Properties

Label 547.8.a.a.1.8
Level $547$
Weight $8$
Character 547.1
Self dual yes
Analytic conductor $170.875$
Analytic rank $1$
Dimension $156$
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] = [547,8,Mod(1,547)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("547.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 547.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: \(170.874608940\)
Analytic rank: \(1\)
Dimension: \(156\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 547.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-20.7147 q^{2} +73.8020 q^{3} +301.098 q^{4} +73.8971 q^{5} -1528.79 q^{6} +1490.12 q^{7} -3585.68 q^{8} +3259.74 q^{9} +O(q^{10})\) \(q-20.7147 q^{2} +73.8020 q^{3} +301.098 q^{4} +73.8971 q^{5} -1528.79 q^{6} +1490.12 q^{7} -3585.68 q^{8} +3259.74 q^{9} -1530.76 q^{10} -3255.54 q^{11} +22221.7 q^{12} -9643.42 q^{13} -30867.4 q^{14} +5453.76 q^{15} +35735.7 q^{16} +23212.3 q^{17} -67524.5 q^{18} -26830.9 q^{19} +22250.3 q^{20} +109974. q^{21} +67437.6 q^{22} -96604.9 q^{23} -264631. q^{24} -72664.2 q^{25} +199760. q^{26} +79170.4 q^{27} +448673. q^{28} -43668.4 q^{29} -112973. q^{30} +84751.2 q^{31} -281286. q^{32} -240266. q^{33} -480836. q^{34} +110116. q^{35} +981503. q^{36} +236077. q^{37} +555793. q^{38} -711704. q^{39} -264971. q^{40} -539942. q^{41} -2.27808e6 q^{42} +481095. q^{43} -980239. q^{44} +240885. q^{45} +2.00114e6 q^{46} +93118.1 q^{47} +2.63737e6 q^{48} +1.39692e6 q^{49} +1.50522e6 q^{50} +1.71312e6 q^{51} -2.90362e6 q^{52} +1.53295e6 q^{53} -1.63999e6 q^{54} -240575. q^{55} -5.34310e6 q^{56} -1.98017e6 q^{57} +904578. q^{58} -2.65658e6 q^{59} +1.64212e6 q^{60} -1.48574e6 q^{61} -1.75559e6 q^{62} +4.85741e6 q^{63} +1.25259e6 q^{64} -712621. q^{65} +4.97703e6 q^{66} -2.58945e6 q^{67} +6.98919e6 q^{68} -7.12964e6 q^{69} -2.28101e6 q^{70} -1.00423e6 q^{71} -1.16884e7 q^{72} -2.40816e6 q^{73} -4.89025e6 q^{74} -5.36277e6 q^{75} -8.07873e6 q^{76} -4.85116e6 q^{77} +1.47427e7 q^{78} -5.50752e6 q^{79} +2.64076e6 q^{80} -1.28612e6 q^{81} +1.11847e7 q^{82} -1.52341e6 q^{83} +3.31130e7 q^{84} +1.71532e6 q^{85} -9.96574e6 q^{86} -3.22282e6 q^{87} +1.16733e7 q^{88} -1.23720e7 q^{89} -4.98987e6 q^{90} -1.43699e7 q^{91} -2.90876e7 q^{92} +6.25481e6 q^{93} -1.92891e6 q^{94} -1.98272e6 q^{95} -2.07595e7 q^{96} +1.34239e7 q^{97} -2.89368e7 q^{98} -1.06122e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 156 q - 56 q^{2} - 284 q^{3} + 9690 q^{4} - 3751 q^{5} - 2322 q^{6} - 2559 q^{7} - 10752 q^{8} + 102594 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 156 q - 56 q^{2} - 284 q^{3} + 9690 q^{4} - 3751 q^{5} - 2322 q^{6} - 2559 q^{7} - 10752 q^{8} + 102594 q^{9} - 10570 q^{10} - 20090 q^{11} - 58311 q^{12} - 63021 q^{13} - 45057 q^{14} - 36391 q^{15} + 574338 q^{16} - 232394 q^{17} - 92277 q^{18} - 43100 q^{19} - 485568 q^{20} - 231868 q^{21} - 225008 q^{22} - 401950 q^{23} - 503569 q^{24} + 2076291 q^{25} - 530768 q^{26} - 959873 q^{27} - 617816 q^{28} - 1275618 q^{29} - 778474 q^{30} - 485945 q^{31} - 1903692 q^{32} - 1050846 q^{33} - 466263 q^{34} - 1826209 q^{35} + 5276156 q^{36} - 2129902 q^{37} - 2480555 q^{38} - 974653 q^{39} - 937648 q^{40} - 2309325 q^{41} - 2803500 q^{42} - 1756918 q^{43} - 3314520 q^{44} - 7492064 q^{45} - 1323786 q^{46} - 6203828 q^{47} - 7957494 q^{48} + 15095175 q^{49} - 5758152 q^{50} - 1556293 q^{51} - 7587898 q^{52} - 13775068 q^{53} - 6848423 q^{54} - 4045669 q^{55} - 8326655 q^{56} - 9421556 q^{57} - 4938892 q^{58} - 7755758 q^{59} - 5358502 q^{60} - 11693582 q^{61} - 14895366 q^{62} - 9477805 q^{63} + 31311690 q^{64} - 15629670 q^{65} - 5969892 q^{66} - 9560716 q^{67} - 34045735 q^{68} - 17825946 q^{69} - 4291177 q^{70} - 13661197 q^{71} - 21516953 q^{72} - 17125972 q^{73} - 19749599 q^{74} - 21752079 q^{75} - 15479244 q^{76} - 55632329 q^{77} - 12746879 q^{78} - 9534338 q^{79} - 61267539 q^{80} + 58468208 q^{81} - 29265046 q^{82} - 38447793 q^{83} - 33520873 q^{84} - 22365109 q^{85} - 21208733 q^{86} - 27018273 q^{87} - 40855385 q^{88} - 62436196 q^{89} - 19477679 q^{90} - 20640165 q^{91} - 78867734 q^{92} - 77801528 q^{93} + 2996793 q^{94} - 30557422 q^{95} - 82397286 q^{96} - 56264748 q^{97} - 72954494 q^{98} - 43444577 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −20.7147 −1.83094 −0.915469 0.402389i \(-0.868180\pi\)
−0.915469 + 0.402389i \(0.868180\pi\)
\(3\) 73.8020 1.57813 0.789067 0.614307i \(-0.210564\pi\)
0.789067 + 0.614307i \(0.210564\pi\)
\(4\) 301.098 2.35233
\(5\) 73.8971 0.264382 0.132191 0.991224i \(-0.457799\pi\)
0.132191 + 0.991224i \(0.457799\pi\)
\(6\) −1528.79 −2.88946
\(7\) 1490.12 1.64202 0.821011 0.570913i \(-0.193411\pi\)
0.821011 + 0.570913i \(0.193411\pi\)
\(8\) −3585.68 −2.47603
\(9\) 3259.74 1.49051
\(10\) −1530.76 −0.484067
\(11\) −3255.54 −0.737478 −0.368739 0.929533i \(-0.620210\pi\)
−0.368739 + 0.929533i \(0.620210\pi\)
\(12\) 22221.7 3.71229
\(13\) −9643.42 −1.21739 −0.608694 0.793405i \(-0.708306\pi\)
−0.608694 + 0.793405i \(0.708306\pi\)
\(14\) −30867.4 −3.00644
\(15\) 5453.76 0.417231
\(16\) 35735.7 2.18113
\(17\) 23212.3 1.14590 0.572951 0.819590i \(-0.305799\pi\)
0.572951 + 0.819590i \(0.305799\pi\)
\(18\) −67524.5 −2.72903
\(19\) −26830.9 −0.897423 −0.448711 0.893677i \(-0.648117\pi\)
−0.448711 + 0.893677i \(0.648117\pi\)
\(20\) 22250.3 0.621915
\(21\) 109974. 2.59133
\(22\) 67437.6 1.35028
\(23\) −96604.9 −1.65559 −0.827793 0.561034i \(-0.810404\pi\)
−0.827793 + 0.561034i \(0.810404\pi\)
\(24\) −264631. −3.90751
\(25\) −72664.2 −0.930102
\(26\) 199760. 2.22896
\(27\) 79170.4 0.774086
\(28\) 448673. 3.86258
\(29\) −43668.4 −0.332487 −0.166243 0.986085i \(-0.553164\pi\)
−0.166243 + 0.986085i \(0.553164\pi\)
\(30\) −112973. −0.763923
\(31\) 84751.2 0.510952 0.255476 0.966815i \(-0.417768\pi\)
0.255476 + 0.966815i \(0.417768\pi\)
\(32\) −281286. −1.51748
\(33\) −240266. −1.16384
\(34\) −480836. −2.09807
\(35\) 110116. 0.434121
\(36\) 981503. 3.50617
\(37\) 236077. 0.766208 0.383104 0.923705i \(-0.374855\pi\)
0.383104 + 0.923705i \(0.374855\pi\)
\(38\) 555793. 1.64312
\(39\) −711704. −1.92120
\(40\) −264971. −0.654620
\(41\) −539942. −1.22350 −0.611749 0.791052i \(-0.709534\pi\)
−0.611749 + 0.791052i \(0.709534\pi\)
\(42\) −2.27808e6 −4.74456
\(43\) 481095. 0.922766 0.461383 0.887201i \(-0.347353\pi\)
0.461383 + 0.887201i \(0.347353\pi\)
\(44\) −980239. −1.73479
\(45\) 240885. 0.394064
\(46\) 2.00114e6 3.03127
\(47\) 93118.1 0.130825 0.0654127 0.997858i \(-0.479164\pi\)
0.0654127 + 0.997858i \(0.479164\pi\)
\(48\) 2.63737e6 3.44212
\(49\) 1.39692e6 1.69623
\(50\) 1.50522e6 1.70296
\(51\) 1.71312e6 1.80839
\(52\) −2.90362e6 −2.86370
\(53\) 1.53295e6 1.41437 0.707185 0.707028i \(-0.249965\pi\)
0.707185 + 0.707028i \(0.249965\pi\)
\(54\) −1.63999e6 −1.41730
\(55\) −240575. −0.194976
\(56\) −5.34310e6 −4.06570
\(57\) −1.98017e6 −1.41625
\(58\) 904578. 0.608763
\(59\) −2.65658e6 −1.68400 −0.841998 0.539480i \(-0.818621\pi\)
−0.841998 + 0.539480i \(0.818621\pi\)
\(60\) 1.64212e6 0.981465
\(61\) −1.48574e6 −0.838087 −0.419044 0.907966i \(-0.637635\pi\)
−0.419044 + 0.907966i \(0.637635\pi\)
\(62\) −1.75559e6 −0.935521
\(63\) 4.85741e6 2.44744
\(64\) 1.25259e6 0.597283
\(65\) −712621. −0.321856
\(66\) 4.97703e6 2.13092
\(67\) −2.58945e6 −1.05183 −0.525916 0.850537i \(-0.676277\pi\)
−0.525916 + 0.850537i \(0.676277\pi\)
\(68\) 6.98919e6 2.69554
\(69\) −7.12964e6 −2.61274
\(70\) −2.28101e6 −0.794849
\(71\) −1.00423e6 −0.332987 −0.166494 0.986043i \(-0.553245\pi\)
−0.166494 + 0.986043i \(0.553245\pi\)
\(72\) −1.16884e7 −3.69055
\(73\) −2.40816e6 −0.724529 −0.362264 0.932075i \(-0.617996\pi\)
−0.362264 + 0.932075i \(0.617996\pi\)
\(74\) −4.89025e6 −1.40288
\(75\) −5.36277e6 −1.46783
\(76\) −8.07873e6 −2.11104
\(77\) −4.85116e6 −1.21095
\(78\) 1.47427e7 3.51760
\(79\) −5.50752e6 −1.25679 −0.628393 0.777896i \(-0.716287\pi\)
−0.628393 + 0.777896i \(0.716287\pi\)
\(80\) 2.64076e6 0.576653
\(81\) −1.28612e6 −0.268895
\(82\) 1.11847e7 2.24015
\(83\) −1.52341e6 −0.292444 −0.146222 0.989252i \(-0.546711\pi\)
−0.146222 + 0.989252i \(0.546711\pi\)
\(84\) 3.31130e7 6.09567
\(85\) 1.71532e6 0.302956
\(86\) −9.96574e6 −1.68953
\(87\) −3.22282e6 −0.524709
\(88\) 1.16733e7 1.82602
\(89\) −1.23720e7 −1.86026 −0.930130 0.367230i \(-0.880306\pi\)
−0.930130 + 0.367230i \(0.880306\pi\)
\(90\) −4.98987e6 −0.721506
\(91\) −1.43699e7 −1.99898
\(92\) −2.90876e7 −3.89449
\(93\) 6.25481e6 0.806350
\(94\) −1.92891e6 −0.239533
\(95\) −1.98272e6 −0.237263
\(96\) −2.07595e7 −2.39479
\(97\) 1.34239e7 1.49340 0.746702 0.665159i \(-0.231636\pi\)
0.746702 + 0.665159i \(0.231636\pi\)
\(98\) −2.89368e7 −3.10570
\(99\) −1.06122e7 −1.09922
\(100\) −2.18791e7 −2.18791
\(101\) 7.79543e6 0.752862 0.376431 0.926445i \(-0.377151\pi\)
0.376431 + 0.926445i \(0.377151\pi\)
\(102\) −3.54867e7 −3.31104
\(103\) −5.82875e6 −0.525588 −0.262794 0.964852i \(-0.584644\pi\)
−0.262794 + 0.964852i \(0.584644\pi\)
\(104\) 3.45782e7 3.01430
\(105\) 8.12676e6 0.685102
\(106\) −3.17546e7 −2.58962
\(107\) −1.11500e7 −0.879897 −0.439949 0.898023i \(-0.645003\pi\)
−0.439949 + 0.898023i \(0.645003\pi\)
\(108\) 2.38381e7 1.82091
\(109\) 1.96029e7 1.44986 0.724932 0.688821i \(-0.241871\pi\)
0.724932 + 0.688821i \(0.241871\pi\)
\(110\) 4.98344e6 0.356989
\(111\) 1.74229e7 1.20918
\(112\) 5.32505e7 3.58147
\(113\) −748612. −0.0488070 −0.0244035 0.999702i \(-0.507769\pi\)
−0.0244035 + 0.999702i \(0.507769\pi\)
\(114\) 4.10187e7 2.59307
\(115\) −7.13882e6 −0.437708
\(116\) −1.31485e7 −0.782120
\(117\) −3.14350e7 −1.81453
\(118\) 5.50303e7 3.08329
\(119\) 3.45892e7 1.88159
\(120\) −1.95554e7 −1.03308
\(121\) −8.88860e6 −0.456126
\(122\) 3.07767e7 1.53448
\(123\) −3.98488e7 −1.93084
\(124\) 2.55185e7 1.20193
\(125\) −1.11429e7 −0.510285
\(126\) −1.00620e8 −4.48112
\(127\) −1.06282e7 −0.460413 −0.230207 0.973142i \(-0.573940\pi\)
−0.230207 + 0.973142i \(0.573940\pi\)
\(128\) 1.00576e7 0.423895
\(129\) 3.55058e7 1.45625
\(130\) 1.47617e7 0.589298
\(131\) −3.82747e7 −1.48752 −0.743760 0.668447i \(-0.766959\pi\)
−0.743760 + 0.668447i \(0.766959\pi\)
\(132\) −7.23436e7 −2.73774
\(133\) −3.99813e7 −1.47359
\(134\) 5.36397e7 1.92584
\(135\) 5.85046e6 0.204655
\(136\) −8.32320e7 −2.83729
\(137\) 6.16888e6 0.204967 0.102484 0.994735i \(-0.467321\pi\)
0.102484 + 0.994735i \(0.467321\pi\)
\(138\) 1.47688e8 4.78376
\(139\) 5.42813e7 1.71435 0.857173 0.515029i \(-0.172219\pi\)
0.857173 + 0.515029i \(0.172219\pi\)
\(140\) 3.31557e7 1.02120
\(141\) 6.87231e6 0.206460
\(142\) 2.08022e7 0.609679
\(143\) 3.13946e7 0.897798
\(144\) 1.16489e8 3.25099
\(145\) −3.22697e6 −0.0879037
\(146\) 4.98843e7 1.32657
\(147\) 1.03096e8 2.67688
\(148\) 7.10823e7 1.80238
\(149\) −5.48491e7 −1.35837 −0.679184 0.733968i \(-0.737666\pi\)
−0.679184 + 0.733968i \(0.737666\pi\)
\(150\) 1.11088e8 2.68750
\(151\) 4.78971e7 1.13211 0.566057 0.824366i \(-0.308468\pi\)
0.566057 + 0.824366i \(0.308468\pi\)
\(152\) 9.62069e7 2.22205
\(153\) 7.56661e7 1.70797
\(154\) 1.00490e8 2.21718
\(155\) 6.26287e6 0.135087
\(156\) −2.14293e8 −4.51931
\(157\) −7.96894e7 −1.64343 −0.821716 0.569897i \(-0.806983\pi\)
−0.821716 + 0.569897i \(0.806983\pi\)
\(158\) 1.14087e8 2.30110
\(159\) 1.13135e8 2.23207
\(160\) −2.07862e7 −0.401196
\(161\) −1.43953e8 −2.71851
\(162\) 2.66415e7 0.492330
\(163\) 5.76508e7 1.04267 0.521337 0.853351i \(-0.325433\pi\)
0.521337 + 0.853351i \(0.325433\pi\)
\(164\) −1.62576e8 −2.87807
\(165\) −1.77549e7 −0.307699
\(166\) 3.15569e7 0.535446
\(167\) 1.91556e6 0.0318264 0.0159132 0.999873i \(-0.494934\pi\)
0.0159132 + 0.999873i \(0.494934\pi\)
\(168\) −3.94332e8 −6.41622
\(169\) 3.02470e7 0.482036
\(170\) −3.55324e7 −0.554694
\(171\) −8.74616e7 −1.33762
\(172\) 1.44857e8 2.17065
\(173\) −8.14785e7 −1.19642 −0.598208 0.801341i \(-0.704120\pi\)
−0.598208 + 0.801341i \(0.704120\pi\)
\(174\) 6.67597e7 0.960709
\(175\) −1.08279e8 −1.52725
\(176\) −1.16339e8 −1.60854
\(177\) −1.96061e8 −2.65757
\(178\) 2.56282e8 3.40602
\(179\) −3.48393e7 −0.454029 −0.227015 0.973891i \(-0.572897\pi\)
−0.227015 + 0.973891i \(0.572897\pi\)
\(180\) 7.25302e7 0.926969
\(181\) 7.40560e7 0.928293 0.464147 0.885758i \(-0.346361\pi\)
0.464147 + 0.885758i \(0.346361\pi\)
\(182\) 2.97668e8 3.66000
\(183\) −1.09651e8 −1.32261
\(184\) 3.46394e8 4.09929
\(185\) 1.74454e7 0.202572
\(186\) −1.29566e8 −1.47638
\(187\) −7.55687e7 −0.845077
\(188\) 2.80377e7 0.307745
\(189\) 1.17974e8 1.27107
\(190\) 4.10715e7 0.434413
\(191\) 1.87132e8 1.94326 0.971632 0.236499i \(-0.0760001\pi\)
0.971632 + 0.236499i \(0.0760001\pi\)
\(192\) 9.24439e7 0.942593
\(193\) 3.37092e7 0.337519 0.168759 0.985657i \(-0.446024\pi\)
0.168759 + 0.985657i \(0.446024\pi\)
\(194\) −2.78072e8 −2.73433
\(195\) −5.25929e7 −0.507932
\(196\) 4.20611e8 3.99010
\(197\) 9.49367e7 0.884713 0.442356 0.896839i \(-0.354143\pi\)
0.442356 + 0.896839i \(0.354143\pi\)
\(198\) 2.19829e8 2.01260
\(199\) 1.01890e8 0.916524 0.458262 0.888817i \(-0.348472\pi\)
0.458262 + 0.888817i \(0.348472\pi\)
\(200\) 2.60551e8 2.30296
\(201\) −1.91107e8 −1.65993
\(202\) −1.61480e8 −1.37844
\(203\) −6.50713e7 −0.545951
\(204\) 5.15817e8 4.25392
\(205\) −3.99001e7 −0.323471
\(206\) 1.20741e8 0.962319
\(207\) −3.14907e8 −2.46766
\(208\) −3.44614e8 −2.65529
\(209\) 8.73490e7 0.661830
\(210\) −1.68343e8 −1.25438
\(211\) −1.09208e8 −0.800321 −0.400161 0.916445i \(-0.631046\pi\)
−0.400161 + 0.916445i \(0.631046\pi\)
\(212\) 4.61570e8 3.32707
\(213\) −7.41140e7 −0.525498
\(214\) 2.30969e8 1.61104
\(215\) 3.55515e7 0.243963
\(216\) −2.83880e8 −1.91666
\(217\) 1.26290e8 0.838994
\(218\) −4.06068e8 −2.65461
\(219\) −1.77727e8 −1.14340
\(220\) −7.24368e7 −0.458649
\(221\) −2.23846e8 −1.39501
\(222\) −3.60911e8 −2.21393
\(223\) 2.85379e8 1.72327 0.861637 0.507524i \(-0.169439\pi\)
0.861637 + 0.507524i \(0.169439\pi\)
\(224\) −4.19151e8 −2.49174
\(225\) −2.36866e8 −1.38632
\(226\) 1.55073e7 0.0893625
\(227\) −6.21605e7 −0.352715 −0.176357 0.984326i \(-0.556431\pi\)
−0.176357 + 0.984326i \(0.556431\pi\)
\(228\) −5.96227e8 −3.33150
\(229\) −3.96137e7 −0.217982 −0.108991 0.994043i \(-0.534762\pi\)
−0.108991 + 0.994043i \(0.534762\pi\)
\(230\) 1.47879e8 0.801415
\(231\) −3.58025e8 −1.91105
\(232\) 1.56581e8 0.823249
\(233\) −2.98931e7 −0.154819 −0.0774096 0.996999i \(-0.524665\pi\)
−0.0774096 + 0.996999i \(0.524665\pi\)
\(234\) 6.51167e8 3.32229
\(235\) 6.88116e6 0.0345879
\(236\) −7.99893e8 −3.96132
\(237\) −4.06466e8 −1.98338
\(238\) −7.16504e8 −3.44508
\(239\) −3.10037e8 −1.46900 −0.734499 0.678609i \(-0.762583\pi\)
−0.734499 + 0.678609i \(0.762583\pi\)
\(240\) 1.94894e8 0.910036
\(241\) −8.93025e7 −0.410964 −0.205482 0.978661i \(-0.565876\pi\)
−0.205482 + 0.978661i \(0.565876\pi\)
\(242\) 1.84125e8 0.835138
\(243\) −2.68064e8 −1.19844
\(244\) −4.47355e8 −1.97146
\(245\) 1.03228e8 0.448454
\(246\) 8.25455e8 3.53525
\(247\) 2.58741e8 1.09251
\(248\) −3.03891e8 −1.26513
\(249\) −1.12430e8 −0.461515
\(250\) 2.30821e8 0.934300
\(251\) −1.04502e8 −0.417126 −0.208563 0.978009i \(-0.566879\pi\)
−0.208563 + 0.978009i \(0.566879\pi\)
\(252\) 1.46256e9 5.75720
\(253\) 3.14502e8 1.22096
\(254\) 2.20160e8 0.842988
\(255\) 1.26594e8 0.478105
\(256\) −3.68671e8 −1.37341
\(257\) −3.65475e8 −1.34305 −0.671523 0.740983i \(-0.734360\pi\)
−0.671523 + 0.740983i \(0.734360\pi\)
\(258\) −7.35492e8 −2.66630
\(259\) 3.51783e8 1.25813
\(260\) −2.14569e8 −0.757112
\(261\) −1.42348e8 −0.495574
\(262\) 7.92849e8 2.72355
\(263\) 4.81180e8 1.63103 0.815516 0.578734i \(-0.196453\pi\)
0.815516 + 0.578734i \(0.196453\pi\)
\(264\) 8.61516e8 2.88171
\(265\) 1.13281e8 0.373935
\(266\) 8.28200e8 2.69805
\(267\) −9.13076e8 −2.93574
\(268\) −7.79680e8 −2.47426
\(269\) −5.61063e8 −1.75743 −0.878717 0.477344i \(-0.841600\pi\)
−0.878717 + 0.477344i \(0.841600\pi\)
\(270\) −1.21190e8 −0.374710
\(271\) −6.31875e7 −0.192858 −0.0964292 0.995340i \(-0.530742\pi\)
−0.0964292 + 0.995340i \(0.530742\pi\)
\(272\) 8.29508e8 2.49936
\(273\) −1.06053e9 −3.15466
\(274\) −1.27787e8 −0.375282
\(275\) 2.36562e8 0.685930
\(276\) −2.14672e9 −6.14602
\(277\) 1.99282e8 0.563365 0.281682 0.959508i \(-0.409108\pi\)
0.281682 + 0.959508i \(0.409108\pi\)
\(278\) −1.12442e9 −3.13886
\(279\) 2.76267e8 0.761577
\(280\) −3.94840e8 −1.07490
\(281\) 3.34142e8 0.898378 0.449189 0.893437i \(-0.351713\pi\)
0.449189 + 0.893437i \(0.351713\pi\)
\(282\) −1.42358e8 −0.378015
\(283\) −3.41530e8 −0.895729 −0.447865 0.894101i \(-0.647815\pi\)
−0.447865 + 0.894101i \(0.647815\pi\)
\(284\) −3.02371e8 −0.783296
\(285\) −1.46329e8 −0.374432
\(286\) −6.50329e8 −1.64381
\(287\) −8.04579e8 −2.00901
\(288\) −9.16920e8 −2.26182
\(289\) 1.28473e8 0.313090
\(290\) 6.68457e7 0.160946
\(291\) 9.90710e8 2.35679
\(292\) −7.25094e8 −1.70433
\(293\) −5.18982e8 −1.20536 −0.602678 0.797985i \(-0.705900\pi\)
−0.602678 + 0.797985i \(0.705900\pi\)
\(294\) −2.13559e9 −4.90121
\(295\) −1.96314e8 −0.445219
\(296\) −8.46495e8 −1.89716
\(297\) −2.57743e8 −0.570872
\(298\) 1.13618e9 2.48709
\(299\) 9.31602e8 2.01549
\(300\) −1.61472e9 −3.45281
\(301\) 7.16891e8 1.51520
\(302\) −9.92175e8 −2.07283
\(303\) 5.75318e8 1.18812
\(304\) −9.58819e8 −1.95740
\(305\) −1.09792e8 −0.221575
\(306\) −1.56740e9 −3.12719
\(307\) 7.56489e8 1.49217 0.746085 0.665850i \(-0.231931\pi\)
0.746085 + 0.665850i \(0.231931\pi\)
\(308\) −1.46068e9 −2.84857
\(309\) −4.30174e8 −0.829448
\(310\) −1.29733e8 −0.247335
\(311\) −7.71933e8 −1.45518 −0.727592 0.686010i \(-0.759361\pi\)
−0.727592 + 0.686010i \(0.759361\pi\)
\(312\) 2.55194e9 4.75696
\(313\) 1.57966e8 0.291177 0.145589 0.989345i \(-0.453492\pi\)
0.145589 + 0.989345i \(0.453492\pi\)
\(314\) 1.65074e9 3.00902
\(315\) 3.58949e8 0.647061
\(316\) −1.65831e9 −2.95638
\(317\) 5.54573e8 0.977803 0.488901 0.872339i \(-0.337398\pi\)
0.488901 + 0.872339i \(0.337398\pi\)
\(318\) −2.34356e9 −4.08677
\(319\) 1.42164e8 0.245202
\(320\) 9.25630e7 0.157911
\(321\) −8.22893e8 −1.38860
\(322\) 2.98194e9 4.97742
\(323\) −6.22807e8 −1.02836
\(324\) −3.87248e8 −0.632530
\(325\) 7.00732e8 1.13230
\(326\) −1.19422e9 −1.90907
\(327\) 1.44673e9 2.28808
\(328\) 1.93606e9 3.02942
\(329\) 1.38757e8 0.214818
\(330\) 3.67788e8 0.563377
\(331\) 3.19945e8 0.484927 0.242464 0.970160i \(-0.422044\pi\)
0.242464 + 0.970160i \(0.422044\pi\)
\(332\) −4.58695e8 −0.687925
\(333\) 7.69548e8 1.14204
\(334\) −3.96802e7 −0.0582722
\(335\) −1.91353e8 −0.278086
\(336\) 3.93000e9 5.65203
\(337\) 1.24573e8 0.177304 0.0886522 0.996063i \(-0.471744\pi\)
0.0886522 + 0.996063i \(0.471744\pi\)
\(338\) −6.26558e8 −0.882577
\(339\) −5.52491e7 −0.0770240
\(340\) 5.16481e8 0.712653
\(341\) −2.75911e8 −0.376816
\(342\) 1.81174e9 2.44909
\(343\) 8.54404e8 1.14323
\(344\) −1.72505e9 −2.28480
\(345\) −5.26860e8 −0.690761
\(346\) 1.68780e9 2.19056
\(347\) 6.62089e8 0.850674 0.425337 0.905035i \(-0.360156\pi\)
0.425337 + 0.905035i \(0.360156\pi\)
\(348\) −9.70386e8 −1.23429
\(349\) −9.31176e8 −1.17258 −0.586291 0.810101i \(-0.699412\pi\)
−0.586291 + 0.810101i \(0.699412\pi\)
\(350\) 2.24296e9 2.79629
\(351\) −7.63473e8 −0.942364
\(352\) 9.15740e8 1.11911
\(353\) 7.55790e8 0.914513 0.457256 0.889335i \(-0.348832\pi\)
0.457256 + 0.889335i \(0.348832\pi\)
\(354\) 4.06135e9 4.86585
\(355\) −7.42094e7 −0.0880359
\(356\) −3.72518e9 −4.37595
\(357\) 2.55275e9 2.96941
\(358\) 7.21686e8 0.831299
\(359\) 8.55845e8 0.976258 0.488129 0.872772i \(-0.337680\pi\)
0.488129 + 0.872772i \(0.337680\pi\)
\(360\) −8.63738e8 −0.975716
\(361\) −1.73977e8 −0.194633
\(362\) −1.53405e9 −1.69965
\(363\) −6.55997e8 −0.719828
\(364\) −4.32675e9 −4.70226
\(365\) −1.77956e8 −0.191553
\(366\) 2.27138e9 2.42162
\(367\) −7.91187e8 −0.835503 −0.417751 0.908561i \(-0.637182\pi\)
−0.417751 + 0.908561i \(0.637182\pi\)
\(368\) −3.45224e9 −3.61105
\(369\) −1.76007e9 −1.82363
\(370\) −3.61376e8 −0.370897
\(371\) 2.28429e9 2.32243
\(372\) 1.88331e9 1.89680
\(373\) 1.38399e9 1.38087 0.690433 0.723396i \(-0.257420\pi\)
0.690433 + 0.723396i \(0.257420\pi\)
\(374\) 1.56538e9 1.54728
\(375\) −8.22368e8 −0.805298
\(376\) −3.33892e8 −0.323928
\(377\) 4.21113e8 0.404766
\(378\) −2.44379e9 −2.32724
\(379\) −1.33706e8 −0.126158 −0.0630788 0.998009i \(-0.520092\pi\)
−0.0630788 + 0.998009i \(0.520092\pi\)
\(380\) −5.96995e8 −0.558121
\(381\) −7.84385e8 −0.726594
\(382\) −3.87638e9 −3.55799
\(383\) 1.90252e9 1.73035 0.865174 0.501471i \(-0.167208\pi\)
0.865174 + 0.501471i \(0.167208\pi\)
\(384\) 7.42269e8 0.668963
\(385\) −3.58487e8 −0.320155
\(386\) −6.98276e8 −0.617976
\(387\) 1.56825e9 1.37539
\(388\) 4.04191e9 3.51298
\(389\) −3.74686e8 −0.322733 −0.161367 0.986895i \(-0.551590\pi\)
−0.161367 + 0.986895i \(0.551590\pi\)
\(390\) 1.08944e9 0.929992
\(391\) −2.24242e9 −1.89714
\(392\) −5.00891e9 −4.19993
\(393\) −2.82475e9 −2.34751
\(394\) −1.96658e9 −1.61985
\(395\) −4.06990e8 −0.332272
\(396\) −3.19532e9 −2.58572
\(397\) 1.86096e8 0.149269 0.0746347 0.997211i \(-0.476221\pi\)
0.0746347 + 0.997211i \(0.476221\pi\)
\(398\) −2.11061e9 −1.67810
\(399\) −2.95070e9 −2.32552
\(400\) −2.59670e9 −2.02868
\(401\) −1.28243e9 −0.993179 −0.496590 0.867986i \(-0.665415\pi\)
−0.496590 + 0.867986i \(0.665415\pi\)
\(402\) 3.95872e9 3.03923
\(403\) −8.17291e8 −0.622027
\(404\) 2.34719e9 1.77098
\(405\) −9.50403e7 −0.0710911
\(406\) 1.34793e9 0.999601
\(407\) −7.68558e8 −0.565062
\(408\) −6.14269e9 −4.47763
\(409\) −7.56893e7 −0.0547020 −0.0273510 0.999626i \(-0.508707\pi\)
−0.0273510 + 0.999626i \(0.508707\pi\)
\(410\) 8.26519e8 0.592256
\(411\) 4.55276e8 0.323466
\(412\) −1.75503e9 −1.23636
\(413\) −3.95863e9 −2.76516
\(414\) 6.52320e9 4.51814
\(415\) −1.12575e8 −0.0773169
\(416\) 2.71256e9 1.84737
\(417\) 4.00607e9 2.70547
\(418\) −1.80941e9 −1.21177
\(419\) −1.16121e9 −0.771189 −0.385594 0.922668i \(-0.626004\pi\)
−0.385594 + 0.922668i \(0.626004\pi\)
\(420\) 2.44696e9 1.61159
\(421\) −8.64920e8 −0.564922 −0.282461 0.959279i \(-0.591151\pi\)
−0.282461 + 0.959279i \(0.591151\pi\)
\(422\) 2.26220e9 1.46534
\(423\) 3.03541e8 0.194996
\(424\) −5.49668e9 −3.50203
\(425\) −1.68670e9 −1.06581
\(426\) 1.53525e9 0.962155
\(427\) −2.21394e9 −1.37616
\(428\) −3.35725e9 −2.06981
\(429\) 2.31698e9 1.41685
\(430\) −7.36439e8 −0.446681
\(431\) −1.29106e9 −0.776742 −0.388371 0.921503i \(-0.626962\pi\)
−0.388371 + 0.921503i \(0.626962\pi\)
\(432\) 2.82921e9 1.68839
\(433\) 2.17469e8 0.128733 0.0643666 0.997926i \(-0.479497\pi\)
0.0643666 + 0.997926i \(0.479497\pi\)
\(434\) −2.61605e9 −1.53614
\(435\) −2.38157e8 −0.138724
\(436\) 5.90240e9 3.41056
\(437\) 2.59199e9 1.48576
\(438\) 3.68156e9 2.09350
\(439\) −9.39396e8 −0.529935 −0.264968 0.964257i \(-0.585361\pi\)
−0.264968 + 0.964257i \(0.585361\pi\)
\(440\) 8.62626e8 0.482768
\(441\) 4.55360e9 2.52825
\(442\) 4.63690e9 2.55417
\(443\) 8.17201e8 0.446597 0.223298 0.974750i \(-0.428318\pi\)
0.223298 + 0.974750i \(0.428318\pi\)
\(444\) 5.24602e9 2.84439
\(445\) −9.14253e8 −0.491820
\(446\) −5.91153e9 −3.15521
\(447\) −4.04798e9 −2.14369
\(448\) 1.86652e9 0.980751
\(449\) 2.04702e8 0.106724 0.0533618 0.998575i \(-0.483006\pi\)
0.0533618 + 0.998575i \(0.483006\pi\)
\(450\) 4.90662e9 2.53827
\(451\) 1.75780e9 0.902303
\(452\) −2.25406e8 −0.114810
\(453\) 3.53491e9 1.78663
\(454\) 1.28764e9 0.645799
\(455\) −1.06189e9 −0.528495
\(456\) 7.10027e9 3.50669
\(457\) 1.45321e9 0.712235 0.356117 0.934441i \(-0.384100\pi\)
0.356117 + 0.934441i \(0.384100\pi\)
\(458\) 8.20586e8 0.399112
\(459\) 1.83773e9 0.887027
\(460\) −2.14949e9 −1.02963
\(461\) −2.51711e9 −1.19660 −0.598301 0.801272i \(-0.704157\pi\)
−0.598301 + 0.801272i \(0.704157\pi\)
\(462\) 7.41638e9 3.49901
\(463\) −2.48995e9 −1.16589 −0.582944 0.812512i \(-0.698099\pi\)
−0.582944 + 0.812512i \(0.698099\pi\)
\(464\) −1.56052e9 −0.725198
\(465\) 4.62212e8 0.213185
\(466\) 6.19226e8 0.283464
\(467\) 1.38614e9 0.629792 0.314896 0.949126i \(-0.398030\pi\)
0.314896 + 0.949126i \(0.398030\pi\)
\(468\) −9.46504e9 −4.26837
\(469\) −3.85860e9 −1.72713
\(470\) −1.42541e8 −0.0633283
\(471\) −5.88124e9 −2.59356
\(472\) 9.52565e9 4.16963
\(473\) −1.56623e9 −0.680520
\(474\) 8.41982e9 3.63144
\(475\) 1.94964e9 0.834695
\(476\) 1.04148e10 4.42613
\(477\) 4.99703e9 2.10813
\(478\) 6.42233e9 2.68964
\(479\) −3.31140e9 −1.37669 −0.688347 0.725381i \(-0.741664\pi\)
−0.688347 + 0.725381i \(0.741664\pi\)
\(480\) −1.53407e9 −0.633140
\(481\) −2.27659e9 −0.932773
\(482\) 1.84987e9 0.752450
\(483\) −1.06240e10 −4.29017
\(484\) −2.67635e9 −1.07296
\(485\) 9.91986e8 0.394830
\(486\) 5.55286e9 2.19427
\(487\) 3.75492e9 1.47316 0.736580 0.676351i \(-0.236440\pi\)
0.736580 + 0.676351i \(0.236440\pi\)
\(488\) 5.32740e9 2.07513
\(489\) 4.25474e9 1.64548
\(490\) −2.13835e9 −0.821091
\(491\) 3.06437e8 0.116830 0.0584151 0.998292i \(-0.481395\pi\)
0.0584151 + 0.998292i \(0.481395\pi\)
\(492\) −1.19984e10 −4.54198
\(493\) −1.01365e9 −0.380997
\(494\) −5.35975e9 −2.00032
\(495\) −7.84213e8 −0.290613
\(496\) 3.02864e9 1.11445
\(497\) −1.49642e9 −0.546772
\(498\) 2.32896e9 0.845006
\(499\) −1.33384e9 −0.480566 −0.240283 0.970703i \(-0.577240\pi\)
−0.240283 + 0.970703i \(0.577240\pi\)
\(500\) −3.35511e9 −1.20036
\(501\) 1.41372e8 0.0502264
\(502\) 2.16473e9 0.763732
\(503\) 8.59977e8 0.301300 0.150650 0.988587i \(-0.451863\pi\)
0.150650 + 0.988587i \(0.451863\pi\)
\(504\) −1.74171e10 −6.05996
\(505\) 5.76060e8 0.199043
\(506\) −6.51480e9 −2.23550
\(507\) 2.23229e9 0.760717
\(508\) −3.20014e9 −1.08304
\(509\) 1.11628e9 0.375198 0.187599 0.982246i \(-0.439929\pi\)
0.187599 + 0.982246i \(0.439929\pi\)
\(510\) −2.62236e9 −0.875381
\(511\) −3.58846e9 −1.18969
\(512\) 6.34954e9 2.09073
\(513\) −2.12421e9 −0.694683
\(514\) 7.57069e9 2.45903
\(515\) −4.30728e8 −0.138956
\(516\) 1.06907e10 3.42558
\(517\) −3.03150e8 −0.0964808
\(518\) −7.28708e9 −2.30356
\(519\) −6.01328e9 −1.88810
\(520\) 2.55523e9 0.796927
\(521\) −4.36175e9 −1.35123 −0.675614 0.737256i \(-0.736121\pi\)
−0.675614 + 0.737256i \(0.736121\pi\)
\(522\) 2.94869e9 0.907365
\(523\) 3.02705e9 0.925259 0.462629 0.886552i \(-0.346906\pi\)
0.462629 + 0.886552i \(0.346906\pi\)
\(524\) −1.15245e10 −3.49914
\(525\) −7.99118e9 −2.41020
\(526\) −9.96750e9 −2.98632
\(527\) 1.96727e9 0.585500
\(528\) −8.58606e9 −2.53849
\(529\) 5.92768e9 1.74097
\(530\) −2.34658e9 −0.684651
\(531\) −8.65976e9 −2.51001
\(532\) −1.20383e10 −3.46637
\(533\) 5.20688e9 1.48947
\(534\) 1.89141e10 5.37516
\(535\) −8.23953e8 −0.232629
\(536\) 9.28495e9 2.60437
\(537\) −2.57121e9 −0.716519
\(538\) 1.16223e10 3.21775
\(539\) −4.54774e9 −1.25094
\(540\) 1.76156e9 0.481416
\(541\) 3.28479e9 0.891903 0.445952 0.895057i \(-0.352865\pi\)
0.445952 + 0.895057i \(0.352865\pi\)
\(542\) 1.30891e9 0.353112
\(543\) 5.46548e9 1.46497
\(544\) −6.52931e9 −1.73889
\(545\) 1.44860e9 0.383318
\(546\) 2.19685e10 5.77598
\(547\) 1.63667e8 0.0427569
\(548\) 1.85744e9 0.482151
\(549\) −4.84313e9 −1.24917
\(550\) −4.90030e9 −1.25589
\(551\) 1.17166e9 0.298381
\(552\) 2.55646e10 6.46923
\(553\) −8.20688e9 −2.06367
\(554\) −4.12807e9 −1.03149
\(555\) 1.28750e9 0.319686
\(556\) 1.63440e10 4.03271
\(557\) 1.55094e9 0.380280 0.190140 0.981757i \(-0.439106\pi\)
0.190140 + 0.981757i \(0.439106\pi\)
\(558\) −5.72278e9 −1.39440
\(559\) −4.63940e9 −1.12336
\(560\) 3.93506e9 0.946876
\(561\) −5.57713e9 −1.33365
\(562\) −6.92165e9 −1.64487
\(563\) 6.31801e9 1.49211 0.746056 0.665884i \(-0.231945\pi\)
0.746056 + 0.665884i \(0.231945\pi\)
\(564\) 2.06924e9 0.485662
\(565\) −5.53202e7 −0.0129037
\(566\) 7.07470e9 1.64002
\(567\) −1.91647e9 −0.441531
\(568\) 3.60084e9 0.824488
\(569\) 3.41487e9 0.777109 0.388554 0.921426i \(-0.372975\pi\)
0.388554 + 0.921426i \(0.372975\pi\)
\(570\) 3.03116e9 0.685562
\(571\) −2.10563e9 −0.473321 −0.236660 0.971592i \(-0.576053\pi\)
−0.236660 + 0.971592i \(0.576053\pi\)
\(572\) 9.45286e9 2.11192
\(573\) 1.38107e10 3.06673
\(574\) 1.66666e10 3.67837
\(575\) 7.01972e9 1.53986
\(576\) 4.08313e9 0.890255
\(577\) −5.35199e9 −1.15985 −0.579923 0.814671i \(-0.696917\pi\)
−0.579923 + 0.814671i \(0.696917\pi\)
\(578\) −2.66128e9 −0.573249
\(579\) 2.48781e9 0.532650
\(580\) −9.71636e8 −0.206779
\(581\) −2.27006e9 −0.480199
\(582\) −2.05223e10 −4.31514
\(583\) −4.99059e9 −1.04307
\(584\) 8.63490e9 1.79396
\(585\) −2.32296e9 −0.479729
\(586\) 1.07505e10 2.20693
\(587\) −4.97991e9 −1.01622 −0.508111 0.861292i \(-0.669656\pi\)
−0.508111 + 0.861292i \(0.669656\pi\)
\(588\) 3.10419e10 6.29692
\(589\) −2.27395e9 −0.458540
\(590\) 4.06658e9 0.815168
\(591\) 7.00652e9 1.39620
\(592\) 8.43636e9 1.67120
\(593\) 2.96746e8 0.0584377 0.0292188 0.999573i \(-0.490698\pi\)
0.0292188 + 0.999573i \(0.490698\pi\)
\(594\) 5.33906e9 1.04523
\(595\) 2.55604e9 0.497460
\(596\) −1.65150e10 −3.19533
\(597\) 7.51966e9 1.44640
\(598\) −1.92978e10 −3.69024
\(599\) 6.91564e9 1.31474 0.657368 0.753570i \(-0.271670\pi\)
0.657368 + 0.753570i \(0.271670\pi\)
\(600\) 1.92292e10 3.63439
\(601\) 2.80612e9 0.527285 0.263642 0.964620i \(-0.415076\pi\)
0.263642 + 0.964620i \(0.415076\pi\)
\(602\) −1.48502e10 −2.77424
\(603\) −8.44094e9 −1.56776
\(604\) 1.44218e10 2.66311
\(605\) −6.56842e8 −0.120592
\(606\) −1.19175e10 −2.17537
\(607\) −9.13664e8 −0.165816 −0.0829079 0.996557i \(-0.526421\pi\)
−0.0829079 + 0.996557i \(0.526421\pi\)
\(608\) 7.54715e9 1.36182
\(609\) −4.80239e9 −0.861583
\(610\) 2.27431e9 0.405691
\(611\) −8.97977e8 −0.159265
\(612\) 2.27830e10 4.01772
\(613\) 1.14941e9 0.201540 0.100770 0.994910i \(-0.467869\pi\)
0.100770 + 0.994910i \(0.467869\pi\)
\(614\) −1.56704e10 −2.73207
\(615\) −2.94471e9 −0.510481
\(616\) 1.73947e10 2.99837
\(617\) −1.14126e10 −1.95608 −0.978041 0.208413i \(-0.933170\pi\)
−0.978041 + 0.208413i \(0.933170\pi\)
\(618\) 8.91092e9 1.51867
\(619\) 6.04418e9 1.02428 0.512142 0.858901i \(-0.328852\pi\)
0.512142 + 0.858901i \(0.328852\pi\)
\(620\) 1.88574e9 0.317769
\(621\) −7.64825e9 −1.28157
\(622\) 1.59903e10 2.66435
\(623\) −1.84357e10 −3.05459
\(624\) −2.54332e10 −4.19040
\(625\) 4.85346e9 0.795192
\(626\) −3.27221e9 −0.533128
\(627\) 6.44654e9 1.04446
\(628\) −2.39944e10 −3.86590
\(629\) 5.47988e9 0.877999
\(630\) −7.43551e9 −1.18473
\(631\) −9.36059e9 −1.48320 −0.741602 0.670841i \(-0.765933\pi\)
−0.741602 + 0.670841i \(0.765933\pi\)
\(632\) 1.97482e10 3.11185
\(633\) −8.05974e9 −1.26301
\(634\) −1.14878e10 −1.79030
\(635\) −7.85395e8 −0.121725
\(636\) 3.40648e10 5.25056
\(637\) −1.34711e10 −2.06498
\(638\) −2.94489e9 −0.448949
\(639\) −3.27352e9 −0.496320
\(640\) 7.43225e8 0.112070
\(641\) −4.42697e9 −0.663901 −0.331950 0.943297i \(-0.607707\pi\)
−0.331950 + 0.943297i \(0.607707\pi\)
\(642\) 1.70460e10 2.54243
\(643\) −1.19909e10 −1.77875 −0.889374 0.457181i \(-0.848859\pi\)
−0.889374 + 0.457181i \(0.848859\pi\)
\(644\) −4.33441e10 −6.39483
\(645\) 2.62378e9 0.385006
\(646\) 1.29012e10 1.88286
\(647\) −8.03624e9 −1.16651 −0.583254 0.812289i \(-0.698221\pi\)
−0.583254 + 0.812289i \(0.698221\pi\)
\(648\) 4.61160e9 0.665794
\(649\) 8.64862e9 1.24191
\(650\) −1.45154e10 −2.07316
\(651\) 9.32043e9 1.32404
\(652\) 1.73586e10 2.45272
\(653\) 8.01372e9 1.12626 0.563129 0.826369i \(-0.309597\pi\)
0.563129 + 0.826369i \(0.309597\pi\)
\(654\) −2.99686e10 −4.18933
\(655\) −2.82839e9 −0.393274
\(656\) −1.92952e10 −2.66861
\(657\) −7.84998e9 −1.07992
\(658\) −2.87432e9 −0.393318
\(659\) 3.08292e9 0.419627 0.209813 0.977741i \(-0.432714\pi\)
0.209813 + 0.977741i \(0.432714\pi\)
\(660\) −5.34599e9 −0.723809
\(661\) −3.22412e8 −0.0434217 −0.0217108 0.999764i \(-0.506911\pi\)
−0.0217108 + 0.999764i \(0.506911\pi\)
\(662\) −6.62755e9 −0.887872
\(663\) −1.65203e10 −2.20151
\(664\) 5.46245e9 0.724101
\(665\) −2.95450e9 −0.389590
\(666\) −1.59410e10 −2.09100
\(667\) 4.21858e9 0.550461
\(668\) 5.76772e8 0.0748663
\(669\) 2.10615e10 2.71956
\(670\) 3.96382e9 0.509157
\(671\) 4.83690e9 0.618071
\(672\) −3.09342e10 −3.93230
\(673\) 1.25539e10 1.58755 0.793775 0.608211i \(-0.208113\pi\)
0.793775 + 0.608211i \(0.208113\pi\)
\(674\) −2.58049e9 −0.324633
\(675\) −5.75285e9 −0.719979
\(676\) 9.10733e9 1.13391
\(677\) 7.04658e9 0.872807 0.436403 0.899751i \(-0.356252\pi\)
0.436403 + 0.899751i \(0.356252\pi\)
\(678\) 1.14447e9 0.141026
\(679\) 2.00032e10 2.45220
\(680\) −6.15060e9 −0.750130
\(681\) −4.58757e9 −0.556631
\(682\) 5.71542e9 0.689926
\(683\) 8.62938e9 1.03635 0.518176 0.855274i \(-0.326611\pi\)
0.518176 + 0.855274i \(0.326611\pi\)
\(684\) −2.63346e10 −3.14651
\(685\) 4.55863e8 0.0541897
\(686\) −1.76987e10 −2.09318
\(687\) −2.92357e9 −0.344005
\(688\) 1.71923e10 2.01267
\(689\) −1.47829e10 −1.72184
\(690\) 1.09137e10 1.26474
\(691\) −6.42519e9 −0.740820 −0.370410 0.928868i \(-0.620783\pi\)
−0.370410 + 0.928868i \(0.620783\pi\)
\(692\) −2.45331e10 −2.81437
\(693\) −1.58135e10 −1.80494
\(694\) −1.37150e10 −1.55753
\(695\) 4.01123e9 0.453243
\(696\) 1.15560e10 1.29920
\(697\) −1.25333e10 −1.40201
\(698\) 1.92890e10 2.14692
\(699\) −2.20617e9 −0.244325
\(700\) −3.26025e10 −3.59259
\(701\) 6.47114e9 0.709525 0.354762 0.934957i \(-0.384562\pi\)
0.354762 + 0.934957i \(0.384562\pi\)
\(702\) 1.58151e10 1.72541
\(703\) −6.33414e9 −0.687613
\(704\) −4.07787e9 −0.440483
\(705\) 5.07844e8 0.0545843
\(706\) −1.56560e10 −1.67442
\(707\) 1.16161e10 1.23621
\(708\) −5.90337e10 −6.25149
\(709\) 7.25651e9 0.764656 0.382328 0.924027i \(-0.375123\pi\)
0.382328 + 0.924027i \(0.375123\pi\)
\(710\) 1.53723e9 0.161188
\(711\) −1.79531e10 −1.87325
\(712\) 4.43619e10 4.60607
\(713\) −8.18738e9 −0.845925
\(714\) −5.28795e10 −5.43680
\(715\) 2.31997e9 0.237362
\(716\) −1.04901e10 −1.06803
\(717\) −2.28814e10 −2.31828
\(718\) −1.77286e10 −1.78747
\(719\) −4.36593e9 −0.438052 −0.219026 0.975719i \(-0.570288\pi\)
−0.219026 + 0.975719i \(0.570288\pi\)
\(720\) 8.60820e9 0.859505
\(721\) −8.68555e9 −0.863026
\(722\) 3.60387e9 0.356360
\(723\) −6.59071e9 −0.648557
\(724\) 2.22981e10 2.18365
\(725\) 3.17313e9 0.309247
\(726\) 1.35888e10 1.31796
\(727\) 4.42118e9 0.426745 0.213372 0.976971i \(-0.431555\pi\)
0.213372 + 0.976971i \(0.431555\pi\)
\(728\) 5.15258e10 4.94954
\(729\) −1.69709e10 −1.62240
\(730\) 3.68631e9 0.350721
\(731\) 1.11673e10 1.05740
\(732\) −3.30157e10 −3.11123
\(733\) −3.24298e9 −0.304144 −0.152072 0.988369i \(-0.548595\pi\)
−0.152072 + 0.988369i \(0.548595\pi\)
\(734\) 1.63892e10 1.52975
\(735\) 7.61847e9 0.707721
\(736\) 2.71736e10 2.51232
\(737\) 8.43007e9 0.775703
\(738\) 3.64593e10 3.33896
\(739\) −1.50780e10 −1.37432 −0.687159 0.726507i \(-0.741143\pi\)
−0.687159 + 0.726507i \(0.741143\pi\)
\(740\) 5.25278e9 0.476516
\(741\) 1.90956e10 1.72413
\(742\) −4.73183e10 −4.25222
\(743\) 5.06652e8 0.0453157 0.0226579 0.999743i \(-0.492787\pi\)
0.0226579 + 0.999743i \(0.492787\pi\)
\(744\) −2.24278e10 −1.99655
\(745\) −4.05319e9 −0.359129
\(746\) −2.86689e10 −2.52828
\(747\) −4.96591e9 −0.435889
\(748\) −2.27536e10 −1.98790
\(749\) −1.66149e10 −1.44481
\(750\) 1.70351e10 1.47445
\(751\) 1.32220e10 1.13908 0.569542 0.821962i \(-0.307120\pi\)
0.569542 + 0.821962i \(0.307120\pi\)
\(752\) 3.32764e9 0.285347
\(753\) −7.71248e9 −0.658281
\(754\) −8.72323e9 −0.741101
\(755\) 3.53946e9 0.299311
\(756\) 3.55216e10 2.98997
\(757\) −1.79649e10 −1.50518 −0.752591 0.658488i \(-0.771196\pi\)
−0.752591 + 0.658488i \(0.771196\pi\)
\(758\) 2.76968e9 0.230987
\(759\) 2.32109e10 1.92684
\(760\) 7.10941e9 0.587471
\(761\) 1.35129e10 1.11148 0.555741 0.831355i \(-0.312434\pi\)
0.555741 + 0.831355i \(0.312434\pi\)
\(762\) 1.62483e10 1.33035
\(763\) 2.92107e10 2.38071
\(764\) 5.63452e10 4.57120
\(765\) 5.59151e9 0.451558
\(766\) −3.94101e10 −3.16816
\(767\) 2.56185e10 2.05008
\(768\) −2.72087e10 −2.16742
\(769\) −1.21372e10 −0.962449 −0.481225 0.876597i \(-0.659808\pi\)
−0.481225 + 0.876597i \(0.659808\pi\)
\(770\) 7.42594e9 0.586184
\(771\) −2.69728e10 −2.11951
\(772\) 1.01498e10 0.793956
\(773\) −7.91784e9 −0.616564 −0.308282 0.951295i \(-0.599754\pi\)
−0.308282 + 0.951295i \(0.599754\pi\)
\(774\) −3.24857e10 −2.51825
\(775\) −6.15838e9 −0.475237
\(776\) −4.81338e10 −3.69772
\(777\) 2.59623e10 1.98550
\(778\) 7.76150e9 0.590905
\(779\) 1.44871e10 1.09799
\(780\) −1.58356e10 −1.19482
\(781\) 3.26930e9 0.245571
\(782\) 4.64511e10 3.47354
\(783\) −3.45725e9 −0.257374
\(784\) 4.99199e10 3.69971
\(785\) −5.88882e9 −0.434495
\(786\) 5.85139e10 4.29813
\(787\) 1.30407e10 0.953649 0.476825 0.878998i \(-0.341788\pi\)
0.476825 + 0.878998i \(0.341788\pi\)
\(788\) 2.85853e10 2.08114
\(789\) 3.55121e10 2.57399
\(790\) 8.43067e9 0.608369
\(791\) −1.11552e9 −0.0801421
\(792\) 3.80521e10 2.72170
\(793\) 1.43276e10 1.02028
\(794\) −3.85492e9 −0.273303
\(795\) 8.36035e9 0.590119
\(796\) 3.06788e10 2.15597
\(797\) −1.26368e10 −0.884168 −0.442084 0.896974i \(-0.645761\pi\)
−0.442084 + 0.896974i \(0.645761\pi\)
\(798\) 6.11228e10 4.25788
\(799\) 2.16149e9 0.149913
\(800\) 2.04395e10 1.41141
\(801\) −4.03294e10 −2.77273
\(802\) 2.65651e10 1.81845
\(803\) 7.83988e9 0.534324
\(804\) −5.75420e10 −3.90471
\(805\) −1.06377e10 −0.718725
\(806\) 1.69299e10 1.13889
\(807\) −4.14076e10 −2.77347
\(808\) −2.79519e10 −1.86411
\(809\) −8.89805e8 −0.0590847 −0.0295424 0.999564i \(-0.509405\pi\)
−0.0295424 + 0.999564i \(0.509405\pi\)
\(810\) 1.96873e9 0.130163
\(811\) −4.11626e9 −0.270976 −0.135488 0.990779i \(-0.543260\pi\)
−0.135488 + 0.990779i \(0.543260\pi\)
\(812\) −1.95929e10 −1.28426
\(813\) −4.66336e9 −0.304356
\(814\) 1.59204e10 1.03459
\(815\) 4.26023e9 0.275665
\(816\) 6.12194e10 3.94433
\(817\) −1.29082e10 −0.828111
\(818\) 1.56788e9 0.100156
\(819\) −4.68420e10 −2.97949
\(820\) −1.20139e10 −0.760912
\(821\) 5.47547e9 0.345319 0.172659 0.984982i \(-0.444764\pi\)
0.172659 + 0.984982i \(0.444764\pi\)
\(822\) −9.43091e9 −0.592246
\(823\) −5.61810e9 −0.351310 −0.175655 0.984452i \(-0.556204\pi\)
−0.175655 + 0.984452i \(0.556204\pi\)
\(824\) 2.09000e10 1.30137
\(825\) 1.74587e10 1.08249
\(826\) 8.20018e10 5.06283
\(827\) 8.60735e9 0.529176 0.264588 0.964362i \(-0.414764\pi\)
0.264588 + 0.964362i \(0.414764\pi\)
\(828\) −9.48180e10 −5.80476
\(829\) −2.40091e10 −1.46364 −0.731822 0.681496i \(-0.761330\pi\)
−0.731822 + 0.681496i \(0.761330\pi\)
\(830\) 2.33196e9 0.141562
\(831\) 1.47074e10 0.889065
\(832\) −1.20793e10 −0.727126
\(833\) 3.24258e10 1.94372
\(834\) −8.29845e10 −4.95354
\(835\) 1.41554e8 0.00841435
\(836\) 2.63007e10 1.55684
\(837\) 6.70978e9 0.395521
\(838\) 2.40541e10 1.41200
\(839\) −1.02346e10 −0.598282 −0.299141 0.954209i \(-0.596700\pi\)
−0.299141 + 0.954209i \(0.596700\pi\)
\(840\) −2.91400e10 −1.69634
\(841\) −1.53429e10 −0.889452
\(842\) 1.79166e10 1.03434
\(843\) 2.46604e10 1.41776
\(844\) −3.28822e10 −1.88262
\(845\) 2.23517e9 0.127442
\(846\) −6.28776e9 −0.357026
\(847\) −1.32451e10 −0.748968
\(848\) 5.47811e10 3.08493
\(849\) −2.52056e10 −1.41358
\(850\) 3.49396e10 1.95142
\(851\) −2.28062e10 −1.26852
\(852\) −2.23156e10 −1.23615
\(853\) 2.74648e9 0.151514 0.0757572 0.997126i \(-0.475863\pi\)
0.0757572 + 0.997126i \(0.475863\pi\)
\(854\) 4.58610e10 2.51966
\(855\) −6.46316e9 −0.353642
\(856\) 3.99804e10 2.17866
\(857\) −1.48574e9 −0.0806324 −0.0403162 0.999187i \(-0.512837\pi\)
−0.0403162 + 0.999187i \(0.512837\pi\)
\(858\) −4.79956e10 −2.59415
\(859\) −2.53060e10 −1.36222 −0.681112 0.732180i \(-0.738503\pi\)
−0.681112 + 0.732180i \(0.738503\pi\)
\(860\) 1.07045e10 0.573882
\(861\) −5.93796e10 −3.17049
\(862\) 2.67439e10 1.42217
\(863\) 2.71902e10 1.44004 0.720020 0.693953i \(-0.244133\pi\)
0.720020 + 0.693953i \(0.244133\pi\)
\(864\) −2.22695e10 −1.17466
\(865\) −6.02103e9 −0.316311
\(866\) −4.50481e9 −0.235702
\(867\) 9.48157e9 0.494098
\(868\) 3.80256e10 1.97359
\(869\) 1.79300e10 0.926852
\(870\) 4.93335e9 0.253995
\(871\) 2.49712e10 1.28049
\(872\) −7.02897e10 −3.58991
\(873\) 4.37584e10 2.22593
\(874\) −5.36923e10 −2.72033
\(875\) −1.66043e10 −0.837899
\(876\) −5.35134e10 −2.68966
\(877\) 1.65208e10 0.827052 0.413526 0.910492i \(-0.364297\pi\)
0.413526 + 0.910492i \(0.364297\pi\)
\(878\) 1.94593e10 0.970278
\(879\) −3.83019e10 −1.90221
\(880\) −8.59712e9 −0.425269
\(881\) 2.13241e10 1.05064 0.525322 0.850904i \(-0.323945\pi\)
0.525322 + 0.850904i \(0.323945\pi\)
\(882\) −9.43264e10 −4.62906
\(883\) −5.74320e9 −0.280732 −0.140366 0.990100i \(-0.544828\pi\)
−0.140366 + 0.990100i \(0.544828\pi\)
\(884\) −6.73997e10 −3.28152
\(885\) −1.44883e10 −0.702615
\(886\) −1.69281e10 −0.817691
\(887\) −9.61464e9 −0.462594 −0.231297 0.972883i \(-0.574297\pi\)
−0.231297 + 0.972883i \(0.574297\pi\)
\(888\) −6.24731e10 −2.99397
\(889\) −1.58374e10 −0.756008
\(890\) 1.89385e10 0.900492
\(891\) 4.18701e9 0.198304
\(892\) 8.59271e10 4.05371
\(893\) −2.49844e9 −0.117406
\(894\) 8.38526e10 3.92496
\(895\) −2.57452e9 −0.120037
\(896\) 1.49870e10 0.696044
\(897\) 6.87541e10 3.18072
\(898\) −4.24034e9 −0.195404
\(899\) −3.70095e9 −0.169885
\(900\) −7.13201e10 −3.26109
\(901\) 3.55834e10 1.62073
\(902\) −3.64124e10 −1.65206
\(903\) 5.29080e10 2.39119
\(904\) 2.68428e9 0.120848
\(905\) 5.47252e9 0.245424
\(906\) −7.32245e10 −3.27120
\(907\) 1.33667e10 0.594837 0.297419 0.954747i \(-0.403874\pi\)
0.297419 + 0.954747i \(0.403874\pi\)
\(908\) −1.87164e10 −0.829702
\(909\) 2.54111e10 1.12215
\(910\) 2.19968e10 0.967640
\(911\) −2.32429e10 −1.01853 −0.509267 0.860609i \(-0.670083\pi\)
−0.509267 + 0.860609i \(0.670083\pi\)
\(912\) −7.07628e10 −3.08904
\(913\) 4.95951e9 0.215671
\(914\) −3.01029e10 −1.30406
\(915\) −8.10288e9 −0.349676
\(916\) −1.19276e10 −0.512767
\(917\) −5.70340e10 −2.44254
\(918\) −3.80680e10 −1.62409
\(919\) 4.33439e9 0.184214 0.0921072 0.995749i \(-0.470640\pi\)
0.0921072 + 0.995749i \(0.470640\pi\)
\(920\) 2.55975e10 1.08378
\(921\) 5.58305e10 2.35485
\(922\) 5.21412e10 2.19090
\(923\) 9.68418e9 0.405375
\(924\) −1.07801e11 −4.49542
\(925\) −1.71543e10 −0.712652
\(926\) 5.15785e10 2.13467
\(927\) −1.90002e10 −0.783393
\(928\) 1.22833e10 0.504543
\(929\) 2.88880e10 1.18212 0.591062 0.806626i \(-0.298709\pi\)
0.591062 + 0.806626i \(0.298709\pi\)
\(930\) −9.57459e9 −0.390328
\(931\) −3.74806e10 −1.52224
\(932\) −9.00076e9 −0.364186
\(933\) −5.69702e10 −2.29648
\(934\) −2.87134e10 −1.15311
\(935\) −5.58431e9 −0.223424
\(936\) 1.12716e11 4.49283
\(937\) −6.06659e9 −0.240911 −0.120455 0.992719i \(-0.538435\pi\)
−0.120455 + 0.992719i \(0.538435\pi\)
\(938\) 7.99297e10 3.16227
\(939\) 1.16582e10 0.459517
\(940\) 2.07191e9 0.0813622
\(941\) 2.80351e9 0.109683 0.0548413 0.998495i \(-0.482535\pi\)
0.0548413 + 0.998495i \(0.482535\pi\)
\(942\) 1.21828e11 4.74864
\(943\) 5.21610e10 2.02561
\(944\) −9.49347e10 −3.67302
\(945\) 8.71790e9 0.336047
\(946\) 3.24439e10 1.24599
\(947\) −4.39767e10 −1.68266 −0.841332 0.540519i \(-0.818228\pi\)
−0.841332 + 0.540519i \(0.818228\pi\)
\(948\) −1.22386e11 −4.66556
\(949\) 2.32229e10 0.882033
\(950\) −4.03863e10 −1.52827
\(951\) 4.09286e10 1.54310
\(952\) −1.24026e11 −4.65889
\(953\) 3.44436e10 1.28909 0.644546 0.764565i \(-0.277046\pi\)
0.644546 + 0.764565i \(0.277046\pi\)
\(954\) −1.03512e11 −3.85985
\(955\) 1.38285e10 0.513764
\(956\) −9.33518e10 −3.45557
\(957\) 1.04920e10 0.386961
\(958\) 6.85947e10 2.52064
\(959\) 9.19239e9 0.336561
\(960\) 6.83134e9 0.249205
\(961\) −2.03298e10 −0.738928
\(962\) 4.71588e10 1.70785
\(963\) −3.63461e10 −1.31149
\(964\) −2.68889e10 −0.966724
\(965\) 2.49101e9 0.0892340
\(966\) 2.20074e11 7.85503
\(967\) 1.37307e10 0.488315 0.244158 0.969736i \(-0.421489\pi\)
0.244158 + 0.969736i \(0.421489\pi\)
\(968\) 3.18717e10 1.12938
\(969\) −4.59644e10 −1.62289
\(970\) −2.05487e10 −0.722908
\(971\) −3.82566e10 −1.34103 −0.670515 0.741896i \(-0.733927\pi\)
−0.670515 + 0.741896i \(0.733927\pi\)
\(972\) −8.07135e10 −2.81913
\(973\) 8.08857e10 2.81499
\(974\) −7.77821e10 −2.69726
\(975\) 5.17154e10 1.78691
\(976\) −5.30940e10 −1.82798
\(977\) 1.53271e10 0.525810 0.262905 0.964822i \(-0.415319\pi\)
0.262905 + 0.964822i \(0.415319\pi\)
\(978\) −8.81357e10 −3.01277
\(979\) 4.02775e10 1.37190
\(980\) 3.10819e10 1.05491
\(981\) 6.39003e10 2.16103
\(982\) −6.34774e9 −0.213909
\(983\) −2.05896e10 −0.691369 −0.345685 0.938351i \(-0.612353\pi\)
−0.345685 + 0.938351i \(0.612353\pi\)
\(984\) 1.42885e11 4.78084
\(985\) 7.01555e9 0.233902
\(986\) 2.09974e10 0.697582
\(987\) 1.02406e10 0.339012
\(988\) 7.79066e10 2.56995
\(989\) −4.64762e10 −1.52772
\(990\) 1.62447e10 0.532095
\(991\) 3.22306e10 1.05199 0.525994 0.850488i \(-0.323693\pi\)
0.525994 + 0.850488i \(0.323693\pi\)
\(992\) −2.38394e10 −0.775360
\(993\) 2.36126e10 0.765281
\(994\) 3.09979e10 1.00111
\(995\) 7.52934e9 0.242313
\(996\) −3.38526e10 −1.08564
\(997\) −5.44583e10 −1.74033 −0.870164 0.492761i \(-0.835988\pi\)
−0.870164 + 0.492761i \(0.835988\pi\)
\(998\) 2.76302e10 0.879887
\(999\) 1.86903e10 0.593111
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 547.8.a.a.1.8 156
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.8.a.a.1.8 156 1.1 even 1 trivial