Properties

Label 169.10.a.a.1.4
Level $169$
Weight $10$
Character 169.1
Self dual yes
Analytic conductor $87.041$
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] = [169,10,Mod(1,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 169.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: \(87.0410563117\)
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: \( x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(36.6235\) of defining polynomial
Character \(\chi\) \(=\) 169.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+44.6235 q^{2} -51.0278 q^{3} +1479.26 q^{4} -151.187 q^{5} -2277.04 q^{6} -5436.58 q^{7} +43162.5 q^{8} -17079.2 q^{9} +O(q^{10})\) \(q+44.6235 q^{2} -51.0278 q^{3} +1479.26 q^{4} -151.187 q^{5} -2277.04 q^{6} -5436.58 q^{7} +43162.5 q^{8} -17079.2 q^{9} -6746.48 q^{10} -33841.6 q^{11} -75483.2 q^{12} -242599. q^{14} +7714.72 q^{15} +1.16868e6 q^{16} -588951. q^{17} -762133. q^{18} +123244. q^{19} -223644. q^{20} +277417. q^{21} -1.51013e6 q^{22} -1.55650e6 q^{23} -2.20248e6 q^{24} -1.93027e6 q^{25} +1.87589e6 q^{27} -8.04211e6 q^{28} +273935. q^{29} +344258. q^{30} +5.07992e6 q^{31} +3.00515e7 q^{32} +1.72686e6 q^{33} -2.62811e7 q^{34} +821938. q^{35} -2.52645e7 q^{36} -8.27832e6 q^{37} +5.49960e6 q^{38} -6.52559e6 q^{40} -1.69055e7 q^{41} +1.23793e7 q^{42} -1.23150e7 q^{43} -5.00604e7 q^{44} +2.58214e6 q^{45} -6.94565e7 q^{46} +3.71528e7 q^{47} -5.96352e7 q^{48} -1.07972e7 q^{49} -8.61353e7 q^{50} +3.00528e7 q^{51} -5.85375e7 q^{53} +8.37089e7 q^{54} +5.11640e6 q^{55} -2.34656e8 q^{56} -6.28889e6 q^{57} +1.22239e7 q^{58} +2.83630e7 q^{59} +1.14121e7 q^{60} +2.89102e7 q^{61} +2.26684e8 q^{62} +9.28523e7 q^{63} +7.42638e8 q^{64} +7.70586e7 q^{66} +9.52452e7 q^{67} -8.71211e8 q^{68} +7.94247e7 q^{69} +3.66778e7 q^{70} +1.35041e7 q^{71} -7.37179e8 q^{72} +1.20684e8 q^{73} -3.69408e8 q^{74} +9.84972e7 q^{75} +1.82310e8 q^{76} +1.83983e8 q^{77} +2.80552e8 q^{79} -1.76689e8 q^{80} +2.40447e8 q^{81} -7.54385e8 q^{82} -5.16031e8 q^{83} +4.10371e8 q^{84} +8.90415e7 q^{85} -5.49538e8 q^{86} -1.39783e7 q^{87} -1.46069e9 q^{88} -1.00683e9 q^{89} +1.15224e8 q^{90} -2.30247e9 q^{92} -2.59217e8 q^{93} +1.65789e9 q^{94} -1.86329e7 q^{95} -1.53346e9 q^{96} -1.99014e8 q^{97} -4.81809e8 q^{98} +5.77986e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 33 q^{2} - 163 q^{3} + 1429 q^{4} - 471 q^{5} + 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 33 q^{2} - 163 q^{3} + 1429 q^{4} - 471 q^{5} + 4529 q^{6} + 11241 q^{7} + 45543 q^{8} - 29953 q^{9} - 67831 q^{10} + 40140 q^{11} - 155479 q^{12} - 277653 q^{14} - 83307 q^{15} + 726609 q^{16} + 78717 q^{17} - 1691026 q^{18} - 209664 q^{19} - 870843 q^{20} - 1138431 q^{21} + 1364090 q^{22} - 4257444 q^{23} - 3561573 q^{24} - 2900157 q^{25} - 2077801 q^{27} - 4035181 q^{28} - 1647936 q^{29} - 744143 q^{30} + 11366002 q^{31} + 29458959 q^{32} + 14413222 q^{33} - 26257659 q^{34} - 13789797 q^{35} - 11587714 q^{36} - 4636891 q^{37} + 25172466 q^{38} + 22536791 q^{40} - 13859538 q^{41} + 75564923 q^{42} - 33368081 q^{43} - 66489222 q^{44} + 17423928 q^{45} - 71369332 q^{46} + 3943005 q^{47} - 620787 q^{48} + 23294923 q^{49} + 4217748 q^{50} - 19664471 q^{51} - 171019326 q^{53} + 64946915 q^{54} - 121160538 q^{55} - 281552967 q^{56} + 47829030 q^{57} - 79964734 q^{58} + 63389388 q^{59} - 37708135 q^{60} + 77050190 q^{61} - 95878740 q^{62} + 155695476 q^{63} + 768962465 q^{64} - 42396374 q^{66} + 41174072 q^{67} - 717615423 q^{68} + 546642556 q^{69} - 409056389 q^{70} - 252460989 q^{71} - 562579254 q^{72} - 594415068 q^{73} - 957058539 q^{74} + 533318748 q^{75} + 326897170 q^{76} + 561950454 q^{77} + 115998984 q^{79} + 509107233 q^{80} + 437803700 q^{81} - 875148240 q^{82} + 79577862 q^{83} - 108899441 q^{84} - 549463469 q^{85} + 589924887 q^{86} - 1087526510 q^{87} - 2327564370 q^{88} + 1152240276 q^{89} + 877550038 q^{90} - 4213481460 q^{92} - 1618266556 q^{93} + 1859909503 q^{94} - 1273705170 q^{95} - 3171454029 q^{96} - 1049098084 q^{97} - 420532254 q^{98} - 2132181050 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 44.6235 1.97210 0.986050 0.166451i \(-0.0532307\pi\)
0.986050 + 0.166451i \(0.0532307\pi\)
\(3\) −51.0278 −0.363715 −0.181857 0.983325i \(-0.558211\pi\)
−0.181857 + 0.983325i \(0.558211\pi\)
\(4\) 1479.26 2.88918
\(5\) −151.187 −0.108180 −0.0540902 0.998536i \(-0.517226\pi\)
−0.0540902 + 0.998536i \(0.517226\pi\)
\(6\) −2277.04 −0.717282
\(7\) −5436.58 −0.855824 −0.427912 0.903820i \(-0.640751\pi\)
−0.427912 + 0.903820i \(0.640751\pi\)
\(8\) 43162.5 3.72564
\(9\) −17079.2 −0.867712
\(10\) −6746.48 −0.213342
\(11\) −33841.6 −0.696921 −0.348461 0.937323i \(-0.613295\pi\)
−0.348461 + 0.937323i \(0.613295\pi\)
\(12\) −75483.2 −1.05084
\(13\) 0 0
\(14\) −242599. −1.68777
\(15\) 7714.72 0.0393468
\(16\) 1.16868e6 4.45816
\(17\) −588951. −1.71025 −0.855124 0.518424i \(-0.826519\pi\)
−0.855124 + 0.518424i \(0.826519\pi\)
\(18\) −762133. −1.71121
\(19\) 123244. 0.216958 0.108479 0.994099i \(-0.465402\pi\)
0.108479 + 0.994099i \(0.465402\pi\)
\(20\) −223644. −0.312552
\(21\) 277417. 0.311276
\(22\) −1.51013e6 −1.37440
\(23\) −1.55650e6 −1.15978 −0.579888 0.814696i \(-0.696904\pi\)
−0.579888 + 0.814696i \(0.696904\pi\)
\(24\) −2.20248e6 −1.35507
\(25\) −1.93027e6 −0.988297
\(26\) 0 0
\(27\) 1.87589e6 0.679314
\(28\) −8.04211e6 −2.47263
\(29\) 273935. 0.0719211 0.0359606 0.999353i \(-0.488551\pi\)
0.0359606 + 0.999353i \(0.488551\pi\)
\(30\) 344258. 0.0775958
\(31\) 5.07992e6 0.987936 0.493968 0.869480i \(-0.335546\pi\)
0.493968 + 0.869480i \(0.335546\pi\)
\(32\) 3.00515e7 5.06630
\(33\) 1.72686e6 0.253480
\(34\) −2.62811e7 −3.37278
\(35\) 821938. 0.0925834
\(36\) −2.52645e7 −2.50697
\(37\) −8.27832e6 −0.726164 −0.363082 0.931757i \(-0.618275\pi\)
−0.363082 + 0.931757i \(0.618275\pi\)
\(38\) 5.49960e6 0.427863
\(39\) 0 0
\(40\) −6.52559e6 −0.403041
\(41\) −1.69055e7 −0.934333 −0.467167 0.884169i \(-0.654725\pi\)
−0.467167 + 0.884169i \(0.654725\pi\)
\(42\) 1.23793e7 0.613867
\(43\) −1.23150e7 −0.549320 −0.274660 0.961541i \(-0.588565\pi\)
−0.274660 + 0.961541i \(0.588565\pi\)
\(44\) −5.00604e7 −2.01353
\(45\) 2.58214e6 0.0938693
\(46\) −6.94565e7 −2.28719
\(47\) 3.71528e7 1.11058 0.555292 0.831655i \(-0.312606\pi\)
0.555292 + 0.831655i \(0.312606\pi\)
\(48\) −5.96352e7 −1.62150
\(49\) −1.07972e7 −0.267565
\(50\) −8.61353e7 −1.94902
\(51\) 3.00528e7 0.622042
\(52\) 0 0
\(53\) −5.85375e7 −1.01904 −0.509521 0.860458i \(-0.670178\pi\)
−0.509521 + 0.860458i \(0.670178\pi\)
\(54\) 8.37089e7 1.33968
\(55\) 5.11640e6 0.0753932
\(56\) −2.34656e8 −3.18850
\(57\) −6.28889e6 −0.0789109
\(58\) 1.22239e7 0.141836
\(59\) 2.83630e7 0.304732 0.152366 0.988324i \(-0.451311\pi\)
0.152366 + 0.988324i \(0.451311\pi\)
\(60\) 1.14121e7 0.113680
\(61\) 2.89102e7 0.267342 0.133671 0.991026i \(-0.457323\pi\)
0.133671 + 0.991026i \(0.457323\pi\)
\(62\) 2.26684e8 1.94831
\(63\) 9.28523e7 0.742609
\(64\) 7.42638e8 5.53308
\(65\) 0 0
\(66\) 7.70586e7 0.499889
\(67\) 9.52452e7 0.577439 0.288720 0.957414i \(-0.406770\pi\)
0.288720 + 0.957414i \(0.406770\pi\)
\(68\) −8.71211e8 −4.94121
\(69\) 7.94247e7 0.421827
\(70\) 3.66778e7 0.182584
\(71\) 1.35041e7 0.0630671 0.0315335 0.999503i \(-0.489961\pi\)
0.0315335 + 0.999503i \(0.489961\pi\)
\(72\) −7.37179e8 −3.23278
\(73\) 1.20684e8 0.497390 0.248695 0.968582i \(-0.419998\pi\)
0.248695 + 0.968582i \(0.419998\pi\)
\(74\) −3.69408e8 −1.43207
\(75\) 9.84972e7 0.359458
\(76\) 1.82310e8 0.626831
\(77\) 1.83983e8 0.596442
\(78\) 0 0
\(79\) 2.80552e8 0.810386 0.405193 0.914231i \(-0.367204\pi\)
0.405193 + 0.914231i \(0.367204\pi\)
\(80\) −1.76689e8 −0.482286
\(81\) 2.40447e8 0.620635
\(82\) −7.54385e8 −1.84260
\(83\) −5.16031e8 −1.19350 −0.596752 0.802426i \(-0.703542\pi\)
−0.596752 + 0.802426i \(0.703542\pi\)
\(84\) 4.10371e8 0.899331
\(85\) 8.90415e7 0.185015
\(86\) −5.49538e8 −1.08331
\(87\) −1.39783e7 −0.0261588
\(88\) −1.46069e9 −2.59648
\(89\) −1.00683e9 −1.70099 −0.850497 0.525980i \(-0.823699\pi\)
−0.850497 + 0.525980i \(0.823699\pi\)
\(90\) 1.15224e8 0.185120
\(91\) 0 0
\(92\) −2.30247e9 −3.35080
\(93\) −2.59217e8 −0.359327
\(94\) 1.65789e9 2.19018
\(95\) −1.86329e7 −0.0234706
\(96\) −1.53346e9 −1.84269
\(97\) −1.99014e8 −0.228250 −0.114125 0.993466i \(-0.536406\pi\)
−0.114125 + 0.993466i \(0.536406\pi\)
\(98\) −4.81809e8 −0.527664
\(99\) 5.77986e8 0.604727
\(100\) −2.85536e9 −2.85536
\(101\) −6.11325e8 −0.584556 −0.292278 0.956333i \(-0.594413\pi\)
−0.292278 + 0.956333i \(0.594413\pi\)
\(102\) 1.34106e9 1.22673
\(103\) −1.23832e9 −1.08409 −0.542047 0.840348i \(-0.682351\pi\)
−0.542047 + 0.840348i \(0.682351\pi\)
\(104\) 0 0
\(105\) −4.19417e7 −0.0336739
\(106\) −2.61215e9 −2.00965
\(107\) 1.86278e8 0.137383 0.0686917 0.997638i \(-0.478118\pi\)
0.0686917 + 0.997638i \(0.478118\pi\)
\(108\) 2.77493e9 1.96266
\(109\) 1.79600e9 1.21868 0.609338 0.792911i \(-0.291435\pi\)
0.609338 + 0.792911i \(0.291435\pi\)
\(110\) 2.28312e8 0.148683
\(111\) 4.22424e8 0.264116
\(112\) −6.35363e9 −3.81541
\(113\) 1.28965e9 0.744081 0.372041 0.928216i \(-0.378658\pi\)
0.372041 + 0.928216i \(0.378658\pi\)
\(114\) −2.80632e8 −0.155620
\(115\) 2.35322e8 0.125465
\(116\) 4.05221e8 0.207793
\(117\) 0 0
\(118\) 1.26566e9 0.600961
\(119\) 3.20188e9 1.46367
\(120\) 3.32986e8 0.146592
\(121\) −1.21269e9 −0.514301
\(122\) 1.29008e9 0.527225
\(123\) 8.62652e8 0.339831
\(124\) 7.51451e9 2.85432
\(125\) 5.87117e8 0.215095
\(126\) 4.14340e9 1.46450
\(127\) 2.03329e9 0.693558 0.346779 0.937947i \(-0.387275\pi\)
0.346779 + 0.937947i \(0.387275\pi\)
\(128\) 1.77528e10 5.84549
\(129\) 6.28406e8 0.199796
\(130\) 0 0
\(131\) −1.89177e8 −0.0561237 −0.0280619 0.999606i \(-0.508934\pi\)
−0.0280619 + 0.999606i \(0.508934\pi\)
\(132\) 2.55447e9 0.732350
\(133\) −6.70028e8 −0.185678
\(134\) 4.25017e9 1.13877
\(135\) −2.83610e8 −0.0734884
\(136\) −2.54206e10 −6.37177
\(137\) −4.31369e9 −1.04618 −0.523090 0.852277i \(-0.675221\pi\)
−0.523090 + 0.852277i \(0.675221\pi\)
\(138\) 3.54421e9 0.831885
\(139\) −4.25633e9 −0.967094 −0.483547 0.875318i \(-0.660652\pi\)
−0.483547 + 0.875318i \(0.660652\pi\)
\(140\) 1.21586e9 0.267490
\(141\) −1.89583e9 −0.403936
\(142\) 6.02600e8 0.124375
\(143\) 0 0
\(144\) −1.99601e10 −3.86840
\(145\) −4.14153e7 −0.00778045
\(146\) 5.38535e9 0.980903
\(147\) 5.50957e8 0.0973172
\(148\) −1.22458e10 −2.09801
\(149\) −4.12743e9 −0.686027 −0.343013 0.939330i \(-0.611448\pi\)
−0.343013 + 0.939330i \(0.611448\pi\)
\(150\) 4.39529e9 0.708887
\(151\) 2.08019e9 0.325617 0.162808 0.986658i \(-0.447945\pi\)
0.162808 + 0.986658i \(0.447945\pi\)
\(152\) 5.31953e9 0.808309
\(153\) 1.00588e10 1.48400
\(154\) 8.20995e9 1.17624
\(155\) −7.68015e8 −0.106875
\(156\) 0 0
\(157\) 8.58336e9 1.12748 0.563740 0.825952i \(-0.309362\pi\)
0.563740 + 0.825952i \(0.309362\pi\)
\(158\) 1.25192e10 1.59816
\(159\) 2.98704e9 0.370641
\(160\) −4.54338e9 −0.548074
\(161\) 8.46204e9 0.992564
\(162\) 1.07296e10 1.22395
\(163\) 1.61852e10 1.79586 0.897931 0.440136i \(-0.145070\pi\)
0.897931 + 0.440136i \(0.145070\pi\)
\(164\) −2.50077e10 −2.69945
\(165\) −2.61078e8 −0.0274216
\(166\) −2.30271e10 −2.35371
\(167\) −6.45435e9 −0.642138 −0.321069 0.947056i \(-0.604042\pi\)
−0.321069 + 0.947056i \(0.604042\pi\)
\(168\) 1.19740e10 1.15970
\(169\) 0 0
\(170\) 3.97335e9 0.364868
\(171\) −2.10491e9 −0.188257
\(172\) −1.82170e10 −1.58708
\(173\) 1.10049e10 0.934067 0.467033 0.884240i \(-0.345323\pi\)
0.467033 + 0.884240i \(0.345323\pi\)
\(174\) −6.23760e8 −0.0515877
\(175\) 1.04941e10 0.845809
\(176\) −3.95500e10 −3.10699
\(177\) −1.44730e9 −0.110835
\(178\) −4.49285e10 −3.35453
\(179\) 6.52622e7 0.00475142 0.00237571 0.999997i \(-0.499244\pi\)
0.00237571 + 0.999997i \(0.499244\pi\)
\(180\) 3.81965e9 0.271205
\(181\) 1.03390e10 0.716017 0.358008 0.933718i \(-0.383456\pi\)
0.358008 + 0.933718i \(0.383456\pi\)
\(182\) 0 0
\(183\) −1.47522e9 −0.0972363
\(184\) −6.71824e10 −4.32091
\(185\) 1.25157e9 0.0785566
\(186\) −1.15672e10 −0.708629
\(187\) 1.99310e10 1.19191
\(188\) 5.49587e10 3.20868
\(189\) −1.01984e10 −0.581374
\(190\) −8.31466e8 −0.0462864
\(191\) −9.97341e9 −0.542243 −0.271121 0.962545i \(-0.587394\pi\)
−0.271121 + 0.962545i \(0.587394\pi\)
\(192\) −3.78952e10 −2.01246
\(193\) −1.88609e10 −0.978488 −0.489244 0.872147i \(-0.662727\pi\)
−0.489244 + 0.872147i \(0.662727\pi\)
\(194\) −8.88071e9 −0.450132
\(195\) 0 0
\(196\) −1.59718e10 −0.773041
\(197\) 2.53795e10 1.20056 0.600282 0.799789i \(-0.295055\pi\)
0.600282 + 0.799789i \(0.295055\pi\)
\(198\) 2.57918e10 1.19258
\(199\) −2.16921e10 −0.980535 −0.490267 0.871572i \(-0.663101\pi\)
−0.490267 + 0.871572i \(0.663101\pi\)
\(200\) −8.33151e10 −3.68204
\(201\) −4.86015e9 −0.210023
\(202\) −2.72795e10 −1.15280
\(203\) −1.48927e9 −0.0615519
\(204\) 4.44559e10 1.79719
\(205\) 2.55589e9 0.101076
\(206\) −5.52584e10 −2.13794
\(207\) 2.65837e10 1.00635
\(208\) 0 0
\(209\) −4.17079e9 −0.151203
\(210\) −1.87158e9 −0.0664084
\(211\) 3.18579e10 1.10649 0.553243 0.833020i \(-0.313390\pi\)
0.553243 + 0.833020i \(0.313390\pi\)
\(212\) −8.65920e10 −2.94419
\(213\) −6.89083e8 −0.0229384
\(214\) 8.31237e9 0.270934
\(215\) 1.86186e9 0.0594257
\(216\) 8.09681e10 2.53088
\(217\) −2.76174e10 −0.845500
\(218\) 8.01440e10 2.40335
\(219\) −6.15824e9 −0.180908
\(220\) 7.56847e9 0.217824
\(221\) 0 0
\(222\) 1.88501e10 0.520864
\(223\) 5.26237e10 1.42498 0.712491 0.701681i \(-0.247567\pi\)
0.712491 + 0.701681i \(0.247567\pi\)
\(224\) −1.63377e11 −4.33586
\(225\) 3.29674e10 0.857557
\(226\) 5.75489e10 1.46740
\(227\) −6.75026e10 −1.68735 −0.843673 0.536857i \(-0.819612\pi\)
−0.843673 + 0.536857i \(0.819612\pi\)
\(228\) −9.30289e9 −0.227987
\(229\) 1.69036e10 0.406180 0.203090 0.979160i \(-0.434902\pi\)
0.203090 + 0.979160i \(0.434902\pi\)
\(230\) 1.05009e10 0.247429
\(231\) −9.38822e9 −0.216935
\(232\) 1.18237e10 0.267953
\(233\) 2.65324e10 0.589760 0.294880 0.955534i \(-0.404720\pi\)
0.294880 + 0.955534i \(0.404720\pi\)
\(234\) 0 0
\(235\) −5.61701e9 −0.120143
\(236\) 4.19561e10 0.880423
\(237\) −1.43160e10 −0.294749
\(238\) 1.42879e11 2.88651
\(239\) 3.72393e10 0.738262 0.369131 0.929377i \(-0.379655\pi\)
0.369131 + 0.929377i \(0.379655\pi\)
\(240\) 9.01604e9 0.175414
\(241\) 3.55835e10 0.679472 0.339736 0.940521i \(-0.389662\pi\)
0.339736 + 0.940521i \(0.389662\pi\)
\(242\) −5.41147e10 −1.01425
\(243\) −4.91926e10 −0.905048
\(244\) 4.27657e10 0.772399
\(245\) 1.63239e9 0.0289452
\(246\) 3.84946e10 0.670180
\(247\) 0 0
\(248\) 2.19262e11 3.68070
\(249\) 2.63319e10 0.434095
\(250\) 2.61992e10 0.424188
\(251\) −9.97075e10 −1.58561 −0.792805 0.609476i \(-0.791380\pi\)
−0.792805 + 0.609476i \(0.791380\pi\)
\(252\) 1.37353e11 2.14553
\(253\) 5.26744e10 0.808272
\(254\) 9.07326e10 1.36776
\(255\) −4.54359e9 −0.0672927
\(256\) 4.11960e11 5.99481
\(257\) 5.47306e10 0.782584 0.391292 0.920267i \(-0.372028\pi\)
0.391292 + 0.920267i \(0.372028\pi\)
\(258\) 2.80417e10 0.394017
\(259\) 4.50058e10 0.621469
\(260\) 0 0
\(261\) −4.67858e9 −0.0624068
\(262\) −8.44172e9 −0.110682
\(263\) 9.60519e10 1.23796 0.618978 0.785408i \(-0.287547\pi\)
0.618978 + 0.785408i \(0.287547\pi\)
\(264\) 7.45356e10 0.944378
\(265\) 8.85008e9 0.110240
\(266\) −2.98990e10 −0.366176
\(267\) 5.13765e10 0.618677
\(268\) 1.40892e11 1.66832
\(269\) −1.50514e11 −1.75264 −0.876319 0.481731i \(-0.840008\pi\)
−0.876319 + 0.481731i \(0.840008\pi\)
\(270\) −1.26557e10 −0.144927
\(271\) −1.10945e11 −1.24953 −0.624765 0.780813i \(-0.714805\pi\)
−0.624765 + 0.780813i \(0.714805\pi\)
\(272\) −6.88296e11 −7.62457
\(273\) 0 0
\(274\) −1.92492e11 −2.06317
\(275\) 6.53233e10 0.688765
\(276\) 1.17490e11 1.21873
\(277\) 6.50996e10 0.664384 0.332192 0.943212i \(-0.392212\pi\)
0.332192 + 0.943212i \(0.392212\pi\)
\(278\) −1.89932e11 −1.90721
\(279\) −8.67607e10 −0.857244
\(280\) 3.54769e10 0.344933
\(281\) 5.78359e10 0.553375 0.276687 0.960960i \(-0.410763\pi\)
0.276687 + 0.960960i \(0.410763\pi\)
\(282\) −8.45985e10 −0.796602
\(283\) 2.07928e10 0.192696 0.0963482 0.995348i \(-0.469284\pi\)
0.0963482 + 0.995348i \(0.469284\pi\)
\(284\) 1.99760e10 0.182212
\(285\) 9.50796e8 0.00853661
\(286\) 0 0
\(287\) 9.19083e10 0.799625
\(288\) −5.13254e11 −4.39609
\(289\) 2.28275e11 1.92495
\(290\) −1.84810e9 −0.0153438
\(291\) 1.01552e10 0.0830179
\(292\) 1.78523e11 1.43705
\(293\) −8.15549e10 −0.646466 −0.323233 0.946319i \(-0.604770\pi\)
−0.323233 + 0.946319i \(0.604770\pi\)
\(294\) 2.45856e10 0.191919
\(295\) −4.28810e9 −0.0329660
\(296\) −3.57313e11 −2.70543
\(297\) −6.34831e10 −0.473428
\(298\) −1.84180e11 −1.35291
\(299\) 0 0
\(300\) 1.45703e11 1.03854
\(301\) 6.69514e10 0.470122
\(302\) 9.28254e10 0.642148
\(303\) 3.11945e10 0.212611
\(304\) 1.44033e11 0.967235
\(305\) −4.37084e9 −0.0289212
\(306\) 4.48859e11 2.92660
\(307\) −2.41080e11 −1.54895 −0.774477 0.632603i \(-0.781987\pi\)
−0.774477 + 0.632603i \(0.781987\pi\)
\(308\) 2.72158e11 1.72323
\(309\) 6.31889e10 0.394301
\(310\) −3.42716e10 −0.210769
\(311\) 6.46351e10 0.391784 0.195892 0.980625i \(-0.437240\pi\)
0.195892 + 0.980625i \(0.437240\pi\)
\(312\) 0 0
\(313\) −2.64577e11 −1.55812 −0.779062 0.626947i \(-0.784304\pi\)
−0.779062 + 0.626947i \(0.784304\pi\)
\(314\) 3.83020e11 2.22350
\(315\) −1.40380e10 −0.0803357
\(316\) 4.15009e11 2.34135
\(317\) 2.92101e11 1.62468 0.812338 0.583187i \(-0.198195\pi\)
0.812338 + 0.583187i \(0.198195\pi\)
\(318\) 1.33292e11 0.730941
\(319\) −9.27039e9 −0.0501234
\(320\) −1.12277e11 −0.598571
\(321\) −9.50534e9 −0.0499684
\(322\) 3.77606e11 1.95743
\(323\) −7.25849e10 −0.371052
\(324\) 3.55683e11 1.79312
\(325\) 0 0
\(326\) 7.22239e11 3.54162
\(327\) −9.16461e10 −0.443250
\(328\) −7.29685e11 −3.48099
\(329\) −2.01984e11 −0.950466
\(330\) −1.16502e10 −0.0540781
\(331\) −6.47388e8 −0.00296441 −0.00148221 0.999999i \(-0.500472\pi\)
−0.00148221 + 0.999999i \(0.500472\pi\)
\(332\) −7.63342e11 −3.44824
\(333\) 1.41387e11 0.630101
\(334\) −2.88016e11 −1.26636
\(335\) −1.43998e10 −0.0624676
\(336\) 3.24211e11 1.38772
\(337\) 1.60029e11 0.675874 0.337937 0.941169i \(-0.390271\pi\)
0.337937 + 0.941169i \(0.390271\pi\)
\(338\) 0 0
\(339\) −6.58082e10 −0.270633
\(340\) 1.31715e11 0.534542
\(341\) −1.71912e11 −0.688514
\(342\) −9.39286e10 −0.371262
\(343\) 2.78085e11 1.08481
\(344\) −5.31545e11 −2.04657
\(345\) −1.20080e10 −0.0456334
\(346\) 4.91077e11 1.84207
\(347\) −2.61745e11 −0.969160 −0.484580 0.874747i \(-0.661027\pi\)
−0.484580 + 0.874747i \(0.661027\pi\)
\(348\) −2.06775e10 −0.0755773
\(349\) −2.80466e11 −1.01196 −0.505982 0.862544i \(-0.668870\pi\)
−0.505982 + 0.862544i \(0.668870\pi\)
\(350\) 4.68282e11 1.66802
\(351\) 0 0
\(352\) −1.01699e12 −3.53081
\(353\) 3.94018e11 1.35061 0.675304 0.737539i \(-0.264012\pi\)
0.675304 + 0.737539i \(0.264012\pi\)
\(354\) −6.45835e10 −0.218578
\(355\) −2.04164e9 −0.00682262
\(356\) −1.48937e12 −4.91447
\(357\) −1.63385e11 −0.532359
\(358\) 2.91223e9 0.00937027
\(359\) 5.24181e10 0.166554 0.0832772 0.996526i \(-0.473461\pi\)
0.0832772 + 0.996526i \(0.473461\pi\)
\(360\) 1.11452e11 0.349724
\(361\) −3.07499e11 −0.952929
\(362\) 4.61361e11 1.41206
\(363\) 6.18811e10 0.187059
\(364\) 0 0
\(365\) −1.82458e10 −0.0538078
\(366\) −6.58297e10 −0.191760
\(367\) 2.64041e11 0.759755 0.379877 0.925037i \(-0.375966\pi\)
0.379877 + 0.925037i \(0.375966\pi\)
\(368\) −1.81905e12 −5.17047
\(369\) 2.88733e11 0.810732
\(370\) 5.58495e10 0.154921
\(371\) 3.18244e11 0.872122
\(372\) −3.83449e11 −1.03816
\(373\) 3.07202e11 0.821740 0.410870 0.911694i \(-0.365225\pi\)
0.410870 + 0.911694i \(0.365225\pi\)
\(374\) 8.89393e11 2.35056
\(375\) −2.99593e10 −0.0782331
\(376\) 1.60361e12 4.13764
\(377\) 0 0
\(378\) −4.55090e11 −1.14653
\(379\) −3.79660e11 −0.945188 −0.472594 0.881280i \(-0.656682\pi\)
−0.472594 + 0.881280i \(0.656682\pi\)
\(380\) −2.75629e10 −0.0678107
\(381\) −1.03754e11 −0.252257
\(382\) −4.45049e11 −1.06936
\(383\) 1.09927e11 0.261041 0.130521 0.991446i \(-0.458335\pi\)
0.130521 + 0.991446i \(0.458335\pi\)
\(384\) −9.05884e11 −2.12609
\(385\) −2.78157e10 −0.0645233
\(386\) −8.41641e11 −1.92968
\(387\) 2.10330e11 0.476652
\(388\) −2.94393e11 −0.659455
\(389\) −5.71954e11 −1.26645 −0.633225 0.773968i \(-0.718269\pi\)
−0.633225 + 0.773968i \(0.718269\pi\)
\(390\) 0 0
\(391\) 9.16702e11 1.98350
\(392\) −4.66034e11 −0.996851
\(393\) 9.65326e9 0.0204130
\(394\) 1.13252e12 2.36763
\(395\) −4.24158e10 −0.0876678
\(396\) 8.54991e11 1.74716
\(397\) −3.88595e11 −0.785127 −0.392563 0.919725i \(-0.628412\pi\)
−0.392563 + 0.919725i \(0.628412\pi\)
\(398\) −9.67978e11 −1.93371
\(399\) 3.41900e10 0.0675339
\(400\) −2.25587e12 −4.40599
\(401\) −2.52996e11 −0.488612 −0.244306 0.969698i \(-0.578560\pi\)
−0.244306 + 0.969698i \(0.578560\pi\)
\(402\) −2.16877e11 −0.414187
\(403\) 0 0
\(404\) −9.04307e11 −1.68888
\(405\) −3.63523e10 −0.0671405
\(406\) −6.64565e10 −0.121386
\(407\) 2.80151e11 0.506079
\(408\) 1.29716e12 2.31751
\(409\) 3.00728e11 0.531397 0.265699 0.964056i \(-0.414397\pi\)
0.265699 + 0.964056i \(0.414397\pi\)
\(410\) 1.14053e11 0.199333
\(411\) 2.20118e11 0.380511
\(412\) −1.83180e12 −3.13214
\(413\) −1.54198e11 −0.260797
\(414\) 1.18626e12 1.98462
\(415\) 7.80169e10 0.129114
\(416\) 0 0
\(417\) 2.17191e11 0.351746
\(418\) −1.86115e11 −0.298187
\(419\) −6.65230e11 −1.05441 −0.527204 0.849739i \(-0.676760\pi\)
−0.527204 + 0.849739i \(0.676760\pi\)
\(420\) −6.20426e10 −0.0972899
\(421\) −3.52570e11 −0.546986 −0.273493 0.961874i \(-0.588179\pi\)
−0.273493 + 0.961874i \(0.588179\pi\)
\(422\) 1.42161e12 2.18210
\(423\) −6.34540e11 −0.963667
\(424\) −2.52662e12 −3.79659
\(425\) 1.13683e12 1.69023
\(426\) −3.07493e10 −0.0452368
\(427\) −1.57173e11 −0.228798
\(428\) 2.75553e11 0.396925
\(429\) 0 0
\(430\) 8.30828e10 0.117193
\(431\) 7.16673e10 0.100040 0.0500200 0.998748i \(-0.484072\pi\)
0.0500200 + 0.998748i \(0.484072\pi\)
\(432\) 2.19232e12 3.02849
\(433\) −7.54601e11 −1.03163 −0.515813 0.856701i \(-0.672510\pi\)
−0.515813 + 0.856701i \(0.672510\pi\)
\(434\) −1.23238e12 −1.66741
\(435\) 2.11333e9 0.00282987
\(436\) 2.65675e12 3.52097
\(437\) −1.91830e11 −0.251623
\(438\) −2.74802e11 −0.356769
\(439\) −9.89110e11 −1.27102 −0.635512 0.772091i \(-0.719211\pi\)
−0.635512 + 0.772091i \(0.719211\pi\)
\(440\) 2.20836e11 0.280888
\(441\) 1.84407e11 0.232169
\(442\) 0 0
\(443\) −1.32070e12 −1.62925 −0.814627 0.579986i \(-0.803058\pi\)
−0.814627 + 0.579986i \(0.803058\pi\)
\(444\) 6.24875e11 0.763079
\(445\) 1.52220e11 0.184014
\(446\) 2.34825e12 2.81021
\(447\) 2.10613e11 0.249518
\(448\) −4.03741e12 −4.73535
\(449\) 1.47544e11 0.171322 0.0856608 0.996324i \(-0.472700\pi\)
0.0856608 + 0.996324i \(0.472700\pi\)
\(450\) 1.47112e12 1.69119
\(451\) 5.72110e11 0.651156
\(452\) 1.90773e12 2.14978
\(453\) −1.06147e11 −0.118432
\(454\) −3.01221e12 −3.32762
\(455\) 0 0
\(456\) −2.71444e11 −0.293994
\(457\) −8.62112e11 −0.924572 −0.462286 0.886731i \(-0.652971\pi\)
−0.462286 + 0.886731i \(0.652971\pi\)
\(458\) 7.54296e11 0.801027
\(459\) −1.10481e12 −1.16180
\(460\) 3.48102e11 0.362490
\(461\) −1.10935e12 −1.14396 −0.571982 0.820266i \(-0.693825\pi\)
−0.571982 + 0.820266i \(0.693825\pi\)
\(462\) −4.18935e11 −0.427817
\(463\) 5.23314e11 0.529234 0.264617 0.964354i \(-0.414754\pi\)
0.264617 + 0.964354i \(0.414754\pi\)
\(464\) 3.20143e11 0.320636
\(465\) 3.91901e10 0.0388721
\(466\) 1.18397e12 1.16306
\(467\) −5.66414e11 −0.551071 −0.275536 0.961291i \(-0.588855\pi\)
−0.275536 + 0.961291i \(0.588855\pi\)
\(468\) 0 0
\(469\) −5.17808e11 −0.494187
\(470\) −2.50651e11 −0.236935
\(471\) −4.37990e11 −0.410081
\(472\) 1.22422e12 1.13532
\(473\) 4.16759e11 0.382833
\(474\) −6.38828e11 −0.581275
\(475\) −2.37895e11 −0.214419
\(476\) 4.73641e12 4.22881
\(477\) 9.99771e11 0.884235
\(478\) 1.66175e12 1.45593
\(479\) 1.28230e12 1.11296 0.556480 0.830861i \(-0.312152\pi\)
0.556480 + 0.830861i \(0.312152\pi\)
\(480\) 2.31839e11 0.199343
\(481\) 0 0
\(482\) 1.58786e12 1.33999
\(483\) −4.31799e11 −0.361010
\(484\) −1.79389e12 −1.48591
\(485\) 3.00883e10 0.0246922
\(486\) −2.19515e12 −1.78485
\(487\) −2.27286e12 −1.83102 −0.915510 0.402295i \(-0.868213\pi\)
−0.915510 + 0.402295i \(0.868213\pi\)
\(488\) 1.24784e12 0.996022
\(489\) −8.25893e11 −0.653182
\(490\) 7.28431e10 0.0570829
\(491\) −1.15965e12 −0.900449 −0.450225 0.892915i \(-0.648656\pi\)
−0.450225 + 0.892915i \(0.648656\pi\)
\(492\) 1.27609e12 0.981831
\(493\) −1.61334e11 −0.123003
\(494\) 0 0
\(495\) −8.73838e10 −0.0654195
\(496\) 5.93680e12 4.40438
\(497\) −7.34160e10 −0.0539743
\(498\) 1.17502e12 0.856079
\(499\) 6.99224e11 0.504852 0.252426 0.967616i \(-0.418772\pi\)
0.252426 + 0.967616i \(0.418772\pi\)
\(500\) 8.68498e11 0.621446
\(501\) 3.29351e11 0.233555
\(502\) −4.44930e12 −3.12698
\(503\) 2.07525e12 1.44549 0.722745 0.691115i \(-0.242880\pi\)
0.722745 + 0.691115i \(0.242880\pi\)
\(504\) 4.00773e12 2.76670
\(505\) 9.24241e10 0.0632374
\(506\) 2.35052e12 1.59399
\(507\) 0 0
\(508\) 3.00776e12 2.00381
\(509\) −1.59826e12 −1.05540 −0.527700 0.849431i \(-0.676945\pi\)
−0.527700 + 0.849431i \(0.676945\pi\)
\(510\) −2.02751e11 −0.132708
\(511\) −6.56109e11 −0.425678
\(512\) 9.29369e12 5.97687
\(513\) 2.31193e11 0.147383
\(514\) 2.44227e12 1.54333
\(515\) 1.87218e11 0.117278
\(516\) 9.29575e11 0.577246
\(517\) −1.25731e12 −0.773990
\(518\) 2.00832e12 1.22560
\(519\) −5.61555e11 −0.339734
\(520\) 0 0
\(521\) 3.94430e11 0.234531 0.117265 0.993101i \(-0.462587\pi\)
0.117265 + 0.993101i \(0.462587\pi\)
\(522\) −2.08775e11 −0.123072
\(523\) −3.33919e12 −1.95157 −0.975783 0.218742i \(-0.929805\pi\)
−0.975783 + 0.218742i \(0.929805\pi\)
\(524\) −2.79841e11 −0.162151
\(525\) −5.35488e11 −0.307633
\(526\) 4.28618e12 2.44137
\(527\) −2.99182e12 −1.68962
\(528\) 2.01815e12 1.13006
\(529\) 6.21539e11 0.345079
\(530\) 3.94922e11 0.217405
\(531\) −4.84416e11 −0.264419
\(532\) −9.91145e11 −0.536457
\(533\) 0 0
\(534\) 2.29260e12 1.22009
\(535\) −2.81627e10 −0.0148622
\(536\) 4.11102e12 2.15133
\(537\) −3.33019e9 −0.00172816
\(538\) −6.71648e12 −3.45638
\(539\) 3.65394e11 0.186471
\(540\) −4.19532e11 −0.212321
\(541\) 4.98286e11 0.250087 0.125043 0.992151i \(-0.460093\pi\)
0.125043 + 0.992151i \(0.460093\pi\)
\(542\) −4.95077e12 −2.46420
\(543\) −5.27574e11 −0.260426
\(544\) −1.76988e13 −8.66463
\(545\) −2.71532e11 −0.131837
\(546\) 0 0
\(547\) −3.77955e12 −1.80508 −0.902542 0.430602i \(-0.858301\pi\)
−0.902542 + 0.430602i \(0.858301\pi\)
\(548\) −6.38106e12 −3.02260
\(549\) −4.93763e11 −0.231976
\(550\) 2.91496e12 1.35831
\(551\) 3.37610e10 0.0156039
\(552\) 3.42817e12 1.57158
\(553\) −1.52525e12 −0.693548
\(554\) 2.90497e12 1.31023
\(555\) −6.38649e10 −0.0285722
\(556\) −6.29621e12 −2.79411
\(557\) 2.05347e12 0.903941 0.451971 0.892033i \(-0.350721\pi\)
0.451971 + 0.892033i \(0.350721\pi\)
\(558\) −3.87157e12 −1.69057
\(559\) 0 0
\(560\) 9.60584e11 0.412752
\(561\) −1.01704e12 −0.433514
\(562\) 2.58084e12 1.09131
\(563\) −2.31654e12 −0.971745 −0.485872 0.874030i \(-0.661498\pi\)
−0.485872 + 0.874030i \(0.661498\pi\)
\(564\) −2.80442e12 −1.16704
\(565\) −1.94978e11 −0.0804950
\(566\) 9.27847e11 0.380016
\(567\) −1.30721e12 −0.531155
\(568\) 5.82870e11 0.234965
\(569\) −2.21221e11 −0.0884752 −0.0442376 0.999021i \(-0.514086\pi\)
−0.0442376 + 0.999021i \(0.514086\pi\)
\(570\) 4.24278e10 0.0168350
\(571\) −1.76805e12 −0.696038 −0.348019 0.937487i \(-0.613146\pi\)
−0.348019 + 0.937487i \(0.613146\pi\)
\(572\) 0 0
\(573\) 5.08921e11 0.197222
\(574\) 4.10127e12 1.57694
\(575\) 3.00446e12 1.14620
\(576\) −1.26836e13 −4.80112
\(577\) 9.40605e10 0.0353278 0.0176639 0.999844i \(-0.494377\pi\)
0.0176639 + 0.999844i \(0.494377\pi\)
\(578\) 1.01865e13 3.79619
\(579\) 9.62432e11 0.355890
\(580\) −6.12639e10 −0.0224791
\(581\) 2.80544e12 1.02143
\(582\) 4.53163e11 0.163720
\(583\) 1.98100e12 0.710192
\(584\) 5.20902e12 1.85310
\(585\) 0 0
\(586\) −3.63926e12 −1.27489
\(587\) −3.14652e12 −1.09385 −0.546927 0.837181i \(-0.684202\pi\)
−0.546927 + 0.837181i \(0.684202\pi\)
\(588\) 8.15008e11 0.281167
\(589\) 6.26071e11 0.214341
\(590\) −1.91350e11 −0.0650122
\(591\) −1.29506e12 −0.436663
\(592\) −9.67472e12 −3.23736
\(593\) −2.16558e12 −0.719165 −0.359583 0.933113i \(-0.617081\pi\)
−0.359583 + 0.933113i \(0.617081\pi\)
\(594\) −2.83284e12 −0.933648
\(595\) −4.84081e11 −0.158340
\(596\) −6.10553e12 −1.98205
\(597\) 1.10690e12 0.356635
\(598\) 0 0
\(599\) −2.13023e12 −0.676094 −0.338047 0.941129i \(-0.609766\pi\)
−0.338047 + 0.941129i \(0.609766\pi\)
\(600\) 4.25138e12 1.33921
\(601\) −5.30559e12 −1.65882 −0.829409 0.558641i \(-0.811323\pi\)
−0.829409 + 0.558641i \(0.811323\pi\)
\(602\) 2.98761e12 0.927127
\(603\) −1.62671e12 −0.501051
\(604\) 3.07714e12 0.940764
\(605\) 1.83343e11 0.0556373
\(606\) 1.39201e12 0.419291
\(607\) 4.05342e12 1.21191 0.605957 0.795497i \(-0.292790\pi\)
0.605957 + 0.795497i \(0.292790\pi\)
\(608\) 3.70368e12 1.09918
\(609\) 7.59941e10 0.0223873
\(610\) −1.95042e11 −0.0570354
\(611\) 0 0
\(612\) 1.48796e13 4.28754
\(613\) 1.53536e12 0.439175 0.219588 0.975593i \(-0.429529\pi\)
0.219588 + 0.975593i \(0.429529\pi\)
\(614\) −1.07578e13 −3.05469
\(615\) −1.30421e11 −0.0367630
\(616\) 7.94114e12 2.22213
\(617\) −3.50959e12 −0.974930 −0.487465 0.873143i \(-0.662078\pi\)
−0.487465 + 0.873143i \(0.662078\pi\)
\(618\) 2.81971e12 0.777601
\(619\) 2.16976e11 0.0594023 0.0297012 0.999559i \(-0.490544\pi\)
0.0297012 + 0.999559i \(0.490544\pi\)
\(620\) −1.13609e12 −0.308782
\(621\) −2.91982e12 −0.787852
\(622\) 2.88425e12 0.772638
\(623\) 5.47373e12 1.45575
\(624\) 0 0
\(625\) 3.68129e12 0.965028
\(626\) −1.18063e13 −3.07278
\(627\) 2.12826e11 0.0549947
\(628\) 1.26970e13 3.25749
\(629\) 4.87552e12 1.24192
\(630\) −6.26426e11 −0.158430
\(631\) −5.59463e12 −1.40488 −0.702441 0.711742i \(-0.747906\pi\)
−0.702441 + 0.711742i \(0.747906\pi\)
\(632\) 1.21093e13 3.01921
\(633\) −1.62564e12 −0.402445
\(634\) 1.30346e13 3.20402
\(635\) −3.07406e11 −0.0750293
\(636\) 4.41860e12 1.07085
\(637\) 0 0
\(638\) −4.13678e11 −0.0988482
\(639\) −2.30638e11 −0.0547240
\(640\) −2.68398e12 −0.632367
\(641\) −5.59530e12 −1.30907 −0.654534 0.756032i \(-0.727135\pi\)
−0.654534 + 0.756032i \(0.727135\pi\)
\(642\) −4.24162e11 −0.0985426
\(643\) 1.31892e11 0.0304278 0.0152139 0.999884i \(-0.495157\pi\)
0.0152139 + 0.999884i \(0.495157\pi\)
\(644\) 1.25175e13 2.86769
\(645\) −9.50066e10 −0.0216140
\(646\) −3.23899e12 −0.731752
\(647\) −7.90835e11 −0.177426 −0.0887128 0.996057i \(-0.528275\pi\)
−0.0887128 + 0.996057i \(0.528275\pi\)
\(648\) 1.03783e13 2.31227
\(649\) −9.59848e11 −0.212374
\(650\) 0 0
\(651\) 1.40925e12 0.307521
\(652\) 2.39421e13 5.18856
\(653\) −8.33141e11 −0.179312 −0.0896560 0.995973i \(-0.528577\pi\)
−0.0896560 + 0.995973i \(0.528577\pi\)
\(654\) −4.08957e12 −0.874133
\(655\) 2.86010e10 0.00607148
\(656\) −1.97572e13 −4.16541
\(657\) −2.06118e12 −0.431591
\(658\) −9.01326e12 −1.87441
\(659\) 3.45121e12 0.712833 0.356416 0.934327i \(-0.383998\pi\)
0.356416 + 0.934327i \(0.383998\pi\)
\(660\) −3.86202e11 −0.0792259
\(661\) −5.01263e12 −1.02131 −0.510656 0.859785i \(-0.670598\pi\)
−0.510656 + 0.859785i \(0.670598\pi\)
\(662\) −2.88887e10 −0.00584611
\(663\) 0 0
\(664\) −2.22732e13 −4.44657
\(665\) 1.01299e11 0.0200867
\(666\) 6.30918e12 1.24262
\(667\) −4.26380e11 −0.0834123
\(668\) −9.54765e12 −1.85525
\(669\) −2.68527e12 −0.518287
\(670\) −6.42570e11 −0.123192
\(671\) −9.78368e11 −0.186316
\(672\) 8.33678e12 1.57702
\(673\) −2.19889e12 −0.413176 −0.206588 0.978428i \(-0.566236\pi\)
−0.206588 + 0.978428i \(0.566236\pi\)
\(674\) 7.14108e12 1.33289
\(675\) −3.62097e12 −0.671364
\(676\) 0 0
\(677\) −4.25317e10 −0.00778151 −0.00389075 0.999992i \(-0.501238\pi\)
−0.00389075 + 0.999992i \(0.501238\pi\)
\(678\) −2.93659e12 −0.533716
\(679\) 1.08196e12 0.195342
\(680\) 3.84325e12 0.689301
\(681\) 3.44451e12 0.613713
\(682\) −7.67134e12 −1.35782
\(683\) 9.09442e12 1.59912 0.799561 0.600585i \(-0.205065\pi\)
0.799561 + 0.600585i \(0.205065\pi\)
\(684\) −3.11371e12 −0.543908
\(685\) 6.52172e11 0.113176
\(686\) 1.24092e13 2.13936
\(687\) −8.62551e11 −0.147734
\(688\) −1.43923e13 −2.44896
\(689\) 0 0
\(690\) −5.35837e11 −0.0899937
\(691\) 6.69654e12 1.11738 0.558688 0.829378i \(-0.311305\pi\)
0.558688 + 0.829378i \(0.311305\pi\)
\(692\) 1.62791e13 2.69868
\(693\) −3.14227e12 −0.517540
\(694\) −1.16800e13 −1.91128
\(695\) 6.43500e11 0.104621
\(696\) −6.03338e11 −0.0974583
\(697\) 9.95654e12 1.59794
\(698\) −1.25154e13 −1.99569
\(699\) −1.35389e12 −0.214504
\(700\) 1.55234e13 2.44369
\(701\) −8.97175e12 −1.40329 −0.701643 0.712529i \(-0.747550\pi\)
−0.701643 + 0.712529i \(0.747550\pi\)
\(702\) 0 0
\(703\) −1.02026e12 −0.157547
\(704\) −2.51320e13 −3.85612
\(705\) 2.86624e11 0.0436979
\(706\) 1.75825e13 2.66353
\(707\) 3.32352e12 0.500277
\(708\) −2.14093e12 −0.320223
\(709\) 5.56970e12 0.827796 0.413898 0.910323i \(-0.364167\pi\)
0.413898 + 0.910323i \(0.364167\pi\)
\(710\) −9.11050e10 −0.0134549
\(711\) −4.79160e12 −0.703181
\(712\) −4.34574e13 −6.33730
\(713\) −7.90689e12 −1.14578
\(714\) −7.29080e12 −1.04986
\(715\) 0 0
\(716\) 9.65397e10 0.0137277
\(717\) −1.90024e12 −0.268517
\(718\) 2.33908e12 0.328462
\(719\) 1.00272e13 1.39927 0.699634 0.714502i \(-0.253346\pi\)
0.699634 + 0.714502i \(0.253346\pi\)
\(720\) 3.01770e12 0.418485
\(721\) 6.73225e12 0.927794
\(722\) −1.37217e13 −1.87927
\(723\) −1.81574e12 −0.247134
\(724\) 1.52940e13 2.06870
\(725\) −5.28768e11 −0.0710794
\(726\) 2.76135e12 0.368899
\(727\) 1.17247e13 1.55668 0.778338 0.627845i \(-0.216063\pi\)
0.778338 + 0.627845i \(0.216063\pi\)
\(728\) 0 0
\(729\) −2.22252e12 −0.291456
\(730\) −8.14192e11 −0.106114
\(731\) 7.25292e12 0.939474
\(732\) −2.18224e12 −0.280933
\(733\) 6.49737e12 0.831323 0.415661 0.909520i \(-0.363550\pi\)
0.415661 + 0.909520i \(0.363550\pi\)
\(734\) 1.17824e13 1.49831
\(735\) −8.32973e10 −0.0105278
\(736\) −4.67751e13 −5.87577
\(737\) −3.22325e12 −0.402430
\(738\) 1.28843e13 1.59884
\(739\) −5.02178e12 −0.619381 −0.309691 0.950837i \(-0.600225\pi\)
−0.309691 + 0.950837i \(0.600225\pi\)
\(740\) 1.85140e12 0.226964
\(741\) 0 0
\(742\) 1.42011e13 1.71991
\(743\) 3.60670e12 0.434170 0.217085 0.976153i \(-0.430345\pi\)
0.217085 + 0.976153i \(0.430345\pi\)
\(744\) −1.11884e13 −1.33872
\(745\) 6.24012e11 0.0742146
\(746\) 1.37084e13 1.62055
\(747\) 8.81337e12 1.03562
\(748\) 2.94832e13 3.44363
\(749\) −1.01271e12 −0.117576
\(750\) −1.33689e12 −0.154283
\(751\) 1.52152e13 1.74541 0.872707 0.488244i \(-0.162362\pi\)
0.872707 + 0.488244i \(0.162362\pi\)
\(752\) 4.34198e13 4.95117
\(753\) 5.08785e12 0.576710
\(754\) 0 0
\(755\) −3.14497e11 −0.0352253
\(756\) −1.50861e13 −1.67969
\(757\) 1.20577e13 1.33454 0.667272 0.744814i \(-0.267462\pi\)
0.667272 + 0.744814i \(0.267462\pi\)
\(758\) −1.69418e13 −1.86401
\(759\) −2.68786e12 −0.293980
\(760\) −8.04242e11 −0.0874431
\(761\) −9.08606e12 −0.982075 −0.491038 0.871138i \(-0.663382\pi\)
−0.491038 + 0.871138i \(0.663382\pi\)
\(762\) −4.62988e12 −0.497476
\(763\) −9.76412e12 −1.04297
\(764\) −1.47533e13 −1.56663
\(765\) −1.52075e12 −0.160540
\(766\) 4.90532e12 0.514799
\(767\) 0 0
\(768\) −2.10214e13 −2.18040
\(769\) −4.28917e12 −0.442288 −0.221144 0.975241i \(-0.570979\pi\)
−0.221144 + 0.975241i \(0.570979\pi\)
\(770\) −1.24123e12 −0.127246
\(771\) −2.79278e12 −0.284637
\(772\) −2.79002e13 −2.82702
\(773\) −4.49046e12 −0.452359 −0.226180 0.974086i \(-0.572624\pi\)
−0.226180 + 0.974086i \(0.572624\pi\)
\(774\) 9.38565e12 0.940005
\(775\) −9.80560e12 −0.976375
\(776\) −8.58994e12 −0.850379
\(777\) −2.29654e12 −0.226037
\(778\) −2.55226e13 −2.49756
\(779\) −2.08351e12 −0.202711
\(780\) 0 0
\(781\) −4.57000e11 −0.0439528
\(782\) 4.09065e13 3.91166
\(783\) 5.13872e11 0.0488570
\(784\) −1.26185e13 −1.19285
\(785\) −1.29769e12 −0.121971
\(786\) 4.30762e11 0.0402565
\(787\) −3.31359e12 −0.307902 −0.153951 0.988078i \(-0.549200\pi\)
−0.153951 + 0.988078i \(0.549200\pi\)
\(788\) 3.75428e13 3.46864
\(789\) −4.90132e12 −0.450263
\(790\) −1.89274e12 −0.172890
\(791\) −7.01131e12 −0.636803
\(792\) 2.49473e13 2.25300
\(793\) 0 0
\(794\) −1.73405e13 −1.54835
\(795\) −4.51600e11 −0.0400961
\(796\) −3.20882e13 −2.83294
\(797\) 8.72292e12 0.765772 0.382886 0.923796i \(-0.374930\pi\)
0.382886 + 0.923796i \(0.374930\pi\)
\(798\) 1.52568e12 0.133183
\(799\) −2.18812e13 −1.89938
\(800\) −5.80074e13 −5.00701
\(801\) 1.71959e13 1.47597
\(802\) −1.12896e13 −0.963592
\(803\) −4.08414e12 −0.346642
\(804\) −7.18941e12 −0.606794
\(805\) −1.27935e12 −0.107376
\(806\) 0 0
\(807\) 7.68041e12 0.637460
\(808\) −2.63863e13 −2.17785
\(809\) 1.32412e13 1.08682 0.543411 0.839467i \(-0.317132\pi\)
0.543411 + 0.839467i \(0.317132\pi\)
\(810\) −1.62217e12 −0.132408
\(811\) 1.09177e13 0.886215 0.443108 0.896468i \(-0.353876\pi\)
0.443108 + 0.896468i \(0.353876\pi\)
\(812\) −2.20301e12 −0.177834
\(813\) 5.66129e12 0.454473
\(814\) 1.25013e13 0.998038
\(815\) −2.44698e12 −0.194277
\(816\) 3.51222e13 2.77317
\(817\) −1.51775e12 −0.119180
\(818\) 1.34196e13 1.04797
\(819\) 0 0
\(820\) 3.78082e12 0.292028
\(821\) 2.10770e13 1.61906 0.809532 0.587075i \(-0.199721\pi\)
0.809532 + 0.587075i \(0.199721\pi\)
\(822\) 9.82244e12 0.750406
\(823\) −2.29237e13 −1.74175 −0.870876 0.491503i \(-0.836448\pi\)
−0.870876 + 0.491503i \(0.836448\pi\)
\(824\) −5.34492e13 −4.03895
\(825\) −3.33330e12 −0.250514
\(826\) −6.88084e12 −0.514317
\(827\) −3.15842e11 −0.0234798 −0.0117399 0.999931i \(-0.503737\pi\)
−0.0117399 + 0.999931i \(0.503737\pi\)
\(828\) 3.93242e13 2.90752
\(829\) 1.47916e13 1.08773 0.543864 0.839174i \(-0.316961\pi\)
0.543864 + 0.839174i \(0.316961\pi\)
\(830\) 3.48139e12 0.254625
\(831\) −3.32189e12 −0.241646
\(832\) 0 0
\(833\) 6.35902e12 0.457602
\(834\) 9.69183e12 0.693679
\(835\) 9.75811e11 0.0694667
\(836\) −6.16967e12 −0.436851
\(837\) 9.52937e12 0.671119
\(838\) −2.96849e13 −2.07940
\(839\) 1.67083e13 1.16413 0.582067 0.813141i \(-0.302244\pi\)
0.582067 + 0.813141i \(0.302244\pi\)
\(840\) −1.81031e12 −0.125457
\(841\) −1.44321e13 −0.994827
\(842\) −1.57329e13 −1.07871
\(843\) −2.95124e12 −0.201271
\(844\) 4.71260e13 3.19683
\(845\) 0 0
\(846\) −2.83154e13 −1.90045
\(847\) 6.59291e12 0.440151
\(848\) −6.84116e13 −4.54306
\(849\) −1.06101e12 −0.0700865
\(850\) 5.07295e13 3.33331
\(851\) 1.28852e13 0.842187
\(852\) −1.01933e12 −0.0662731
\(853\) −1.47950e13 −0.956850 −0.478425 0.878128i \(-0.658792\pi\)
−0.478425 + 0.878128i \(0.658792\pi\)
\(854\) −7.01360e12 −0.451212
\(855\) 3.18235e11 0.0203657
\(856\) 8.04021e12 0.511842
\(857\) 7.22513e12 0.457543 0.228771 0.973480i \(-0.426529\pi\)
0.228771 + 0.973480i \(0.426529\pi\)
\(858\) 0 0
\(859\) −9.20444e11 −0.0576804 −0.0288402 0.999584i \(-0.509181\pi\)
−0.0288402 + 0.999584i \(0.509181\pi\)
\(860\) 2.75417e12 0.171691
\(861\) −4.68988e12 −0.290835
\(862\) 3.19805e12 0.197289
\(863\) −2.54308e11 −0.0156067 −0.00780334 0.999970i \(-0.502484\pi\)
−0.00780334 + 0.999970i \(0.502484\pi\)
\(864\) 5.63733e13 3.44161
\(865\) −1.66379e12 −0.101048
\(866\) −3.36730e13 −2.03447
\(867\) −1.16484e13 −0.700132
\(868\) −4.08532e13 −2.44280
\(869\) −9.49433e12 −0.564775
\(870\) 9.43042e10 0.00558078
\(871\) 0 0
\(872\) 7.75200e13 4.54035
\(873\) 3.39900e12 0.198055
\(874\) −8.56013e12 −0.496225
\(875\) −3.19191e12 −0.184083
\(876\) −9.10962e12 −0.522675
\(877\) 2.42430e13 1.38385 0.691925 0.721970i \(-0.256763\pi\)
0.691925 + 0.721970i \(0.256763\pi\)
\(878\) −4.41375e13 −2.50659
\(879\) 4.16156e12 0.235129
\(880\) 5.97943e12 0.336115
\(881\) 2.82293e13 1.57873 0.789366 0.613923i \(-0.210409\pi\)
0.789366 + 0.613923i \(0.210409\pi\)
\(882\) 8.22890e12 0.457860
\(883\) 8.84255e12 0.489502 0.244751 0.969586i \(-0.421294\pi\)
0.244751 + 0.969586i \(0.421294\pi\)
\(884\) 0 0
\(885\) 2.18812e11 0.0119902
\(886\) −5.89344e13 −3.21305
\(887\) −1.51600e13 −0.822323 −0.411161 0.911563i \(-0.634877\pi\)
−0.411161 + 0.911563i \(0.634877\pi\)
\(888\) 1.82329e13 0.984004
\(889\) −1.10541e13 −0.593564
\(890\) 6.79258e12 0.362894
\(891\) −8.13710e12 −0.432534
\(892\) 7.78440e13 4.11702
\(893\) 4.57888e12 0.240950
\(894\) 9.39831e12 0.492074
\(895\) −9.86678e9 −0.000514010 0
\(896\) −9.65143e13 −5.00271
\(897\) 0 0
\(898\) 6.58392e12 0.337863
\(899\) 1.39157e12 0.0710535
\(900\) 4.87672e13 2.47763
\(901\) 3.44757e13 1.74282
\(902\) 2.55296e13 1.28415
\(903\) −3.41638e12 −0.170990
\(904\) 5.56647e13 2.77218
\(905\) −1.56311e12 −0.0774589
\(906\) −4.73667e12 −0.233559
\(907\) −2.96544e13 −1.45498 −0.727488 0.686120i \(-0.759313\pi\)
−0.727488 + 0.686120i \(0.759313\pi\)
\(908\) −9.98538e13 −4.87504
\(909\) 1.04409e13 0.507226
\(910\) 0 0
\(911\) −7.54006e12 −0.362695 −0.181348 0.983419i \(-0.558046\pi\)
−0.181348 + 0.983419i \(0.558046\pi\)
\(912\) −7.34970e12 −0.351798
\(913\) 1.74633e13 0.831778
\(914\) −3.84705e13 −1.82335
\(915\) 2.23034e11 0.0105191
\(916\) 2.50047e13 1.17353
\(917\) 1.02847e12 0.0480321
\(918\) −4.93004e13 −2.29118
\(919\) −9.55982e12 −0.442109 −0.221055 0.975261i \(-0.570950\pi\)
−0.221055 + 0.975261i \(0.570950\pi\)
\(920\) 1.01571e13 0.467437
\(921\) 1.23018e13 0.563377
\(922\) −4.95029e13 −2.25601
\(923\) 0 0
\(924\) −1.38876e13 −0.626763
\(925\) 1.59794e13 0.717665
\(926\) 2.33521e13 1.04370
\(927\) 2.11496e13 0.940681
\(928\) 8.23215e12 0.364374
\(929\) 1.17249e13 0.516460 0.258230 0.966083i \(-0.416861\pi\)
0.258230 + 0.966083i \(0.416861\pi\)
\(930\) 1.74880e12 0.0766597
\(931\) −1.33069e12 −0.0580503
\(932\) 3.92483e13 1.70392
\(933\) −3.29819e12 −0.142498
\(934\) −2.52754e13 −1.08677
\(935\) −3.01331e12 −0.128941
\(936\) 0 0
\(937\) −3.12797e12 −0.132567 −0.0662833 0.997801i \(-0.521114\pi\)
−0.0662833 + 0.997801i \(0.521114\pi\)
\(938\) −2.31064e13 −0.974585
\(939\) 1.35008e13 0.566713
\(940\) −8.30901e12 −0.347116
\(941\) −4.12131e13 −1.71349 −0.856745 0.515740i \(-0.827517\pi\)
−0.856745 + 0.515740i \(0.827517\pi\)
\(942\) −1.95446e13 −0.808721
\(943\) 2.63135e13 1.08362
\(944\) 3.31473e13 1.35854
\(945\) 1.54187e12 0.0628932
\(946\) 1.85972e13 0.754985
\(947\) 3.89950e13 1.57556 0.787778 0.615959i \(-0.211231\pi\)
0.787778 + 0.615959i \(0.211231\pi\)
\(948\) −2.11770e13 −0.851583
\(949\) 0 0
\(950\) −1.06157e13 −0.422856
\(951\) −1.49053e13 −0.590918
\(952\) 1.38201e14 5.45312
\(953\) −1.29321e13 −0.507869 −0.253935 0.967221i \(-0.581725\pi\)
−0.253935 + 0.967221i \(0.581725\pi\)
\(954\) 4.46133e13 1.74380
\(955\) 1.50785e12 0.0586600
\(956\) 5.50865e13 2.13297
\(957\) 4.73047e11 0.0182306
\(958\) 5.72207e13 2.19487
\(959\) 2.34517e13 0.895347
\(960\) 5.72924e12 0.217709
\(961\) −6.34069e11 −0.0239818
\(962\) 0 0
\(963\) −3.18147e12 −0.119209
\(964\) 5.26371e13 1.96311
\(965\) 2.85152e12 0.105853
\(966\) −1.92684e13 −0.711948
\(967\) −5.25218e12 −0.193161 −0.0965807 0.995325i \(-0.530791\pi\)
−0.0965807 + 0.995325i \(0.530791\pi\)
\(968\) −5.23429e13 −1.91610
\(969\) 3.70385e12 0.134957
\(970\) 1.34264e12 0.0486954
\(971\) −3.59385e13 −1.29740 −0.648699 0.761045i \(-0.724686\pi\)
−0.648699 + 0.761045i \(0.724686\pi\)
\(972\) −7.27686e13 −2.61484
\(973\) 2.31399e13 0.827663
\(974\) −1.01423e14 −3.61095
\(975\) 0 0
\(976\) 3.37868e13 1.19186
\(977\) 7.49430e12 0.263151 0.131576 0.991306i \(-0.457996\pi\)
0.131576 + 0.991306i \(0.457996\pi\)
\(978\) −3.68543e13 −1.28814
\(979\) 3.40729e13 1.18546
\(980\) 2.41473e12 0.0836279
\(981\) −3.06743e13 −1.05746
\(982\) −5.17475e13 −1.77578
\(983\) 4.10408e12 0.140193 0.0700963 0.997540i \(-0.477669\pi\)
0.0700963 + 0.997540i \(0.477669\pi\)
\(984\) 3.72342e13 1.26609
\(985\) −3.83704e12 −0.129877
\(986\) −7.19930e12 −0.242574
\(987\) 1.03068e13 0.345698
\(988\) 0 0
\(989\) 1.91683e13 0.637088
\(990\) −3.89937e12 −0.129014
\(991\) −5.13439e13 −1.69105 −0.845527 0.533933i \(-0.820714\pi\)
−0.845527 + 0.533933i \(0.820714\pi\)
\(992\) 1.52659e14 5.00518
\(993\) 3.30347e10 0.00107820
\(994\) −3.27608e12 −0.106443
\(995\) 3.27956e12 0.106075
\(996\) 3.89517e13 1.25418
\(997\) 5.42449e12 0.173872 0.0869362 0.996214i \(-0.472292\pi\)
0.0869362 + 0.996214i \(0.472292\pi\)
\(998\) 3.12018e13 0.995618
\(999\) −1.55292e13 −0.493293
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 169.10.a.a.1.4 4
13.12 even 2 13.10.a.a.1.1 4
39.38 odd 2 117.10.a.c.1.4 4
52.51 odd 2 208.10.a.g.1.3 4
65.64 even 2 325.10.a.a.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.a.1.1 4 13.12 even 2
117.10.a.c.1.4 4 39.38 odd 2
169.10.a.a.1.4 4 1.1 even 1 trivial
208.10.a.g.1.3 4 52.51 odd 2
325.10.a.a.1.4 4 65.64 even 2