Properties

Label 27.8.a.e.1.1
Level $27$
Weight $8$
Character 27.1
Self dual yes
Analytic conductor $8.434$
Analytic rank $0$
Dimension $2$
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] = [27,8,Mod(1,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 27.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.59339 q^{2} -70.3405 q^{4} +65.8132 q^{5} -738.405 q^{7} +1506.08 q^{8} -499.745 q^{10} +4961.26 q^{11} +5967.24 q^{13} +5606.99 q^{14} -2432.64 q^{16} +36651.6 q^{17} +22378.9 q^{19} -4629.33 q^{20} -37672.8 q^{22} -51473.6 q^{23} -73793.6 q^{25} -45311.5 q^{26} +51939.7 q^{28} -68495.7 q^{29} +150655. q^{31} -174306. q^{32} -278310. q^{34} -48596.8 q^{35} +489027. q^{37} -169932. q^{38} +99119.7 q^{40} +590635. q^{41} -842643. q^{43} -348978. q^{44} +390859. q^{46} +1.22637e6 q^{47} -278301. q^{49} +560343. q^{50} -419738. q^{52} +958904. q^{53} +326517. q^{55} -1.11209e6 q^{56} +520114. q^{58} -316269. q^{59} -29722.9 q^{61} -1.14398e6 q^{62} +1.63495e6 q^{64} +392723. q^{65} +293025. q^{67} -2.57809e6 q^{68} +369014. q^{70} +714537. q^{71} -3.96273e6 q^{73} -3.71337e6 q^{74} -1.57415e6 q^{76} -3.66342e6 q^{77} +2.53805e6 q^{79} -160100. q^{80} -4.48492e6 q^{82} +1.66311e6 q^{83} +2.41216e6 q^{85} +6.39851e6 q^{86} +7.47204e6 q^{88} +4.64819e6 q^{89} -4.40624e6 q^{91} +3.62068e6 q^{92} -9.31228e6 q^{94} +1.47283e6 q^{95} +1.47010e7 q^{97} +2.11325e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.59339 −0.671167 −0.335583 0.942010i \(-0.608933\pi\)
−0.335583 + 0.942010i \(0.608933\pi\)
\(3\) 0 0
\(4\) −70.3405 −0.549535
\(5\) 65.8132 0.235461 0.117730 0.993046i \(-0.462438\pi\)
0.117730 + 0.993046i \(0.462438\pi\)
\(6\) 0 0
\(7\) −738.405 −0.813676 −0.406838 0.913500i \(-0.633369\pi\)
−0.406838 + 0.913500i \(0.633369\pi\)
\(8\) 1506.08 1.04000
\(9\) 0 0
\(10\) −499.745 −0.158033
\(11\) 4961.26 1.12387 0.561937 0.827180i \(-0.310056\pi\)
0.561937 + 0.827180i \(0.310056\pi\)
\(12\) 0 0
\(13\) 5967.24 0.753306 0.376653 0.926354i \(-0.377075\pi\)
0.376653 + 0.926354i \(0.377075\pi\)
\(14\) 5606.99 0.546112
\(15\) 0 0
\(16\) −2432.64 −0.148476
\(17\) 36651.6 1.80935 0.904674 0.426104i \(-0.140114\pi\)
0.904674 + 0.426104i \(0.140114\pi\)
\(18\) 0 0
\(19\) 22378.9 0.748518 0.374259 0.927324i \(-0.377897\pi\)
0.374259 + 0.927324i \(0.377897\pi\)
\(20\) −4629.33 −0.129394
\(21\) 0 0
\(22\) −37672.8 −0.754308
\(23\) −51473.6 −0.882139 −0.441069 0.897473i \(-0.645401\pi\)
−0.441069 + 0.897473i \(0.645401\pi\)
\(24\) 0 0
\(25\) −73793.6 −0.944558
\(26\) −45311.5 −0.505594
\(27\) 0 0
\(28\) 51939.7 0.447143
\(29\) −68495.7 −0.521519 −0.260760 0.965404i \(-0.583973\pi\)
−0.260760 + 0.965404i \(0.583973\pi\)
\(30\) 0 0
\(31\) 150655. 0.908275 0.454137 0.890932i \(-0.349948\pi\)
0.454137 + 0.890932i \(0.349948\pi\)
\(32\) −174306. −0.940344
\(33\) 0 0
\(34\) −278310. −1.21437
\(35\) −48596.8 −0.191589
\(36\) 0 0
\(37\) 489027. 1.58718 0.793591 0.608451i \(-0.208209\pi\)
0.793591 + 0.608451i \(0.208209\pi\)
\(38\) −169932. −0.502380
\(39\) 0 0
\(40\) 99119.7 0.244878
\(41\) 590635. 1.33837 0.669184 0.743096i \(-0.266644\pi\)
0.669184 + 0.743096i \(0.266644\pi\)
\(42\) 0 0
\(43\) −842643. −1.61623 −0.808117 0.589023i \(-0.799513\pi\)
−0.808117 + 0.589023i \(0.799513\pi\)
\(44\) −348978. −0.617609
\(45\) 0 0
\(46\) 390859. 0.592062
\(47\) 1.22637e6 1.72297 0.861486 0.507782i \(-0.169534\pi\)
0.861486 + 0.507782i \(0.169534\pi\)
\(48\) 0 0
\(49\) −278301. −0.337932
\(50\) 560343. 0.633956
\(51\) 0 0
\(52\) −419738. −0.413968
\(53\) 958904. 0.884727 0.442364 0.896836i \(-0.354140\pi\)
0.442364 + 0.896836i \(0.354140\pi\)
\(54\) 0 0
\(55\) 326517. 0.264628
\(56\) −1.11209e6 −0.846220
\(57\) 0 0
\(58\) 520114. 0.350027
\(59\) −316269. −0.200482 −0.100241 0.994963i \(-0.531961\pi\)
−0.100241 + 0.994963i \(0.531961\pi\)
\(60\) 0 0
\(61\) −29722.9 −0.0167663 −0.00838315 0.999965i \(-0.502668\pi\)
−0.00838315 + 0.999965i \(0.502668\pi\)
\(62\) −1.14398e6 −0.609604
\(63\) 0 0
\(64\) 1.63495e6 0.779604
\(65\) 392723. 0.177374
\(66\) 0 0
\(67\) 293025. 0.119026 0.0595130 0.998228i \(-0.481045\pi\)
0.0595130 + 0.998228i \(0.481045\pi\)
\(68\) −2.57809e6 −0.994300
\(69\) 0 0
\(70\) 369014. 0.128588
\(71\) 714537. 0.236930 0.118465 0.992958i \(-0.462203\pi\)
0.118465 + 0.992958i \(0.462203\pi\)
\(72\) 0 0
\(73\) −3.96273e6 −1.19224 −0.596121 0.802894i \(-0.703292\pi\)
−0.596121 + 0.802894i \(0.703292\pi\)
\(74\) −3.71337e6 −1.06526
\(75\) 0 0
\(76\) −1.57415e6 −0.411337
\(77\) −3.66342e6 −0.914470
\(78\) 0 0
\(79\) 2.53805e6 0.579168 0.289584 0.957153i \(-0.406483\pi\)
0.289584 + 0.957153i \(0.406483\pi\)
\(80\) −160100. −0.0349603
\(81\) 0 0
\(82\) −4.48492e6 −0.898269
\(83\) 1.66311e6 0.319263 0.159631 0.987177i \(-0.448969\pi\)
0.159631 + 0.987177i \(0.448969\pi\)
\(84\) 0 0
\(85\) 2.41216e6 0.426030
\(86\) 6.39851e6 1.08476
\(87\) 0 0
\(88\) 7.47204e6 1.16883
\(89\) 4.64819e6 0.698906 0.349453 0.936954i \(-0.386368\pi\)
0.349453 + 0.936954i \(0.386368\pi\)
\(90\) 0 0
\(91\) −4.40624e6 −0.612947
\(92\) 3.62068e6 0.484766
\(93\) 0 0
\(94\) −9.31228e6 −1.15640
\(95\) 1.47283e6 0.176246
\(96\) 0 0
\(97\) 1.47010e7 1.63548 0.817738 0.575590i \(-0.195228\pi\)
0.817738 + 0.575590i \(0.195228\pi\)
\(98\) 2.11325e6 0.226809
\(99\) 0 0
\(100\) 5.19068e6 0.519068
\(101\) 9.28380e6 0.896604 0.448302 0.893882i \(-0.352029\pi\)
0.448302 + 0.893882i \(0.352029\pi\)
\(102\) 0 0
\(103\) 4.11049e6 0.370650 0.185325 0.982677i \(-0.440666\pi\)
0.185325 + 0.982677i \(0.440666\pi\)
\(104\) 8.98711e6 0.783436
\(105\) 0 0
\(106\) −7.28133e6 −0.593800
\(107\) −1.77931e7 −1.40414 −0.702068 0.712110i \(-0.747740\pi\)
−0.702068 + 0.712110i \(0.747740\pi\)
\(108\) 0 0
\(109\) −1.72244e7 −1.27394 −0.636972 0.770887i \(-0.719813\pi\)
−0.636972 + 0.770887i \(0.719813\pi\)
\(110\) −2.47937e6 −0.177610
\(111\) 0 0
\(112\) 1.79627e6 0.120812
\(113\) −2.16751e6 −0.141315 −0.0706573 0.997501i \(-0.522510\pi\)
−0.0706573 + 0.997501i \(0.522510\pi\)
\(114\) 0 0
\(115\) −3.38764e6 −0.207709
\(116\) 4.81802e6 0.286593
\(117\) 0 0
\(118\) 2.40156e6 0.134557
\(119\) −2.70638e7 −1.47222
\(120\) 0 0
\(121\) 5.12697e6 0.263095
\(122\) 225698. 0.0112530
\(123\) 0 0
\(124\) −1.05971e7 −0.499129
\(125\) −9.99825e6 −0.457867
\(126\) 0 0
\(127\) 9.66827e6 0.418828 0.209414 0.977827i \(-0.432844\pi\)
0.209414 + 0.977827i \(0.432844\pi\)
\(128\) 9.89634e6 0.417100
\(129\) 0 0
\(130\) −2.98210e6 −0.119048
\(131\) −4.22431e7 −1.64175 −0.820873 0.571111i \(-0.806513\pi\)
−0.820873 + 0.571111i \(0.806513\pi\)
\(132\) 0 0
\(133\) −1.65247e7 −0.609051
\(134\) −2.22505e6 −0.0798864
\(135\) 0 0
\(136\) 5.52002e7 1.88172
\(137\) −8.01451e6 −0.266290 −0.133145 0.991097i \(-0.542508\pi\)
−0.133145 + 0.991097i \(0.542508\pi\)
\(138\) 0 0
\(139\) 1.46010e7 0.461138 0.230569 0.973056i \(-0.425941\pi\)
0.230569 + 0.973056i \(0.425941\pi\)
\(140\) 3.41832e6 0.105285
\(141\) 0 0
\(142\) −5.42576e6 −0.159020
\(143\) 2.96050e7 0.846622
\(144\) 0 0
\(145\) −4.50792e6 −0.122797
\(146\) 3.00906e7 0.800194
\(147\) 0 0
\(148\) −3.43984e7 −0.872212
\(149\) 4.07087e7 1.00817 0.504086 0.863653i \(-0.331829\pi\)
0.504086 + 0.863653i \(0.331829\pi\)
\(150\) 0 0
\(151\) −5.29940e7 −1.25259 −0.626293 0.779588i \(-0.715428\pi\)
−0.626293 + 0.779588i \(0.715428\pi\)
\(152\) 3.37044e7 0.778456
\(153\) 0 0
\(154\) 2.78178e7 0.613762
\(155\) 9.91508e6 0.213863
\(156\) 0 0
\(157\) 4.10669e7 0.846922 0.423461 0.905914i \(-0.360815\pi\)
0.423461 + 0.905914i \(0.360815\pi\)
\(158\) −1.92724e7 −0.388718
\(159\) 0 0
\(160\) −1.14716e7 −0.221414
\(161\) 3.80083e7 0.717775
\(162\) 0 0
\(163\) −4.33772e7 −0.784522 −0.392261 0.919854i \(-0.628307\pi\)
−0.392261 + 0.919854i \(0.628307\pi\)
\(164\) −4.15456e7 −0.735480
\(165\) 0 0
\(166\) −1.26287e7 −0.214279
\(167\) 3.86845e7 0.642731 0.321365 0.946955i \(-0.395858\pi\)
0.321365 + 0.946955i \(0.395858\pi\)
\(168\) 0 0
\(169\) −2.71406e7 −0.432529
\(170\) −1.83165e7 −0.285937
\(171\) 0 0
\(172\) 5.92719e7 0.888177
\(173\) 5.30547e7 0.779045 0.389523 0.921017i \(-0.372640\pi\)
0.389523 + 0.921017i \(0.372640\pi\)
\(174\) 0 0
\(175\) 5.44896e7 0.768564
\(176\) −1.20689e7 −0.166869
\(177\) 0 0
\(178\) −3.52955e7 −0.469082
\(179\) −1.34132e8 −1.74802 −0.874012 0.485904i \(-0.838491\pi\)
−0.874012 + 0.485904i \(0.838491\pi\)
\(180\) 0 0
\(181\) 6.35105e6 0.0796105 0.0398053 0.999207i \(-0.487326\pi\)
0.0398053 + 0.999207i \(0.487326\pi\)
\(182\) 3.34583e7 0.411390
\(183\) 0 0
\(184\) −7.75231e7 −0.917422
\(185\) 3.21845e7 0.373719
\(186\) 0 0
\(187\) 1.81839e8 2.03348
\(188\) −8.62633e7 −0.946833
\(189\) 0 0
\(190\) −1.11838e7 −0.118291
\(191\) −1.15843e8 −1.20297 −0.601483 0.798885i \(-0.705423\pi\)
−0.601483 + 0.798885i \(0.705423\pi\)
\(192\) 0 0
\(193\) 5.73856e7 0.574582 0.287291 0.957843i \(-0.407245\pi\)
0.287291 + 0.957843i \(0.407245\pi\)
\(194\) −1.11630e8 −1.09768
\(195\) 0 0
\(196\) 1.95759e7 0.185705
\(197\) 1.04846e8 0.977059 0.488530 0.872547i \(-0.337533\pi\)
0.488530 + 0.872547i \(0.337533\pi\)
\(198\) 0 0
\(199\) −2.10623e8 −1.89461 −0.947304 0.320335i \(-0.896205\pi\)
−0.947304 + 0.320335i \(0.896205\pi\)
\(200\) −1.11139e8 −0.982337
\(201\) 0 0
\(202\) −7.04955e7 −0.601771
\(203\) 5.05776e7 0.424348
\(204\) 0 0
\(205\) 3.88716e7 0.315133
\(206\) −3.12125e7 −0.248768
\(207\) 0 0
\(208\) −1.45161e7 −0.111848
\(209\) 1.11028e8 0.841240
\(210\) 0 0
\(211\) −6.49193e7 −0.475757 −0.237878 0.971295i \(-0.576452\pi\)
−0.237878 + 0.971295i \(0.576452\pi\)
\(212\) −6.74497e7 −0.486189
\(213\) 0 0
\(214\) 1.35110e8 0.942410
\(215\) −5.54570e7 −0.380559
\(216\) 0 0
\(217\) −1.11244e8 −0.739041
\(218\) 1.30791e8 0.855030
\(219\) 0 0
\(220\) −2.29674e7 −0.145422
\(221\) 2.18709e8 1.36299
\(222\) 0 0
\(223\) 1.40879e8 0.850705 0.425352 0.905028i \(-0.360150\pi\)
0.425352 + 0.905028i \(0.360150\pi\)
\(224\) 1.28708e8 0.765135
\(225\) 0 0
\(226\) 1.64588e7 0.0948457
\(227\) −3.40163e6 −0.0193017 −0.00965086 0.999953i \(-0.503072\pi\)
−0.00965086 + 0.999953i \(0.503072\pi\)
\(228\) 0 0
\(229\) −2.76169e7 −0.151967 −0.0759836 0.997109i \(-0.524210\pi\)
−0.0759836 + 0.997109i \(0.524210\pi\)
\(230\) 2.57237e7 0.139407
\(231\) 0 0
\(232\) −1.03160e8 −0.542378
\(233\) 3.51458e8 1.82023 0.910117 0.414351i \(-0.135992\pi\)
0.910117 + 0.414351i \(0.135992\pi\)
\(234\) 0 0
\(235\) 8.07112e7 0.405692
\(236\) 2.22465e7 0.110172
\(237\) 0 0
\(238\) 2.05506e8 0.988107
\(239\) −3.79195e8 −1.79668 −0.898339 0.439302i \(-0.855226\pi\)
−0.898339 + 0.439302i \(0.855226\pi\)
\(240\) 0 0
\(241\) 2.99276e7 0.137725 0.0688624 0.997626i \(-0.478063\pi\)
0.0688624 + 0.997626i \(0.478063\pi\)
\(242\) −3.89311e7 −0.176581
\(243\) 0 0
\(244\) 2.09072e6 0.00921367
\(245\) −1.83159e7 −0.0795696
\(246\) 0 0
\(247\) 1.33540e8 0.563863
\(248\) 2.26898e8 0.944602
\(249\) 0 0
\(250\) 7.59206e7 0.307305
\(251\) −1.71456e7 −0.0684374 −0.0342187 0.999414i \(-0.510894\pi\)
−0.0342187 + 0.999414i \(0.510894\pi\)
\(252\) 0 0
\(253\) −2.55374e8 −0.991414
\(254\) −7.34149e7 −0.281103
\(255\) 0 0
\(256\) −2.84420e8 −1.05955
\(257\) −9.79449e7 −0.359928 −0.179964 0.983673i \(-0.557598\pi\)
−0.179964 + 0.983673i \(0.557598\pi\)
\(258\) 0 0
\(259\) −3.61100e8 −1.29145
\(260\) −2.76243e7 −0.0974732
\(261\) 0 0
\(262\) 3.20768e8 1.10189
\(263\) −8.92540e7 −0.302540 −0.151270 0.988492i \(-0.548336\pi\)
−0.151270 + 0.988492i \(0.548336\pi\)
\(264\) 0 0
\(265\) 6.31085e7 0.208318
\(266\) 1.25479e8 0.408775
\(267\) 0 0
\(268\) −2.06115e7 −0.0654090
\(269\) −5.34653e8 −1.67471 −0.837354 0.546662i \(-0.815898\pi\)
−0.837354 + 0.546662i \(0.815898\pi\)
\(270\) 0 0
\(271\) 2.11501e8 0.645535 0.322768 0.946478i \(-0.395387\pi\)
0.322768 + 0.946478i \(0.395387\pi\)
\(272\) −8.91601e7 −0.268645
\(273\) 0 0
\(274\) 6.08573e7 0.178725
\(275\) −3.66110e8 −1.06157
\(276\) 0 0
\(277\) −3.24017e8 −0.915986 −0.457993 0.888956i \(-0.651432\pi\)
−0.457993 + 0.888956i \(0.651432\pi\)
\(278\) −1.10871e8 −0.309501
\(279\) 0 0
\(280\) −7.31905e7 −0.199251
\(281\) 4.91210e7 0.132067 0.0660336 0.997817i \(-0.478966\pi\)
0.0660336 + 0.997817i \(0.478966\pi\)
\(282\) 0 0
\(283\) −4.03017e8 −1.05699 −0.528495 0.848936i \(-0.677244\pi\)
−0.528495 + 0.848936i \(0.677244\pi\)
\(284\) −5.02609e7 −0.130202
\(285\) 0 0
\(286\) −2.24803e8 −0.568225
\(287\) −4.36128e8 −1.08900
\(288\) 0 0
\(289\) 9.33004e8 2.27374
\(290\) 3.42304e7 0.0824174
\(291\) 0 0
\(292\) 2.78740e8 0.655179
\(293\) 4.85906e8 1.12854 0.564268 0.825592i \(-0.309159\pi\)
0.564268 + 0.825592i \(0.309159\pi\)
\(294\) 0 0
\(295\) −2.08147e7 −0.0472056
\(296\) 7.36512e8 1.65066
\(297\) 0 0
\(298\) −3.09117e8 −0.676652
\(299\) −3.07155e8 −0.664521
\(300\) 0 0
\(301\) 6.22212e8 1.31509
\(302\) 4.02404e8 0.840694
\(303\) 0 0
\(304\) −5.44398e7 −0.111137
\(305\) −1.95616e6 −0.00394780
\(306\) 0 0
\(307\) 2.82215e8 0.556668 0.278334 0.960484i \(-0.410218\pi\)
0.278334 + 0.960484i \(0.410218\pi\)
\(308\) 2.57687e8 0.502533
\(309\) 0 0
\(310\) −7.52890e7 −0.143538
\(311\) 7.04440e7 0.132795 0.0663977 0.997793i \(-0.478849\pi\)
0.0663977 + 0.997793i \(0.478849\pi\)
\(312\) 0 0
\(313\) −4.78096e8 −0.881273 −0.440636 0.897686i \(-0.645247\pi\)
−0.440636 + 0.897686i \(0.645247\pi\)
\(314\) −3.11837e8 −0.568426
\(315\) 0 0
\(316\) −1.78527e8 −0.318273
\(317\) 6.89470e8 1.21565 0.607824 0.794072i \(-0.292043\pi\)
0.607824 + 0.794072i \(0.292043\pi\)
\(318\) 0 0
\(319\) −3.39825e8 −0.586123
\(320\) 1.07601e8 0.183566
\(321\) 0 0
\(322\) −2.88612e8 −0.481747
\(323\) 8.20225e8 1.35433
\(324\) 0 0
\(325\) −4.40344e8 −0.711542
\(326\) 3.29380e8 0.526545
\(327\) 0 0
\(328\) 8.89541e8 1.39190
\(329\) −9.05555e8 −1.40194
\(330\) 0 0
\(331\) 5.36833e8 0.813657 0.406829 0.913504i \(-0.366635\pi\)
0.406829 + 0.913504i \(0.366635\pi\)
\(332\) −1.16984e8 −0.175446
\(333\) 0 0
\(334\) −2.93746e8 −0.431379
\(335\) 1.92849e7 0.0280259
\(336\) 0 0
\(337\) −1.24011e9 −1.76505 −0.882524 0.470268i \(-0.844157\pi\)
−0.882524 + 0.470268i \(0.844157\pi\)
\(338\) 2.06089e8 0.290299
\(339\) 0 0
\(340\) −1.69673e8 −0.234119
\(341\) 7.47438e8 1.02079
\(342\) 0 0
\(343\) 8.13607e8 1.08864
\(344\) −1.26908e9 −1.68088
\(345\) 0 0
\(346\) −4.02865e8 −0.522869
\(347\) 4.86766e7 0.0625413 0.0312706 0.999511i \(-0.490045\pi\)
0.0312706 + 0.999511i \(0.490045\pi\)
\(348\) 0 0
\(349\) 1.23665e9 1.55725 0.778626 0.627488i \(-0.215917\pi\)
0.778626 + 0.627488i \(0.215917\pi\)
\(350\) −4.13760e8 −0.515835
\(351\) 0 0
\(352\) −8.64777e8 −1.05683
\(353\) 6.06858e7 0.0734304 0.0367152 0.999326i \(-0.488311\pi\)
0.0367152 + 0.999326i \(0.488311\pi\)
\(354\) 0 0
\(355\) 4.70260e7 0.0557878
\(356\) −3.26956e8 −0.384073
\(357\) 0 0
\(358\) 1.01852e9 1.17322
\(359\) −5.36061e8 −0.611482 −0.305741 0.952115i \(-0.598904\pi\)
−0.305741 + 0.952115i \(0.598904\pi\)
\(360\) 0 0
\(361\) −3.93055e8 −0.439722
\(362\) −4.82260e7 −0.0534320
\(363\) 0 0
\(364\) 3.09937e8 0.336836
\(365\) −2.60800e8 −0.280726
\(366\) 0 0
\(367\) −4.31327e8 −0.455487 −0.227743 0.973721i \(-0.573135\pi\)
−0.227743 + 0.973721i \(0.573135\pi\)
\(368\) 1.25216e8 0.130977
\(369\) 0 0
\(370\) −2.44389e8 −0.250828
\(371\) −7.08059e8 −0.719881
\(372\) 0 0
\(373\) 2.69689e8 0.269080 0.134540 0.990908i \(-0.457044\pi\)
0.134540 + 0.990908i \(0.457044\pi\)
\(374\) −1.38077e9 −1.36481
\(375\) 0 0
\(376\) 1.84700e9 1.79188
\(377\) −4.08730e8 −0.392864
\(378\) 0 0
\(379\) −8.03807e8 −0.758429 −0.379214 0.925309i \(-0.623806\pi\)
−0.379214 + 0.925309i \(0.623806\pi\)
\(380\) −1.03600e8 −0.0968535
\(381\) 0 0
\(382\) 8.79642e8 0.807392
\(383\) 9.83923e8 0.894882 0.447441 0.894314i \(-0.352336\pi\)
0.447441 + 0.894314i \(0.352336\pi\)
\(384\) 0 0
\(385\) −2.41102e8 −0.215322
\(386\) −4.35751e8 −0.385641
\(387\) 0 0
\(388\) −1.03407e9 −0.898752
\(389\) −2.16250e8 −0.186265 −0.0931326 0.995654i \(-0.529688\pi\)
−0.0931326 + 0.995654i \(0.529688\pi\)
\(390\) 0 0
\(391\) −1.88659e9 −1.59610
\(392\) −4.19143e8 −0.351448
\(393\) 0 0
\(394\) −7.96138e8 −0.655770
\(395\) 1.67037e8 0.136371
\(396\) 0 0
\(397\) 4.33533e8 0.347741 0.173870 0.984769i \(-0.444373\pi\)
0.173870 + 0.984769i \(0.444373\pi\)
\(398\) 1.59934e9 1.27160
\(399\) 0 0
\(400\) 1.79513e8 0.140245
\(401\) 2.42713e9 1.87970 0.939849 0.341591i \(-0.110966\pi\)
0.939849 + 0.341591i \(0.110966\pi\)
\(402\) 0 0
\(403\) 8.98993e8 0.684209
\(404\) −6.53027e8 −0.492715
\(405\) 0 0
\(406\) −3.84055e8 −0.284808
\(407\) 2.42619e9 1.78379
\(408\) 0 0
\(409\) 9.18976e8 0.664160 0.332080 0.943251i \(-0.392250\pi\)
0.332080 + 0.943251i \(0.392250\pi\)
\(410\) −2.95167e8 −0.211507
\(411\) 0 0
\(412\) −2.89134e8 −0.203685
\(413\) 2.33535e8 0.163127
\(414\) 0 0
\(415\) 1.09455e8 0.0751738
\(416\) −1.04012e9 −0.708367
\(417\) 0 0
\(418\) −8.43077e8 −0.564612
\(419\) 1.10024e9 0.730699 0.365349 0.930870i \(-0.380950\pi\)
0.365349 + 0.930870i \(0.380950\pi\)
\(420\) 0 0
\(421\) −2.14909e9 −1.40367 −0.701837 0.712337i \(-0.747637\pi\)
−0.701837 + 0.712337i \(0.747637\pi\)
\(422\) 4.92957e8 0.319312
\(423\) 0 0
\(424\) 1.44418e9 0.920113
\(425\) −2.70466e9 −1.70904
\(426\) 0 0
\(427\) 2.19476e7 0.0136423
\(428\) 1.25158e9 0.771622
\(429\) 0 0
\(430\) 4.21107e8 0.255419
\(431\) 1.61850e9 0.973737 0.486869 0.873475i \(-0.338139\pi\)
0.486869 + 0.873475i \(0.338139\pi\)
\(432\) 0 0
\(433\) 1.16527e9 0.689794 0.344897 0.938641i \(-0.387914\pi\)
0.344897 + 0.938641i \(0.387914\pi\)
\(434\) 8.44721e8 0.496020
\(435\) 0 0
\(436\) 1.21157e9 0.700077
\(437\) −1.15192e9 −0.660296
\(438\) 0 0
\(439\) 2.08096e9 1.17392 0.586958 0.809617i \(-0.300325\pi\)
0.586958 + 0.809617i \(0.300325\pi\)
\(440\) 4.91759e8 0.275212
\(441\) 0 0
\(442\) −1.66074e9 −0.914796
\(443\) 2.23100e9 1.21923 0.609616 0.792697i \(-0.291324\pi\)
0.609616 + 0.792697i \(0.291324\pi\)
\(444\) 0 0
\(445\) 3.05912e8 0.164565
\(446\) −1.06975e9 −0.570965
\(447\) 0 0
\(448\) −1.20725e9 −0.634345
\(449\) −2.52631e9 −1.31712 −0.658559 0.752529i \(-0.728834\pi\)
−0.658559 + 0.752529i \(0.728834\pi\)
\(450\) 0 0
\(451\) 2.93030e9 1.50416
\(452\) 1.52464e8 0.0776573
\(453\) 0 0
\(454\) 2.58299e7 0.0129547
\(455\) −2.89989e8 −0.144325
\(456\) 0 0
\(457\) −1.87627e9 −0.919579 −0.459790 0.888028i \(-0.652075\pi\)
−0.459790 + 0.888028i \(0.652075\pi\)
\(458\) 2.09705e8 0.101995
\(459\) 0 0
\(460\) 2.38288e8 0.114143
\(461\) −1.97321e9 −0.938036 −0.469018 0.883189i \(-0.655392\pi\)
−0.469018 + 0.883189i \(0.655392\pi\)
\(462\) 0 0
\(463\) −1.60344e9 −0.750790 −0.375395 0.926865i \(-0.622493\pi\)
−0.375395 + 0.926865i \(0.622493\pi\)
\(464\) 1.66625e8 0.0774333
\(465\) 0 0
\(466\) −2.66875e9 −1.22168
\(467\) −1.32390e9 −0.601515 −0.300757 0.953701i \(-0.597239\pi\)
−0.300757 + 0.953701i \(0.597239\pi\)
\(468\) 0 0
\(469\) −2.16371e8 −0.0968486
\(470\) −6.12871e8 −0.272287
\(471\) 0 0
\(472\) −4.76326e8 −0.208501
\(473\) −4.18057e9 −1.81644
\(474\) 0 0
\(475\) −1.65142e9 −0.707018
\(476\) 1.90368e9 0.809038
\(477\) 0 0
\(478\) 2.87938e9 1.20587
\(479\) −2.44225e9 −1.01535 −0.507675 0.861548i \(-0.669495\pi\)
−0.507675 + 0.861548i \(0.669495\pi\)
\(480\) 0 0
\(481\) 2.91814e9 1.19563
\(482\) −2.27252e8 −0.0924363
\(483\) 0 0
\(484\) −3.60634e8 −0.144580
\(485\) 9.67517e8 0.385090
\(486\) 0 0
\(487\) −1.37309e7 −0.00538701 −0.00269350 0.999996i \(-0.500857\pi\)
−0.00269350 + 0.999996i \(0.500857\pi\)
\(488\) −4.47650e7 −0.0174369
\(489\) 0 0
\(490\) 1.39080e8 0.0534045
\(491\) −6.32242e8 −0.241045 −0.120522 0.992711i \(-0.538457\pi\)
−0.120522 + 0.992711i \(0.538457\pi\)
\(492\) 0 0
\(493\) −2.51048e9 −0.943610
\(494\) −1.01402e9 −0.378446
\(495\) 0 0
\(496\) −3.66488e8 −0.134857
\(497\) −5.27618e8 −0.192784
\(498\) 0 0
\(499\) −2.43365e9 −0.876812 −0.438406 0.898777i \(-0.644457\pi\)
−0.438406 + 0.898777i \(0.644457\pi\)
\(500\) 7.03282e8 0.251614
\(501\) 0 0
\(502\) 1.30193e8 0.0459329
\(503\) 2.00281e9 0.701700 0.350850 0.936432i \(-0.385893\pi\)
0.350850 + 0.936432i \(0.385893\pi\)
\(504\) 0 0
\(505\) 6.10997e8 0.211115
\(506\) 1.93915e9 0.665404
\(507\) 0 0
\(508\) −6.80071e8 −0.230161
\(509\) −5.28309e8 −0.177573 −0.0887863 0.996051i \(-0.528299\pi\)
−0.0887863 + 0.996051i \(0.528299\pi\)
\(510\) 0 0
\(511\) 2.92610e9 0.970099
\(512\) 8.92980e8 0.294034
\(513\) 0 0
\(514\) 7.43734e8 0.241572
\(515\) 2.70525e8 0.0872733
\(516\) 0 0
\(517\) 6.08433e9 1.93640
\(518\) 2.74197e9 0.866780
\(519\) 0 0
\(520\) 5.91471e8 0.184468
\(521\) −1.76283e9 −0.546109 −0.273055 0.961999i \(-0.588034\pi\)
−0.273055 + 0.961999i \(0.588034\pi\)
\(522\) 0 0
\(523\) −3.39318e9 −1.03717 −0.518586 0.855026i \(-0.673541\pi\)
−0.518586 + 0.855026i \(0.673541\pi\)
\(524\) 2.97140e9 0.902197
\(525\) 0 0
\(526\) 6.77740e8 0.203055
\(527\) 5.52175e9 1.64339
\(528\) 0 0
\(529\) −7.55295e8 −0.221831
\(530\) −4.79208e8 −0.139816
\(531\) 0 0
\(532\) 1.16236e9 0.334695
\(533\) 3.52446e9 1.00820
\(534\) 0 0
\(535\) −1.17102e9 −0.330619
\(536\) 4.41317e8 0.123787
\(537\) 0 0
\(538\) 4.05983e9 1.12401
\(539\) −1.38073e9 −0.379793
\(540\) 0 0
\(541\) −3.83907e9 −1.04240 −0.521202 0.853433i \(-0.674516\pi\)
−0.521202 + 0.853433i \(0.674516\pi\)
\(542\) −1.60601e9 −0.433262
\(543\) 0 0
\(544\) −6.38859e9 −1.70141
\(545\) −1.13359e9 −0.299964
\(546\) 0 0
\(547\) 4.31235e9 1.12657 0.563285 0.826263i \(-0.309537\pi\)
0.563285 + 0.826263i \(0.309537\pi\)
\(548\) 5.63744e8 0.146336
\(549\) 0 0
\(550\) 2.78001e9 0.712488
\(551\) −1.53286e9 −0.390366
\(552\) 0 0
\(553\) −1.87410e9 −0.471255
\(554\) 2.46039e9 0.614779
\(555\) 0 0
\(556\) −1.02704e9 −0.253412
\(557\) −1.84109e9 −0.451421 −0.225710 0.974194i \(-0.572470\pi\)
−0.225710 + 0.974194i \(0.572470\pi\)
\(558\) 0 0
\(559\) −5.02825e9 −1.21752
\(560\) 1.18218e8 0.0284464
\(561\) 0 0
\(562\) −3.72994e8 −0.0886391
\(563\) −1.16055e9 −0.274085 −0.137042 0.990565i \(-0.543760\pi\)
−0.137042 + 0.990565i \(0.543760\pi\)
\(564\) 0 0
\(565\) −1.42651e8 −0.0332740
\(566\) 3.06027e9 0.709417
\(567\) 0 0
\(568\) 1.07615e9 0.246407
\(569\) 7.02920e8 0.159960 0.0799802 0.996796i \(-0.474514\pi\)
0.0799802 + 0.996796i \(0.474514\pi\)
\(570\) 0 0
\(571\) −7.95929e8 −0.178915 −0.0894577 0.995991i \(-0.528513\pi\)
−0.0894577 + 0.995991i \(0.528513\pi\)
\(572\) −2.08243e9 −0.465249
\(573\) 0 0
\(574\) 3.31169e9 0.730900
\(575\) 3.79842e9 0.833232
\(576\) 0 0
\(577\) −4.44712e9 −0.963749 −0.481874 0.876240i \(-0.660044\pi\)
−0.481874 + 0.876240i \(0.660044\pi\)
\(578\) −7.08466e9 −1.52606
\(579\) 0 0
\(580\) 3.17090e8 0.0674814
\(581\) −1.22805e9 −0.259776
\(582\) 0 0
\(583\) 4.75737e9 0.994323
\(584\) −5.96818e9 −1.23993
\(585\) 0 0
\(586\) −3.68967e9 −0.757436
\(587\) 5.50846e9 1.12408 0.562039 0.827111i \(-0.310017\pi\)
0.562039 + 0.827111i \(0.310017\pi\)
\(588\) 0 0
\(589\) 3.37150e9 0.679859
\(590\) 1.58054e8 0.0316828
\(591\) 0 0
\(592\) −1.18962e9 −0.235659
\(593\) −8.05850e9 −1.58695 −0.793475 0.608603i \(-0.791730\pi\)
−0.793475 + 0.608603i \(0.791730\pi\)
\(594\) 0 0
\(595\) −1.78115e9 −0.346650
\(596\) −2.86347e9 −0.554026
\(597\) 0 0
\(598\) 2.33235e9 0.446004
\(599\) −3.39566e9 −0.645551 −0.322775 0.946476i \(-0.604616\pi\)
−0.322775 + 0.946476i \(0.604616\pi\)
\(600\) 0 0
\(601\) 1.35177e9 0.254005 0.127002 0.991902i \(-0.459464\pi\)
0.127002 + 0.991902i \(0.459464\pi\)
\(602\) −4.72469e9 −0.882645
\(603\) 0 0
\(604\) 3.72762e9 0.688339
\(605\) 3.37423e8 0.0619485
\(606\) 0 0
\(607\) 4.91297e9 0.891627 0.445814 0.895126i \(-0.352914\pi\)
0.445814 + 0.895126i \(0.352914\pi\)
\(608\) −3.90078e9 −0.703864
\(609\) 0 0
\(610\) 1.48539e7 0.00264963
\(611\) 7.31802e9 1.29793
\(612\) 0 0
\(613\) −2.06110e9 −0.361399 −0.180700 0.983538i \(-0.557836\pi\)
−0.180700 + 0.983538i \(0.557836\pi\)
\(614\) −2.14297e9 −0.373617
\(615\) 0 0
\(616\) −5.51739e9 −0.951045
\(617\) −3.72277e9 −0.638071 −0.319035 0.947743i \(-0.603359\pi\)
−0.319035 + 0.947743i \(0.603359\pi\)
\(618\) 0 0
\(619\) 3.30066e9 0.559349 0.279675 0.960095i \(-0.409773\pi\)
0.279675 + 0.960095i \(0.409773\pi\)
\(620\) −6.97432e8 −0.117525
\(621\) 0 0
\(622\) −5.34909e8 −0.0891278
\(623\) −3.43224e9 −0.568683
\(624\) 0 0
\(625\) 5.10711e9 0.836749
\(626\) 3.63037e9 0.591481
\(627\) 0 0
\(628\) −2.88867e9 −0.465413
\(629\) 1.79236e10 2.87177
\(630\) 0 0
\(631\) 5.12709e9 0.812397 0.406199 0.913785i \(-0.366854\pi\)
0.406199 + 0.913785i \(0.366854\pi\)
\(632\) 3.82249e9 0.602333
\(633\) 0 0
\(634\) −5.23541e9 −0.815903
\(635\) 6.36300e8 0.0986174
\(636\) 0 0
\(637\) −1.66069e9 −0.254566
\(638\) 2.58043e9 0.393386
\(639\) 0 0
\(640\) 6.51310e8 0.0982105
\(641\) −4.51842e9 −0.677615 −0.338808 0.940856i \(-0.610024\pi\)
−0.338808 + 0.940856i \(0.610024\pi\)
\(642\) 0 0
\(643\) −9.70600e9 −1.43980 −0.719900 0.694078i \(-0.755812\pi\)
−0.719900 + 0.694078i \(0.755812\pi\)
\(644\) −2.67353e9 −0.394443
\(645\) 0 0
\(646\) −6.22829e9 −0.908981
\(647\) −6.12536e9 −0.889132 −0.444566 0.895746i \(-0.646642\pi\)
−0.444566 + 0.895746i \(0.646642\pi\)
\(648\) 0 0
\(649\) −1.56910e9 −0.225317
\(650\) 3.34370e9 0.477563
\(651\) 0 0
\(652\) 3.05117e9 0.431122
\(653\) 1.30103e10 1.82849 0.914243 0.405166i \(-0.132786\pi\)
0.914243 + 0.405166i \(0.132786\pi\)
\(654\) 0 0
\(655\) −2.78015e9 −0.386566
\(656\) −1.43680e9 −0.198716
\(657\) 0 0
\(658\) 6.87623e9 0.940936
\(659\) 1.23137e10 1.67606 0.838032 0.545621i \(-0.183706\pi\)
0.838032 + 0.545621i \(0.183706\pi\)
\(660\) 0 0
\(661\) 6.97776e9 0.939746 0.469873 0.882734i \(-0.344300\pi\)
0.469873 + 0.882734i \(0.344300\pi\)
\(662\) −4.07638e9 −0.546100
\(663\) 0 0
\(664\) 2.50477e9 0.332032
\(665\) −1.08755e9 −0.143407
\(666\) 0 0
\(667\) 3.52572e9 0.460053
\(668\) −2.72108e9 −0.353203
\(669\) 0 0
\(670\) −1.46438e8 −0.0188101
\(671\) −1.47463e8 −0.0188432
\(672\) 0 0
\(673\) −1.06462e10 −1.34630 −0.673152 0.739504i \(-0.735060\pi\)
−0.673152 + 0.739504i \(0.735060\pi\)
\(674\) 9.41665e9 1.18464
\(675\) 0 0
\(676\) 1.90908e9 0.237690
\(677\) 4.33348e9 0.536756 0.268378 0.963314i \(-0.413512\pi\)
0.268378 + 0.963314i \(0.413512\pi\)
\(678\) 0 0
\(679\) −1.08553e10 −1.33075
\(680\) 3.63290e9 0.443070
\(681\) 0 0
\(682\) −5.67559e9 −0.685118
\(683\) −6.16980e9 −0.740966 −0.370483 0.928839i \(-0.620808\pi\)
−0.370483 + 0.928839i \(0.620808\pi\)
\(684\) 0 0
\(685\) −5.27461e8 −0.0627008
\(686\) −6.17803e9 −0.730661
\(687\) 0 0
\(688\) 2.04984e9 0.239972
\(689\) 5.72201e9 0.666471
\(690\) 0 0
\(691\) 7.57273e9 0.873131 0.436565 0.899672i \(-0.356195\pi\)
0.436565 + 0.899672i \(0.356195\pi\)
\(692\) −3.73190e9 −0.428113
\(693\) 0 0
\(694\) −3.69620e8 −0.0419756
\(695\) 9.60939e8 0.108580
\(696\) 0 0
\(697\) 2.16477e10 2.42158
\(698\) −9.39039e9 −1.04518
\(699\) 0 0
\(700\) −3.83282e9 −0.422353
\(701\) −4.42696e9 −0.485391 −0.242696 0.970102i \(-0.578032\pi\)
−0.242696 + 0.970102i \(0.578032\pi\)
\(702\) 0 0
\(703\) 1.09439e10 1.18803
\(704\) 8.11141e9 0.876178
\(705\) 0 0
\(706\) −4.60811e8 −0.0492841
\(707\) −6.85520e9 −0.729545
\(708\) 0 0
\(709\) −1.13232e10 −1.19319 −0.596594 0.802544i \(-0.703480\pi\)
−0.596594 + 0.802544i \(0.703480\pi\)
\(710\) −3.57087e8 −0.0374429
\(711\) 0 0
\(712\) 7.00052e9 0.726859
\(713\) −7.75474e9 −0.801224
\(714\) 0 0
\(715\) 1.94840e9 0.199346
\(716\) 9.43492e9 0.960601
\(717\) 0 0
\(718\) 4.07052e9 0.410407
\(719\) −1.02156e10 −1.02497 −0.512487 0.858695i \(-0.671276\pi\)
−0.512487 + 0.858695i \(0.671276\pi\)
\(720\) 0 0
\(721\) −3.03521e9 −0.301589
\(722\) 2.98462e9 0.295127
\(723\) 0 0
\(724\) −4.46736e8 −0.0437488
\(725\) 5.05455e9 0.492605
\(726\) 0 0
\(727\) −1.95355e10 −1.88562 −0.942811 0.333329i \(-0.891828\pi\)
−0.942811 + 0.333329i \(0.891828\pi\)
\(728\) −6.63613e9 −0.637463
\(729\) 0 0
\(730\) 1.98036e9 0.188414
\(731\) −3.08842e10 −2.92433
\(732\) 0 0
\(733\) −1.32745e10 −1.24496 −0.622478 0.782637i \(-0.713874\pi\)
−0.622478 + 0.782637i \(0.713874\pi\)
\(734\) 3.27523e9 0.305708
\(735\) 0 0
\(736\) 8.97214e9 0.829514
\(737\) 1.45377e9 0.133770
\(738\) 0 0
\(739\) 1.69495e9 0.154490 0.0772452 0.997012i \(-0.475388\pi\)
0.0772452 + 0.997012i \(0.475388\pi\)
\(740\) −2.26387e9 −0.205372
\(741\) 0 0
\(742\) 5.37657e9 0.483160
\(743\) −9.34493e9 −0.835824 −0.417912 0.908487i \(-0.637238\pi\)
−0.417912 + 0.908487i \(0.637238\pi\)
\(744\) 0 0
\(745\) 2.67917e9 0.237385
\(746\) −2.04785e9 −0.180598
\(747\) 0 0
\(748\) −1.27906e10 −1.11747
\(749\) 1.31385e10 1.14251
\(750\) 0 0
\(751\) −1.31194e10 −1.13025 −0.565124 0.825006i \(-0.691172\pi\)
−0.565124 + 0.825006i \(0.691172\pi\)
\(752\) −2.98330e9 −0.255820
\(753\) 0 0
\(754\) 3.10365e9 0.263677
\(755\) −3.48770e9 −0.294934
\(756\) 0 0
\(757\) 5.72593e9 0.479745 0.239873 0.970804i \(-0.422894\pi\)
0.239873 + 0.970804i \(0.422894\pi\)
\(758\) 6.10362e9 0.509032
\(759\) 0 0
\(760\) 2.21819e9 0.183296
\(761\) −1.47621e10 −1.21423 −0.607115 0.794614i \(-0.707673\pi\)
−0.607115 + 0.794614i \(0.707673\pi\)
\(762\) 0 0
\(763\) 1.27186e10 1.03658
\(764\) 8.14847e9 0.661072
\(765\) 0 0
\(766\) −7.47131e9 −0.600615
\(767\) −1.88726e9 −0.151024
\(768\) 0 0
\(769\) 5.63551e9 0.446880 0.223440 0.974718i \(-0.428271\pi\)
0.223440 + 0.974718i \(0.428271\pi\)
\(770\) 1.83078e9 0.144517
\(771\) 0 0
\(772\) −4.03653e9 −0.315753
\(773\) 4.65338e9 0.362360 0.181180 0.983450i \(-0.442008\pi\)
0.181180 + 0.983450i \(0.442008\pi\)
\(774\) 0 0
\(775\) −1.11174e10 −0.857918
\(776\) 2.21408e10 1.70089
\(777\) 0 0
\(778\) 1.64207e9 0.125015
\(779\) 1.32178e10 1.00179
\(780\) 0 0
\(781\) 3.54501e9 0.266280
\(782\) 1.43256e10 1.07125
\(783\) 0 0
\(784\) 6.77006e8 0.0501749
\(785\) 2.70275e9 0.199417
\(786\) 0 0
\(787\) −1.23587e10 −0.903780 −0.451890 0.892074i \(-0.649250\pi\)
−0.451890 + 0.892074i \(0.649250\pi\)
\(788\) −7.37493e9 −0.536928
\(789\) 0 0
\(790\) −1.26838e9 −0.0915279
\(791\) 1.60050e9 0.114984
\(792\) 0 0
\(793\) −1.77364e8 −0.0126302
\(794\) −3.29198e9 −0.233392
\(795\) 0 0
\(796\) 1.48153e10 1.04115
\(797\) 1.70444e10 1.19256 0.596278 0.802778i \(-0.296646\pi\)
0.596278 + 0.802778i \(0.296646\pi\)
\(798\) 0 0
\(799\) 4.49484e10 3.11746
\(800\) 1.28627e10 0.888210
\(801\) 0 0
\(802\) −1.84301e10 −1.26159
\(803\) −1.96602e10 −1.33993
\(804\) 0 0
\(805\) 2.50145e9 0.169008
\(806\) −6.82640e9 −0.459218
\(807\) 0 0
\(808\) 1.39821e10 0.932465
\(809\) 1.00497e10 0.667315 0.333658 0.942694i \(-0.391717\pi\)
0.333658 + 0.942694i \(0.391717\pi\)
\(810\) 0 0
\(811\) 5.26821e9 0.346809 0.173404 0.984851i \(-0.444523\pi\)
0.173404 + 0.984851i \(0.444523\pi\)
\(812\) −3.55765e9 −0.233194
\(813\) 0 0
\(814\) −1.84230e10 −1.19722
\(815\) −2.85480e9 −0.184724
\(816\) 0 0
\(817\) −1.88575e10 −1.20978
\(818\) −6.97814e9 −0.445762
\(819\) 0 0
\(820\) −2.73425e9 −0.173177
\(821\) 1.82202e10 1.14909 0.574543 0.818474i \(-0.305180\pi\)
0.574543 + 0.818474i \(0.305180\pi\)
\(822\) 0 0
\(823\) −1.35868e9 −0.0849603 −0.0424802 0.999097i \(-0.513526\pi\)
−0.0424802 + 0.999097i \(0.513526\pi\)
\(824\) 6.19071e9 0.385474
\(825\) 0 0
\(826\) −1.77332e9 −0.109486
\(827\) 8.90171e9 0.547273 0.273637 0.961833i \(-0.411773\pi\)
0.273637 + 0.961833i \(0.411773\pi\)
\(828\) 0 0
\(829\) 6.14205e9 0.374432 0.187216 0.982319i \(-0.440054\pi\)
0.187216 + 0.982319i \(0.440054\pi\)
\(830\) −8.31133e8 −0.0504542
\(831\) 0 0
\(832\) 9.75613e9 0.587281
\(833\) −1.02002e10 −0.611436
\(834\) 0 0
\(835\) 2.54595e9 0.151338
\(836\) −7.80975e9 −0.462291
\(837\) 0 0
\(838\) −8.35455e9 −0.490421
\(839\) 1.05464e10 0.616504 0.308252 0.951305i \(-0.400256\pi\)
0.308252 + 0.951305i \(0.400256\pi\)
\(840\) 0 0
\(841\) −1.25582e10 −0.728018
\(842\) 1.63188e10 0.942100
\(843\) 0 0
\(844\) 4.56645e9 0.261445
\(845\) −1.78621e9 −0.101844
\(846\) 0 0
\(847\) −3.78578e9 −0.214074
\(848\) −2.33266e9 −0.131361
\(849\) 0 0
\(850\) 2.05375e10 1.14705
\(851\) −2.51720e10 −1.40012
\(852\) 0 0
\(853\) 3.41878e9 0.188604 0.0943018 0.995544i \(-0.469938\pi\)
0.0943018 + 0.995544i \(0.469938\pi\)
\(854\) −1.66656e8 −0.00915628
\(855\) 0 0
\(856\) −2.67978e10 −1.46030
\(857\) −2.17037e10 −1.17788 −0.588941 0.808176i \(-0.700455\pi\)
−0.588941 + 0.808176i \(0.700455\pi\)
\(858\) 0 0
\(859\) −2.19211e10 −1.18001 −0.590005 0.807400i \(-0.700874\pi\)
−0.590005 + 0.807400i \(0.700874\pi\)
\(860\) 3.90088e9 0.209131
\(861\) 0 0
\(862\) −1.22899e10 −0.653540
\(863\) −1.35347e10 −0.716822 −0.358411 0.933564i \(-0.616681\pi\)
−0.358411 + 0.933564i \(0.616681\pi\)
\(864\) 0 0
\(865\) 3.49170e9 0.183434
\(866\) −8.84836e9 −0.462967
\(867\) 0 0
\(868\) 7.82497e9 0.406129
\(869\) 1.25919e10 0.650912
\(870\) 0 0
\(871\) 1.74855e9 0.0896631
\(872\) −2.59412e10 −1.32490
\(873\) 0 0
\(874\) 8.74701e9 0.443169
\(875\) 7.38276e9 0.372555
\(876\) 0 0
\(877\) 2.45197e10 1.22749 0.613743 0.789506i \(-0.289663\pi\)
0.613743 + 0.789506i \(0.289663\pi\)
\(878\) −1.58015e10 −0.787894
\(879\) 0 0
\(880\) −7.94296e8 −0.0392910
\(881\) −2.13536e10 −1.05210 −0.526048 0.850455i \(-0.676327\pi\)
−0.526048 + 0.850455i \(0.676327\pi\)
\(882\) 0 0
\(883\) −1.64504e9 −0.0804109 −0.0402054 0.999191i \(-0.512801\pi\)
−0.0402054 + 0.999191i \(0.512801\pi\)
\(884\) −1.53841e10 −0.749013
\(885\) 0 0
\(886\) −1.69408e10 −0.818309
\(887\) 9.65301e9 0.464441 0.232220 0.972663i \(-0.425401\pi\)
0.232220 + 0.972663i \(0.425401\pi\)
\(888\) 0 0
\(889\) −7.13909e9 −0.340790
\(890\) −2.32291e9 −0.110450
\(891\) 0 0
\(892\) −9.90949e9 −0.467492
\(893\) 2.74448e10 1.28967
\(894\) 0 0
\(895\) −8.82767e9 −0.411591
\(896\) −7.30751e9 −0.339384
\(897\) 0 0
\(898\) 1.91833e10 0.884006
\(899\) −1.03192e10 −0.473683
\(900\) 0 0
\(901\) 3.51454e10 1.60078
\(902\) −2.22509e10 −1.00954
\(903\) 0 0
\(904\) −3.26444e9 −0.146967
\(905\) 4.17983e8 0.0187451
\(906\) 0 0
\(907\) −8.05417e9 −0.358423 −0.179211 0.983811i \(-0.557355\pi\)
−0.179211 + 0.983811i \(0.557355\pi\)
\(908\) 2.39272e8 0.0106070
\(909\) 0 0
\(910\) 2.20200e9 0.0968661
\(911\) 5.73799e9 0.251446 0.125723 0.992065i \(-0.459875\pi\)
0.125723 + 0.992065i \(0.459875\pi\)
\(912\) 0 0
\(913\) 8.25114e9 0.358811
\(914\) 1.42473e10 0.617191
\(915\) 0 0
\(916\) 1.94258e9 0.0835113
\(917\) 3.11925e10 1.33585
\(918\) 0 0
\(919\) 3.67326e10 1.56116 0.780579 0.625057i \(-0.214924\pi\)
0.780579 + 0.625057i \(0.214924\pi\)
\(920\) −5.10205e9 −0.216017
\(921\) 0 0
\(922\) 1.49833e10 0.629579
\(923\) 4.26381e9 0.178481
\(924\) 0 0
\(925\) −3.60871e10 −1.49919
\(926\) 1.21755e10 0.503906
\(927\) 0 0
\(928\) 1.19392e10 0.490408
\(929\) −3.50768e9 −0.143537 −0.0717687 0.997421i \(-0.522864\pi\)
−0.0717687 + 0.997421i \(0.522864\pi\)
\(930\) 0 0
\(931\) −6.22809e9 −0.252948
\(932\) −2.47217e10 −1.00028
\(933\) 0 0
\(934\) 1.00529e10 0.403717
\(935\) 1.19674e10 0.478805
\(936\) 0 0
\(937\) −1.68124e10 −0.667638 −0.333819 0.942637i \(-0.608337\pi\)
−0.333819 + 0.942637i \(0.608337\pi\)
\(938\) 1.64299e9 0.0650016
\(939\) 0 0
\(940\) −5.67726e9 −0.222942
\(941\) 3.51454e10 1.37501 0.687503 0.726182i \(-0.258707\pi\)
0.687503 + 0.726182i \(0.258707\pi\)
\(942\) 0 0
\(943\) −3.04021e10 −1.18063
\(944\) 7.69368e8 0.0297668
\(945\) 0 0
\(946\) 3.17447e10 1.21914
\(947\) 3.30590e10 1.26493 0.632463 0.774591i \(-0.282044\pi\)
0.632463 + 0.774591i \(0.282044\pi\)
\(948\) 0 0
\(949\) −2.36466e10 −0.898124
\(950\) 1.25399e10 0.474527
\(951\) 0 0
\(952\) −4.07601e10 −1.53111
\(953\) 6.24805e9 0.233840 0.116920 0.993141i \(-0.462698\pi\)
0.116920 + 0.993141i \(0.462698\pi\)
\(954\) 0 0
\(955\) −7.62401e9 −0.283251
\(956\) 2.66728e10 0.987338
\(957\) 0 0
\(958\) 1.85450e10 0.681470
\(959\) 5.91795e9 0.216674
\(960\) 0 0
\(961\) −4.81574e9 −0.175037
\(962\) −2.21586e10 −0.802471
\(963\) 0 0
\(964\) −2.10512e9 −0.0756846
\(965\) 3.77673e9 0.135291
\(966\) 0 0
\(967\) 2.21264e10 0.786899 0.393449 0.919346i \(-0.371282\pi\)
0.393449 + 0.919346i \(0.371282\pi\)
\(968\) 7.72161e9 0.273618
\(969\) 0 0
\(970\) −7.34673e9 −0.258460
\(971\) 5.45446e9 0.191199 0.0955993 0.995420i \(-0.469523\pi\)
0.0955993 + 0.995420i \(0.469523\pi\)
\(972\) 0 0
\(973\) −1.07815e10 −0.375217
\(974\) 1.04264e8 0.00361558
\(975\) 0 0
\(976\) 7.23050e7 0.00248940
\(977\) −4.95373e10 −1.69942 −0.849711 0.527248i \(-0.823224\pi\)
−0.849711 + 0.527248i \(0.823224\pi\)
\(978\) 0 0
\(979\) 2.30609e10 0.785482
\(980\) 1.28835e9 0.0437263
\(981\) 0 0
\(982\) 4.80086e9 0.161781
\(983\) −8.72042e9 −0.292820 −0.146410 0.989224i \(-0.546772\pi\)
−0.146410 + 0.989224i \(0.546772\pi\)
\(984\) 0 0
\(985\) 6.90027e9 0.230059
\(986\) 1.90631e10 0.633320
\(987\) 0 0
\(988\) −9.39330e9 −0.309862
\(989\) 4.33739e10 1.42574
\(990\) 0 0
\(991\) 1.64359e10 0.536458 0.268229 0.963355i \(-0.413562\pi\)
0.268229 + 0.963355i \(0.413562\pi\)
\(992\) −2.62600e10 −0.854091
\(993\) 0 0
\(994\) 4.00641e9 0.129391
\(995\) −1.38618e10 −0.446106
\(996\) 0 0
\(997\) −3.18717e10 −1.01853 −0.509264 0.860611i \(-0.670082\pi\)
−0.509264 + 0.860611i \(0.670082\pi\)
\(998\) 1.84797e10 0.588487
\(999\) 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 27.8.a.e.1.1 yes 2
3.2 odd 2 27.8.a.b.1.2 2
4.3 odd 2 432.8.a.q.1.1 2
9.2 odd 6 81.8.c.h.28.1 4
9.4 even 3 81.8.c.d.55.2 4
9.5 odd 6 81.8.c.h.55.1 4
9.7 even 3 81.8.c.d.28.2 4
12.11 even 2 432.8.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.2 2 3.2 odd 2
27.8.a.e.1.1 yes 2 1.1 even 1 trivial
81.8.c.d.28.2 4 9.7 even 3
81.8.c.d.55.2 4 9.4 even 3
81.8.c.h.28.1 4 9.2 odd 6
81.8.c.h.55.1 4 9.5 odd 6
432.8.a.j.1.2 2 12.11 even 2
432.8.a.q.1.1 2 4.3 odd 2