Properties

Label 547.8.a.a.1.12
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 1.12
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)
 
\(f(q)\) \(=\) \(q-20.1873 q^{2} +34.9985 q^{3} +279.528 q^{4} -313.532 q^{5} -706.526 q^{6} +1440.96 q^{7} -3058.93 q^{8} -962.103 q^{9} +O(q^{10})\) \(q-20.1873 q^{2} +34.9985 q^{3} +279.528 q^{4} -313.532 q^{5} -706.526 q^{6} +1440.96 q^{7} -3058.93 q^{8} -962.103 q^{9} +6329.36 q^{10} -893.205 q^{11} +9783.05 q^{12} +10490.3 q^{13} -29089.1 q^{14} -10973.1 q^{15} +25972.1 q^{16} -13076.6 q^{17} +19422.3 q^{18} +40332.1 q^{19} -87640.7 q^{20} +50431.5 q^{21} +18031.4 q^{22} -90822.6 q^{23} -107058. q^{24} +20177.1 q^{25} -211771. q^{26} -110214. q^{27} +402788. q^{28} +185391. q^{29} +221518. q^{30} -43656.9 q^{31} -132764. q^{32} -31260.8 q^{33} +263982. q^{34} -451787. q^{35} -268934. q^{36} -80252.9 q^{37} -814197. q^{38} +367146. q^{39} +959073. q^{40} +69266.3 q^{41} -1.01808e6 q^{42} -505879. q^{43} -249675. q^{44} +301650. q^{45} +1.83346e6 q^{46} +341684. q^{47} +908986. q^{48} +1.25282e6 q^{49} -407322. q^{50} -457663. q^{51} +2.93233e6 q^{52} -524279. q^{53} +2.22492e6 q^{54} +280048. q^{55} -4.40780e6 q^{56} +1.41156e6 q^{57} -3.74254e6 q^{58} +1.46495e6 q^{59} -3.06730e6 q^{60} +1.06050e6 q^{61} +881315. q^{62} -1.38635e6 q^{63} -644289. q^{64} -3.28905e6 q^{65} +631072. q^{66} -3.43501e6 q^{67} -3.65528e6 q^{68} -3.17866e6 q^{69} +9.12036e6 q^{70} -1.63429e6 q^{71} +2.94301e6 q^{72} -5.51772e6 q^{73} +1.62009e6 q^{74} +706169. q^{75} +1.12739e7 q^{76} -1.28707e6 q^{77} -7.41168e6 q^{78} +3.54074e6 q^{79} -8.14308e6 q^{80} -1.75321e6 q^{81} -1.39830e6 q^{82} -7.05178e6 q^{83} +1.40970e7 q^{84} +4.09994e6 q^{85} +1.02123e7 q^{86} +6.48840e6 q^{87} +2.73225e6 q^{88} -1.13603e7 q^{89} -6.08950e6 q^{90} +1.51161e7 q^{91} -2.53874e7 q^{92} -1.52793e6 q^{93} -6.89769e6 q^{94} -1.26454e7 q^{95} -4.64653e6 q^{96} -3.92765e6 q^{97} -2.52911e7 q^{98} +859355. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 156 q - 56 q^{2} - 284 q^{3} + 9690 q^{4} - 3751 q^{5} - 2322 q^{6} - 2559 q^{7} - 10752 q^{8} + 102594 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −20.1873 −1.78432 −0.892162 0.451717i \(-0.850812\pi\)
−0.892162 + 0.451717i \(0.850812\pi\)
\(3\) 34.9985 0.748385 0.374193 0.927351i \(-0.377920\pi\)
0.374193 + 0.927351i \(0.377920\pi\)
\(4\) 279.528 2.18381
\(5\) −313.532 −1.12173 −0.560863 0.827909i \(-0.689530\pi\)
−0.560863 + 0.827909i \(0.689530\pi\)
\(6\) −706.526 −1.33536
\(7\) 1440.96 1.58785 0.793924 0.608017i \(-0.208035\pi\)
0.793924 + 0.608017i \(0.208035\pi\)
\(8\) −3058.93 −2.11230
\(9\) −962.103 −0.439919
\(10\) 6329.36 2.00152
\(11\) −893.205 −0.202338 −0.101169 0.994869i \(-0.532258\pi\)
−0.101169 + 0.994869i \(0.532258\pi\)
\(12\) 9783.05 1.63433
\(13\) 10490.3 1.32430 0.662151 0.749371i \(-0.269644\pi\)
0.662151 + 0.749371i \(0.269644\pi\)
\(14\) −29089.1 −2.83323
\(15\) −10973.1 −0.839483
\(16\) 25972.1 1.58521
\(17\) −13076.6 −0.645542 −0.322771 0.946477i \(-0.604614\pi\)
−0.322771 + 0.946477i \(0.604614\pi\)
\(18\) 19422.3 0.784958
\(19\) 40332.1 1.34900 0.674502 0.738273i \(-0.264358\pi\)
0.674502 + 0.738273i \(0.264358\pi\)
\(20\) −87640.7 −2.44963
\(21\) 50431.5 1.18832
\(22\) 18031.4 0.361036
\(23\) −90822.6 −1.55649 −0.778245 0.627961i \(-0.783890\pi\)
−0.778245 + 0.627961i \(0.783890\pi\)
\(24\) −107058. −1.58081
\(25\) 20177.1 0.258267
\(26\) −211771. −2.36298
\(27\) −110214. −1.07761
\(28\) 402788. 3.46755
\(29\) 185391. 1.41155 0.705773 0.708438i \(-0.250600\pi\)
0.705773 + 0.708438i \(0.250600\pi\)
\(30\) 221518. 1.49791
\(31\) −43656.9 −0.263201 −0.131600 0.991303i \(-0.542012\pi\)
−0.131600 + 0.991303i \(0.542012\pi\)
\(32\) −132764. −0.716233
\(33\) −31260.8 −0.151427
\(34\) 263982. 1.15186
\(35\) −451787. −1.78113
\(36\) −268934. −0.960699
\(37\) −80252.9 −0.260468 −0.130234 0.991483i \(-0.541573\pi\)
−0.130234 + 0.991483i \(0.541573\pi\)
\(38\) −814197. −2.40706
\(39\) 367146. 0.991088
\(40\) 959073. 2.36942
\(41\) 69266.3 0.156956 0.0784781 0.996916i \(-0.474994\pi\)
0.0784781 + 0.996916i \(0.474994\pi\)
\(42\) −1.01808e6 −2.12035
\(43\) −505879. −0.970301 −0.485151 0.874431i \(-0.661235\pi\)
−0.485151 + 0.874431i \(0.661235\pi\)
\(44\) −249675. −0.441867
\(45\) 301650. 0.493468
\(46\) 1.83346e6 2.77728
\(47\) 341684. 0.480046 0.240023 0.970767i \(-0.422845\pi\)
0.240023 + 0.970767i \(0.422845\pi\)
\(48\) 908986. 1.18635
\(49\) 1.25282e6 1.52126
\(50\) −407322. −0.460832
\(51\) −457663. −0.483114
\(52\) 2.93233e6 2.89202
\(53\) −524279. −0.483723 −0.241861 0.970311i \(-0.577758\pi\)
−0.241861 + 0.970311i \(0.577758\pi\)
\(54\) 2.22492e6 1.92281
\(55\) 280048. 0.226967
\(56\) −4.40780e6 −3.35401
\(57\) 1.41156e6 1.00958
\(58\) −3.74254e6 −2.51865
\(59\) 1.46495e6 0.928626 0.464313 0.885671i \(-0.346301\pi\)
0.464313 + 0.885671i \(0.346301\pi\)
\(60\) −3.06730e6 −1.83327
\(61\) 1.06050e6 0.598212 0.299106 0.954220i \(-0.403312\pi\)
0.299106 + 0.954220i \(0.403312\pi\)
\(62\) 881315. 0.469635
\(63\) −1.38635e6 −0.698525
\(64\) −644289. −0.307221
\(65\) −3.28905e6 −1.48550
\(66\) 631072. 0.270194
\(67\) −3.43501e6 −1.39530 −0.697648 0.716441i \(-0.745770\pi\)
−0.697648 + 0.716441i \(0.745770\pi\)
\(68\) −3.65528e6 −1.40974
\(69\) −3.17866e6 −1.16485
\(70\) 9.12036e6 3.17811
\(71\) −1.63429e6 −0.541908 −0.270954 0.962592i \(-0.587339\pi\)
−0.270954 + 0.962592i \(0.587339\pi\)
\(72\) 2.94301e6 0.929240
\(73\) −5.51772e6 −1.66008 −0.830041 0.557702i \(-0.811683\pi\)
−0.830041 + 0.557702i \(0.811683\pi\)
\(74\) 1.62009e6 0.464759
\(75\) 706169. 0.193283
\(76\) 1.12739e7 2.94597
\(77\) −1.28707e6 −0.321281
\(78\) −7.41168e6 −1.76842
\(79\) 3.54074e6 0.807978 0.403989 0.914764i \(-0.367623\pi\)
0.403989 + 0.914764i \(0.367623\pi\)
\(80\) −8.14308e6 −1.77817
\(81\) −1.75321e6 −0.366552
\(82\) −1.39830e6 −0.280060
\(83\) −7.05178e6 −1.35371 −0.676855 0.736116i \(-0.736658\pi\)
−0.676855 + 0.736116i \(0.736658\pi\)
\(84\) 1.40970e7 2.59507
\(85\) 4.09994e6 0.724121
\(86\) 1.02123e7 1.73133
\(87\) 6.48840e6 1.05638
\(88\) 2.73225e6 0.427397
\(89\) −1.13603e7 −1.70814 −0.854072 0.520154i \(-0.825874\pi\)
−0.854072 + 0.520154i \(0.825874\pi\)
\(90\) −6.08950e6 −0.880507
\(91\) 1.51161e7 2.10279
\(92\) −2.53874e7 −3.39908
\(93\) −1.52793e6 −0.196975
\(94\) −6.89769e6 −0.856557
\(95\) −1.26454e7 −1.51321
\(96\) −4.64653e6 −0.536018
\(97\) −3.92765e6 −0.436950 −0.218475 0.975843i \(-0.570108\pi\)
−0.218475 + 0.975843i \(0.570108\pi\)
\(98\) −2.52911e7 −2.71442
\(99\) 859355. 0.0890122
\(100\) 5.64006e6 0.564006
\(101\) −1.82904e7 −1.76644 −0.883218 0.468962i \(-0.844628\pi\)
−0.883218 + 0.468962i \(0.844628\pi\)
\(102\) 9.23898e6 0.862032
\(103\) 1.88696e7 1.70151 0.850753 0.525566i \(-0.176146\pi\)
0.850753 + 0.525566i \(0.176146\pi\)
\(104\) −3.20892e7 −2.79732
\(105\) −1.58119e7 −1.33297
\(106\) 1.05838e7 0.863118
\(107\) 1.12587e7 0.888475 0.444237 0.895909i \(-0.353475\pi\)
0.444237 + 0.895909i \(0.353475\pi\)
\(108\) −3.08078e7 −2.35330
\(109\) 1.72287e7 1.27427 0.637133 0.770754i \(-0.280120\pi\)
0.637133 + 0.770754i \(0.280120\pi\)
\(110\) −5.65342e6 −0.404983
\(111\) −2.80873e6 −0.194931
\(112\) 3.74248e7 2.51707
\(113\) 1.18251e7 0.770955 0.385478 0.922717i \(-0.374037\pi\)
0.385478 + 0.922717i \(0.374037\pi\)
\(114\) −2.84957e7 −1.80141
\(115\) 2.84758e7 1.74595
\(116\) 5.18218e7 3.08255
\(117\) −1.00928e7 −0.582586
\(118\) −2.95734e7 −1.65697
\(119\) −1.88429e7 −1.02502
\(120\) 3.35661e7 1.77324
\(121\) −1.86894e7 −0.959060
\(122\) −2.14086e7 −1.06740
\(123\) 2.42422e6 0.117464
\(124\) −1.22033e7 −0.574780
\(125\) 1.81685e7 0.832020
\(126\) 2.79867e7 1.24639
\(127\) −4.54097e6 −0.196714 −0.0983570 0.995151i \(-0.531359\pi\)
−0.0983570 + 0.995151i \(0.531359\pi\)
\(128\) 3.00002e7 1.26441
\(129\) −1.77050e7 −0.726159
\(130\) 6.63970e7 2.65062
\(131\) 2.84820e7 1.10693 0.553466 0.832872i \(-0.313305\pi\)
0.553466 + 0.832872i \(0.313305\pi\)
\(132\) −8.73827e6 −0.330687
\(133\) 5.81169e7 2.14201
\(134\) 6.93436e7 2.48966
\(135\) 3.45556e7 1.20879
\(136\) 4.00005e7 1.36358
\(137\) 2.88141e6 0.0957377 0.0478688 0.998854i \(-0.484757\pi\)
0.0478688 + 0.998854i \(0.484757\pi\)
\(138\) 6.41685e7 2.07848
\(139\) 4.54316e7 1.43485 0.717424 0.696637i \(-0.245321\pi\)
0.717424 + 0.696637i \(0.245321\pi\)
\(140\) −1.26287e8 −3.88964
\(141\) 1.19585e7 0.359259
\(142\) 3.29919e7 0.966938
\(143\) −9.37000e6 −0.267956
\(144\) −2.49879e7 −0.697365
\(145\) −5.81259e7 −1.58337
\(146\) 1.11388e8 2.96212
\(147\) 4.38469e7 1.13849
\(148\) −2.24329e7 −0.568813
\(149\) −3.84291e7 −0.951719 −0.475859 0.879521i \(-0.657863\pi\)
−0.475859 + 0.879521i \(0.657863\pi\)
\(150\) −1.42557e7 −0.344880
\(151\) 3.91346e7 0.925000 0.462500 0.886619i \(-0.346953\pi\)
0.462500 + 0.886619i \(0.346953\pi\)
\(152\) −1.23373e8 −2.84950
\(153\) 1.25811e7 0.283986
\(154\) 2.59825e7 0.573269
\(155\) 1.36878e7 0.295239
\(156\) 1.02627e8 2.16435
\(157\) 2.65725e7 0.548005 0.274002 0.961729i \(-0.411652\pi\)
0.274002 + 0.961729i \(0.411652\pi\)
\(158\) −7.14780e7 −1.44169
\(159\) −1.83490e7 −0.362011
\(160\) 4.16256e7 0.803417
\(161\) −1.30872e8 −2.47147
\(162\) 3.53925e7 0.654047
\(163\) −8.26037e7 −1.49397 −0.746987 0.664839i \(-0.768500\pi\)
−0.746987 + 0.664839i \(0.768500\pi\)
\(164\) 1.93618e7 0.342762
\(165\) 9.80127e6 0.169859
\(166\) 1.42357e8 2.41546
\(167\) 4.39922e7 0.730916 0.365458 0.930828i \(-0.380912\pi\)
0.365458 + 0.930828i \(0.380912\pi\)
\(168\) −1.54266e8 −2.51009
\(169\) 4.72982e7 0.753774
\(170\) −8.27667e7 −1.29207
\(171\) −3.88037e7 −0.593453
\(172\) −1.41407e8 −2.11895
\(173\) 9.93656e7 1.45906 0.729532 0.683946i \(-0.239738\pi\)
0.729532 + 0.683946i \(0.239738\pi\)
\(174\) −1.30983e8 −1.88492
\(175\) 2.90744e7 0.410089
\(176\) −2.31984e7 −0.320748
\(177\) 5.12711e7 0.694970
\(178\) 2.29334e8 3.04788
\(179\) 9.08490e7 1.18395 0.591977 0.805955i \(-0.298348\pi\)
0.591977 + 0.805955i \(0.298348\pi\)
\(180\) 8.43194e7 1.07764
\(181\) −6.48032e7 −0.812309 −0.406155 0.913804i \(-0.633131\pi\)
−0.406155 + 0.913804i \(0.633131\pi\)
\(182\) −3.05154e8 −3.75205
\(183\) 3.71159e7 0.447693
\(184\) 2.77820e8 3.28777
\(185\) 2.51618e7 0.292174
\(186\) 3.08447e7 0.351468
\(187\) 1.16801e7 0.130617
\(188\) 9.55102e7 1.04833
\(189\) −1.58814e8 −1.71109
\(190\) 2.55277e8 2.70006
\(191\) −1.41189e8 −1.46617 −0.733085 0.680136i \(-0.761920\pi\)
−0.733085 + 0.680136i \(0.761920\pi\)
\(192\) −2.25492e7 −0.229920
\(193\) 7.55673e6 0.0756629 0.0378315 0.999284i \(-0.487955\pi\)
0.0378315 + 0.999284i \(0.487955\pi\)
\(194\) 7.92887e7 0.779660
\(195\) −1.15112e8 −1.11173
\(196\) 3.50198e8 3.32214
\(197\) 4.96606e7 0.462786 0.231393 0.972860i \(-0.425672\pi\)
0.231393 + 0.972860i \(0.425672\pi\)
\(198\) −1.73481e7 −0.158827
\(199\) −1.06500e8 −0.957994 −0.478997 0.877817i \(-0.658999\pi\)
−0.478997 + 0.877817i \(0.658999\pi\)
\(200\) −6.17205e7 −0.545537
\(201\) −1.20220e8 −1.04422
\(202\) 3.69234e8 3.15189
\(203\) 2.67141e8 2.24132
\(204\) −1.27929e8 −1.05503
\(205\) −2.17172e7 −0.176062
\(206\) −3.80927e8 −3.03604
\(207\) 8.73807e7 0.684730
\(208\) 2.72456e8 2.09930
\(209\) −3.60248e7 −0.272954
\(210\) 3.19199e8 2.37845
\(211\) −1.73101e8 −1.26856 −0.634280 0.773103i \(-0.718704\pi\)
−0.634280 + 0.773103i \(0.718704\pi\)
\(212\) −1.46550e8 −1.05636
\(213\) −5.71978e7 −0.405556
\(214\) −2.27283e8 −1.58533
\(215\) 1.58609e8 1.08841
\(216\) 3.37137e8 2.27624
\(217\) −6.29078e7 −0.417922
\(218\) −3.47801e8 −2.27370
\(219\) −1.93112e8 −1.24238
\(220\) 7.82811e7 0.495653
\(221\) −1.37178e8 −0.854892
\(222\) 5.67008e7 0.347819
\(223\) −2.79295e8 −1.68654 −0.843270 0.537491i \(-0.819372\pi\)
−0.843270 + 0.537491i \(0.819372\pi\)
\(224\) −1.91307e8 −1.13727
\(225\) −1.94125e7 −0.113617
\(226\) −2.38716e8 −1.37563
\(227\) −5.76230e6 −0.0326968 −0.0163484 0.999866i \(-0.505204\pi\)
−0.0163484 + 0.999866i \(0.505204\pi\)
\(228\) 3.94571e8 2.20472
\(229\) 3.01983e8 1.66172 0.830860 0.556481i \(-0.187849\pi\)
0.830860 + 0.556481i \(0.187849\pi\)
\(230\) −5.74849e8 −3.11535
\(231\) −4.50456e7 −0.240442
\(232\) −5.67098e8 −2.98161
\(233\) 4.30976e7 0.223207 0.111603 0.993753i \(-0.464401\pi\)
0.111603 + 0.993753i \(0.464401\pi\)
\(234\) 2.03746e8 1.03952
\(235\) −1.07129e8 −0.538480
\(236\) 4.09494e8 2.02794
\(237\) 1.23921e8 0.604679
\(238\) 3.80387e8 1.82897
\(239\) 1.34579e8 0.637652 0.318826 0.947813i \(-0.396711\pi\)
0.318826 + 0.947813i \(0.396711\pi\)
\(240\) −2.84996e8 −1.33076
\(241\) −1.58485e8 −0.729336 −0.364668 0.931138i \(-0.618817\pi\)
−0.364668 + 0.931138i \(0.618817\pi\)
\(242\) 3.77288e8 1.71127
\(243\) 1.79678e8 0.803292
\(244\) 2.96438e8 1.30638
\(245\) −3.92799e8 −1.70643
\(246\) −4.89384e7 −0.209593
\(247\) 4.23097e8 1.78649
\(248\) 1.33543e8 0.555958
\(249\) −2.46802e8 −1.01310
\(250\) −3.66773e8 −1.48459
\(251\) 2.38896e8 0.953564 0.476782 0.879021i \(-0.341803\pi\)
0.476782 + 0.879021i \(0.341803\pi\)
\(252\) −3.87524e8 −1.52544
\(253\) 8.11231e7 0.314936
\(254\) 9.16699e7 0.351001
\(255\) 1.43492e8 0.541921
\(256\) −5.23155e8 −1.94890
\(257\) −1.21568e8 −0.446739 −0.223370 0.974734i \(-0.571706\pi\)
−0.223370 + 0.974734i \(0.571706\pi\)
\(258\) 3.57416e8 1.29570
\(259\) −1.15641e8 −0.413584
\(260\) −9.19379e8 −3.24405
\(261\) −1.78365e8 −0.620966
\(262\) −5.74975e8 −1.97513
\(263\) −4.45794e8 −1.51108 −0.755542 0.655100i \(-0.772627\pi\)
−0.755542 + 0.655100i \(0.772627\pi\)
\(264\) 9.56248e7 0.319858
\(265\) 1.64378e8 0.542604
\(266\) −1.17322e9 −3.82204
\(267\) −3.97594e8 −1.27835
\(268\) −9.60180e8 −3.04706
\(269\) 4.13312e8 1.29463 0.647314 0.762224i \(-0.275892\pi\)
0.647314 + 0.762224i \(0.275892\pi\)
\(270\) −6.97584e8 −2.15687
\(271\) 4.45149e8 1.35867 0.679334 0.733830i \(-0.262269\pi\)
0.679334 + 0.733830i \(0.262269\pi\)
\(272\) −3.39628e8 −1.02332
\(273\) 5.29042e8 1.57370
\(274\) −5.81679e7 −0.170827
\(275\) −1.80223e7 −0.0522572
\(276\) −8.88522e8 −2.54382
\(277\) −4.55821e7 −0.128859 −0.0644296 0.997922i \(-0.520523\pi\)
−0.0644296 + 0.997922i \(0.520523\pi\)
\(278\) −9.17141e8 −2.56023
\(279\) 4.20024e7 0.115787
\(280\) 1.38198e9 3.76227
\(281\) −6.46666e8 −1.73863 −0.869317 0.494255i \(-0.835441\pi\)
−0.869317 + 0.494255i \(0.835441\pi\)
\(282\) −2.41409e8 −0.641035
\(283\) −1.71067e8 −0.448656 −0.224328 0.974514i \(-0.572019\pi\)
−0.224328 + 0.974514i \(0.572019\pi\)
\(284\) −4.56829e8 −1.18342
\(285\) −4.42570e8 −1.13247
\(286\) 1.89155e8 0.478120
\(287\) 9.98099e7 0.249222
\(288\) 1.27732e8 0.315085
\(289\) −2.39341e8 −0.583276
\(290\) 1.17341e9 2.82524
\(291\) −1.37462e8 −0.327007
\(292\) −1.54235e9 −3.62530
\(293\) −7.86048e8 −1.82563 −0.912814 0.408376i \(-0.866095\pi\)
−0.912814 + 0.408376i \(0.866095\pi\)
\(294\) −8.85152e8 −2.03143
\(295\) −4.59308e8 −1.04166
\(296\) 2.45488e8 0.550186
\(297\) 9.84436e7 0.218042
\(298\) 7.75781e8 1.69817
\(299\) −9.52757e8 −2.06126
\(300\) 1.97394e8 0.422094
\(301\) −7.28951e8 −1.54069
\(302\) −7.90023e8 −1.65050
\(303\) −6.40137e8 −1.32198
\(304\) 1.04751e9 2.13846
\(305\) −3.32500e8 −0.671030
\(306\) −2.53978e8 −0.506723
\(307\) −6.05296e8 −1.19394 −0.596971 0.802263i \(-0.703629\pi\)
−0.596971 + 0.802263i \(0.703629\pi\)
\(308\) −3.59772e8 −0.701617
\(309\) 6.60410e8 1.27338
\(310\) −2.76320e8 −0.526801
\(311\) −2.15847e8 −0.406898 −0.203449 0.979086i \(-0.565215\pi\)
−0.203449 + 0.979086i \(0.565215\pi\)
\(312\) −1.12307e9 −2.09347
\(313\) −1.22640e8 −0.226062 −0.113031 0.993591i \(-0.536056\pi\)
−0.113031 + 0.993591i \(0.536056\pi\)
\(314\) −5.36428e8 −0.977817
\(315\) 4.34665e8 0.783552
\(316\) 9.89735e8 1.76447
\(317\) 3.18524e8 0.561610 0.280805 0.959765i \(-0.409399\pi\)
0.280805 + 0.959765i \(0.409399\pi\)
\(318\) 3.70417e8 0.645945
\(319\) −1.65592e8 −0.285609
\(320\) 2.02005e8 0.344617
\(321\) 3.94038e8 0.664922
\(322\) 2.64195e9 4.40990
\(323\) −5.27408e8 −0.870839
\(324\) −4.90069e8 −0.800479
\(325\) 2.11664e8 0.342024
\(326\) 1.66755e9 2.66573
\(327\) 6.02979e8 0.953642
\(328\) −2.11881e8 −0.331538
\(329\) 4.92354e8 0.762240
\(330\) −1.97861e8 −0.303083
\(331\) −8.17220e8 −1.23863 −0.619314 0.785143i \(-0.712589\pi\)
−0.619314 + 0.785143i \(0.712589\pi\)
\(332\) −1.97117e9 −2.95624
\(333\) 7.72116e7 0.114585
\(334\) −8.88083e8 −1.30419
\(335\) 1.07698e9 1.56514
\(336\) 1.30981e9 1.88374
\(337\) −8.51210e8 −1.21152 −0.605762 0.795646i \(-0.707132\pi\)
−0.605762 + 0.795646i \(0.707132\pi\)
\(338\) −9.54823e8 −1.34498
\(339\) 4.13860e8 0.576972
\(340\) 1.14605e9 1.58134
\(341\) 3.89945e7 0.0532554
\(342\) 7.83341e8 1.05891
\(343\) 6.18574e8 0.827680
\(344\) 1.54745e9 2.04956
\(345\) 9.96609e8 1.30665
\(346\) −2.00592e9 −2.60344
\(347\) 1.05847e8 0.135996 0.0679979 0.997685i \(-0.478339\pi\)
0.0679979 + 0.997685i \(0.478339\pi\)
\(348\) 1.81369e9 2.30693
\(349\) −5.35227e8 −0.673983 −0.336991 0.941508i \(-0.609409\pi\)
−0.336991 + 0.941508i \(0.609409\pi\)
\(350\) −5.86934e8 −0.731731
\(351\) −1.15618e9 −1.42709
\(352\) 1.18585e8 0.144921
\(353\) 1.15795e9 1.40113 0.700564 0.713590i \(-0.252932\pi\)
0.700564 + 0.713590i \(0.252932\pi\)
\(354\) −1.03503e9 −1.24005
\(355\) 5.12402e8 0.607871
\(356\) −3.17552e9 −3.73026
\(357\) −6.59473e8 −0.767112
\(358\) −1.83400e9 −2.11256
\(359\) 2.73189e8 0.311626 0.155813 0.987787i \(-0.450200\pi\)
0.155813 + 0.987787i \(0.450200\pi\)
\(360\) −9.22727e8 −1.04235
\(361\) 7.32807e8 0.819813
\(362\) 1.30820e9 1.44942
\(363\) −6.54100e8 −0.717746
\(364\) 4.22537e9 4.59209
\(365\) 1.72998e9 1.86216
\(366\) −7.49270e8 −0.798830
\(367\) 8.77227e8 0.926362 0.463181 0.886264i \(-0.346708\pi\)
0.463181 + 0.886264i \(0.346708\pi\)
\(368\) −2.35885e9 −2.46737
\(369\) −6.66413e7 −0.0690480
\(370\) −5.07950e8 −0.521332
\(371\) −7.55465e8 −0.768078
\(372\) −4.27097e8 −0.430157
\(373\) −3.96814e8 −0.395919 −0.197959 0.980210i \(-0.563431\pi\)
−0.197959 + 0.980210i \(0.563431\pi\)
\(374\) −2.35790e8 −0.233064
\(375\) 6.35871e8 0.622672
\(376\) −1.04519e9 −1.01400
\(377\) 1.94481e9 1.86931
\(378\) 3.20603e9 3.05313
\(379\) 6.97685e7 0.0658298 0.0329149 0.999458i \(-0.489521\pi\)
0.0329149 + 0.999458i \(0.489521\pi\)
\(380\) −3.53474e9 −3.30457
\(381\) −1.58927e8 −0.147218
\(382\) 2.85023e9 2.61612
\(383\) −1.78953e9 −1.62759 −0.813793 0.581155i \(-0.802601\pi\)
−0.813793 + 0.581155i \(0.802601\pi\)
\(384\) 1.04996e9 0.946269
\(385\) 4.03538e8 0.360389
\(386\) −1.52550e8 −0.135007
\(387\) 4.86707e8 0.426854
\(388\) −1.09789e9 −0.954215
\(389\) −6.69935e8 −0.577044 −0.288522 0.957473i \(-0.593164\pi\)
−0.288522 + 0.957473i \(0.593164\pi\)
\(390\) 2.32380e9 1.98368
\(391\) 1.18765e9 1.00478
\(392\) −3.83230e9 −3.21335
\(393\) 9.96829e8 0.828412
\(394\) −1.00251e9 −0.825759
\(395\) −1.11013e9 −0.906329
\(396\) 2.40213e8 0.194386
\(397\) 3.64916e8 0.292703 0.146351 0.989233i \(-0.453247\pi\)
0.146351 + 0.989233i \(0.453247\pi\)
\(398\) 2.14994e9 1.70937
\(399\) 2.03401e9 1.60305
\(400\) 5.24042e8 0.409408
\(401\) 5.75325e6 0.00445562 0.00222781 0.999998i \(-0.499291\pi\)
0.00222781 + 0.999998i \(0.499291\pi\)
\(402\) 2.42692e9 1.86322
\(403\) −4.57974e8 −0.348557
\(404\) −5.11267e9 −3.85756
\(405\) 5.49686e8 0.411170
\(406\) −5.39285e9 −3.99924
\(407\) 7.16823e7 0.0527025
\(408\) 1.39996e9 1.02048
\(409\) −2.70795e9 −1.95708 −0.978541 0.206054i \(-0.933938\pi\)
−0.978541 + 0.206054i \(0.933938\pi\)
\(410\) 4.38411e8 0.314151
\(411\) 1.00845e8 0.0716487
\(412\) 5.27459e9 3.71576
\(413\) 2.11093e9 1.47452
\(414\) −1.76398e9 −1.22178
\(415\) 2.21096e9 1.51849
\(416\) −1.39273e9 −0.948508
\(417\) 1.59004e9 1.07382
\(418\) 7.27244e8 0.487039
\(419\) −1.91901e9 −1.27447 −0.637234 0.770670i \(-0.719922\pi\)
−0.637234 + 0.770670i \(0.719922\pi\)
\(420\) −4.41985e9 −2.91095
\(421\) 1.43923e9 0.940029 0.470015 0.882659i \(-0.344249\pi\)
0.470015 + 0.882659i \(0.344249\pi\)
\(422\) 3.49445e9 2.26352
\(423\) −3.28736e8 −0.211181
\(424\) 1.60373e9 1.02177
\(425\) −2.63849e8 −0.166722
\(426\) 1.15467e9 0.723643
\(427\) 1.52814e9 0.949870
\(428\) 3.14712e9 1.94026
\(429\) −3.27936e8 −0.200534
\(430\) −3.20189e9 −1.94208
\(431\) 5.52326e8 0.332296 0.166148 0.986101i \(-0.446867\pi\)
0.166148 + 0.986101i \(0.446867\pi\)
\(432\) −2.86249e9 −1.70825
\(433\) −2.10681e9 −1.24715 −0.623573 0.781765i \(-0.714320\pi\)
−0.623573 + 0.781765i \(0.714320\pi\)
\(434\) 1.26994e9 0.745708
\(435\) −2.03432e9 −1.18497
\(436\) 4.81590e9 2.78275
\(437\) −3.66307e9 −2.09971
\(438\) 3.89841e9 2.21681
\(439\) −5.02898e8 −0.283697 −0.141848 0.989888i \(-0.545305\pi\)
−0.141848 + 0.989888i \(0.545305\pi\)
\(440\) −8.56648e8 −0.479422
\(441\) −1.20534e9 −0.669231
\(442\) 2.76925e9 1.52540
\(443\) −5.97939e7 −0.0326771 −0.0163386 0.999867i \(-0.505201\pi\)
−0.0163386 + 0.999867i \(0.505201\pi\)
\(444\) −7.85118e8 −0.425691
\(445\) 3.56181e9 1.91607
\(446\) 5.63822e9 3.00933
\(447\) −1.34496e9 −0.712253
\(448\) −9.28395e8 −0.487820
\(449\) −1.06534e9 −0.555426 −0.277713 0.960664i \(-0.589576\pi\)
−0.277713 + 0.960664i \(0.589576\pi\)
\(450\) 3.91886e8 0.202729
\(451\) −6.18689e7 −0.0317581
\(452\) 3.30543e9 1.68362
\(453\) 1.36965e9 0.692257
\(454\) 1.16325e8 0.0583416
\(455\) −4.73938e9 −2.35875
\(456\) −4.31788e9 −2.13252
\(457\) 4.31135e8 0.211303 0.105652 0.994403i \(-0.466307\pi\)
0.105652 + 0.994403i \(0.466307\pi\)
\(458\) −6.09622e9 −2.96505
\(459\) 1.44123e9 0.695645
\(460\) 7.95976e9 3.81283
\(461\) −1.10900e8 −0.0527202 −0.0263601 0.999653i \(-0.508392\pi\)
−0.0263601 + 0.999653i \(0.508392\pi\)
\(462\) 9.09350e8 0.429027
\(463\) −3.05318e9 −1.42961 −0.714807 0.699322i \(-0.753485\pi\)
−0.714807 + 0.699322i \(0.753485\pi\)
\(464\) 4.81499e9 2.23760
\(465\) 4.79053e8 0.220952
\(466\) −8.70024e8 −0.398273
\(467\) 1.24433e9 0.565360 0.282680 0.959214i \(-0.408776\pi\)
0.282680 + 0.959214i \(0.408776\pi\)
\(468\) −2.82121e9 −1.27226
\(469\) −4.94971e9 −2.21552
\(470\) 2.16264e9 0.960821
\(471\) 9.30000e8 0.410119
\(472\) −4.48118e9 −1.96153
\(473\) 4.51853e8 0.196328
\(474\) −2.50163e9 −1.07894
\(475\) 8.13786e8 0.348404
\(476\) −5.26711e9 −2.23845
\(477\) 5.04410e8 0.212799
\(478\) −2.71678e9 −1.13778
\(479\) 6.52185e8 0.271142 0.135571 0.990768i \(-0.456713\pi\)
0.135571 + 0.990768i \(0.456713\pi\)
\(480\) 1.45684e9 0.601265
\(481\) −8.41878e8 −0.344938
\(482\) 3.19938e9 1.30137
\(483\) −4.58032e9 −1.84961
\(484\) −5.22419e9 −2.09440
\(485\) 1.23144e9 0.490138
\(486\) −3.62722e9 −1.43333
\(487\) −1.97337e8 −0.0774207 −0.0387103 0.999250i \(-0.512325\pi\)
−0.0387103 + 0.999250i \(0.512325\pi\)
\(488\) −3.24399e9 −1.26360
\(489\) −2.89101e9 −1.11807
\(490\) 7.92957e9 3.04483
\(491\) −3.67817e9 −1.40232 −0.701160 0.713004i \(-0.747334\pi\)
−0.701160 + 0.713004i \(0.747334\pi\)
\(492\) 6.77635e8 0.256518
\(493\) −2.42429e9 −0.911212
\(494\) −8.54118e9 −3.18767
\(495\) −2.69435e8 −0.0998472
\(496\) −1.13386e9 −0.417229
\(497\) −2.35495e9 −0.860467
\(498\) 4.98227e9 1.80769
\(499\) 1.32419e9 0.477088 0.238544 0.971132i \(-0.423330\pi\)
0.238544 + 0.971132i \(0.423330\pi\)
\(500\) 5.07859e9 1.81697
\(501\) 1.53966e9 0.547007
\(502\) −4.82266e9 −1.70147
\(503\) 6.74501e8 0.236317 0.118158 0.992995i \(-0.462301\pi\)
0.118158 + 0.992995i \(0.462301\pi\)
\(504\) 4.24076e9 1.47549
\(505\) 5.73462e9 1.98146
\(506\) −1.63766e9 −0.561948
\(507\) 1.65537e9 0.564113
\(508\) −1.26932e9 −0.429586
\(509\) 5.04001e9 1.69402 0.847011 0.531576i \(-0.178400\pi\)
0.847011 + 0.531576i \(0.178400\pi\)
\(510\) −2.89671e9 −0.966963
\(511\) −7.95081e9 −2.63596
\(512\) 6.72106e9 2.21306
\(513\) −4.44516e9 −1.45371
\(514\) 2.45414e9 0.797127
\(515\) −5.91623e9 −1.90862
\(516\) −4.94904e9 −1.58579
\(517\) −3.05194e8 −0.0971313
\(518\) 2.33449e9 0.737967
\(519\) 3.47765e9 1.09194
\(520\) 1.00610e10 3.13782
\(521\) −5.26761e9 −1.63185 −0.815927 0.578155i \(-0.803773\pi\)
−0.815927 + 0.578155i \(0.803773\pi\)
\(522\) 3.60071e9 1.10800
\(523\) −8.68988e8 −0.265618 −0.132809 0.991142i \(-0.542400\pi\)
−0.132809 + 0.991142i \(0.542400\pi\)
\(524\) 7.96151e9 2.41733
\(525\) 1.01756e9 0.306905
\(526\) 8.99937e9 2.69626
\(527\) 5.70885e8 0.169907
\(528\) −8.11910e8 −0.240043
\(529\) 4.84391e9 1.42266
\(530\) −3.31835e9 −0.968181
\(531\) −1.40943e9 −0.408520
\(532\) 1.62453e10 4.67775
\(533\) 7.26625e8 0.207857
\(534\) 8.02635e9 2.28099
\(535\) −3.52996e9 −0.996625
\(536\) 1.05075e10 2.94728
\(537\) 3.17958e9 0.886054
\(538\) −8.34365e9 −2.31003
\(539\) −1.11903e9 −0.307808
\(540\) 9.65923e9 2.63976
\(541\) 5.32013e9 1.44455 0.722273 0.691608i \(-0.243097\pi\)
0.722273 + 0.691608i \(0.243097\pi\)
\(542\) −8.98636e9 −2.42430
\(543\) −2.26802e9 −0.607921
\(544\) 1.73610e9 0.462358
\(545\) −5.40174e9 −1.42938
\(546\) −1.06799e10 −2.80798
\(547\) 1.63667e8 0.0427569
\(548\) 8.05433e8 0.209073
\(549\) −1.02031e9 −0.263165
\(550\) 3.63822e8 0.0932436
\(551\) 7.47720e9 1.90418
\(552\) 9.72330e9 2.46052
\(553\) 5.10207e9 1.28295
\(554\) 9.20180e8 0.229926
\(555\) 8.80627e8 0.218659
\(556\) 1.26994e10 3.13343
\(557\) −6.55254e9 −1.60663 −0.803316 0.595554i \(-0.796933\pi\)
−0.803316 + 0.595554i \(0.796933\pi\)
\(558\) −8.47916e8 −0.206601
\(559\) −5.30682e9 −1.28497
\(560\) −1.17339e10 −2.82347
\(561\) 4.08786e8 0.0977522
\(562\) 1.30545e10 3.10229
\(563\) −2.00044e9 −0.472440 −0.236220 0.971700i \(-0.575909\pi\)
−0.236220 + 0.971700i \(0.575909\pi\)
\(564\) 3.34272e9 0.784554
\(565\) −3.70753e9 −0.864800
\(566\) 3.45338e9 0.800547
\(567\) −2.52630e9 −0.582028
\(568\) 4.99919e9 1.14467
\(569\) −8.17447e9 −1.86023 −0.930115 0.367268i \(-0.880293\pi\)
−0.930115 + 0.367268i \(0.880293\pi\)
\(570\) 8.93430e9 2.02068
\(571\) −8.10920e9 −1.82285 −0.911426 0.411464i \(-0.865018\pi\)
−0.911426 + 0.411464i \(0.865018\pi\)
\(572\) −2.61917e9 −0.585164
\(573\) −4.94141e9 −1.09726
\(574\) −2.01489e9 −0.444693
\(575\) −1.83254e9 −0.401990
\(576\) 6.19873e8 0.135152
\(577\) −6.89467e9 −1.49416 −0.747081 0.664733i \(-0.768545\pi\)
−0.747081 + 0.664733i \(0.768545\pi\)
\(578\) 4.83164e9 1.04075
\(579\) 2.64474e8 0.0566250
\(580\) −1.62478e10 −3.45777
\(581\) −1.01613e10 −2.14949
\(582\) 2.77499e9 0.583486
\(583\) 4.68288e8 0.0978753
\(584\) 1.68783e10 3.50659
\(585\) 3.16440e9 0.653501
\(586\) 1.58682e10 3.25751
\(587\) −2.85927e9 −0.583474 −0.291737 0.956499i \(-0.594233\pi\)
−0.291737 + 0.956499i \(0.594233\pi\)
\(588\) 1.22564e10 2.48624
\(589\) −1.76077e9 −0.355059
\(590\) 9.27220e9 1.85866
\(591\) 1.73805e9 0.346342
\(592\) −2.08434e9 −0.412897
\(593\) 4.00405e9 0.788512 0.394256 0.919001i \(-0.371002\pi\)
0.394256 + 0.919001i \(0.371002\pi\)
\(594\) −1.98731e9 −0.389057
\(595\) 5.90784e9 1.14979
\(596\) −1.07420e10 −2.07837
\(597\) −3.72733e9 −0.716948
\(598\) 1.92336e10 3.67796
\(599\) 4.04548e9 0.769088 0.384544 0.923107i \(-0.374359\pi\)
0.384544 + 0.923107i \(0.374359\pi\)
\(600\) −2.16013e9 −0.408272
\(601\) −7.25054e9 −1.36242 −0.681208 0.732090i \(-0.738545\pi\)
−0.681208 + 0.732090i \(0.738545\pi\)
\(602\) 1.47156e10 2.74909
\(603\) 3.30483e9 0.613817
\(604\) 1.09392e10 2.02002
\(605\) 5.85971e9 1.07580
\(606\) 1.29226e10 2.35883
\(607\) 1.51128e9 0.274274 0.137137 0.990552i \(-0.456210\pi\)
0.137137 + 0.990552i \(0.456210\pi\)
\(608\) −5.35464e9 −0.966201
\(609\) 9.34953e9 1.67737
\(610\) 6.71228e9 1.19733
\(611\) 3.58438e9 0.635725
\(612\) 3.51675e9 0.620172
\(613\) −6.33593e9 −1.11096 −0.555480 0.831530i \(-0.687466\pi\)
−0.555480 + 0.831530i \(0.687466\pi\)
\(614\) 1.22193e10 2.13038
\(615\) −7.60069e8 −0.131762
\(616\) 3.93707e9 0.678641
\(617\) 1.06446e9 0.182445 0.0912224 0.995831i \(-0.470923\pi\)
0.0912224 + 0.995831i \(0.470923\pi\)
\(618\) −1.33319e10 −2.27213
\(619\) 9.54226e9 1.61709 0.808544 0.588435i \(-0.200256\pi\)
0.808544 + 0.588435i \(0.200256\pi\)
\(620\) 3.82612e9 0.644745
\(621\) 1.00099e10 1.67730
\(622\) 4.35738e9 0.726038
\(623\) −1.63697e10 −2.71227
\(624\) 9.53554e9 1.57108
\(625\) −7.27274e9 −1.19157
\(626\) 2.47577e9 0.403367
\(627\) −1.26082e9 −0.204275
\(628\) 7.42776e9 1.19674
\(629\) 1.04944e9 0.168143
\(630\) −8.77472e9 −1.39811
\(631\) −9.18516e9 −1.45540 −0.727702 0.685893i \(-0.759412\pi\)
−0.727702 + 0.685893i \(0.759412\pi\)
\(632\) −1.08309e10 −1.70669
\(633\) −6.05828e9 −0.949373
\(634\) −6.43014e9 −1.00209
\(635\) 1.42374e9 0.220659
\(636\) −5.12905e9 −0.790563
\(637\) 1.31425e10 2.01461
\(638\) 3.34285e9 0.509618
\(639\) 1.57236e9 0.238396
\(640\) −9.40602e9 −1.41833
\(641\) −8.43980e9 −1.26570 −0.632848 0.774276i \(-0.718114\pi\)
−0.632848 + 0.774276i \(0.718114\pi\)
\(642\) −7.95457e9 −1.18644
\(643\) −4.30752e9 −0.638983 −0.319491 0.947589i \(-0.603512\pi\)
−0.319491 + 0.947589i \(0.603512\pi\)
\(644\) −3.65822e10 −5.39721
\(645\) 5.55108e9 0.814551
\(646\) 1.06469e10 1.55386
\(647\) −2.03241e9 −0.295016 −0.147508 0.989061i \(-0.547125\pi\)
−0.147508 + 0.989061i \(0.547125\pi\)
\(648\) 5.36294e9 0.774267
\(649\) −1.30850e9 −0.187896
\(650\) −4.27293e9 −0.610280
\(651\) −2.20168e9 −0.312767
\(652\) −2.30900e10 −3.26255
\(653\) 4.80229e9 0.674920 0.337460 0.941340i \(-0.390432\pi\)
0.337460 + 0.941340i \(0.390432\pi\)
\(654\) −1.21725e10 −1.70160
\(655\) −8.93001e9 −1.24167
\(656\) 1.79899e9 0.248809
\(657\) 5.30862e9 0.730302
\(658\) −9.93930e9 −1.36008
\(659\) 2.92321e9 0.397888 0.198944 0.980011i \(-0.436249\pi\)
0.198944 + 0.980011i \(0.436249\pi\)
\(660\) 2.73972e9 0.370939
\(661\) −1.05017e10 −1.41434 −0.707169 0.707044i \(-0.750028\pi\)
−0.707169 + 0.707044i \(0.750028\pi\)
\(662\) 1.64975e10 2.21011
\(663\) −4.80102e9 −0.639789
\(664\) 2.15709e10 2.85944
\(665\) −1.82215e10 −2.40275
\(666\) −1.55869e9 −0.204457
\(667\) −1.68377e10 −2.19706
\(668\) 1.22970e10 1.59618
\(669\) −9.77492e9 −1.26218
\(670\) −2.17414e10 −2.79271
\(671\) −9.47242e8 −0.121041
\(672\) −6.69547e9 −0.851115
\(673\) 1.54011e9 0.194760 0.0973801 0.995247i \(-0.468954\pi\)
0.0973801 + 0.995247i \(0.468954\pi\)
\(674\) 1.71836e10 2.16175
\(675\) −2.22380e9 −0.278312
\(676\) 1.32211e10 1.64610
\(677\) −2.05812e9 −0.254924 −0.127462 0.991843i \(-0.540683\pi\)
−0.127462 + 0.991843i \(0.540683\pi\)
\(678\) −8.35472e9 −1.02950
\(679\) −5.65958e9 −0.693810
\(680\) −1.25414e10 −1.52956
\(681\) −2.01672e8 −0.0244698
\(682\) −7.87195e8 −0.0950248
\(683\) −1.30589e10 −1.56832 −0.784158 0.620562i \(-0.786905\pi\)
−0.784158 + 0.620562i \(0.786905\pi\)
\(684\) −1.08467e10 −1.29599
\(685\) −9.03413e8 −0.107391
\(686\) −1.24873e10 −1.47685
\(687\) 1.05690e10 1.24361
\(688\) −1.31387e10 −1.53813
\(689\) −5.49985e9 −0.640595
\(690\) −2.01189e10 −2.33148
\(691\) −2.65141e8 −0.0305706 −0.0152853 0.999883i \(-0.504866\pi\)
−0.0152853 + 0.999883i \(0.504866\pi\)
\(692\) 2.77754e10 3.18632
\(693\) 1.23830e9 0.141338
\(694\) −2.13677e9 −0.242661
\(695\) −1.42442e10 −1.60950
\(696\) −1.98476e10 −2.23139
\(697\) −9.05769e8 −0.101322
\(698\) 1.08048e10 1.20260
\(699\) 1.50835e9 0.167045
\(700\) 8.12710e9 0.895556
\(701\) 5.87882e9 0.644580 0.322290 0.946641i \(-0.395547\pi\)
0.322290 + 0.946641i \(0.395547\pi\)
\(702\) 2.33401e10 2.54638
\(703\) −3.23677e9 −0.351373
\(704\) 5.75482e8 0.0621623
\(705\) −3.74935e9 −0.402990
\(706\) −2.33759e10 −2.50006
\(707\) −2.63557e10 −2.80483
\(708\) 1.43317e10 1.51768
\(709\) 8.43565e9 0.888908 0.444454 0.895802i \(-0.353398\pi\)
0.444454 + 0.895802i \(0.353398\pi\)
\(710\) −1.03440e10 −1.08464
\(711\) −3.40656e9 −0.355445
\(712\) 3.47504e10 3.60811
\(713\) 3.96503e9 0.409669
\(714\) 1.33130e10 1.36878
\(715\) 2.93779e9 0.300573
\(716\) 2.53948e10 2.58553
\(717\) 4.71006e9 0.477209
\(718\) −5.51496e9 −0.556041
\(719\) 2.77509e8 0.0278437 0.0139218 0.999903i \(-0.495568\pi\)
0.0139218 + 0.999903i \(0.495568\pi\)
\(720\) 7.83448e9 0.782252
\(721\) 2.71904e10 2.70173
\(722\) −1.47934e10 −1.46281
\(723\) −5.54673e9 −0.545824
\(724\) −1.81143e10 −1.77393
\(725\) 3.74065e9 0.364556
\(726\) 1.32045e10 1.28069
\(727\) 4.44199e9 0.428753 0.214376 0.976751i \(-0.431228\pi\)
0.214376 + 0.976751i \(0.431228\pi\)
\(728\) −4.62392e10 −4.44171
\(729\) 1.01227e10 0.967724
\(730\) −3.49236e10 −3.32269
\(731\) 6.61518e9 0.626370
\(732\) 1.03749e10 0.977677
\(733\) 2.19158e8 0.0205539 0.0102769 0.999947i \(-0.496729\pi\)
0.0102769 + 0.999947i \(0.496729\pi\)
\(734\) −1.77089e10 −1.65293
\(735\) −1.37474e10 −1.27707
\(736\) 1.20579e10 1.11481
\(737\) 3.06817e9 0.282321
\(738\) 1.34531e9 0.123204
\(739\) −5.72237e9 −0.521580 −0.260790 0.965396i \(-0.583983\pi\)
−0.260790 + 0.965396i \(0.583983\pi\)
\(740\) 7.03342e9 0.638052
\(741\) 1.48078e10 1.33698
\(742\) 1.52508e10 1.37050
\(743\) 1.28569e9 0.114994 0.0574971 0.998346i \(-0.481688\pi\)
0.0574971 + 0.998346i \(0.481688\pi\)
\(744\) 4.67382e9 0.416071
\(745\) 1.20488e10 1.06757
\(746\) 8.01061e9 0.706447
\(747\) 6.78455e9 0.595523
\(748\) 3.26491e9 0.285243
\(749\) 1.62233e10 1.41076
\(750\) −1.28365e10 −1.11105
\(751\) 1.10781e10 0.954389 0.477195 0.878798i \(-0.341654\pi\)
0.477195 + 0.878798i \(0.341654\pi\)
\(752\) 8.87427e9 0.760975
\(753\) 8.36099e9 0.713634
\(754\) −3.92604e10 −3.33546
\(755\) −1.22699e10 −1.03760
\(756\) −4.43929e10 −3.73669
\(757\) 1.55439e10 1.30234 0.651172 0.758930i \(-0.274278\pi\)
0.651172 + 0.758930i \(0.274278\pi\)
\(758\) −1.40844e9 −0.117462
\(759\) 2.83919e9 0.235694
\(760\) 3.86814e10 3.19635
\(761\) −4.93995e7 −0.00406328 −0.00203164 0.999998i \(-0.500647\pi\)
−0.00203164 + 0.999998i \(0.500647\pi\)
\(762\) 3.20831e9 0.262684
\(763\) 2.48259e10 2.02334
\(764\) −3.94663e10 −3.20184
\(765\) −3.94456e9 −0.318555
\(766\) 3.61259e10 2.90414
\(767\) 1.53678e10 1.22978
\(768\) −1.83096e10 −1.45853
\(769\) −1.42651e10 −1.13118 −0.565590 0.824687i \(-0.691352\pi\)
−0.565590 + 0.824687i \(0.691352\pi\)
\(770\) −8.14634e9 −0.643051
\(771\) −4.25471e9 −0.334333
\(772\) 2.11231e9 0.165233
\(773\) 1.33979e10 1.04329 0.521647 0.853161i \(-0.325318\pi\)
0.521647 + 0.853161i \(0.325318\pi\)
\(774\) −9.82531e9 −0.761646
\(775\) −8.80870e8 −0.0679761
\(776\) 1.20144e10 0.922968
\(777\) −4.04727e9 −0.309520
\(778\) 1.35242e10 1.02963
\(779\) 2.79365e9 0.211734
\(780\) −3.21769e10 −2.42780
\(781\) 1.45976e9 0.109648
\(782\) −2.39755e10 −1.79285
\(783\) −2.04327e10 −1.52110
\(784\) 3.25384e10 2.41152
\(785\) −8.33133e9 −0.614711
\(786\) −2.01233e10 −1.47816
\(787\) −1.07376e10 −0.785227 −0.392614 0.919703i \(-0.628429\pi\)
−0.392614 + 0.919703i \(0.628429\pi\)
\(788\) 1.38815e10 1.01064
\(789\) −1.56021e10 −1.13087
\(790\) 2.24106e10 1.61718
\(791\) 1.70394e10 1.22416
\(792\) −2.62871e9 −0.188020
\(793\) 1.11250e10 0.792213
\(794\) −7.36668e9 −0.522276
\(795\) 5.75299e9 0.406077
\(796\) −2.97696e10 −2.09207
\(797\) 2.14499e10 1.50079 0.750397 0.660987i \(-0.229862\pi\)
0.750397 + 0.660987i \(0.229862\pi\)
\(798\) −4.10611e10 −2.86036
\(799\) −4.46808e9 −0.309890
\(800\) −2.67879e9 −0.184979
\(801\) 1.09298e10 0.751446
\(802\) −1.16143e8 −0.00795026
\(803\) 4.92845e9 0.335897
\(804\) −3.36049e10 −2.28037
\(805\) 4.10324e10 2.77231
\(806\) 9.24527e9 0.621938
\(807\) 1.44653e10 0.968881
\(808\) 5.59491e10 3.73124
\(809\) 3.44487e9 0.228745 0.114373 0.993438i \(-0.463514\pi\)
0.114373 + 0.993438i \(0.463514\pi\)
\(810\) −1.10967e10 −0.733661
\(811\) 1.37994e10 0.908421 0.454210 0.890894i \(-0.349921\pi\)
0.454210 + 0.890894i \(0.349921\pi\)
\(812\) 7.46732e10 4.89461
\(813\) 1.55796e10 1.01681
\(814\) −1.44707e9 −0.0940383
\(815\) 2.58989e10 1.67583
\(816\) −1.18865e10 −0.765839
\(817\) −2.04031e10 −1.30894
\(818\) 5.46662e10 3.49207
\(819\) −1.45433e10 −0.925057
\(820\) −6.07055e9 −0.384485
\(821\) 4.91296e9 0.309844 0.154922 0.987927i \(-0.450487\pi\)
0.154922 + 0.987927i \(0.450487\pi\)
\(822\) −2.03579e9 −0.127844
\(823\) −1.69039e10 −1.05703 −0.528516 0.848923i \(-0.677251\pi\)
−0.528516 + 0.848923i \(0.677251\pi\)
\(824\) −5.77210e10 −3.59409
\(825\) −6.30754e8 −0.0391085
\(826\) −4.26141e10 −2.63101
\(827\) 7.46121e9 0.458712 0.229356 0.973343i \(-0.426338\pi\)
0.229356 + 0.973343i \(0.426338\pi\)
\(828\) 2.44253e10 1.49532
\(829\) 2.06299e10 1.25764 0.628821 0.777550i \(-0.283538\pi\)
0.628821 + 0.777550i \(0.283538\pi\)
\(830\) −4.46333e10 −2.70948
\(831\) −1.59531e9 −0.0964363
\(832\) −6.75879e9 −0.406853
\(833\) −1.63827e10 −0.982037
\(834\) −3.20986e10 −1.91604
\(835\) −1.37929e10 −0.819887
\(836\) −1.00699e10 −0.596080
\(837\) 4.81160e9 0.283629
\(838\) 3.87397e10 2.27406
\(839\) −1.91337e10 −1.11849 −0.559246 0.829002i \(-0.688909\pi\)
−0.559246 + 0.829002i \(0.688909\pi\)
\(840\) 4.83674e10 2.81563
\(841\) 1.71199e10 0.992463
\(842\) −2.90541e10 −1.67732
\(843\) −2.26324e10 −1.30117
\(844\) −4.83865e10 −2.77029
\(845\) −1.48295e10 −0.845527
\(846\) 6.63629e9 0.376816
\(847\) −2.69306e10 −1.52284
\(848\) −1.36166e10 −0.766803
\(849\) −5.98709e9 −0.335767
\(850\) 5.32640e9 0.297486
\(851\) 7.28878e9 0.405416
\(852\) −1.59884e10 −0.885656
\(853\) 2.25524e10 1.24414 0.622072 0.782960i \(-0.286291\pi\)
0.622072 + 0.782960i \(0.286291\pi\)
\(854\) −3.08489e10 −1.69487
\(855\) 1.21662e10 0.665691
\(856\) −3.44396e10 −1.87672
\(857\) −3.11415e10 −1.69008 −0.845039 0.534705i \(-0.820423\pi\)
−0.845039 + 0.534705i \(0.820423\pi\)
\(858\) 6.62015e9 0.357818
\(859\) −4.11638e9 −0.221585 −0.110792 0.993844i \(-0.535339\pi\)
−0.110792 + 0.993844i \(0.535339\pi\)
\(860\) 4.43356e10 2.37688
\(861\) 3.49320e9 0.186514
\(862\) −1.11500e10 −0.592923
\(863\) −2.25000e10 −1.19164 −0.595821 0.803117i \(-0.703173\pi\)
−0.595821 + 0.803117i \(0.703173\pi\)
\(864\) 1.46324e10 0.771823
\(865\) −3.11543e10 −1.63667
\(866\) 4.25308e10 2.22531
\(867\) −8.37656e9 −0.436515
\(868\) −1.75845e10 −0.912662
\(869\) −3.16261e9 −0.163484
\(870\) 4.10675e10 2.11437
\(871\) −3.60343e10 −1.84779
\(872\) −5.27014e10 −2.69163
\(873\) 3.77880e9 0.192223
\(874\) 7.39474e10 3.74656
\(875\) 2.61801e10 1.32112
\(876\) −5.39801e10 −2.71312
\(877\) 3.89462e10 1.94969 0.974846 0.222879i \(-0.0715453\pi\)
0.974846 + 0.222879i \(0.0715453\pi\)
\(878\) 1.01522e10 0.506207
\(879\) −2.75105e10 −1.36627
\(880\) 7.27344e9 0.359791
\(881\) 2.20801e10 1.08789 0.543945 0.839121i \(-0.316930\pi\)
0.543945 + 0.839121i \(0.316930\pi\)
\(882\) 2.43327e10 1.19412
\(883\) 1.12408e10 0.549457 0.274728 0.961522i \(-0.411412\pi\)
0.274728 + 0.961522i \(0.411412\pi\)
\(884\) −3.83450e10 −1.86692
\(885\) −1.60751e10 −0.779565
\(886\) 1.20708e9 0.0583066
\(887\) −1.64151e10 −0.789788 −0.394894 0.918727i \(-0.629219\pi\)
−0.394894 + 0.918727i \(0.629219\pi\)
\(888\) 8.59173e9 0.411751
\(889\) −6.54335e9 −0.312352
\(890\) −7.19034e10 −3.41889
\(891\) 1.56597e9 0.0741672
\(892\) −7.80707e10 −3.68308
\(893\) 1.37809e10 0.647584
\(894\) 2.71512e10 1.27089
\(895\) −2.84841e10 −1.32807
\(896\) 4.32291e10 2.00770
\(897\) −3.33451e10 −1.54262
\(898\) 2.15064e10 0.991059
\(899\) −8.09358e9 −0.371520
\(900\) −5.42632e9 −0.248117
\(901\) 6.85580e9 0.312264
\(902\) 1.24897e9 0.0566667
\(903\) −2.55122e10 −1.15303
\(904\) −3.61721e10 −1.62849
\(905\) 2.03179e10 0.911188
\(906\) −2.76496e10 −1.23521
\(907\) −1.54682e10 −0.688357 −0.344178 0.938904i \(-0.611842\pi\)
−0.344178 + 0.938904i \(0.611842\pi\)
\(908\) −1.61072e9 −0.0714035
\(909\) 1.75972e10 0.777089
\(910\) 9.56754e10 4.20877
\(911\) 1.93795e10 0.849237 0.424618 0.905372i \(-0.360408\pi\)
0.424618 + 0.905372i \(0.360408\pi\)
\(912\) 3.66613e10 1.60039
\(913\) 6.29869e9 0.273906
\(914\) −8.70346e9 −0.377034
\(915\) −1.16370e10 −0.502189
\(916\) 8.44125e10 3.62888
\(917\) 4.10414e10 1.75764
\(918\) −2.90945e10 −1.24126
\(919\) −3.48975e10 −1.48317 −0.741583 0.670861i \(-0.765925\pi\)
−0.741583 + 0.670861i \(0.765925\pi\)
\(920\) −8.71054e10 −3.68797
\(921\) −2.11845e10 −0.893529
\(922\) 2.23877e9 0.0940699
\(923\) −1.71442e10 −0.717649
\(924\) −1.25915e10 −0.525080
\(925\) −1.61927e9 −0.0672704
\(926\) 6.16355e10 2.55089
\(927\) −1.81546e10 −0.748525
\(928\) −2.46132e10 −1.01100
\(929\) 3.05212e10 1.24896 0.624478 0.781042i \(-0.285312\pi\)
0.624478 + 0.781042i \(0.285312\pi\)
\(930\) −9.67080e9 −0.394250
\(931\) 5.05290e10 2.05219
\(932\) 1.20470e10 0.487440
\(933\) −7.55434e9 −0.304517
\(934\) −2.51196e10 −1.00879
\(935\) −3.66208e9 −0.146517
\(936\) 3.08731e10 1.23059
\(937\) −9.96636e8 −0.0395775 −0.0197887 0.999804i \(-0.506299\pi\)
−0.0197887 + 0.999804i \(0.506299\pi\)
\(938\) 9.99213e10 3.95320
\(939\) −4.29222e9 −0.169181
\(940\) −2.99455e10 −1.17594
\(941\) −1.05816e10 −0.413989 −0.206994 0.978342i \(-0.566368\pi\)
−0.206994 + 0.978342i \(0.566368\pi\)
\(942\) −1.87742e10 −0.731784
\(943\) −6.29094e9 −0.244301
\(944\) 3.80478e10 1.47207
\(945\) 4.97932e10 1.91937
\(946\) −9.12170e9 −0.350313
\(947\) −1.60623e10 −0.614588 −0.307294 0.951615i \(-0.599423\pi\)
−0.307294 + 0.951615i \(0.599423\pi\)
\(948\) 3.46392e10 1.32050
\(949\) −5.78826e10 −2.19845
\(950\) −1.64281e10 −0.621664
\(951\) 1.11479e10 0.420301
\(952\) 5.76392e10 2.16515
\(953\) −9.37744e9 −0.350961 −0.175481 0.984483i \(-0.556148\pi\)
−0.175481 + 0.984483i \(0.556148\pi\)
\(954\) −1.01827e10 −0.379702
\(955\) 4.42673e10 1.64464
\(956\) 3.76184e10 1.39251
\(957\) −5.79547e9 −0.213746
\(958\) −1.31659e10 −0.483805
\(959\) 4.15200e9 0.152017
\(960\) 7.06988e9 0.257907
\(961\) −2.56067e10 −0.930725
\(962\) 1.69953e10 0.615482
\(963\) −1.08320e10 −0.390857
\(964\) −4.43008e10 −1.59273
\(965\) −2.36927e9 −0.0848730
\(966\) 9.24643e10 3.30030
\(967\) 3.72823e10 1.32590 0.662949 0.748665i \(-0.269305\pi\)
0.662949 + 0.748665i \(0.269305\pi\)
\(968\) 5.71695e10 2.02582
\(969\) −1.84585e10 −0.651723
\(970\) −2.48595e10 −0.874564
\(971\) 3.77401e10 1.32293 0.661463 0.749977i \(-0.269936\pi\)
0.661463 + 0.749977i \(0.269936\pi\)
\(972\) 5.02250e10 1.75424
\(973\) 6.54651e10 2.27832
\(974\) 3.98370e9 0.138144
\(975\) 7.40794e9 0.255965
\(976\) 2.75434e10 0.948294
\(977\) 1.12330e9 0.0385360 0.0192680 0.999814i \(-0.493866\pi\)
0.0192680 + 0.999814i \(0.493866\pi\)
\(978\) 5.83617e10 1.99500
\(979\) 1.01471e10 0.345622
\(980\) −1.09798e11 −3.72653
\(981\) −1.65758e10 −0.560574
\(982\) 7.42524e10 2.50219
\(983\) −2.01354e10 −0.676119 −0.338060 0.941125i \(-0.609771\pi\)
−0.338060 + 0.941125i \(0.609771\pi\)
\(984\) −7.41552e9 −0.248118
\(985\) −1.55702e10 −0.519118
\(986\) 4.89398e10 1.62590
\(987\) 1.72317e10 0.570449
\(988\) 1.18267e11 3.90135
\(989\) 4.59452e10 1.51026
\(990\) 5.43917e9 0.178160
\(991\) −1.41141e10 −0.460676 −0.230338 0.973111i \(-0.573983\pi\)
−0.230338 + 0.973111i \(0.573983\pi\)
\(992\) 5.79605e9 0.188513
\(993\) −2.86015e10 −0.926971
\(994\) 4.75401e10 1.53535
\(995\) 3.33910e10 1.07461
\(996\) −6.89880e10 −2.21241
\(997\) 2.51409e10 0.803430 0.401715 0.915765i \(-0.368414\pi\)
0.401715 + 0.915765i \(0.368414\pi\)
\(998\) −2.67318e10 −0.851279
\(999\) 8.84499e9 0.280684
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.1.12 156
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.8.a.a.1.12 156 1.1 even 1 trivial