Properties

Label 2.14.a.a.1.1
Level $2$
Weight $14$
Character 2.1
Self dual yes
Analytic conductor $2.145$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2,14,Mod(1,2)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2, base_ring=CyclotomicField(1))
 
chi = DirichletCharacter(H, H._module([]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 2.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: \(2.14461857904\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-64.0000 q^{2} -1836.00 q^{3} +4096.00 q^{4} +3990.00 q^{5} +117504. q^{6} -433432. q^{7} -262144. q^{8} +1.77657e6 q^{9} +O(q^{10})\) \(q-64.0000 q^{2} -1836.00 q^{3} +4096.00 q^{4} +3990.00 q^{5} +117504. q^{6} -433432. q^{7} -262144. q^{8} +1.77657e6 q^{9} -255360. q^{10} +1.61977e6 q^{11} -7.52026e6 q^{12} -1.08785e7 q^{13} +2.77396e7 q^{14} -7.32564e6 q^{15} +1.67772e7 q^{16} +6.05693e7 q^{17} -1.13701e8 q^{18} -2.43132e8 q^{19} +1.63430e7 q^{20} +7.95781e8 q^{21} -1.03665e8 q^{22} -6.06096e8 q^{23} +4.81296e8 q^{24} -1.20478e9 q^{25} +6.96222e8 q^{26} -3.34611e8 q^{27} -1.77534e9 q^{28} +5.25864e9 q^{29} +4.68841e8 q^{30} -1.82431e9 q^{31} -1.07374e9 q^{32} -2.97390e9 q^{33} -3.87644e9 q^{34} -1.72939e9 q^{35} +7.27684e9 q^{36} -3.00588e9 q^{37} +1.55604e10 q^{38} +1.99729e10 q^{39} -1.04595e9 q^{40} -4.97049e10 q^{41} -5.09300e10 q^{42} +5.87667e10 q^{43} +6.63459e9 q^{44} +7.08853e9 q^{45} +3.87902e10 q^{46} -4.20959e10 q^{47} -3.08030e10 q^{48} +9.09743e10 q^{49} +7.71061e10 q^{50} -1.11205e11 q^{51} -4.45582e10 q^{52} -1.81141e11 q^{53} +2.14151e10 q^{54} +6.46289e9 q^{55} +1.13622e11 q^{56} +4.46390e11 q^{57} -3.36553e11 q^{58} +2.06731e11 q^{59} -3.00058e10 q^{60} -1.24479e11 q^{61} +1.16756e11 q^{62} -7.70024e11 q^{63} +6.87195e10 q^{64} -4.34051e10 q^{65} +1.90330e11 q^{66} +9.56651e10 q^{67} +2.48092e11 q^{68} +1.11279e12 q^{69} +1.10681e11 q^{70} -3.71436e11 q^{71} -4.65718e11 q^{72} -1.80058e12 q^{73} +1.92376e11 q^{74} +2.21198e12 q^{75} -9.95868e11 q^{76} -7.02061e11 q^{77} -1.27826e12 q^{78} +1.55793e12 q^{79} +6.69411e10 q^{80} -2.21809e12 q^{81} +3.18111e12 q^{82} +2.49279e12 q^{83} +3.25952e12 q^{84} +2.41671e11 q^{85} -3.76107e12 q^{86} -9.65486e12 q^{87} -4.24614e11 q^{88} +2.99424e12 q^{89} -4.53666e11 q^{90} +4.71508e12 q^{91} -2.48257e12 q^{92} +3.34944e12 q^{93} +2.69414e12 q^{94} -9.70096e11 q^{95} +1.97139e12 q^{96} +4.38249e12 q^{97} -5.82235e12 q^{98} +2.87764e12 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −64.0000 −0.707107
\(3\) −1836.00 −1.45407 −0.727034 0.686602i \(-0.759102\pi\)
−0.727034 + 0.686602i \(0.759102\pi\)
\(4\) 4096.00 0.500000
\(5\) 3990.00 0.114200 0.0571002 0.998368i \(-0.481815\pi\)
0.0571002 + 0.998368i \(0.481815\pi\)
\(6\) 117504. 1.02818
\(7\) −433432. −1.39246 −0.696232 0.717817i \(-0.745141\pi\)
−0.696232 + 0.717817i \(0.745141\pi\)
\(8\) −262144. −0.353553
\(9\) 1.77657e6 1.11431
\(10\) −255360. −0.0807519
\(11\) 1.61977e6 0.275678 0.137839 0.990455i \(-0.455984\pi\)
0.137839 + 0.990455i \(0.455984\pi\)
\(12\) −7.52026e6 −0.727034
\(13\) −1.08785e7 −0.625080 −0.312540 0.949905i \(-0.601180\pi\)
−0.312540 + 0.949905i \(0.601180\pi\)
\(14\) 2.77396e7 0.984620
\(15\) −7.32564e6 −0.166055
\(16\) 1.67772e7 0.250000
\(17\) 6.05693e7 0.608604 0.304302 0.952576i \(-0.401577\pi\)
0.304302 + 0.952576i \(0.401577\pi\)
\(18\) −1.13701e8 −0.787937
\(19\) −2.43132e8 −1.18561 −0.592807 0.805345i \(-0.701980\pi\)
−0.592807 + 0.805345i \(0.701980\pi\)
\(20\) 1.63430e7 0.0571002
\(21\) 7.95781e8 2.02474
\(22\) −1.03665e8 −0.194933
\(23\) −6.06096e8 −0.853711 −0.426855 0.904320i \(-0.640379\pi\)
−0.426855 + 0.904320i \(0.640379\pi\)
\(24\) 4.81296e8 0.514090
\(25\) −1.20478e9 −0.986958
\(26\) 6.96222e8 0.441999
\(27\) −3.34611e8 −0.166217
\(28\) −1.77534e9 −0.696232
\(29\) 5.25864e9 1.64167 0.820836 0.571164i \(-0.193508\pi\)
0.820836 + 0.571164i \(0.193508\pi\)
\(30\) 4.68841e8 0.117419
\(31\) −1.82431e9 −0.369189 −0.184594 0.982815i \(-0.559097\pi\)
−0.184594 + 0.982815i \(0.559097\pi\)
\(32\) −1.07374e9 −0.176777
\(33\) −2.97390e9 −0.400854
\(34\) −3.87644e9 −0.430348
\(35\) −1.72939e9 −0.159020
\(36\) 7.27684e9 0.557156
\(37\) −3.00588e9 −0.192602 −0.0963008 0.995352i \(-0.530701\pi\)
−0.0963008 + 0.995352i \(0.530701\pi\)
\(38\) 1.55604e10 0.838356
\(39\) 1.99729e10 0.908909
\(40\) −1.04595e9 −0.0403760
\(41\) −4.97049e10 −1.63420 −0.817098 0.576499i \(-0.804418\pi\)
−0.817098 + 0.576499i \(0.804418\pi\)
\(42\) −5.09300e10 −1.43170
\(43\) 5.87667e10 1.41771 0.708853 0.705356i \(-0.249213\pi\)
0.708853 + 0.705356i \(0.249213\pi\)
\(44\) 6.63459e9 0.137839
\(45\) 7.08853e9 0.127255
\(46\) 3.87902e10 0.603665
\(47\) −4.20959e10 −0.569644 −0.284822 0.958580i \(-0.591934\pi\)
−0.284822 + 0.958580i \(0.591934\pi\)
\(48\) −3.08030e10 −0.363517
\(49\) 9.09743e10 0.938954
\(50\) 7.71061e10 0.697885
\(51\) −1.11205e11 −0.884951
\(52\) −4.45582e10 −0.312540
\(53\) −1.81141e11 −1.12259 −0.561297 0.827614i \(-0.689698\pi\)
−0.561297 + 0.827614i \(0.689698\pi\)
\(54\) 2.14151e10 0.117533
\(55\) 6.46289e9 0.0314825
\(56\) 1.13622e11 0.492310
\(57\) 4.46390e11 1.72396
\(58\) −3.36553e11 −1.16084
\(59\) 2.06731e11 0.638068 0.319034 0.947743i \(-0.396642\pi\)
0.319034 + 0.947743i \(0.396642\pi\)
\(60\) −3.00058e10 −0.0830276
\(61\) −1.24479e11 −0.309351 −0.154676 0.987965i \(-0.549433\pi\)
−0.154676 + 0.987965i \(0.549433\pi\)
\(62\) 1.16756e11 0.261056
\(63\) −7.70024e11 −1.55164
\(64\) 6.87195e10 0.125000
\(65\) −4.34051e10 −0.0713845
\(66\) 1.90330e11 0.283446
\(67\) 9.56651e10 0.129201 0.0646007 0.997911i \(-0.479423\pi\)
0.0646007 + 0.997911i \(0.479423\pi\)
\(68\) 2.48092e11 0.304302
\(69\) 1.11279e12 1.24135
\(70\) 1.10681e11 0.112444
\(71\) −3.71436e11 −0.344116 −0.172058 0.985087i \(-0.555042\pi\)
−0.172058 + 0.985087i \(0.555042\pi\)
\(72\) −4.65718e11 −0.393969
\(73\) −1.80058e12 −1.39256 −0.696278 0.717772i \(-0.745162\pi\)
−0.696278 + 0.717772i \(0.745162\pi\)
\(74\) 1.92376e11 0.136190
\(75\) 2.21198e12 1.43510
\(76\) −9.95868e11 −0.592807
\(77\) −7.02061e11 −0.383871
\(78\) −1.27826e12 −0.642696
\(79\) 1.55793e12 0.721062 0.360531 0.932747i \(-0.382596\pi\)
0.360531 + 0.932747i \(0.382596\pi\)
\(80\) 6.69411e10 0.0285501
\(81\) −2.21809e12 −0.872621
\(82\) 3.18111e12 1.15555
\(83\) 2.49279e12 0.836909 0.418455 0.908238i \(-0.362572\pi\)
0.418455 + 0.908238i \(0.362572\pi\)
\(84\) 3.25952e12 1.01237
\(85\) 2.41671e11 0.0695028
\(86\) −3.76107e12 −1.00247
\(87\) −9.65486e12 −2.38710
\(88\) −4.24614e11 −0.0974667
\(89\) 2.99424e12 0.638632 0.319316 0.947648i \(-0.396547\pi\)
0.319316 + 0.947648i \(0.396547\pi\)
\(90\) −4.53666e11 −0.0899828
\(91\) 4.71508e12 0.870401
\(92\) −2.48257e12 −0.426855
\(93\) 3.34944e12 0.536825
\(94\) 2.69414e12 0.402799
\(95\) −9.70096e11 −0.135398
\(96\) 1.97139e12 0.257045
\(97\) 4.38249e12 0.534201 0.267101 0.963669i \(-0.413934\pi\)
0.267101 + 0.963669i \(0.413934\pi\)
\(98\) −5.82235e12 −0.663940
\(99\) 2.87764e12 0.307191
\(100\) −4.93479e12 −0.493479
\(101\) −5.66110e12 −0.530655 −0.265327 0.964158i \(-0.585480\pi\)
−0.265327 + 0.964158i \(0.585480\pi\)
\(102\) 7.11713e12 0.625755
\(103\) −2.16104e13 −1.78329 −0.891644 0.452737i \(-0.850448\pi\)
−0.891644 + 0.452737i \(0.850448\pi\)
\(104\) 2.85172e12 0.220999
\(105\) 3.17517e12 0.231226
\(106\) 1.15930e13 0.793794
\(107\) 1.09348e13 0.704397 0.352198 0.935925i \(-0.385434\pi\)
0.352198 + 0.935925i \(0.385434\pi\)
\(108\) −1.37057e12 −0.0831086
\(109\) −4.20682e12 −0.240260 −0.120130 0.992758i \(-0.538331\pi\)
−0.120130 + 0.992758i \(0.538331\pi\)
\(110\) −4.13625e11 −0.0222615
\(111\) 5.51879e12 0.280056
\(112\) −7.27178e12 −0.348116
\(113\) 2.87650e13 1.29974 0.649868 0.760047i \(-0.274824\pi\)
0.649868 + 0.760047i \(0.274824\pi\)
\(114\) −2.85690e13 −1.21903
\(115\) −2.41832e12 −0.0974942
\(116\) 2.15394e13 0.820836
\(117\) −1.93264e13 −0.696534
\(118\) −1.32308e13 −0.451182
\(119\) −2.62527e13 −0.847458
\(120\) 1.92037e12 0.0587094
\(121\) −3.18991e13 −0.924002
\(122\) 7.96666e12 0.218745
\(123\) 9.12582e13 2.37623
\(124\) −7.47239e12 −0.184594
\(125\) −9.67769e12 −0.226912
\(126\) 4.92815e13 1.09717
\(127\) 4.78013e13 1.01092 0.505458 0.862851i \(-0.331323\pi\)
0.505458 + 0.862851i \(0.331323\pi\)
\(128\) −4.39805e12 −0.0883883
\(129\) −1.07896e14 −2.06144
\(130\) 2.77793e12 0.0504764
\(131\) −1.03694e14 −1.79262 −0.896312 0.443425i \(-0.853763\pi\)
−0.896312 + 0.443425i \(0.853763\pi\)
\(132\) −1.21811e13 −0.200427
\(133\) 1.05381e14 1.65092
\(134\) −6.12257e12 −0.0913592
\(135\) −1.33510e12 −0.0189821
\(136\) −1.58779e13 −0.215174
\(137\) 5.05383e13 0.653035 0.326518 0.945191i \(-0.394125\pi\)
0.326518 + 0.945191i \(0.394125\pi\)
\(138\) −7.12188e13 −0.877769
\(139\) 2.14878e13 0.252694 0.126347 0.991986i \(-0.459675\pi\)
0.126347 + 0.991986i \(0.459675\pi\)
\(140\) −7.08360e12 −0.0795100
\(141\) 7.72880e13 0.828301
\(142\) 2.37719e13 0.243327
\(143\) −1.76206e13 −0.172321
\(144\) 2.98059e13 0.278578
\(145\) 2.09820e13 0.187480
\(146\) 1.15237e14 0.984686
\(147\) −1.67029e14 −1.36530
\(148\) −1.23121e13 −0.0963008
\(149\) −8.79741e13 −0.658634 −0.329317 0.944219i \(-0.606818\pi\)
−0.329317 + 0.944219i \(0.606818\pi\)
\(150\) −1.41567e14 −1.01477
\(151\) −2.07435e13 −0.142407 −0.0712037 0.997462i \(-0.522684\pi\)
−0.0712037 + 0.997462i \(0.522684\pi\)
\(152\) 6.37355e13 0.419178
\(153\) 1.07606e14 0.678174
\(154\) 4.49319e13 0.271438
\(155\) −7.27901e12 −0.0421615
\(156\) 8.18088e13 0.454454
\(157\) −2.27523e14 −1.21249 −0.606243 0.795279i \(-0.707324\pi\)
−0.606243 + 0.795279i \(0.707324\pi\)
\(158\) −9.97077e13 −0.509868
\(159\) 3.32574e14 1.63233
\(160\) −4.28423e12 −0.0201880
\(161\) 2.62702e14 1.18876
\(162\) 1.41957e14 0.617036
\(163\) −1.41533e14 −0.591070 −0.295535 0.955332i \(-0.595498\pi\)
−0.295535 + 0.955332i \(0.595498\pi\)
\(164\) −2.03591e14 −0.817098
\(165\) −1.18659e13 −0.0457777
\(166\) −1.59539e14 −0.591784
\(167\) 5.58000e13 0.199057 0.0995284 0.995035i \(-0.468267\pi\)
0.0995284 + 0.995035i \(0.468267\pi\)
\(168\) −2.08609e14 −0.715852
\(169\) −1.84534e14 −0.609275
\(170\) −1.54670e13 −0.0491459
\(171\) −4.31941e14 −1.32114
\(172\) 2.40708e14 0.708853
\(173\) 1.26777e14 0.359535 0.179767 0.983709i \(-0.442465\pi\)
0.179767 + 0.983709i \(0.442465\pi\)
\(174\) 6.17911e14 1.68794
\(175\) 5.22192e14 1.37430
\(176\) 2.71753e13 0.0689194
\(177\) −3.79557e14 −0.927793
\(178\) −1.91631e14 −0.451581
\(179\) 3.08976e14 0.702070 0.351035 0.936362i \(-0.385830\pi\)
0.351035 + 0.936362i \(0.385830\pi\)
\(180\) 2.90346e13 0.0636275
\(181\) −4.74336e14 −1.00271 −0.501355 0.865242i \(-0.667165\pi\)
−0.501355 + 0.865242i \(0.667165\pi\)
\(182\) −3.01765e14 −0.615467
\(183\) 2.28543e14 0.449818
\(184\) 1.58885e14 0.301832
\(185\) −1.19934e13 −0.0219952
\(186\) −2.14364e14 −0.379593
\(187\) 9.81085e13 0.167778
\(188\) −1.72425e14 −0.284822
\(189\) 1.45031e14 0.231451
\(190\) 6.20861e13 0.0957406
\(191\) −6.58450e14 −0.981310 −0.490655 0.871354i \(-0.663242\pi\)
−0.490655 + 0.871354i \(0.663242\pi\)
\(192\) −1.26169e14 −0.181758
\(193\) 2.10092e14 0.292609 0.146305 0.989240i \(-0.453262\pi\)
0.146305 + 0.989240i \(0.453262\pi\)
\(194\) −2.80480e14 −0.377737
\(195\) 7.96917e13 0.103798
\(196\) 3.72631e14 0.469477
\(197\) 1.56559e15 1.90831 0.954153 0.299320i \(-0.0967599\pi\)
0.954153 + 0.299320i \(0.0967599\pi\)
\(198\) −1.84169e14 −0.217217
\(199\) 3.66804e14 0.418686 0.209343 0.977842i \(-0.432867\pi\)
0.209343 + 0.977842i \(0.432867\pi\)
\(200\) 3.15827e14 0.348942
\(201\) −1.75641e14 −0.187868
\(202\) 3.62311e14 0.375230
\(203\) −2.27926e15 −2.28597
\(204\) −4.55497e14 −0.442475
\(205\) −1.98322e14 −0.186626
\(206\) 1.38307e15 1.26097
\(207\) −1.07677e15 −0.951300
\(208\) −1.82510e14 −0.156270
\(209\) −3.93818e14 −0.326847
\(210\) −2.03211e14 −0.163501
\(211\) 1.34485e15 1.04915 0.524574 0.851365i \(-0.324224\pi\)
0.524574 + 0.851365i \(0.324224\pi\)
\(212\) −7.41953e14 −0.561297
\(213\) 6.81957e14 0.500368
\(214\) −6.99829e14 −0.498084
\(215\) 2.34479e14 0.161903
\(216\) 8.77163e13 0.0587666
\(217\) 7.90716e14 0.514082
\(218\) 2.69237e14 0.169890
\(219\) 3.30586e15 2.02487
\(220\) 2.64720e13 0.0157413
\(221\) −6.58901e14 −0.380426
\(222\) −3.53202e14 −0.198029
\(223\) −2.04083e15 −1.11128 −0.555642 0.831421i \(-0.687528\pi\)
−0.555642 + 0.831421i \(0.687528\pi\)
\(224\) 4.65394e14 0.246155
\(225\) −2.14038e15 −1.09978
\(226\) −1.84096e15 −0.919052
\(227\) 2.91994e14 0.141647 0.0708233 0.997489i \(-0.477437\pi\)
0.0708233 + 0.997489i \(0.477437\pi\)
\(228\) 1.82841e15 0.861981
\(229\) −2.81485e15 −1.28981 −0.644903 0.764264i \(-0.723102\pi\)
−0.644903 + 0.764264i \(0.723102\pi\)
\(230\) 1.54773e14 0.0689388
\(231\) 1.28898e15 0.558174
\(232\) −1.37852e15 −0.580419
\(233\) 4.39377e15 1.79897 0.899486 0.436950i \(-0.143941\pi\)
0.899486 + 0.436950i \(0.143941\pi\)
\(234\) 1.23689e15 0.492524
\(235\) −1.67963e14 −0.0650536
\(236\) 8.46768e14 0.319034
\(237\) −2.86036e15 −1.04847
\(238\) 1.68017e15 0.599243
\(239\) −4.40953e15 −1.53040 −0.765202 0.643791i \(-0.777361\pi\)
−0.765202 + 0.643791i \(0.777361\pi\)
\(240\) −1.22904e14 −0.0415138
\(241\) 1.92940e15 0.634324 0.317162 0.948371i \(-0.397270\pi\)
0.317162 + 0.948371i \(0.397270\pi\)
\(242\) 2.04154e15 0.653368
\(243\) 4.60588e15 1.43507
\(244\) −5.09866e14 −0.154676
\(245\) 3.62987e14 0.107229
\(246\) −5.84052e15 −1.68025
\(247\) 2.64490e15 0.741104
\(248\) 4.78233e14 0.130528
\(249\) −4.57676e15 −1.21692
\(250\) 6.19372e14 0.160451
\(251\) 3.46331e14 0.0874202 0.0437101 0.999044i \(-0.486082\pi\)
0.0437101 + 0.999044i \(0.486082\pi\)
\(252\) −3.15402e15 −0.775819
\(253\) −9.81738e14 −0.235349
\(254\) −3.05928e15 −0.714826
\(255\) −4.43709e14 −0.101062
\(256\) 2.81475e14 0.0625000
\(257\) 4.59506e14 0.0994778 0.0497389 0.998762i \(-0.484161\pi\)
0.0497389 + 0.998762i \(0.484161\pi\)
\(258\) 6.90532e15 1.45766
\(259\) 1.30284e15 0.268191
\(260\) −1.77787e14 −0.0356922
\(261\) 9.34236e15 1.82933
\(262\) 6.63640e15 1.26758
\(263\) −8.08966e15 −1.50736 −0.753681 0.657240i \(-0.771724\pi\)
−0.753681 + 0.657240i \(0.771724\pi\)
\(264\) 7.79590e14 0.141723
\(265\) −7.22752e14 −0.128201
\(266\) −6.74439e15 −1.16738
\(267\) −5.49742e15 −0.928615
\(268\) 3.91844e14 0.0646007
\(269\) 5.06059e15 0.814350 0.407175 0.913350i \(-0.366514\pi\)
0.407175 + 0.913350i \(0.366514\pi\)
\(270\) 8.54463e13 0.0134224
\(271\) −4.91453e14 −0.0753671 −0.0376835 0.999290i \(-0.511998\pi\)
−0.0376835 + 0.999290i \(0.511998\pi\)
\(272\) 1.01618e15 0.152151
\(273\) −8.65688e15 −1.26562
\(274\) −3.23445e15 −0.461766
\(275\) −1.95147e15 −0.272082
\(276\) 4.55800e15 0.620677
\(277\) 3.92917e15 0.522615 0.261308 0.965256i \(-0.415846\pi\)
0.261308 + 0.965256i \(0.415846\pi\)
\(278\) −1.37522e15 −0.178682
\(279\) −3.24103e15 −0.411391
\(280\) 4.53350e14 0.0562220
\(281\) 1.14417e16 1.38644 0.693219 0.720727i \(-0.256192\pi\)
0.693219 + 0.720727i \(0.256192\pi\)
\(282\) −4.94643e15 −0.585697
\(283\) 2.12246e15 0.245599 0.122800 0.992431i \(-0.460813\pi\)
0.122800 + 0.992431i \(0.460813\pi\)
\(284\) −1.52140e15 −0.172058
\(285\) 1.78110e15 0.196877
\(286\) 1.12772e15 0.121849
\(287\) 2.15437e16 2.27556
\(288\) −1.90758e15 −0.196984
\(289\) −6.23594e15 −0.629602
\(290\) −1.34285e15 −0.132568
\(291\) −8.04626e15 −0.776765
\(292\) −7.37516e15 −0.696278
\(293\) −1.99050e16 −1.83790 −0.918951 0.394372i \(-0.870962\pi\)
−0.918951 + 0.394372i \(0.870962\pi\)
\(294\) 1.06898e16 0.965414
\(295\) 8.24855e14 0.0728676
\(296\) 7.87972e14 0.0680949
\(297\) −5.41994e14 −0.0458223
\(298\) 5.63034e15 0.465724
\(299\) 6.59340e15 0.533638
\(300\) 9.06028e15 0.717552
\(301\) −2.54714e16 −1.97410
\(302\) 1.32759e15 0.100697
\(303\) 1.03938e16 0.771608
\(304\) −4.07907e15 −0.296403
\(305\) −4.96671e14 −0.0353281
\(306\) −6.88677e15 −0.479542
\(307\) −1.26611e16 −0.863118 −0.431559 0.902085i \(-0.642036\pi\)
−0.431559 + 0.902085i \(0.642036\pi\)
\(308\) −2.87564e15 −0.191935
\(309\) 3.96768e16 2.59302
\(310\) 4.65857e14 0.0298127
\(311\) 1.08678e15 0.0681081 0.0340541 0.999420i \(-0.489158\pi\)
0.0340541 + 0.999420i \(0.489158\pi\)
\(312\) −5.23577e15 −0.321348
\(313\) 2.13383e16 1.28269 0.641346 0.767252i \(-0.278376\pi\)
0.641346 + 0.767252i \(0.278376\pi\)
\(314\) 1.45615e16 0.857358
\(315\) −3.07239e15 −0.177198
\(316\) 6.38129e15 0.360531
\(317\) −3.00971e16 −1.66586 −0.832932 0.553375i \(-0.813340\pi\)
−0.832932 + 0.553375i \(0.813340\pi\)
\(318\) −2.12848e16 −1.15423
\(319\) 8.51780e15 0.452572
\(320\) 2.74191e14 0.0142751
\(321\) −2.00763e16 −1.02424
\(322\) −1.68129e16 −0.840581
\(323\) −1.47263e16 −0.721569
\(324\) −9.08528e15 −0.436310
\(325\) 1.31062e16 0.616928
\(326\) 9.05813e15 0.417950
\(327\) 7.72373e15 0.349355
\(328\) 1.30298e16 0.577775
\(329\) 1.82457e16 0.793208
\(330\) 7.59415e14 0.0323697
\(331\) 1.00468e16 0.419900 0.209950 0.977712i \(-0.432670\pi\)
0.209950 + 0.977712i \(0.432670\pi\)
\(332\) 1.02105e16 0.418455
\(333\) −5.34016e15 −0.214618
\(334\) −3.57120e15 −0.140754
\(335\) 3.81704e14 0.0147549
\(336\) 1.33510e16 0.506184
\(337\) 4.00487e15 0.148934 0.0744671 0.997223i \(-0.476274\pi\)
0.0744671 + 0.997223i \(0.476274\pi\)
\(338\) 1.18102e16 0.430822
\(339\) −5.28126e16 −1.88990
\(340\) 9.89886e14 0.0347514
\(341\) −2.95497e15 −0.101777
\(342\) 2.76442e16 0.934190
\(343\) 2.56363e15 0.0850048
\(344\) −1.54053e16 −0.501235
\(345\) 4.44004e15 0.141763
\(346\) −8.11373e15 −0.254229
\(347\) 6.06273e13 0.00186435 0.000932173 1.00000i \(-0.499703\pi\)
0.000932173 1.00000i \(0.499703\pi\)
\(348\) −3.95463e16 −1.19355
\(349\) 3.07578e16 0.911151 0.455576 0.890197i \(-0.349433\pi\)
0.455576 + 0.890197i \(0.349433\pi\)
\(350\) −3.34203e16 −0.971779
\(351\) 3.64005e15 0.103899
\(352\) −1.73922e15 −0.0487334
\(353\) −2.08897e16 −0.574640 −0.287320 0.957835i \(-0.592764\pi\)
−0.287320 + 0.957835i \(0.592764\pi\)
\(354\) 2.42917e16 0.656049
\(355\) −1.48203e15 −0.0392982
\(356\) 1.22644e16 0.319316
\(357\) 4.81999e16 1.23226
\(358\) −1.97745e16 −0.496439
\(359\) −2.46366e16 −0.607390 −0.303695 0.952769i \(-0.598220\pi\)
−0.303695 + 0.952769i \(0.598220\pi\)
\(360\) −1.85821e15 −0.0449914
\(361\) 1.70601e16 0.405680
\(362\) 3.03575e16 0.709023
\(363\) 5.85667e16 1.34356
\(364\) 1.93129e16 0.435201
\(365\) −7.18430e15 −0.159031
\(366\) −1.46268e16 −0.318069
\(367\) −5.28651e16 −1.12938 −0.564689 0.825304i \(-0.691004\pi\)
−0.564689 + 0.825304i \(0.691004\pi\)
\(368\) −1.01686e16 −0.213428
\(369\) −8.83043e16 −1.82100
\(370\) 7.67580e14 0.0155529
\(371\) 7.85122e16 1.56317
\(372\) 1.37193e16 0.268413
\(373\) −6.44569e15 −0.123926 −0.0619630 0.998078i \(-0.519736\pi\)
−0.0619630 + 0.998078i \(0.519736\pi\)
\(374\) −6.27894e15 −0.118637
\(375\) 1.77682e16 0.329945
\(376\) 1.10352e16 0.201400
\(377\) −5.72059e16 −1.02618
\(378\) −9.28199e15 −0.163661
\(379\) −6.58787e16 −1.14180 −0.570900 0.821019i \(-0.693406\pi\)
−0.570900 + 0.821019i \(0.693406\pi\)
\(380\) −3.97351e15 −0.0676988
\(381\) −8.77632e16 −1.46994
\(382\) 4.21408e16 0.693891
\(383\) 8.87457e16 1.43666 0.718332 0.695700i \(-0.244906\pi\)
0.718332 + 0.695700i \(0.244906\pi\)
\(384\) 8.07481e15 0.128523
\(385\) −2.80122e15 −0.0438382
\(386\) −1.34459e16 −0.206906
\(387\) 1.04403e17 1.57977
\(388\) 1.79507e16 0.267101
\(389\) 6.27464e16 0.918156 0.459078 0.888396i \(-0.348180\pi\)
0.459078 + 0.888396i \(0.348180\pi\)
\(390\) −5.10027e15 −0.0733961
\(391\) −3.67108e16 −0.519572
\(392\) −2.38484e16 −0.331970
\(393\) 1.90382e17 2.60659
\(394\) −1.00198e17 −1.34938
\(395\) 6.21615e15 0.0823456
\(396\) 1.17868e16 0.153595
\(397\) −9.86261e16 −1.26431 −0.632155 0.774842i \(-0.717829\pi\)
−0.632155 + 0.774842i \(0.717829\pi\)
\(398\) −2.34754e16 −0.296056
\(399\) −1.93480e17 −2.40055
\(400\) −2.02129e16 −0.246740
\(401\) −3.66088e15 −0.0439691 −0.0219845 0.999758i \(-0.506998\pi\)
−0.0219845 + 0.999758i \(0.506998\pi\)
\(402\) 1.12410e16 0.132842
\(403\) 1.98457e16 0.230773
\(404\) −2.31879e16 −0.265327
\(405\) −8.85016e15 −0.0996537
\(406\) 1.45873e17 1.61642
\(407\) −4.86883e15 −0.0530959
\(408\) 2.91518e16 0.312877
\(409\) 1.41675e17 1.49655 0.748273 0.663390i \(-0.230883\pi\)
0.748273 + 0.663390i \(0.230883\pi\)
\(410\) 1.26926e16 0.131964
\(411\) −9.27882e16 −0.949557
\(412\) −8.85163e16 −0.891644
\(413\) −8.96037e16 −0.888486
\(414\) 6.89136e16 0.672671
\(415\) 9.94624e15 0.0955754
\(416\) 1.16807e16 0.110500
\(417\) −3.94515e16 −0.367434
\(418\) 2.52044e16 0.231116
\(419\) 1.44390e17 1.30360 0.651801 0.758390i \(-0.274014\pi\)
0.651801 + 0.758390i \(0.274014\pi\)
\(420\) 1.30055e16 0.115613
\(421\) 1.32169e17 1.15690 0.578451 0.815717i \(-0.303657\pi\)
0.578451 + 0.815717i \(0.303657\pi\)
\(422\) −8.60702e16 −0.741860
\(423\) −7.47864e16 −0.634761
\(424\) 4.74850e16 0.396897
\(425\) −7.29729e16 −0.600666
\(426\) −4.36453e16 −0.353814
\(427\) 5.39532e16 0.430761
\(428\) 4.47890e16 0.352198
\(429\) 3.23515e16 0.250566
\(430\) −1.50067e16 −0.114483
\(431\) −1.80535e17 −1.35662 −0.678312 0.734774i \(-0.737288\pi\)
−0.678312 + 0.734774i \(0.737288\pi\)
\(432\) −5.61384e15 −0.0415543
\(433\) −1.33563e17 −0.973900 −0.486950 0.873430i \(-0.661891\pi\)
−0.486950 + 0.873430i \(0.661891\pi\)
\(434\) −5.06058e16 −0.363511
\(435\) −3.85229e16 −0.272608
\(436\) −1.72311e16 −0.120130
\(437\) 1.47361e17 1.01217
\(438\) −2.11575e17 −1.43180
\(439\) 2.34588e17 1.56418 0.782090 0.623165i \(-0.214154\pi\)
0.782090 + 0.623165i \(0.214154\pi\)
\(440\) −1.69421e15 −0.0111307
\(441\) 1.61622e17 1.04629
\(442\) 4.21697e16 0.269002
\(443\) −1.26522e17 −0.795318 −0.397659 0.917533i \(-0.630177\pi\)
−0.397659 + 0.917533i \(0.630177\pi\)
\(444\) 2.26050e16 0.140028
\(445\) 1.19470e16 0.0729321
\(446\) 1.30613e17 0.785797
\(447\) 1.61520e17 0.957698
\(448\) −2.97852e16 −0.174058
\(449\) −1.91607e17 −1.10359 −0.551797 0.833978i \(-0.686058\pi\)
−0.551797 + 0.833978i \(0.686058\pi\)
\(450\) 1.36985e17 0.777661
\(451\) −8.05106e16 −0.450511
\(452\) 1.17822e17 0.649868
\(453\) 3.80851e16 0.207070
\(454\) −1.86876e16 −0.100159
\(455\) 1.88132e16 0.0994002
\(456\) −1.17018e17 −0.609513
\(457\) −2.62243e17 −1.34663 −0.673316 0.739355i \(-0.735131\pi\)
−0.673316 + 0.739355i \(0.735131\pi\)
\(458\) 1.80150e17 0.912031
\(459\) −2.02672e16 −0.101160
\(460\) −9.90546e15 −0.0487471
\(461\) −6.94961e16 −0.337213 −0.168607 0.985683i \(-0.553927\pi\)
−0.168607 + 0.985683i \(0.553927\pi\)
\(462\) −8.24950e16 −0.394689
\(463\) −2.31198e17 −1.09071 −0.545353 0.838207i \(-0.683604\pi\)
−0.545353 + 0.838207i \(0.683604\pi\)
\(464\) 8.82253e16 0.410418
\(465\) 1.33643e16 0.0613057
\(466\) −2.81202e17 −1.27206
\(467\) 1.70771e17 0.761825 0.380913 0.924611i \(-0.375610\pi\)
0.380913 + 0.924611i \(0.375610\pi\)
\(468\) −7.91609e16 −0.348267
\(469\) −4.14643e16 −0.179908
\(470\) 1.07496e16 0.0459998
\(471\) 4.17732e17 1.76304
\(472\) −5.41932e16 −0.225591
\(473\) 9.51886e16 0.390830
\(474\) 1.83063e17 0.741382
\(475\) 2.92921e17 1.17015
\(476\) −1.07531e17 −0.423729
\(477\) −3.21810e17 −1.25092
\(478\) 2.82210e17 1.08216
\(479\) 3.22880e17 1.22141 0.610703 0.791859i \(-0.290887\pi\)
0.610703 + 0.791859i \(0.290887\pi\)
\(480\) 7.86585e15 0.0293547
\(481\) 3.26993e16 0.120391
\(482\) −1.23481e17 −0.448535
\(483\) −4.82320e17 −1.72854
\(484\) −1.30659e17 −0.462001
\(485\) 1.74861e16 0.0610060
\(486\) −2.94776e17 −1.01475
\(487\) −2.41562e16 −0.0820521 −0.0410261 0.999158i \(-0.513063\pi\)
−0.0410261 + 0.999158i \(0.513063\pi\)
\(488\) 3.26314e16 0.109372
\(489\) 2.59855e17 0.859456
\(490\) −2.32312e16 −0.0758223
\(491\) −4.61064e17 −1.48502 −0.742508 0.669837i \(-0.766364\pi\)
−0.742508 + 0.669837i \(0.766364\pi\)
\(492\) 3.73793e17 1.18812
\(493\) 3.18512e17 0.999128
\(494\) −1.69274e17 −0.524040
\(495\) 1.14818e16 0.0350813
\(496\) −3.06069e16 −0.0922972
\(497\) 1.60992e17 0.479169
\(498\) 2.92913e17 0.860494
\(499\) −5.85158e17 −1.69676 −0.848379 0.529390i \(-0.822421\pi\)
−0.848379 + 0.529390i \(0.822421\pi\)
\(500\) −3.96398e16 −0.113456
\(501\) −1.02449e17 −0.289442
\(502\) −2.21652e16 −0.0618154
\(503\) −5.03073e17 −1.38497 −0.692483 0.721434i \(-0.743483\pi\)
−0.692483 + 0.721434i \(0.743483\pi\)
\(504\) 2.01857e17 0.548587
\(505\) −2.25878e16 −0.0606010
\(506\) 6.28312e16 0.166417
\(507\) 3.38805e17 0.885926
\(508\) 1.95794e17 0.505458
\(509\) 2.39181e17 0.609622 0.304811 0.952413i \(-0.401407\pi\)
0.304811 + 0.952413i \(0.401407\pi\)
\(510\) 2.83974e16 0.0714615
\(511\) 7.80427e17 1.93908
\(512\) −1.80144e16 −0.0441942
\(513\) 8.13546e16 0.197069
\(514\) −2.94084e16 −0.0703415
\(515\) −8.62256e16 −0.203652
\(516\) −4.41941e17 −1.03072
\(517\) −6.81857e16 −0.157038
\(518\) −8.33819e16 −0.189639
\(519\) −2.32763e17 −0.522788
\(520\) 1.13784e16 0.0252382
\(521\) 5.84939e17 1.28134 0.640671 0.767815i \(-0.278656\pi\)
0.640671 + 0.767815i \(0.278656\pi\)
\(522\) −5.97911e17 −1.29353
\(523\) −1.38091e17 −0.295055 −0.147528 0.989058i \(-0.547132\pi\)
−0.147528 + 0.989058i \(0.547132\pi\)
\(524\) −4.24729e17 −0.896312
\(525\) −9.58744e17 −1.99833
\(526\) 5.17738e17 1.06587
\(527\) −1.10497e17 −0.224690
\(528\) −4.98938e16 −0.100213
\(529\) −1.36683e17 −0.271178
\(530\) 4.62561e16 0.0906517
\(531\) 3.67272e17 0.711006
\(532\) 4.31641e17 0.825462
\(533\) 5.40713e17 1.02150
\(534\) 3.51835e17 0.656630
\(535\) 4.36299e16 0.0804424
\(536\) −2.50780e16 −0.0456796
\(537\) −5.67281e17 −1.02086
\(538\) −3.23878e17 −0.575833
\(539\) 1.47358e17 0.258848
\(540\) −5.46856e15 −0.00949104
\(541\) 2.87308e16 0.0492680 0.0246340 0.999697i \(-0.492158\pi\)
0.0246340 + 0.999697i \(0.492158\pi\)
\(542\) 3.14530e16 0.0532926
\(543\) 8.70880e17 1.45801
\(544\) −6.50358e16 −0.107587
\(545\) −1.67852e16 −0.0274378
\(546\) 5.54040e17 0.894930
\(547\) 1.77510e17 0.283338 0.141669 0.989914i \(-0.454753\pi\)
0.141669 + 0.989914i \(0.454753\pi\)
\(548\) 2.07005e17 0.326518
\(549\) −2.21146e17 −0.344714
\(550\) 1.24894e17 0.192391
\(551\) −1.27854e18 −1.94639
\(552\) −2.91712e17 −0.438885
\(553\) −6.75258e17 −1.00405
\(554\) −2.51467e17 −0.369545
\(555\) 2.20200e16 0.0319825
\(556\) 8.80138e16 0.126347
\(557\) 6.70558e17 0.951432 0.475716 0.879599i \(-0.342189\pi\)
0.475716 + 0.879599i \(0.342189\pi\)
\(558\) 2.07426e17 0.290898
\(559\) −6.39291e17 −0.886180
\(560\) −2.90144e16 −0.0397550
\(561\) −1.80127e17 −0.243961
\(562\) −7.32271e17 −0.980360
\(563\) 3.10511e17 0.410934 0.205467 0.978664i \(-0.434129\pi\)
0.205467 + 0.978664i \(0.434129\pi\)
\(564\) 3.16572e17 0.414150
\(565\) 1.14772e17 0.148430
\(566\) −1.35837e17 −0.173665
\(567\) 9.61389e17 1.21509
\(568\) 9.73698e16 0.121663
\(569\) −2.58365e17 −0.319157 −0.159578 0.987185i \(-0.551013\pi\)
−0.159578 + 0.987185i \(0.551013\pi\)
\(570\) −1.13990e17 −0.139213
\(571\) 9.91259e17 1.19688 0.598442 0.801166i \(-0.295787\pi\)
0.598442 + 0.801166i \(0.295787\pi\)
\(572\) −7.21741e16 −0.0861603
\(573\) 1.20891e18 1.42689
\(574\) −1.37880e18 −1.60906
\(575\) 7.30215e17 0.842577
\(576\) 1.22085e17 0.139289
\(577\) −9.99359e17 −1.12740 −0.563701 0.825979i \(-0.690623\pi\)
−0.563701 + 0.825979i \(0.690623\pi\)
\(578\) 3.99100e17 0.445196
\(579\) −3.85730e17 −0.425473
\(580\) 8.59422e16 0.0937399
\(581\) −1.08046e18 −1.16536
\(582\) 5.14960e17 0.549256
\(583\) −2.93407e17 −0.309474
\(584\) 4.72010e17 0.492343
\(585\) −7.71123e16 −0.0795446
\(586\) 1.27392e18 1.29959
\(587\) 7.79668e17 0.786614 0.393307 0.919407i \(-0.371331\pi\)
0.393307 + 0.919407i \(0.371331\pi\)
\(588\) −6.84150e17 −0.682651
\(589\) 4.43548e17 0.437715
\(590\) −5.27907e16 −0.0515252
\(591\) −2.87443e18 −2.77480
\(592\) −5.04302e16 −0.0481504
\(593\) 8.86852e17 0.837521 0.418761 0.908097i \(-0.362465\pi\)
0.418761 + 0.908097i \(0.362465\pi\)
\(594\) 3.46876e16 0.0324013
\(595\) −1.04748e17 −0.0967801
\(596\) −3.60342e17 −0.329317
\(597\) −6.73452e17 −0.608798
\(598\) −4.21978e17 −0.377339
\(599\) −1.56613e18 −1.38533 −0.692665 0.721260i \(-0.743563\pi\)
−0.692665 + 0.721260i \(0.743563\pi\)
\(600\) −5.79858e17 −0.507386
\(601\) 1.44186e18 1.24807 0.624036 0.781396i \(-0.285492\pi\)
0.624036 + 0.781396i \(0.285492\pi\)
\(602\) 1.63017e18 1.39590
\(603\) 1.69956e17 0.143971
\(604\) −8.49655e16 −0.0712037
\(605\) −1.27277e17 −0.105521
\(606\) −6.65202e17 −0.545609
\(607\) 4.08280e17 0.331307 0.165654 0.986184i \(-0.447027\pi\)
0.165654 + 0.986184i \(0.447027\pi\)
\(608\) 2.61061e17 0.209589
\(609\) 4.18473e18 3.32395
\(610\) 3.17870e16 0.0249807
\(611\) 4.57939e17 0.356073
\(612\) 4.40753e17 0.339087
\(613\) −1.01386e18 −0.771768 −0.385884 0.922547i \(-0.626104\pi\)
−0.385884 + 0.922547i \(0.626104\pi\)
\(614\) 8.10307e17 0.610317
\(615\) 3.64120e17 0.271367
\(616\) 1.84041e17 0.135719
\(617\) −2.38764e18 −1.74227 −0.871133 0.491047i \(-0.836614\pi\)
−0.871133 + 0.491047i \(0.836614\pi\)
\(618\) −2.53931e18 −1.83354
\(619\) −8.05525e17 −0.575559 −0.287780 0.957697i \(-0.592917\pi\)
−0.287780 + 0.957697i \(0.592917\pi\)
\(620\) −2.98148e16 −0.0210808
\(621\) 2.02807e17 0.141901
\(622\) −6.95539e16 −0.0481597
\(623\) −1.29780e18 −0.889272
\(624\) 3.35089e17 0.227227
\(625\) 1.43207e18 0.961045
\(626\) −1.36565e18 −0.907001
\(627\) 7.23050e17 0.475258
\(628\) −9.31933e17 −0.606243
\(629\) −1.82064e17 −0.117218
\(630\) 1.96633e17 0.125298
\(631\) 2.64035e18 1.66521 0.832607 0.553865i \(-0.186847\pi\)
0.832607 + 0.553865i \(0.186847\pi\)
\(632\) −4.08403e17 −0.254934
\(633\) −2.46914e18 −1.52553
\(634\) 1.92622e18 1.17794
\(635\) 1.90727e17 0.115447
\(636\) 1.36222e18 0.816164
\(637\) −9.89661e17 −0.586921
\(638\) −5.45139e17 −0.320017
\(639\) −6.59884e17 −0.383453
\(640\) −1.75482e16 −0.0100940
\(641\) −1.06429e18 −0.606015 −0.303007 0.952988i \(-0.597991\pi\)
−0.303007 + 0.952988i \(0.597991\pi\)
\(642\) 1.28489e18 0.724247
\(643\) 6.81097e17 0.380047 0.190024 0.981780i \(-0.439144\pi\)
0.190024 + 0.981780i \(0.439144\pi\)
\(644\) 1.07603e18 0.594380
\(645\) −4.30504e17 −0.235417
\(646\) 9.42484e17 0.510226
\(647\) 2.04457e18 1.09578 0.547891 0.836550i \(-0.315431\pi\)
0.547891 + 0.836550i \(0.315431\pi\)
\(648\) 5.81458e17 0.308518
\(649\) 3.34856e17 0.175901
\(650\) −8.38796e17 −0.436234
\(651\) −1.45175e18 −0.747509
\(652\) −5.79720e17 −0.295535
\(653\) 2.08196e18 1.05084 0.525419 0.850844i \(-0.323909\pi\)
0.525419 + 0.850844i \(0.323909\pi\)
\(654\) −4.94318e17 −0.247031
\(655\) −4.13738e17 −0.204718
\(656\) −8.33910e17 −0.408549
\(657\) −3.19885e18 −1.55174
\(658\) −1.16772e18 −0.560883
\(659\) 2.11908e18 1.00784 0.503920 0.863750i \(-0.331891\pi\)
0.503920 + 0.863750i \(0.331891\pi\)
\(660\) −4.86026e16 −0.0228888
\(661\) −1.46459e18 −0.682980 −0.341490 0.939885i \(-0.610931\pi\)
−0.341490 + 0.939885i \(0.610931\pi\)
\(662\) −6.42996e17 −0.296914
\(663\) 1.20974e18 0.553165
\(664\) −6.53470e17 −0.295892
\(665\) 4.20470e17 0.188536
\(666\) 3.41770e17 0.151758
\(667\) −3.18724e18 −1.40151
\(668\) 2.28557e17 0.0995284
\(669\) 3.74696e18 1.61588
\(670\) −2.44290e16 −0.0104333
\(671\) −2.01628e17 −0.0852813
\(672\) −8.54464e17 −0.357926
\(673\) 7.25358e17 0.300922 0.150461 0.988616i \(-0.451924\pi\)
0.150461 + 0.988616i \(0.451924\pi\)
\(674\) −2.56312e17 −0.105312
\(675\) 4.03134e17 0.164049
\(676\) −7.55852e17 −0.304637
\(677\) 4.22902e17 0.168816 0.0844079 0.996431i \(-0.473100\pi\)
0.0844079 + 0.996431i \(0.473100\pi\)
\(678\) 3.38001e18 1.33636
\(679\) −1.89951e18 −0.743856
\(680\) −6.33527e16 −0.0245730
\(681\) −5.36102e17 −0.205964
\(682\) 1.89118e17 0.0719672
\(683\) 1.37469e18 0.518168 0.259084 0.965855i \(-0.416579\pi\)
0.259084 + 0.965855i \(0.416579\pi\)
\(684\) −1.76923e18 −0.660572
\(685\) 2.01648e17 0.0745769
\(686\) −1.64072e17 −0.0601075
\(687\) 5.16806e18 1.87546
\(688\) 9.85942e17 0.354427
\(689\) 1.97053e18 0.701712
\(690\) −2.84163e17 −0.100242
\(691\) −3.23918e18 −1.13195 −0.565976 0.824422i \(-0.691500\pi\)
−0.565976 + 0.824422i \(0.691500\pi\)
\(692\) 5.19279e17 0.179767
\(693\) −1.24726e18 −0.427752
\(694\) −3.88014e15 −0.00131829
\(695\) 8.57361e16 0.0288578
\(696\) 2.53096e18 0.843968
\(697\) −3.01059e18 −0.994577
\(698\) −1.96850e18 −0.644281
\(699\) −8.06697e18 −2.61583
\(700\) 2.13890e18 0.687152
\(701\) −4.32730e18 −1.37737 −0.688683 0.725062i \(-0.741811\pi\)
−0.688683 + 0.725062i \(0.741811\pi\)
\(702\) −2.32963e17 −0.0734677
\(703\) 7.30824e17 0.228351
\(704\) 1.11310e17 0.0344597
\(705\) 3.08379e17 0.0945923
\(706\) 1.33694e18 0.406332
\(707\) 2.45370e18 0.738917
\(708\) −1.55467e18 −0.463897
\(709\) 3.72211e18 1.10050 0.550248 0.835001i \(-0.314533\pi\)
0.550248 + 0.835001i \(0.314533\pi\)
\(710\) 9.48500e16 0.0277881
\(711\) 2.76778e18 0.803488
\(712\) −7.84921e17 −0.225791
\(713\) 1.10571e18 0.315180
\(714\) −3.08479e18 −0.871340
\(715\) −7.03063e16 −0.0196791
\(716\) 1.26557e18 0.351035
\(717\) 8.09590e18 2.22531
\(718\) 1.57674e18 0.429489
\(719\) −2.69743e18 −0.728135 −0.364068 0.931373i \(-0.618612\pi\)
−0.364068 + 0.931373i \(0.618612\pi\)
\(720\) 1.18926e17 0.0318137
\(721\) 9.36665e18 2.48316
\(722\) −1.09184e18 −0.286859
\(723\) −3.54237e18 −0.922349
\(724\) −1.94288e18 −0.501355
\(725\) −6.33552e18 −1.62026
\(726\) −3.74827e18 −0.950041
\(727\) −1.30338e18 −0.327414 −0.163707 0.986509i \(-0.552345\pi\)
−0.163707 + 0.986509i \(0.552345\pi\)
\(728\) −1.23603e18 −0.307733
\(729\) −4.92006e18 −1.21406
\(730\) 4.59795e17 0.112452
\(731\) 3.55946e18 0.862821
\(732\) 9.36114e17 0.224909
\(733\) 1.23214e18 0.293417 0.146708 0.989180i \(-0.453132\pi\)
0.146708 + 0.989180i \(0.453132\pi\)
\(734\) 3.38336e18 0.798591
\(735\) −6.66445e17 −0.155918
\(736\) 6.50791e17 0.150916
\(737\) 1.54956e17 0.0356179
\(738\) 5.65148e18 1.28764
\(739\) −2.37998e18 −0.537508 −0.268754 0.963209i \(-0.586612\pi\)
−0.268754 + 0.963209i \(0.586612\pi\)
\(740\) −4.91251e16 −0.0109976
\(741\) −4.85604e18 −1.07762
\(742\) −5.02478e18 −1.10533
\(743\) −1.71342e18 −0.373626 −0.186813 0.982396i \(-0.559816\pi\)
−0.186813 + 0.982396i \(0.559816\pi\)
\(744\) −8.78035e17 −0.189796
\(745\) −3.51017e17 −0.0752163
\(746\) 4.12524e17 0.0876289
\(747\) 4.42863e18 0.932578
\(748\) 4.01852e17 0.0838892
\(749\) −4.73950e18 −0.980846
\(750\) −1.13717e18 −0.233306
\(751\) 8.15576e18 1.65884 0.829421 0.558624i \(-0.188671\pi\)
0.829421 + 0.558624i \(0.188671\pi\)
\(752\) −7.06252e17 −0.142411
\(753\) −6.35863e17 −0.127115
\(754\) 3.66118e18 0.725617
\(755\) −8.27667e16 −0.0162630
\(756\) 5.94047e17 0.115726
\(757\) −4.08819e18 −0.789601 −0.394800 0.918767i \(-0.629186\pi\)
−0.394800 + 0.918767i \(0.629186\pi\)
\(758\) 4.21624e18 0.807375
\(759\) 1.80247e18 0.342213
\(760\) 2.54305e17 0.0478703
\(761\) 4.83340e18 0.902094 0.451047 0.892500i \(-0.351051\pi\)
0.451047 + 0.892500i \(0.351051\pi\)
\(762\) 5.61684e18 1.03941
\(763\) 1.82337e18 0.334554
\(764\) −2.69701e18 −0.490655
\(765\) 4.29347e17 0.0774478
\(766\) −5.67972e18 −1.01587
\(767\) −2.24891e18 −0.398844
\(768\) −5.16788e17 −0.0908792
\(769\) −5.13768e18 −0.895873 −0.447936 0.894065i \(-0.647841\pi\)
−0.447936 + 0.894065i \(0.647841\pi\)
\(770\) 1.79278e17 0.0309983
\(771\) −8.43654e17 −0.144647
\(772\) 8.60539e17 0.146305
\(773\) −2.81948e18 −0.475337 −0.237669 0.971346i \(-0.576383\pi\)
−0.237669 + 0.971346i \(0.576383\pi\)
\(774\) −6.68181e18 −1.11706
\(775\) 2.19790e18 0.364374
\(776\) −1.14884e18 −0.188869
\(777\) −2.39202e18 −0.389967
\(778\) −4.01577e18 −0.649235
\(779\) 1.20848e19 1.93752
\(780\) 3.26417e17 0.0518989
\(781\) −6.01642e17 −0.0948652
\(782\) 2.34949e18 0.367393
\(783\) −1.75960e18 −0.272874
\(784\) 1.52630e18 0.234738
\(785\) −9.07816e17 −0.138467
\(786\) −1.21844e19 −1.84314
\(787\) −5.90600e18 −0.886050 −0.443025 0.896509i \(-0.646095\pi\)
−0.443025 + 0.896509i \(0.646095\pi\)
\(788\) 6.41266e18 0.954153
\(789\) 1.48526e19 2.19181
\(790\) −3.97834e17 −0.0582271
\(791\) −1.24677e19 −1.80983
\(792\) −7.54357e17 −0.108608
\(793\) 1.35414e18 0.193370
\(794\) 6.31207e18 0.894002
\(795\) 1.32697e18 0.186413
\(796\) 1.50243e18 0.209343
\(797\) 6.00465e18 0.829867 0.414933 0.909852i \(-0.363805\pi\)
0.414933 + 0.909852i \(0.363805\pi\)
\(798\) 1.23827e19 1.69745
\(799\) −2.54972e18 −0.346687
\(800\) 1.29363e18 0.174471
\(801\) 5.31948e18 0.711636
\(802\) 2.34296e17 0.0310908
\(803\) −2.91652e18 −0.383897
\(804\) −7.19426e17 −0.0939338
\(805\) 1.04818e18 0.135757
\(806\) −1.27013e18 −0.163181
\(807\) −9.29124e18 −1.18412
\(808\) 1.48402e18 0.187615
\(809\) −1.03281e19 −1.29525 −0.647624 0.761960i \(-0.724237\pi\)
−0.647624 + 0.761960i \(0.724237\pi\)
\(810\) 5.66410e17 0.0704658
\(811\) 2.37489e18 0.293094 0.146547 0.989204i \(-0.453184\pi\)
0.146547 + 0.989204i \(0.453184\pi\)
\(812\) −9.33586e18 −1.14298
\(813\) 9.02308e17 0.109589
\(814\) 3.11605e17 0.0375445
\(815\) −5.64718e17 −0.0675005
\(816\) −1.86571e18 −0.221238
\(817\) −1.42880e19 −1.68085
\(818\) −9.06717e18 −1.05822
\(819\) 8.37668e18 0.969899
\(820\) −8.12329e17 −0.0933129
\(821\) 6.17741e18 0.704006 0.352003 0.935999i \(-0.385501\pi\)
0.352003 + 0.935999i \(0.385501\pi\)
\(822\) 5.93845e18 0.671438
\(823\) −3.84246e18 −0.431033 −0.215516 0.976500i \(-0.569143\pi\)
−0.215516 + 0.976500i \(0.569143\pi\)
\(824\) 5.66504e18 0.630487
\(825\) 3.58291e18 0.395626
\(826\) 5.73463e18 0.628254
\(827\) 1.35926e19 1.47746 0.738731 0.674001i \(-0.235426\pi\)
0.738731 + 0.674001i \(0.235426\pi\)
\(828\) −4.41047e18 −0.475650
\(829\) 4.07966e17 0.0436536 0.0218268 0.999762i \(-0.493052\pi\)
0.0218268 + 0.999762i \(0.493052\pi\)
\(830\) −6.36559e17 −0.0675820
\(831\) −7.21395e18 −0.759917
\(832\) −7.47562e17 −0.0781350
\(833\) 5.51025e18 0.571451
\(834\) 2.52490e18 0.259815
\(835\) 2.22642e17 0.0227324
\(836\) −1.61308e18 −0.163424
\(837\) 6.10435e17 0.0613655
\(838\) −9.24094e18 −0.921785
\(839\) 4.60930e18 0.456229 0.228114 0.973634i \(-0.426744\pi\)
0.228114 + 0.973634i \(0.426744\pi\)
\(840\) −8.32351e17 −0.0817506
\(841\) 1.73927e19 1.69509
\(842\) −8.45883e18 −0.818054
\(843\) −2.10070e19 −2.01598
\(844\) 5.50849e18 0.524574
\(845\) −7.36291e17 −0.0695794
\(846\) 4.78633e18 0.448844
\(847\) 1.38261e19 1.28664
\(848\) −3.03904e18 −0.280649
\(849\) −3.89683e18 −0.357118
\(850\) 4.67026e18 0.424735
\(851\) 1.82185e18 0.164426
\(852\) 2.79330e18 0.250184
\(853\) −5.13496e18 −0.456424 −0.228212 0.973612i \(-0.573288\pi\)
−0.228212 + 0.973612i \(0.573288\pi\)
\(854\) −3.45300e18 −0.304594
\(855\) −1.72345e18 −0.150875
\(856\) −2.86650e18 −0.249042
\(857\) 8.29128e18 0.714901 0.357451 0.933932i \(-0.383646\pi\)
0.357451 + 0.933932i \(0.383646\pi\)
\(858\) −2.07050e18 −0.177177
\(859\) −6.79619e17 −0.0577179 −0.0288589 0.999583i \(-0.509187\pi\)
−0.0288589 + 0.999583i \(0.509187\pi\)
\(860\) 9.60426e17 0.0809514
\(861\) −3.95542e19 −3.30881
\(862\) 1.15543e19 0.959279
\(863\) −2.11875e19 −1.74586 −0.872929 0.487846i \(-0.837783\pi\)
−0.872929 + 0.487846i \(0.837783\pi\)
\(864\) 3.59286e17 0.0293833
\(865\) 5.05840e17 0.0410590
\(866\) 8.54802e18 0.688651
\(867\) 1.14492e19 0.915483
\(868\) 3.23877e18 0.257041
\(869\) 2.52349e18 0.198781
\(870\) 2.46547e18 0.192763
\(871\) −1.04069e18 −0.0807613
\(872\) 1.10279e18 0.0849448
\(873\) 7.78582e18 0.595267
\(874\) −9.43112e18 −0.715713
\(875\) 4.19462e18 0.315966
\(876\) 1.35408e19 1.01244
\(877\) 5.29243e18 0.392788 0.196394 0.980525i \(-0.437077\pi\)
0.196394 + 0.980525i \(0.437077\pi\)
\(878\) −1.50136e19 −1.10604
\(879\) 3.65455e19 2.67243
\(880\) 1.08429e17 0.00787063
\(881\) 2.31832e18 0.167043 0.0835217 0.996506i \(-0.473383\pi\)
0.0835217 + 0.996506i \(0.473383\pi\)
\(882\) −1.03438e19 −0.739837
\(883\) 1.38961e19 0.986615 0.493307 0.869855i \(-0.335788\pi\)
0.493307 + 0.869855i \(0.335788\pi\)
\(884\) −2.69886e18 −0.190213
\(885\) −1.51443e18 −0.105954
\(886\) 8.09739e18 0.562375
\(887\) −1.68484e19 −1.16159 −0.580797 0.814048i \(-0.697259\pi\)
−0.580797 + 0.814048i \(0.697259\pi\)
\(888\) −1.44672e18 −0.0990146
\(889\) −2.07186e19 −1.40766
\(890\) −7.64608e17 −0.0515708
\(891\) −3.59279e18 −0.240562
\(892\) −8.35924e18 −0.555642
\(893\) 1.02348e19 0.675378
\(894\) −1.03373e19 −0.677195
\(895\) 1.23282e18 0.0801768
\(896\) 1.90625e18 0.123078
\(897\) −1.21055e19 −0.775945
\(898\) 1.22628e19 0.780359
\(899\) −9.59340e18 −0.606087
\(900\) −8.76702e18 −0.549890
\(901\) −1.09716e19 −0.683215
\(902\) 5.15268e18 0.318559
\(903\) 4.67654e19 2.87048
\(904\) −7.54058e18 −0.459526
\(905\) −1.89260e18 −0.114510
\(906\) −2.43745e18 −0.146420
\(907\) −2.54854e19 −1.52000 −0.760000 0.649923i \(-0.774801\pi\)
−0.760000 + 0.649923i \(0.774801\pi\)
\(908\) 1.19601e18 0.0708233
\(909\) −1.00574e19 −0.591315
\(910\) −1.20404e18 −0.0702866
\(911\) −1.75641e19 −1.01802 −0.509010 0.860761i \(-0.669988\pi\)
−0.509010 + 0.860761i \(0.669988\pi\)
\(912\) 7.48918e18 0.430991
\(913\) 4.03775e18 0.230717
\(914\) 1.67836e19 0.952213
\(915\) 9.11888e17 0.0513694
\(916\) −1.15296e19 −0.644903
\(917\) 4.49442e19 2.49616
\(918\) 1.29710e18 0.0715312
\(919\) 2.76291e18 0.151292 0.0756460 0.997135i \(-0.475898\pi\)
0.0756460 + 0.997135i \(0.475898\pi\)
\(920\) 6.33949e17 0.0344694
\(921\) 2.32457e19 1.25503
\(922\) 4.44775e18 0.238446
\(923\) 4.04066e18 0.215100
\(924\) 5.27968e18 0.279087
\(925\) 3.62143e18 0.190090
\(926\) 1.47967e19 0.771245
\(927\) −3.83925e19 −1.98714
\(928\) −5.64642e18 −0.290209
\(929\) −2.02252e19 −1.03226 −0.516131 0.856510i \(-0.672628\pi\)
−0.516131 + 0.856510i \(0.672628\pi\)
\(930\) −8.55313e17 −0.0433497
\(931\) −2.21187e19 −1.11324
\(932\) 1.79969e19 0.899486
\(933\) −1.99533e18 −0.0990338
\(934\) −1.09294e19 −0.538692
\(935\) 3.91453e17 0.0191604
\(936\) 5.06630e18 0.246262
\(937\) 9.12409e18 0.440436 0.220218 0.975451i \(-0.429323\pi\)
0.220218 + 0.975451i \(0.429323\pi\)
\(938\) 2.65372e18 0.127214
\(939\) −3.91772e19 −1.86512
\(940\) −6.87975e17 −0.0325268
\(941\) 9.47005e18 0.444651 0.222326 0.974972i \(-0.428635\pi\)
0.222326 + 0.974972i \(0.428635\pi\)
\(942\) −2.67348e19 −1.24666
\(943\) 3.01260e19 1.39513
\(944\) 3.46836e18 0.159517
\(945\) 5.78674e17 0.0264318
\(946\) −6.09207e18 −0.276358
\(947\) −1.07060e19 −0.482340 −0.241170 0.970483i \(-0.577531\pi\)
−0.241170 + 0.970483i \(0.577531\pi\)
\(948\) −1.17160e19 −0.524236
\(949\) 1.95875e19 0.870460
\(950\) −1.87469e19 −0.827422
\(951\) 5.52583e19 2.42228
\(952\) 6.88198e18 0.299622
\(953\) −1.17296e17 −0.00507199 −0.00253599 0.999997i \(-0.500807\pi\)
−0.00253599 + 0.999997i \(0.500807\pi\)
\(954\) 2.05958e19 0.884534
\(955\) −2.62722e18 −0.112066
\(956\) −1.80614e19 −0.765202
\(957\) −1.56387e19 −0.658070
\(958\) −2.06643e19 −0.863665
\(959\) −2.19049e19 −0.909328
\(960\) −5.03414e17 −0.0207569
\(961\) −2.10894e19 −0.863700
\(962\) −2.09276e18 −0.0851296
\(963\) 1.94265e19 0.784917
\(964\) 7.90281e18 0.317162
\(965\) 8.38269e17 0.0334161
\(966\) 3.08685e19 1.22226
\(967\) 1.48196e19 0.582860 0.291430 0.956592i \(-0.405869\pi\)
0.291430 + 0.956592i \(0.405869\pi\)
\(968\) 8.36214e18 0.326684
\(969\) 2.70375e19 1.04921
\(970\) −1.11911e18 −0.0431378
\(971\) 3.03687e19 1.16279 0.581395 0.813622i \(-0.302507\pi\)
0.581395 + 0.813622i \(0.302507\pi\)
\(972\) 1.88657e19 0.717533
\(973\) −9.31348e18 −0.351867
\(974\) 1.54599e18 0.0580196
\(975\) −2.40630e19 −0.897055
\(976\) −2.08841e18 −0.0773379
\(977\) −3.42204e19 −1.25884 −0.629419 0.777066i \(-0.716707\pi\)
−0.629419 + 0.777066i \(0.716707\pi\)
\(978\) −1.66307e19 −0.607727
\(979\) 4.84998e18 0.176057
\(980\) 1.48680e18 0.0536145
\(981\) −7.47373e18 −0.267725
\(982\) 2.95081e19 1.05007
\(983\) −2.19778e19 −0.776939 −0.388469 0.921462i \(-0.626996\pi\)
−0.388469 + 0.921462i \(0.626996\pi\)
\(984\) −2.39228e19 −0.840124
\(985\) 6.24671e18 0.217929
\(986\) −2.03848e19 −0.706490
\(987\) −3.34991e19 −1.15338
\(988\) 1.08335e19 0.370552
\(989\) −3.56183e19 −1.21031
\(990\) −7.34835e17 −0.0248062
\(991\) 1.72348e19 0.578001 0.289000 0.957329i \(-0.406677\pi\)
0.289000 + 0.957329i \(0.406677\pi\)
\(992\) 1.95884e18 0.0652639
\(993\) −1.84459e19 −0.610564
\(994\) −1.03035e19 −0.338824
\(995\) 1.46355e18 0.0478142
\(996\) −1.87464e19 −0.608461
\(997\) 5.30366e19 1.71024 0.855119 0.518431i \(-0.173484\pi\)
0.855119 + 0.518431i \(0.173484\pi\)
\(998\) 3.74501e19 1.19979
\(999\) 1.00580e18 0.0320137
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 2.14.a.a.1.1 1
3.2 odd 2 18.14.a.d.1.1 1
4.3 odd 2 16.14.a.d.1.1 1
5.2 odd 4 50.14.b.e.49.1 2
5.3 odd 4 50.14.b.e.49.2 2
5.4 even 2 50.14.a.e.1.1 1
7.2 even 3 98.14.c.h.67.1 2
7.3 odd 6 98.14.c.e.79.1 2
7.4 even 3 98.14.c.h.79.1 2
7.5 odd 6 98.14.c.e.67.1 2
7.6 odd 2 98.14.a.b.1.1 1
8.3 odd 2 64.14.a.a.1.1 1
8.5 even 2 64.14.a.i.1.1 1
12.11 even 2 144.14.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.14.a.a.1.1 1 1.1 even 1 trivial
16.14.a.d.1.1 1 4.3 odd 2
18.14.a.d.1.1 1 3.2 odd 2
50.14.a.e.1.1 1 5.4 even 2
50.14.b.e.49.1 2 5.2 odd 4
50.14.b.e.49.2 2 5.3 odd 4
64.14.a.a.1.1 1 8.3 odd 2
64.14.a.i.1.1 1 8.5 even 2
98.14.a.b.1.1 1 7.6 odd 2
98.14.c.e.67.1 2 7.5 odd 6
98.14.c.e.79.1 2 7.3 odd 6
98.14.c.h.67.1 2 7.2 even 3
98.14.c.h.79.1 2 7.4 even 3
144.14.a.d.1.1 1 12.11 even 2