Properties

Label 15.18.a.a.1.1
Level $15$
Weight $18$
Character 15.1
Self dual yes
Analytic conductor $27.483$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [15,18,Mod(1,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.1");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 15.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: \(27.4833131017\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{849}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 212 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(15.0688\) of defining polynomial
Character \(\chi\) \(=\) 15.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-702.477 q^{2} +6561.00 q^{3} +362402. q^{4} +390625. q^{5} -4.60895e6 q^{6} -2.67481e6 q^{7} -1.62504e8 q^{8} +4.30467e7 q^{9} -2.74405e8 q^{10} -9.86334e8 q^{11} +2.37772e9 q^{12} +2.81488e9 q^{13} +1.87899e9 q^{14} +2.56289e9 q^{15} +6.66544e10 q^{16} +1.28501e10 q^{17} -3.02393e10 q^{18} -1.22560e11 q^{19} +1.41563e11 q^{20} -1.75494e10 q^{21} +6.92877e11 q^{22} +1.14646e11 q^{23} -1.06619e12 q^{24} +1.52588e11 q^{25} -1.97739e12 q^{26} +2.82430e11 q^{27} -9.69355e11 q^{28} -4.90027e12 q^{29} -1.80037e12 q^{30} +7.64406e12 q^{31} -2.55235e13 q^{32} -6.47134e12 q^{33} -9.02688e12 q^{34} -1.04485e12 q^{35} +1.56002e13 q^{36} +3.18983e12 q^{37} +8.60956e13 q^{38} +1.84684e13 q^{39} -6.34781e13 q^{40} -5.53137e12 q^{41} +1.23281e13 q^{42} +2.42435e13 q^{43} -3.57449e14 q^{44} +1.68151e13 q^{45} -8.05365e13 q^{46} -5.82161e13 q^{47} +4.37320e14 q^{48} -2.25476e14 q^{49} -1.07189e14 q^{50} +8.43094e13 q^{51} +1.02012e15 q^{52} -6.45746e14 q^{53} -1.98400e14 q^{54} -3.85287e14 q^{55} +4.34667e14 q^{56} -8.04117e14 q^{57} +3.44233e15 q^{58} -1.04433e12 q^{59} +9.28796e14 q^{60} -1.56088e15 q^{61} -5.36978e15 q^{62} -1.15142e14 q^{63} +9.19314e15 q^{64} +1.09956e15 q^{65} +4.54597e15 q^{66} +1.31151e15 q^{67} +4.65689e15 q^{68} +7.52195e14 q^{69} +7.33981e14 q^{70} -7.09230e15 q^{71} -6.99526e15 q^{72} -1.17619e16 q^{73} -2.24078e15 q^{74} +1.00113e15 q^{75} -4.44160e16 q^{76} +2.63826e15 q^{77} -1.29737e16 q^{78} -7.14674e15 q^{79} +2.60369e16 q^{80} +1.85302e15 q^{81} +3.88566e15 q^{82} +9.04707e15 q^{83} -6.35994e15 q^{84} +5.01956e15 q^{85} -1.70305e16 q^{86} -3.21507e16 q^{87} +1.60283e17 q^{88} -1.16786e16 q^{89} -1.18122e16 q^{90} -7.52927e15 q^{91} +4.15481e16 q^{92} +5.01527e16 q^{93} +4.08955e16 q^{94} -4.78750e16 q^{95} -1.67460e17 q^{96} -8.32538e16 q^{97} +1.58392e17 q^{98} -4.24585e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 356 q^{2} + 13122 q^{3} + 351376 q^{4} + 781250 q^{5} - 2335716 q^{6} - 20754552 q^{7} - 211737408 q^{8} + 86093442 q^{9} - 139062500 q^{10} - 1131629912 q^{11} + 2305377936 q^{12} - 446672524 q^{13}+ \cdots - 48\!\cdots\!52 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −702.477 −1.94034 −0.970168 0.242432i \(-0.922055\pi\)
−0.970168 + 0.242432i \(0.922055\pi\)
\(3\) 6561.00 0.577350
\(4\) 362402. 2.76491
\(5\) 390625. 0.447214
\(6\) −4.60895e6 −1.12025
\(7\) −2.67481e6 −0.175372 −0.0876858 0.996148i \(-0.527947\pi\)
−0.0876858 + 0.996148i \(0.527947\pi\)
\(8\) −1.62504e8 −3.42451
\(9\) 4.30467e7 0.333333
\(10\) −2.74405e8 −0.867745
\(11\) −9.86334e8 −1.38735 −0.693675 0.720288i \(-0.744010\pi\)
−0.693675 + 0.720288i \(0.744010\pi\)
\(12\) 2.37772e9 1.59632
\(13\) 2.81488e9 0.957065 0.478533 0.878070i \(-0.341169\pi\)
0.478533 + 0.878070i \(0.341169\pi\)
\(14\) 1.87899e9 0.340280
\(15\) 2.56289e9 0.258199
\(16\) 6.66544e10 3.87980
\(17\) 1.28501e10 0.446776 0.223388 0.974730i \(-0.428288\pi\)
0.223388 + 0.974730i \(0.428288\pi\)
\(18\) −3.02393e10 −0.646779
\(19\) −1.22560e11 −1.65555 −0.827777 0.561057i \(-0.810395\pi\)
−0.827777 + 0.561057i \(0.810395\pi\)
\(20\) 1.41563e11 1.23650
\(21\) −1.75494e10 −0.101251
\(22\) 6.92877e11 2.69193
\(23\) 1.14646e11 0.305263 0.152631 0.988283i \(-0.451225\pi\)
0.152631 + 0.988283i \(0.451225\pi\)
\(24\) −1.06619e12 −1.97714
\(25\) 1.52588e11 0.200000
\(26\) −1.97739e12 −1.85703
\(27\) 2.82430e11 0.192450
\(28\) −9.69355e11 −0.484886
\(29\) −4.90027e12 −1.81902 −0.909510 0.415682i \(-0.863543\pi\)
−0.909510 + 0.415682i \(0.863543\pi\)
\(30\) −1.80037e12 −0.500993
\(31\) 7.64406e12 1.60972 0.804859 0.593466i \(-0.202241\pi\)
0.804859 + 0.593466i \(0.202241\pi\)
\(32\) −2.55235e13 −4.10361
\(33\) −6.47134e12 −0.800987
\(34\) −9.02688e12 −0.866896
\(35\) −1.04485e12 −0.0784286
\(36\) 1.56002e13 0.921635
\(37\) 3.18983e12 0.149298 0.0746488 0.997210i \(-0.476216\pi\)
0.0746488 + 0.997210i \(0.476216\pi\)
\(38\) 8.60956e13 3.21233
\(39\) 1.84684e13 0.552562
\(40\) −6.34781e13 −1.53149
\(41\) −5.53137e12 −0.108186 −0.0540928 0.998536i \(-0.517227\pi\)
−0.0540928 + 0.998536i \(0.517227\pi\)
\(42\) 1.23281e13 0.196461
\(43\) 2.42435e13 0.316311 0.158155 0.987414i \(-0.449445\pi\)
0.158155 + 0.987414i \(0.449445\pi\)
\(44\) −3.57449e14 −3.83589
\(45\) 1.68151e13 0.149071
\(46\) −8.05365e13 −0.592313
\(47\) −5.82161e13 −0.356625 −0.178312 0.983974i \(-0.557064\pi\)
−0.178312 + 0.983974i \(0.557064\pi\)
\(48\) 4.37320e14 2.24000
\(49\) −2.25476e14 −0.969245
\(50\) −1.07189e14 −0.388067
\(51\) 8.43094e13 0.257946
\(52\) 1.02012e15 2.64620
\(53\) −6.45746e14 −1.42468 −0.712339 0.701835i \(-0.752364\pi\)
−0.712339 + 0.701835i \(0.752364\pi\)
\(54\) −1.98400e14 −0.373418
\(55\) −3.85287e14 −0.620442
\(56\) 4.34667e14 0.600562
\(57\) −8.04117e14 −0.955835
\(58\) 3.44233e15 3.52951
\(59\) −1.04433e12 −0.000925970 0 −0.000462985 1.00000i \(-0.500147\pi\)
−0.000462985 1.00000i \(0.500147\pi\)
\(60\) 9.28796e14 0.713896
\(61\) −1.56088e15 −1.04247 −0.521237 0.853412i \(-0.674529\pi\)
−0.521237 + 0.853412i \(0.674529\pi\)
\(62\) −5.36978e15 −3.12340
\(63\) −1.15142e14 −0.0584572
\(64\) 9.19314e15 4.08258
\(65\) 1.09956e15 0.428013
\(66\) 4.54597e15 1.55419
\(67\) 1.31151e15 0.394581 0.197290 0.980345i \(-0.436786\pi\)
0.197290 + 0.980345i \(0.436786\pi\)
\(68\) 4.65689e15 1.23529
\(69\) 7.52195e14 0.176244
\(70\) 7.33981e14 0.152178
\(71\) −7.09230e15 −1.30344 −0.651720 0.758459i \(-0.725952\pi\)
−0.651720 + 0.758459i \(0.725952\pi\)
\(72\) −6.99526e15 −1.14150
\(73\) −1.17619e16 −1.70700 −0.853501 0.521091i \(-0.825525\pi\)
−0.853501 + 0.521091i \(0.825525\pi\)
\(74\) −2.24078e15 −0.289688
\(75\) 1.00113e15 0.115470
\(76\) −4.44160e16 −4.57745
\(77\) 2.63826e15 0.243302
\(78\) −1.29737e16 −1.07216
\(79\) −7.14674e15 −0.530003 −0.265001 0.964248i \(-0.585372\pi\)
−0.265001 + 0.964248i \(0.585372\pi\)
\(80\) 2.60369e16 1.73510
\(81\) 1.85302e15 0.111111
\(82\) 3.88566e15 0.209917
\(83\) 9.04707e15 0.440904 0.220452 0.975398i \(-0.429247\pi\)
0.220452 + 0.975398i \(0.429247\pi\)
\(84\) −6.35994e15 −0.279949
\(85\) 5.01956e15 0.199804
\(86\) −1.70305e16 −0.613749
\(87\) −3.21507e16 −1.05021
\(88\) 1.60283e17 4.75100
\(89\) −1.16786e16 −0.314467 −0.157233 0.987561i \(-0.550257\pi\)
−0.157233 + 0.987561i \(0.550257\pi\)
\(90\) −1.18122e16 −0.289248
\(91\) −7.52927e15 −0.167842
\(92\) 4.15481e16 0.844023
\(93\) 5.01527e16 0.929371
\(94\) 4.08955e16 0.691972
\(95\) −4.78750e16 −0.740386
\(96\) −1.67460e17 −2.36922
\(97\) −8.32538e16 −1.07856 −0.539280 0.842126i \(-0.681304\pi\)
−0.539280 + 0.842126i \(0.681304\pi\)
\(98\) 1.58392e17 1.88066
\(99\) −4.24585e16 −0.462450
\(100\) 5.52981e16 0.552981
\(101\) −1.36210e16 −0.125164 −0.0625818 0.998040i \(-0.519933\pi\)
−0.0625818 + 0.998040i \(0.519933\pi\)
\(102\) −5.92254e16 −0.500503
\(103\) 1.19030e17 0.925847 0.462923 0.886398i \(-0.346801\pi\)
0.462923 + 0.886398i \(0.346801\pi\)
\(104\) −4.57429e17 −3.27748
\(105\) −6.85524e15 −0.0452808
\(106\) 4.53622e17 2.76436
\(107\) 3.05375e17 1.71819 0.859096 0.511814i \(-0.171026\pi\)
0.859096 + 0.511814i \(0.171026\pi\)
\(108\) 1.02353e17 0.532106
\(109\) −2.94297e17 −1.41469 −0.707344 0.706869i \(-0.750107\pi\)
−0.707344 + 0.706869i \(0.750107\pi\)
\(110\) 2.70655e17 1.20387
\(111\) 2.09285e16 0.0861970
\(112\) −1.78288e17 −0.680407
\(113\) −7.25034e16 −0.256562 −0.128281 0.991738i \(-0.540946\pi\)
−0.128281 + 0.991738i \(0.540946\pi\)
\(114\) 5.64873e17 1.85464
\(115\) 4.47838e16 0.136518
\(116\) −1.77587e18 −5.02942
\(117\) 1.21171e17 0.319022
\(118\) 7.33620e14 0.00179669
\(119\) −3.43715e16 −0.0783519
\(120\) −4.16480e17 −0.884205
\(121\) 4.67408e17 0.924742
\(122\) 1.09648e18 2.02275
\(123\) −3.62913e16 −0.0624610
\(124\) 2.77022e18 4.45072
\(125\) 5.96046e16 0.0894427
\(126\) 8.08844e16 0.113427
\(127\) −1.00107e18 −1.31261 −0.656304 0.754497i \(-0.727881\pi\)
−0.656304 + 0.754497i \(0.727881\pi\)
\(128\) −3.11255e18 −3.81797
\(129\) 1.59062e17 0.182622
\(130\) −7.72418e17 −0.830489
\(131\) 6.36150e17 0.640846 0.320423 0.947275i \(-0.396175\pi\)
0.320423 + 0.947275i \(0.396175\pi\)
\(132\) −2.34522e18 −2.21465
\(133\) 3.27825e17 0.290337
\(134\) −9.21306e17 −0.765620
\(135\) 1.10324e17 0.0860663
\(136\) −2.08819e18 −1.52999
\(137\) 2.75383e17 0.189588 0.0947942 0.995497i \(-0.469781\pi\)
0.0947942 + 0.995497i \(0.469781\pi\)
\(138\) −5.28400e17 −0.341972
\(139\) 3.61048e17 0.219755 0.109878 0.993945i \(-0.464954\pi\)
0.109878 + 0.993945i \(0.464954\pi\)
\(140\) −3.78654e17 −0.216848
\(141\) −3.81956e17 −0.205897
\(142\) 4.98218e18 2.52911
\(143\) −2.77641e18 −1.32779
\(144\) 2.86926e18 1.29327
\(145\) −1.91417e18 −0.813490
\(146\) 8.26248e18 3.31216
\(147\) −1.47935e18 −0.559594
\(148\) 1.15600e18 0.412794
\(149\) −4.02824e18 −1.35841 −0.679207 0.733947i \(-0.737676\pi\)
−0.679207 + 0.733947i \(0.737676\pi\)
\(150\) −7.03270e17 −0.224051
\(151\) −3.02464e18 −0.910689 −0.455345 0.890315i \(-0.650484\pi\)
−0.455345 + 0.890315i \(0.650484\pi\)
\(152\) 1.99165e19 5.66947
\(153\) 5.53154e17 0.148925
\(154\) −1.85331e18 −0.472088
\(155\) 2.98596e18 0.719888
\(156\) 6.69300e18 1.52778
\(157\) −3.80909e18 −0.823520 −0.411760 0.911292i \(-0.635086\pi\)
−0.411760 + 0.911292i \(0.635086\pi\)
\(158\) 5.02042e18 1.02838
\(159\) −4.23674e18 −0.822539
\(160\) −9.97012e18 −1.83519
\(161\) −3.06657e17 −0.0535345
\(162\) −1.30170e18 −0.215593
\(163\) −7.01276e18 −1.10229 −0.551143 0.834411i \(-0.685808\pi\)
−0.551143 + 0.834411i \(0.685808\pi\)
\(164\) −2.00458e18 −0.299123
\(165\) −2.52787e18 −0.358212
\(166\) −6.35535e18 −0.855502
\(167\) −5.20936e18 −0.666337 −0.333169 0.942867i \(-0.608118\pi\)
−0.333169 + 0.942867i \(0.608118\pi\)
\(168\) 2.85185e18 0.346735
\(169\) −7.26857e17 −0.0840256
\(170\) −3.52613e18 −0.387688
\(171\) −5.27581e18 −0.551851
\(172\) 8.78590e18 0.874570
\(173\) 4.55714e18 0.431818 0.215909 0.976413i \(-0.430729\pi\)
0.215909 + 0.976413i \(0.430729\pi\)
\(174\) 2.25851e19 2.03776
\(175\) −4.08143e17 −0.0350743
\(176\) −6.57436e19 −5.38264
\(177\) −6.85187e15 −0.000534609 0
\(178\) 8.20393e18 0.610171
\(179\) 1.27111e19 0.901428 0.450714 0.892668i \(-0.351169\pi\)
0.450714 + 0.892668i \(0.351169\pi\)
\(180\) 6.09383e18 0.412168
\(181\) 1.67839e19 1.08299 0.541494 0.840704i \(-0.317859\pi\)
0.541494 + 0.840704i \(0.317859\pi\)
\(182\) 5.28914e18 0.325670
\(183\) −1.02409e19 −0.601872
\(184\) −1.86305e19 −1.04538
\(185\) 1.24603e18 0.0667679
\(186\) −3.52311e19 −1.80329
\(187\) −1.26745e19 −0.619835
\(188\) −2.10976e19 −0.986034
\(189\) −7.55445e17 −0.0337503
\(190\) 3.36311e19 1.43660
\(191\) 1.93607e19 0.790930 0.395465 0.918481i \(-0.370583\pi\)
0.395465 + 0.918481i \(0.370583\pi\)
\(192\) 6.03162e19 2.35708
\(193\) 5.41103e18 0.202322 0.101161 0.994870i \(-0.467744\pi\)
0.101161 + 0.994870i \(0.467744\pi\)
\(194\) 5.84839e19 2.09277
\(195\) 7.21423e18 0.247113
\(196\) −8.17129e19 −2.67987
\(197\) 1.69857e19 0.533483 0.266741 0.963768i \(-0.414053\pi\)
0.266741 + 0.963768i \(0.414053\pi\)
\(198\) 2.98261e19 0.897309
\(199\) −8.38703e18 −0.241745 −0.120872 0.992668i \(-0.538569\pi\)
−0.120872 + 0.992668i \(0.538569\pi\)
\(200\) −2.47961e19 −0.684902
\(201\) 8.60482e18 0.227811
\(202\) 9.56845e18 0.242860
\(203\) 1.31073e19 0.319004
\(204\) 3.05539e19 0.713197
\(205\) −2.16069e18 −0.0483821
\(206\) −8.36156e19 −1.79645
\(207\) 4.93515e18 0.101754
\(208\) 1.87624e20 3.71322
\(209\) 1.20885e20 2.29683
\(210\) 4.81565e18 0.0878599
\(211\) 2.60946e19 0.457245 0.228623 0.973515i \(-0.426578\pi\)
0.228623 + 0.973515i \(0.426578\pi\)
\(212\) −2.34020e20 −3.93910
\(213\) −4.65326e19 −0.752542
\(214\) −2.14519e20 −3.33387
\(215\) 9.47013e18 0.141459
\(216\) −4.58959e19 −0.659048
\(217\) −2.04464e19 −0.282299
\(218\) 2.06737e20 2.74497
\(219\) −7.71700e19 −0.985539
\(220\) −1.39629e20 −1.71546
\(221\) 3.61715e19 0.427594
\(222\) −1.47018e19 −0.167251
\(223\) 1.19753e19 0.131128 0.0655641 0.997848i \(-0.479115\pi\)
0.0655641 + 0.997848i \(0.479115\pi\)
\(224\) 6.82705e19 0.719656
\(225\) 6.56841e18 0.0666667
\(226\) 5.09320e19 0.497816
\(227\) 1.24200e20 1.16924 0.584619 0.811308i \(-0.301244\pi\)
0.584619 + 0.811308i \(0.301244\pi\)
\(228\) −2.91413e20 −2.64279
\(229\) 1.03099e20 0.900853 0.450426 0.892814i \(-0.351272\pi\)
0.450426 + 0.892814i \(0.351272\pi\)
\(230\) −3.14596e19 −0.264890
\(231\) 1.73096e19 0.140470
\(232\) 7.96313e20 6.22925
\(233\) 1.13357e20 0.854919 0.427459 0.904035i \(-0.359409\pi\)
0.427459 + 0.904035i \(0.359409\pi\)
\(234\) −8.51201e19 −0.619010
\(235\) −2.27407e19 −0.159487
\(236\) −3.78468e17 −0.00256022
\(237\) −4.68898e19 −0.305997
\(238\) 2.41452e19 0.152029
\(239\) −8.29065e19 −0.503740 −0.251870 0.967761i \(-0.581046\pi\)
−0.251870 + 0.967761i \(0.581046\pi\)
\(240\) 1.70828e20 1.00176
\(241\) −1.62688e20 −0.920896 −0.460448 0.887687i \(-0.652311\pi\)
−0.460448 + 0.887687i \(0.652311\pi\)
\(242\) −3.28343e20 −1.79431
\(243\) 1.21577e19 0.0641500
\(244\) −5.65664e20 −2.88234
\(245\) −8.80765e19 −0.433459
\(246\) 2.54938e19 0.121195
\(247\) −3.44992e20 −1.58447
\(248\) −1.24219e21 −5.51250
\(249\) 5.93578e19 0.254556
\(250\) −4.18709e19 −0.173549
\(251\) 4.10409e20 1.64433 0.822167 0.569247i \(-0.192765\pi\)
0.822167 + 0.569247i \(0.192765\pi\)
\(252\) −4.17276e19 −0.161629
\(253\) −1.13080e20 −0.423507
\(254\) 7.03232e20 2.54690
\(255\) 3.29333e19 0.115357
\(256\) 9.81533e20 3.32556
\(257\) −1.69582e20 −0.555839 −0.277920 0.960604i \(-0.589645\pi\)
−0.277920 + 0.960604i \(0.589645\pi\)
\(258\) −1.11737e20 −0.354348
\(259\) −8.53218e18 −0.0261826
\(260\) 3.98484e20 1.18341
\(261\) −2.10941e20 −0.606340
\(262\) −4.46881e20 −1.24346
\(263\) −3.50624e20 −0.944534 −0.472267 0.881456i \(-0.656564\pi\)
−0.472267 + 0.881456i \(0.656564\pi\)
\(264\) 1.05162e21 2.74299
\(265\) −2.52245e20 −0.637136
\(266\) −2.30289e20 −0.563352
\(267\) −7.66231e19 −0.181557
\(268\) 4.75294e20 1.09098
\(269\) 2.92843e20 0.651240 0.325620 0.945501i \(-0.394427\pi\)
0.325620 + 0.945501i \(0.394427\pi\)
\(270\) −7.75001e19 −0.166998
\(271\) 5.64043e20 1.17780 0.588902 0.808204i \(-0.299560\pi\)
0.588902 + 0.808204i \(0.299560\pi\)
\(272\) 8.56515e20 1.73340
\(273\) −4.93995e19 −0.0969037
\(274\) −1.93450e20 −0.367865
\(275\) −1.50503e20 −0.277470
\(276\) 2.72597e20 0.487297
\(277\) 7.92843e20 1.37439 0.687194 0.726474i \(-0.258842\pi\)
0.687194 + 0.726474i \(0.258842\pi\)
\(278\) −2.53628e20 −0.426399
\(279\) 3.29052e20 0.536573
\(280\) 1.69792e20 0.268580
\(281\) −4.07531e20 −0.625399 −0.312700 0.949852i \(-0.601233\pi\)
−0.312700 + 0.949852i \(0.601233\pi\)
\(282\) 2.68315e20 0.399510
\(283\) 9.05800e20 1.30872 0.654362 0.756182i \(-0.272937\pi\)
0.654362 + 0.756182i \(0.272937\pi\)
\(284\) −2.57026e21 −3.60389
\(285\) −3.14108e20 −0.427462
\(286\) 1.95037e21 2.57635
\(287\) 1.47953e19 0.0189727
\(288\) −1.09870e21 −1.36787
\(289\) −6.62116e20 −0.800391
\(290\) 1.34466e21 1.57845
\(291\) −5.46228e20 −0.622707
\(292\) −4.26254e21 −4.71970
\(293\) 1.22920e21 1.32205 0.661025 0.750364i \(-0.270122\pi\)
0.661025 + 0.750364i \(0.270122\pi\)
\(294\) 1.03921e21 1.08580
\(295\) −4.07943e17 −0.000414106 0
\(296\) −5.18359e20 −0.511271
\(297\) −2.78570e20 −0.266996
\(298\) 2.82974e21 2.63578
\(299\) 3.22716e20 0.292157
\(300\) 3.62811e20 0.319264
\(301\) −6.48468e19 −0.0554719
\(302\) 2.12474e21 1.76704
\(303\) −8.93675e19 −0.0722632
\(304\) −8.16918e21 −6.42322
\(305\) −6.09717e20 −0.466208
\(306\) −3.88578e20 −0.288965
\(307\) −1.33769e21 −0.967565 −0.483783 0.875188i \(-0.660738\pi\)
−0.483783 + 0.875188i \(0.660738\pi\)
\(308\) 9.56108e20 0.672707
\(309\) 7.80954e20 0.534538
\(310\) −2.09757e21 −1.39683
\(311\) −8.26556e19 −0.0535561 −0.0267781 0.999641i \(-0.508525\pi\)
−0.0267781 + 0.999641i \(0.508525\pi\)
\(312\) −3.00119e21 −1.89226
\(313\) −2.88314e20 −0.176905 −0.0884523 0.996080i \(-0.528192\pi\)
−0.0884523 + 0.996080i \(0.528192\pi\)
\(314\) 2.67580e21 1.59791
\(315\) −4.49772e19 −0.0261429
\(316\) −2.58999e21 −1.46541
\(317\) −3.17162e21 −1.74694 −0.873468 0.486882i \(-0.838134\pi\)
−0.873468 + 0.486882i \(0.838134\pi\)
\(318\) 2.97621e21 1.59600
\(319\) 4.83331e21 2.52362
\(320\) 3.59107e21 1.82578
\(321\) 2.00357e21 0.991999
\(322\) 2.15420e20 0.103875
\(323\) −1.57491e21 −0.739662
\(324\) 6.71538e20 0.307212
\(325\) 4.29517e20 0.191413
\(326\) 4.92630e21 2.13881
\(327\) −1.93088e21 −0.816771
\(328\) 8.98868e20 0.370483
\(329\) 1.55717e20 0.0625419
\(330\) 1.77577e21 0.695053
\(331\) −1.58984e21 −0.606478 −0.303239 0.952915i \(-0.598068\pi\)
−0.303239 + 0.952915i \(0.598068\pi\)
\(332\) 3.27867e21 1.21906
\(333\) 1.37312e20 0.0497659
\(334\) 3.65945e21 1.29292
\(335\) 5.12309e20 0.176462
\(336\) −1.16975e21 −0.392833
\(337\) −3.01551e20 −0.0987432 −0.0493716 0.998780i \(-0.515722\pi\)
−0.0493716 + 0.998780i \(0.515722\pi\)
\(338\) 5.10600e20 0.163038
\(339\) −4.75695e20 −0.148126
\(340\) 1.81910e21 0.552440
\(341\) −7.53960e21 −2.23324
\(342\) 3.70613e21 1.07078
\(343\) 1.22535e21 0.345350
\(344\) −3.93967e21 −1.08321
\(345\) 2.93826e20 0.0788185
\(346\) −3.20129e21 −0.837873
\(347\) −3.81902e21 −0.975331 −0.487665 0.873031i \(-0.662151\pi\)
−0.487665 + 0.873031i \(0.662151\pi\)
\(348\) −1.16515e22 −2.90374
\(349\) −3.13624e20 −0.0762769 −0.0381384 0.999272i \(-0.512143\pi\)
−0.0381384 + 0.999272i \(0.512143\pi\)
\(350\) 2.86711e20 0.0680560
\(351\) 7.95006e20 0.184187
\(352\) 2.51747e22 5.69314
\(353\) −4.65586e21 −1.02781 −0.513907 0.857846i \(-0.671802\pi\)
−0.513907 + 0.857846i \(0.671802\pi\)
\(354\) 4.81328e18 0.00103732
\(355\) −2.77043e21 −0.582916
\(356\) −4.23234e21 −0.869471
\(357\) −2.25511e20 −0.0452365
\(358\) −8.92923e21 −1.74907
\(359\) 3.57780e21 0.684405 0.342203 0.939626i \(-0.388827\pi\)
0.342203 + 0.939626i \(0.388827\pi\)
\(360\) −2.73252e21 −0.510496
\(361\) 9.54059e21 1.74086
\(362\) −1.17903e22 −2.10136
\(363\) 3.06667e21 0.533900
\(364\) −2.72862e21 −0.464068
\(365\) −4.59450e21 −0.763395
\(366\) 7.19400e21 1.16783
\(367\) −5.72348e21 −0.907818 −0.453909 0.891048i \(-0.649971\pi\)
−0.453909 + 0.891048i \(0.649971\pi\)
\(368\) 7.64170e21 1.18436
\(369\) −2.38107e20 −0.0360619
\(370\) −8.75305e20 −0.129552
\(371\) 1.72725e21 0.249848
\(372\) 1.81754e22 2.56962
\(373\) 7.79326e21 1.07695 0.538473 0.842643i \(-0.319001\pi\)
0.538473 + 0.842643i \(0.319001\pi\)
\(374\) 8.90352e21 1.20269
\(375\) 3.91066e20 0.0516398
\(376\) 9.46034e21 1.22127
\(377\) −1.37937e22 −1.74092
\(378\) 5.30683e20 0.0654869
\(379\) 4.41273e21 0.532444 0.266222 0.963912i \(-0.414225\pi\)
0.266222 + 0.963912i \(0.414225\pi\)
\(380\) −1.73500e22 −2.04710
\(381\) −6.56805e21 −0.757834
\(382\) −1.36005e22 −1.53467
\(383\) 7.42094e21 0.818973 0.409486 0.912316i \(-0.365708\pi\)
0.409486 + 0.912316i \(0.365708\pi\)
\(384\) −2.04215e22 −2.20430
\(385\) 1.03057e21 0.108808
\(386\) −3.80112e21 −0.392572
\(387\) 1.04360e21 0.105437
\(388\) −3.01713e22 −2.98212
\(389\) 4.44977e21 0.430295 0.215147 0.976582i \(-0.430977\pi\)
0.215147 + 0.976582i \(0.430977\pi\)
\(390\) −5.06783e21 −0.479483
\(391\) 1.47322e21 0.136384
\(392\) 3.66407e22 3.31919
\(393\) 4.17378e21 0.369993
\(394\) −1.19321e22 −1.03514
\(395\) −2.79170e21 −0.237024
\(396\) −1.53870e22 −1.27863
\(397\) −4.94222e20 −0.0401979 −0.0200989 0.999798i \(-0.506398\pi\)
−0.0200989 + 0.999798i \(0.506398\pi\)
\(398\) 5.89170e21 0.469066
\(399\) 2.15086e21 0.167626
\(400\) 1.01707e22 0.775960
\(401\) 2.49341e22 1.86237 0.931187 0.364542i \(-0.118775\pi\)
0.931187 + 0.364542i \(0.118775\pi\)
\(402\) −6.04469e21 −0.442031
\(403\) 2.15171e22 1.54061
\(404\) −4.93628e21 −0.346066
\(405\) 7.23836e20 0.0496904
\(406\) −9.20757e21 −0.618976
\(407\) −3.14624e21 −0.207128
\(408\) −1.37006e22 −0.883340
\(409\) 7.82942e21 0.494404 0.247202 0.968964i \(-0.420489\pi\)
0.247202 + 0.968964i \(0.420489\pi\)
\(410\) 1.51783e21 0.0938775
\(411\) 1.80679e21 0.109459
\(412\) 4.31366e22 2.55988
\(413\) 2.79339e18 0.000162389 0
\(414\) −3.46683e21 −0.197438
\(415\) 3.53401e21 0.197178
\(416\) −7.18457e22 −3.92742
\(417\) 2.36884e21 0.126876
\(418\) −8.49191e22 −4.45663
\(419\) −1.37314e20 −0.00706150 −0.00353075 0.999994i \(-0.501124\pi\)
−0.00353075 + 0.999994i \(0.501124\pi\)
\(420\) −2.48435e21 −0.125197
\(421\) −1.84158e21 −0.0909481 −0.0454740 0.998966i \(-0.514480\pi\)
−0.0454740 + 0.998966i \(0.514480\pi\)
\(422\) −1.83308e22 −0.887209
\(423\) −2.50601e21 −0.118875
\(424\) 1.04936e23 4.87883
\(425\) 1.96077e21 0.0893552
\(426\) 3.26881e22 1.46018
\(427\) 4.17505e21 0.182820
\(428\) 1.10669e23 4.75064
\(429\) −1.82161e22 −0.766597
\(430\) −6.65255e21 −0.274477
\(431\) −3.30639e21 −0.133751 −0.0668756 0.997761i \(-0.521303\pi\)
−0.0668756 + 0.997761i \(0.521303\pi\)
\(432\) 1.88252e22 0.746668
\(433\) 1.49902e22 0.582990 0.291495 0.956572i \(-0.405847\pi\)
0.291495 + 0.956572i \(0.405847\pi\)
\(434\) 1.43631e22 0.547755
\(435\) −1.25589e22 −0.469669
\(436\) −1.06654e23 −3.91148
\(437\) −1.40511e22 −0.505379
\(438\) 5.42102e22 1.91228
\(439\) 5.27352e22 1.82453 0.912267 0.409597i \(-0.134331\pi\)
0.912267 + 0.409597i \(0.134331\pi\)
\(440\) 6.26106e22 2.12471
\(441\) −9.70600e21 −0.323082
\(442\) −2.54096e22 −0.829676
\(443\) 2.21332e22 0.708944 0.354472 0.935067i \(-0.384661\pi\)
0.354472 + 0.935067i \(0.384661\pi\)
\(444\) 7.58452e21 0.238327
\(445\) −4.56194e21 −0.140634
\(446\) −8.41239e21 −0.254433
\(447\) −2.64293e22 −0.784280
\(448\) −2.45899e22 −0.715968
\(449\) 1.01171e22 0.289041 0.144521 0.989502i \(-0.453836\pi\)
0.144521 + 0.989502i \(0.453836\pi\)
\(450\) −4.61416e21 −0.129356
\(451\) 5.45578e21 0.150091
\(452\) −2.62754e22 −0.709369
\(453\) −1.98447e22 −0.525787
\(454\) −8.72479e22 −2.26872
\(455\) −2.94112e21 −0.0750613
\(456\) 1.30672e23 3.27327
\(457\) −7.09706e22 −1.74498 −0.872490 0.488632i \(-0.837496\pi\)
−0.872490 + 0.488632i \(0.837496\pi\)
\(458\) −7.24248e22 −1.74796
\(459\) 3.62924e21 0.0859821
\(460\) 1.62297e22 0.377459
\(461\) 1.09889e22 0.250899 0.125449 0.992100i \(-0.459963\pi\)
0.125449 + 0.992100i \(0.459963\pi\)
\(462\) −1.21596e22 −0.272560
\(463\) 1.78028e22 0.391787 0.195893 0.980625i \(-0.437239\pi\)
0.195893 + 0.980625i \(0.437239\pi\)
\(464\) −3.26625e23 −7.05743
\(465\) 1.95909e22 0.415628
\(466\) −7.96310e22 −1.65883
\(467\) 7.82704e22 1.60105 0.800524 0.599301i \(-0.204555\pi\)
0.800524 + 0.599301i \(0.204555\pi\)
\(468\) 4.39127e22 0.882065
\(469\) −3.50804e21 −0.0691983
\(470\) 1.59748e22 0.309459
\(471\) −2.49914e22 −0.475459
\(472\) 1.69708e20 0.00317100
\(473\) −2.39122e22 −0.438834
\(474\) 3.29390e22 0.593737
\(475\) −1.87012e22 −0.331111
\(476\) −1.24563e22 −0.216636
\(477\) −2.77973e22 −0.474893
\(478\) 5.82399e22 0.977425
\(479\) −1.17366e23 −1.93504 −0.967522 0.252785i \(-0.918653\pi\)
−0.967522 + 0.252785i \(0.918653\pi\)
\(480\) −6.54140e22 −1.05955
\(481\) 8.97899e21 0.142888
\(482\) 1.14285e23 1.78685
\(483\) −2.01198e21 −0.0309081
\(484\) 1.69390e23 2.55683
\(485\) −3.25210e22 −0.482347
\(486\) −8.54048e21 −0.124473
\(487\) 4.40441e22 0.630800 0.315400 0.948959i \(-0.397861\pi\)
0.315400 + 0.948959i \(0.397861\pi\)
\(488\) 2.53648e23 3.56996
\(489\) −4.60107e22 −0.636405
\(490\) 6.18717e22 0.841057
\(491\) −9.19512e21 −0.122847 −0.0614235 0.998112i \(-0.519564\pi\)
−0.0614235 + 0.998112i \(0.519564\pi\)
\(492\) −1.31520e22 −0.172699
\(493\) −6.29689e22 −0.812695
\(494\) 2.42349e23 3.07441
\(495\) −1.65853e22 −0.206814
\(496\) 5.09511e23 6.24539
\(497\) 1.89705e22 0.228586
\(498\) −4.16975e22 −0.493924
\(499\) −9.12259e22 −1.06234 −0.531170 0.847265i \(-0.678247\pi\)
−0.531170 + 0.847265i \(0.678247\pi\)
\(500\) 2.16008e22 0.247301
\(501\) −3.41786e22 −0.384710
\(502\) −2.88303e23 −3.19056
\(503\) −4.20493e22 −0.457542 −0.228771 0.973480i \(-0.573471\pi\)
−0.228771 + 0.973480i \(0.573471\pi\)
\(504\) 1.87110e22 0.200187
\(505\) −5.32071e21 −0.0559749
\(506\) 7.94359e22 0.821746
\(507\) −4.76891e21 −0.0485122
\(508\) −3.62791e23 −3.62924
\(509\) 3.12071e22 0.307010 0.153505 0.988148i \(-0.450944\pi\)
0.153505 + 0.988148i \(0.450944\pi\)
\(510\) −2.31349e22 −0.223832
\(511\) 3.14609e22 0.299360
\(512\) −2.81536e23 −2.63475
\(513\) −3.46146e22 −0.318612
\(514\) 1.19128e23 1.07852
\(515\) 4.64960e22 0.414051
\(516\) 5.76443e22 0.504933
\(517\) 5.74206e22 0.494764
\(518\) 5.99366e21 0.0508030
\(519\) 2.98994e22 0.249310
\(520\) −1.78683e23 −1.46573
\(521\) −1.71736e23 −1.38593 −0.692965 0.720971i \(-0.743696\pi\)
−0.692965 + 0.720971i \(0.743696\pi\)
\(522\) 1.48181e23 1.17650
\(523\) −3.54202e22 −0.276686 −0.138343 0.990384i \(-0.544178\pi\)
−0.138343 + 0.990384i \(0.544178\pi\)
\(524\) 2.30542e23 1.77188
\(525\) −2.67783e21 −0.0202502
\(526\) 2.46305e23 1.83271
\(527\) 9.82268e22 0.719184
\(528\) −4.31344e23 −3.10767
\(529\) −1.27906e23 −0.906815
\(530\) 1.77196e23 1.23626
\(531\) −4.49551e19 −0.000308657 0
\(532\) 1.18804e23 0.802755
\(533\) −1.55701e22 −0.103541
\(534\) 5.38260e22 0.352283
\(535\) 1.19287e23 0.768399
\(536\) −2.13126e23 −1.35125
\(537\) 8.33973e22 0.520440
\(538\) −2.05716e23 −1.26363
\(539\) 2.22395e23 1.34468
\(540\) 3.99816e22 0.237965
\(541\) −1.03474e23 −0.606257 −0.303128 0.952950i \(-0.598031\pi\)
−0.303128 + 0.952950i \(0.598031\pi\)
\(542\) −3.96227e23 −2.28534
\(543\) 1.10119e23 0.625264
\(544\) −3.27979e23 −1.83339
\(545\) −1.14960e23 −0.632668
\(546\) 3.47020e22 0.188026
\(547\) 2.96616e23 1.58235 0.791176 0.611589i \(-0.209469\pi\)
0.791176 + 0.611589i \(0.209469\pi\)
\(548\) 9.97992e22 0.524194
\(549\) −6.71906e22 −0.347491
\(550\) 1.05725e23 0.538385
\(551\) 6.00578e23 3.01149
\(552\) −1.22235e23 −0.603548
\(553\) 1.91162e22 0.0929474
\(554\) −5.56954e23 −2.66678
\(555\) 8.17518e21 0.0385485
\(556\) 1.30845e23 0.607603
\(557\) −5.46640e22 −0.249996 −0.124998 0.992157i \(-0.539892\pi\)
−0.124998 + 0.992157i \(0.539892\pi\)
\(558\) −2.31151e23 −1.04113
\(559\) 6.82427e22 0.302730
\(560\) −6.96437e22 −0.304287
\(561\) −8.31572e22 −0.357862
\(562\) 2.86281e23 1.21349
\(563\) −1.18516e23 −0.494829 −0.247414 0.968910i \(-0.579581\pi\)
−0.247414 + 0.968910i \(0.579581\pi\)
\(564\) −1.38422e23 −0.569287
\(565\) −2.83217e22 −0.114738
\(566\) −6.36304e23 −2.53936
\(567\) −4.95647e21 −0.0194857
\(568\) 1.15253e24 4.46365
\(569\) −4.39785e23 −1.67798 −0.838988 0.544149i \(-0.816853\pi\)
−0.838988 + 0.544149i \(0.816853\pi\)
\(570\) 2.20654e23 0.829421
\(571\) −4.89721e22 −0.181360 −0.0906800 0.995880i \(-0.528904\pi\)
−0.0906800 + 0.995880i \(0.528904\pi\)
\(572\) −1.00618e24 −3.67120
\(573\) 1.27026e23 0.456644
\(574\) −1.03934e22 −0.0368134
\(575\) 1.74937e22 0.0610526
\(576\) 3.95735e23 1.36086
\(577\) 5.15333e23 1.74620 0.873099 0.487543i \(-0.162107\pi\)
0.873099 + 0.487543i \(0.162107\pi\)
\(578\) 4.65121e23 1.55303
\(579\) 3.55018e22 0.116811
\(580\) −6.93698e23 −2.24922
\(581\) −2.41992e22 −0.0773220
\(582\) 3.83713e23 1.20826
\(583\) 6.36922e23 1.97653
\(584\) 1.91136e24 5.84565
\(585\) 4.73326e22 0.142671
\(586\) −8.63485e23 −2.56522
\(587\) −3.59992e22 −0.105407 −0.0527035 0.998610i \(-0.516784\pi\)
−0.0527035 + 0.998610i \(0.516784\pi\)
\(588\) −5.36118e23 −1.54722
\(589\) −9.36857e23 −2.66498
\(590\) 2.86570e20 0.000803506 0
\(591\) 1.11443e23 0.308006
\(592\) 2.12616e23 0.579245
\(593\) −3.67652e23 −0.987351 −0.493676 0.869646i \(-0.664347\pi\)
−0.493676 + 0.869646i \(0.664347\pi\)
\(594\) 1.95689e23 0.518062
\(595\) −1.34264e22 −0.0350400
\(596\) −1.45984e24 −3.75588
\(597\) −5.50273e22 −0.139571
\(598\) −2.26701e23 −0.566882
\(599\) −6.97132e23 −1.71865 −0.859324 0.511432i \(-0.829115\pi\)
−0.859324 + 0.511432i \(0.829115\pi\)
\(600\) −1.62687e23 −0.395429
\(601\) 3.44330e23 0.825166 0.412583 0.910920i \(-0.364627\pi\)
0.412583 + 0.910920i \(0.364627\pi\)
\(602\) 4.55534e22 0.107634
\(603\) 5.64562e22 0.131527
\(604\) −1.09614e24 −2.51797
\(605\) 1.82581e23 0.413557
\(606\) 6.27786e22 0.140215
\(607\) −3.08234e23 −0.678855 −0.339427 0.940632i \(-0.610233\pi\)
−0.339427 + 0.940632i \(0.610233\pi\)
\(608\) 3.12816e24 6.79374
\(609\) 8.59970e22 0.184177
\(610\) 4.28312e23 0.904601
\(611\) −1.63872e23 −0.341313
\(612\) 2.00464e23 0.411765
\(613\) −4.35176e23 −0.881558 −0.440779 0.897616i \(-0.645298\pi\)
−0.440779 + 0.897616i \(0.645298\pi\)
\(614\) 9.39699e23 1.87740
\(615\) −1.41763e22 −0.0279334
\(616\) −4.28727e23 −0.833190
\(617\) 6.04231e23 1.15819 0.579094 0.815261i \(-0.303406\pi\)
0.579094 + 0.815261i \(0.303406\pi\)
\(618\) −5.48602e23 −1.03718
\(619\) −3.33503e23 −0.621913 −0.310956 0.950424i \(-0.600649\pi\)
−0.310956 + 0.950424i \(0.600649\pi\)
\(620\) 1.08212e24 1.99042
\(621\) 3.23795e22 0.0587479
\(622\) 5.80637e22 0.103917
\(623\) 3.12379e22 0.0551485
\(624\) 1.23100e24 2.14383
\(625\) 2.32831e22 0.0400000
\(626\) 2.02534e23 0.343255
\(627\) 7.93128e23 1.32608
\(628\) −1.38042e24 −2.27695
\(629\) 4.09896e22 0.0667026
\(630\) 3.15955e22 0.0507259
\(631\) 7.67831e23 1.21623 0.608116 0.793848i \(-0.291926\pi\)
0.608116 + 0.793848i \(0.291926\pi\)
\(632\) 1.16137e24 1.81500
\(633\) 1.71206e23 0.263991
\(634\) 2.22799e24 3.38964
\(635\) −3.91045e23 −0.587016
\(636\) −1.53540e24 −2.27424
\(637\) −6.34688e23 −0.927631
\(638\) −3.39529e24 −4.89667
\(639\) −3.05300e23 −0.434480
\(640\) −1.21584e24 −1.70745
\(641\) −1.34739e24 −1.86724 −0.933620 0.358264i \(-0.883369\pi\)
−0.933620 + 0.358264i \(0.883369\pi\)
\(642\) −1.40746e24 −1.92481
\(643\) 1.07291e24 1.44800 0.723999 0.689801i \(-0.242302\pi\)
0.723999 + 0.689801i \(0.242302\pi\)
\(644\) −1.11133e23 −0.148018
\(645\) 6.21335e22 0.0816711
\(646\) 1.10634e24 1.43519
\(647\) 2.56767e23 0.328740 0.164370 0.986399i \(-0.447441\pi\)
0.164370 + 0.986399i \(0.447441\pi\)
\(648\) −3.01123e23 −0.380501
\(649\) 1.03006e21 0.00128465
\(650\) −3.01726e23 −0.371406
\(651\) −1.34149e23 −0.162985
\(652\) −2.54144e24 −3.04772
\(653\) 9.55573e23 1.13110 0.565551 0.824713i \(-0.308664\pi\)
0.565551 + 0.824713i \(0.308664\pi\)
\(654\) 1.35640e24 1.58481
\(655\) 2.48496e23 0.286595
\(656\) −3.68690e23 −0.419739
\(657\) −5.06312e23 −0.569001
\(658\) −1.09388e23 −0.121352
\(659\) −4.48249e23 −0.490901 −0.245450 0.969409i \(-0.578936\pi\)
−0.245450 + 0.969409i \(0.578936\pi\)
\(660\) −9.16103e23 −0.990424
\(661\) 3.31071e23 0.353353 0.176677 0.984269i \(-0.443465\pi\)
0.176677 + 0.984269i \(0.443465\pi\)
\(662\) 1.11682e24 1.17677
\(663\) 2.37321e23 0.246872
\(664\) −1.47018e24 −1.50988
\(665\) 1.28057e23 0.129843
\(666\) −9.64583e22 −0.0965625
\(667\) −5.61799e23 −0.555279
\(668\) −1.88788e24 −1.84236
\(669\) 7.85701e22 0.0757069
\(670\) −3.59885e23 −0.342396
\(671\) 1.53955e24 1.44628
\(672\) 4.47923e23 0.415494
\(673\) −9.61434e23 −0.880626 −0.440313 0.897844i \(-0.645132\pi\)
−0.440313 + 0.897844i \(0.645132\pi\)
\(674\) 2.11833e23 0.191595
\(675\) 4.30953e22 0.0384900
\(676\) −2.63414e23 −0.232323
\(677\) 1.08092e24 0.941432 0.470716 0.882285i \(-0.343996\pi\)
0.470716 + 0.882285i \(0.343996\pi\)
\(678\) 3.34165e23 0.287414
\(679\) 2.22688e23 0.189149
\(680\) −8.15698e23 −0.684232
\(681\) 8.14879e23 0.675060
\(682\) 5.29640e24 4.33325
\(683\) 4.56949e23 0.369226 0.184613 0.982811i \(-0.440897\pi\)
0.184613 + 0.982811i \(0.440897\pi\)
\(684\) −1.91196e24 −1.52582
\(685\) 1.07571e23 0.0847865
\(686\) −8.60778e23 −0.670095
\(687\) 6.76434e23 0.520107
\(688\) 1.61594e24 1.22722
\(689\) −1.81770e24 −1.36351
\(690\) −2.06406e23 −0.152934
\(691\) −1.38418e24 −1.01305 −0.506523 0.862226i \(-0.669070\pi\)
−0.506523 + 0.862226i \(0.669070\pi\)
\(692\) 1.65152e24 1.19394
\(693\) 1.13568e23 0.0811007
\(694\) 2.68278e24 1.89247
\(695\) 1.41035e23 0.0982776
\(696\) 5.22461e24 3.59646
\(697\) −7.10785e22 −0.0483348
\(698\) 2.20314e23 0.148003
\(699\) 7.43739e23 0.493588
\(700\) −1.47912e23 −0.0969772
\(701\) 9.18978e22 0.0595254 0.0297627 0.999557i \(-0.490525\pi\)
0.0297627 + 0.999557i \(0.490525\pi\)
\(702\) −5.58473e23 −0.357385
\(703\) −3.90946e23 −0.247170
\(704\) −9.06751e24 −5.66396
\(705\) −1.49202e23 −0.0920801
\(706\) 3.27063e24 1.99431
\(707\) 3.64336e22 0.0219501
\(708\) −2.48313e21 −0.00147814
\(709\) −1.54021e24 −0.905913 −0.452956 0.891533i \(-0.649631\pi\)
−0.452956 + 0.891533i \(0.649631\pi\)
\(710\) 1.94616e24 1.13105
\(711\) −3.07644e23 −0.176668
\(712\) 1.89781e24 1.07690
\(713\) 8.76365e23 0.491387
\(714\) 1.58417e23 0.0877740
\(715\) −1.08454e24 −0.593804
\(716\) 4.60651e24 2.49236
\(717\) −5.43949e23 −0.290834
\(718\) −2.51332e24 −1.32798
\(719\) 3.23522e24 1.68931 0.844654 0.535313i \(-0.179806\pi\)
0.844654 + 0.535313i \(0.179806\pi\)
\(720\) 1.12080e24 0.578366
\(721\) −3.18382e23 −0.162367
\(722\) −6.70204e24 −3.37786
\(723\) −1.06740e24 −0.531679
\(724\) 6.08250e24 2.99436
\(725\) −7.47722e23 −0.363804
\(726\) −2.15426e24 −1.03595
\(727\) −2.33512e24 −1.10986 −0.554929 0.831898i \(-0.687255\pi\)
−0.554929 + 0.831898i \(0.687255\pi\)
\(728\) 1.22354e24 0.574777
\(729\) 7.97664e22 0.0370370
\(730\) 3.22753e24 1.48124
\(731\) 3.11531e23 0.141320
\(732\) −3.71132e24 −1.66412
\(733\) 1.81465e24 0.804282 0.402141 0.915578i \(-0.368266\pi\)
0.402141 + 0.915578i \(0.368266\pi\)
\(734\) 4.02061e24 1.76147
\(735\) −5.77870e23 −0.250258
\(736\) −2.92618e24 −1.25268
\(737\) −1.29359e24 −0.547422
\(738\) 1.67265e23 0.0699722
\(739\) −1.21901e24 −0.504114 −0.252057 0.967712i \(-0.581107\pi\)
−0.252057 + 0.967712i \(0.581107\pi\)
\(740\) 4.51562e23 0.184607
\(741\) −2.26349e24 −0.914797
\(742\) −1.21335e24 −0.484790
\(743\) 4.70556e23 0.185869 0.0929344 0.995672i \(-0.470375\pi\)
0.0929344 + 0.995672i \(0.470375\pi\)
\(744\) −8.15001e24 −3.18264
\(745\) −1.57353e24 −0.607501
\(746\) −5.47458e24 −2.08964
\(747\) 3.89447e23 0.146968
\(748\) −4.59325e24 −1.71379
\(749\) −8.16820e23 −0.301322
\(750\) −2.74715e23 −0.100199
\(751\) −3.20027e24 −1.15411 −0.577056 0.816705i \(-0.695798\pi\)
−0.577056 + 0.816705i \(0.695798\pi\)
\(752\) −3.88036e24 −1.38363
\(753\) 2.69269e24 0.949356
\(754\) 9.68975e24 3.37797
\(755\) −1.18150e24 −0.407273
\(756\) −2.73775e23 −0.0933164
\(757\) −5.33142e24 −1.79692 −0.898458 0.439060i \(-0.855312\pi\)
−0.898458 + 0.439060i \(0.855312\pi\)
\(758\) −3.09984e24 −1.03312
\(759\) −7.41916e23 −0.244512
\(760\) 7.77988e24 2.53546
\(761\) 3.19775e24 1.03057 0.515283 0.857020i \(-0.327687\pi\)
0.515283 + 0.857020i \(0.327687\pi\)
\(762\) 4.61390e24 1.47045
\(763\) 7.87188e23 0.248096
\(764\) 7.01636e24 2.18685
\(765\) 2.16076e23 0.0666014
\(766\) −5.21304e24 −1.58908
\(767\) −2.93967e21 −0.000886214 0
\(768\) 6.43984e24 1.92001
\(769\) −3.24975e24 −0.958245 −0.479122 0.877748i \(-0.659045\pi\)
−0.479122 + 0.877748i \(0.659045\pi\)
\(770\) −7.23950e23 −0.211124
\(771\) −1.11263e24 −0.320914
\(772\) 1.96097e24 0.559401
\(773\) 4.35391e24 1.22844 0.614220 0.789135i \(-0.289471\pi\)
0.614220 + 0.789135i \(0.289471\pi\)
\(774\) −7.33108e23 −0.204583
\(775\) 1.16639e24 0.321944
\(776\) 1.35291e25 3.69354
\(777\) −5.59797e22 −0.0151165
\(778\) −3.12586e24 −0.834917
\(779\) 6.77925e23 0.179107
\(780\) 2.61445e24 0.683245
\(781\) 6.99538e24 1.80833
\(782\) −1.03490e24 −0.264631
\(783\) −1.38398e24 −0.350070
\(784\) −1.50290e25 −3.76048
\(785\) −1.48792e24 −0.368289
\(786\) −2.93199e24 −0.717910
\(787\) 1.67282e24 0.405194 0.202597 0.979262i \(-0.435062\pi\)
0.202597 + 0.979262i \(0.435062\pi\)
\(788\) 6.15565e24 1.47503
\(789\) −2.30044e24 −0.545327
\(790\) 1.96110e24 0.459907
\(791\) 1.93933e23 0.0449936
\(792\) 6.89966e24 1.58367
\(793\) −4.39368e24 −0.997715
\(794\) 3.47180e23 0.0779974
\(795\) −1.65498e24 −0.367850
\(796\) −3.03948e24 −0.668402
\(797\) 9.07285e23 0.197400 0.0987002 0.995117i \(-0.468532\pi\)
0.0987002 + 0.995117i \(0.468532\pi\)
\(798\) −1.51093e24 −0.325251
\(799\) −7.48082e23 −0.159331
\(800\) −3.89458e24 −0.820721
\(801\) −5.02724e23 −0.104822
\(802\) −1.75156e25 −3.61363
\(803\) 1.16012e25 2.36821
\(804\) 3.11840e24 0.629877
\(805\) −1.19788e23 −0.0239413
\(806\) −1.51153e25 −2.98929
\(807\) 1.92135e24 0.375994
\(808\) 2.21347e24 0.428624
\(809\) −9.92795e23 −0.190238 −0.0951189 0.995466i \(-0.530323\pi\)
−0.0951189 + 0.995466i \(0.530323\pi\)
\(810\) −5.08478e23 −0.0964161
\(811\) 5.63576e24 1.05749 0.528744 0.848781i \(-0.322663\pi\)
0.528744 + 0.848781i \(0.322663\pi\)
\(812\) 4.75011e24 0.882017
\(813\) 3.70068e24 0.680006
\(814\) 2.21016e24 0.401898
\(815\) −2.73936e24 −0.492957
\(816\) 5.61959e24 1.00078
\(817\) −2.97129e24 −0.523670
\(818\) −5.49999e24 −0.959310
\(819\) −3.24110e23 −0.0559474
\(820\) −7.83038e23 −0.133772
\(821\) −9.18781e24 −1.55344 −0.776721 0.629845i \(-0.783118\pi\)
−0.776721 + 0.629845i \(0.783118\pi\)
\(822\) −1.26923e24 −0.212387
\(823\) 6.83078e24 1.13128 0.565642 0.824651i \(-0.308628\pi\)
0.565642 + 0.824651i \(0.308628\pi\)
\(824\) −1.93428e25 −3.17057
\(825\) −9.87448e23 −0.160197
\(826\) −1.96229e21 −0.000315089 0
\(827\) −9.93808e24 −1.57945 −0.789724 0.613462i \(-0.789776\pi\)
−0.789724 + 0.613462i \(0.789776\pi\)
\(828\) 1.78851e24 0.281341
\(829\) 1.00372e25 1.56278 0.781391 0.624042i \(-0.214511\pi\)
0.781391 + 0.624042i \(0.214511\pi\)
\(830\) −2.48256e24 −0.382592
\(831\) 5.20185e24 0.793503
\(832\) 2.58776e25 3.90729
\(833\) −2.89738e24 −0.433035
\(834\) −1.66405e24 −0.246182
\(835\) −2.03491e24 −0.297995
\(836\) 4.38090e25 6.35053
\(837\) 2.15891e24 0.309790
\(838\) 9.64602e22 0.0137017
\(839\) 1.16269e25 1.63489 0.817446 0.576005i \(-0.195389\pi\)
0.817446 + 0.576005i \(0.195389\pi\)
\(840\) 1.11400e24 0.155065
\(841\) 1.67555e25 2.30883
\(842\) 1.29367e24 0.176470
\(843\) −2.67381e24 −0.361074
\(844\) 9.45671e24 1.26424
\(845\) −2.83928e23 −0.0375774
\(846\) 1.76042e24 0.230657
\(847\) −1.25023e24 −0.162174
\(848\) −4.30419e25 −5.52747
\(849\) 5.94296e24 0.755592
\(850\) −1.37739e24 −0.173379
\(851\) 3.65703e23 0.0455750
\(852\) −1.68635e25 −2.08071
\(853\) 1.09678e23 0.0133984 0.00669922 0.999978i \(-0.497868\pi\)
0.00669922 + 0.999978i \(0.497868\pi\)
\(854\) −2.93287e24 −0.354733
\(855\) −2.06086e24 −0.246795
\(856\) −4.96246e25 −5.88397
\(857\) 1.52818e25 1.79406 0.897030 0.441970i \(-0.145720\pi\)
0.897030 + 0.441970i \(0.145720\pi\)
\(858\) 1.27964e25 1.48746
\(859\) −1.62639e25 −1.87190 −0.935951 0.352130i \(-0.885457\pi\)
−0.935951 + 0.352130i \(0.885457\pi\)
\(860\) 3.43199e24 0.391119
\(861\) 9.70723e22 0.0109539
\(862\) 2.32266e24 0.259522
\(863\) −8.64833e24 −0.956842 −0.478421 0.878130i \(-0.658791\pi\)
−0.478421 + 0.878130i \(0.658791\pi\)
\(864\) −7.20859e24 −0.789739
\(865\) 1.78013e24 0.193115
\(866\) −1.05303e25 −1.13120
\(867\) −4.34414e24 −0.462106
\(868\) −7.40981e24 −0.780530
\(869\) 7.04908e24 0.735299
\(870\) 8.82231e24 0.911316
\(871\) 3.69175e24 0.377640
\(872\) 4.78244e25 4.84462
\(873\) −3.58381e24 −0.359520
\(874\) 9.87056e24 0.980606
\(875\) −1.59431e23 −0.0156857
\(876\) −2.79665e25 −2.72492
\(877\) 1.02051e25 0.984740 0.492370 0.870386i \(-0.336131\pi\)
0.492370 + 0.870386i \(0.336131\pi\)
\(878\) −3.70452e25 −3.54021
\(879\) 8.06479e24 0.763286
\(880\) −2.56811e25 −2.40719
\(881\) −9.18419e23 −0.0852601 −0.0426301 0.999091i \(-0.513574\pi\)
−0.0426301 + 0.999091i \(0.513574\pi\)
\(882\) 6.81824e24 0.626887
\(883\) 8.44090e24 0.768640 0.384320 0.923200i \(-0.374436\pi\)
0.384320 + 0.923200i \(0.374436\pi\)
\(884\) 1.31086e25 1.18226
\(885\) −2.67651e21 −0.000239084 0
\(886\) −1.55480e25 −1.37559
\(887\) −1.20407e25 −1.05512 −0.527558 0.849519i \(-0.676892\pi\)
−0.527558 + 0.849519i \(0.676892\pi\)
\(888\) −3.40096e24 −0.295183
\(889\) 2.67768e24 0.230194
\(890\) 3.20466e24 0.272877
\(891\) −1.82770e24 −0.154150
\(892\) 4.33988e24 0.362557
\(893\) 7.13498e24 0.590412
\(894\) 1.85660e25 1.52177
\(895\) 4.96526e24 0.403131
\(896\) 8.32548e24 0.669563
\(897\) 2.11734e24 0.168677
\(898\) −7.10700e24 −0.560838
\(899\) −3.74580e25 −2.92811
\(900\) 2.38040e24 0.184327
\(901\) −8.29789e24 −0.636512
\(902\) −3.83256e24 −0.291228
\(903\) −4.25460e23 −0.0320267
\(904\) 1.17821e25 0.878599
\(905\) 6.55620e24 0.484327
\(906\) 1.39404e25 1.02020
\(907\) −2.51467e25 −1.82314 −0.911568 0.411150i \(-0.865127\pi\)
−0.911568 + 0.411150i \(0.865127\pi\)
\(908\) 4.50104e25 3.23284
\(909\) −5.86340e23 −0.0417212
\(910\) 2.06607e24 0.145644
\(911\) −8.57548e24 −0.598897 −0.299448 0.954112i \(-0.596803\pi\)
−0.299448 + 0.954112i \(0.596803\pi\)
\(912\) −5.35980e25 −3.70845
\(913\) −8.92343e24 −0.611688
\(914\) 4.98552e25 3.38585
\(915\) −4.00036e24 −0.269165
\(916\) 3.73633e25 2.49077
\(917\) −1.70158e24 −0.112386
\(918\) −2.54946e24 −0.166834
\(919\) 1.26569e25 0.820626 0.410313 0.911945i \(-0.365419\pi\)
0.410313 + 0.911945i \(0.365419\pi\)
\(920\) −7.27753e24 −0.467507
\(921\) −8.77661e24 −0.558624
\(922\) −7.71948e24 −0.486828
\(923\) −1.99640e25 −1.24748
\(924\) 6.27303e24 0.388388
\(925\) 4.86729e23 0.0298595
\(926\) −1.25060e25 −0.760198
\(927\) 5.12384e24 0.308616
\(928\) 1.25072e26 7.46454
\(929\) −1.61298e25 −0.953881 −0.476941 0.878936i \(-0.658254\pi\)
−0.476941 + 0.878936i \(0.658254\pi\)
\(930\) −1.37622e25 −0.806457
\(931\) 2.76343e25 1.60464
\(932\) 4.10810e25 2.36377
\(933\) −5.42304e23 −0.0309206
\(934\) −5.49832e25 −3.10657
\(935\) −4.95097e24 −0.277199
\(936\) −1.96908e25 −1.09249
\(937\) 2.27726e24 0.125206 0.0626032 0.998038i \(-0.480060\pi\)
0.0626032 + 0.998038i \(0.480060\pi\)
\(938\) 2.46432e24 0.134268
\(939\) −1.89163e24 −0.102136
\(940\) −8.24126e24 −0.440968
\(941\) −1.42919e25 −0.757839 −0.378919 0.925430i \(-0.623704\pi\)
−0.378919 + 0.925430i \(0.623704\pi\)
\(942\) 1.75559e25 0.922551
\(943\) −6.34151e23 −0.0330251
\(944\) −6.96095e22 −0.00359258
\(945\) −2.95096e23 −0.0150936
\(946\) 1.67978e25 0.851486
\(947\) −1.82863e24 −0.0918653 −0.0459326 0.998945i \(-0.514626\pi\)
−0.0459326 + 0.998945i \(0.514626\pi\)
\(948\) −1.69929e25 −0.846053
\(949\) −3.31084e25 −1.63371
\(950\) 1.31372e25 0.642467
\(951\) −2.08090e25 −1.00859
\(952\) 5.58550e24 0.268317
\(953\) 1.66599e25 0.793201 0.396601 0.917991i \(-0.370190\pi\)
0.396601 + 0.917991i \(0.370190\pi\)
\(954\) 1.95269e25 0.921452
\(955\) 7.56278e24 0.353715
\(956\) −3.00455e25 −1.39279
\(957\) 3.17113e25 1.45701
\(958\) 8.24470e25 3.75464
\(959\) −7.36596e23 −0.0332484
\(960\) 2.35610e25 1.05412
\(961\) 3.58816e25 1.59119
\(962\) −6.30753e24 −0.277250
\(963\) 1.31454e25 0.572731
\(964\) −5.89584e25 −2.54619
\(965\) 2.11368e24 0.0904810
\(966\) 1.41337e24 0.0599722
\(967\) −3.59794e25 −1.51331 −0.756657 0.653812i \(-0.773169\pi\)
−0.756657 + 0.653812i \(0.773169\pi\)
\(968\) −7.59556e25 −3.16679
\(969\) −1.03330e25 −0.427044
\(970\) 2.28453e25 0.935915
\(971\) −2.68946e25 −1.09220 −0.546099 0.837721i \(-0.683888\pi\)
−0.546099 + 0.837721i \(0.683888\pi\)
\(972\) 4.40596e24 0.177369
\(973\) −9.65735e23 −0.0385389
\(974\) −3.09400e25 −1.22396
\(975\) 2.81806e24 0.110512
\(976\) −1.04039e26 −4.04459
\(977\) 1.26185e25 0.486301 0.243151 0.969989i \(-0.421819\pi\)
0.243151 + 0.969989i \(0.421819\pi\)
\(978\) 3.23215e25 1.23484
\(979\) 1.15190e25 0.436276
\(980\) −3.19191e25 −1.19847
\(981\) −1.26685e25 −0.471563
\(982\) 6.45936e24 0.238365
\(983\) −2.72481e25 −0.996852 −0.498426 0.866932i \(-0.666088\pi\)
−0.498426 + 0.866932i \(0.666088\pi\)
\(984\) 5.89747e24 0.213898
\(985\) 6.63504e24 0.238581
\(986\) 4.42342e25 1.57690
\(987\) 1.02166e24 0.0361086
\(988\) −1.25026e26 −4.38092
\(989\) 2.77944e24 0.0965580
\(990\) 1.16508e25 0.401289
\(991\) −2.34443e25 −0.800593 −0.400296 0.916386i \(-0.631093\pi\)
−0.400296 + 0.916386i \(0.631093\pi\)
\(992\) −1.95103e26 −6.60565
\(993\) −1.04309e25 −0.350150
\(994\) −1.33264e25 −0.443535
\(995\) −3.27619e24 −0.108112
\(996\) 2.15114e25 0.703823
\(997\) 5.52750e25 1.79316 0.896582 0.442877i \(-0.146042\pi\)
0.896582 + 0.442877i \(0.146042\pi\)
\(998\) 6.40841e25 2.06130
\(999\) 9.00902e23 0.0287323
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 15.18.a.a.1.1 2
3.2 odd 2 45.18.a.b.1.2 2
5.2 odd 4 75.18.b.b.49.1 4
5.3 odd 4 75.18.b.b.49.4 4
5.4 even 2 75.18.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.a.1.1 2 1.1 even 1 trivial
45.18.a.b.1.2 2 3.2 odd 2
75.18.a.c.1.2 2 5.4 even 2
75.18.b.b.49.1 4 5.2 odd 4
75.18.b.b.49.4 4 5.3 odd 4