Properties

Label 81.10.a.c.1.7
Level $81$
Weight $10$
Character 81.1
Self dual yes
Analytic conductor $41.718$
Analytic rank $1$
Dimension $8$
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] = [81,10,Mod(1,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 81.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: \(41.7179027293\)
Analytic rank: \(1\)
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: \( x^{8} - x^{7} - 2930 x^{6} - 1276 x^{5} + 2487472 x^{4} + 3423248 x^{3} - 586568096 x^{2} + \cdots + 965565184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{18}\cdot 17 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(27.6540\) of defining polynomial
Character \(\chi\) \(=\) 81.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+25.6540 q^{2} +146.127 q^{4} +706.451 q^{5} -4568.42 q^{7} -9386.09 q^{8} +18123.3 q^{10} +71433.4 q^{11} -26402.1 q^{13} -117198. q^{14} -315608. q^{16} -314759. q^{17} -904019. q^{19} +103232. q^{20} +1.83255e6 q^{22} -111842. q^{23} -1.45405e6 q^{25} -677318. q^{26} -667571. q^{28} +4.38803e6 q^{29} -9.88785e6 q^{31} -3.29093e6 q^{32} -8.07483e6 q^{34} -3.22736e6 q^{35} -6.32122e6 q^{37} -2.31917e7 q^{38} -6.63082e6 q^{40} +5.36038e6 q^{41} +2.24550e7 q^{43} +1.04384e7 q^{44} -2.86920e6 q^{46} -4.47288e7 q^{47} -1.94832e7 q^{49} -3.73022e7 q^{50} -3.85806e6 q^{52} -4.00972e7 q^{53} +5.04642e7 q^{55} +4.28796e7 q^{56} +1.12571e8 q^{58} -9.58236e7 q^{59} +2.02225e7 q^{61} -2.53663e8 q^{62} +7.71659e7 q^{64} -1.86518e7 q^{65} +1.82498e8 q^{67} -4.59950e7 q^{68} -8.27948e7 q^{70} +1.38543e8 q^{71} -2.17970e8 q^{73} -1.62165e8 q^{74} -1.32102e8 q^{76} -3.26337e8 q^{77} +3.94587e8 q^{79} -2.22962e8 q^{80} +1.37515e8 q^{82} -4.73277e8 q^{83} -2.22362e8 q^{85} +5.76060e8 q^{86} -6.70481e8 q^{88} +2.60507e8 q^{89} +1.20616e8 q^{91} -1.63432e7 q^{92} -1.14747e9 q^{94} -6.38646e8 q^{95} +5.74842e8 q^{97} -4.99822e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 15 q^{2} + 1793 q^{4} - 453 q^{5} + 343 q^{7} - 7239 q^{8} + 510 q^{10} - 99150 q^{11} - 32435 q^{13} - 394824 q^{14} + 328193 q^{16} - 415539 q^{17} - 85277 q^{19} - 1855164 q^{20} - 529359 q^{22}+ \cdots + 2413650159 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 25.6540 1.13376 0.566879 0.823801i \(-0.308151\pi\)
0.566879 + 0.823801i \(0.308151\pi\)
\(3\) 0 0
\(4\) 146.127 0.285405
\(5\) 706.451 0.505495 0.252748 0.967532i \(-0.418666\pi\)
0.252748 + 0.967532i \(0.418666\pi\)
\(6\) 0 0
\(7\) −4568.42 −0.719158 −0.359579 0.933115i \(-0.617080\pi\)
−0.359579 + 0.933115i \(0.617080\pi\)
\(8\) −9386.09 −0.810177
\(9\) 0 0
\(10\) 18123.3 0.573109
\(11\) 71433.4 1.47107 0.735537 0.677485i \(-0.236930\pi\)
0.735537 + 0.677485i \(0.236930\pi\)
\(12\) 0 0
\(13\) −26402.1 −0.256385 −0.128193 0.991749i \(-0.540918\pi\)
−0.128193 + 0.991749i \(0.540918\pi\)
\(14\) −117198. −0.815351
\(15\) 0 0
\(16\) −315608. −1.20395
\(17\) −314759. −0.914026 −0.457013 0.889460i \(-0.651081\pi\)
−0.457013 + 0.889460i \(0.651081\pi\)
\(18\) 0 0
\(19\) −904019. −1.59143 −0.795713 0.605674i \(-0.792904\pi\)
−0.795713 + 0.605674i \(0.792904\pi\)
\(20\) 103232. 0.144271
\(21\) 0 0
\(22\) 1.83255e6 1.66784
\(23\) −111842. −0.0833357 −0.0416678 0.999132i \(-0.513267\pi\)
−0.0416678 + 0.999132i \(0.513267\pi\)
\(24\) 0 0
\(25\) −1.45405e6 −0.744474
\(26\) −677318. −0.290678
\(27\) 0 0
\(28\) −667571. −0.205251
\(29\) 4.38803e6 1.15207 0.576035 0.817425i \(-0.304599\pi\)
0.576035 + 0.817425i \(0.304599\pi\)
\(30\) 0 0
\(31\) −9.88785e6 −1.92298 −0.961488 0.274845i \(-0.911373\pi\)
−0.961488 + 0.274845i \(0.911373\pi\)
\(32\) −3.29093e6 −0.554809
\(33\) 0 0
\(34\) −8.07483e6 −1.03628
\(35\) −3.22736e6 −0.363531
\(36\) 0 0
\(37\) −6.32122e6 −0.554489 −0.277245 0.960799i \(-0.589421\pi\)
−0.277245 + 0.960799i \(0.589421\pi\)
\(38\) −2.31917e7 −1.80429
\(39\) 0 0
\(40\) −6.63082e6 −0.409541
\(41\) 5.36038e6 0.296257 0.148128 0.988968i \(-0.452675\pi\)
0.148128 + 0.988968i \(0.452675\pi\)
\(42\) 0 0
\(43\) 2.24550e7 1.00162 0.500812 0.865556i \(-0.333035\pi\)
0.500812 + 0.865556i \(0.333035\pi\)
\(44\) 1.04384e7 0.419852
\(45\) 0 0
\(46\) −2.86920e6 −0.0944824
\(47\) −4.47288e7 −1.33705 −0.668524 0.743691i \(-0.733074\pi\)
−0.668524 + 0.743691i \(0.733074\pi\)
\(48\) 0 0
\(49\) −1.94832e7 −0.482812
\(50\) −3.73022e7 −0.844053
\(51\) 0 0
\(52\) −3.85806e6 −0.0731736
\(53\) −4.00972e7 −0.698027 −0.349013 0.937118i \(-0.613483\pi\)
−0.349013 + 0.937118i \(0.613483\pi\)
\(54\) 0 0
\(55\) 5.04642e7 0.743621
\(56\) 4.28796e7 0.582645
\(57\) 0 0
\(58\) 1.12571e8 1.30617
\(59\) −9.58236e7 −1.02953 −0.514764 0.857332i \(-0.672121\pi\)
−0.514764 + 0.857332i \(0.672121\pi\)
\(60\) 0 0
\(61\) 2.02225e7 0.187004 0.0935020 0.995619i \(-0.470194\pi\)
0.0935020 + 0.995619i \(0.470194\pi\)
\(62\) −2.53663e8 −2.18019
\(63\) 0 0
\(64\) 7.71659e7 0.574931
\(65\) −1.86518e7 −0.129601
\(66\) 0 0
\(67\) 1.82498e8 1.10642 0.553211 0.833041i \(-0.313402\pi\)
0.553211 + 0.833041i \(0.313402\pi\)
\(68\) −4.59950e7 −0.260868
\(69\) 0 0
\(70\) −8.27948e7 −0.412156
\(71\) 1.38543e8 0.647025 0.323512 0.946224i \(-0.395136\pi\)
0.323512 + 0.946224i \(0.395136\pi\)
\(72\) 0 0
\(73\) −2.17970e8 −0.898347 −0.449173 0.893445i \(-0.648281\pi\)
−0.449173 + 0.893445i \(0.648281\pi\)
\(74\) −1.62165e8 −0.628656
\(75\) 0 0
\(76\) −1.32102e8 −0.454201
\(77\) −3.26337e8 −1.05793
\(78\) 0 0
\(79\) 3.94587e8 1.13978 0.569890 0.821721i \(-0.306986\pi\)
0.569890 + 0.821721i \(0.306986\pi\)
\(80\) −2.22962e8 −0.608591
\(81\) 0 0
\(82\) 1.37515e8 0.335883
\(83\) −4.73277e8 −1.09462 −0.547310 0.836930i \(-0.684348\pi\)
−0.547310 + 0.836930i \(0.684348\pi\)
\(84\) 0 0
\(85\) −2.22362e8 −0.462036
\(86\) 5.76060e8 1.13560
\(87\) 0 0
\(88\) −6.70481e8 −1.19183
\(89\) 2.60507e8 0.440113 0.220057 0.975487i \(-0.429376\pi\)
0.220057 + 0.975487i \(0.429376\pi\)
\(90\) 0 0
\(91\) 1.20616e8 0.184381
\(92\) −1.63432e7 −0.0237844
\(93\) 0 0
\(94\) −1.14747e9 −1.51589
\(95\) −6.38646e8 −0.804459
\(96\) 0 0
\(97\) 5.74842e8 0.659289 0.329644 0.944105i \(-0.393071\pi\)
0.329644 + 0.944105i \(0.393071\pi\)
\(98\) −4.99822e8 −0.547391
\(99\) 0 0
\(100\) −2.12477e8 −0.212477
\(101\) 1.40772e8 0.134608 0.0673039 0.997733i \(-0.478560\pi\)
0.0673039 + 0.997733i \(0.478560\pi\)
\(102\) 0 0
\(103\) −3.41637e7 −0.0299087 −0.0149544 0.999888i \(-0.504760\pi\)
−0.0149544 + 0.999888i \(0.504760\pi\)
\(104\) 2.47812e8 0.207717
\(105\) 0 0
\(106\) −1.02865e9 −0.791393
\(107\) 1.61837e9 1.19358 0.596790 0.802397i \(-0.296443\pi\)
0.596790 + 0.802397i \(0.296443\pi\)
\(108\) 0 0
\(109\) −8.56262e8 −0.581015 −0.290508 0.956873i \(-0.593824\pi\)
−0.290508 + 0.956873i \(0.593824\pi\)
\(110\) 1.29461e9 0.843085
\(111\) 0 0
\(112\) 1.44183e9 0.865830
\(113\) 1.93262e9 1.11505 0.557524 0.830160i \(-0.311751\pi\)
0.557524 + 0.830160i \(0.311751\pi\)
\(114\) 0 0
\(115\) −7.90112e7 −0.0421258
\(116\) 6.41212e8 0.328807
\(117\) 0 0
\(118\) −2.45826e9 −1.16724
\(119\) 1.43795e9 0.657329
\(120\) 0 0
\(121\) 2.74478e9 1.16406
\(122\) 5.18789e8 0.212017
\(123\) 0 0
\(124\) −1.44489e9 −0.548827
\(125\) −2.40700e9 −0.881824
\(126\) 0 0
\(127\) 4.98705e9 1.70109 0.850545 0.525903i \(-0.176272\pi\)
0.850545 + 0.525903i \(0.176272\pi\)
\(128\) 3.66457e9 1.20664
\(129\) 0 0
\(130\) −4.78492e8 −0.146937
\(131\) −1.91672e8 −0.0568641 −0.0284320 0.999596i \(-0.509051\pi\)
−0.0284320 + 0.999596i \(0.509051\pi\)
\(132\) 0 0
\(133\) 4.12994e9 1.14449
\(134\) 4.68180e9 1.25441
\(135\) 0 0
\(136\) 2.95436e9 0.740523
\(137\) 1.00077e9 0.242712 0.121356 0.992609i \(-0.461276\pi\)
0.121356 + 0.992609i \(0.461276\pi\)
\(138\) 0 0
\(139\) −1.35024e9 −0.306792 −0.153396 0.988165i \(-0.549021\pi\)
−0.153396 + 0.988165i \(0.549021\pi\)
\(140\) −4.71606e8 −0.103754
\(141\) 0 0
\(142\) 3.55417e9 0.733569
\(143\) −1.88599e9 −0.377161
\(144\) 0 0
\(145\) 3.09993e9 0.582366
\(146\) −5.59180e9 −1.01851
\(147\) 0 0
\(148\) −9.23704e8 −0.158254
\(149\) 3.52992e9 0.586714 0.293357 0.956003i \(-0.405227\pi\)
0.293357 + 0.956003i \(0.405227\pi\)
\(150\) 0 0
\(151\) 5.72646e8 0.0896376 0.0448188 0.998995i \(-0.485729\pi\)
0.0448188 + 0.998995i \(0.485729\pi\)
\(152\) 8.48521e9 1.28934
\(153\) 0 0
\(154\) −8.37186e9 −1.19944
\(155\) −6.98528e9 −0.972056
\(156\) 0 0
\(157\) 8.40171e8 0.110362 0.0551810 0.998476i \(-0.482426\pi\)
0.0551810 + 0.998476i \(0.482426\pi\)
\(158\) 1.01227e10 1.29223
\(159\) 0 0
\(160\) −2.32488e9 −0.280453
\(161\) 5.10942e8 0.0599315
\(162\) 0 0
\(163\) −7.00425e9 −0.777173 −0.388586 0.921412i \(-0.627037\pi\)
−0.388586 + 0.921412i \(0.627037\pi\)
\(164\) 7.83299e8 0.0845532
\(165\) 0 0
\(166\) −1.21414e10 −1.24103
\(167\) −9.27017e9 −0.922281 −0.461141 0.887327i \(-0.652560\pi\)
−0.461141 + 0.887327i \(0.652560\pi\)
\(168\) 0 0
\(169\) −9.90743e9 −0.934267
\(170\) −5.70448e9 −0.523836
\(171\) 0 0
\(172\) 3.28129e9 0.285869
\(173\) 8.27334e9 0.702220 0.351110 0.936334i \(-0.385804\pi\)
0.351110 + 0.936334i \(0.385804\pi\)
\(174\) 0 0
\(175\) 6.64271e9 0.535395
\(176\) −2.25450e10 −1.77110
\(177\) 0 0
\(178\) 6.68304e9 0.498981
\(179\) −2.36310e10 −1.72046 −0.860229 0.509907i \(-0.829680\pi\)
−0.860229 + 0.509907i \(0.829680\pi\)
\(180\) 0 0
\(181\) 1.40618e10 0.973841 0.486921 0.873446i \(-0.338120\pi\)
0.486921 + 0.873446i \(0.338120\pi\)
\(182\) 3.09427e9 0.209044
\(183\) 0 0
\(184\) 1.04976e9 0.0675166
\(185\) −4.46563e9 −0.280292
\(186\) 0 0
\(187\) −2.24843e10 −1.34460
\(188\) −6.53611e9 −0.381600
\(189\) 0 0
\(190\) −1.63838e10 −0.912061
\(191\) −9.75192e9 −0.530201 −0.265100 0.964221i \(-0.585405\pi\)
−0.265100 + 0.964221i \(0.585405\pi\)
\(192\) 0 0
\(193\) 9.95176e9 0.516288 0.258144 0.966106i \(-0.416889\pi\)
0.258144 + 0.966106i \(0.416889\pi\)
\(194\) 1.47470e10 0.747473
\(195\) 0 0
\(196\) −2.84703e9 −0.137797
\(197\) −3.23117e9 −0.152849 −0.0764244 0.997075i \(-0.524350\pi\)
−0.0764244 + 0.997075i \(0.524350\pi\)
\(198\) 0 0
\(199\) −2.90155e10 −1.31157 −0.655785 0.754947i \(-0.727662\pi\)
−0.655785 + 0.754947i \(0.727662\pi\)
\(200\) 1.36479e10 0.603156
\(201\) 0 0
\(202\) 3.61137e9 0.152613
\(203\) −2.00464e10 −0.828521
\(204\) 0 0
\(205\) 3.78685e9 0.149757
\(206\) −8.76437e8 −0.0339092
\(207\) 0 0
\(208\) 8.33270e9 0.308675
\(209\) −6.45772e10 −2.34110
\(210\) 0 0
\(211\) −4.25536e9 −0.147797 −0.0738985 0.997266i \(-0.523544\pi\)
−0.0738985 + 0.997266i \(0.523544\pi\)
\(212\) −5.85929e9 −0.199220
\(213\) 0 0
\(214\) 4.15177e10 1.35323
\(215\) 1.58634e10 0.506317
\(216\) 0 0
\(217\) 4.51718e10 1.38292
\(218\) −2.19665e10 −0.658730
\(219\) 0 0
\(220\) 7.37421e9 0.212233
\(221\) 8.31029e9 0.234343
\(222\) 0 0
\(223\) 4.23518e10 1.14683 0.573416 0.819265i \(-0.305618\pi\)
0.573416 + 0.819265i \(0.305618\pi\)
\(224\) 1.50343e10 0.398995
\(225\) 0 0
\(226\) 4.95795e10 1.26419
\(227\) 5.92602e10 1.48131 0.740656 0.671884i \(-0.234515\pi\)
0.740656 + 0.671884i \(0.234515\pi\)
\(228\) 0 0
\(229\) −6.15045e10 −1.47791 −0.738954 0.673756i \(-0.764680\pi\)
−0.738954 + 0.673756i \(0.764680\pi\)
\(230\) −2.02695e9 −0.0477604
\(231\) 0 0
\(232\) −4.11865e10 −0.933381
\(233\) −4.69389e10 −1.04335 −0.521677 0.853143i \(-0.674693\pi\)
−0.521677 + 0.853143i \(0.674693\pi\)
\(234\) 0 0
\(235\) −3.15987e10 −0.675872
\(236\) −1.40025e10 −0.293833
\(237\) 0 0
\(238\) 3.68892e10 0.745251
\(239\) 1.04488e10 0.207146 0.103573 0.994622i \(-0.466973\pi\)
0.103573 + 0.994622i \(0.466973\pi\)
\(240\) 0 0
\(241\) 2.99400e10 0.571708 0.285854 0.958273i \(-0.407723\pi\)
0.285854 + 0.958273i \(0.407723\pi\)
\(242\) 7.04147e10 1.31976
\(243\) 0 0
\(244\) 2.95507e9 0.0533719
\(245\) −1.37639e10 −0.244059
\(246\) 0 0
\(247\) 2.38680e10 0.408018
\(248\) 9.28082e10 1.55795
\(249\) 0 0
\(250\) −6.17493e10 −0.999774
\(251\) −1.50652e9 −0.0239576 −0.0119788 0.999928i \(-0.503813\pi\)
−0.0119788 + 0.999928i \(0.503813\pi\)
\(252\) 0 0
\(253\) −7.98928e9 −0.122593
\(254\) 1.27938e11 1.92862
\(255\) 0 0
\(256\) 5.45019e10 0.793106
\(257\) −8.71580e10 −1.24626 −0.623130 0.782119i \(-0.714139\pi\)
−0.623130 + 0.782119i \(0.714139\pi\)
\(258\) 0 0
\(259\) 2.88780e10 0.398765
\(260\) −2.72553e9 −0.0369889
\(261\) 0 0
\(262\) −4.91715e9 −0.0644700
\(263\) −4.21286e10 −0.542970 −0.271485 0.962443i \(-0.587515\pi\)
−0.271485 + 0.962443i \(0.587515\pi\)
\(264\) 0 0
\(265\) −2.83267e10 −0.352849
\(266\) 1.05949e11 1.29757
\(267\) 0 0
\(268\) 2.66679e10 0.315779
\(269\) 6.32241e10 0.736202 0.368101 0.929786i \(-0.380008\pi\)
0.368101 + 0.929786i \(0.380008\pi\)
\(270\) 0 0
\(271\) −1.63811e11 −1.84493 −0.922467 0.386075i \(-0.873831\pi\)
−0.922467 + 0.386075i \(0.873831\pi\)
\(272\) 9.93406e10 1.10044
\(273\) 0 0
\(274\) 2.56738e10 0.275177
\(275\) −1.03868e11 −1.09518
\(276\) 0 0
\(277\) −1.72604e11 −1.76154 −0.880771 0.473542i \(-0.842975\pi\)
−0.880771 + 0.473542i \(0.842975\pi\)
\(278\) −3.46390e10 −0.347827
\(279\) 0 0
\(280\) 3.02923e10 0.294525
\(281\) −2.55429e10 −0.244394 −0.122197 0.992506i \(-0.538994\pi\)
−0.122197 + 0.992506i \(0.538994\pi\)
\(282\) 0 0
\(283\) −7.38934e10 −0.684805 −0.342402 0.939553i \(-0.611241\pi\)
−0.342402 + 0.939553i \(0.611241\pi\)
\(284\) 2.02449e10 0.184664
\(285\) 0 0
\(286\) −4.83831e10 −0.427609
\(287\) −2.44885e10 −0.213056
\(288\) 0 0
\(289\) −1.95145e10 −0.164557
\(290\) 7.95256e10 0.660262
\(291\) 0 0
\(292\) −3.18514e10 −0.256393
\(293\) 2.03535e11 1.61337 0.806687 0.590979i \(-0.201258\pi\)
0.806687 + 0.590979i \(0.201258\pi\)
\(294\) 0 0
\(295\) −6.76947e10 −0.520422
\(296\) 5.93316e10 0.449234
\(297\) 0 0
\(298\) 9.05566e10 0.665192
\(299\) 2.95287e9 0.0213660
\(300\) 0 0
\(301\) −1.02584e11 −0.720326
\(302\) 1.46907e10 0.101627
\(303\) 0 0
\(304\) 2.85316e11 1.91600
\(305\) 1.42862e10 0.0945297
\(306\) 0 0
\(307\) 8.22523e10 0.528476 0.264238 0.964457i \(-0.414880\pi\)
0.264238 + 0.964457i \(0.414880\pi\)
\(308\) −4.76869e10 −0.301940
\(309\) 0 0
\(310\) −1.79200e11 −1.10208
\(311\) −2.32094e11 −1.40683 −0.703415 0.710780i \(-0.748342\pi\)
−0.703415 + 0.710780i \(0.748342\pi\)
\(312\) 0 0
\(313\) 6.30080e10 0.371062 0.185531 0.982638i \(-0.440600\pi\)
0.185531 + 0.982638i \(0.440600\pi\)
\(314\) 2.15537e10 0.125124
\(315\) 0 0
\(316\) 5.76600e10 0.325299
\(317\) −2.31470e11 −1.28744 −0.643722 0.765260i \(-0.722611\pi\)
−0.643722 + 0.765260i \(0.722611\pi\)
\(318\) 0 0
\(319\) 3.13452e11 1.69478
\(320\) 5.45140e10 0.290625
\(321\) 0 0
\(322\) 1.31077e10 0.0679478
\(323\) 2.84548e11 1.45460
\(324\) 0 0
\(325\) 3.83899e10 0.190872
\(326\) −1.79687e11 −0.881125
\(327\) 0 0
\(328\) −5.03131e10 −0.240021
\(329\) 2.04340e11 0.961549
\(330\) 0 0
\(331\) 1.54455e11 0.707254 0.353627 0.935386i \(-0.384948\pi\)
0.353627 + 0.935386i \(0.384948\pi\)
\(332\) −6.91587e10 −0.312410
\(333\) 0 0
\(334\) −2.37817e11 −1.04564
\(335\) 1.28926e11 0.559291
\(336\) 0 0
\(337\) 1.35250e11 0.571218 0.285609 0.958346i \(-0.407804\pi\)
0.285609 + 0.958346i \(0.407804\pi\)
\(338\) −2.54165e11 −1.05923
\(339\) 0 0
\(340\) −3.24932e10 −0.131867
\(341\) −7.06323e11 −2.82884
\(342\) 0 0
\(343\) 2.73359e11 1.06638
\(344\) −2.10765e11 −0.811493
\(345\) 0 0
\(346\) 2.12244e11 0.796147
\(347\) −2.83506e11 −1.04973 −0.524867 0.851184i \(-0.675885\pi\)
−0.524867 + 0.851184i \(0.675885\pi\)
\(348\) 0 0
\(349\) 9.73597e10 0.351289 0.175645 0.984454i \(-0.443799\pi\)
0.175645 + 0.984454i \(0.443799\pi\)
\(350\) 1.70412e11 0.607008
\(351\) 0 0
\(352\) −2.35082e11 −0.816164
\(353\) 3.83723e10 0.131532 0.0657660 0.997835i \(-0.479051\pi\)
0.0657660 + 0.997835i \(0.479051\pi\)
\(354\) 0 0
\(355\) 9.78737e10 0.327068
\(356\) 3.80672e10 0.125611
\(357\) 0 0
\(358\) −6.06231e11 −1.95058
\(359\) −4.48036e11 −1.42360 −0.711800 0.702383i \(-0.752120\pi\)
−0.711800 + 0.702383i \(0.752120\pi\)
\(360\) 0 0
\(361\) 4.94563e11 1.53264
\(362\) 3.60742e11 1.10410
\(363\) 0 0
\(364\) 1.76252e10 0.0526234
\(365\) −1.53985e11 −0.454110
\(366\) 0 0
\(367\) −7.48073e10 −0.215252 −0.107626 0.994191i \(-0.534325\pi\)
−0.107626 + 0.994191i \(0.534325\pi\)
\(368\) 3.52983e10 0.100332
\(369\) 0 0
\(370\) −1.14561e11 −0.317783
\(371\) 1.83180e11 0.501992
\(372\) 0 0
\(373\) −2.79392e11 −0.747351 −0.373675 0.927560i \(-0.621903\pi\)
−0.373675 + 0.927560i \(0.621903\pi\)
\(374\) −5.76813e11 −1.52445
\(375\) 0 0
\(376\) 4.19829e11 1.08325
\(377\) −1.15853e11 −0.295374
\(378\) 0 0
\(379\) −6.40616e11 −1.59485 −0.797427 0.603415i \(-0.793806\pi\)
−0.797427 + 0.603415i \(0.793806\pi\)
\(380\) −9.33236e10 −0.229597
\(381\) 0 0
\(382\) −2.50176e11 −0.601119
\(383\) 3.33056e11 0.790903 0.395451 0.918487i \(-0.370588\pi\)
0.395451 + 0.918487i \(0.370588\pi\)
\(384\) 0 0
\(385\) −2.30542e11 −0.534781
\(386\) 2.55302e11 0.585345
\(387\) 0 0
\(388\) 8.40002e10 0.188164
\(389\) −1.63159e11 −0.361275 −0.180638 0.983550i \(-0.557816\pi\)
−0.180638 + 0.983550i \(0.557816\pi\)
\(390\) 0 0
\(391\) 3.52034e10 0.0761709
\(392\) 1.82871e11 0.391163
\(393\) 0 0
\(394\) −8.28925e10 −0.173293
\(395\) 2.78757e11 0.576153
\(396\) 0 0
\(397\) 2.79119e11 0.563939 0.281969 0.959423i \(-0.409012\pi\)
0.281969 + 0.959423i \(0.409012\pi\)
\(398\) −7.44364e11 −1.48700
\(399\) 0 0
\(400\) 4.58910e11 0.896309
\(401\) 7.59932e11 1.46766 0.733830 0.679333i \(-0.237731\pi\)
0.733830 + 0.679333i \(0.237731\pi\)
\(402\) 0 0
\(403\) 2.61059e11 0.493023
\(404\) 2.05707e10 0.0384178
\(405\) 0 0
\(406\) −5.14269e11 −0.939341
\(407\) −4.51546e11 −0.815694
\(408\) 0 0
\(409\) −3.79941e10 −0.0671369 −0.0335685 0.999436i \(-0.510687\pi\)
−0.0335685 + 0.999436i \(0.510687\pi\)
\(410\) 9.71478e10 0.169788
\(411\) 0 0
\(412\) −4.99226e9 −0.00853610
\(413\) 4.37762e11 0.740394
\(414\) 0 0
\(415\) −3.34347e11 −0.553326
\(416\) 8.68872e10 0.142245
\(417\) 0 0
\(418\) −1.65666e12 −2.65424
\(419\) −7.33645e11 −1.16285 −0.581424 0.813601i \(-0.697504\pi\)
−0.581424 + 0.813601i \(0.697504\pi\)
\(420\) 0 0
\(421\) −3.21432e11 −0.498677 −0.249339 0.968416i \(-0.580213\pi\)
−0.249339 + 0.968416i \(0.580213\pi\)
\(422\) −1.09167e11 −0.167566
\(423\) 0 0
\(424\) 3.76356e11 0.565525
\(425\) 4.57676e11 0.680469
\(426\) 0 0
\(427\) −9.23849e10 −0.134486
\(428\) 2.36489e11 0.340654
\(429\) 0 0
\(430\) 4.06959e11 0.574040
\(431\) 3.86098e11 0.538952 0.269476 0.963007i \(-0.413150\pi\)
0.269476 + 0.963007i \(0.413150\pi\)
\(432\) 0 0
\(433\) 5.48808e11 0.750282 0.375141 0.926968i \(-0.377594\pi\)
0.375141 + 0.926968i \(0.377594\pi\)
\(434\) 1.15884e12 1.56790
\(435\) 0 0
\(436\) −1.25123e11 −0.165825
\(437\) 1.01108e11 0.132623
\(438\) 0 0
\(439\) −3.00100e11 −0.385635 −0.192817 0.981235i \(-0.561762\pi\)
−0.192817 + 0.981235i \(0.561762\pi\)
\(440\) −4.73662e11 −0.602465
\(441\) 0 0
\(442\) 2.13192e11 0.265687
\(443\) 4.22660e11 0.521404 0.260702 0.965419i \(-0.416046\pi\)
0.260702 + 0.965419i \(0.416046\pi\)
\(444\) 0 0
\(445\) 1.84036e11 0.222475
\(446\) 1.08649e12 1.30023
\(447\) 0 0
\(448\) −3.52526e11 −0.413466
\(449\) 1.63406e12 1.89741 0.948704 0.316165i \(-0.102395\pi\)
0.948704 + 0.316165i \(0.102395\pi\)
\(450\) 0 0
\(451\) 3.82911e11 0.435816
\(452\) 2.82409e11 0.318241
\(453\) 0 0
\(454\) 1.52026e12 1.67945
\(455\) 8.52090e10 0.0932040
\(456\) 0 0
\(457\) 1.24637e12 1.33667 0.668334 0.743862i \(-0.267008\pi\)
0.668334 + 0.743862i \(0.267008\pi\)
\(458\) −1.57784e12 −1.67559
\(459\) 0 0
\(460\) −1.15457e10 −0.0120229
\(461\) 2.88051e11 0.297040 0.148520 0.988909i \(-0.452549\pi\)
0.148520 + 0.988909i \(0.452549\pi\)
\(462\) 0 0
\(463\) 1.57266e12 1.59045 0.795225 0.606315i \(-0.207353\pi\)
0.795225 + 0.606315i \(0.207353\pi\)
\(464\) −1.38490e12 −1.38703
\(465\) 0 0
\(466\) −1.20417e12 −1.18291
\(467\) 2.12882e11 0.207116 0.103558 0.994623i \(-0.466977\pi\)
0.103558 + 0.994623i \(0.466977\pi\)
\(468\) 0 0
\(469\) −8.33726e11 −0.795693
\(470\) −8.10634e11 −0.766274
\(471\) 0 0
\(472\) 8.99409e11 0.834100
\(473\) 1.60404e12 1.47346
\(474\) 0 0
\(475\) 1.31449e12 1.18478
\(476\) 2.10124e11 0.187605
\(477\) 0 0
\(478\) 2.68053e11 0.234853
\(479\) 1.52484e12 1.32347 0.661733 0.749739i \(-0.269821\pi\)
0.661733 + 0.749739i \(0.269821\pi\)
\(480\) 0 0
\(481\) 1.66893e11 0.142163
\(482\) 7.68079e11 0.648178
\(483\) 0 0
\(484\) 4.01088e11 0.332228
\(485\) 4.06098e11 0.333267
\(486\) 0 0
\(487\) −1.81008e12 −1.45820 −0.729101 0.684406i \(-0.760062\pi\)
−0.729101 + 0.684406i \(0.760062\pi\)
\(488\) −1.89810e11 −0.151506
\(489\) 0 0
\(490\) −3.53100e11 −0.276704
\(491\) −1.15383e12 −0.895931 −0.447966 0.894051i \(-0.647851\pi\)
−0.447966 + 0.894051i \(0.647851\pi\)
\(492\) 0 0
\(493\) −1.38117e12 −1.05302
\(494\) 6.12309e11 0.462593
\(495\) 0 0
\(496\) 3.12068e12 2.31517
\(497\) −6.32921e11 −0.465313
\(498\) 0 0
\(499\) −2.01943e12 −1.45806 −0.729032 0.684479i \(-0.760030\pi\)
−0.729032 + 0.684479i \(0.760030\pi\)
\(500\) −3.51729e11 −0.251677
\(501\) 0 0
\(502\) −3.86482e10 −0.0271621
\(503\) −7.16624e11 −0.499155 −0.249578 0.968355i \(-0.580292\pi\)
−0.249578 + 0.968355i \(0.580292\pi\)
\(504\) 0 0
\(505\) 9.94486e10 0.0680437
\(506\) −2.04957e11 −0.138991
\(507\) 0 0
\(508\) 7.28745e11 0.485500
\(509\) −1.58134e12 −1.04423 −0.522114 0.852876i \(-0.674856\pi\)
−0.522114 + 0.852876i \(0.674856\pi\)
\(510\) 0 0
\(511\) 9.95778e11 0.646053
\(512\) −4.78068e11 −0.307450
\(513\) 0 0
\(514\) −2.23595e12 −1.41296
\(515\) −2.41350e10 −0.0151187
\(516\) 0 0
\(517\) −3.19513e12 −1.96690
\(518\) 7.40835e11 0.452103
\(519\) 0 0
\(520\) 1.75067e11 0.105000
\(521\) 1.03065e12 0.612834 0.306417 0.951897i \(-0.400870\pi\)
0.306417 + 0.951897i \(0.400870\pi\)
\(522\) 0 0
\(523\) 2.03811e12 1.19116 0.595579 0.803297i \(-0.296923\pi\)
0.595579 + 0.803297i \(0.296923\pi\)
\(524\) −2.80085e10 −0.0162293
\(525\) 0 0
\(526\) −1.08077e12 −0.615596
\(527\) 3.11229e12 1.75765
\(528\) 0 0
\(529\) −1.78864e12 −0.993055
\(530\) −7.26693e11 −0.400046
\(531\) 0 0
\(532\) 6.03497e11 0.326642
\(533\) −1.41525e11 −0.0759559
\(534\) 0 0
\(535\) 1.14330e12 0.603349
\(536\) −1.71294e12 −0.896398
\(537\) 0 0
\(538\) 1.62195e12 0.834675
\(539\) −1.39175e12 −0.710251
\(540\) 0 0
\(541\) −1.53998e12 −0.772905 −0.386453 0.922309i \(-0.626300\pi\)
−0.386453 + 0.922309i \(0.626300\pi\)
\(542\) −4.20240e12 −2.09171
\(543\) 0 0
\(544\) 1.03585e12 0.507109
\(545\) −6.04907e11 −0.293700
\(546\) 0 0
\(547\) −1.91921e12 −0.916598 −0.458299 0.888798i \(-0.651541\pi\)
−0.458299 + 0.888798i \(0.651541\pi\)
\(548\) 1.46240e11 0.0692713
\(549\) 0 0
\(550\) −2.66463e12 −1.24166
\(551\) −3.96687e12 −1.83343
\(552\) 0 0
\(553\) −1.80264e12 −0.819682
\(554\) −4.42799e12 −1.99716
\(555\) 0 0
\(556\) −1.97307e11 −0.0875599
\(557\) −2.71058e11 −0.119320 −0.0596601 0.998219i \(-0.519002\pi\)
−0.0596601 + 0.998219i \(0.519002\pi\)
\(558\) 0 0
\(559\) −5.92858e11 −0.256802
\(560\) 1.01858e12 0.437673
\(561\) 0 0
\(562\) −6.55277e11 −0.277084
\(563\) 3.08234e11 0.129298 0.0646492 0.997908i \(-0.479407\pi\)
0.0646492 + 0.997908i \(0.479407\pi\)
\(564\) 0 0
\(565\) 1.36530e12 0.563652
\(566\) −1.89566e12 −0.776402
\(567\) 0 0
\(568\) −1.30037e12 −0.524205
\(569\) 3.58165e12 1.43244 0.716222 0.697872i \(-0.245870\pi\)
0.716222 + 0.697872i \(0.245870\pi\)
\(570\) 0 0
\(571\) −8.90822e10 −0.0350694 −0.0175347 0.999846i \(-0.505582\pi\)
−0.0175347 + 0.999846i \(0.505582\pi\)
\(572\) −2.75595e11 −0.107644
\(573\) 0 0
\(574\) −6.28227e11 −0.241553
\(575\) 1.62624e11 0.0620413
\(576\) 0 0
\(577\) −4.29975e11 −0.161492 −0.0807462 0.996735i \(-0.525730\pi\)
−0.0807462 + 0.996735i \(0.525730\pi\)
\(578\) −5.00624e11 −0.186568
\(579\) 0 0
\(580\) 4.52985e11 0.166210
\(581\) 2.16212e12 0.787205
\(582\) 0 0
\(583\) −2.86428e12 −1.02685
\(584\) 2.04589e12 0.727820
\(585\) 0 0
\(586\) 5.22149e12 1.82917
\(587\) 2.27007e12 0.789164 0.394582 0.918861i \(-0.370889\pi\)
0.394582 + 0.918861i \(0.370889\pi\)
\(588\) 0 0
\(589\) 8.93880e12 3.06028
\(590\) −1.73664e12 −0.590032
\(591\) 0 0
\(592\) 1.99503e12 0.667577
\(593\) −4.09944e12 −1.36138 −0.680688 0.732573i \(-0.738319\pi\)
−0.680688 + 0.732573i \(0.738319\pi\)
\(594\) 0 0
\(595\) 1.01584e12 0.332277
\(596\) 5.15818e11 0.167451
\(597\) 0 0
\(598\) 7.57528e10 0.0242239
\(599\) −1.61831e12 −0.513618 −0.256809 0.966462i \(-0.582671\pi\)
−0.256809 + 0.966462i \(0.582671\pi\)
\(600\) 0 0
\(601\) 1.72161e12 0.538269 0.269134 0.963103i \(-0.413262\pi\)
0.269134 + 0.963103i \(0.413262\pi\)
\(602\) −2.63168e12 −0.816675
\(603\) 0 0
\(604\) 8.36793e10 0.0255830
\(605\) 1.93906e12 0.588425
\(606\) 0 0
\(607\) −4.14056e12 −1.23797 −0.618985 0.785402i \(-0.712456\pi\)
−0.618985 + 0.785402i \(0.712456\pi\)
\(608\) 2.97506e12 0.882937
\(609\) 0 0
\(610\) 3.66499e11 0.107174
\(611\) 1.18093e12 0.342799
\(612\) 0 0
\(613\) 1.88290e12 0.538585 0.269292 0.963058i \(-0.413210\pi\)
0.269292 + 0.963058i \(0.413210\pi\)
\(614\) 2.11010e12 0.599163
\(615\) 0 0
\(616\) 3.06303e12 0.857114
\(617\) −2.91305e12 −0.809217 −0.404609 0.914490i \(-0.632592\pi\)
−0.404609 + 0.914490i \(0.632592\pi\)
\(618\) 0 0
\(619\) −2.47495e12 −0.677577 −0.338789 0.940863i \(-0.610017\pi\)
−0.338789 + 0.940863i \(0.610017\pi\)
\(620\) −1.02074e12 −0.277430
\(621\) 0 0
\(622\) −5.95413e12 −1.59500
\(623\) −1.19010e12 −0.316511
\(624\) 0 0
\(625\) 1.13951e12 0.298716
\(626\) 1.61641e12 0.420694
\(627\) 0 0
\(628\) 1.22772e11 0.0314979
\(629\) 1.98966e12 0.506817
\(630\) 0 0
\(631\) 5.10645e12 1.28229 0.641146 0.767419i \(-0.278459\pi\)
0.641146 + 0.767419i \(0.278459\pi\)
\(632\) −3.70363e12 −0.923423
\(633\) 0 0
\(634\) −5.93813e12 −1.45965
\(635\) 3.52311e12 0.859893
\(636\) 0 0
\(637\) 5.14396e11 0.123786
\(638\) 8.04130e12 1.92147
\(639\) 0 0
\(640\) 2.58884e12 0.609951
\(641\) −6.06494e11 −0.141894 −0.0709472 0.997480i \(-0.522602\pi\)
−0.0709472 + 0.997480i \(0.522602\pi\)
\(642\) 0 0
\(643\) −2.50984e12 −0.579024 −0.289512 0.957174i \(-0.593493\pi\)
−0.289512 + 0.957174i \(0.593493\pi\)
\(644\) 7.46627e10 0.0171048
\(645\) 0 0
\(646\) 7.29980e12 1.64917
\(647\) 2.81592e12 0.631758 0.315879 0.948799i \(-0.397701\pi\)
0.315879 + 0.948799i \(0.397701\pi\)
\(648\) 0 0
\(649\) −6.84501e12 −1.51451
\(650\) 9.84856e11 0.216403
\(651\) 0 0
\(652\) −1.02351e12 −0.221809
\(653\) 1.08372e11 0.0233242 0.0116621 0.999932i \(-0.496288\pi\)
0.0116621 + 0.999932i \(0.496288\pi\)
\(654\) 0 0
\(655\) −1.35407e11 −0.0287445
\(656\) −1.69178e12 −0.356678
\(657\) 0 0
\(658\) 5.24213e12 1.09016
\(659\) −6.52712e12 −1.34815 −0.674073 0.738665i \(-0.735457\pi\)
−0.674073 + 0.738665i \(0.735457\pi\)
\(660\) 0 0
\(661\) 4.28485e12 0.873029 0.436515 0.899697i \(-0.356213\pi\)
0.436515 + 0.899697i \(0.356213\pi\)
\(662\) 3.96238e12 0.801855
\(663\) 0 0
\(664\) 4.44222e12 0.886836
\(665\) 2.91760e12 0.578533
\(666\) 0 0
\(667\) −4.90768e11 −0.0960085
\(668\) −1.35463e12 −0.263224
\(669\) 0 0
\(670\) 3.30746e12 0.634101
\(671\) 1.44456e12 0.275097
\(672\) 0 0
\(673\) 4.97112e12 0.934086 0.467043 0.884235i \(-0.345319\pi\)
0.467043 + 0.884235i \(0.345319\pi\)
\(674\) 3.46970e12 0.647623
\(675\) 0 0
\(676\) −1.44775e12 −0.266644
\(677\) 7.64288e12 1.39833 0.699163 0.714963i \(-0.253556\pi\)
0.699163 + 0.714963i \(0.253556\pi\)
\(678\) 0 0
\(679\) −2.62612e12 −0.474133
\(680\) 2.08711e12 0.374331
\(681\) 0 0
\(682\) −1.81200e13 −3.20722
\(683\) −4.20646e12 −0.739646 −0.369823 0.929102i \(-0.620582\pi\)
−0.369823 + 0.929102i \(0.620582\pi\)
\(684\) 0 0
\(685\) 7.06995e11 0.122690
\(686\) 7.01276e12 1.20901
\(687\) 0 0
\(688\) −7.08698e12 −1.20590
\(689\) 1.05865e12 0.178964
\(690\) 0 0
\(691\) −2.33689e12 −0.389931 −0.194966 0.980810i \(-0.562460\pi\)
−0.194966 + 0.980810i \(0.562460\pi\)
\(692\) 1.20896e12 0.200417
\(693\) 0 0
\(694\) −7.27305e12 −1.19014
\(695\) −9.53877e11 −0.155082
\(696\) 0 0
\(697\) −1.68723e12 −0.270786
\(698\) 2.49767e12 0.398277
\(699\) 0 0
\(700\) 9.70682e11 0.152804
\(701\) 1.10126e12 0.172250 0.0861251 0.996284i \(-0.472552\pi\)
0.0861251 + 0.996284i \(0.472552\pi\)
\(702\) 0 0
\(703\) 5.71450e12 0.882429
\(704\) 5.51222e12 0.845765
\(705\) 0 0
\(706\) 9.84403e11 0.149125
\(707\) −6.43105e11 −0.0968043
\(708\) 0 0
\(709\) 1.31550e12 0.195516 0.0977578 0.995210i \(-0.468833\pi\)
0.0977578 + 0.995210i \(0.468833\pi\)
\(710\) 2.51085e12 0.370816
\(711\) 0 0
\(712\) −2.44514e12 −0.356570
\(713\) 1.10588e12 0.160253
\(714\) 0 0
\(715\) −1.33236e12 −0.190653
\(716\) −3.45314e12 −0.491028
\(717\) 0 0
\(718\) −1.14939e13 −1.61402
\(719\) −1.32450e13 −1.84830 −0.924151 0.382027i \(-0.875226\pi\)
−0.924151 + 0.382027i \(0.875226\pi\)
\(720\) 0 0
\(721\) 1.56074e11 0.0215091
\(722\) 1.26875e13 1.73764
\(723\) 0 0
\(724\) 2.05482e12 0.277939
\(725\) −6.38043e12 −0.857687
\(726\) 0 0
\(727\) 4.71696e12 0.626264 0.313132 0.949710i \(-0.398622\pi\)
0.313132 + 0.949710i \(0.398622\pi\)
\(728\) −1.13211e12 −0.149382
\(729\) 0 0
\(730\) −3.95034e12 −0.514851
\(731\) −7.06792e12 −0.915511
\(732\) 0 0
\(733\) 1.65912e12 0.212280 0.106140 0.994351i \(-0.466151\pi\)
0.106140 + 0.994351i \(0.466151\pi\)
\(734\) −1.91911e12 −0.244043
\(735\) 0 0
\(736\) 3.68065e11 0.0462353
\(737\) 1.30364e13 1.62763
\(738\) 0 0
\(739\) −5.72028e12 −0.705533 −0.352767 0.935711i \(-0.614759\pi\)
−0.352767 + 0.935711i \(0.614759\pi\)
\(740\) −6.52552e11 −0.0799967
\(741\) 0 0
\(742\) 4.69931e12 0.569137
\(743\) −6.71743e12 −0.808637 −0.404318 0.914618i \(-0.632491\pi\)
−0.404318 + 0.914618i \(0.632491\pi\)
\(744\) 0 0
\(745\) 2.49372e12 0.296581
\(746\) −7.16752e12 −0.847314
\(747\) 0 0
\(748\) −3.28558e12 −0.383755
\(749\) −7.39340e12 −0.858373
\(750\) 0 0
\(751\) 9.11015e12 1.04507 0.522536 0.852618i \(-0.324986\pi\)
0.522536 + 0.852618i \(0.324986\pi\)
\(752\) 1.41168e13 1.60974
\(753\) 0 0
\(754\) −2.97209e12 −0.334882
\(755\) 4.04547e11 0.0453114
\(756\) 0 0
\(757\) −4.86606e12 −0.538575 −0.269288 0.963060i \(-0.586788\pi\)
−0.269288 + 0.963060i \(0.586788\pi\)
\(758\) −1.64343e13 −1.80818
\(759\) 0 0
\(760\) 5.99439e12 0.651754
\(761\) 3.71178e12 0.401191 0.200596 0.979674i \(-0.435712\pi\)
0.200596 + 0.979674i \(0.435712\pi\)
\(762\) 0 0
\(763\) 3.91176e12 0.417842
\(764\) −1.42502e12 −0.151322
\(765\) 0 0
\(766\) 8.54423e12 0.896692
\(767\) 2.52994e12 0.263956
\(768\) 0 0
\(769\) −6.63302e12 −0.683979 −0.341989 0.939704i \(-0.611101\pi\)
−0.341989 + 0.939704i \(0.611101\pi\)
\(770\) −5.91431e12 −0.606312
\(771\) 0 0
\(772\) 1.45422e12 0.147351
\(773\) −7.34772e12 −0.740193 −0.370096 0.928993i \(-0.620675\pi\)
−0.370096 + 0.928993i \(0.620675\pi\)
\(774\) 0 0
\(775\) 1.43774e13 1.43161
\(776\) −5.39552e12 −0.534140
\(777\) 0 0
\(778\) −4.18568e12 −0.409598
\(779\) −4.84589e12 −0.471471
\(780\) 0 0
\(781\) 9.89658e12 0.951821
\(782\) 9.03108e11 0.0863593
\(783\) 0 0
\(784\) 6.14905e12 0.581281
\(785\) 5.93540e11 0.0557875
\(786\) 0 0
\(787\) −3.81365e12 −0.354368 −0.177184 0.984178i \(-0.556699\pi\)
−0.177184 + 0.984178i \(0.556699\pi\)
\(788\) −4.72163e11 −0.0436238
\(789\) 0 0
\(790\) 7.15122e12 0.653218
\(791\) −8.82902e12 −0.801897
\(792\) 0 0
\(793\) −5.33916e11 −0.0479451
\(794\) 7.16051e12 0.639369
\(795\) 0 0
\(796\) −4.23996e12 −0.374329
\(797\) −1.78732e13 −1.56906 −0.784529 0.620092i \(-0.787095\pi\)
−0.784529 + 0.620092i \(0.787095\pi\)
\(798\) 0 0
\(799\) 1.40788e13 1.22210
\(800\) 4.78518e12 0.413041
\(801\) 0 0
\(802\) 1.94953e13 1.66397
\(803\) −1.55703e13 −1.32153
\(804\) 0 0
\(805\) 3.60956e11 0.0302951
\(806\) 6.69722e12 0.558968
\(807\) 0 0
\(808\) −1.32130e12 −0.109056
\(809\) 1.16949e13 0.959908 0.479954 0.877294i \(-0.340653\pi\)
0.479954 + 0.877294i \(0.340653\pi\)
\(810\) 0 0
\(811\) −3.78089e12 −0.306902 −0.153451 0.988156i \(-0.549039\pi\)
−0.153451 + 0.988156i \(0.549039\pi\)
\(812\) −2.92932e12 −0.236464
\(813\) 0 0
\(814\) −1.15840e13 −0.924799
\(815\) −4.94817e12 −0.392857
\(816\) 0 0
\(817\) −2.02997e13 −1.59401
\(818\) −9.74701e11 −0.0761170
\(819\) 0 0
\(820\) 5.53363e11 0.0427413
\(821\) 7.32598e12 0.562757 0.281379 0.959597i \(-0.409208\pi\)
0.281379 + 0.959597i \(0.409208\pi\)
\(822\) 0 0
\(823\) 6.46115e12 0.490920 0.245460 0.969407i \(-0.421061\pi\)
0.245460 + 0.969407i \(0.421061\pi\)
\(824\) 3.20664e11 0.0242314
\(825\) 0 0
\(826\) 1.12303e13 0.839427
\(827\) −1.17187e13 −0.871176 −0.435588 0.900146i \(-0.643460\pi\)
−0.435588 + 0.900146i \(0.643460\pi\)
\(828\) 0 0
\(829\) −2.81041e12 −0.206669 −0.103334 0.994647i \(-0.532951\pi\)
−0.103334 + 0.994647i \(0.532951\pi\)
\(830\) −8.57733e12 −0.627337
\(831\) 0 0
\(832\) −2.03734e12 −0.147404
\(833\) 6.13251e12 0.441302
\(834\) 0 0
\(835\) −6.54892e12 −0.466209
\(836\) −9.43650e12 −0.668163
\(837\) 0 0
\(838\) −1.88209e13 −1.31839
\(839\) −1.60278e13 −1.11672 −0.558361 0.829598i \(-0.688570\pi\)
−0.558361 + 0.829598i \(0.688570\pi\)
\(840\) 0 0
\(841\) 4.74769e12 0.327266
\(842\) −8.24601e12 −0.565379
\(843\) 0 0
\(844\) −6.21825e11 −0.0421820
\(845\) −6.99912e12 −0.472268
\(846\) 0 0
\(847\) −1.25393e13 −0.837141
\(848\) 1.26550e13 0.840389
\(849\) 0 0
\(850\) 1.17412e13 0.771486
\(851\) 7.06980e11 0.0462087
\(852\) 0 0
\(853\) −1.79323e13 −1.15975 −0.579875 0.814705i \(-0.696899\pi\)
−0.579875 + 0.814705i \(0.696899\pi\)
\(854\) −2.37004e12 −0.152474
\(855\) 0 0
\(856\) −1.51902e13 −0.967011
\(857\) −4.36164e12 −0.276208 −0.138104 0.990418i \(-0.544101\pi\)
−0.138104 + 0.990418i \(0.544101\pi\)
\(858\) 0 0
\(859\) 7.35160e12 0.460694 0.230347 0.973109i \(-0.426014\pi\)
0.230347 + 0.973109i \(0.426014\pi\)
\(860\) 2.31807e12 0.144505
\(861\) 0 0
\(862\) 9.90496e12 0.611041
\(863\) 1.41639e13 0.869231 0.434615 0.900616i \(-0.356884\pi\)
0.434615 + 0.900616i \(0.356884\pi\)
\(864\) 0 0
\(865\) 5.84471e12 0.354969
\(866\) 1.40791e13 0.850638
\(867\) 0 0
\(868\) 6.60084e12 0.394694
\(869\) 2.81867e13 1.67670
\(870\) 0 0
\(871\) −4.81832e12 −0.283670
\(872\) 8.03695e12 0.470725
\(873\) 0 0
\(874\) 2.59381e12 0.150362
\(875\) 1.09962e13 0.634171
\(876\) 0 0
\(877\) −4.68660e12 −0.267522 −0.133761 0.991014i \(-0.542706\pi\)
−0.133761 + 0.991014i \(0.542706\pi\)
\(878\) −7.69877e12 −0.437216
\(879\) 0 0
\(880\) −1.59269e13 −0.895282
\(881\) 1.22598e13 0.685635 0.342817 0.939402i \(-0.388619\pi\)
0.342817 + 0.939402i \(0.388619\pi\)
\(882\) 0 0
\(883\) −6.77194e12 −0.374878 −0.187439 0.982276i \(-0.560019\pi\)
−0.187439 + 0.982276i \(0.560019\pi\)
\(884\) 1.21436e12 0.0668826
\(885\) 0 0
\(886\) 1.08429e13 0.591145
\(887\) −2.17402e13 −1.17925 −0.589627 0.807675i \(-0.700725\pi\)
−0.589627 + 0.807675i \(0.700725\pi\)
\(888\) 0 0
\(889\) −2.27829e13 −1.22335
\(890\) 4.72125e12 0.252233
\(891\) 0 0
\(892\) 6.18875e12 0.327311
\(893\) 4.04357e13 2.12781
\(894\) 0 0
\(895\) −1.66942e13 −0.869684
\(896\) −1.67413e13 −0.867765
\(897\) 0 0
\(898\) 4.19203e13 2.15120
\(899\) −4.33882e13 −2.21540
\(900\) 0 0
\(901\) 1.26210e13 0.638015
\(902\) 9.82318e12 0.494109
\(903\) 0 0
\(904\) −1.81398e13 −0.903387
\(905\) 9.93399e12 0.492272
\(906\) 0 0
\(907\) −2.27896e13 −1.11816 −0.559079 0.829114i \(-0.688845\pi\)
−0.559079 + 0.829114i \(0.688845\pi\)
\(908\) 8.65954e12 0.422774
\(909\) 0 0
\(910\) 2.18595e12 0.105671
\(911\) −1.12776e13 −0.542482 −0.271241 0.962511i \(-0.587434\pi\)
−0.271241 + 0.962511i \(0.587434\pi\)
\(912\) 0 0
\(913\) −3.38078e13 −1.61027
\(914\) 3.19743e13 1.51546
\(915\) 0 0
\(916\) −8.98750e12 −0.421803
\(917\) 8.75637e11 0.0408942
\(918\) 0 0
\(919\) −2.27954e12 −0.105421 −0.0527105 0.998610i \(-0.516786\pi\)
−0.0527105 + 0.998610i \(0.516786\pi\)
\(920\) 7.41606e11 0.0341294
\(921\) 0 0
\(922\) 7.38966e12 0.336771
\(923\) −3.65781e12 −0.165888
\(924\) 0 0
\(925\) 9.19138e12 0.412803
\(926\) 4.03450e13 1.80318
\(927\) 0 0
\(928\) −1.44407e13 −0.639179
\(929\) −1.88735e13 −0.831344 −0.415672 0.909514i \(-0.636454\pi\)
−0.415672 + 0.909514i \(0.636454\pi\)
\(930\) 0 0
\(931\) 1.76132e13 0.768359
\(932\) −6.85906e12 −0.297778
\(933\) 0 0
\(934\) 5.46128e12 0.234819
\(935\) −1.58841e13 −0.679689
\(936\) 0 0
\(937\) −1.08798e13 −0.461096 −0.230548 0.973061i \(-0.574052\pi\)
−0.230548 + 0.973061i \(0.574052\pi\)
\(938\) −2.13884e13 −0.902122
\(939\) 0 0
\(940\) −4.61744e12 −0.192897
\(941\) 2.83800e13 1.17994 0.589970 0.807426i \(-0.299140\pi\)
0.589970 + 0.807426i \(0.299140\pi\)
\(942\) 0 0
\(943\) −5.99518e11 −0.0246888
\(944\) 3.02427e13 1.23950
\(945\) 0 0
\(946\) 4.11500e13 1.67055
\(947\) 3.58374e12 0.144798 0.0723988 0.997376i \(-0.476935\pi\)
0.0723988 + 0.997376i \(0.476935\pi\)
\(948\) 0 0
\(949\) 5.75486e12 0.230323
\(950\) 3.37219e13 1.34325
\(951\) 0 0
\(952\) −1.34967e13 −0.532553
\(953\) 2.91182e12 0.114353 0.0571763 0.998364i \(-0.481790\pi\)
0.0571763 + 0.998364i \(0.481790\pi\)
\(954\) 0 0
\(955\) −6.88926e12 −0.268014
\(956\) 1.52686e12 0.0591204
\(957\) 0 0
\(958\) 3.91181e13 1.50049
\(959\) −4.57193e12 −0.174549
\(960\) 0 0
\(961\) 7.13299e13 2.69784
\(962\) 4.28148e12 0.161178
\(963\) 0 0
\(964\) 4.37505e12 0.163168
\(965\) 7.03043e12 0.260981
\(966\) 0 0
\(967\) −2.68326e13 −0.986833 −0.493417 0.869793i \(-0.664252\pi\)
−0.493417 + 0.869793i \(0.664252\pi\)
\(968\) −2.57628e13 −0.943092
\(969\) 0 0
\(970\) 1.04180e13 0.377844
\(971\) 5.18797e13 1.87288 0.936442 0.350824i \(-0.114098\pi\)
0.936442 + 0.350824i \(0.114098\pi\)
\(972\) 0 0
\(973\) 6.16844e12 0.220632
\(974\) −4.64358e13 −1.65325
\(975\) 0 0
\(976\) −6.38239e12 −0.225143
\(977\) 5.55949e13 1.95213 0.976067 0.217469i \(-0.0697801\pi\)
0.976067 + 0.217469i \(0.0697801\pi\)
\(978\) 0 0
\(979\) 1.86089e13 0.647439
\(980\) −2.01129e12 −0.0696557
\(981\) 0 0
\(982\) −2.96003e13 −1.01577
\(983\) 4.02241e13 1.37403 0.687015 0.726644i \(-0.258921\pi\)
0.687015 + 0.726644i \(0.258921\pi\)
\(984\) 0 0
\(985\) −2.28267e12 −0.0772644
\(986\) −3.54326e13 −1.19387
\(987\) 0 0
\(988\) 3.48776e12 0.116450
\(989\) −2.51142e12 −0.0834710
\(990\) 0 0
\(991\) 1.34682e13 0.443588 0.221794 0.975094i \(-0.428809\pi\)
0.221794 + 0.975094i \(0.428809\pi\)
\(992\) 3.25402e13 1.06688
\(993\) 0 0
\(994\) −1.62369e13 −0.527552
\(995\) −2.04981e13 −0.662993
\(996\) 0 0
\(997\) −3.74584e13 −1.20066 −0.600331 0.799752i \(-0.704964\pi\)
−0.600331 + 0.799752i \(0.704964\pi\)
\(998\) −5.18065e13 −1.65309
\(999\) 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 81.10.a.c.1.7 8
3.2 odd 2 81.10.a.d.1.2 8
9.2 odd 6 27.10.c.a.10.7 16
9.4 even 3 9.10.c.a.7.2 yes 16
9.5 odd 6 27.10.c.a.19.7 16
9.7 even 3 9.10.c.a.4.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.10.c.a.4.2 16 9.7 even 3
9.10.c.a.7.2 yes 16 9.4 even 3
27.10.c.a.10.7 16 9.2 odd 6
27.10.c.a.19.7 16 9.5 odd 6
81.10.a.c.1.7 8 1.1 even 1 trivial
81.10.a.d.1.2 8 3.2 odd 2