Properties

Label 5.28.a.a.1.2
Level $5$
Weight $28$
Character 5.1
Self dual yes
Analytic conductor $23.093$
Analytic rank $1$
Dimension $4$
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] = [5,28,Mod(1,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.1");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(23.0927787419\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 19275662x^{2} - 30468026939x + 4134032404260 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2324.55\) of defining polynomial
Character \(\chi\) \(=\) 5.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7959.76 q^{2} +829899. q^{3} -7.08599e7 q^{4} +1.22070e9 q^{5} -6.60580e9 q^{6} -2.73788e10 q^{7} +1.63237e12 q^{8} -6.93687e12 q^{9} +O(q^{10})\) \(q-7959.76 q^{2} +829899. q^{3} -7.08599e7 q^{4} +1.22070e9 q^{5} -6.60580e9 q^{6} -2.73788e10 q^{7} +1.63237e12 q^{8} -6.93687e12 q^{9} -9.71651e12 q^{10} +6.42529e13 q^{11} -5.88066e13 q^{12} +1.71367e15 q^{13} +2.17929e14 q^{14} +1.01306e15 q^{15} -3.48262e15 q^{16} +1.17307e15 q^{17} +5.52158e16 q^{18} -2.79305e17 q^{19} -8.64989e16 q^{20} -2.27217e16 q^{21} -5.11438e17 q^{22} -3.09362e18 q^{23} +1.35470e18 q^{24} +1.49012e18 q^{25} -1.36404e19 q^{26} -1.20854e19 q^{27} +1.94006e18 q^{28} -7.42209e19 q^{29} -8.06372e18 q^{30} -1.27785e20 q^{31} -1.91372e20 q^{32} +5.33234e19 q^{33} -9.33735e18 q^{34} -3.34214e19 q^{35} +4.91545e20 q^{36} +8.47020e20 q^{37} +2.22320e21 q^{38} +1.42217e21 q^{39} +1.99264e21 q^{40} -1.17207e21 q^{41} +1.80859e20 q^{42} +3.56147e21 q^{43} -4.55295e21 q^{44} -8.46785e21 q^{45} +2.46245e22 q^{46} +9.14938e21 q^{47} -2.89022e21 q^{48} -6.49628e22 q^{49} -1.18610e22 q^{50} +9.73528e20 q^{51} -1.21430e23 q^{52} -2.22609e23 q^{53} +9.61967e22 q^{54} +7.84337e22 q^{55} -4.46924e22 q^{56} -2.31795e23 q^{57} +5.90781e23 q^{58} -6.15031e23 q^{59} -7.17853e22 q^{60} +1.92253e24 q^{61} +1.01714e24 q^{62} +1.89923e23 q^{63} +1.99071e24 q^{64} +2.09188e24 q^{65} -4.24442e23 q^{66} -5.40282e24 q^{67} -8.31235e22 q^{68} -2.56739e24 q^{69} +2.66027e23 q^{70} +1.94780e24 q^{71} -1.13235e25 q^{72} -1.12164e25 q^{73} -6.74208e24 q^{74} +1.23665e24 q^{75} +1.97915e25 q^{76} -1.75917e24 q^{77} -1.13201e25 q^{78} -1.22236e25 q^{79} -4.25125e24 q^{80} +4.28681e25 q^{81} +9.32937e24 q^{82} +4.79523e25 q^{83} +1.61006e24 q^{84} +1.43197e24 q^{85} -2.83485e25 q^{86} -6.15958e25 q^{87} +1.04884e26 q^{88} +1.66993e26 q^{89} +6.74021e25 q^{90} -4.69182e25 q^{91} +2.19214e26 q^{92} -1.06049e26 q^{93} -7.28269e25 q^{94} -3.40948e26 q^{95} -1.58819e26 q^{96} -5.46723e26 q^{97} +5.17088e26 q^{98} -4.45714e26 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 11550 q^{2} - 2473800 q^{3} + 358631812 q^{4} + 4882812500 q^{5} - 160951168752 q^{6} - 215015185000 q^{7} + 183282059400 q^{8} + 25791757037748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 11550 q^{2} - 2473800 q^{3} + 358631812 q^{4} + 4882812500 q^{5} - 160951168752 q^{6} - 215015185000 q^{7} + 183282059400 q^{8} + 25791757037748 q^{9} - 14099121093750 q^{10} - 107427307660512 q^{11} + 13\!\cdots\!00 q^{12}+ \cdots - 34\!\cdots\!44 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7959.76 −0.687061 −0.343530 0.939142i \(-0.611623\pi\)
−0.343530 + 0.939142i \(0.611623\pi\)
\(3\) 829899. 0.300530 0.150265 0.988646i \(-0.451987\pi\)
0.150265 + 0.988646i \(0.451987\pi\)
\(4\) −7.08599e7 −0.527947
\(5\) 1.22070e9 0.447214
\(6\) −6.60580e9 −0.206483
\(7\) −2.73788e10 −0.106805 −0.0534025 0.998573i \(-0.517007\pi\)
−0.0534025 + 0.998573i \(0.517007\pi\)
\(8\) 1.63237e12 1.04979
\(9\) −6.93687e12 −0.909682
\(10\) −9.71651e12 −0.307263
\(11\) 6.42529e13 0.561166 0.280583 0.959830i \(-0.409472\pi\)
0.280583 + 0.959830i \(0.409472\pi\)
\(12\) −5.88066e13 −0.158664
\(13\) 1.71367e15 1.56925 0.784623 0.619973i \(-0.212857\pi\)
0.784623 + 0.619973i \(0.212857\pi\)
\(14\) 2.17929e14 0.0733815
\(15\) 1.01306e15 0.134401
\(16\) −3.48262e15 −0.193324
\(17\) 1.17307e15 0.0287252 0.0143626 0.999897i \(-0.495428\pi\)
0.0143626 + 0.999897i \(0.495428\pi\)
\(18\) 5.52158e16 0.625007
\(19\) −2.79305e17 −1.52372 −0.761860 0.647742i \(-0.775713\pi\)
−0.761860 + 0.647742i \(0.775713\pi\)
\(20\) −8.64989e16 −0.236105
\(21\) −2.27217e16 −0.0320981
\(22\) −5.11438e17 −0.385555
\(23\) −3.09362e18 −1.27980 −0.639899 0.768459i \(-0.721024\pi\)
−0.639899 + 0.768459i \(0.721024\pi\)
\(24\) 1.35470e18 0.315495
\(25\) 1.49012e18 0.200000
\(26\) −1.36404e19 −1.07817
\(27\) −1.20854e19 −0.573917
\(28\) 1.94006e18 0.0563874
\(29\) −7.42209e19 −1.34324 −0.671619 0.740896i \(-0.734401\pi\)
−0.671619 + 0.740896i \(0.734401\pi\)
\(30\) −8.06372e18 −0.0923418
\(31\) −1.27785e20 −0.939937 −0.469968 0.882683i \(-0.655735\pi\)
−0.469968 + 0.882683i \(0.655735\pi\)
\(32\) −1.91372e20 −0.916967
\(33\) 5.33234e19 0.168647
\(34\) −9.33735e18 −0.0197360
\(35\) −3.34214e19 −0.0477646
\(36\) 4.91545e20 0.480264
\(37\) 8.47020e20 0.571703 0.285851 0.958274i \(-0.407724\pi\)
0.285851 + 0.958274i \(0.407724\pi\)
\(38\) 2.22320e21 1.04689
\(39\) 1.42217e21 0.471606
\(40\) 1.99264e21 0.469482
\(41\) −1.17207e21 −0.197866 −0.0989328 0.995094i \(-0.531543\pi\)
−0.0989328 + 0.995094i \(0.531543\pi\)
\(42\) 1.80859e20 0.0220534
\(43\) 3.56147e21 0.316086 0.158043 0.987432i \(-0.449481\pi\)
0.158043 + 0.987432i \(0.449481\pi\)
\(44\) −4.55295e21 −0.296266
\(45\) −8.46785e21 −0.406822
\(46\) 2.46245e22 0.879299
\(47\) 9.14938e21 0.244383 0.122191 0.992507i \(-0.461008\pi\)
0.122191 + 0.992507i \(0.461008\pi\)
\(48\) −2.89022e21 −0.0580998
\(49\) −6.49628e22 −0.988593
\(50\) −1.18610e22 −0.137412
\(51\) 9.73528e20 0.00863279
\(52\) −1.21430e23 −0.828479
\(53\) −2.22609e23 −1.17441 −0.587204 0.809439i \(-0.699772\pi\)
−0.587204 + 0.809439i \(0.699772\pi\)
\(54\) 9.61967e22 0.394316
\(55\) 7.84337e22 0.250961
\(56\) −4.46924e22 −0.112123
\(57\) −2.31795e23 −0.457924
\(58\) 5.90781e23 0.922887
\(59\) −6.15031e23 −0.762773 −0.381386 0.924416i \(-0.624553\pi\)
−0.381386 + 0.924416i \(0.624553\pi\)
\(60\) −7.17853e22 −0.0709568
\(61\) 1.92253e24 1.52027 0.760134 0.649767i \(-0.225133\pi\)
0.760134 + 0.649767i \(0.225133\pi\)
\(62\) 1.01714e24 0.645794
\(63\) 1.89923e23 0.0971585
\(64\) 1.99071e24 0.823337
\(65\) 2.09188e24 0.701788
\(66\) −4.24442e23 −0.115871
\(67\) −5.40282e24 −1.20396 −0.601978 0.798513i \(-0.705620\pi\)
−0.601978 + 0.798513i \(0.705620\pi\)
\(68\) −8.31235e22 −0.0151654
\(69\) −2.56739e24 −0.384618
\(70\) 2.66027e23 0.0328172
\(71\) 1.94780e24 0.198407 0.0992034 0.995067i \(-0.468371\pi\)
0.0992034 + 0.995067i \(0.468371\pi\)
\(72\) −1.13235e25 −0.954977
\(73\) −1.12164e25 −0.785227 −0.392614 0.919703i \(-0.628429\pi\)
−0.392614 + 0.919703i \(0.628429\pi\)
\(74\) −6.74208e24 −0.392795
\(75\) 1.23665e24 0.0601061
\(76\) 1.97915e25 0.804443
\(77\) −1.75917e24 −0.0599353
\(78\) −1.13201e25 −0.324022
\(79\) −1.22236e25 −0.294600 −0.147300 0.989092i \(-0.547058\pi\)
−0.147300 + 0.989092i \(0.547058\pi\)
\(80\) −4.25125e24 −0.0864573
\(81\) 4.28681e25 0.737202
\(82\) 9.32937e24 0.135946
\(83\) 4.79523e25 0.593274 0.296637 0.954990i \(-0.404135\pi\)
0.296637 + 0.954990i \(0.404135\pi\)
\(84\) 1.61006e24 0.0169461
\(85\) 1.43197e24 0.0128463
\(86\) −2.83485e25 −0.217171
\(87\) −6.15958e25 −0.403684
\(88\) 1.04884e26 0.589108
\(89\) 1.66993e26 0.805255 0.402627 0.915364i \(-0.368097\pi\)
0.402627 + 0.915364i \(0.368097\pi\)
\(90\) 6.74021e25 0.279511
\(91\) −4.69182e25 −0.167603
\(92\) 2.19214e26 0.675666
\(93\) −1.06049e26 −0.282480
\(94\) −7.28269e25 −0.167906
\(95\) −3.40948e26 −0.681428
\(96\) −1.58819e26 −0.275576
\(97\) −5.46723e26 −0.824800 −0.412400 0.911003i \(-0.635309\pi\)
−0.412400 + 0.911003i \(0.635309\pi\)
\(98\) 5.17088e26 0.679223
\(99\) −4.45714e26 −0.510482
\(100\) −1.05589e26 −0.105589
\(101\) −1.97130e27 −1.72351 −0.861754 0.507326i \(-0.830634\pi\)
−0.861754 + 0.507326i \(0.830634\pi\)
\(102\) −7.74906e24 −0.00593125
\(103\) −1.51530e27 −1.01671 −0.508354 0.861148i \(-0.669746\pi\)
−0.508354 + 0.861148i \(0.669746\pi\)
\(104\) 2.79734e27 1.64738
\(105\) −2.77364e25 −0.0143547
\(106\) 1.77192e27 0.806890
\(107\) 8.61493e26 0.345597 0.172799 0.984957i \(-0.444719\pi\)
0.172799 + 0.984957i \(0.444719\pi\)
\(108\) 8.56368e26 0.302998
\(109\) 2.11934e27 0.662131 0.331065 0.943608i \(-0.392592\pi\)
0.331065 + 0.943608i \(0.392592\pi\)
\(110\) −6.24314e26 −0.172426
\(111\) 7.02941e26 0.171814
\(112\) 9.53502e25 0.0206480
\(113\) 3.78513e27 0.726978 0.363489 0.931598i \(-0.381585\pi\)
0.363489 + 0.931598i \(0.381585\pi\)
\(114\) 1.84503e27 0.314622
\(115\) −3.77639e27 −0.572343
\(116\) 5.25928e27 0.709159
\(117\) −1.18875e28 −1.42751
\(118\) 4.89550e27 0.524071
\(119\) −3.21173e25 −0.00306799
\(120\) 1.65369e27 0.141093
\(121\) −8.98156e27 −0.685093
\(122\) −1.53029e28 −1.04452
\(123\) −9.72697e26 −0.0594646
\(124\) 9.05486e27 0.496237
\(125\) 1.81899e27 0.0894427
\(126\) −1.51174e27 −0.0667538
\(127\) −4.38836e28 −1.74161 −0.870807 0.491624i \(-0.836403\pi\)
−0.870807 + 0.491624i \(0.836403\pi\)
\(128\) 9.83997e27 0.351285
\(129\) 2.95566e27 0.0949935
\(130\) −1.66509e28 −0.482171
\(131\) 6.40410e28 1.67223 0.836114 0.548556i \(-0.184822\pi\)
0.836114 + 0.548556i \(0.184822\pi\)
\(132\) −3.77849e27 −0.0890369
\(133\) 7.64704e27 0.162741
\(134\) 4.30052e28 0.827191
\(135\) −1.47527e28 −0.256664
\(136\) 1.91488e27 0.0301555
\(137\) −1.22682e29 −1.75005 −0.875027 0.484074i \(-0.839157\pi\)
−0.875027 + 0.484074i \(0.839157\pi\)
\(138\) 2.04358e28 0.264256
\(139\) 1.28776e29 1.51055 0.755273 0.655410i \(-0.227504\pi\)
0.755273 + 0.655410i \(0.227504\pi\)
\(140\) 2.36824e27 0.0252172
\(141\) 7.59306e27 0.0734444
\(142\) −1.55040e28 −0.136318
\(143\) 1.10108e29 0.880607
\(144\) 2.41585e28 0.175864
\(145\) −9.06017e28 −0.600715
\(146\) 8.92800e28 0.539499
\(147\) −5.39125e28 −0.297102
\(148\) −6.00197e28 −0.301829
\(149\) 3.62979e29 1.66674 0.833369 0.552717i \(-0.186409\pi\)
0.833369 + 0.552717i \(0.186409\pi\)
\(150\) −9.84341e27 −0.0412965
\(151\) −3.99824e29 −1.53349 −0.766744 0.641953i \(-0.778124\pi\)
−0.766744 + 0.641953i \(0.778124\pi\)
\(152\) −4.55929e29 −1.59959
\(153\) −8.13742e27 −0.0261308
\(154\) 1.40026e28 0.0411792
\(155\) −1.55988e29 −0.420353
\(156\) −1.00775e29 −0.248983
\(157\) 4.28378e28 0.0970917 0.0485458 0.998821i \(-0.484541\pi\)
0.0485458 + 0.998821i \(0.484541\pi\)
\(158\) 9.72969e28 0.202408
\(159\) −1.84743e29 −0.352945
\(160\) −2.33608e29 −0.410080
\(161\) 8.46998e28 0.136689
\(162\) −3.41220e29 −0.506503
\(163\) 1.17374e30 1.60339 0.801696 0.597732i \(-0.203931\pi\)
0.801696 + 0.597732i \(0.203931\pi\)
\(164\) 8.30525e28 0.104463
\(165\) 6.50920e28 0.0754214
\(166\) −3.81689e29 −0.407616
\(167\) 6.93075e29 0.682508 0.341254 0.939971i \(-0.389148\pi\)
0.341254 + 0.939971i \(0.389148\pi\)
\(168\) −3.70902e28 −0.0336964
\(169\) 1.74412e30 1.46253
\(170\) −1.13981e28 −0.00882619
\(171\) 1.93750e30 1.38610
\(172\) −2.52366e29 −0.166877
\(173\) −1.66313e30 −1.01696 −0.508479 0.861074i \(-0.669792\pi\)
−0.508479 + 0.861074i \(0.669792\pi\)
\(174\) 4.90288e29 0.277355
\(175\) −4.07977e28 −0.0213610
\(176\) −2.23769e29 −0.108487
\(177\) −5.10414e29 −0.229236
\(178\) −1.32923e30 −0.553259
\(179\) −3.59477e30 −1.38725 −0.693625 0.720336i \(-0.743987\pi\)
−0.693625 + 0.720336i \(0.743987\pi\)
\(180\) 6.00031e29 0.214781
\(181\) −1.98619e29 −0.0659722 −0.0329861 0.999456i \(-0.510502\pi\)
−0.0329861 + 0.999456i \(0.510502\pi\)
\(182\) 3.73458e29 0.115154
\(183\) 1.59550e30 0.456887
\(184\) −5.04993e30 −1.34352
\(185\) 1.03396e30 0.255673
\(186\) 8.44125e29 0.194081
\(187\) 7.53730e28 0.0161196
\(188\) −6.48324e29 −0.129021
\(189\) 3.30884e29 0.0612972
\(190\) 2.71387e30 0.468182
\(191\) −3.73765e30 −0.600688 −0.300344 0.953831i \(-0.597101\pi\)
−0.300344 + 0.953831i \(0.597101\pi\)
\(192\) 1.65209e30 0.247438
\(193\) −7.33944e30 −1.02480 −0.512400 0.858747i \(-0.671244\pi\)
−0.512400 + 0.858747i \(0.671244\pi\)
\(194\) 4.35178e30 0.566688
\(195\) 1.73605e30 0.210909
\(196\) 4.60325e30 0.521925
\(197\) 8.12445e30 0.860003 0.430002 0.902828i \(-0.358513\pi\)
0.430002 + 0.902828i \(0.358513\pi\)
\(198\) 3.54777e30 0.350732
\(199\) −1.52784e31 −1.41111 −0.705557 0.708653i \(-0.749303\pi\)
−0.705557 + 0.708653i \(0.749303\pi\)
\(200\) 2.43242e30 0.209959
\(201\) −4.48379e30 −0.361825
\(202\) 1.56911e31 1.18416
\(203\) 2.03208e30 0.143465
\(204\) −6.89841e28 −0.00455766
\(205\) −1.43074e30 −0.0884881
\(206\) 1.20614e31 0.698540
\(207\) 2.14600e31 1.16421
\(208\) −5.96805e30 −0.303373
\(209\) −1.79461e31 −0.855059
\(210\) 2.20775e29 0.00986257
\(211\) −2.23121e31 −0.934818 −0.467409 0.884041i \(-0.654812\pi\)
−0.467409 + 0.884041i \(0.654812\pi\)
\(212\) 1.57741e31 0.620026
\(213\) 1.61648e30 0.0596273
\(214\) −6.85728e30 −0.237447
\(215\) 4.34750e30 0.141358
\(216\) −1.97278e31 −0.602494
\(217\) 3.49862e30 0.100390
\(218\) −1.68695e31 −0.454924
\(219\) −9.30849e30 −0.235985
\(220\) −5.55780e30 −0.132494
\(221\) 2.01025e30 0.0450769
\(222\) −5.59524e30 −0.118047
\(223\) −5.43506e31 −1.07917 −0.539583 0.841932i \(-0.681418\pi\)
−0.539583 + 0.841932i \(0.681418\pi\)
\(224\) 5.23955e30 0.0979367
\(225\) −1.03367e31 −0.181936
\(226\) −3.01287e31 −0.499478
\(227\) 5.55710e31 0.867959 0.433979 0.900923i \(-0.357109\pi\)
0.433979 + 0.900923i \(0.357109\pi\)
\(228\) 1.64250e31 0.241760
\(229\) −1.14629e31 −0.159043 −0.0795217 0.996833i \(-0.525339\pi\)
−0.0795217 + 0.996833i \(0.525339\pi\)
\(230\) 3.00592e31 0.393235
\(231\) −1.45993e30 −0.0180124
\(232\) −1.21156e32 −1.41012
\(233\) 1.27501e32 1.40026 0.700128 0.714018i \(-0.253126\pi\)
0.700128 + 0.714018i \(0.253126\pi\)
\(234\) 9.46215e31 0.980789
\(235\) 1.11687e31 0.109291
\(236\) 4.35811e31 0.402704
\(237\) −1.01443e31 −0.0885363
\(238\) 2.55646e29 0.00210790
\(239\) 9.40042e31 0.732446 0.366223 0.930527i \(-0.380651\pi\)
0.366223 + 0.930527i \(0.380651\pi\)
\(240\) −3.52811e30 −0.0259830
\(241\) 1.81502e32 1.26372 0.631862 0.775081i \(-0.282291\pi\)
0.631862 + 0.775081i \(0.282291\pi\)
\(242\) 7.14911e31 0.470700
\(243\) 1.27734e32 0.795469
\(244\) −1.36230e32 −0.802621
\(245\) −7.93002e31 −0.442112
\(246\) 7.74244e30 0.0408558
\(247\) −4.78635e32 −2.39109
\(248\) −2.08593e32 −0.986739
\(249\) 3.97956e31 0.178297
\(250\) −1.44787e31 −0.0614526
\(251\) 8.28027e31 0.333004 0.166502 0.986041i \(-0.446753\pi\)
0.166502 + 0.986041i \(0.446753\pi\)
\(252\) −1.34579e31 −0.0512946
\(253\) −1.98774e32 −0.718179
\(254\) 3.49303e32 1.19660
\(255\) 1.18839e30 0.00386070
\(256\) −3.45512e32 −1.06469
\(257\) −2.07555e32 −0.606786 −0.303393 0.952866i \(-0.598119\pi\)
−0.303393 + 0.952866i \(0.598119\pi\)
\(258\) −2.35264e31 −0.0652663
\(259\) −2.31904e31 −0.0610607
\(260\) −1.48230e32 −0.370507
\(261\) 5.14860e32 1.22192
\(262\) −5.09751e32 −1.14892
\(263\) 9.03345e32 1.93397 0.966987 0.254824i \(-0.0820176\pi\)
0.966987 + 0.254824i \(0.0820176\pi\)
\(264\) 8.70435e31 0.177045
\(265\) −2.71740e32 −0.525212
\(266\) −6.08687e31 −0.111813
\(267\) 1.38587e32 0.242003
\(268\) 3.82843e32 0.635625
\(269\) −8.06210e31 −0.127290 −0.0636448 0.997973i \(-0.520272\pi\)
−0.0636448 + 0.997973i \(0.520272\pi\)
\(270\) 1.17428e32 0.176344
\(271\) −6.60901e32 −0.944171 −0.472086 0.881553i \(-0.656499\pi\)
−0.472086 + 0.881553i \(0.656499\pi\)
\(272\) −4.08535e30 −0.00555328
\(273\) −3.89374e31 −0.0503699
\(274\) 9.76516e32 1.20239
\(275\) 9.57443e31 0.112233
\(276\) 1.81925e32 0.203058
\(277\) 7.48899e32 0.796061 0.398030 0.917372i \(-0.369694\pi\)
0.398030 + 0.917372i \(0.369694\pi\)
\(278\) −1.02502e33 −1.03784
\(279\) 8.86431e32 0.855043
\(280\) −5.45561e31 −0.0501430
\(281\) −2.28413e32 −0.200071 −0.100036 0.994984i \(-0.531896\pi\)
−0.100036 + 0.994984i \(0.531896\pi\)
\(282\) −6.04390e31 −0.0504608
\(283\) −1.63098e32 −0.129817 −0.0649084 0.997891i \(-0.520676\pi\)
−0.0649084 + 0.997891i \(0.520676\pi\)
\(284\) −1.38021e32 −0.104748
\(285\) −2.82953e32 −0.204790
\(286\) −8.76434e32 −0.605031
\(287\) 3.20898e31 0.0211330
\(288\) 1.32752e33 0.834148
\(289\) −1.66634e33 −0.999175
\(290\) 7.21168e32 0.412727
\(291\) −4.53725e32 −0.247877
\(292\) 7.94794e32 0.414559
\(293\) −4.06308e32 −0.202368 −0.101184 0.994868i \(-0.532263\pi\)
−0.101184 + 0.994868i \(0.532263\pi\)
\(294\) 4.29131e32 0.204127
\(295\) −7.50771e32 −0.341122
\(296\) 1.38265e33 0.600169
\(297\) −7.76520e32 −0.322063
\(298\) −2.88923e33 −1.14515
\(299\) −5.30143e33 −2.00832
\(300\) −8.76286e31 −0.0317328
\(301\) −9.75090e31 −0.0337596
\(302\) 3.18251e33 1.05360
\(303\) −1.63598e33 −0.517967
\(304\) 9.72713e32 0.294572
\(305\) 2.34684e33 0.679884
\(306\) 6.47719e31 0.0179534
\(307\) 1.37932e32 0.0365844 0.0182922 0.999833i \(-0.494177\pi\)
0.0182922 + 0.999833i \(0.494177\pi\)
\(308\) 1.24655e32 0.0316427
\(309\) −1.25755e33 −0.305552
\(310\) 1.24163e33 0.288808
\(311\) 4.42328e33 0.985097 0.492549 0.870285i \(-0.336065\pi\)
0.492549 + 0.870285i \(0.336065\pi\)
\(312\) 2.32151e33 0.495089
\(313\) 2.96295e33 0.605167 0.302584 0.953123i \(-0.402151\pi\)
0.302584 + 0.953123i \(0.402151\pi\)
\(314\) −3.40979e32 −0.0667079
\(315\) 2.31840e32 0.0434506
\(316\) 8.66162e32 0.155533
\(317\) 5.27424e33 0.907527 0.453764 0.891122i \(-0.350081\pi\)
0.453764 + 0.891122i \(0.350081\pi\)
\(318\) 1.47051e33 0.242495
\(319\) −4.76891e33 −0.753780
\(320\) 2.43006e33 0.368207
\(321\) 7.14952e32 0.103863
\(322\) −6.74190e32 −0.0939135
\(323\) −3.27644e32 −0.0437691
\(324\) −3.03763e33 −0.389204
\(325\) 2.55356e33 0.313849
\(326\) −9.34271e33 −1.10163
\(327\) 1.75884e33 0.198990
\(328\) −1.91325e33 −0.207718
\(329\) −2.50499e32 −0.0261013
\(330\) −5.18117e32 −0.0518191
\(331\) 9.41253e32 0.0903709 0.0451855 0.998979i \(-0.485612\pi\)
0.0451855 + 0.998979i \(0.485612\pi\)
\(332\) −3.39790e33 −0.313218
\(333\) −5.87566e33 −0.520067
\(334\) −5.51671e33 −0.468925
\(335\) −6.59524e33 −0.538425
\(336\) 7.91310e31 0.00620535
\(337\) 7.47402e33 0.563055 0.281527 0.959553i \(-0.409159\pi\)
0.281527 + 0.959553i \(0.409159\pi\)
\(338\) −1.38828e34 −1.00485
\(339\) 3.14127e33 0.218479
\(340\) −1.01469e32 −0.00678217
\(341\) −8.21059e33 −0.527461
\(342\) −1.54220e34 −0.952334
\(343\) 3.57773e33 0.212392
\(344\) 5.81364e33 0.331825
\(345\) −3.13403e33 −0.172006
\(346\) 1.32381e34 0.698713
\(347\) 2.12986e34 1.08119 0.540596 0.841282i \(-0.318199\pi\)
0.540596 + 0.841282i \(0.318199\pi\)
\(348\) 4.36467e33 0.213124
\(349\) 3.74540e34 1.75936 0.879680 0.475565i \(-0.157756\pi\)
0.879680 + 0.475565i \(0.157756\pi\)
\(350\) 3.24740e32 0.0146763
\(351\) −2.07103e34 −0.900617
\(352\) −1.22962e34 −0.514571
\(353\) 2.00261e34 0.806563 0.403282 0.915076i \(-0.367869\pi\)
0.403282 + 0.915076i \(0.367869\pi\)
\(354\) 4.06277e33 0.157499
\(355\) 2.37769e33 0.0887302
\(356\) −1.18331e34 −0.425132
\(357\) −2.66541e31 −0.000922025 0
\(358\) 2.86135e34 0.953125
\(359\) −9.41455e33 −0.302012 −0.151006 0.988533i \(-0.548251\pi\)
−0.151006 + 0.988533i \(0.548251\pi\)
\(360\) −1.38227e34 −0.427079
\(361\) 4.44106e34 1.32172
\(362\) 1.58096e33 0.0453269
\(363\) −7.45379e33 −0.205891
\(364\) 3.32462e33 0.0884857
\(365\) −1.36919e34 −0.351164
\(366\) −1.26998e34 −0.313909
\(367\) −7.58151e34 −1.80619 −0.903096 0.429439i \(-0.858711\pi\)
−0.903096 + 0.429439i \(0.858711\pi\)
\(368\) 1.07739e34 0.247416
\(369\) 8.13047e33 0.179995
\(370\) −8.23008e33 −0.175663
\(371\) 6.09479e33 0.125433
\(372\) 7.51462e33 0.149134
\(373\) 3.75622e34 0.718922 0.359461 0.933160i \(-0.382961\pi\)
0.359461 + 0.933160i \(0.382961\pi\)
\(374\) −5.99952e32 −0.0110751
\(375\) 1.50958e33 0.0268803
\(376\) 1.49352e34 0.256551
\(377\) −1.27190e35 −2.10787
\(378\) −2.63375e33 −0.0421149
\(379\) 9.15706e34 1.41295 0.706475 0.707738i \(-0.250284\pi\)
0.706475 + 0.707738i \(0.250284\pi\)
\(380\) 2.41596e34 0.359758
\(381\) −3.64190e34 −0.523408
\(382\) 2.97508e34 0.412709
\(383\) −1.31082e35 −1.75534 −0.877670 0.479266i \(-0.840903\pi\)
−0.877670 + 0.479266i \(0.840903\pi\)
\(384\) 8.16618e33 0.105572
\(385\) −2.14742e33 −0.0268039
\(386\) 5.84202e34 0.704100
\(387\) −2.47055e34 −0.287538
\(388\) 3.87407e34 0.435451
\(389\) −7.23296e34 −0.785229 −0.392614 0.919703i \(-0.628429\pi\)
−0.392614 + 0.919703i \(0.628429\pi\)
\(390\) −1.38185e34 −0.144907
\(391\) −3.62903e33 −0.0367624
\(392\) −1.06043e35 −1.03782
\(393\) 5.31475e34 0.502555
\(394\) −6.46687e34 −0.590875
\(395\) −1.49214e34 −0.131749
\(396\) 3.15832e34 0.269508
\(397\) 1.87264e35 1.54448 0.772239 0.635332i \(-0.219137\pi\)
0.772239 + 0.635332i \(0.219137\pi\)
\(398\) 1.21613e35 0.969522
\(399\) 6.34627e33 0.0489085
\(400\) −5.18951e33 −0.0386649
\(401\) 1.46014e35 1.05183 0.525916 0.850537i \(-0.323723\pi\)
0.525916 + 0.850537i \(0.323723\pi\)
\(402\) 3.56899e34 0.248596
\(403\) −2.18982e35 −1.47499
\(404\) 1.39686e35 0.909922
\(405\) 5.23292e34 0.329687
\(406\) −1.61749e34 −0.0985689
\(407\) 5.44235e34 0.320820
\(408\) 1.58916e33 0.00906264
\(409\) 1.50577e35 0.830795 0.415398 0.909640i \(-0.363642\pi\)
0.415398 + 0.909640i \(0.363642\pi\)
\(410\) 1.13884e34 0.0607967
\(411\) −1.01813e35 −0.525944
\(412\) 1.07374e35 0.536768
\(413\) 1.68388e34 0.0814679
\(414\) −1.70817e35 −0.799882
\(415\) 5.85356e34 0.265320
\(416\) −3.27948e35 −1.43895
\(417\) 1.06871e35 0.453965
\(418\) 1.42847e35 0.587478
\(419\) −3.49644e35 −1.39231 −0.696157 0.717890i \(-0.745108\pi\)
−0.696157 + 0.717890i \(0.745108\pi\)
\(420\) 1.96540e33 0.00757854
\(421\) −1.36815e35 −0.510888 −0.255444 0.966824i \(-0.582222\pi\)
−0.255444 + 0.966824i \(0.582222\pi\)
\(422\) 1.77599e35 0.642277
\(423\) −6.34680e34 −0.222310
\(424\) −3.63381e35 −1.23289
\(425\) 1.74801e33 0.00574504
\(426\) −1.28668e34 −0.0409676
\(427\) −5.26366e34 −0.162372
\(428\) −6.10453e34 −0.182457
\(429\) 9.13785e34 0.264649
\(430\) −3.46051e34 −0.0971216
\(431\) −4.89720e34 −0.133200 −0.0666001 0.997780i \(-0.521215\pi\)
−0.0666001 + 0.997780i \(0.521215\pi\)
\(432\) 4.20888e34 0.110952
\(433\) −4.40990e35 −1.12679 −0.563393 0.826189i \(-0.690504\pi\)
−0.563393 + 0.826189i \(0.690504\pi\)
\(434\) −2.78482e34 −0.0689740
\(435\) −7.51902e34 −0.180533
\(436\) −1.50176e35 −0.349570
\(437\) 8.64064e35 1.95005
\(438\) 7.40934e34 0.162136
\(439\) −3.19792e35 −0.678573 −0.339287 0.940683i \(-0.610186\pi\)
−0.339287 + 0.940683i \(0.610186\pi\)
\(440\) 1.28033e35 0.263457
\(441\) 4.50638e35 0.899305
\(442\) −1.60011e34 −0.0309706
\(443\) 2.30373e35 0.432496 0.216248 0.976338i \(-0.430618\pi\)
0.216248 + 0.976338i \(0.430618\pi\)
\(444\) −4.98103e34 −0.0907087
\(445\) 2.03849e35 0.360121
\(446\) 4.32618e35 0.741453
\(447\) 3.01236e35 0.500905
\(448\) −5.45032e34 −0.0879365
\(449\) 8.08913e35 1.26642 0.633208 0.773982i \(-0.281738\pi\)
0.633208 + 0.773982i \(0.281738\pi\)
\(450\) 8.22780e34 0.125001
\(451\) −7.53086e34 −0.111035
\(452\) −2.68214e35 −0.383806
\(453\) −3.31814e35 −0.460859
\(454\) −4.42332e35 −0.596340
\(455\) −5.72732e34 −0.0749545
\(456\) −3.78375e35 −0.480725
\(457\) −4.51820e35 −0.557309 −0.278655 0.960391i \(-0.589888\pi\)
−0.278655 + 0.960391i \(0.589888\pi\)
\(458\) 9.12420e34 0.109273
\(459\) −1.41770e34 −0.0164859
\(460\) 2.67595e35 0.302167
\(461\) 1.24872e36 1.36932 0.684658 0.728865i \(-0.259952\pi\)
0.684658 + 0.728865i \(0.259952\pi\)
\(462\) 1.16207e34 0.0123756
\(463\) −3.60640e35 −0.373018 −0.186509 0.982453i \(-0.559717\pi\)
−0.186509 + 0.982453i \(0.559717\pi\)
\(464\) 2.58483e35 0.259681
\(465\) −1.29454e35 −0.126329
\(466\) −1.01487e36 −0.962061
\(467\) −1.86337e36 −1.71602 −0.858009 0.513634i \(-0.828299\pi\)
−0.858009 + 0.513634i \(0.828299\pi\)
\(468\) 8.42345e35 0.753652
\(469\) 1.47923e35 0.128588
\(470\) −8.89000e34 −0.0750898
\(471\) 3.55511e34 0.0291790
\(472\) −1.00396e36 −0.800753
\(473\) 2.28835e35 0.177377
\(474\) 8.07466e34 0.0608298
\(475\) −4.16197e35 −0.304744
\(476\) 2.27583e33 0.00161974
\(477\) 1.54421e36 1.06834
\(478\) −7.48251e35 −0.503235
\(479\) −9.56809e35 −0.625599 −0.312799 0.949819i \(-0.601267\pi\)
−0.312799 + 0.949819i \(0.601267\pi\)
\(480\) −1.93871e35 −0.123242
\(481\) 1.45151e36 0.897142
\(482\) −1.44472e36 −0.868255
\(483\) 7.02923e34 0.0410791
\(484\) 6.36432e35 0.361693
\(485\) −6.67386e35 −0.368862
\(486\) −1.01674e36 −0.546536
\(487\) −1.57059e36 −0.821150 −0.410575 0.911827i \(-0.634672\pi\)
−0.410575 + 0.911827i \(0.634672\pi\)
\(488\) 3.13828e36 1.59597
\(489\) 9.74087e35 0.481868
\(490\) 6.31211e35 0.303758
\(491\) −1.55981e36 −0.730249 −0.365124 0.930959i \(-0.618974\pi\)
−0.365124 + 0.930959i \(0.618974\pi\)
\(492\) 6.89252e34 0.0313942
\(493\) −8.70662e34 −0.0385848
\(494\) 3.80982e36 1.64282
\(495\) −5.44084e35 −0.228295
\(496\) 4.45029e35 0.181713
\(497\) −5.33285e34 −0.0211908
\(498\) −3.16764e35 −0.122501
\(499\) 1.75600e36 0.660948 0.330474 0.943815i \(-0.392791\pi\)
0.330474 + 0.943815i \(0.392791\pi\)
\(500\) −1.28893e35 −0.0472210
\(501\) 5.75182e35 0.205114
\(502\) −6.59090e35 −0.228794
\(503\) −2.95639e36 −0.999066 −0.499533 0.866295i \(-0.666495\pi\)
−0.499533 + 0.866295i \(0.666495\pi\)
\(504\) 3.10025e35 0.101996
\(505\) −2.40637e36 −0.770777
\(506\) 1.58219e36 0.493433
\(507\) 1.44744e36 0.439535
\(508\) 3.10959e36 0.919481
\(509\) −1.51597e36 −0.436517 −0.218258 0.975891i \(-0.570038\pi\)
−0.218258 + 0.975891i \(0.570038\pi\)
\(510\) −9.45930e33 −0.00265254
\(511\) 3.07092e35 0.0838662
\(512\) 1.42949e36 0.380223
\(513\) 3.37550e36 0.874489
\(514\) 1.65209e36 0.416899
\(515\) −1.84973e36 −0.454686
\(516\) −2.09438e35 −0.0501516
\(517\) 5.87874e35 0.137139
\(518\) 1.84590e35 0.0419524
\(519\) −1.38023e36 −0.305627
\(520\) 3.41472e36 0.736732
\(521\) −3.10378e36 −0.652501 −0.326250 0.945283i \(-0.605785\pi\)
−0.326250 + 0.945283i \(0.605785\pi\)
\(522\) −4.09817e36 −0.839533
\(523\) −7.80473e36 −1.55806 −0.779032 0.626984i \(-0.784289\pi\)
−0.779032 + 0.626984i \(0.784289\pi\)
\(524\) −4.53793e36 −0.882848
\(525\) −3.38579e34 −0.00641963
\(526\) −7.19041e36 −1.32876
\(527\) −1.49901e35 −0.0269999
\(528\) −1.85705e35 −0.0326037
\(529\) 3.72728e36 0.637883
\(530\) 2.16299e36 0.360852
\(531\) 4.26639e36 0.693880
\(532\) −5.41869e35 −0.0859186
\(533\) −2.00853e36 −0.310500
\(534\) −1.10312e36 −0.166271
\(535\) 1.05163e36 0.154556
\(536\) −8.81940e36 −1.26390
\(537\) −2.98329e36 −0.416911
\(538\) 6.41724e35 0.0874556
\(539\) −4.17404e36 −0.554765
\(540\) 1.04537e36 0.135505
\(541\) 1.44148e37 1.82241 0.911205 0.411952i \(-0.135153\pi\)
0.911205 + 0.411952i \(0.135153\pi\)
\(542\) 5.26061e36 0.648703
\(543\) −1.64834e35 −0.0198267
\(544\) −2.24493e35 −0.0263401
\(545\) 2.58709e36 0.296114
\(546\) 3.09932e35 0.0346072
\(547\) 1.71311e37 1.86619 0.933096 0.359626i \(-0.117096\pi\)
0.933096 + 0.359626i \(0.117096\pi\)
\(548\) 8.69320e36 0.923936
\(549\) −1.33363e37 −1.38296
\(550\) −7.62102e35 −0.0771110
\(551\) 2.07303e37 2.04672
\(552\) −4.19093e36 −0.403769
\(553\) 3.34668e35 0.0314648
\(554\) −5.96106e36 −0.546942
\(555\) 8.58082e35 0.0768376
\(556\) −9.12503e36 −0.797489
\(557\) 1.10383e37 0.941578 0.470789 0.882246i \(-0.343969\pi\)
0.470789 + 0.882246i \(0.343969\pi\)
\(558\) −7.05578e36 −0.587467
\(559\) 6.10318e36 0.496017
\(560\) 1.16394e35 0.00923407
\(561\) 6.25520e34 0.00484443
\(562\) 1.81811e36 0.137461
\(563\) −1.63003e37 −1.20319 −0.601593 0.798803i \(-0.705467\pi\)
−0.601593 + 0.798803i \(0.705467\pi\)
\(564\) −5.38043e35 −0.0387748
\(565\) 4.62052e36 0.325115
\(566\) 1.29822e36 0.0891920
\(567\) −1.17368e36 −0.0787368
\(568\) 3.17953e36 0.208286
\(569\) −1.16009e37 −0.742120 −0.371060 0.928609i \(-0.621006\pi\)
−0.371060 + 0.928609i \(0.621006\pi\)
\(570\) 2.25224e36 0.140703
\(571\) −1.57490e37 −0.960872 −0.480436 0.877030i \(-0.659522\pi\)
−0.480436 + 0.877030i \(0.659522\pi\)
\(572\) −7.80224e36 −0.464914
\(573\) −3.10188e36 −0.180525
\(574\) −2.55427e35 −0.0145197
\(575\) −4.60986e36 −0.255960
\(576\) −1.38093e37 −0.748974
\(577\) 1.14306e37 0.605611 0.302805 0.953052i \(-0.402077\pi\)
0.302805 + 0.953052i \(0.402077\pi\)
\(578\) 1.32636e37 0.686494
\(579\) −6.09099e36 −0.307984
\(580\) 6.42002e36 0.317146
\(581\) −1.31288e36 −0.0633647
\(582\) 3.61154e36 0.170307
\(583\) −1.43033e37 −0.659038
\(584\) −1.83093e37 −0.824326
\(585\) −1.45111e37 −0.638404
\(586\) 3.23411e36 0.139039
\(587\) 1.18730e37 0.498824 0.249412 0.968397i \(-0.419763\pi\)
0.249412 + 0.968397i \(0.419763\pi\)
\(588\) 3.82024e36 0.156854
\(589\) 3.56911e37 1.43220
\(590\) 5.97596e36 0.234372
\(591\) 6.74248e36 0.258457
\(592\) −2.94985e36 −0.110524
\(593\) −3.28233e37 −1.20211 −0.601054 0.799209i \(-0.705252\pi\)
−0.601054 + 0.799209i \(0.705252\pi\)
\(594\) 6.18092e36 0.221277
\(595\) −3.92056e34 −0.00137205
\(596\) −2.57207e37 −0.879950
\(597\) −1.26796e37 −0.424083
\(598\) 4.21982e37 1.37984
\(599\) 1.12882e37 0.360881 0.180441 0.983586i \(-0.442248\pi\)
0.180441 + 0.983586i \(0.442248\pi\)
\(600\) 2.01866e36 0.0630989
\(601\) −7.84746e36 −0.239841 −0.119921 0.992783i \(-0.538264\pi\)
−0.119921 + 0.992783i \(0.538264\pi\)
\(602\) 7.76149e35 0.0231949
\(603\) 3.74786e37 1.09522
\(604\) 2.83315e37 0.809600
\(605\) −1.09638e37 −0.306383
\(606\) 1.30220e37 0.355875
\(607\) −2.64580e37 −0.707148 −0.353574 0.935407i \(-0.615034\pi\)
−0.353574 + 0.935407i \(0.615034\pi\)
\(608\) 5.34511e37 1.39720
\(609\) 1.68642e36 0.0431155
\(610\) −1.86803e37 −0.467122
\(611\) 1.56790e37 0.383496
\(612\) 5.76617e35 0.0137957
\(613\) −7.75347e37 −1.81459 −0.907297 0.420490i \(-0.861858\pi\)
−0.907297 + 0.420490i \(0.861858\pi\)
\(614\) −1.09790e36 −0.0251357
\(615\) −1.18737e36 −0.0265934
\(616\) −2.87161e36 −0.0629197
\(617\) −2.09704e37 −0.449528 −0.224764 0.974413i \(-0.572161\pi\)
−0.224764 + 0.974413i \(0.572161\pi\)
\(618\) 1.00098e37 0.209933
\(619\) 6.54119e37 1.34225 0.671124 0.741345i \(-0.265812\pi\)
0.671124 + 0.741345i \(0.265812\pi\)
\(620\) 1.10533e37 0.221924
\(621\) 3.73876e37 0.734498
\(622\) −3.52083e37 −0.676822
\(623\) −4.57208e36 −0.0860052
\(624\) −4.95288e36 −0.0911729
\(625\) 2.22045e36 0.0400000
\(626\) −2.35844e37 −0.415787
\(627\) −1.48935e37 −0.256971
\(628\) −3.03548e36 −0.0512593
\(629\) 9.93612e35 0.0164223
\(630\) −1.84539e36 −0.0298532
\(631\) 6.54067e37 1.03568 0.517841 0.855477i \(-0.326736\pi\)
0.517841 + 0.855477i \(0.326736\pi\)
\(632\) −1.99534e37 −0.309269
\(633\) −1.85168e37 −0.280941
\(634\) −4.19817e37 −0.623527
\(635\) −5.35689e37 −0.778874
\(636\) 1.30909e37 0.186337
\(637\) −1.11324e38 −1.55134
\(638\) 3.79594e37 0.517893
\(639\) −1.35116e37 −0.180487
\(640\) 1.20117e37 0.157099
\(641\) −1.66460e37 −0.213170 −0.106585 0.994304i \(-0.533992\pi\)
−0.106585 + 0.994304i \(0.533992\pi\)
\(642\) −5.69085e36 −0.0713599
\(643\) −4.51668e37 −0.554588 −0.277294 0.960785i \(-0.589438\pi\)
−0.277294 + 0.960785i \(0.589438\pi\)
\(644\) −6.00182e36 −0.0721645
\(645\) 3.60799e36 0.0424824
\(646\) 2.60797e36 0.0300721
\(647\) 5.82402e36 0.0657681 0.0328840 0.999459i \(-0.489531\pi\)
0.0328840 + 0.999459i \(0.489531\pi\)
\(648\) 6.99766e37 0.773909
\(649\) −3.95175e37 −0.428042
\(650\) −2.03257e37 −0.215633
\(651\) 2.90350e36 0.0301702
\(652\) −8.31712e37 −0.846507
\(653\) 7.78394e37 0.776017 0.388009 0.921656i \(-0.373163\pi\)
0.388009 + 0.921656i \(0.373163\pi\)
\(654\) −1.40000e37 −0.136718
\(655\) 7.81750e37 0.747843
\(656\) 4.08186e36 0.0382522
\(657\) 7.78067e37 0.714307
\(658\) 1.99392e36 0.0179332
\(659\) 1.70605e38 1.50328 0.751638 0.659576i \(-0.229264\pi\)
0.751638 + 0.659576i \(0.229264\pi\)
\(660\) −4.61241e36 −0.0398185
\(661\) −1.16463e38 −0.985072 −0.492536 0.870292i \(-0.663930\pi\)
−0.492536 + 0.870292i \(0.663930\pi\)
\(662\) −7.49216e36 −0.0620903
\(663\) 1.66830e36 0.0135470
\(664\) 7.82759e37 0.622815
\(665\) 9.33477e36 0.0727799
\(666\) 4.67689e37 0.357318
\(667\) 2.29611e38 1.71907
\(668\) −4.91112e37 −0.360328
\(669\) −4.51055e37 −0.324322
\(670\) 5.24965e37 0.369931
\(671\) 1.23528e38 0.853123
\(672\) 4.34829e36 0.0294329
\(673\) −1.98332e38 −1.31580 −0.657898 0.753107i \(-0.728554\pi\)
−0.657898 + 0.753107i \(0.728554\pi\)
\(674\) −5.94914e37 −0.386853
\(675\) −1.80086e37 −0.114783
\(676\) −1.23588e38 −0.772140
\(677\) 1.56476e38 0.958299 0.479149 0.877733i \(-0.340945\pi\)
0.479149 + 0.877733i \(0.340945\pi\)
\(678\) −2.50038e37 −0.150108
\(679\) 1.49686e37 0.0880927
\(680\) 2.33750e36 0.0134860
\(681\) 4.61183e37 0.260848
\(682\) 6.53543e37 0.362398
\(683\) 1.36864e38 0.744061 0.372030 0.928221i \(-0.378662\pi\)
0.372030 + 0.928221i \(0.378662\pi\)
\(684\) −1.37291e38 −0.731787
\(685\) −1.49758e38 −0.782648
\(686\) −2.84779e37 −0.145926
\(687\) −9.51306e36 −0.0477974
\(688\) −1.24033e37 −0.0611072
\(689\) −3.81478e38 −1.84294
\(690\) 2.49461e37 0.118179
\(691\) −3.59443e38 −1.66984 −0.834922 0.550368i \(-0.814487\pi\)
−0.834922 + 0.550368i \(0.814487\pi\)
\(692\) 1.17849e38 0.536901
\(693\) 1.22031e37 0.0545221
\(694\) −1.69532e38 −0.742845
\(695\) 1.57197e38 0.675537
\(696\) −1.00547e38 −0.423785
\(697\) −1.37491e36 −0.00568372
\(698\) −2.98125e38 −1.20879
\(699\) 1.05813e38 0.420819
\(700\) 2.89092e36 0.0112775
\(701\) 2.57026e38 0.983519 0.491760 0.870731i \(-0.336354\pi\)
0.491760 + 0.870731i \(0.336354\pi\)
\(702\) 1.64849e38 0.618779
\(703\) −2.36577e38 −0.871114
\(704\) 1.27909e38 0.462029
\(705\) 9.26887e36 0.0328453
\(706\) −1.59403e38 −0.554158
\(707\) 5.39718e37 0.184079
\(708\) 3.61679e37 0.121025
\(709\) 1.31686e38 0.432330 0.216165 0.976357i \(-0.430645\pi\)
0.216165 + 0.976357i \(0.430645\pi\)
\(710\) −1.89258e37 −0.0609631
\(711\) 8.47934e37 0.267992
\(712\) 2.72594e38 0.845351
\(713\) 3.95320e38 1.20293
\(714\) 2.12160e35 0.000633487 0
\(715\) 1.34409e38 0.393820
\(716\) 2.54725e38 0.732395
\(717\) 7.80140e37 0.220122
\(718\) 7.49376e37 0.207501
\(719\) −4.19294e38 −1.13941 −0.569703 0.821851i \(-0.692942\pi\)
−0.569703 + 0.821851i \(0.692942\pi\)
\(720\) 2.94903e37 0.0786486
\(721\) 4.14872e37 0.108589
\(722\) −3.53498e38 −0.908102
\(723\) 1.50629e38 0.379787
\(724\) 1.40742e37 0.0348299
\(725\) −1.10598e38 −0.268648
\(726\) 5.93304e37 0.141460
\(727\) −5.58729e36 −0.0130764 −0.00653818 0.999979i \(-0.502081\pi\)
−0.00653818 + 0.999979i \(0.502081\pi\)
\(728\) −7.65878e37 −0.175949
\(729\) −2.20888e38 −0.498139
\(730\) 1.08984e38 0.241271
\(731\) 4.17785e36 0.00907964
\(732\) −1.13057e38 −0.241212
\(733\) 3.94722e37 0.0826776 0.0413388 0.999145i \(-0.486838\pi\)
0.0413388 + 0.999145i \(0.486838\pi\)
\(734\) 6.03470e38 1.24096
\(735\) −6.58112e37 −0.132868
\(736\) 5.92033e38 1.17353
\(737\) −3.47147e38 −0.675619
\(738\) −6.47166e37 −0.123667
\(739\) −7.95586e37 −0.149275 −0.0746376 0.997211i \(-0.523780\pi\)
−0.0746376 + 0.997211i \(0.523780\pi\)
\(740\) −7.32663e37 −0.134982
\(741\) −3.97219e38 −0.718595
\(742\) −4.85131e37 −0.0861799
\(743\) 4.62071e37 0.0806045 0.0403023 0.999188i \(-0.487168\pi\)
0.0403023 + 0.999188i \(0.487168\pi\)
\(744\) −1.73111e38 −0.296545
\(745\) 4.43090e38 0.745388
\(746\) −2.98986e38 −0.493943
\(747\) −3.32639e38 −0.539691
\(748\) −5.34092e36 −0.00851030
\(749\) −2.35867e37 −0.0369115
\(750\) −1.20159e37 −0.0184684
\(751\) 3.67388e38 0.554607 0.277303 0.960782i \(-0.410559\pi\)
0.277303 + 0.960782i \(0.410559\pi\)
\(752\) −3.18638e37 −0.0472451
\(753\) 6.87179e37 0.100078
\(754\) 1.01240e39 1.44824
\(755\) −4.88067e38 −0.685796
\(756\) −2.34464e37 −0.0323617
\(757\) −1.04932e39 −1.42271 −0.711354 0.702834i \(-0.751918\pi\)
−0.711354 + 0.702834i \(0.751918\pi\)
\(758\) −7.28880e38 −0.970782
\(759\) −1.64962e38 −0.215835
\(760\) −5.56554e38 −0.715358
\(761\) −5.58189e38 −0.704836 −0.352418 0.935843i \(-0.614641\pi\)
−0.352418 + 0.935843i \(0.614641\pi\)
\(762\) 2.89886e38 0.359613
\(763\) −5.80252e37 −0.0707188
\(764\) 2.64850e38 0.317131
\(765\) −9.93337e36 −0.0116860
\(766\) 1.04339e39 1.20602
\(767\) −1.05396e39 −1.19698
\(768\) −2.86740e38 −0.319972
\(769\) 6.88068e38 0.754442 0.377221 0.926123i \(-0.376880\pi\)
0.377221 + 0.926123i \(0.376880\pi\)
\(770\) 1.70930e37 0.0184159
\(771\) −1.72249e38 −0.182358
\(772\) 5.20072e38 0.541041
\(773\) 8.57010e38 0.876118 0.438059 0.898946i \(-0.355666\pi\)
0.438059 + 0.898946i \(0.355666\pi\)
\(774\) 1.96650e38 0.197556
\(775\) −1.90415e38 −0.187987
\(776\) −8.92453e38 −0.865869
\(777\) −1.92457e37 −0.0183506
\(778\) 5.75726e38 0.539500
\(779\) 3.27364e38 0.301491
\(780\) −1.23016e38 −0.111349
\(781\) 1.25152e38 0.111339
\(782\) 2.88862e37 0.0252580
\(783\) 8.96987e38 0.770908
\(784\) 2.26241e38 0.191119
\(785\) 5.22923e37 0.0434207
\(786\) −4.23042e38 −0.345286
\(787\) 2.50661e38 0.201107 0.100553 0.994932i \(-0.467939\pi\)
0.100553 + 0.994932i \(0.467939\pi\)
\(788\) −5.75698e38 −0.454036
\(789\) 7.49685e38 0.581218
\(790\) 1.18771e38 0.0905197
\(791\) −1.03632e38 −0.0776449
\(792\) −7.27569e38 −0.535901
\(793\) 3.29457e39 2.38567
\(794\) −1.49058e39 −1.06115
\(795\) −2.25517e38 −0.157842
\(796\) 1.08263e39 0.744994
\(797\) 2.39896e39 1.62306 0.811531 0.584309i \(-0.198634\pi\)
0.811531 + 0.584309i \(0.198634\pi\)
\(798\) −5.05148e37 −0.0336031
\(799\) 1.07328e37 0.00701994
\(800\) −2.85167e38 −0.183393
\(801\) −1.15841e39 −0.732525
\(802\) −1.16224e39 −0.722673
\(803\) −7.20687e38 −0.440643
\(804\) 3.17721e38 0.191025
\(805\) 1.03393e38 0.0611291
\(806\) 1.74304e39 1.01341
\(807\) −6.69073e37 −0.0382544
\(808\) −3.21788e39 −1.80933
\(809\) 1.25830e39 0.695791 0.347896 0.937533i \(-0.386896\pi\)
0.347896 + 0.937533i \(0.386896\pi\)
\(810\) −4.16528e38 −0.226515
\(811\) −3.51523e39 −1.88006 −0.940030 0.341092i \(-0.889203\pi\)
−0.940030 + 0.341092i \(0.889203\pi\)
\(812\) −1.43993e38 −0.0757417
\(813\) −5.48481e38 −0.283752
\(814\) −4.33198e38 −0.220423
\(815\) 1.43279e39 0.717059
\(816\) −3.39043e36 −0.00166893
\(817\) −9.94737e38 −0.481627
\(818\) −1.19856e39 −0.570807
\(819\) 3.25465e38 0.152466
\(820\) 1.01382e38 0.0467171
\(821\) 1.79305e39 0.812755 0.406377 0.913705i \(-0.366792\pi\)
0.406377 + 0.913705i \(0.366792\pi\)
\(822\) 8.10410e38 0.361356
\(823\) 3.11454e37 0.0136614 0.00683071 0.999977i \(-0.497826\pi\)
0.00683071 + 0.999977i \(0.497826\pi\)
\(824\) −2.47353e39 −1.06733
\(825\) 7.94581e37 0.0337295
\(826\) −1.34033e38 −0.0559734
\(827\) −1.45069e39 −0.596005 −0.298003 0.954565i \(-0.596320\pi\)
−0.298003 + 0.954565i \(0.596320\pi\)
\(828\) −1.52066e39 −0.614641
\(829\) −4.64859e39 −1.84857 −0.924283 0.381707i \(-0.875336\pi\)
−0.924283 + 0.381707i \(0.875336\pi\)
\(830\) −4.65929e38 −0.182291
\(831\) 6.21510e38 0.239240
\(832\) 3.41141e39 1.29202
\(833\) −7.62058e37 −0.0283975
\(834\) −8.50666e38 −0.311902
\(835\) 8.46039e38 0.305227
\(836\) 1.27166e39 0.451426
\(837\) 1.54434e39 0.539446
\(838\) 2.78309e39 0.956604
\(839\) 6.25511e38 0.211567 0.105784 0.994389i \(-0.466265\pi\)
0.105784 + 0.994389i \(0.466265\pi\)
\(840\) −4.52761e37 −0.0150695
\(841\) 2.45560e39 0.804290
\(842\) 1.08902e39 0.351011
\(843\) −1.89559e38 −0.0601275
\(844\) 1.58103e39 0.493535
\(845\) 2.12905e39 0.654064
\(846\) 5.05190e38 0.152741
\(847\) 2.45905e38 0.0731713
\(848\) 7.75265e38 0.227042
\(849\) −1.35355e38 −0.0390139
\(850\) −1.39137e37 −0.00394719
\(851\) −2.62036e39 −0.731664
\(852\) −1.14543e38 −0.0314801
\(853\) −1.68444e39 −0.455662 −0.227831 0.973701i \(-0.573163\pi\)
−0.227831 + 0.973701i \(0.573163\pi\)
\(854\) 4.18975e38 0.111560
\(855\) 2.36511e39 0.619882
\(856\) 1.40627e39 0.362806
\(857\) 4.47378e39 1.13614 0.568072 0.822979i \(-0.307689\pi\)
0.568072 + 0.822979i \(0.307689\pi\)
\(858\) −7.27351e38 −0.181830
\(859\) −2.74806e39 −0.676267 −0.338134 0.941098i \(-0.609796\pi\)
−0.338134 + 0.941098i \(0.609796\pi\)
\(860\) −3.08064e38 −0.0746296
\(861\) 2.66313e37 0.00635111
\(862\) 3.89806e38 0.0915166
\(863\) −4.77950e39 −1.10468 −0.552341 0.833618i \(-0.686265\pi\)
−0.552341 + 0.833618i \(0.686265\pi\)
\(864\) 2.31280e39 0.526263
\(865\) −2.03019e39 −0.454798
\(866\) 3.51017e39 0.774171
\(867\) −1.38289e39 −0.300282
\(868\) −2.47912e38 −0.0530006
\(869\) −7.85401e38 −0.165320
\(870\) 5.98496e38 0.124037
\(871\) −9.25863e39 −1.88930
\(872\) 3.45955e39 0.695100
\(873\) 3.79254e39 0.750305
\(874\) −6.87774e39 −1.33980
\(875\) −4.98018e37 −0.00955293
\(876\) 6.59599e38 0.124587
\(877\) −2.09448e39 −0.389566 −0.194783 0.980846i \(-0.562400\pi\)
−0.194783 + 0.980846i \(0.562400\pi\)
\(878\) 2.54547e39 0.466221
\(879\) −3.37194e38 −0.0608178
\(880\) −2.73155e38 −0.0485169
\(881\) −3.84071e39 −0.671796 −0.335898 0.941898i \(-0.609040\pi\)
−0.335898 + 0.941898i \(0.609040\pi\)
\(882\) −3.58697e39 −0.617877
\(883\) 5.26228e39 0.892697 0.446349 0.894859i \(-0.352724\pi\)
0.446349 + 0.894859i \(0.352724\pi\)
\(884\) −1.42446e38 −0.0237982
\(885\) −6.23064e38 −0.102518
\(886\) −1.83372e39 −0.297151
\(887\) 1.78125e38 0.0284287 0.0142143 0.999899i \(-0.495475\pi\)
0.0142143 + 0.999899i \(0.495475\pi\)
\(888\) 1.14746e39 0.180369
\(889\) 1.20148e39 0.186013
\(890\) −1.62259e39 −0.247425
\(891\) 2.75440e39 0.413693
\(892\) 3.85127e39 0.569743
\(893\) −2.55547e39 −0.372371
\(894\) −2.39777e39 −0.344152
\(895\) −4.38814e39 −0.620397
\(896\) −2.69407e38 −0.0375190
\(897\) −4.39966e39 −0.603560
\(898\) −6.43876e39 −0.870105
\(899\) 9.48435e39 1.26256
\(900\) 7.32460e38 0.0960528
\(901\) −2.61136e38 −0.0337351
\(902\) 5.99439e38 0.0762881
\(903\) −8.09227e37 −0.0101458
\(904\) 6.17873e39 0.763177
\(905\) −2.42455e38 −0.0295037
\(906\) 2.64116e39 0.316639
\(907\) −4.51361e39 −0.533121 −0.266561 0.963818i \(-0.585887\pi\)
−0.266561 + 0.963818i \(0.585887\pi\)
\(908\) −3.93775e39 −0.458236
\(909\) 1.36746e40 1.56784
\(910\) 4.55881e38 0.0514983
\(911\) −3.61144e38 −0.0401960 −0.0200980 0.999798i \(-0.506398\pi\)
−0.0200980 + 0.999798i \(0.506398\pi\)
\(912\) 8.07254e38 0.0885278
\(913\) 3.08108e39 0.332925
\(914\) 3.59638e39 0.382905
\(915\) 1.94764e39 0.204326
\(916\) 8.12260e38 0.0839666
\(917\) −1.75337e39 −0.178602
\(918\) 1.12845e38 0.0113268
\(919\) −1.21557e40 −1.20233 −0.601163 0.799126i \(-0.705296\pi\)
−0.601163 + 0.799126i \(0.705296\pi\)
\(920\) −6.16447e39 −0.600842
\(921\) 1.14469e38 0.0109947
\(922\) −9.93954e39 −0.940803
\(923\) 3.33788e39 0.311349
\(924\) 1.03451e38 0.00950959
\(925\) 1.26216e39 0.114341
\(926\) 2.87061e39 0.256286
\(927\) 1.05114e40 0.924881
\(928\) 1.42038e40 1.23171
\(929\) 7.00396e39 0.598593 0.299296 0.954160i \(-0.403248\pi\)
0.299296 + 0.954160i \(0.403248\pi\)
\(930\) 1.03043e39 0.0867955
\(931\) 1.81444e40 1.50634
\(932\) −9.03468e39 −0.739261
\(933\) 3.67088e39 0.296052
\(934\) 1.48320e40 1.17901
\(935\) 9.20081e37 0.00720890
\(936\) −1.94047e40 −1.49859
\(937\) −1.68667e40 −1.28394 −0.641971 0.766729i \(-0.721883\pi\)
−0.641971 + 0.766729i \(0.721883\pi\)
\(938\) −1.17743e39 −0.0883481
\(939\) 2.45895e39 0.181871
\(940\) −7.91411e38 −0.0577000
\(941\) 8.43439e39 0.606169 0.303085 0.952964i \(-0.401983\pi\)
0.303085 + 0.952964i \(0.401983\pi\)
\(942\) −2.82978e38 −0.0200477
\(943\) 3.62593e39 0.253228
\(944\) 2.14192e39 0.147463
\(945\) 4.03911e38 0.0274129
\(946\) −1.82147e39 −0.121869
\(947\) −2.44897e40 −1.61532 −0.807659 0.589649i \(-0.799266\pi\)
−0.807659 + 0.589649i \(0.799266\pi\)
\(948\) 7.18827e38 0.0467425
\(949\) −1.92212e40 −1.23221
\(950\) 3.31283e39 0.209378
\(951\) 4.37709e39 0.272740
\(952\) −5.24272e37 −0.00322076
\(953\) −1.31618e40 −0.797191 −0.398596 0.917127i \(-0.630502\pi\)
−0.398596 + 0.917127i \(0.630502\pi\)
\(954\) −1.22916e40 −0.734013
\(955\) −4.56257e39 −0.268636
\(956\) −6.66113e39 −0.386693
\(957\) −3.95771e39 −0.226534
\(958\) 7.61597e39 0.429824
\(959\) 3.35888e39 0.186915
\(960\) 2.01671e39 0.110657
\(961\) −2.15358e39 −0.116519
\(962\) −1.15537e40 −0.616391
\(963\) −5.97606e39 −0.314384
\(964\) −1.28612e40 −0.667179
\(965\) −8.95928e39 −0.458305
\(966\) −5.59510e38 −0.0282239
\(967\) 3.47387e40 1.72805 0.864025 0.503449i \(-0.167936\pi\)
0.864025 + 0.503449i \(0.167936\pi\)
\(968\) −1.46612e40 −0.719205
\(969\) −2.71911e38 −0.0131539
\(970\) 5.31223e39 0.253430
\(971\) 3.09835e40 1.45771 0.728854 0.684669i \(-0.240053\pi\)
0.728854 + 0.684669i \(0.240053\pi\)
\(972\) −9.05124e39 −0.419966
\(973\) −3.52573e39 −0.161334
\(974\) 1.25015e40 0.564180
\(975\) 2.11920e39 0.0943212
\(976\) −6.69544e39 −0.293905
\(977\) 3.13699e40 1.35812 0.679058 0.734084i \(-0.262388\pi\)
0.679058 + 0.734084i \(0.262388\pi\)
\(978\) −7.75350e39 −0.331073
\(979\) 1.07298e40 0.451882
\(980\) 5.61921e39 0.233412
\(981\) −1.47016e40 −0.602328
\(982\) 1.24157e40 0.501725
\(983\) 7.16907e39 0.285752 0.142876 0.989741i \(-0.454365\pi\)
0.142876 + 0.989741i \(0.454365\pi\)
\(984\) −1.58780e39 −0.0624255
\(985\) 9.91754e39 0.384605
\(986\) 6.93026e38 0.0265101
\(987\) −2.07889e38 −0.00784423
\(988\) 3.39160e40 1.26237
\(989\) −1.10179e40 −0.404527
\(990\) 4.33078e39 0.156852
\(991\) −1.14112e40 −0.407698 −0.203849 0.979002i \(-0.565345\pi\)
−0.203849 + 0.979002i \(0.565345\pi\)
\(992\) 2.44546e40 0.861891
\(993\) 7.81145e38 0.0271592
\(994\) 4.24482e38 0.0145594
\(995\) −1.86504e40 −0.631070
\(996\) −2.81991e39 −0.0941314
\(997\) 1.49086e40 0.490968 0.245484 0.969401i \(-0.421053\pi\)
0.245484 + 0.969401i \(0.421053\pi\)
\(998\) −1.39773e40 −0.454111
\(999\) −1.02366e40 −0.328110
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 5.28.a.a.1.2 4
3.2 odd 2 45.28.a.b.1.3 4
5.2 odd 4 25.28.b.b.24.3 8
5.3 odd 4 25.28.b.b.24.6 8
5.4 even 2 25.28.a.b.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.28.a.a.1.2 4 1.1 even 1 trivial
25.28.a.b.1.3 4 5.4 even 2
25.28.b.b.24.3 8 5.2 odd 4
25.28.b.b.24.6 8 5.3 odd 4
45.28.a.b.1.3 4 3.2 odd 2