524e7 −0.752580 −0.376290 0.926502i \(-0.622800\pi\)
−0.376290 + 0.926502i \(0.622800\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.51981e6 0.0486659
\(177\) 0 0
\(178\) 2.81480e7 0.374091
\(179\) 5.01627e7 0.653725 0.326862 0.945072i \(-0.394009\pi\)
0.326862 + 0.945072i \(0.394009\pi\)
\(180\) 0 0
\(181\) 6.90817e7 0.865940 0.432970 0.901408i \(-0.357466\pi\)
0.432970 + 0.901408i \(0.357466\pi\)
\(182\) 8.88878e7 1.09293
\(183\) 0 0
\(184\) 1.08965e7 0.128951
\(185\) 0 0
\(186\) 0 0
\(187\) 2.10769e6 0.0235701
\(188\) −2.69881e7 −0.296224
\(189\) 0 0
\(190\) 0 0
\(191\) −1.54745e8 −1.60695 −0.803473 0.595342i \(-0.797017\pi\)
−0.803473 + 0.595342i \(0.797017\pi\)
\(192\) 0 0
\(193\) 1.59406e7 0.159607 0.0798037 0.996811i \(-0.474571\pi\)
0.0798037 + 0.996811i \(0.474571\pi\)
\(194\) −6.83471e7 −0.672069
\(195\) 0 0
\(196\) 1.27785e8 1.21223
\(197\) −1.68188e8 −1.56734 −0.783670 0.621177i \(-0.786655\pi\)
−0.783670 + 0.621177i \(0.786655\pi\)
\(198\) 0 0
\(199\) 1.77773e8 1.59911 0.799556 0.600591i \(-0.205068\pi\)
0.799556 + 0.600591i \(0.205068\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.36174e8 1.16242
\(203\) 1.34167e8 1.12566
\(204\) 0 0
\(205\) 0 0
\(206\) 2.28411e8 1.82046
\(207\) 0 0
\(208\) 7.90320e7 0.608949
\(209\) 4.46168e6 0.0338054
\(210\) 0 0
\(211\) −1.61996e8 −1.18718 −0.593590 0.804767i \(-0.702290\pi\)
−0.593590 + 0.804767i \(0.702290\pi\)
\(212\) −8.70739e7 −0.627642
\(213\) 0 0
\(214\) 5.72564e7 0.399370
\(215\) 0 0
\(216\) 0 0
\(217\) −2.57924e8 −1.71349
\(218\) 3.75645e8 2.45572
\(219\) 0 0
\(220\) 0 0
\(221\) 4.73249e7 0.294929
\(222\) 0 0
\(223\) −1.75932e7 −0.106237 −0.0531187 0.998588i \(-0.516916\pi\)
−0.0531187 + 0.998588i \(0.516916\pi\)
\(224\) 2.94237e8 1.74916
\(225\) 0 0
\(226\) 2.44735e8 1.41031
\(227\) −2.03036e8 −1.15208 −0.576039 0.817422i \(-0.695403\pi\)
−0.576039 + 0.817422i \(0.695403\pi\)
\(228\) 0 0
\(229\) 1.59559e8 0.878005 0.439003 0.898486i \(-0.355332\pi\)
0.439003 + 0.898486i \(0.355332\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.85524e7 0.360425
\(233\) −1.94985e7 −0.100985 −0.0504924 0.998724i \(-0.516079\pi\)
−0.0504924 + 0.998724i \(0.516079\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 9.09446e7 0.450386
\(237\) 0 0
\(238\) 2.82038e8 1.35609
\(239\) 1.60220e8 0.759146 0.379573 0.925162i \(-0.376071\pi\)
0.379573 + 0.925162i \(0.376071\pi\)
\(240\) 0 0
\(241\) −3.78779e8 −1.74311 −0.871557 0.490294i \(-0.836890\pi\)
−0.871557 + 0.490294i \(0.836890\pi\)
\(242\) 2.72406e8 1.23556
\(243\) 0 0
\(244\) −6.28305e7 −0.276889
\(245\) 0 0
\(246\) 0 0
\(247\) 1.00180e8 0.423002
\(248\) −1.31786e8 −0.548641
\(249\) 0 0
\(250\) 0 0
\(251\) −1.61304e8 −0.643855 −0.321927 0.946764i \(-0.604331\pi\)
−0.321927 + 0.946764i \(0.604331\pi\)
\(252\) 0 0
\(253\) −2.23118e6 −0.00866191
\(254\) −1.75025e8 −0.670164
\(255\) 0 0
\(256\) 3.28458e8 1.22360
\(257\) 2.27387e8 0.835603 0.417801 0.908538i \(-0.362801\pi\)
0.417801 + 0.908538i \(0.362801\pi\)
\(258\) 0 0
\(259\) −1.81047e8 −0.647504
\(260\) 0 0
\(261\) 0 0
\(262\) −1.08587e8 −0.373011
\(263\) 4.57728e8 1.55154 0.775768 0.631018i \(-0.217363\pi\)
0.775768 + 0.631018i \(0.217363\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 5.97035e8 1.94498
\(267\) 0 0
\(268\) 5.42167e7 0.172053
\(269\) −4.67286e8 −1.46369 −0.731847 0.681469i \(-0.761341\pi\)
−0.731847 + 0.681469i \(0.761341\pi\)
\(270\) 0 0
\(271\) −4.45932e7 −0.136106 −0.0680528 0.997682i \(-0.521679\pi\)
−0.0680528 + 0.997682i \(0.521679\pi\)
\(272\) 2.50766e8 0.755574
\(273\) 0 0
\(274\) −5.06408e8 −1.48722
\(275\) 0 0
\(276\) 0 0
\(277\) −3.16657e8 −0.895179 −0.447590 0.894239i \(-0.647717\pi\)
−0.447590 + 0.894239i \(0.647717\pi\)
\(278\) −1.52729e8 −0.426349
\(279\) 0 0
\(280\) 0 0
\(281\) 2.25818e8 0.607136 0.303568 0.952810i \(-0.401822\pi\)
0.303568 + 0.952810i \(0.401822\pi\)
\(282\) 0 0
\(283\) −2.08210e7 −0.0546072 −0.0273036 0.999627i \(-0.508692\pi\)
−0.0273036 + 0.999627i \(0.508692\pi\)
\(284\) −3.47031e8 −0.898987
\(285\) 0 0
\(286\) −9.29970e6 −0.0235065
\(287\) −7.69198e8 −1.92066
\(288\) 0 0
\(289\) −2.60178e8 −0.634057
\(290\) 0 0
\(291\) 0 0
\(292\) 2.90189e8 0.682088
\(293\) −1.78825e8 −0.415329 −0.207665 0.978200i \(-0.566586\pi\)
−0.207665 + 0.978200i \(0.566586\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −9.25058e7 −0.207323
\(297\) 0 0
\(298\) −5.10413e8 −1.11729
\(299\) −5.00979e7 −0.108385
\(300\) 0 0
\(301\) 8.20698e8 1.73461
\(302\) −1.00654e8 −0.210284
\(303\) 0 0
\(304\) 5.30836e8 1.08368
\(305\) 0 0
\(306\) 0 0
\(307\) 8.55159e7 0.168680 0.0843398 0.996437i \(-0.473122\pi\)
0.0843398 + 0.996437i \(0.473122\pi\)
\(308\) −1.92282e7 −0.0374983
\(309\) 0 0
\(310\) 0 0
\(311\) 4.84706e8 0.913728 0.456864 0.889537i \(-0.348973\pi\)
0.456864 + 0.889537i \(0.348973\pi\)
\(312\) 0 0
\(313\) −5.70821e8 −1.05219 −0.526096 0.850425i \(-0.676345\pi\)
−0.526096 + 0.850425i \(0.676345\pi\)
\(314\) 1.23190e9 2.24555
\(315\) 0 0
\(316\) −65280.0 −0.000116379 0
\(317\) −5.50191e8 −0.970076 −0.485038 0.874493i \(-0.661194\pi\)
−0.485038 + 0.874493i \(0.661194\pi\)
\(318\) 0 0
\(319\) −1.40369e7 −0.0242105
\(320\) 0 0
\(321\) 0 0
\(322\) −2.98564e8 −0.498358
\(323\) 3.17869e8 0.524855
\(324\) 0 0
\(325\) 0 0
\(326\) −7.68425e8 −1.22840
\(327\) 0 0
\(328\) −3.93021e8 −0.614975
\(329\) −6.52477e8 −1.01014
\(330\) 0 0
\(331\) −9.39839e8 −1.42448 −0.712238 0.701938i \(-0.752319\pi\)
−0.712238 + 0.701938i \(0.752319\pi\)
\(332\) 4.17577e8 0.626257
\(333\) 0 0
\(334\) −1.20604e8 −0.177113
\(335\) 0 0
\(336\) 0 0
\(337\) −5.33632e8 −0.759516 −0.379758 0.925086i \(-0.623993\pi\)
−0.379758 + 0.925086i \(0.623993\pi\)
\(338\) 6.69669e8 0.943304
\(339\) 0 0
\(340\) 0 0
\(341\) 2.69847e7 0.0368534
\(342\) 0 0
\(343\) 1.73549e9 2.32216
\(344\) 4.19335e8 0.555401
\(345\) 0 0
\(346\) 7.17533e8 0.931270
\(347\) 1.07934e9 1.38677 0.693385 0.720567i \(-0.256118\pi\)
0.693385 + 0.720567i \(0.256118\pi\)
\(348\) 0 0
\(349\) −4.27217e8 −0.537972 −0.268986 0.963144i \(-0.586689\pi\)
−0.268986 + 0.963144i \(0.586689\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.07839e7 −0.0376204
\(353\) −1.48966e9 −1.80250 −0.901250 0.433299i \(-0.857350\pi\)
−0.901250 + 0.433299i \(0.857350\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.36719e8 −0.160603
\(357\) 0 0
\(358\) −7.02277e8 −0.808943
\(359\) −8.41275e8 −0.959638 −0.479819 0.877367i \(-0.659298\pi\)
−0.479819 + 0.877367i \(0.659298\pi\)
\(360\) 0 0
\(361\) −2.20988e8 −0.247226
\(362\) −9.67143e8 −1.07155
\(363\) 0 0
\(364\) −4.31741e8 −0.469211
\(365\) 0 0
\(366\) 0 0
\(367\) 7.50462e8 0.792496 0.396248 0.918143i \(-0.370312\pi\)
0.396248 + 0.918143i \(0.370312\pi\)
\(368\) −2.65459e8 −0.277671
\(369\) 0 0
\(370\) 0 0
\(371\) −2.10514e9 −2.14029
\(372\) 0 0
\(373\) −1.71074e8 −0.170688 −0.0853439 0.996352i \(-0.527199\pi\)
−0.0853439 + 0.996352i \(0.527199\pi\)
\(374\) −2.95076e7 −0.0291665
\(375\) 0 0
\(376\) −3.33383e8 −0.323434
\(377\) −3.15178e8 −0.302943
\(378\) 0 0
\(379\) 4.66239e7 0.0439918 0.0219959 0.999758i \(-0.492998\pi\)
0.0219959 + 0.999758i \(0.492998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.16644e9 1.98849
\(383\) −4.42266e8 −0.402242 −0.201121 0.979566i \(-0.564458\pi\)
−0.201121 + 0.979566i \(0.564458\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.23168e8 −0.197504
\(387\) 0 0
\(388\) 3.31972e8 0.288529
\(389\) 4.64033e8 0.399691 0.199846 0.979827i \(-0.435956\pi\)
0.199846 + 0.979827i \(0.435956\pi\)
\(390\) 0 0
\(391\) −1.58959e8 −0.134483
\(392\) 1.57852e9 1.32358
\(393\) 0 0
\(394\) 2.35463e9 1.93948
\(395\) 0 0
\(396\) 0 0
\(397\) 3.17792e8 0.254904 0.127452 0.991845i \(-0.459320\pi\)
0.127452 + 0.991845i \(0.459320\pi\)
\(398\) −2.48882e9 −1.97880
\(399\) 0 0
\(400\) 0 0
\(401\) 1.19563e9 0.925958 0.462979 0.886369i \(-0.346781\pi\)
0.462979 + 0.886369i \(0.346781\pi\)
\(402\) 0 0
\(403\) 6.05901e8 0.461142
\(404\) −6.61416e8 −0.499045
\(405\) 0 0
\(406\) −1.87834e9 −1.39294
\(407\) 1.89417e7 0.0139264
\(408\) 0 0
\(409\) 2.21305e9 1.59941 0.799704 0.600395i \(-0.204990\pi\)
0.799704 + 0.600395i \(0.204990\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.10942e9 −0.781551
\(413\) 2.19872e9 1.53583
\(414\) 0 0
\(415\) 0 0
\(416\) −6.91205e8 −0.470739
\(417\) 0 0
\(418\) −6.24635e7 −0.0418321
\(419\) 8.02299e8 0.532828 0.266414 0.963859i \(-0.414161\pi\)
0.266414 + 0.963859i \(0.414161\pi\)
\(420\) 0 0
\(421\) 3.44713e7 0.0225149 0.0112575 0.999937i \(-0.496417\pi\)
0.0112575 + 0.999937i \(0.496417\pi\)
\(422\) 2.26795e9 1.46906
\(423\) 0 0
\(424\) −1.07562e9 −0.685295
\(425\) 0 0
\(426\) 0 0
\(427\) −1.51902e9 −0.944204
\(428\) −2.78103e8 −0.171456
\(429\) 0 0
\(430\) 0 0
\(431\) −1.72692e9 −1.03897 −0.519485 0.854480i \(-0.673876\pi\)
−0.519485 + 0.854480i \(0.673876\pi\)
\(432\) 0 0
\(433\) −4.88308e8 −0.289059 −0.144529 0.989501i \(-0.546167\pi\)
−0.144529 + 0.989501i \(0.546167\pi\)
\(434\) 3.61093e9 2.12034
\(435\) 0 0
\(436\) −1.82456e9 −1.05428
\(437\) −3.36494e8 −0.192882
\(438\) 0 0
\(439\) −2.88640e9 −1.62828 −0.814142 0.580665i \(-0.802793\pi\)
−0.814142 + 0.580665i \(0.802793\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.62549e8 −0.364956
\(443\) 9.26583e8 0.506374 0.253187 0.967417i \(-0.418521\pi\)
0.253187 + 0.967417i \(0.418521\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 2.46304e8 0.131462
\(447\) 0 0
\(448\) 1.86969e8 0.0982418
\(449\) −1.35535e9 −0.706627 −0.353313 0.935505i \(-0.614945\pi\)
−0.353313 + 0.935505i \(0.614945\pi\)
\(450\) 0 0
\(451\) 8.04757e7 0.0413092
\(452\) −1.18871e9 −0.605469
\(453\) 0 0
\(454\) 2.84250e9 1.42563
\(455\) 0 0
\(456\) 0 0
\(457\) −4.63429e7 −0.0227131 −0.0113566 0.999936i \(-0.503615\pi\)
−0.0113566 + 0.999936i \(0.503615\pi\)
\(458\) −2.23383e9 −1.08648
\(459\) 0 0
\(460\) 0 0
\(461\) 1.52117e8 0.0723144 0.0361572 0.999346i \(-0.488488\pi\)
0.0361572 + 0.999346i \(0.488488\pi\)
\(462\) 0 0
\(463\) −1.63450e9 −0.765337 −0.382668 0.923886i \(-0.624995\pi\)
−0.382668 + 0.923886i \(0.624995\pi\)
\(464\) −1.67007e9 −0.776106
\(465\) 0 0
\(466\) 2.72979e8 0.124962
\(467\) −1.11380e9 −0.506057 −0.253029 0.967459i \(-0.581427\pi\)
−0.253029 + 0.967459i \(0.581427\pi\)
\(468\) 0 0
\(469\) 1.31077e9 0.586706
\(470\) 0 0
\(471\) 0 0
\(472\) 1.12343e9 0.491756
\(473\) −8.58638e7 −0.0373075
\(474\) 0 0
\(475\) 0 0
\(476\) −1.36990e9 −0.582189
\(477\) 0 0
\(478\) −2.24309e9 −0.939395
\(479\) −1.27745e9 −0.531091 −0.265546 0.964098i \(-0.585552\pi\)
−0.265546 + 0.964098i \(0.585552\pi\)
\(480\) 0 0
\(481\) 4.25307e8 0.174259
\(482\) 5.30290e9 2.15699
\(483\) 0 0
\(484\) −1.32312e9 −0.530443
\(485\) 0 0
\(486\) 0 0
\(487\) −9.79673e8 −0.384352 −0.192176 0.981360i \(-0.561555\pi\)
−0.192176 + 0.981360i \(0.561555\pi\)
\(488\) −7.76142e8 −0.302323
\(489\) 0 0
\(490\) 0 0
\(491\) 4.92125e9 1.87625 0.938124 0.346298i \(-0.112562\pi\)
0.938124 + 0.346298i \(0.112562\pi\)
\(492\) 0 0
\(493\) −1.00005e9 −0.375887
\(494\) −1.40252e9 −0.523439
\(495\) 0 0
\(496\) 3.21056e9 1.18139
\(497\) −8.38998e9 −3.06559
\(498\) 0 0
\(499\) −3.65786e9 −1.31788 −0.658940 0.752196i \(-0.728995\pi\)
−0.658940 + 0.752196i \(0.728995\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.25826e9 0.796729
\(503\) −3.88358e9 −1.36064 −0.680322 0.732914i \(-0.738160\pi\)
−0.680322 + 0.732914i \(0.738160\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.12366e7 0.0107186
\(507\) 0 0
\(508\) 8.50120e8 0.287711
\(509\) −3.90072e9 −1.31109 −0.655545 0.755156i \(-0.727561\pi\)
−0.655545 + 0.755156i \(0.727561\pi\)
\(510\) 0 0
\(511\) 7.01573e9 2.32595
\(512\) −1.46228e9 −0.481487
\(513\) 0 0
\(514\) −3.18342e9 −1.03401
\(515\) 0 0
\(516\) 0 0
\(517\) 6.82640e7 0.0217258
\(518\) 2.53466e9 0.801245
\(519\) 0 0
\(520\) 0 0
\(521\) −2.88399e9 −0.893431 −0.446716 0.894676i \(-0.647406\pi\)
−0.446716 + 0.894676i \(0.647406\pi\)
\(522\) 0 0
\(523\) −8.77188e8 −0.268125 −0.134062 0.990973i \(-0.542802\pi\)
−0.134062 + 0.990973i \(0.542802\pi\)
\(524\) 5.27421e8 0.160139
\(525\) 0 0
\(526\) −6.40819e9 −1.91993
\(527\) 1.92251e9 0.572177
\(528\) 0 0
\(529\) −3.23655e9 −0.950578
\(530\) 0 0
\(531\) 0 0
\(532\) −2.89988e9 −0.835007
\(533\) 1.80696e9 0.516896
\(534\) 0 0
\(535\) 0 0
\(536\) 6.69735e8 0.187857
\(537\) 0 0
\(538\) 6.54201e9 1.81123
\(539\) −3.23221e8 −0.0889077
\(540\) 0 0
\(541\) −6.53485e8 −0.177437 −0.0887187 0.996057i \(-0.528277\pi\)
−0.0887187 + 0.996057i \(0.528277\pi\)
\(542\) 6.24305e8 0.168422
\(543\) 0 0
\(544\) −2.19317e9 −0.584086
\(545\) 0 0
\(546\) 0 0
\(547\) 4.59299e9 1.19988 0.599942 0.800043i \(-0.295190\pi\)
0.599942 + 0.800043i \(0.295190\pi\)
\(548\) 2.45970e9 0.638484
\(549\) 0 0
\(550\) 0 0
\(551\) −2.11696e9 −0.539117
\(552\) 0 0
\(553\) −1.57824e6 −0.000396858 0
\(554\) 4.43320e9 1.10773
\(555\) 0 0
\(556\) 7.41827e8 0.183038
\(557\) 6.83164e9 1.67507 0.837533 0.546387i \(-0.183997\pi\)
0.837533 + 0.546387i \(0.183997\pi\)
\(558\) 0 0
\(559\) −1.92794e9 −0.466823
\(560\) 0 0
\(561\) 0 0
\(562\) −3.16145e9 −0.751292
\(563\) 3.42509e9 0.808897 0.404449 0.914561i \(-0.367463\pi\)
0.404449 + 0.914561i \(0.367463\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.91494e8 0.0675729
\(567\) 0 0
\(568\) −4.28685e9 −0.981565
\(569\) 7.50930e9 1.70886 0.854430 0.519566i \(-0.173906\pi\)
0.854430 + 0.519566i \(0.173906\pi\)
\(570\) 0 0
\(571\) −1.35841e8 −0.0305355 −0.0152677 0.999883i \(-0.504860\pi\)
−0.0152677 + 0.999883i \(0.504860\pi\)
\(572\) 4.51700e7 0.0100917
\(573\) 0 0
\(574\) 1.07688e10 2.37670
\(575\) 0 0
\(576\) 0 0
\(577\) −1.63775e9 −0.354922 −0.177461 0.984128i \(-0.556788\pi\)
−0.177461 + 0.984128i \(0.556788\pi\)
\(578\) 3.64249e9 0.784606
\(579\) 0 0
\(580\) 0 0
\(581\) 1.00955e10 2.13556
\(582\) 0 0
\(583\) 2.20246e8 0.0460328
\(584\) 3.58468e9 0.744742
\(585\) 0 0
\(586\) 2.50356e9 0.513944
\(587\) −5.97205e9 −1.21868 −0.609341 0.792909i \(-0.708566\pi\)
−0.609341 + 0.792909i \(0.708566\pi\)
\(588\) 0 0
\(589\) 4.06967e9 0.820647
\(590\) 0 0
\(591\) 0 0
\(592\) 2.25362e9 0.446431
\(593\) 8.31347e9 1.63716 0.818579 0.574394i \(-0.194762\pi\)
0.818579 + 0.574394i \(0.194762\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.47915e9 0.479668
\(597\) 0 0
\(598\) 7.01370e8 0.134120
\(599\) −9.78368e9 −1.85998 −0.929990 0.367585i \(-0.880185\pi\)
−0.929990 + 0.367585i \(0.880185\pi\)
\(600\) 0 0
\(601\) 5.40159e9 1.01499 0.507494 0.861655i \(-0.330572\pi\)
0.507494 + 0.861655i \(0.330572\pi\)
\(602\) −1.14898e10 −2.14646
\(603\) 0 0
\(604\) 4.88890e8 0.0902779
\(605\) 0 0
\(606\) 0 0
\(607\) 2.84439e9 0.516214 0.258107 0.966116i \(-0.416901\pi\)
0.258107 + 0.966116i \(0.416901\pi\)
\(608\) −4.64264e9 −0.837726
\(609\) 0 0
\(610\) 0 0
\(611\) 1.53277e9 0.271851
\(612\) 0 0
\(613\) 7.02106e9 1.23109 0.615547 0.788101i \(-0.288935\pi\)
0.615547 + 0.788101i \(0.288935\pi\)
\(614\) −1.19722e9 −0.208730
\(615\) 0 0
\(616\) −2.37525e8 −0.0409428
\(617\) 3.35166e9 0.574462 0.287231 0.957861i \(-0.407265\pi\)
0.287231 + 0.957861i \(0.407265\pi\)
\(618\) 0 0
\(619\) −3.92362e9 −0.664921 −0.332461 0.943117i \(-0.607879\pi\)
−0.332461 + 0.943117i \(0.607879\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.78588e9 −1.13068
\(623\) −3.30538e9 −0.547662
\(624\) 0 0
\(625\) 0 0
\(626\) 7.99150e9 1.30202
\(627\) 0 0
\(628\) −5.98354e9 −0.964049
\(629\) 1.34948e9 0.216217
\(630\) 0 0
\(631\) −6.81545e8 −0.107992 −0.0539960 0.998541i \(-0.517196\pi\)
−0.0539960 + 0.998541i \(0.517196\pi\)
\(632\) −806400. −0.000127069 0
\(633\) 0 0
\(634\) 7.70267e9 1.20041
\(635\) 0 0
\(636\) 0 0
\(637\) −7.25744e9 −1.11249
\(638\) 1.96517e8 0.0299590
\(639\) 0 0
\(640\) 0 0
\(641\) −9.65199e9 −1.44748 −0.723742 0.690071i \(-0.757579\pi\)
−0.723742 + 0.690071i \(0.757579\pi\)
\(642\) 0 0
\(643\) 5.07826e9 0.753315 0.376657 0.926353i \(-0.377073\pi\)
0.376657 + 0.926353i \(0.377073\pi\)
\(644\) 1.45017e9 0.213952
\(645\) 0 0
\(646\) −4.45016e9 −0.649474
\(647\) −2.08330e9 −0.302404 −0.151202 0.988503i \(-0.548314\pi\)
−0.151202 + 0.988503i \(0.548314\pi\)
\(648\) 0 0
\(649\) −2.30036e8 −0.0330324
\(650\) 0 0
\(651\) 0 0
\(652\) 3.73235e9 0.527370
\(653\) 6.84881e9 0.962540 0.481270 0.876572i \(-0.340176\pi\)
0.481270 + 0.876572i \(0.340176\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 9.57474e9 1.32423
\(657\) 0 0
\(658\) 9.13468e9 1.24998
\(659\) −6.99913e9 −0.952676 −0.476338 0.879262i \(-0.658036\pi\)
−0.476338 + 0.879262i \(0.658036\pi\)
\(660\) 0 0
\(661\) −1.99594e9 −0.268809 −0.134404 0.990927i \(-0.542912\pi\)
−0.134404 + 0.990927i \(0.542912\pi\)
\(662\) 1.31577e10 1.76270
\(663\) 0 0
\(664\) 5.15830e9 0.683783
\(665\) 0 0
\(666\) 0 0
\(667\) 1.05864e9 0.138137
\(668\) 5.85793e8 0.0760372
\(669\) 0 0
\(670\) 0 0
\(671\) 1.58924e8 0.0203077
\(672\) 0 0
\(673\) 1.17939e10 1.49144 0.745718 0.666261i \(-0.232106\pi\)
0.745718 + 0.666261i \(0.232106\pi\)
\(674\) 7.47085e9 0.939853
\(675\) 0 0
\(676\) −3.25268e9 −0.404974
\(677\) 7.80222e9 0.966402 0.483201 0.875509i \(-0.339474\pi\)
0.483201 + 0.875509i \(0.339474\pi\)
\(678\) 0 0
\(679\) 8.02590e9 0.983897
\(680\) 0 0
\(681\) 0 0
\(682\) −3.77786e8 −0.0456038
\(683\) 4.51153e9 0.541815 0.270908 0.962605i \(-0.412676\pi\)
0.270908 + 0.962605i \(0.412676\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.42968e10 −2.87353
\(687\) 0 0
\(688\) −1.02158e10 −1.19595
\(689\) 4.94528e9 0.576002
\(690\) 0 0
\(691\) −1.06331e10 −1.22599 −0.612997 0.790086i \(-0.710036\pi\)
−0.612997 + 0.790086i \(0.710036\pi\)
\(692\) −3.48516e9 −0.399808
\(693\) 0 0
\(694\) −1.51107e10 −1.71604
\(695\) 0 0
\(696\) 0 0
\(697\) 5.73343e9 0.641356
\(698\) 5.98104e9 0.665707
\(699\) 0 0
\(700\) 0 0
\(701\) 4.38514e9 0.480807 0.240403 0.970673i \(-0.422720\pi\)
0.240403 + 0.970673i \(0.422720\pi\)
\(702\) 0 0
\(703\) 2.85667e9 0.310110
\(704\) −1.95612e7 −0.00211296
\(705\) 0 0
\(706\) 2.08552e10 2.23048
\(707\) −1.59907e10 −1.70176
\(708\) 0 0
\(709\) −5.98805e9 −0.630992 −0.315496 0.948927i \(-0.602171\pi\)
−0.315496 + 0.948927i \(0.602171\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.68888e9 −0.175355
\(713\) −2.03515e9 −0.210273
\(714\) 0 0
\(715\) 0 0
\(716\) 3.41106e9 0.347291
\(717\) 0 0
\(718\) 1.17779e10 1.18749
\(719\) 1.17768e10 1.18161 0.590807 0.806813i \(-0.298809\pi\)
0.590807 + 0.806813i \(0.298809\pi\)
\(720\) 0 0
\(721\) −2.68219e10 −2.66512
\(722\) 3.09383e9 0.305926
\(723\) 0 0
\(724\) 4.69755e9 0.460031
\(725\) 0 0
\(726\) 0 0
\(727\) −8.41051e9 −0.811805 −0.405902 0.913916i \(-0.633043\pi\)
−0.405902 + 0.913916i \(0.633043\pi\)
\(728\) −5.33327e9 −0.512311
\(729\) 0 0
\(730\) 0 0
\(731\) −6.11729e9 −0.579227
\(732\) 0 0
\(733\) −1.44084e10 −1.35130 −0.675650 0.737223i \(-0.736137\pi\)
−0.675650 + 0.737223i \(0.736137\pi\)
\(734\) −1.05065e10 −0.980664
\(735\) 0 0
\(736\) 2.32168e9 0.214649
\(737\) −1.37136e8 −0.0126187
\(738\) 0 0
\(739\) 8.21708e9 0.748966 0.374483 0.927234i \(-0.377820\pi\)
0.374483 + 0.927234i \(0.377820\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.94719e10 2.64847
\(743\) −1.72531e10 −1.54314 −0.771570 0.636144i \(-0.780528\pi\)
−0.771570 + 0.636144i \(0.780528\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.39503e9 0.211215
\(747\) 0 0
\(748\) 1.43323e8 0.0125216
\(749\) −6.72354e9 −0.584671
\(750\) 0 0
\(751\) 1.58498e10 1.36548 0.682739 0.730662i \(-0.260789\pi\)
0.682739 + 0.730662i \(0.260789\pi\)
\(752\) 8.12183e9 0.696453
\(753\) 0 0
\(754\) 4.41249e9 0.374873
\(755\) 0 0
\(756\) 0 0
\(757\) −7.13856e9 −0.598102 −0.299051 0.954237i \(-0.596670\pi\)
−0.299051 + 0.954237i \(0.596670\pi\)
\(758\) −6.52735e8 −0.0544371
\(759\) 0 0
\(760\) 0 0
\(761\) 2.59993e9 0.213853 0.106926 0.994267i \(-0.465899\pi\)
0.106926 + 0.994267i \(0.465899\pi\)
\(762\) 0 0
\(763\) −4.41114e10 −3.59514
\(764\) −1.05227e10 −0.853690
\(765\) 0 0
\(766\) 6.19172e9 0.497749
\(767\) −5.16512e9 −0.413329
\(768\) 0 0
\(769\) −4.96477e9 −0.393692 −0.196846 0.980434i \(-0.563070\pi\)
−0.196846 + 0.980434i \(0.563070\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.08396e9 0.0847914
\(773\) −1.49681e10 −1.16557 −0.582786 0.812626i \(-0.698037\pi\)
−0.582786 + 0.812626i \(0.698037\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 4.10082e9 0.315032
\(777\) 0 0
\(778\) −6.49646e9 −0.494593
\(779\) 1.21369e10 0.919867
\(780\) 0 0
\(781\) 8.77783e8 0.0659339
\(782\) 2.22542e9 0.166414
\(783\) 0 0
\(784\) −3.84558e10 −2.85007
\(785\) 0 0
\(786\) 0 0
\(787\) 9.79990e9 0.716655 0.358328 0.933596i \(-0.383347\pi\)
0.358328 + 0.933596i \(0.383347\pi\)
\(788\) −1.14368e10 −0.832650
\(789\) 0 0
\(790\) 0 0
\(791\) −2.87388e10 −2.06467
\(792\) 0 0
\(793\) 3.56840e9 0.254108
\(794\) −4.44909e9 −0.315428
\(795\) 0 0
\(796\) 1.20885e10 0.849529
\(797\) 3.93169e9 0.275090 0.137545 0.990495i \(-0.456079\pi\)
0.137545 + 0.990495i \(0.456079\pi\)
\(798\) 0 0
\(799\) 4.86342e9 0.337309
\(800\) 0 0
\(801\) 0 0
\(802\) −1.67388e10 −1.14581
\(803\) −7.34006e8 −0.0500259
\(804\) 0 0
\(805\) 0 0
\(806\) −8.48262e9 −0.570634
\(807\) 0 0
\(808\) −8.17043e9 −0.544885
\(809\) −1.26324e10 −0.838816 −0.419408 0.907798i \(-0.637762\pi\)
−0.419408 + 0.907798i \(0.637762\pi\)
\(810\) 0 0
\(811\) 1.16653e10 0.767934 0.383967 0.923347i \(-0.374558\pi\)
0.383967 + 0.923347i \(0.374558\pi\)
\(812\) 9.12335e9 0.598009
\(813\) 0 0
\(814\) −2.65183e8 −0.0172330
\(815\) 0 0
\(816\) 0 0
\(817\) −1.29495e10 −0.830758
\(818\) −3.09827e10 −1.97917
\(819\) 0 0
\(820\) 0 0
\(821\) 8.17500e9 0.515569 0.257784 0.966202i \(-0.417008\pi\)
0.257784 + 0.966202i \(0.417008\pi\)
\(822\) 0 0
\(823\) 1.75211e10 1.09563 0.547813 0.836601i \(-0.315460\pi\)
0.547813 + 0.836601i \(0.315460\pi\)
\(824\) −1.37046e10 −0.853341
\(825\) 0 0
\(826\) −3.07821e10 −1.90050
\(827\) 1.22225e10 0.751437 0.375718 0.926734i \(-0.377396\pi\)
0.375718 + 0.926734i \(0.377396\pi\)
\(828\) 0 0
\(829\) −1.06634e10 −0.650063 −0.325032 0.945703i \(-0.605375\pi\)
−0.325032 + 0.945703i \(0.605375\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.39218e8 −0.0264392
\(833\) −2.30276e10 −1.38036
\(834\) 0 0
\(835\) 0 0
\(836\) 3.03394e8 0.0179591
\(837\) 0 0
\(838\) −1.12322e10 −0.659341
\(839\) 2.31400e9 0.135268 0.0676342 0.997710i \(-0.478455\pi\)
0.0676342 + 0.997710i \(0.478455\pi\)
\(840\) 0 0
\(841\) −1.05897e10 −0.613899
\(842\) −4.82599e8 −0.0278608
\(843\) 0 0
\(844\) −1.10158e10 −0.630690
\(845\) 0 0
\(846\) 0 0
\(847\) −3.19883e10 −1.80883
\(848\) 2.62041e10 1.47565
\(849\) 0 0
\(850\) 0 0
\(851\) −1.42855e9 −0.0794590
\(852\) 0 0
\(853\) 4.22377e9 0.233012 0.116506 0.993190i \(-0.462831\pi\)
0.116506 + 0.993190i \(0.462831\pi\)
\(854\) 2.12663e10 1.16839
\(855\) 0 0
\(856\) −3.43538e9 −0.187205
\(857\) −3.52104e9 −0.191090 −0.0955450 0.995425i \(-0.530459\pi\)
−0.0955450 + 0.995425i \(0.530459\pi\)
\(858\) 0 0
\(859\) 2.44930e10 1.31846 0.659229 0.751943i \(-0.270883\pi\)
0.659229 + 0.751943i \(0.270883\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.41769e10 1.28566
\(863\) −5.40573e9 −0.286297 −0.143148 0.989701i \(-0.545723\pi\)
−0.143148 + 0.989701i \(0.545723\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6.83631e9 0.357692
\(867\) 0 0
\(868\) −1.75388e10 −0.910293
\(869\) 165120. 8.53553e−6 0
\(870\) 0 0
\(871\) −3.07919e9 −0.157897
\(872\) −2.25387e10 −1.15112
\(873\) 0 0
\(874\) 4.71091e9 0.238679
\(875\) 0 0
\(876\) 0 0
\(877\) 2.89155e10 1.44755 0.723773 0.690039i \(-0.242406\pi\)
0.723773 + 0.690039i \(0.242406\pi\)
\(878\) 4.04096e10 2.01490
\(879\) 0 0
\(880\) 0 0
\(881\) −7.80643e8 −0.0384624 −0.0192312 0.999815i \(-0.506122\pi\)
−0.0192312 + 0.999815i \(0.506122\pi\)
\(882\) 0 0
\(883\) 1.36907e10 0.669211 0.334605 0.942358i \(-0.391397\pi\)
0.334605 + 0.942358i \(0.391397\pi\)
\(884\) 3.21810e9 0.156681
\(885\) 0 0
\(886\) −1.29722e10 −0.626606
\(887\) 3.58403e9 0.172441 0.0862203 0.996276i \(-0.472521\pi\)
0.0862203 + 0.996276i \(0.472521\pi\)
\(888\) 0 0
\(889\) 2.05529e10 0.981108
\(890\) 0 0
\(891\) 0 0
\(892\) −1.19634e9 −0.0564386
\(893\) 1.02952e10 0.483786
\(894\) 0 0
\(895\) 0 0
\(896\) −4.02798e10 −1.87072
\(897\) 0 0
\(898\) 1.89749e10 0.874406
\(899\) −1.28036e10 −0.587725
\(900\) 0 0
\(901\) 1.56912e10 0.714694
\(902\) −1.12666e9 −0.0511175
\(903\) 0 0
\(904\) −1.46841e10 −0.661085
\(905\) 0 0
\(906\) 0 0
\(907\) 3.01108e10 1.33998 0.669988 0.742372i \(-0.266299\pi\)
0.669988 + 0.742372i \(0.266299\pi\)
\(908\) −1.38064e10 −0.612042
\(909\) 0 0
\(910\) 0 0
\(911\) 2.48800e10 1.09027 0.545137 0.838347i \(-0.316477\pi\)
0.545137 + 0.838347i \(0.316477\pi\)
\(912\) 0 0
\(913\) −1.05622e9 −0.0459312
\(914\) 6.48801e8 0.0281060
\(915\) 0 0
\(916\) 1.08500e10 0.466440
\(917\) 1.27512e10 0.546082
\(918\) 0 0
\(919\) 1.08420e10 0.460794 0.230397 0.973097i \(-0.425997\pi\)
0.230397 + 0.973097i \(0.425997\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.12964e9 −0.0894846
\(923\) 1.97093e10 0.825021
\(924\) 0 0
\(925\) 0 0
\(926\) 2.28831e10 0.947056
\(927\) 0 0
\(928\) 1.46062e10 0.599957
\(929\) −1.07045e10 −0.438038 −0.219019 0.975721i \(-0.570286\pi\)
−0.219019 + 0.975721i \(0.570286\pi\)
\(930\) 0 0
\(931\) −4.87463e10 −1.97978
\(932\) −1.32590e9 −0.0536482
\(933\) 0 0
\(934\) 1.55933e10 0.626214
\(935\) 0 0
\(936\) 0 0
\(937\) −3.42787e10 −1.36124 −0.680621 0.732635i \(-0.738290\pi\)
−0.680621 + 0.732635i \(0.738290\pi\)
\(938\) −1.83507e10 −0.726012
\(939\) 0 0
\(940\) 0 0
\(941\) 3.73695e9 0.146202 0.0731010 0.997325i \(-0.476710\pi\)
0.0731010 + 0.997325i \(0.476710\pi\)
\(942\) 0 0
\(943\) −6.06937e9 −0.235696
\(944\) −2.73690e10 −1.05890
\(945\) 0 0
\(946\) 1.20209e9 0.0461657
\(947\) 3.32150e10 1.27089 0.635447 0.772145i \(-0.280816\pi\)
0.635447 + 0.772145i \(0.280816\pi\)
\(948\) 0 0
\(949\) −1.64810e10 −0.625968
\(950\) 0 0
\(951\) 0 0
\(952\) −1.69223e10 −0.635667
\(953\) 4.69895e10 1.75864 0.879318 0.476235i \(-0.157999\pi\)
0.879318 + 0.476235i \(0.157999\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.08950e10 0.403296
\(957\) 0 0
\(958\) 1.78843e10 0.657192
\(959\) 5.94668e10 2.17726
\(960\) 0 0
\(961\) −2.89877e9 −0.105361
\(962\) −5.95429e9 −0.215634
\(963\) 0 0
\(964\) −2.57570e10 −0.926030
\(965\) 0 0
\(966\) 0 0
\(967\) 1.42294e10 0.506050 0.253025 0.967460i \(-0.418575\pi\)
0.253025 + 0.967460i \(0.418575\pi\)
\(968\) −1.63444e10 −0.579168
\(969\) 0 0
\(970\) 0 0
\(971\) −5.45474e8 −0.0191208 −0.00956041 0.999954i \(-0.503043\pi\)
−0.00956041 + 0.999954i \(0.503043\pi\)
\(972\) 0 0
\(973\) 1.79348e10 0.624167
\(974\) 1.37154e10 0.475612
\(975\) 0 0
\(976\) 1.89083e10 0.650995
\(977\) −1.97916e10 −0.678968 −0.339484 0.940612i \(-0.610252\pi\)
−0.339484 + 0.940612i \(0.610252\pi\)
\(978\) 0 0
\(979\) 3.45818e8 0.0117790
\(980\) 0 0
\(981\) 0 0
\(982\) −6.88975e10 −2.32174
\(983\) −4.71503e10 −1.58324 −0.791620 0.611013i \(-0.790762\pi\)
−0.791620 + 0.611013i \(0.790762\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.40007e10 0.465136
\(987\) 0 0
\(988\) 6.81226e9 0.224720
\(989\) 6.47573e9 0.212864
\(990\) 0 0
\(991\) 3.87968e10 1.26631 0.633153 0.774027i \(-0.281761\pi\)
0.633153 + 0.774027i \(0.281761\pi\)
\(992\) −2.80792e10 −0.913258
\(993\) 0 0
\(994\) 1.17460e11 3.79347
\(995\) 0 0
\(996\) 0 0
\(997\) 5.66394e10 1.81003 0.905015 0.425380i \(-0.139860\pi\)
0.905015 + 0.425380i \(0.139860\pi\)
\(998\) 5.12101e10 1.63079
\(999\) 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 225.8.a.b.1.1 1
3.2 odd 2 25.8.a.a.1.1 1
5.2 odd 4 225.8.b.b.199.1 2
5.3 odd 4 225.8.b.b.199.2 2
5.4 even 2 45.8.a.f.1.1 1
12.11 even 2 400.8.a.e.1.1 1
15.2 even 4 25.8.b.a.24.2 2
15.8 even 4 25.8.b.a.24.1 2
15.14 odd 2 5.8.a.a.1.1 1
60.23 odd 4 400.8.c.e.49.1 2
60.47 odd 4 400.8.c.e.49.2 2
60.59 even 2 80.8.a.d.1.1 1
105.104 even 2 245.8.a.a.1.1 1
120.29 odd 2 320.8.a.h.1.1 1
120.59 even 2 320.8.a.a.1.1 1
165.164 even 2 605.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.8.a.a.1.1 1 15.14 odd 2
25.8.a.a.1.1 1 3.2 odd 2
25.8.b.a.24.1 2 15.8 even 4
25.8.b.a.24.2 2 15.2 even 4
45.8.a.f.1.1 1 5.4 even 2
80.8.a.d.1.1 1 60.59 even 2
225.8.a.b.1.1 1 1.1 even 1 trivial
225.8.b.b.199.1 2 5.2 odd 4
225.8.b.b.199.2 2 5.3 odd 4
245.8.a.a.1.1 1 105.104 even 2
320.8.a.a.1.1 1 120.59 even 2
320.8.a.h.1.1 1 120.29 odd 2
400.8.a.e.1.1 1 12.11 even 2
400.8.c.e.49.1 2 60.23 odd 4
400.8.c.e.49.2 2 60.47 odd 4
605.8.a.c.1.1 1 165.164 even 2