Properties

Label 27.10.a.c.1.3
Level $27$
Weight $10$
Character 27.1
Self dual yes
Analytic conductor $13.906$
Analytic rank $1$
Dimension $3$
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] = [27,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
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: \(13.9059675764\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.177113.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 118x + 136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(10.7777\) 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)
 
\(f(q)\) \(=\) \(q+32.3331 q^{2} +533.432 q^{4} -2071.63 q^{5} -10517.1 q^{7} +692.954 q^{8} -66982.2 q^{10} +31620.5 q^{11} +47622.2 q^{13} -340051. q^{14} -250712. q^{16} -34440.3 q^{17} +763166. q^{19} -1.10507e6 q^{20} +1.02239e6 q^{22} -1.48534e6 q^{23} +2.33851e6 q^{25} +1.53977e6 q^{26} -5.61016e6 q^{28} -4.80506e6 q^{29} -7.82305e6 q^{31} -8.46109e6 q^{32} -1.11356e6 q^{34} +2.17875e7 q^{35} +9.43456e6 q^{37} +2.46755e7 q^{38} -1.43554e6 q^{40} -1.22899e7 q^{41} +3.74423e7 q^{43} +1.68674e7 q^{44} -4.80258e7 q^{46} -1.98234e7 q^{47} +7.02560e7 q^{49} +7.56113e7 q^{50} +2.54032e7 q^{52} -5.78978e7 q^{53} -6.55059e7 q^{55} -7.28788e6 q^{56} -1.55363e8 q^{58} +2.84988e7 q^{59} +4.45141e7 q^{61} -2.52944e8 q^{62} -1.45209e8 q^{64} -9.86553e7 q^{65} -1.81934e8 q^{67} -1.83715e7 q^{68} +7.04459e8 q^{70} -1.53722e8 q^{71} +1.55706e7 q^{73} +3.05049e8 q^{74} +4.07097e8 q^{76} -3.32557e8 q^{77} -3.26170e8 q^{79} +5.19381e8 q^{80} -3.97371e8 q^{82} -5.01181e7 q^{83} +7.13473e7 q^{85} +1.21063e9 q^{86} +2.19116e7 q^{88} +1.59066e8 q^{89} -5.00848e8 q^{91} -7.92329e8 q^{92} -6.40953e8 q^{94} -1.58099e9 q^{95} -1.49373e8 q^{97} +2.27160e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 597 q^{4} - 1983 q^{5} - 3693 q^{7} - 4503 q^{8} - 18981 q^{10} - 16863 q^{11} + 116916 q^{13} - 503463 q^{14} - 239919 q^{16} - 1014048 q^{17} - 15222 q^{19} - 2548407 q^{20} + 305721 q^{22}+ \cdots + 2592182286 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 32.3331 1.42894 0.714468 0.699668i \(-0.246669\pi\)
0.714468 + 0.699668i \(0.246669\pi\)
\(3\) 0 0
\(4\) 533.432 1.04186
\(5\) −2071.63 −1.48233 −0.741167 0.671320i \(-0.765728\pi\)
−0.741167 + 0.671320i \(0.765728\pi\)
\(6\) 0 0
\(7\) −10517.1 −1.65560 −0.827800 0.561024i \(-0.810408\pi\)
−0.827800 + 0.561024i \(0.810408\pi\)
\(8\) 692.954 0.0598136
\(9\) 0 0
\(10\) −66982.2 −2.11816
\(11\) 31620.5 0.651182 0.325591 0.945511i \(-0.394437\pi\)
0.325591 + 0.945511i \(0.394437\pi\)
\(12\) 0 0
\(13\) 47622.2 0.462449 0.231225 0.972900i \(-0.425727\pi\)
0.231225 + 0.972900i \(0.425727\pi\)
\(14\) −340051. −2.36575
\(15\) 0 0
\(16\) −250712. −0.956389
\(17\) −34440.3 −0.100011 −0.0500053 0.998749i \(-0.515924\pi\)
−0.0500053 + 0.998749i \(0.515924\pi\)
\(18\) 0 0
\(19\) 763166. 1.34347 0.671735 0.740792i \(-0.265550\pi\)
0.671735 + 0.740792i \(0.265550\pi\)
\(20\) −1.10507e6 −1.54438
\(21\) 0 0
\(22\) 1.02239e6 0.930497
\(23\) −1.48534e6 −1.10675 −0.553377 0.832931i \(-0.686661\pi\)
−0.553377 + 0.832931i \(0.686661\pi\)
\(24\) 0 0
\(25\) 2.33851e6 1.19732
\(26\) 1.53977e6 0.660811
\(27\) 0 0
\(28\) −5.61016e6 −1.72490
\(29\) −4.80506e6 −1.26156 −0.630780 0.775962i \(-0.717265\pi\)
−0.630780 + 0.775962i \(0.717265\pi\)
\(30\) 0 0
\(31\) −7.82305e6 −1.52142 −0.760708 0.649094i \(-0.775148\pi\)
−0.760708 + 0.649094i \(0.775148\pi\)
\(32\) −8.46109e6 −1.42643
\(33\) 0 0
\(34\) −1.11356e6 −0.142909
\(35\) 2.17875e7 2.45415
\(36\) 0 0
\(37\) 9.43456e6 0.827587 0.413794 0.910371i \(-0.364204\pi\)
0.413794 + 0.910371i \(0.364204\pi\)
\(38\) 2.46755e7 1.91973
\(39\) 0 0
\(40\) −1.43554e6 −0.0886637
\(41\) −1.22899e7 −0.679237 −0.339618 0.940563i \(-0.610298\pi\)
−0.339618 + 0.940563i \(0.610298\pi\)
\(42\) 0 0
\(43\) 3.74423e7 1.67014 0.835072 0.550140i \(-0.185426\pi\)
0.835072 + 0.550140i \(0.185426\pi\)
\(44\) 1.68674e7 0.678439
\(45\) 0 0
\(46\) −4.80258e7 −1.58148
\(47\) −1.98234e7 −0.592568 −0.296284 0.955100i \(-0.595747\pi\)
−0.296284 + 0.955100i \(0.595747\pi\)
\(48\) 0 0
\(49\) 7.02560e7 1.74101
\(50\) 7.56113e7 1.71089
\(51\) 0 0
\(52\) 2.54032e7 0.481807
\(53\) −5.78978e7 −1.00791 −0.503953 0.863731i \(-0.668122\pi\)
−0.503953 + 0.863731i \(0.668122\pi\)
\(54\) 0 0
\(55\) −6.55059e7 −0.965269
\(56\) −7.28788e6 −0.0990273
\(57\) 0 0
\(58\) −1.55363e8 −1.80269
\(59\) 2.84988e7 0.306191 0.153095 0.988211i \(-0.451076\pi\)
0.153095 + 0.988211i \(0.451076\pi\)
\(60\) 0 0
\(61\) 4.45141e7 0.411636 0.205818 0.978590i \(-0.434014\pi\)
0.205818 + 0.978590i \(0.434014\pi\)
\(62\) −2.52944e8 −2.17401
\(63\) 0 0
\(64\) −1.45209e8 −1.08189
\(65\) −9.86553e7 −0.685505
\(66\) 0 0
\(67\) −1.81934e8 −1.10301 −0.551503 0.834173i \(-0.685946\pi\)
−0.551503 + 0.834173i \(0.685946\pi\)
\(68\) −1.83715e7 −0.104197
\(69\) 0 0
\(70\) 7.04459e8 3.50683
\(71\) −1.53722e8 −0.717915 −0.358958 0.933354i \(-0.616868\pi\)
−0.358958 + 0.933354i \(0.616868\pi\)
\(72\) 0 0
\(73\) 1.55706e7 0.0641729 0.0320864 0.999485i \(-0.489785\pi\)
0.0320864 + 0.999485i \(0.489785\pi\)
\(74\) 3.05049e8 1.18257
\(75\) 0 0
\(76\) 4.07097e8 1.39971
\(77\) −3.32557e8 −1.07810
\(78\) 0 0
\(79\) −3.26170e8 −0.942154 −0.471077 0.882092i \(-0.656135\pi\)
−0.471077 + 0.882092i \(0.656135\pi\)
\(80\) 5.19381e8 1.41769
\(81\) 0 0
\(82\) −3.97371e8 −0.970586
\(83\) −5.01181e7 −0.115916 −0.0579580 0.998319i \(-0.518459\pi\)
−0.0579580 + 0.998319i \(0.518459\pi\)
\(84\) 0 0
\(85\) 7.13473e7 0.148249
\(86\) 1.21063e9 2.38653
\(87\) 0 0
\(88\) 2.19116e7 0.0389495
\(89\) 1.59066e8 0.268735 0.134367 0.990932i \(-0.457100\pi\)
0.134367 + 0.990932i \(0.457100\pi\)
\(90\) 0 0
\(91\) −5.00848e8 −0.765631
\(92\) −7.92329e8 −1.15308
\(93\) 0 0
\(94\) −6.40953e8 −0.846742
\(95\) −1.58099e9 −1.99147
\(96\) 0 0
\(97\) −1.49373e8 −0.171317 −0.0856583 0.996325i \(-0.527299\pi\)
−0.0856583 + 0.996325i \(0.527299\pi\)
\(98\) 2.27160e9 2.48779
\(99\) 0 0
\(100\) 1.24743e9 1.24743
\(101\) 1.84265e9 1.76196 0.880981 0.473152i \(-0.156884\pi\)
0.880981 + 0.473152i \(0.156884\pi\)
\(102\) 0 0
\(103\) −3.50430e8 −0.306785 −0.153392 0.988165i \(-0.549020\pi\)
−0.153392 + 0.988165i \(0.549020\pi\)
\(104\) 3.30000e7 0.0276607
\(105\) 0 0
\(106\) −1.87202e9 −1.44023
\(107\) −7.93460e8 −0.585192 −0.292596 0.956236i \(-0.594519\pi\)
−0.292596 + 0.956236i \(0.594519\pi\)
\(108\) 0 0
\(109\) 1.09944e9 0.746020 0.373010 0.927827i \(-0.378326\pi\)
0.373010 + 0.927827i \(0.378326\pi\)
\(110\) −2.11801e9 −1.37931
\(111\) 0 0
\(112\) 2.63676e9 1.58340
\(113\) −2.91370e9 −1.68109 −0.840547 0.541739i \(-0.817766\pi\)
−0.840547 + 0.541739i \(0.817766\pi\)
\(114\) 0 0
\(115\) 3.07707e9 1.64058
\(116\) −2.56317e9 −1.31437
\(117\) 0 0
\(118\) 9.21455e8 0.437527
\(119\) 3.62212e8 0.165578
\(120\) 0 0
\(121\) −1.35809e9 −0.575962
\(122\) 1.43928e9 0.588202
\(123\) 0 0
\(124\) −4.17306e9 −1.58510
\(125\) −7.98371e8 −0.292489
\(126\) 0 0
\(127\) 1.58765e9 0.541551 0.270776 0.962643i \(-0.412720\pi\)
0.270776 + 0.962643i \(0.412720\pi\)
\(128\) −3.62989e8 −0.119522
\(129\) 0 0
\(130\) −3.18984e9 −0.979543
\(131\) 3.75999e9 1.11549 0.557745 0.830012i \(-0.311667\pi\)
0.557745 + 0.830012i \(0.311667\pi\)
\(132\) 0 0
\(133\) −8.02630e9 −2.22425
\(134\) −5.88251e9 −1.57613
\(135\) 0 0
\(136\) −2.38655e7 −0.00598199
\(137\) −9.88347e8 −0.239699 −0.119850 0.992792i \(-0.538241\pi\)
−0.119850 + 0.992792i \(0.538241\pi\)
\(138\) 0 0
\(139\) 4.71601e9 1.07154 0.535769 0.844364i \(-0.320022\pi\)
0.535769 + 0.844364i \(0.320022\pi\)
\(140\) 1.16222e10 2.55688
\(141\) 0 0
\(142\) −4.97031e9 −1.02585
\(143\) 1.50584e9 0.301139
\(144\) 0 0
\(145\) 9.95428e9 1.87005
\(146\) 5.03445e8 0.0916990
\(147\) 0 0
\(148\) 5.03269e9 0.862229
\(149\) −4.36878e9 −0.726143 −0.363071 0.931761i \(-0.618272\pi\)
−0.363071 + 0.931761i \(0.618272\pi\)
\(150\) 0 0
\(151\) 6.47508e9 1.01356 0.506779 0.862076i \(-0.330836\pi\)
0.506779 + 0.862076i \(0.330836\pi\)
\(152\) 5.28839e8 0.0803577
\(153\) 0 0
\(154\) −1.07526e10 −1.54053
\(155\) 1.62064e10 2.25525
\(156\) 0 0
\(157\) −8.50860e8 −0.111766 −0.0558830 0.998437i \(-0.517797\pi\)
−0.0558830 + 0.998437i \(0.517797\pi\)
\(158\) −1.05461e10 −1.34628
\(159\) 0 0
\(160\) 1.75282e10 2.11445
\(161\) 1.56215e10 1.83234
\(162\) 0 0
\(163\) −8.54283e9 −0.947889 −0.473945 0.880555i \(-0.657170\pi\)
−0.473945 + 0.880555i \(0.657170\pi\)
\(164\) −6.55582e9 −0.707669
\(165\) 0 0
\(166\) −1.62048e9 −0.165637
\(167\) −4.57212e8 −0.0454876 −0.0227438 0.999741i \(-0.507240\pi\)
−0.0227438 + 0.999741i \(0.507240\pi\)
\(168\) 0 0
\(169\) −8.33663e9 −0.786141
\(170\) 2.30688e9 0.211839
\(171\) 0 0
\(172\) 1.99729e10 1.74005
\(173\) −9.13108e9 −0.775023 −0.387511 0.921865i \(-0.626665\pi\)
−0.387511 + 0.921865i \(0.626665\pi\)
\(174\) 0 0
\(175\) −2.45944e10 −1.98228
\(176\) −7.92764e9 −0.622783
\(177\) 0 0
\(178\) 5.14312e9 0.384005
\(179\) −1.84541e9 −0.134355 −0.0671777 0.997741i \(-0.521399\pi\)
−0.0671777 + 0.997741i \(0.521399\pi\)
\(180\) 0 0
\(181\) −1.02105e10 −0.707123 −0.353562 0.935411i \(-0.615030\pi\)
−0.353562 + 0.935411i \(0.615030\pi\)
\(182\) −1.61940e10 −1.09404
\(183\) 0 0
\(184\) −1.02927e9 −0.0661989
\(185\) −1.95449e10 −1.22676
\(186\) 0 0
\(187\) −1.08902e9 −0.0651251
\(188\) −1.05744e10 −0.617372
\(189\) 0 0
\(190\) −5.11185e10 −2.84569
\(191\) −1.00230e10 −0.544938 −0.272469 0.962164i \(-0.587840\pi\)
−0.272469 + 0.962164i \(0.587840\pi\)
\(192\) 0 0
\(193\) 5.87244e9 0.304657 0.152328 0.988330i \(-0.451323\pi\)
0.152328 + 0.988330i \(0.451323\pi\)
\(194\) −4.82970e9 −0.244801
\(195\) 0 0
\(196\) 3.74768e10 1.81389
\(197\) 2.73628e10 1.29438 0.647191 0.762328i \(-0.275944\pi\)
0.647191 + 0.762328i \(0.275944\pi\)
\(198\) 0 0
\(199\) 2.21639e9 0.100186 0.0500930 0.998745i \(-0.484048\pi\)
0.0500930 + 0.998745i \(0.484048\pi\)
\(200\) 1.62048e9 0.0716158
\(201\) 0 0
\(202\) 5.95786e10 2.51773
\(203\) 5.05353e10 2.08864
\(204\) 0 0
\(205\) 2.54601e10 1.00686
\(206\) −1.13305e10 −0.438376
\(207\) 0 0
\(208\) −1.19394e10 −0.442281
\(209\) 2.41317e10 0.874843
\(210\) 0 0
\(211\) −1.35207e10 −0.469599 −0.234799 0.972044i \(-0.575443\pi\)
−0.234799 + 0.972044i \(0.575443\pi\)
\(212\) −3.08845e10 −1.05010
\(213\) 0 0
\(214\) −2.56551e10 −0.836202
\(215\) −7.75664e10 −2.47571
\(216\) 0 0
\(217\) 8.22758e10 2.51886
\(218\) 3.55482e10 1.06601
\(219\) 0 0
\(220\) −3.49429e10 −1.00567
\(221\) −1.64012e9 −0.0462499
\(222\) 0 0
\(223\) −2.97283e9 −0.0805004 −0.0402502 0.999190i \(-0.512815\pi\)
−0.0402502 + 0.999190i \(0.512815\pi\)
\(224\) 8.89862e10 2.36160
\(225\) 0 0
\(226\) −9.42090e10 −2.40218
\(227\) 3.34599e10 0.836388 0.418194 0.908358i \(-0.362663\pi\)
0.418194 + 0.908358i \(0.362663\pi\)
\(228\) 0 0
\(229\) −7.60785e10 −1.82811 −0.914055 0.405590i \(-0.867066\pi\)
−0.914055 + 0.405590i \(0.867066\pi\)
\(230\) 9.94914e10 2.34429
\(231\) 0 0
\(232\) −3.32969e9 −0.0754584
\(233\) −6.49552e10 −1.44382 −0.721909 0.691988i \(-0.756735\pi\)
−0.721909 + 0.691988i \(0.756735\pi\)
\(234\) 0 0
\(235\) 4.10667e10 0.878384
\(236\) 1.52022e10 0.319008
\(237\) 0 0
\(238\) 1.17115e10 0.236600
\(239\) 3.98471e10 0.789963 0.394981 0.918689i \(-0.370751\pi\)
0.394981 + 0.918689i \(0.370751\pi\)
\(240\) 0 0
\(241\) 1.18614e10 0.226496 0.113248 0.993567i \(-0.463875\pi\)
0.113248 + 0.993567i \(0.463875\pi\)
\(242\) −4.39113e10 −0.823014
\(243\) 0 0
\(244\) 2.37452e10 0.428867
\(245\) −1.45544e11 −2.58076
\(246\) 0 0
\(247\) 3.63436e10 0.621286
\(248\) −5.42101e9 −0.0910014
\(249\) 0 0
\(250\) −2.58138e10 −0.417948
\(251\) 6.51966e10 1.03680 0.518398 0.855139i \(-0.326529\pi\)
0.518398 + 0.855139i \(0.326529\pi\)
\(252\) 0 0
\(253\) −4.69673e10 −0.720698
\(254\) 5.13339e10 0.773842
\(255\) 0 0
\(256\) 6.26105e10 0.911102
\(257\) −3.80164e10 −0.543591 −0.271796 0.962355i \(-0.587617\pi\)
−0.271796 + 0.962355i \(0.587617\pi\)
\(258\) 0 0
\(259\) −9.92243e10 −1.37015
\(260\) −5.26259e10 −0.714199
\(261\) 0 0
\(262\) 1.21572e11 1.59397
\(263\) −3.53924e10 −0.456152 −0.228076 0.973643i \(-0.573243\pi\)
−0.228076 + 0.973643i \(0.573243\pi\)
\(264\) 0 0
\(265\) 1.19943e11 1.49406
\(266\) −2.59515e11 −3.17831
\(267\) 0 0
\(268\) −9.70495e10 −1.14918
\(269\) 6.44475e9 0.0750448 0.0375224 0.999296i \(-0.488053\pi\)
0.0375224 + 0.999296i \(0.488053\pi\)
\(270\) 0 0
\(271\) 1.68603e11 1.89890 0.949452 0.313912i \(-0.101640\pi\)
0.949452 + 0.313912i \(0.101640\pi\)
\(272\) 8.63458e9 0.0956491
\(273\) 0 0
\(274\) −3.19563e10 −0.342515
\(275\) 7.39449e10 0.779671
\(276\) 0 0
\(277\) −4.87122e10 −0.497140 −0.248570 0.968614i \(-0.579961\pi\)
−0.248570 + 0.968614i \(0.579961\pi\)
\(278\) 1.52483e11 1.53116
\(279\) 0 0
\(280\) 1.50978e10 0.146792
\(281\) −1.51361e11 −1.44822 −0.724110 0.689684i \(-0.757749\pi\)
−0.724110 + 0.689684i \(0.757749\pi\)
\(282\) 0 0
\(283\) −7.14282e10 −0.661959 −0.330979 0.943638i \(-0.607379\pi\)
−0.330979 + 0.943638i \(0.607379\pi\)
\(284\) −8.20001e10 −0.747966
\(285\) 0 0
\(286\) 4.86885e10 0.430308
\(287\) 1.29254e11 1.12454
\(288\) 0 0
\(289\) −1.17402e11 −0.989998
\(290\) 3.21853e11 2.67219
\(291\) 0 0
\(292\) 8.30584e9 0.0668591
\(293\) −2.22271e11 −1.76189 −0.880945 0.473218i \(-0.843092\pi\)
−0.880945 + 0.473218i \(0.843092\pi\)
\(294\) 0 0
\(295\) −5.90388e10 −0.453878
\(296\) 6.53772e9 0.0495009
\(297\) 0 0
\(298\) −1.41256e11 −1.03761
\(299\) −7.07352e10 −0.511818
\(300\) 0 0
\(301\) −3.93784e11 −2.76509
\(302\) 2.09360e11 1.44831
\(303\) 0 0
\(304\) −1.91335e11 −1.28488
\(305\) −9.22166e10 −0.610183
\(306\) 0 0
\(307\) 2.24657e11 1.44343 0.721717 0.692188i \(-0.243353\pi\)
0.721717 + 0.692188i \(0.243353\pi\)
\(308\) −1.77396e11 −1.12322
\(309\) 0 0
\(310\) 5.24005e11 3.22261
\(311\) 2.99889e11 1.81777 0.908886 0.417046i \(-0.136934\pi\)
0.908886 + 0.417046i \(0.136934\pi\)
\(312\) 0 0
\(313\) 1.50502e11 0.886325 0.443162 0.896441i \(-0.353857\pi\)
0.443162 + 0.896441i \(0.353857\pi\)
\(314\) −2.75110e10 −0.159706
\(315\) 0 0
\(316\) −1.73989e11 −0.981591
\(317\) 2.49896e11 1.38993 0.694966 0.719043i \(-0.255420\pi\)
0.694966 + 0.719043i \(0.255420\pi\)
\(318\) 0 0
\(319\) −1.51939e11 −0.821504
\(320\) 3.00819e11 1.60373
\(321\) 0 0
\(322\) 5.05092e11 2.61830
\(323\) −2.62836e10 −0.134361
\(324\) 0 0
\(325\) 1.11365e11 0.553698
\(326\) −2.76216e11 −1.35447
\(327\) 0 0
\(328\) −8.51634e9 −0.0406276
\(329\) 2.08485e11 0.981055
\(330\) 0 0
\(331\) 5.32449e10 0.243810 0.121905 0.992542i \(-0.461100\pi\)
0.121905 + 0.992542i \(0.461100\pi\)
\(332\) −2.67346e10 −0.120768
\(333\) 0 0
\(334\) −1.47831e10 −0.0649989
\(335\) 3.76900e11 1.63502
\(336\) 0 0
\(337\) 3.54762e10 0.149831 0.0749156 0.997190i \(-0.476131\pi\)
0.0749156 + 0.997190i \(0.476131\pi\)
\(338\) −2.69549e11 −1.12334
\(339\) 0 0
\(340\) 3.80589e10 0.154455
\(341\) −2.47369e11 −0.990719
\(342\) 0 0
\(343\) −3.14487e11 −1.22681
\(344\) 2.59458e10 0.0998973
\(345\) 0 0
\(346\) −2.95236e11 −1.10746
\(347\) −3.92158e11 −1.45204 −0.726019 0.687675i \(-0.758632\pi\)
−0.726019 + 0.687675i \(0.758632\pi\)
\(348\) 0 0
\(349\) −8.30145e10 −0.299530 −0.149765 0.988722i \(-0.547852\pi\)
−0.149765 + 0.988722i \(0.547852\pi\)
\(350\) −7.95213e11 −2.83255
\(351\) 0 0
\(352\) −2.67544e11 −0.928867
\(353\) −1.85625e11 −0.636281 −0.318141 0.948044i \(-0.603058\pi\)
−0.318141 + 0.948044i \(0.603058\pi\)
\(354\) 0 0
\(355\) 3.18454e11 1.06419
\(356\) 8.48511e10 0.279984
\(357\) 0 0
\(358\) −5.96680e10 −0.191985
\(359\) 3.92754e11 1.24794 0.623972 0.781447i \(-0.285518\pi\)
0.623972 + 0.781447i \(0.285518\pi\)
\(360\) 0 0
\(361\) 2.59734e11 0.804909
\(362\) −3.30139e11 −1.01043
\(363\) 0 0
\(364\) −2.67168e11 −0.797679
\(365\) −3.22564e10 −0.0951257
\(366\) 0 0
\(367\) 4.97459e10 0.143140 0.0715698 0.997436i \(-0.477199\pi\)
0.0715698 + 0.997436i \(0.477199\pi\)
\(368\) 3.72393e11 1.05849
\(369\) 0 0
\(370\) −6.31947e11 −1.75296
\(371\) 6.08917e11 1.66869
\(372\) 0 0
\(373\) 4.03156e11 1.07841 0.539204 0.842175i \(-0.318725\pi\)
0.539204 + 0.842175i \(0.318725\pi\)
\(374\) −3.52114e10 −0.0930596
\(375\) 0 0
\(376\) −1.37367e10 −0.0354436
\(377\) −2.28827e11 −0.583407
\(378\) 0 0
\(379\) −3.41222e11 −0.849494 −0.424747 0.905312i \(-0.639637\pi\)
−0.424747 + 0.905312i \(0.639637\pi\)
\(380\) −8.43352e11 −2.07483
\(381\) 0 0
\(382\) −3.24075e11 −0.778682
\(383\) −3.16998e11 −0.752770 −0.376385 0.926463i \(-0.622833\pi\)
−0.376385 + 0.926463i \(0.622833\pi\)
\(384\) 0 0
\(385\) 6.88933e11 1.59810
\(386\) 1.89874e11 0.435335
\(387\) 0 0
\(388\) −7.96803e10 −0.178488
\(389\) 1.11909e11 0.247795 0.123898 0.992295i \(-0.460461\pi\)
0.123898 + 0.992295i \(0.460461\pi\)
\(390\) 0 0
\(391\) 5.11556e10 0.110687
\(392\) 4.86842e10 0.104136
\(393\) 0 0
\(394\) 8.84725e11 1.84959
\(395\) 6.75702e11 1.39659
\(396\) 0 0
\(397\) 3.48788e11 0.704700 0.352350 0.935868i \(-0.385383\pi\)
0.352350 + 0.935868i \(0.385383\pi\)
\(398\) 7.16627e10 0.143159
\(399\) 0 0
\(400\) −5.86291e11 −1.14510
\(401\) −2.54977e11 −0.492437 −0.246218 0.969214i \(-0.579188\pi\)
−0.246218 + 0.969214i \(0.579188\pi\)
\(402\) 0 0
\(403\) −3.72550e11 −0.703578
\(404\) 9.82928e11 1.83572
\(405\) 0 0
\(406\) 1.63397e12 2.98453
\(407\) 2.98326e11 0.538910
\(408\) 0 0
\(409\) 2.68334e10 0.0474156 0.0237078 0.999719i \(-0.492453\pi\)
0.0237078 + 0.999719i \(0.492453\pi\)
\(410\) 8.23204e11 1.43873
\(411\) 0 0
\(412\) −1.86930e11 −0.319626
\(413\) −2.99725e11 −0.506930
\(414\) 0 0
\(415\) 1.03826e11 0.171826
\(416\) −4.02935e11 −0.659653
\(417\) 0 0
\(418\) 7.80254e11 1.25009
\(419\) 8.67220e9 0.0137457 0.00687284 0.999976i \(-0.497812\pi\)
0.00687284 + 0.999976i \(0.497812\pi\)
\(420\) 0 0
\(421\) −9.68105e11 −1.50194 −0.750971 0.660335i \(-0.770414\pi\)
−0.750971 + 0.660335i \(0.770414\pi\)
\(422\) −4.37166e11 −0.671027
\(423\) 0 0
\(424\) −4.01205e10 −0.0602865
\(425\) −8.05389e10 −0.119744
\(426\) 0 0
\(427\) −4.68160e11 −0.681504
\(428\) −4.23257e11 −0.609687
\(429\) 0 0
\(430\) −2.50796e12 −3.53764
\(431\) 1.38264e12 1.93001 0.965006 0.262227i \(-0.0844570\pi\)
0.965006 + 0.262227i \(0.0844570\pi\)
\(432\) 0 0
\(433\) −1.26247e12 −1.72594 −0.862971 0.505254i \(-0.831399\pi\)
−0.862971 + 0.505254i \(0.831399\pi\)
\(434\) 2.66024e12 3.59929
\(435\) 0 0
\(436\) 5.86474e11 0.777247
\(437\) −1.13356e12 −1.48689
\(438\) 0 0
\(439\) 4.36242e11 0.560579 0.280290 0.959915i \(-0.409570\pi\)
0.280290 + 0.959915i \(0.409570\pi\)
\(440\) −4.53926e10 −0.0577362
\(441\) 0 0
\(442\) −5.30302e10 −0.0660881
\(443\) −5.07995e11 −0.626675 −0.313337 0.949642i \(-0.601447\pi\)
−0.313337 + 0.949642i \(0.601447\pi\)
\(444\) 0 0
\(445\) −3.29526e11 −0.398355
\(446\) −9.61208e10 −0.115030
\(447\) 0 0
\(448\) 1.52718e12 1.79118
\(449\) −1.47077e12 −1.70780 −0.853899 0.520439i \(-0.825768\pi\)
−0.853899 + 0.520439i \(0.825768\pi\)
\(450\) 0 0
\(451\) −3.88613e11 −0.442307
\(452\) −1.55426e12 −1.75146
\(453\) 0 0
\(454\) 1.08186e12 1.19515
\(455\) 1.03757e12 1.13492
\(456\) 0 0
\(457\) −1.11991e12 −1.20105 −0.600526 0.799605i \(-0.705042\pi\)
−0.600526 + 0.799605i \(0.705042\pi\)
\(458\) −2.45986e12 −2.61225
\(459\) 0 0
\(460\) 1.64141e12 1.70925
\(461\) −1.72669e12 −1.78057 −0.890286 0.455402i \(-0.849496\pi\)
−0.890286 + 0.455402i \(0.849496\pi\)
\(462\) 0 0
\(463\) 2.74673e11 0.277780 0.138890 0.990308i \(-0.455647\pi\)
0.138890 + 0.990308i \(0.455647\pi\)
\(464\) 1.20468e12 1.20654
\(465\) 0 0
\(466\) −2.10021e12 −2.06312
\(467\) −1.14495e12 −1.11394 −0.556971 0.830532i \(-0.688036\pi\)
−0.556971 + 0.830532i \(0.688036\pi\)
\(468\) 0 0
\(469\) 1.91342e12 1.82614
\(470\) 1.32782e12 1.25516
\(471\) 0 0
\(472\) 1.97484e10 0.0183144
\(473\) 1.18394e12 1.08757
\(474\) 0 0
\(475\) 1.78467e12 1.60856
\(476\) 1.93215e11 0.172508
\(477\) 0 0
\(478\) 1.28838e12 1.12881
\(479\) −9.32652e11 −0.809487 −0.404744 0.914430i \(-0.632639\pi\)
−0.404744 + 0.914430i \(0.632639\pi\)
\(480\) 0 0
\(481\) 4.49294e11 0.382717
\(482\) 3.83517e11 0.323648
\(483\) 0 0
\(484\) −7.24448e11 −0.600071
\(485\) 3.09445e11 0.253949
\(486\) 0 0
\(487\) −1.02912e12 −0.829059 −0.414530 0.910036i \(-0.636054\pi\)
−0.414530 + 0.910036i \(0.636054\pi\)
\(488\) 3.08462e10 0.0246214
\(489\) 0 0
\(490\) −4.70590e12 −3.68774
\(491\) −1.72889e12 −1.34246 −0.671228 0.741251i \(-0.734233\pi\)
−0.671228 + 0.741251i \(0.734233\pi\)
\(492\) 0 0
\(493\) 1.65487e11 0.126169
\(494\) 1.17510e12 0.887779
\(495\) 0 0
\(496\) 1.96133e12 1.45507
\(497\) 1.61671e12 1.18858
\(498\) 0 0
\(499\) 2.39976e12 1.73267 0.866336 0.499462i \(-0.166469\pi\)
0.866336 + 0.499462i \(0.166469\pi\)
\(500\) −4.25876e11 −0.304732
\(501\) 0 0
\(502\) 2.10801e12 1.48152
\(503\) −1.98201e12 −1.38054 −0.690271 0.723551i \(-0.742509\pi\)
−0.690271 + 0.723551i \(0.742509\pi\)
\(504\) 0 0
\(505\) −3.81728e12 −2.61182
\(506\) −1.51860e12 −1.02983
\(507\) 0 0
\(508\) 8.46905e11 0.564220
\(509\) 2.25559e12 1.48946 0.744731 0.667364i \(-0.232578\pi\)
0.744731 + 0.667364i \(0.232578\pi\)
\(510\) 0 0
\(511\) −1.63757e11 −0.106245
\(512\) 2.21024e12 1.42143
\(513\) 0 0
\(514\) −1.22919e12 −0.776757
\(515\) 7.25959e11 0.454757
\(516\) 0 0
\(517\) −6.26827e11 −0.385869
\(518\) −3.20823e12 −1.95786
\(519\) 0 0
\(520\) −6.83636e10 −0.0410025
\(521\) 7.71315e11 0.458630 0.229315 0.973352i \(-0.426351\pi\)
0.229315 + 0.973352i \(0.426351\pi\)
\(522\) 0 0
\(523\) −1.54315e12 −0.901886 −0.450943 0.892553i \(-0.648912\pi\)
−0.450943 + 0.892553i \(0.648912\pi\)
\(524\) 2.00570e12 1.16218
\(525\) 0 0
\(526\) −1.14435e12 −0.651812
\(527\) 2.69428e11 0.152158
\(528\) 0 0
\(529\) 4.05089e11 0.224905
\(530\) 3.87812e12 2.13491
\(531\) 0 0
\(532\) −4.28148e12 −2.31735
\(533\) −5.85272e11 −0.314113
\(534\) 0 0
\(535\) 1.64375e12 0.867450
\(536\) −1.26072e11 −0.0659747
\(537\) 0 0
\(538\) 2.08379e11 0.107234
\(539\) 2.22153e12 1.13371
\(540\) 0 0
\(541\) 3.58683e12 1.80021 0.900105 0.435674i \(-0.143490\pi\)
0.900105 + 0.435674i \(0.143490\pi\)
\(542\) 5.45146e12 2.71341
\(543\) 0 0
\(544\) 2.91402e11 0.142658
\(545\) −2.27762e12 −1.10585
\(546\) 0 0
\(547\) −2.61010e12 −1.24656 −0.623282 0.781997i \(-0.714201\pi\)
−0.623282 + 0.781997i \(0.714201\pi\)
\(548\) −5.27215e11 −0.249733
\(549\) 0 0
\(550\) 2.39087e12 1.11410
\(551\) −3.66706e12 −1.69487
\(552\) 0 0
\(553\) 3.43036e12 1.55983
\(554\) −1.57502e12 −0.710382
\(555\) 0 0
\(556\) 2.51567e12 1.11639
\(557\) 1.44707e12 0.637002 0.318501 0.947923i \(-0.396821\pi\)
0.318501 + 0.947923i \(0.396821\pi\)
\(558\) 0 0
\(559\) 1.78308e12 0.772357
\(560\) −5.46238e12 −2.34712
\(561\) 0 0
\(562\) −4.89397e12 −2.06942
\(563\) 3.02936e12 1.27076 0.635380 0.772200i \(-0.280844\pi\)
0.635380 + 0.772200i \(0.280844\pi\)
\(564\) 0 0
\(565\) 6.03610e12 2.49194
\(566\) −2.30950e12 −0.945897
\(567\) 0 0
\(568\) −1.06522e11 −0.0429411
\(569\) 8.23594e11 0.329388 0.164694 0.986345i \(-0.447336\pi\)
0.164694 + 0.986345i \(0.447336\pi\)
\(570\) 0 0
\(571\) −7.54932e11 −0.297198 −0.148599 0.988898i \(-0.547476\pi\)
−0.148599 + 0.988898i \(0.547476\pi\)
\(572\) 8.03262e11 0.313744
\(573\) 0 0
\(574\) 4.17920e12 1.60690
\(575\) −3.47349e12 −1.32514
\(576\) 0 0
\(577\) −4.22193e12 −1.58569 −0.792847 0.609420i \(-0.791402\pi\)
−0.792847 + 0.609420i \(0.791402\pi\)
\(578\) −3.79597e12 −1.41464
\(579\) 0 0
\(580\) 5.30993e12 1.94833
\(581\) 5.27098e11 0.191910
\(582\) 0 0
\(583\) −1.83076e12 −0.656330
\(584\) 1.07897e10 0.00383841
\(585\) 0 0
\(586\) −7.18673e12 −2.51763
\(587\) 2.52727e12 0.878578 0.439289 0.898346i \(-0.355230\pi\)
0.439289 + 0.898346i \(0.355230\pi\)
\(588\) 0 0
\(589\) −5.97028e12 −2.04398
\(590\) −1.90891e12 −0.648562
\(591\) 0 0
\(592\) −2.36535e12 −0.791495
\(593\) 2.59659e12 0.862299 0.431150 0.902280i \(-0.358108\pi\)
0.431150 + 0.902280i \(0.358108\pi\)
\(594\) 0 0
\(595\) −7.50368e11 −0.245441
\(596\) −2.33045e12 −0.756538
\(597\) 0 0
\(598\) −2.28709e12 −0.731355
\(599\) 3.56005e12 1.12989 0.564944 0.825129i \(-0.308898\pi\)
0.564944 + 0.825129i \(0.308898\pi\)
\(600\) 0 0
\(601\) 1.93346e12 0.604506 0.302253 0.953228i \(-0.402261\pi\)
0.302253 + 0.953228i \(0.402261\pi\)
\(602\) −1.27323e13 −3.95114
\(603\) 0 0
\(604\) 3.45401e12 1.05598
\(605\) 2.81345e12 0.853769
\(606\) 0 0
\(607\) 1.32290e12 0.395528 0.197764 0.980250i \(-0.436632\pi\)
0.197764 + 0.980250i \(0.436632\pi\)
\(608\) −6.45721e12 −1.91637
\(609\) 0 0
\(610\) −2.98165e12 −0.871912
\(611\) −9.44034e11 −0.274033
\(612\) 0 0
\(613\) −9.74984e11 −0.278885 −0.139443 0.990230i \(-0.544531\pi\)
−0.139443 + 0.990230i \(0.544531\pi\)
\(614\) 7.26386e12 2.06257
\(615\) 0 0
\(616\) −2.30447e11 −0.0644848
\(617\) 2.87228e12 0.797890 0.398945 0.916975i \(-0.369376\pi\)
0.398945 + 0.916975i \(0.369376\pi\)
\(618\) 0 0
\(619\) −4.16143e12 −1.13929 −0.569645 0.821891i \(-0.692919\pi\)
−0.569645 + 0.821891i \(0.692919\pi\)
\(620\) 8.64502e12 2.34965
\(621\) 0 0
\(622\) 9.69636e12 2.59748
\(623\) −1.67292e12 −0.444917
\(624\) 0 0
\(625\) −2.91347e12 −0.763750
\(626\) 4.86620e12 1.26650
\(627\) 0 0
\(628\) −4.53876e11 −0.116444
\(629\) −3.24929e11 −0.0827675
\(630\) 0 0
\(631\) −3.22906e12 −0.810855 −0.405428 0.914127i \(-0.632877\pi\)
−0.405428 + 0.914127i \(0.632877\pi\)
\(632\) −2.26021e11 −0.0563536
\(633\) 0 0
\(634\) 8.07993e12 1.98612
\(635\) −3.28903e12 −0.802760
\(636\) 0 0
\(637\) 3.34574e12 0.805129
\(638\) −4.91265e12 −1.17388
\(639\) 0 0
\(640\) 7.51978e11 0.177172
\(641\) −4.91306e12 −1.14945 −0.574727 0.818345i \(-0.694892\pi\)
−0.574727 + 0.818345i \(0.694892\pi\)
\(642\) 0 0
\(643\) 1.18575e12 0.273553 0.136777 0.990602i \(-0.456326\pi\)
0.136777 + 0.990602i \(0.456326\pi\)
\(644\) 8.33301e12 1.90904
\(645\) 0 0
\(646\) −8.49832e11 −0.191994
\(647\) −6.72898e12 −1.50966 −0.754832 0.655919i \(-0.772281\pi\)
−0.754832 + 0.655919i \(0.772281\pi\)
\(648\) 0 0
\(649\) 9.01147e11 0.199386
\(650\) 3.60078e12 0.791199
\(651\) 0 0
\(652\) −4.55702e12 −0.987567
\(653\) −7.24926e11 −0.156021 −0.0780107 0.996953i \(-0.524857\pi\)
−0.0780107 + 0.996953i \(0.524857\pi\)
\(654\) 0 0
\(655\) −7.78929e12 −1.65353
\(656\) 3.08122e12 0.649615
\(657\) 0 0
\(658\) 6.74098e12 1.40187
\(659\) 6.02523e12 1.24448 0.622242 0.782825i \(-0.286222\pi\)
0.622242 + 0.782825i \(0.286222\pi\)
\(660\) 0 0
\(661\) 1.74974e12 0.356506 0.178253 0.983985i \(-0.442956\pi\)
0.178253 + 0.983985i \(0.442956\pi\)
\(662\) 1.72158e12 0.348390
\(663\) 0 0
\(664\) −3.47296e10 −0.00693335
\(665\) 1.66275e13 3.29708
\(666\) 0 0
\(667\) 7.13716e12 1.39624
\(668\) −2.43891e11 −0.0473917
\(669\) 0 0
\(670\) 1.21864e13 2.33635
\(671\) 1.40756e12 0.268050
\(672\) 0 0
\(673\) −3.59466e12 −0.675445 −0.337722 0.941246i \(-0.609656\pi\)
−0.337722 + 0.941246i \(0.609656\pi\)
\(674\) 1.14706e12 0.214099
\(675\) 0 0
\(676\) −4.44702e12 −0.819048
\(677\) 2.33912e12 0.427960 0.213980 0.976838i \(-0.431357\pi\)
0.213980 + 0.976838i \(0.431357\pi\)
\(678\) 0 0
\(679\) 1.57097e12 0.283632
\(680\) 4.94404e10 0.00886732
\(681\) 0 0
\(682\) −7.99821e12 −1.41567
\(683\) −4.03165e11 −0.0708907 −0.0354453 0.999372i \(-0.511285\pi\)
−0.0354453 + 0.999372i \(0.511285\pi\)
\(684\) 0 0
\(685\) 2.04748e12 0.355315
\(686\) −1.01683e13 −1.75304
\(687\) 0 0
\(688\) −9.38721e12 −1.59731
\(689\) −2.75722e12 −0.466106
\(690\) 0 0
\(691\) 7.79472e12 1.30062 0.650308 0.759670i \(-0.274640\pi\)
0.650308 + 0.759670i \(0.274640\pi\)
\(692\) −4.87081e12 −0.807465
\(693\) 0 0
\(694\) −1.26797e13 −2.07487
\(695\) −9.76980e12 −1.58838
\(696\) 0 0
\(697\) 4.23268e11 0.0679309
\(698\) −2.68412e12 −0.428009
\(699\) 0 0
\(700\) −1.31194e13 −2.06525
\(701\) 2.40116e12 0.375569 0.187784 0.982210i \(-0.439869\pi\)
0.187784 + 0.982210i \(0.439869\pi\)
\(702\) 0 0
\(703\) 7.20013e12 1.11184
\(704\) −4.59159e12 −0.704508
\(705\) 0 0
\(706\) −6.00182e12 −0.909206
\(707\) −1.93793e13 −2.91710
\(708\) 0 0
\(709\) −5.36135e12 −0.796830 −0.398415 0.917205i \(-0.630440\pi\)
−0.398415 + 0.917205i \(0.630440\pi\)
\(710\) 1.02966e13 1.52066
\(711\) 0 0
\(712\) 1.10226e11 0.0160740
\(713\) 1.16199e13 1.68383
\(714\) 0 0
\(715\) −3.11953e12 −0.446388
\(716\) −9.84402e11 −0.139979
\(717\) 0 0
\(718\) 1.26990e13 1.78323
\(719\) −5.40289e12 −0.753956 −0.376978 0.926222i \(-0.623037\pi\)
−0.376978 + 0.926222i \(0.623037\pi\)
\(720\) 0 0
\(721\) 3.68551e12 0.507912
\(722\) 8.39803e12 1.15016
\(723\) 0 0
\(724\) −5.44662e12 −0.736722
\(725\) −1.12367e13 −1.51049
\(726\) 0 0
\(727\) 3.15572e12 0.418980 0.209490 0.977811i \(-0.432820\pi\)
0.209490 + 0.977811i \(0.432820\pi\)
\(728\) −3.47065e11 −0.0457951
\(729\) 0 0
\(730\) −1.04295e12 −0.135929
\(731\) −1.28952e12 −0.167032
\(732\) 0 0
\(733\) −1.20641e13 −1.54357 −0.771784 0.635885i \(-0.780635\pi\)
−0.771784 + 0.635885i \(0.780635\pi\)
\(734\) 1.60844e12 0.204537
\(735\) 0 0
\(736\) 1.25676e13 1.57871
\(737\) −5.75286e12 −0.718257
\(738\) 0 0
\(739\) 8.39060e12 1.03489 0.517444 0.855717i \(-0.326884\pi\)
0.517444 + 0.855717i \(0.326884\pi\)
\(740\) −1.04259e13 −1.27811
\(741\) 0 0
\(742\) 1.96882e13 2.38445
\(743\) −7.40995e11 −0.0892001 −0.0446001 0.999005i \(-0.514201\pi\)
−0.0446001 + 0.999005i \(0.514201\pi\)
\(744\) 0 0
\(745\) 9.05048e12 1.07639
\(746\) 1.30353e13 1.54098
\(747\) 0 0
\(748\) −5.80918e11 −0.0678512
\(749\) 8.34491e12 0.968843
\(750\) 0 0
\(751\) 1.41125e13 1.61891 0.809457 0.587180i \(-0.199762\pi\)
0.809457 + 0.587180i \(0.199762\pi\)
\(752\) 4.96996e12 0.566726
\(753\) 0 0
\(754\) −7.39871e12 −0.833652
\(755\) −1.34139e13 −1.50243
\(756\) 0 0
\(757\) 5.66623e12 0.627138 0.313569 0.949565i \(-0.398475\pi\)
0.313569 + 0.949565i \(0.398475\pi\)
\(758\) −1.10328e13 −1.21387
\(759\) 0 0
\(760\) −1.09556e12 −0.119117
\(761\) 1.34131e13 1.44977 0.724883 0.688872i \(-0.241894\pi\)
0.724883 + 0.688872i \(0.241894\pi\)
\(762\) 0 0
\(763\) −1.15629e13 −1.23511
\(764\) −5.34658e12 −0.567749
\(765\) 0 0
\(766\) −1.02495e13 −1.07566
\(767\) 1.35717e12 0.141598
\(768\) 0 0
\(769\) −6.64729e12 −0.685450 −0.342725 0.939436i \(-0.611350\pi\)
−0.342725 + 0.939436i \(0.611350\pi\)
\(770\) 2.22754e13 2.28358
\(771\) 0 0
\(772\) 3.13255e12 0.317409
\(773\) −1.79549e13 −1.80874 −0.904369 0.426752i \(-0.859658\pi\)
−0.904369 + 0.426752i \(0.859658\pi\)
\(774\) 0 0
\(775\) −1.82943e13 −1.82162
\(776\) −1.03509e11 −0.0102471
\(777\) 0 0
\(778\) 3.61838e12 0.354083
\(779\) −9.37924e12 −0.912534
\(780\) 0 0
\(781\) −4.86077e12 −0.467493
\(782\) 1.65402e12 0.158165
\(783\) 0 0
\(784\) −1.76140e13 −1.66508
\(785\) 1.76266e12 0.165675
\(786\) 0 0
\(787\) 9.36005e12 0.869745 0.434873 0.900492i \(-0.356793\pi\)
0.434873 + 0.900492i \(0.356793\pi\)
\(788\) 1.45962e13 1.34856
\(789\) 0 0
\(790\) 2.18476e13 1.99563
\(791\) 3.06437e13 2.78322
\(792\) 0 0
\(793\) 2.11986e12 0.190361
\(794\) 1.12774e13 1.00697
\(795\) 0 0
\(796\) 1.18229e12 0.104380
\(797\) 1.60113e13 1.40561 0.702803 0.711384i \(-0.251931\pi\)
0.702803 + 0.711384i \(0.251931\pi\)
\(798\) 0 0
\(799\) 6.82724e11 0.0592631
\(800\) −1.97863e13 −1.70789
\(801\) 0 0
\(802\) −8.24419e12 −0.703661
\(803\) 4.92350e11 0.0417882
\(804\) 0 0
\(805\) −3.23619e13 −2.71614
\(806\) −1.20457e13 −1.00537
\(807\) 0 0
\(808\) 1.27687e12 0.105389
\(809\) −1.53912e13 −1.26329 −0.631645 0.775258i \(-0.717620\pi\)
−0.631645 + 0.775258i \(0.717620\pi\)
\(810\) 0 0
\(811\) −1.81531e13 −1.47352 −0.736760 0.676154i \(-0.763645\pi\)
−0.736760 + 0.676154i \(0.763645\pi\)
\(812\) 2.69571e13 2.17606
\(813\) 0 0
\(814\) 9.64581e12 0.770068
\(815\) 1.76975e13 1.40509
\(816\) 0 0
\(817\) 2.85747e13 2.24379
\(818\) 8.67609e11 0.0677539
\(819\) 0 0
\(820\) 1.35812e13 1.04900
\(821\) −2.20209e13 −1.69157 −0.845786 0.533523i \(-0.820868\pi\)
−0.845786 + 0.533523i \(0.820868\pi\)
\(822\) 0 0
\(823\) −1.72414e13 −1.31000 −0.655002 0.755627i \(-0.727332\pi\)
−0.655002 + 0.755627i \(0.727332\pi\)
\(824\) −2.42832e11 −0.0183499
\(825\) 0 0
\(826\) −9.69105e12 −0.724370
\(827\) 3.54008e12 0.263171 0.131585 0.991305i \(-0.457993\pi\)
0.131585 + 0.991305i \(0.457993\pi\)
\(828\) 0 0
\(829\) 1.85206e13 1.36194 0.680971 0.732310i \(-0.261558\pi\)
0.680971 + 0.732310i \(0.261558\pi\)
\(830\) 3.35702e12 0.245529
\(831\) 0 0
\(832\) −6.91517e12 −0.500320
\(833\) −2.41964e12 −0.174120
\(834\) 0 0
\(835\) 9.47171e11 0.0674279
\(836\) 1.28726e13 0.911462
\(837\) 0 0
\(838\) 2.80399e11 0.0196417
\(839\) −1.10025e13 −0.766591 −0.383295 0.923626i \(-0.625211\pi\)
−0.383295 + 0.923626i \(0.625211\pi\)
\(840\) 0 0
\(841\) 8.58144e12 0.591532
\(842\) −3.13019e13 −2.14618
\(843\) 0 0
\(844\) −7.21235e12 −0.489256
\(845\) 1.72704e13 1.16532
\(846\) 0 0
\(847\) 1.42832e13 0.953563
\(848\) 1.45156e13 0.963951
\(849\) 0 0
\(850\) −2.60407e12 −0.171107
\(851\) −1.40135e13 −0.915936
\(852\) 0 0
\(853\) −1.48235e13 −0.958693 −0.479347 0.877626i \(-0.659126\pi\)
−0.479347 + 0.877626i \(0.659126\pi\)
\(854\) −1.51371e13 −0.973826
\(855\) 0 0
\(856\) −5.49832e11 −0.0350024
\(857\) 7.68399e12 0.486601 0.243301 0.969951i \(-0.421770\pi\)
0.243301 + 0.969951i \(0.421770\pi\)
\(858\) 0 0
\(859\) −1.90428e12 −0.119333 −0.0596667 0.998218i \(-0.519004\pi\)
−0.0596667 + 0.998218i \(0.519004\pi\)
\(860\) −4.13764e13 −2.57934
\(861\) 0 0
\(862\) 4.47050e13 2.75786
\(863\) −3.09626e12 −0.190015 −0.0950077 0.995477i \(-0.530288\pi\)
−0.0950077 + 0.995477i \(0.530288\pi\)
\(864\) 0 0
\(865\) 1.89162e13 1.14884
\(866\) −4.08197e13 −2.46626
\(867\) 0 0
\(868\) 4.38885e13 2.62429
\(869\) −1.03137e13 −0.613513
\(870\) 0 0
\(871\) −8.66411e12 −0.510084
\(872\) 7.61858e11 0.0446221
\(873\) 0 0
\(874\) −3.66516e13 −2.12467
\(875\) 8.39656e12 0.484245
\(876\) 0 0
\(877\) 4.14925e12 0.236849 0.118425 0.992963i \(-0.462216\pi\)
0.118425 + 0.992963i \(0.462216\pi\)
\(878\) 1.41051e13 0.801032
\(879\) 0 0
\(880\) 1.64231e13 0.923173
\(881\) 1.36244e13 0.761948 0.380974 0.924586i \(-0.375589\pi\)
0.380974 + 0.924586i \(0.375589\pi\)
\(882\) 0 0
\(883\) −1.83775e13 −1.01734 −0.508668 0.860963i \(-0.669862\pi\)
−0.508668 + 0.860963i \(0.669862\pi\)
\(884\) −8.74892e11 −0.0481858
\(885\) 0 0
\(886\) −1.64251e13 −0.895479
\(887\) 2.97823e13 1.61548 0.807741 0.589538i \(-0.200690\pi\)
0.807741 + 0.589538i \(0.200690\pi\)
\(888\) 0 0
\(889\) −1.66975e13 −0.896592
\(890\) −1.06546e13 −0.569223
\(891\) 0 0
\(892\) −1.58580e12 −0.0838700
\(893\) −1.51286e13 −0.796097
\(894\) 0 0
\(895\) 3.82301e12 0.199160
\(896\) 3.81760e12 0.197881
\(897\) 0 0
\(898\) −4.75546e13 −2.44033
\(899\) 3.75902e13 1.91936
\(900\) 0 0
\(901\) 1.99401e12 0.100801
\(902\) −1.25651e13 −0.632028
\(903\) 0 0
\(904\) −2.01906e12 −0.100552
\(905\) 2.11524e13 1.04819
\(906\) 0 0
\(907\) −3.49542e13 −1.71501 −0.857504 0.514477i \(-0.827986\pi\)
−0.857504 + 0.514477i \(0.827986\pi\)
\(908\) 1.78486e13 0.871398
\(909\) 0 0
\(910\) 3.35479e13 1.62173
\(911\) −2.02274e13 −0.972989 −0.486494 0.873684i \(-0.661725\pi\)
−0.486494 + 0.873684i \(0.661725\pi\)
\(912\) 0 0
\(913\) −1.58476e12 −0.0754824
\(914\) −3.62104e13 −1.71623
\(915\) 0 0
\(916\) −4.05827e13 −1.90463
\(917\) −3.95442e13 −1.84681
\(918\) 0 0
\(919\) 4.43929e12 0.205302 0.102651 0.994717i \(-0.467268\pi\)
0.102651 + 0.994717i \(0.467268\pi\)
\(920\) 2.13227e12 0.0981290
\(921\) 0 0
\(922\) −5.58292e13 −2.54432
\(923\) −7.32057e12 −0.331999
\(924\) 0 0
\(925\) 2.20628e13 0.990884
\(926\) 8.88104e12 0.396930
\(927\) 0 0
\(928\) 4.06560e13 1.79953
\(929\) 1.16451e13 0.512948 0.256474 0.966551i \(-0.417439\pi\)
0.256474 + 0.966551i \(0.417439\pi\)
\(930\) 0 0
\(931\) 5.36170e13 2.33899
\(932\) −3.46492e13 −1.50425
\(933\) 0 0
\(934\) −3.70200e13 −1.59175
\(935\) 2.25604e12 0.0965372
\(936\) 0 0
\(937\) −6.68375e12 −0.283265 −0.141632 0.989919i \(-0.545235\pi\)
−0.141632 + 0.989919i \(0.545235\pi\)
\(938\) 6.18670e13 2.60943
\(939\) 0 0
\(940\) 2.19063e13 0.915152
\(941\) 2.77843e13 1.15517 0.577586 0.816330i \(-0.303995\pi\)
0.577586 + 0.816330i \(0.303995\pi\)
\(942\) 0 0
\(943\) 1.82547e13 0.751748
\(944\) −7.14498e12 −0.292838
\(945\) 0 0
\(946\) 3.82806e13 1.55406
\(947\) 2.02254e13 0.817189 0.408595 0.912716i \(-0.366019\pi\)
0.408595 + 0.912716i \(0.366019\pi\)
\(948\) 0 0
\(949\) 7.41504e11 0.0296767
\(950\) 5.77040e13 2.29853
\(951\) 0 0
\(952\) 2.50996e11 0.00990379
\(953\) 1.95165e13 0.766450 0.383225 0.923655i \(-0.374813\pi\)
0.383225 + 0.923655i \(0.374813\pi\)
\(954\) 0 0
\(955\) 2.07639e13 0.807781
\(956\) 2.12557e13 0.823030
\(957\) 0 0
\(958\) −3.01556e13 −1.15671
\(959\) 1.03946e13 0.396846
\(960\) 0 0
\(961\) 3.47604e13 1.31471
\(962\) 1.45271e13 0.546878
\(963\) 0 0
\(964\) 6.32727e12 0.235977
\(965\) −1.21655e13 −0.451603
\(966\) 0 0
\(967\) 4.20323e13 1.54584 0.772919 0.634504i \(-0.218796\pi\)
0.772919 + 0.634504i \(0.218796\pi\)
\(968\) −9.41094e11 −0.0344504
\(969\) 0 0
\(970\) 1.00053e13 0.362876
\(971\) −8.15237e12 −0.294305 −0.147152 0.989114i \(-0.547011\pi\)
−0.147152 + 0.989114i \(0.547011\pi\)
\(972\) 0 0
\(973\) −4.95988e13 −1.77404
\(974\) −3.32747e13 −1.18467
\(975\) 0 0
\(976\) −1.11602e13 −0.393684
\(977\) 1.48575e13 0.521699 0.260849 0.965379i \(-0.415997\pi\)
0.260849 + 0.965379i \(0.415997\pi\)
\(978\) 0 0
\(979\) 5.02977e12 0.174995
\(980\) −7.76379e13 −2.68879
\(981\) 0 0
\(982\) −5.59004e13 −1.91828
\(983\) 8.38233e12 0.286335 0.143167 0.989698i \(-0.454271\pi\)
0.143167 + 0.989698i \(0.454271\pi\)
\(984\) 0 0
\(985\) −5.66855e13 −1.91871
\(986\) 5.35073e12 0.180288
\(987\) 0 0
\(988\) 1.93868e13 0.647293
\(989\) −5.56146e13 −1.84844
\(990\) 0 0
\(991\) 3.54621e13 1.16797 0.583986 0.811763i \(-0.301492\pi\)
0.583986 + 0.811763i \(0.301492\pi\)
\(992\) 6.61915e13 2.17020
\(993\) 0 0
\(994\) 5.22733e13 1.69840
\(995\) −4.59152e12 −0.148509
\(996\) 0 0
\(997\) 4.17955e13 1.33968 0.669841 0.742505i \(-0.266362\pi\)
0.669841 + 0.742505i \(0.266362\pi\)
\(998\) 7.75919e13 2.47588
\(999\) 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 27.10.a.c.1.3 yes 3
3.2 odd 2 27.10.a.b.1.1 3
9.2 odd 6 81.10.c.h.28.3 6
9.4 even 3 81.10.c.g.55.1 6
9.5 odd 6 81.10.c.h.55.3 6
9.7 even 3 81.10.c.g.28.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.10.a.b.1.1 3 3.2 odd 2
27.10.a.c.1.3 yes 3 1.1 even 1 trivial
81.10.c.g.28.1 6 9.7 even 3
81.10.c.g.55.1 6 9.4 even 3
81.10.c.h.28.3 6 9.2 odd 6
81.10.c.h.55.3 6 9.5 odd 6